permanent magnets for nmr and mri, by cedric hugon

Université de Versailles Saint-Quentin-en-Yvelines

École Doctorale Sciences et Technologies de Versailles - STVLaboratoire de Structure et Dynamique par Résonance Dynamique

CEA/DSM/IRAMIS/SIS2M/LSDRM

THÈSE DE DOCTORATDE L’UNIVERSITÉ DE VERSAILLES SAINT-QUENTIN-EN-YVELINES

Spécialité : Physique

Pour obtenir le grade de Docteur de l’Université de Versailles Saint-Quentin-en-Yvelines

Présentée et soutenue publiquement par

Cédric HUGONle 6 Octobre 2010

Aimants permanents pour la RMN et l’IRM

Directeur de thèse: Dimitrios Sakellariou

Devant le jury composé de :M. Peter BLÜMLER, Research Scientist, Université Johannes Gutenberg - Mainz (Allemagne) RapporteurM. Lucio FRYDMAN, Professeur, Weizmann Institute of Sciences - Rehovot (Israël) RapporteurM. Dimitrios SAKELLARIOU, Directeur de Recherche, CEA - Gif/Yvette Directeur de thèseM. Guy AUBERT, Professeur Emérite, Université de Poitiers - Poitiers ExaminateurM. Luc DARRASSE, Directeur de Recherche, Université Paris Sud 11 - Orsay ExaminateurM. Christian JEANDEY, CEA -Grenoble ExaminateurM. Francis TAULELLE, Université Versailles Saint-Quentin-en-Yvelines - Versailles Examinateur

Numéro national d’enregistrement :

À Laura,

et à mes parents,

Remerciements

Je n’aurais pas pu accomplir ce travail sans l’environnement qui m’a accompagné tout le long de laroute. Je ne procéderai pas ici par ordre d’apparition et aimerai tout d’abord remercier les membresdu jury, messieurs Guy Aubert, Peter Blümler, Luc Darrasse, Lucio Frydman, Christian Jeandey, etFrancis Taulelle pour leur lecture pleine d’intérêt du manuscrit, leurs corrections et commentaires etleurs remarques illuminantes. J’aimerai tout spécialement remercier monsieur Christian Jeandey poursa lecture particulièrement détaillée et vigilante.

Je citerai à nouveau Guy Aubert, lui aussi membre du jury, qui a tenu un rôle tout particulierau cours de ma thèse puisqu’il m’a entre autres inculqué l’utilisation des harmoniques sphériquesdans la conception des aimants permanents, et plus généralement, le magnétisme. Ce travail n’auraitcertainement pas été possible sans Guy, ce puits de science qui a bien voulu partager sa connaissanceavec nous d’une manière si didactique. Guy, tu es un admirable professeur, et ton rôle peut se résumerpar cette phrase que nous avons si souvent prononcée, sous une forme ou une autre : "Voici unproblème de magnétisme pour lequel nous n’avons pas la moindre idée du début d’une solution. Guyl’a probablement traité dans sa jeunesse !"

Une autre personne d’une importance toute particulière est Dimitrios Sakellariou, mon directeur dethèse. C’est grâce aux idées un peu folles de Dimitris et à son obstination dans leur accomplissementque cette thèse a eu lieu. Dimitris a un talent reconnu pour obtenir des financements, et ceci estcertainement dû à sa capacité à faire rêver avec la science, et à partager son enthousiasme pour elle.Ceci a sans doute joué un rôle important pour me convaincre de me lancer dans ce sujet auquel maformation en optique ne me destinait pas particulièrement (ma fascination d’enfant pour ces objets quel’on colle sur le frigo a aussi beaucoup joué...). Toujours présent avec le bon mot d’encouragement dansles périodes de doute, toujours présent pour aider à réfléchir lorsque l’on semble être dans l’impasse,Dimitris a su aussi me laisser de l’espace pour développer mes idées, la juste dose de liberté entrel’étou!ement et l’oubli. Tout ceci, il l’a accompli tout en gérant les nombreux problèmes administratifsattenant à la gestion de son groupe au sein du CEA et en fournissant le "leadership" qui nous motivetous. Dimitris, tu es loin de ne m’avoir qu’enseigné un peu de RMN, tu m’as surtout montré une visionde la recherche scientifique, et une manière de travailler. Pour finir, plus qu’un "boss", je me plais àcroire que j’ai trouvé un ami en mon directeur de thèse.

Au cours de ma thèse, le groupe de Dimitris n’a cessé de s’accroître. nous étions trois à monarrivée, en la personne de Jacques-François, Dimitris, et moi-même. Nous sommes aujourd’hui huit,avec, dans le désordre, Alan, Pedro, Aurore, Angelo, et Birgit, en plus des trois précédents. Nousavons de plus eu la visite de di!érents invités comme Francesca qui a travaillé sur les projets d’aimantspermanents. J’aimerais remercier toutes ces personnes pour leur présence, leur discussion, scientifiqueou non, qui a participé à faire avancer ce projet. Certains d’entre eux ont de plus été mis à contributionpour mon déménagement au cinquième étage sans ascenseur, et ceci su"t déjà à leur décerner unemédaille du mérite. Autour de ce groupe se trouvent aussi tous les membres (passés et présents)du LSDRM, Thibault, Anne, Olivier, Patrick, Marie-Anne, Denis, Nawal, Céline, Thierry, Francine,

Gaspard, Steven, Estelle... À eux aussi va un grand merci.De nombreuses autres personnes au CEA ont contribué d’une manière ou une autre à ce travail :

Michel du SPAM pour la conception mécanique de l’aimant ex-situ ; Stéphane, Thierry et Jean-Marcdu SACM pour les mesures de stabilité de champ par RMN de l’hélium 3 ; Sandrine, Philippe, Jean-Christophe, Denis et Philippe de l’IRFU pour la mécanique, l’électrotechnique et autres réalisationspour le projet d’aimant tournant ; Hélène, Christine, Michèle et Stéphanie du SIS2M pour leurs bonso"ces au secrétariat.

L’aimantation du système ex-situ a été faite au Laboratoire National des Champs MagnétiquesIntenses et je remercie monsieur François Debray pour son aide en la matière.

J’aimerai aussi mentionner le professeur Kurtis Thome, ancien chef du "Remote Sensing Group" àl’université d’Arizona. Bien qu’il n’ait pas grand chose à voir avec la RMN et les aimants, il m’a unpeu lancé sur la voie de ce travail en me donnant une belle opportunité de me confronter à la rechercheacadémique et à la fabrication d’instruments, au cours de mon master. Il a joué un excellent rôle dementor à cette occasion et m’a permis d’acquérir la confiance en moi dont j’avais besoin pour me lancerdans l’aventure de cette thèse.

Il me reste à remercier mes amis et ma famille pour leur compagnie et leur soutien. Leur présencem’a permis de me rappeler tout au long de ces trois ans que ma vie ne se résumait pas qu’à la thèse...Une pensée toute particulière va à mes parents qui me supportent dans tous les sens du terme depuisvingt-six ans. Il est probablement inutile de mentionner que j’ai pu arriver jusqu’ici grâce à eux et cequ’ils m’ont inculqué.

Enfin, Laura, je te remercie, toi qui m’a rencontré au début de ma thèse, n’a pas pris peur, estrestée et m’a accompagné, supporté, sans vraiment te plaindre de mes horaires ou de mon incapacité àoublier mes aimants pendant les vacances. Tu es pour beaucoup dans l’accomplissement de ce travail,même si tu "n’y comprends pas grand chose".

Résumé

Ce travail porte sur l’utilisation de matériaux magnétiques permanents pour la fabrication d’aimantsadaptés à des expériences de Résonance Magnétique Nucléaire (RMN).

Chapitre 1 : Introduction à la RMN et aux aimants permanents

Nous présentons dans ce premier chapitre les bases nécessaires pour comprendre les points clés del’expérience de RMN. Celle-ci se base sur un phénomène physique propre aux noyaux atomiques despin non nul. D’un point de vue classique , lorsqu’un ensemble de noyaux d’un type donné est plongédans un champ magnétique (on supposera pour simplifier que ce champ est homogène et de directionparallèle à l’axe Oz), une aimantation globale de l’échantillon apparaît, ainsi qu’un moment angulairequi lui est associé. Cette aimantation interagit avec le champ magnétique extérieur, donnant lieu à uncouple qui résulte en la précession de l’aimantation autour de l’axe. Cette précession s’e!ectue à lafréquence !0 dite de Larmor, donnée par

!0 = "B0, (1)

où " est le rapport gyromagnétique du noyau considéré et B0 est le champ magnétique externe auquell’ensemble de noyaux est soumis.

On observe par ailleurs des phénomènes de relaxation qui tendent d’une part à ramener l’aimantationà une position d’équilibre (parallèle à

!"B0) et d’autre part à faire disparaître la composante transverse

de l’aimantation. Le premier phénomène est appelé relaxation longitudinale et le second relaxationtransverse. Ces processus sont très di!érents par leur nature. Le premier résulte de la thermalisationdu système qui transfère son excédent d’énergie à son environnement appelé le réseau. L’évolution dela composante longitudinale de l’aimantation Mz suit alors une loi exponentielle du type

Mz(t) = M0

!1! exp(! t

T1)"

, (2)

où M0 est l’aimantation à l’équilibre et T1 est la constante de relaxation longitudinale. Le secondphénomène résulte de l’interaction des noyaux (ou spins) entre eux. Aucun échange énergétique avecle réseau n’est mis en jeu, si bien que la composante transverse de l’aimantation peu disparaîtrealors que la composante longitudinale n’a pas encore atteint l’équilibre. Ceci s’explique par une pertede cohérence des spins très bien décrite par l’approche quantique du phénomène. La perte de co-hérence se traduit par une décroissance exponentielle de la composante transverse. Comme on l’adit, l’aimantation placée dans le champ magnétique externe précesse autour de celui-ci, c’est à direautour de Oz dans ce cas précis. On a donc l’évolution suivante pour les composantes transverses de

1

2

l’aimantation :

Mx(t) = M0 cos !0t exp!! t

T2

"(3)

My(t) = M0 sin!0t exp!! t

T2

", (4)

où T2 est la constante de relaxation transverse.L’expérience de RMN consiste à déplacer l’aimantation hors équilibre et à observer son évolution

dans le champ environnant jusqu’à son retour au repos. Cette évolution, où précession libre, est appeléeen anglais Free Induction Decay ou FID. Cette précession libre peut être détectée, par exemple, parinduction à l’aide d’une bobine dans le plan xOy, puisque le mouvement de l’aimantation induit decette manière une tension oscillante aux bornes de celle-ci. Ce signal en tension, ou signal de précessionlibre, peut être analysé à l’aide d’un spectromètre pour en révéler le contenu spectral et l’évolutiontemporelle. Par ailleurs, le déplacement hors équilibre peut être lui aussi accompli à l’aide d’une bobineplacée dans le plan xOy. L’opération consiste à appliquer une impulsion radio-fréquence (ou RF) à lafréquence de résonance. Ceci a pour e!et de générer un champ statique dans le référentiel tournant del’aimantation, causant une précession de l’aimantation autour de ce dernier. Ainsi, l’aimantation aurepos peut être basculée dans le plan xOy afin d’observer sa précession libre.

Une approche quantique du phénomène de RMN permet de rendre compte de certains aspectsimpossibles à saisir par l’approche classique, notamment les interactions entre spins. L’interaction duchamp magnétique avec les spins est appelée e!et Zeeman et résulte en la séparation des di!érentsniveaux énergétiques du noyau en plusieurs, séparés par une énergie h̄"!0. Le phénomène de RMN con-siste en des échanges de population entre les niveaux issus de l’état fondamental. Comme l’échantillonen RMN est un ensemble de très nombreux spins, on a recours au formalisme de la matrice densitépourdécrire le problème. Ce formalisme permet de bien isoler les aspects énergétiques (liés aux populationsqui sont les termes diagonaux de la matrice densité) des aspects de cohérence (termes non diagonaux,appelés cohérences). Un résultat principal est que l’espérance d’une observable peut être calculée grâceà la matrice densité. On a en e!et

#A$ = TrA#, (5)

où A est une observable. On peut de plus calculer l’évolution temporelle de la matrice densité à partirde l’Hamiltonien du système suivant

d#

dt= !i[H,#], (6)

où # est la matrice densité et H est l’Hamiltonien du système. Cet Hamiltonien inclut toutes les inter-actions auxquelles le système est soumis. On ne considère en général que les interactions du systèmeavec le champ magnétique externe, le champ radio-fréquence et les interactions internes (entre spins),résumant toutes les interactions avec l’environnement (en particulier le réseau) à des phénomènes derelaxation. Ainsi, on peut montrer que l’application d’un champ radiofréquence induit un échangede population entre les di!érents niveaux et une création de cohérences. On trouve deux impul-sions particulières : l’impulsion" !

2 " qui permet d’égaliser les populations et d’obtenir des cohérencesmaximums, et l’impulsion "$" qui permet d’inverser les populations. Ces cohérences correspondentdans la représentation classique à l’aimantation transverse en précession. L’observation d’un signal deprécession libre dans le plan xOy par une bobine correspond à l’enregistrement de l’évolution tem-porelle de l’espérance d’une observable particulière. Cette espérance est directement proportionnelleaux cohérences. La perte de cohérence induit donc une perte de signal. Une distribution de champinhomogène peut notamment induire une telle perte de cohérence. Ce type de destruction de co-hérence est dit réversible alors que la perte de cohérence due à la relaxation transverse est irréversible.

3

L’expérience RMN la plus simple consiste à appliquer une impulsion !2 et à observer le signal induit

dans la bobine de détection.Nous donnons par la suite des exemples d’expériences et de séquences. L’écho de Hahn revêt une

importance toute particulière pour ce travail puisqu’il permet de "réanimer" le signal après une pertede cohérence réversible. Il consiste en l’application d’une impulsion !

2 suivie d’une impulsion $ aprèsun temps %1 fixé. Cette dernière impulsion a pour e!et d’inverser l’état du système en transformantl’avance des spins soumis à un champ plus fort en retard et le retard des spins soumis à un champplus faible en avance. Le signal se "refocalise" ainsi un temps %1 après l’impulsion $. Une améliora-tion de cette séquence consiste en une série d’impulsions $ intercalées d’observations. On a ainsi untrain d’échos dont l’amplitude décroît avec la relaxation transverse (constante de temps T2). Cetteamélioration est appelée la séquence CPMG (Carr-Purcell-Meiboom-Gill, du nom de ses inventeurs).

Nous terminons notre revue des bases de RMN par des considérations concernant le rapport signalsur bruit du signal de RMN. Ces considérations sont basées sur l’estimation de la polarisation del’échantillon à l’équilibre thermique et la sensibilité de la bobine de détection. On donne en particulierune figure de mérite pour les détecteurs par induction, dérivée du principe de réciprocité et du bruitthermique de Johnson. Cette figure de mérite est le rapport entre l’intensité du champ généré dans leplan perpendiculaire à

!"B0 et la racine de la puissance dissipée dans la bobine à cet e!et.

Il est nécessaire d’e!ectuer les expériences de RMN dans un champ contrôlé. La spectroscopieRMN consiste à observer des décalages de fréquence de l’ordre de quelques ppm dus à de très faiblesinteractions entre les noyaux. Il faut donc que le champ magnétique ait une valeur constante avecune tolérance meilleure que 1 ppm sur le volume de l’échantillon. On exige généralement 1 ppb. Demême l’imagerie par résonance magnétique, bien qu’elle fasse usage de gradients de champ magnétiquepour l’encodage spatial des spins, nécessite un champ statique extrêmement homogène (10 ppm dansune sphère de diamètre 60 cm pour un imageur standard). Il faut donc apporter une attention touteparticulière à l’homogénéité du champ lors de la conception d’un aimant pour la RMN.

Nous discutons par ailleurs dans ce chapitre certains des paramètres des matériaux magnétiquesessentiels à ce travail. Nous nous intéressons ici aux matériaux dits permanents, c’est à dire quimaintiennent une aimantation en l’absence de champ extérieur. Ces matériaux sont souvent anisotropes(ils admettent un axe de facile aimantation) et suivent une courbe d’hysteresis après leur premièreaimantation. On peut alors définir :

• la rémanence du matériau qui consiste en l’aimantation persistante lorsque le champ extérieurest ramené à zéro. On la note généralement Br (Br = µ0Mr où Mr est l’aimantation rémanente).

• la coercivité qui correspond à la valeur du champ extérieur H opposé à l’aimantation rémanentepour laquelle l’aimantation s’annule. On la note HCi. On parle aussi de coercivité intrinsèque. Ilexiste aussi une autre définition de coercivité qui consiste en la valeur du champ externe H telleque l’induction magnétique dans le matériaux soit annulée (

!"B = µ0(

!"H +

!"M)). Cette définition

de coercivité est généralement notée HCB.

• l’aimantation à saturation consiste en l’aimantation maximum que le matériau peut atteindredans un champ externe. Il est généralement nécessaire d’atteindre la saturation au moment del’aimantation du matériaux afin d’obtenir la meilleure coercivité possible.

• le champ d’anisotropie est une quantité correspondant à l’intensité du champ magnétiqueorthogonal à l’axe d’aimantation nécessaire pour incliner l’aimantation de 90º. Cette quantitéest généralement très élevée dans les matériaux modernes (plus de 20 T). On peut donc en règlegénérale négliger l’inclinaison de l’aimantation.

4

Les propriétés de deux matériaux modernes (NdFeB et SmCo) sont données et le choix du matériauxfinal est expliqué. Nous avons retenu le NdFeB pour ses meilleures rémanence et coercivité ainsi queson prix plus bas et sa moins grande fragilité. Par contre, sa sensibilité à la température est trois foisplus élevée que celle du SmCo et peut être un problème dans le contexte de la RMN (variation derémanence de l’ordre de 1000 ppm/K).

Finalement, nous donnons une revue de l’état de l’art dans le domaine de l’utilisation des aimantspermanents en vue d’expériences de RMN. Nous commençons par remarquer que le développementinitial de la RMN s’est accompagné de l’augmentation progressive du champ magnétique externe.L’intensité du champ magnétique externe détermine en e!et la polarisation de l’échantillon, et doncle signal sur bruit, mais a!ecte aussi la résolution. Il y a donc un double intérêt dans l’accroissementdu champ. Ceci a stimulé le développement des aimants supraconducteurs qui permettent d’atteindredes champs très importants à des coûts d’opération bien moindres qu’avec des aimants résistifs. Ona ainsi atteint récemment la fréquence de Larmor de 1 GHz pour le proton (environ 23.5 T). Onnote par contre que ces aimants sont devenus de plus en plus volumineux, coûteux à l’achat et àl’installation, nécessitent des réapprovisionnements réguliers en azote et hélium liquide, et n’o!rentpas un accès facile de part leur géométrie et l’impossibilité de les déplacer. On voit alors l’intérêt desmatériaux aimantés qui ne nécessitent pas de maintenance, sont compacts et peuvent être déplacésplus facilement. Ces matériaux o!rent aussi une plus grande flexibilité sur la géométrie de l’aimantet ses propriétés magnétiques (position de la région d’intérêt, direction du champ). On peut en e!etenvisager des aimants dits in situ de géométrie semblable aux aimants supraconducteurs et nécessitantd’insérer le sujet d’analyse au centre de l’aimant. Cette configuration permet d’obtenir des champsplus intenses. On peut aussi considérer des configurations dites ex situ générant un champ utile à laRMN en dehors de l’aimant et o!rant donc un accès beaucoup plus aisé.

Les aimants permanents ont d’ailleurs fait l’objet de nombreux travaux depuis les quinze dernièresannées en vue du développement de la RMN portable. Ce dernier concept a été développé depuisle début des années 50 pour la diagraphie, mais ce n’est que depuis 1995 que des prototypes de trèspetite taille méritant le nom d’appareils portables ont été développés. Diverses configurations ont étéexplorées, dont la structure de Halbachqui permet d’obtenir des champs plus intenses avec moins dematériaux, mais aussi des structures ex situ qui permettent d’e!ectuer des analyses RMN de surfaces.Ces appareils n’ont pour la plupart pas permis d’obtenir l’homogénéité nécessaire à la RMN hauterésolution et se sont limités à des analyses de temps de relaxation, de constante de di!usion et à del’imagerie basse résolution. De plus, très peu d’éléments ont été donnés pour la mise en place deméthodes systématiques pour la conception, la fabrication et la caractérisation de ce type d’aimants.

Ce travail se place dans le contexte de l’amélioration des performances RMN d’aimants à basede matériaux aimantés. L’obtention de l’homogénéité est l’objet principal de notre attention et nousaspirons à mettre en place une méthode générale systématique pour concevoir des aimants adaptésà la RMN. Cette méthode doit s’étendre à la fabrication et à la caractérisation afin de garantir unecohérence entre la conception et l’objet final.

Chapitre 2 : Méthode générale de conception d’aimants

Nous décrivons dans ce chapitre une approche générale pour la conception et la fabrication d’aimants.Largement inspirée de la méthode utilisée pour les aimants supraconducteurs, elle se base sur la Dé-composition en Harmoniques Sphériques (DHS) du potentiel magnétique et des composantes du champ

5

qui en dérivent. On peut alors écrire le potentiel et chaque composante du champ sous la forme

&!(r, ',&) =1µ0

#Z0 +

"$

n=1

rn

%ZnPn(cos ') +

n$

m=1

(Xmn cos m& + Y m

n sinm&)Pmn (cos ')

&', (7)

où nous avons utilisé les coordonnées sphériques. Les termes Zn, dits axiaux et les termes Xmn et Y m

n

dits non-axiaux sont déterminés par la géométrie de l’aimant et peuvent donc être contrôlés par leconcepteur. Par ailleurs, le terme en rn donne la vitesse de variation associée à un degré. Si nousposons r0 comme une dimension caractéristique de l’aimant (par exemple le rayon du trou d’accèsau centre magnétique), on s’aperçoit que les termes axiaux et non-axiaux sont en 1

rn0, si bien que la

dépendance radiale s’écrit(

rr0

)navec r < r0. Ainsi, les termes de degré plus élevé ont une variation

plus lente à partir du centre. Le DHS des composantes du champ peut se déduire directement de celuidu potentiel par des relations que nous donnons.

Cette dernière remarque est capitale pour l’obtention de l’homogénéité. Nous désirons idéalementobtenir un champ constant, c’est à dire ne contenant que le terme de degré zéro. Bien que cela nesoit pas physiquement possible, l’annulation des termes de degré supérieur jusqu’à un degré n0 donnépermet de garantir une tolérance de variation dans un volume donné. Les variations du champ sontalors dominées par le degré n0. Ainsi, la réalisation de l’homogénéité revient à annuler un nombresu"sant de termes dans le DHS du champ. En pratique il nous faut spécifier le rayon du volumed’homogénéité désiré, ainsi que la tolérance de variation exigée. Ceci permet de déterminer le degréminimum nécessaire des premiers termes non nuls. On aura ainsi pour un rayon désiré R0 et unevariation maximum du champ !B0

n0 %ln !B0

ln rR0

. (8)

Nous devons alors discuter de la pertinence de ces remarques du point de vue de la RMN. En e!et,les DHS ne peuvent concerner qu’une quantité scalaire (un potentiel ou une composante du champ),or, la RMN est sensible au module du champ, c’est à dire les trois composantes à la fois. Cettesituation se résout en notant que toutes les composantes du champ dérivent du potentiel et qu’ainsi laconnaissance d’une composante est su"sante pour toutes les connaître. D’autre part, dans le contexted’un champ homogène, nous démontrons que les variations de la composante à l’origine dominent lesvariations du module (les fluctuations des autres composantes ne contribuent qu’à l’ordre deux alorsque la composante principale contribue à l’ordre un). Ainsi, il su"t de contrôler les variations de lacomposante principale pour s’assurer de l’homogénéité du module.

Il se peut que le profil désiré ne soit pas un champ uniforme mais une forte variation linéairedu module dans une direction donnée. Ceci permet d’encoder l’espace par la fréquence RMN. Onest alors confronté à un champ largement inhomogène, si bien que les variations de la composanteprincipale ne su"sent plus à décrire celles du module. Il faut alors considérer les autres composantes.Nous proposons un formalisme approprié dans le cas d’une configuration axi-symmétrique générantun gradient intense le long de son axe. Ce formalisme met en évidence l’impossibilité de maintenirla constance du gradient le long de l’axe tout en maintenant l’uniformité du champ dans les planstransverses. Un choix doit être fait entre l’une ou l’autre de ces propriétés. La RMN trouve une plusgrande utilité en l’uniformité du champ dans des plans transverses et c’est donc cette propriété quel’on retiendra.

Le travail d’homogénéisation se résume donc à l’annulation d’un nombre de termes dans le DHSde la composante principale du champ. Nous pouvons nous servir des propriétés de symétrie del’équation 7 afin de nous simplifier la tâche. Il est notable qu’une symétrie axiale d’ordre k permetd’annuler tous les termes non-axiaux jusqu’à l’ordre k, alors qu’un symétrie plane contenant l’origine

6

permet de se débarrasser d’un terme axial sur deux. Ces symétries nous permettent donc de simplifierle problème de manière significative. Les quelques termes qu’il reste à annuler peuvent être traitéspar des méthodes de minimisation standards en se basant sur des expressions analytiques des termes.Nous avons calculé de telles expressions pour di!érentes géométries remarquables telles que des anneauxcylindriques ou polygonaux aimantés selon l’axe ou radialement, mais aussi pour la configuration deHalbach. Ces formules permettent de trouver les paramètres optimaux pour la géométrie satisfaisant lesconditions d’homogénéité et générant le maximum de champ pour un volume de matériau donné. Nousdonnons aussi les expressions analytiques du DHS du potentiel généré par un dipôle. Ces dernières serévéleront d’une utilité toute particulière pour l’approximation rapide d’une structure ou l’ajustement("shimming") d’un aimant par des petites pièces aimantées situées à une distance importante de la zoned’intérêt. Nous utilisons en outre de nos calculs analytiques un logiciel de simulation numérique appeléRadia et distribué par l’ESRFqui nous a permis de comparer nos calculs, modéliser des structures ete!ectuer des analyses d’erreurs et de désaimantation.

Le contrôle précis de l’orientation de l’aimantation n’est pas facilement réalisable en pratique. Il estdonc plus réaliste d’optimiser la géométrie de la structure en se basant sur des blocs dont l’orientationd’aimantation est fixée a priori. Le choix de cette orientation pour chaque aimant ne doit pas être faitau hasard et nous donnons une analyse de l’e"cacité de deux orientations d’aimantation particulières(longitudinale et radiale) suivant leur position dans l’espace pour générer un champ longitudinal dansle cadre d’un aimant axi-symmétrique. Il en résulte un découpage de l’espace en régions privilégiéespour chaque orientation.

Une autre nécessité pour une approche générale de la fabrication de ce type d’aimant est de sedoter de moyens de mesure capables de caractériser précisément les variations du champ et d’obtenirles valeurs expérimentales du DHS. Ceci permet d’avoir une cohérence entre la conception théorique,l’étude des imperfections, la caractérisation de l’aimant réel et les corrections possibles. Nous décrivonsles écueils liés au calcul des termes du DHS à partir d’un ensemble de points de mesure et proposons uneméthode pour échantillonner dans l’espace les variations du champ, à partir de micro-bobines RMN.Deux situations particulières sont discutées, l’une concernant la mesure de champ homogène et l’autrela mesure d’un champ comportant un fort gradient dans une direction donnée. On souligne ainsi ladi"culté grandissante pour e!ectuer des mesures précises lorsque les variations du champ augmentent.Une solution pour maintenir une grande résolution RMN malgré un fort gradient est aussi discutée,basée sur l’utilisation d’une bobine de compensation générant localement un gradient opposé à celuique l’on mesure. Les erreurs de mesure sont estimées et indiquent que l’on peut espérer une précisionsu"sante pour caractériser et corriger un aimant soumis aux tolérances de fabrication.

Chapitre 3 : Aimants in situ

Ce chapitre est consacré à l’utilisation des considérations données au chapitre 2 pour la conceptiond’aimants du type in situ. Cette configuration, présente deux avantages importants : elle permet deschamps plus intenses puisqu’elle autorise l’échantillon à être entouré de matériaux magnétiques maiselle permet aussi de simplifier le problème d’homogénéité en supprimant un terme axial sur deux.C’est donc la configuration à choisir dans le cas où l’application n’a pas des exigences d’accessibilitéextrêmes.

La réalisation expérimentale d’un premier prototype très simple et bon marché nous a permis demettre à l’épreuve les éléments théoriques décrits dans le chapitre précédent, tout en accumulant del’expérience en matière de manipulation, assemblage et mesures. Basé sur un principe proposé audébut des années 90 par Aubert, il crée un champ longitudinal à partir de deux couronnes aimantéesde manière radiale. Le modèle théorique peut atteindre le degré quatre et fournir environ 15 ppm

7

d’homogénéité dans un diamètre de 3 mm pour un champ de 0.12 T. Les couronnes sont réaliséesà partir de cubes achetés en gros. Nous avons pu ainsi mettre en application notre méthodologiede mesure et mettre en place une procédure de caractérisation des cubes en vue de les trier. Lesmatériaux magnétiques ont en e!et une mauvaise répétabilité qui induit des imperfections sévères dansla structure, lorsque l’on compare au modèle théorique. Nous avons ainsi été amenés à "shimmer" cetaimant à l’aide de cubes aimantés de plus petite taille. Ceci a été fait à partir de mesures expérimentalesdu DHS du champ et du calcul des positions nécessaires de chaque cube de "shim". Une homogénéitéd’environ 12 ppm dans un volume de 2 mm de long par 1 mm de diamètre a pu être atteinte, avecun champ d’environ 0.12 T (environ 5 MHz pour le proton). Nous avons ici mis à contribution lesformules du dipôle démontrée dans le chapitre précédent.

Suivant la discussion de ce premier prototype, nous proposons le calcul théorique d’une structurebasée sur le même principe mais beaucoup plus complexe. Cette structure est hautement homogène,le premier terme apparaissant au degré douze. Elle o!re un champ de 0.92 T (environ 39 MHz pour leproton) avec une homogénéite de l’ordre de 0.5 ppm dans une sphère de diamètre 20 mm. La longueurde l’aimant est de 200 mm pour un diamètre extérieur de 200 mm. Le trou a un diamètre de 50 mmet la structure pèse environ 35kg. Ce modèle théorique démontre ainsi la puissance de notre méthodepour atteindre une homogénéité théorique désirée.

Parmi les aimants in situ, une structure a particulièrement attiré l’attention des travaux e!ectuésjusqu’à présent à cause de son e"cacité supérieure. Cette structure est dite "de Halbach", du nom deson inventeur. Elle génère un champ transverse dont l’intensité est supérieure à la plupart des autresstructures pour un même volume de matériau aimanté et, permet en théorie (en deux dimensions)d’obtenir un champ homogène en son centre et un champ nul à l’extérieur. Nous revisitons ici cettestructure à l’aide des éléments fournis au chapitre deux et donnons un modèle d’aimant de typeHalbach en trois dimensions homogène au degré six (premier degré non nul). Cet aimant est capablede fournir 0.635 T (environ 27 MHz pour le proton) en théorie avec un trou intérieur de diamètre70 mm, un diamètre extérieur de 120 mm et une longueur de 160 mm (environ 9 kg de NdFeB).L’homogénéité théorique dans une sphère de diamètre 10 mm est meilleur que 1 ppm. Nous nousattendons par contre à ce que l’homogénéité d’un tel aimant soit "victime" des champs démagnétisants.Nous discutons à l’occasion de ce modèle du problème du passage du modèle parfait à aimantationchangeant continûment d’orientation dans la structure à un modèle segmenté dont l’aimantation changed’orientation de manière discrète. Nous montrons qu’une structure de section cylindrique segmentéeen douze ne di!ère pas de la structure parfaite de manière visible. Il faut par contre légèrement ajusterles paramètres géométriques de l’aimant pour réaliser une structure segmentée de section polygonale.

La délimitation en trois dimensions de l’aimant et la variation de l’orientation de l’aimantation enson sein le soumettent à un champ opposé à son aimantation (d’où le terme "démagnétisant") qui variedans la structure. Ceci résulte en une distribution de points de fonctionnement (position sur la courbed’hystérésis) dans la structure et donc une aimantation inhomogène en module (alors que le modèlethéorique prévoit un module identique dans toute la structure). Ceci a!ecte directement l’intensité duchamp au centre, ainsi que son homogénéité. On s’attend ainsi à une dégradation de l’homogénéité duchamp de plus de 400 ppm. Nous montrons alors que ceci peut être compensé par l’utilisation de blocsdont le module de l’aimantation initiale di!ère. Nous proposons ainsi une structure à douze segmentscompensant la désaimantation et réalisant une homogénéité meilleure que 10 ppm dans un centimètrede diamètre.

Nous avons ainsi rapidement couvert les deux cas de figure principaux pour les aimants in situ,démontrant le potentiel théorique de la méthode décrite au chapitre deux. La réalisation d’un premierprototype simple a permis de vérifier la pertinence expérimentale de nos développements théoriques. Ceprototype o!re des performances excellentes étant donné son coût et permet d’envisager la réalisation

8

de prototypes plus complexes.

Chapitre 4 : Un système de RMN/IRM ouvert. Partie 1 : l’aimant

Un des aspects attrayants des aimants permanents est la possibilité d’explorer des configurations plusoriginales, o!rant un accès plus facile à la région d’intérêt en laissant la moitié de l’espace libre. Lesstructures o!rant cette possibilité sont dites ex situ.

Nous considérons dans ce chapitre des aimants axi-symmétriques dont l’aimantation est exclusive-ment longitudinale. Ceci permet de simplifier le problème et revêt un aspect pratique : l’aimantpeut être aimanté une fois assemblé. Ceci simplifie les opérations de montage et permet de "ranimer"l’aimant en cas d’accident et de désaimantation critique. L’aimant s’apparente alors à un cylindreaimanté de manière uniforme selon son axe. Nous donnons le rapport rayon sur hauteur donnant lemaximum de champ à une distance donnée du cylindre. Ces considérations permettent de remarquerque le champ passe par un maximum lorsque l’on augmente le rayon. Par ailleurs, on s’aperçoit quel’augmentation du rayon permet d’améliorer l’homogénéité du champ généré par le cylindre.

Nous nous attachons ensuite à utiliser les développements du chapitre deux afin de contrôler le profildu champ généré par un tel aimant. Nous avons abouti à des modèles d’aimants basés sur un cylindrecoi!é d’anneaux polygonaux. Ces anneaux permettent la correction des di!érents termes non désirésdans le DHS. Le travail est doublé par rapport à un aimant in situ puisque nous ne pouvons pas faireusage de la symétrie plane. Deux systèmes théoriques d’aimants générant un champ perpendiculaireà leur surface sont envisagés : l’un produit un champ uniforme à 30 ppm près dans un diamètre de9 mm à 18 mm de la surface de l’aimant, avec un champ d’environ 150 mT (environ 6.4 MHz pourle proton), l’autre produit un gradient de champ de 3.3 T m#1 avec un champ de 330 mT (environ14 MHz pour le proton). L’aimant créant un gradient de champ peut être conçu pour obtenir ungradient extrêmement constant (à 3 ppm près sur 1 cm) le long de l’axe Oz, mais avec des variationsde champ dans le plan xOy de l’ordre de 350 ppm dans un diamètre de 1 cm. Il peut aussi être conçupour obtenir un gradient variant légèrement le long de l’axe Oz (environ 0.1% sur 1 cm) mais avecun champ constant dans les plans transverses (moins de 1 ppm dans un diamètre de 1 cm). Un telgradient trouve un usage dans la RMN localisée et permet de faire des images en une dimension. Legradient permet en e!et d’adresser l’espace en fréquence de Larmor. Le gradient très intense associéà de très faibles variations dans les plans transverses permet d’obtenir une résolution très importante.Ainsi, on peut espérer un résolution de 30 µm sur un volume de 1 cm de diamètre par 1 cm de hautdans le cas du gradient constant le long de Oz et meilleure que 1 µm dans le même volume pour l’autrecas. On s’attend à des distorsions de l’image mineures dans le dernier cas puisque le gradient varie de0.1 % sur le volume d’intérêt.

Nous e!ectuons ensuite une analyse d’erreurs. Nous nous attendons en e!et à ce que l’aimantfabriqué ne corresponde pas au modèle théorique du fait des imperfections de fabrication. Ces imper-fections sont magnétiques (l’aimantation des di!érents blocs est connue avec une certaine tolérance surle module et l’orientation) et géométriques (les pièces sont fabriquées et positionnées avec une certainetolérance). L’aimant est de plus sujet à la désaimantation qui induit un point de fonctionnementinhomogène dans l’aimant. Nous avons estimé l’e!et de ces imperfections sur le profil du champ enutilisant notre modèle d’erreurs et le logiciel Radia pour les calculs magnétiques. Les résultats ontété traduits en termes de DHS du champ et ont montré que l’on doit s’attendre à des imperfectionsassez importantes (plusieurs centaines de ppm de variation de champ dans le volume d’intérêt). Nousmontrons alors qu’il est possible de corriger ces imperfections par des mouvements mineurs des piècessupérieures de l’aimant. La procédure de correction est discutée et testée numériquement, donnant desrésultats concluants.

9

Forts des résultats analytiques et numériques, nous avons pu aborder la fabrication d’un prototyped’aimant ex situ générant un gradient avec plus de confiance. Les pièces magnétiques ont été sous-traitées à un industriel et nous avons conçu une mécanique adaptée afin de contenir, protéger etdéplacer les aimants. Nous avons aussi fabriqué un système d’assemblage permettant d’une part dedéplacer les pièces supérieures de manière micrométrique, et d’autre part de les enlever et remettreune fois le système aimanté (on s’attend à ce que des forces très importantes, de l’ordre de 100 kg,s’appliquent sur les pièces supérieures du fait de leur proximité avec la base cylindrique. Les aimantssupérieurs sont collés dans des montures qui permettent de les raccorder au système d’assemblage etde les désaccoupler une fois les réglages achevés en les fixant sur la structure de l’aimant.

Nous avons pu mettre en application la méthode de mesure décrite au chapitre deux afin de car-actériser l’aimant final et de l’ajuster. Nous avons pu ainsi retrouver en partie le profil initialementsouhaité. Nous avons par contre remarqué la présence anormale de certains termes du DHS. L’originede se problème est di"cile à trouver précisément mais l’explication réside probablement dans une com-binaison d’erreurs de mesure (mauvais centrage de la mesure) et d’erreurs de fabrication de l’aimant.Le choix initial du concept s’était porté sur le gradient constant le long de l’axe Oz et s’est avéréultérieurement peu utile pour la RMN. Ceci a pu être partiellement compensé par le déplacement despièces supérieures qui a permis de placer l’aimant dans une configuration rendant les variations trans-verses du champ très faible, quoique dans une moindre mesure que si l’aimant avait été conçu pour.Nous sommes ainsi parvenus à obtenir des variations de quelques dizaines de ppm dans un plan de1 cm de côté au centre de la région d’intérêt. Les performances de l’aimant pourraient être amélioréespar le remplacement des pièces supérieures par des blocs plus adaptés au profil de champ désiré.

Nous avons aussi vérifié la stabilité de l’aimant au cours du temps et avec la température. LeNdFeB est en e!et très sensible à la température (sa rémanence varie d’environ 1000 ppm/K). Il estdonc important de vérifier que le profil de champ est stable et que la variation temporelle du champnominal se corrèle bien avec les variations de température. Nous avons donc enregistré la températureau cours du temps en même temps que nous avons cartographié de manière répétée l’aimant. Ceci apermis de vérifier qu’il n’y avait pas de variations notable des termes du DHS sur de longues durées(huit heures) et qu’on observait bien une très bonne corrélation du champ nominal avec la température.Nous avons ainsi trouvé un coe"cient de 900 ppm/K en utilisant la température du châssis de l’aimantcomme référence.

Nous avons donc décrit théoriquement des modèles d’aimants ex situ et e!ectué une analysed’erreurs détaillée en vue de la fabrication d’un de ces modèles. Suivant ces bases, nous avons réaliséun prototype expérimental et avons ainsi pu montrer la pertinence de la méthode de conception parun prototype fonctionnel vérifiant raisonnablement les performances initialement annoncées.

Chapitre 5 : Un système de RMN/IRM ouvert. Partie 2 : détectionde signaux RMN

En vue d’une utilisation en RMN, le prototype d’aimant ex situ doit être intégré dans un système deRMN qui inclut un spectromètre et des amplificateurs (amplificateur de puissance pour les impulsionsRF, et préamplificateur bas bruit pour les signaux RMN) , disponibles dans le commerce spécialisé.Il faut en plus une sonde RMN, c’est à dire une bobine adaptée à la détection dans la configurationqui nous intéresse et un circuit d’accord. Nos avons décidé de réaliser nous-mêmes cette partie de lachaîne de détection. En se basant sur le concept bien connu de la fonction flux, nous proposons uneméthode analytique de calcul pour la conception de bobines aux propriétés diverses dans la limite del’approximation quasi-statique. Le contours de la fonction flux permettent de dessiner directement lespistes d’un circuit imprimé qui permettent d’obtenir la distribution de courants nécessaire pour obtenir

10

les propriétés désirées. Nous donnons l’exemple d’une bobine permettant d’optimiser la sensibilité dedétection du signal RMN. Cette bobine est plate puisque réalisée à l’aide de la technologie de circuitsimprimés et permet de générer un champ B1 parallèle à la surface de l’aimant, permettant donc ladétection du signal RMN lorsqu’elle est conjuguée avec l’aimant fabriqué au chapitre quatre. L’allureet la taille de la bobine sont déterminées par deux simples paramètres qui sont le rayon maximumde la bobine et la distance sur l’axe de la bobine à laquelle on désire e!ectuer la détection. On serend compte qu’en théorie, il est important d’adapter la bobine à la distance de détection car unebobine définie pour une longue distance de détection, fournit une moins bonne sensibilité à courtedistance qu’une bobine conçue pour la détection de proximité. Nous avons fabriqué quelques bobinesà partir de ces résultats et e!ectué quelques vérifications expérimentales. La valeur de B1 déduite desimpulsions RF nécessaire à un !

2 correspond bien aux mesures e!ectuées à l’aide d’une sonde de Hall enrégime statique (un courant continu alimente la bobine), indiquant que l’approximation quasi-statiqueest valable. On a par contre une réduction d’un facteur deux en sensibilité par rapport au calculthéorique. Ceci peut sans doute s’expliquer par l’ablation d’une partie de la bobine pour faciliter lafabrication, ainsi que par la perturbation de la distribution de courants due à la connexion entre lespistes. Enfin, on remarque que l’augmentation de la taille de la bobine n’améliore pas le signal surbruit e!ectif lors de la détection RMN. Nous suspectons que des sources de bruit dont nous n’avonspas tenu compte n’auraient pas dues être négligées mais nous n’avons pas poussé plus avant l’étude deces bobines.

En e!et, le but de la fabrication de ces bobines est avant tout d’obtenir un détecteur e"cacepour e!ectuer des expériences de RMN ex situ. Nous avons ainsi pu démontrer la possibilité demesurer des temps de relaxation T1 pour des échantillons comme de l’eau et de l’huile alimentaire.Nous nous sommes concentré principalement sur les capacités d’imagerie 1D de l’appareil et avonsmesuré la réponse percusionnelle de l’instrument, obtenant une résolution de l’ordre de 15 µm, bienque l’alignement de l’échantillon avec l’appareil soit très délicat. Enfin, nous avons pu faire l’imageen un spectre d’un empilement de lamelles de microscope espacées par des films d’huile, démontrantla possibilité d’e!ectuer des images dans une épaisseur de 1 mm en un seul spectre, sans déplacerl’échantillon ni ré-accorder la sonde. La limite de profondeur de l’image est principalement donnée parla bande passante du détecteur accordé. Ainsi, il su"t de changer l’accord de la sonde, sans déplacerl’échantillon pour observer une autre région de celui-ci.

Nous avons donc pu intégrer l’aimant ex situ avec une sonde RMN plane et nous avons pu détecterdes signaux RMN avec un rapport signal sur bruit raisonnable, démontrant une excellente résolutionpour l’imagerie 1D. Nous avons aussi démontré la possibilité d’e!ectuer de simples expériences deRMN comme la mesure de temps de relaxation. Bien d’autres expériences sont possibles à l’aide decet appareil a priori. Il reste beaucoup à faire afin de terminer l’étude de ses capacités et de le mettreen situation d’applications pratiques concrètes. Les bobines décrites dans ce travail nécessitent aussiune étude plus détaillée pour mieux comprendre leur performances.

Conclusions et perspectives

Nous avons discuté dans ce travail des aspects théoriques et pratiques de la conception et de la fabri-cation d’aimants permanents pour la RMN. Nous avons proposé plusieurs modèles d’aimants à hauteperformance en homogénéité et en champ dans les configurations in situ et ex situ. Basés sur la théoriedu DHS, nous avons mis en place un formalisme pour l’obtention de l’homogénéité du champ et lecontrôle des variations spatiales d’un gradient de champ. Enfin, ce formalisme a aussi été mis en appli-cation pour la mesure des aimants et l’analyse des erreurs de fabrications. Nous avons ainsi pu aborderla fabrication de prototypes dotés de tous les outils nécessaires à une réalisation pratique réussie. Nous

11

avons ainsi démontré les performances d’homogénéité raisonnables d’un petit prototype in situ générantun champ longitudinal pour un très bas coût. Nous avons aussi fabriqué un prototype plus complexepermettant la RMN ex situ. Ce prototype génère un fort gradient de champ, permettant d’e!ectuer del’imagerie en une dimension. Nous avons pu démontrer la possibilité de contrôler en pratique le profildu champ. Cet aimant a été intégré avec des bobines de surface conçues spécifiquement à l’aide d’uneméthode que nous décrivons. Nous avons ainsi pu réaliser quelques expériences de RMN démontrantles capacités du système, notamment une résolution de 15 µm sur plusieurs millimètres de profondeur.On peut envisager d’utiliser ce système en combinaison avec des gradients plats conçus à l’aide de lamême méthode utilisée pour la bobine RF, afin de réaliser des images 3D.

Ce travail a permis d’établir une approche systématique et cohérente de la fabrication d’aimantspermanents. Ceux-ci semblent avoir un intérêt grandissant pour la RMN où ils peuvent trouver toutessortes d’applications. On peut citer entre autres l’application à la rhéologie qui bénéficierait desinformations de flot apportées par la RMN. Le principal inconvénient des aimants permanents, lemanque de sensibilité RMN, pourrait sans doute être en partie résolu par l’utilisation de techniquesd’hyperpolarisation, comme la Polarisation Dynamique Nucléaire (en anglais Dynamic Nuclear Polar-ization, ou DNP) qui semble avoir un maximum d’e"cacité dans les valeurs de champ des aimantspermanents.

Enfin, dans un futur lointain, on peut imaginer qu’une fois les problèmes de sensibilité résolus, desappareils d’IRM et de RMN basés sur des aimants permanents compacts pourront être fabriqués enmasse, rendant ces techniques beaucoup plus abordables financièrement. On pourrait ainsi, qui sait,un jour trouver un appareil d’IRM dans le coin du cabinet du médecin de famille.

Contents

1 Introduction to NMR and permanent magnet based NMR 171.1 Basics of NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.1.1 Classical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.1.2 Quantum description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.1.2.1 Zeeman Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.1.2.2 Density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.1.2.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1.3 Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.1.3.1 !

2 pulse-acquisition (neglecting relaxation) . . . . . . . . . . . . . . . . 241.1.3.2 Hahn echo and CPMG . . . . . . . . . . . . . . . . . . . . . . . . . . 241.1.3.3 Nutation echoes and composite pulses . . . . . . . . . . . . . . . . . . 26

1.1.4 Self-di!usion and field gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.1.5 NMR and magnetic field homogeneity . . . . . . . . . . . . . . . . . . . . . . . 271.1.6 Signal and noise in NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.1.6.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.1.6.2 Detection and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.2 Magnetic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2.1 Magnetostatics and materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2.2 Materials of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.3 The development of permanent magnets in NMR . . . . . . . . . . . . . . . . . . . . . 351.3.1 From low-field to high-field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.3.2 Back to low-field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.3.3 Magnets for low-field NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.3.3.1 In situ magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.3.3.2 Ex situ magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.3.4 Motivation for this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2 General approach to magnet design and fabrication 472.1 Preliminary remarks on field homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . 472.2 Homogeneity and Spherical Harmonic Expansions (SHE) . . . . . . . . . . . . . . . . . 48

2.2.1 Theoretical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.2.2 Design methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3 Analytical bricks for axisymmetric magnets . . . . . . . . . . . . . . . . . . . . . . . . 532.3.1 Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.2 Creation of a longitudinal field with rings . . . . . . . . . . . . . . . . . . . . . 55

2.3.2.1 Cylindrical rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.3.2.2 Polygonal rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

13

14 CONTENTS

2.3.2.3 E"ciency of longitudinal and radial magnetization . . . . . . . . . . . 592.4 Generation of a transverse field with Halbach rings of finite length . . . . . . . . . . . 602.5 Field modulus profile control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.6 Conclusions on the theory for magnet design . . . . . . . . . . . . . . . . . . . . . . . . 642.7 Magnetic field measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.7.1 Magnetic field measurement devices . . . . . . . . . . . . . . . . . . . . . . . . 642.7.1.1 Hall probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.7.1.2 NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.7.2 Retrieving SHE terms from field measurements or computations . . . . . . . . . 682.7.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.7.2.2 Experimental errors in the extraction of the SHE terms . . . . . . . . 72

2.7.3 Measurements in a strong field gradient . . . . . . . . . . . . . . . . . . . . . . 762.7.3.1 High precision measurements of B0 with NMR . . . . . . . . . . . . . 762.7.3.2 Measuring Bz with NMR and extracting SHE terms . . . . . . . . . . 79

2.8 Conclusions on magnetic field measurements . . . . . . . . . . . . . . . . . . . . . . . . 81

3 In-situ magnets 833.1 Longitudinal field - Aubert configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.1.1 A simple and low-cost test-magnet . . . . . . . . . . . . . . . . . . . . . . . . . 833.1.1.1 Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1.1.2 Part measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.1.1.3 Screening of parts and simulations based on measurements . . . . . . 903.1.1.4 Characterization of the magnet and correction system . . . . . . . . . 913.1.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.1.2 A theoretical highly homogeneous magnet . . . . . . . . . . . . . . . . . . . . . 963.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.2 Transverse field - Halbach magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.2.1 Theoretical design : from analytical to numerical calculations . . . . . . . . . . 993.2.2 Demagnetization considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.3 Conclusions on in situ magnets - Magic-Angle Magnets . . . . . . . . . . . . . . . . . . 102

4 An open NMR/MRI system: The magnet 1054.1 Single-sided magnets in theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.1.1 Preliminary considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.1.1.1 Size and field strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.1.1.2 Size and homogeneity : from a cylinder to a tailored magnet . . . . . 108

4.1.2 Theoretical design examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.1.2.1 Homogeneous field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.1.2.2 Constant gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.1.3 Imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.1.3.1 Demagnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.1.3.2 Definition of imperfection models . . . . . . . . . . . . . . . . . . . . . 1234.1.3.3 E!ects of imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.1.4 Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.1.4.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.1.4.2 E"ciency of the considered correction system . . . . . . . . . . . . . . 1304.1.4.3 Linearity and correction simulations . . . . . . . . . . . . . . . . . . . 133

CONTENTS 15

4.2 Fabrication of an open tomograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.2.1 Fabrication, assembly and magnetization . . . . . . . . . . . . . . . . . . . . . . 1364.2.2 Magnet characterization and adjustment . . . . . . . . . . . . . . . . . . . . . . 139

4.2.2.1 Initial measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.2.2.2 NMR measurements, adjustments and final performance . . . . . . . . 139

4.2.3 Field stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.3 Conclusions on single-sided magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5 An open NMR/MRI system: Coils and NMR experiments 1495.1 Single-sided coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.1.1 Design theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.1.2 RF coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.1.2.1 Optimal sensitivity in one point . . . . . . . . . . . . . . . . . . . . . 1525.1.2.2 Practical realization and experimental study . . . . . . . . . . . . . . 1555.1.2.3 NMR detection performances . . . . . . . . . . . . . . . . . . . . . . . 156

5.2 NMR experiments with an open tomograph . . . . . . . . . . . . . . . . . . . . . . . . 1585.2.1 Relaxation times and di!usion measurements . . . . . . . . . . . . . . . . . . . 159

5.2.1.1 T1 measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.2.2 1-D profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.2.2.1 Point Spread Function and NMR simulations . . . . . . . . . . . . . . 1605.2.2.2 Experimental PSF and layer profiles . . . . . . . . . . . . . . . . . . . 161

6 Conclusions and perspectives 165

Appendices 171

A Magnetic materials 171A.0.3 Magnetic materials modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

A.0.3.1 "Soft" materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173A.0.3.2 "Hard" materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

B Legendre functions and associated Legendre functions 177

C Derivation of SHE for dipoles 179

16 CONTENTS

Chapter 1

Introduction to NMR and permanentmagnet based NMR

1.1 Basics of NMR

1.1.1 Classical description

Nuclear magnetic resonance (NMR) can in principle be observed when a number of atoms for whichthe nucleus has a non-zero spin is placed in a magnetic field

!"B0. An ensemble of a given type of nuclei

exhibits a bulk angular momentum!"I and a bulk magnetization (magnetic moment)

!"M such that

!"M = "h̄

!"I , (1.1)

where " is the gyromagnetic ratio of the considered type of nucleus.There exists a Larmor precession having a pulsation !0 such that

!"!0 = !"!"B0. (1.2)

This Larmor precession is the result of the interaction of the bulk magnetization!"M of the material

with the external field!"B0. There indeed exists a torque

!"T such that

!"T =

!"M & !"B0 (1.3)

It is then possible to write the time variation of the angular momentum as

d!"I

dt=!"T . (1.4)

Hence,d!"M

dt= "

!"M & !"B0, (1.5)

which is the equation of motion for the precession of!"M around

!"B0 with pulsation !0.

The considered system of nuclei (or spins) is however not isolated and is embedded in a latticewith which it is coupled. There is therefore an exchange of energy between the system of spins andthe lattice, resulting in an evolution towards thermal equilibrium between the spin system and thelattice. This is called the spin-lattice relaxation, or longitudinal relaxation. When thermal equilibrium

17

18 CHAPTER 1. INTRODUCTION TO NMR AND PERMANENT MAGNET BASED NMR

is reached, the bulk magnetization is aligned with the external field. We shall now consider that fieldto be along the Oz axis (we will say "longitudinal"). Hence, the spin system will be said to be excitedwhen the bulk magnetization exhibits components along Ox and Oy, and of course when it exhibits acomponent along Oz opposite to the field. When the spin system is excited, a coupling between thespins results in an additional relaxation phenomenon which does not involve an exchange of energywith the lattice. This is called the spin-spin relaxation, or transverse relaxation. It corresponds toa loss of transverse components of the magnetization, without a gain in longitudinal magnetization.This loss of bulk magnetization originates in a loss of coherence between the spins, as we shall see lateron. These two relaxation phenomena can be described by exponential decays with time constants T1

and T2 respectively :

dMz

dt= ! 1

T1(Mz !M0) (1.6)

dMxy

dt= ! 1

T2Mxy. (1.7)

By combining equations 1.5, 1.6 and 1.7, the well known Bloch equation (1) can be obtained,describing the time evolution of the magnetization:

d!"M

dt= "

!"M & !"B0 !

1T1

(Mz !M0)!"uz !1T2

(Mx!"ux + My

!"uy) (1.8)

where !"ux, !"uy and !"uz are unitary vectors respectively in direction Ox, Oy and Oz.If we assume

!"M to be initially along Ox with a magnitude M0, we can give the solution of 1.8 as

Mx(t) = M0 cos !0t exp!! t

T2

"(1.9)

My(t) = M0 sin!0t exp!! t

T2

"(1.10)

Mz(t) = M0

*1! exp

!! t

T1

"+. (1.11)

The evolution of the magnetization as described by these equations is called the free inductiondecay or FID. If a coil of axis parallel to Ox is placed around the sample, the time-varying transversemagnetization will induce a voltage V1 across the coil :

V1(t) ' !!0M0 sin !0t exp!! t

T2

"(1.12)

Similarly, a coil placed along Oy would pickup a signal V2

V2(t) ' !0M0 cos !0t exp!! t

T2

"(1.13)

The signal obtained from this second coil is equivalent to the signal obtained with the first one withan added !

2 phase. The complex combination of both signal V1 + iV2 gives the full record of the FID.Both real and imaginary part are necessary in order to establish what is the direction of precession.The typical shape of a recorded FID is shown on figure 1.1.

When the spin system is at rest, that is to say!"M is parallel to Oz, it is possible to "tip" it through

the use of a radiofrequency (RF) field in the transverse plane, usually called B1. If we superimpose on

1.1. BASICS OF NMR 19

! !"# !"$ !"% !"& !"' !"( !") !"* !"+ #!"#$

!"#%

!"#&

!"#'

!

!"$

!"&

!"(

!"*

# ,-./01.2"

314/5678-9:3

";"<

=51-9:><

Figure 1.1: Typical FID record.

!"B0 a transverse RF field linearly polarized

!"B$ = 2B1 cos(!t)!"uy. B$ can be decomposed into two fields

of amplitude B1 rotating around B0 with opposite frequencies ! and !!. If we now use a referenceframe rotating around B0 at frequency !, the first component of B$ looks static with modulus B1

while the second component is rotating with frequency !2!. !! is usually very far from !0 and thelatter component has no e!ect on the magnetization. In a rotating frame rotating at frequency !"! , wecan write #

d!"M

dt

'

rot

=

#d!"M

dt

'

lab

+!"M & !"! . (1.14)

Equation 1.5 can hence be rewritten in the rotating frame#

d!"M

dt

'

rot

=!"M & ("

!"B0 + "

!"B1 + !) (1.15)

and we can define an e!ective field!!"Beff such that

!!"Beff =

!"B0 +

!"B1 +

!"!"

(1.16)

The magnetization is hence going to rotate around Oy with frequency !1 = !"B1 and around Oz withpulsation !offset = !0 ! !. This is equivalent to a rotation around the axis defined by

!"B eff , with

pulsation !eff =,

!21 + !2

offset. If we apply a pulse of RF along Oy for a time %pw with pulsation!0, the magnetization is going to rotate around the Oy axis with an angle !1%pw. Hence, a completelytransverse magnetization can be obtained through the use of a RF pulse at pulsation !0 for a duration%pw = !

2"1. The resulting magnetization is along Ox and its evolution after the RF pulse is governed by

the Bloch equation, giving rise to a FID. This FID can be observed with the same coil that producedthe RF pulse.

1.1.2 Quantum description

The classical description of NMR brings some insight to the phenomenon and remains very useful inmany cases. Bloch equations are for example widely used in magnetic resonance imaging (MRI). This


description is however too limited when trying to explain, for example, interactions between spins. Itis hence necessary to use a quantum description of the phenomenon to understand NMR spectroscopy.We shall summarize in this section some well known results that are derived in more details in (1–3).

1.1.2.1 Zeeman Hamiltonian

NMR is the result of the Zeeman e!ect on the nucleus. A nucleus in a particular level of energycan indeed be considered as a particle with an intrinsic angular momentum, the so-called spin of thenucleus (see chapter 14 §11 in (4) and section 3.1.2 in (3)). The state of an isolated nucleus in a givenenergy level of spin I is (2I + 1)-fold degenerate in the absence of an internal or external magneticfield.

The nucleus also possesses a magnetic moment !"µ to which corresponds a quantum mechanicaloperator. When the nucleus is submitted to an external magnetic field

!"B0, an additional term arises

in the Hamiltonian of the system. This new term HZ is called the Zeeman Hamiltonian :

h̄HZ = !!"µ ·!"B0 (1.17)

This additional term is in most cases very small compared to the original Hamiltonian of thenucleus. As a result, the Zeeman Hamiltonian can be treated as a perturbation and one can neglectthe coupling between di!erent energy levels due to !"µ . In this context, the magnetic moment operatorcan be considered as proportional to the spin operator

!"I (see (3), section 3.5) :

!"µ = "h̄!"I (1.18)

where " is the gyromagnetic ratio of the nucleus.The spin operator

!"I has three components which, in a Cartesian reference frame can be noted Iz,

Ix and Iy. These components can be seen as the generators of rotations around respectively Oz, Oxand Oy within the spin space. In the very common case of spin 1

2 (dimension 2), these operators canbe represented by the following matrices :

Iz =*

12 00 !1

2

+Ix =

*0 1

212 0

+Iy =

*0 ! i

2i2 0

+, (1.19)

where the axes have been chosen so that the basis kets of the spin space are the eigenkets of Iz. Wewill name these eigenkets respectively |+$ and |!$. We have the following commutation rule betweenthe Cartesian components of the spin operator :

[Ix, Iy] = iIz (1.20)

All other commutation rules can be found by permutation of the indices. It is also convenient to definetwo additional composite operators :

I+ = Ix + iIy (1.21)I# = Ix ! iIy. (1.22)

Another consequence of the perturbation nature of the Zeeman e!ect is that the Hamiltonianrelevant to NMR is only constituted of Zeeman terms (main magnetic field and radiofrequency field)and interaction terms between nuclear spins :

H = HZ + HRF + HInteractions. (1.23)

We will not detail here the diverse nuclear spin interactions relevant to NMR and will introduce lateronly the ones that are of interest for this work. Good descriptions of these interactions can be foundin (2) and (1).


1.1.2.2 Density matrix

The NMR experiment is most of the time performed on a very large number of spins (typically >1018

spins), mainly because of its poor sensitivity. The observed signal is hence the average response of theensemble of spins, as indicated by the central limit theorem. As a result, the quantum description ofNMR can adopt a statistical formalism, a quantum version of statistical mechanics. The main elementof this formalism is the density matrix, which we shall introduce briefly. A detailed introduction tothe density matrix formalism can be found in (3).

Definition and elementary properties For a given state |($, the expectation value of an operatorA is #(|A|($. As a result, for a statistical distribution of states with a probability law P((), we have

#A$ =-

P(()#(|A|($d% (1.24)

If we introduce an orthonormal basis, the closure theorem allows to write

#A$ =$

i,j

-P(()#(|i$#i|A|j$#j|($d%

=$

i,j

#i|A|j$-

P(()#j|($#(|i$d%. (1.25)

Therefore, we can introduce the operator density matrix # such that

# =-

P(()|($#(|d% (1.26)

and

#A$ =$

i,j

#i|A|j$#j|#|i$d%

= TrA# (1.27)

The density matrix is at the core of NMR response calculation. Its matrix representation varieswith the basis used. A ket |($ can be expanded on the basis as

|($ =$

i

ai|i$, (1.28)

hence, the elements of the density matrix can be written as

#i|#|j$ = aia!j , (1.29)

where the bar represents an ensemble average. If we have a pure state such that

|($ =$

j

)j exp(i&j)|j$, (1.30)

we have

#i|#|i$ = )2i (1.31)

#j|#|j$ = )2j (1.32)

#i|#|j$ = )i)j exp[i(&i ! &j)]. (1.33)


The diagonal terms #k|#|k$ are called the population terms. In a pure state, if neither )i nor )j

vanishes, then the corresponding cross-terms do not vanish either. However, in a statistical mixture,there can be a mixture of random phases &i and &j such that the resulting cross-terms vanish while thepopulations do not. The existence of cross-terms in the density matrix of a statistical mixture meansthat there exists a coherence between the states (the phases are correlated). Therefore, the cross-termsare traditionally called the coherence terms.

Density matrix and time evolution It is possible to show (3) that the time evolution of thedensity matrix is given by

d#

dt= !i[H, #], (1.34)

where H is the total Hamiltonian of the system.It follows that for a time-independent Hamiltonian, the populations are constant in time while the

coherences oscillate. An important result is that, for a time-independent Hamiltonian, it is possible towrite

#(t) = exp(!iHt)#(0) exp(iHt) (1.35)

If the Hamiltonian depends explicitly on time, equation 1.35 is not valid anymore but it is stillpossible to write

#(t) = U(t)#(0)U †(t), (1.36)

where U(t) is a unitary operator which can be calculated by piece wise multiplications of infinitesimalpropagation.

For an observable quantity Q, one can write

#Q$(t) = Tr[QU(t)#(0)U †(t)] (1.37)= Tr[U †(t)QU(t)#(0)], (1.38)

where U(t) can be replaced by exp(!iHt) if H is time independent.

RF pulse and change of representation In the case of a radio-frequency pulse B1 orthogonal toa main field B0 along the z axis, the Hamiltonian is time-dependent :

H = !0Iz + !1(Ix cos !t + Iy sin!t), (1.39)

with !0 = !"B0 and !1 = !"B1.In order to find the time evolution of the density matrix, it is possible to simplify the Hamiltonian

by a change of representation, equivalent to a change of reference frame in classical mechanics. Onecan define a time dependent unitary operator U(t) and a Hermitian operator A that may depend ontime such that

dU(t)dt

= iAU(t). (1.40)

We can associate to an operator Q an operator Q̃ such that

Q̃ = U(t)QU †(t). (1.41)

Starting from equation 1.34, it is possible to show that

d#̃

dt= !i[(H̃!A), #̃]. (1.42)


Equation 1.41 defines a change of representation and this last result indicates that, in the new repre-sentation, the evolution of the density matrix is the same as if the Hamiltonian of the system was thee!ective Hamiltonian Heff

Heff = H̃!A (1.43)

In the case of a Hamiltonian such as in equation 1.39, if we choose A such that

A = !Iz, (1.44)

we define a new representation corresponding to the rotating frame and we obtain a time-independente!ective Hamiltonian

Heff = (!0 ! !)Iz + !1Ix. (1.45)

This corresponds to an interaction of the Zeeman type with an e!ective field!"B eff with the following

components:

Bz = B0 +!

"(1.46)

Bx = B1. (1.47)

If ! = !0, the e!ective field is purely transverse and we obtain the same result as in equation 1.16.

1.1.2.3 Measurement

What is observed in NMR is the evolution of the bulk magnetization (the average magnetization ofthe ensemble of spins). According to equation 1.18, this average magnetization is proportional to theaverage spin vector. Hence, the relevant observables are Iz, Ix and Iy. Their average expectationvalue at a given time when the system is subjected solely to the Zeeman Hamiltonian can be given byequation 1.38 so that we have :

#Ik$(t) = Tr [exp(i!0Iz)Ik exp(!i!0Iz)#(0)] , (1.48)

with k standing for x, y or z. In the case of Iz, we can write

#Iz$ = Tr [Iz#(0)] (1.49)= #Iz$(0). (1.50)

Hence, neglecting relaxation, the z-component of the spin is time-independent when the system issubjected to B0.

The expectation value of Ix and Iy can be treated in the same time by computing the expectationvalue of I+. It can be shown (3) that

exp(i&Iz)I+ exp(!i&Iz) = exp(i&)I+, (1.51)

Hence,#I+$(t) = exp(i!0t)#I+$(0). (1.52)

This corresponds to a magnetization rotating around the Oz axis with pulsation !0 and verifies theclassical description.

In most NMR experiments, a coil orthogonal to B0 (i.e. in the xOy plane) is used to excite anddetect. The electromotive force (emf, or voltage) picked up by a coil along the Ox axis is given by theLenz law so that

*(t) ' Mx(t) ' #Ix$(t) (1.53)


We can define a complex signal corresponding to the signal of a coil along the Ox axis with the signalof a coil along Oy so that

*̂(t) ' #I+$(t) (1.54)

Hence the FID defined in the classical description corresponds, in the absence of relaxation, to thetime evolution of #I+$ in the quantum description.

1.1.3 Important examples

1.1.3.1 !2 pulse-acquisition (neglecting relaxation)

From the quantum description point of view, the simple experiment !2 pulse-acquisition corresponds

to a calculation in two steps. First, the pulse is applied to the system and the original longitudinalmagnetization is converted into transverse magnetization. The calculation at this stage only concernsthe density matrix. Assuming the initial density matrix is such that # ' Iz and using the rotatingframe Hamiltonian 1.45 and equation 1.34, we can write that, in the rotating frame, the density matrixright after the pulse of duration % and pulsation !0 is such that

#̃(%) ' exp(!i!1%Iy)Iz exp(i!1%Iy) (1.55)

which can be rewritten#̃(%) ' cos(!1%)Iz + sin(!1%)Ix, (1.56)

or, in the laboratory frame :

#(%) ' cos(!1%)Iz + sin(!1%)[cos(!0%)Ix + sin(!0%)Iy] (1.57)

Hence, for % = !2"1

, we have a completely transverse magnetization (# is in the xOy plane) and thepulse is a !

2 pulse.Right after the pulse, the measurement is performed and the evolution of #I+$ can be computed

using equation 1.52 with #I+$(0) = Tr[I+#(%)]. We hence obtain the same result as in the classicaldescription.

1.1.3.2 Hahn echo and CPMG

Hahn echo We now consider an ensemble of spin located in a region of space where the externalfield B0 is not uniform. Each spin will hence "see" a field B(i)

0 . We now assume a perfect !2 pulse along

Oy was performed for all spins so that we initially have #(0) = M0#h̄ Ix. The spins then evolve according

to their local Zeeman Hamiltonian H(i)0 . As a result, each spin precesses around Oz with a di!erent

pulsation !(i)0 , leading to a loss of coherence between the spins and a loss of bulk magnetization :

quickly, the signal disappears as the di!erent spins are not in phase anymore. This loss of coherence ishowever not irreversible. The spins can be "refocused" through the use of a $ pulse along Ox. Indeed,if we consider all spins are precessing in the xOy plane, the $ pulse will keep all of them in the xOyplane while mirroring their orientation with respect to Ox. If we consider two spins having precessedwith pulsation !(1)

0 and !(2)0 for a duration %1. We have

#I(1)+ $(%1) =

M0

"h̄[cos(!(1)

0 %1)Ix + sin(!(1)0 %1)Iy]

#I(2)+ $(%1) =

M0

"h̄[cos(!(2)

0 %1)Ix + sin(!(2)0 %1)Iy].


We apply right away a $ pulse so that we can now write

#I(1)+ $(%1) =

M0

"h̄[cos(!(1)

0 %1)Ix ! sin(!(1)0 %1)Iy]

=M0

"h̄[cos(!!(1)

0 %1)Ix + sin(!!(1)0 %1)Iy]

#I(2)+ $(%1) =

M0

"h̄[cos(!(2)

0 %1 + $)Ix ! sin(!(2)0 %1 + $)Iy]

=M0

"h̄[cos(!!(2)

0 %1)Ix + sin(!!(2)0 %1)Iy],

and, after a time %1 of free precession, we obtain

#I(1)+ $(2%1) =

M0

"h̄Ix

#I(2)+ $(2%1) =

M0

"h̄Ix,

and the spins are back in phase.This result indicates that, at a time %1 after the $ pulse, the signal will have regained its whole

amplitude, as if it was "reborn". This e!ect was first described by Hahn in 1950 (5) and is thereforecalled the "Hahn echo". Figure 1.2 gives a visual description of the e!ect.

z

y

x

z

y

x

z

y

x

z

y

x

a) b)

c) d)

Figure 1.2: Visual explanation of the Hahn echo. In the rotating frame, a) all spins are first in-phase alongOy. b) After a time %1, due to local variations of the field, some spins lag behind the rotating frame while somerotate faster, resulting in a loss of coherence and of total magnetization. c) A $ pulse is applied along Oy. Allmagnetizations are mirrored around Oy, transforming lag into advance and advance into lag. d) The spins keepprecessing at their own pace and "refocus" after a time %1.


Carr-Purcell-Meiboom-Gill sequence After the refocusing obtained through the Hahn echo, thespins obvisouly dephase again and the signal "dies out" again. Carr and Purcell proposed in 1954 (6)to use a succession of $ pulses in order to obtain a train of echoes. Such a train could be used tomeasure the true T2 of the signal, or the coe"cient of di!usion of the studied specie, by inspection ofthe decay of the echos. The sequence introduced by Carr and Purcell was modified by Meiboom andGill in 1958 (7). The resulting sequence is the now well-known Carr-Purcell-Meiboom-Gill (CPMG)sequence. The modification of Meiboom and Gill was to introduce the use of coherent pulses for whichthe phase can be controlled. This makes possible to apply a particular phase on the !

2 pulse andanother phase on the $ pulses. A di!erence in phase of 90º between the two types of pulses increasestremendously the robustness of the sequence, so that it is not necessary to apply exact $ pulses. Inaddition, all echoes are in phase. The CPMG sequence is shown on figure 1.3.

!1 !1 !1 !1 !1 !1 !1

t

...

Figure 1.3: CPMG sequence. A !2 pulse is applied along Ox to put the magnetization in the xOy plane.

Repeated $ pulses are applied along Oy to refocus the signal. The signal is acquired between each $ pulse anddecreases as time advances due to relaxation and irreversible decoherence.

1.1.3.3 Nutation echoes and composite pulses

The previous Hahn and CPMG sequences refocus the spins de-phasing due to all inhomogeneities ofthe external field B0, wether they are due to the magnet generating the field or they are due to theenvironment of the spins. However, de-phasing due to the environment of the spins is of interestas it provides the chemical structure information. An important example is the chemical shift : theneighbors of a given spin change locally the e!ective static field (they "shield" the spin) so that a slightfrequency shift occurs in the signal given that particular spin. A concept was proposed by Meriles et alin 2001 (8) to refocus de-phasings due to static field inhomogeneities while preserving de-phasings suchas chemical shift. It relies on the correlation of the static field B0 and the RF field B1. It had indeedbeen shown a few years before that the correlation of these fields may induce echoes (9–11). Compositeor adiabatic pulses (12; 13) can be used to "rewind" the dephasing due to B0 inhomogeneities with B1

inhomogeneities. This can be done if the following condition is fulfilled at all locations in the sample :

B$1 ' B0, (1.58)

where B$1 is the

!"B1 component orthogonal to

!"B0. Once the initial !

2 has been applied, if we let thesystem of spin evolve during %free, each spin will acquire a phase !0(!"x )%free which depends on thelocation in space. If we apply to the system of spin a composite !z pulse of duration %rewind (forexample,

.!2

/#y

[!1(!"x )%rewind]x.

!2

/y), the phase acquired by the spins will be !!1(!"x )%rewind. As B1

is correlated to B0, we can write !0 = k!1. Hence, if %rewind = k%free, all spins are in phase right afterthe composite pulse, and if %rewind = 2k%free, a nutation echo arises a time %free after the compositepulse. However, the evolution due to chemical shift remains unchanged. One can make a single pointacquisition at the time the echo arises and reiterate the operation in order to obtain a collection ofpoints at di!erent times which will show the modulation due to chemical shift and other interactions.


1.1.4 Self-di!usion and field gradients

Spins in liquid samples (or liquid-like, such as polymers) are not immobile and di!use in space. Theirability to move is characterized by the di!usion coe"cient which appears in the law governing thedi!usion of the local magnetization m (14). Self-di!usion finds a particular importance in the contextof inhomogeneous fields.

We consider here that the sample is placed in a strong field gradient G = $B0$z in one direction so

that we have for the Larmor frequency

!(z) = !0 + Gz (1.59)

As a result, the e!ect of di!usion can be assumed to take place in only one dimension. Themagnetization’s evolution is hence governed by an equation of the type of the Fick’s second law :

+M

+t= D

+2M

+z2. (1.60)

It the case of NMR, neglecting relaxation, we have M = A(t) exp(iGzt). Hence, from equation 1.60,we obtain

dA

dt= !DG2t2A, (1.61)

and

A(t) = A(0) exp(!DG2t3

3). (1.62)

It can then be shown (3) that in the case of a CPMG sequence with a time %1 between each $pulse, the apparent transverse relaxation time T %

2 can be expressed as

1T %

2

=1T2

+13DG2%2

1 , (1.63)

where T2 is the real transverse relaxation time. Hence, knowing the gradient and using several CPMGexperiments with di!erent %1 times, it is possible to retrieve both T2 and D.

1.1.5 NMR and magnetic field homogeneity

It is necessary to underline the necessity of extremely homogeneous fields for NMR spectroscopy (MRS)or imaging (MRI). On the one hand, MRS is interested in the HInteractions terms seen in equation 1.23.This term describes the modifications of the magnetic environment of the nucleus studied, thus givingaccess to structural information. Some of these interactions are proportional to the external field (e.g.chemical shift), some are not (e.g. dipolar coupling), but always remain in the order of the part permillion (ppm) of the main Zeeman Hamiltonian (H0). Hence MRS requires sub-ppm homogeneity overthe volume of the sample in order to be useful. On the other hand, MRI is mostly concerned withthe main term H0 and specifically with its spatial variations induced by gradient coils. The main fieldmust also be homogeneous over the volume of sample so that the spatial variations are well-controlled(the main field variations must be negligible in front of the applied gradients over a voxel, the elementof volume of an image). The requirement on homogeneity is usually less stringent (up to 20 ppm whenonly images are recorded), but the necessary volume is much greater (typically a diameter 20 to 30cm, while NMR requires a diameter of about a centimeter).


1.1.6 Signal and noise in NMR

1.1.6.1 Polarization

The signal originates from the precession of the bulk magnetization of the sample. This bulk magneti-zation corresponds to the average orientation of the spins. In an external field, the populations of thedi!erent Zeeman energy levels verify the Boltzmann law of statistical mechanics and we can write thenet magnetization of a system of spins as (2)

M = N"h̄

0Im=#I m exp(#h̄mB0

kBT )0I

m=#I exp(#h̄mB0kBT )

, (1.64)

where " and I are respectively the gyromagnetic ratio and the spin of the nucleus studied, N isthe number of spins per unit volume, h̄ is the Plank constant, kB the Boltzmann constant, T thetemperature and B0 the modulus of the external magnetic field.

In most experimental conditions, #h̄H0kBT is a very small number and equation 1.64 can be expanded

so that M can be written as

M =N"2h̄2I(I + 1)

3kBTB0 =

,

µ0B0. (1.65)

, is the nuclear paramagnetic susceptibility. For the proton in water (spin 12) at 300K, , is about

4.04( 10#9. Thus, the induced polarization is extremely small, even for fields of 20 T. As a result, theNMR signal is always very weak and requires highly sensitive detectors in order to be observed, suchas, for example, tuned circuits with high-gain, low-noise preamplifiers.

1.1.6.2 Detection and noise

The detection of the NMR signal can be done using many devices such as an inductive coil, a Su-perconducting Quantum Interference Device (SQUID) (15–17), a mixed-sensor (18; 19), a cantilever(force detection) (20–22), or a laser (23)... The most commonly used device is the inductive coil. It isthe most simple to use and has a very good intrinsic sensitivity. Most other devices feature much moreinvolved experimental conditions, requiring cryogenics and special care (SQUIDs and mixed sensorscannot operate in a strong magnetic field, while cantilevers are only suitable for nano-samples, forexample). We shall hence concentrate here on inductive coils and not go further into the comparisonof detection systems.

The evaluation of the quality of the detection of the NMR signal is done through the computationof the Signal to Noise Ratio (SNR). This involves the evaluation of the ability of the system to detectthe signal, and the evaluation of the noise this system is subjected to.

Signal and principle of reciprocity The signal obtained with an inductive coil can easily becomputed thanks to the principle of reciprocity (24). We shall state here this principle and give itsproof, which is not documented in many texts (25). This will in the mean time persuade the readerthat this principle is largely general and does not rely on specific assumptions.

We consider an arbitrary loop where a unit current circulates. The arbitrary point P in space isthe location of our calculations. The surface S bounded by the loop is equivalent to an ensemble ofelementary loops of surface dS bounded by an elementary loop of unit current. P is located in freespace and there exist a magnetic pseudoscalar potential (a pseudoscalar is a quantity that behaves likescalars under proper rotations but reverses sign under inversion (26)). The potential at P produced


by an elementary loop is

d" = d!"S ·!")

!1r

", (1.66)

where r is the distance from P to the elementary loop and the derivatives are taken on the coordinatesof point P . The total field created by the loop is hence

!"B1 = !µ0

-

S

!")*d!"S ·)(

1r)+

. (1.67)

We now consider the elementary magnetic flux dF through the area d!"S generated by a dipole !"m

located in P .

dF = d!"S ·!"B = µ0d

!"S ·!") [!"m ·!")(

1r)] (1.68)

We shall consider that !"m is constant through space. As d!"S is also constant, we can write

dF = µ0!"m ·!")

*d!"S ·!")

!1r

"+(1.69)

thus the electromotive force (EMF) induced in the loop is

* =+F

+t= µ0

+

+t

!!"m ·

-

S

!")*d!"S ·!")

!1r

"+". (1.70)

and

* = ! +

+t

!"B1 ·!"m (1.71)

This result is the so-called principle of reciprocity. It shows that a coil e!cient at generatinga field in a location of space is also e!cient at detecting the field generated by a dipolein this same location. This formulation is valid in the case of the near-field approximation, orquasi-static approximation, when the scale and distance of the sample to be detected are much smallerthan the wavelength, so that the temporal terms of the retarded potential can be neglected. It haseven been proven that this principle of reciprocity is also valid for the intermediate and radiation zonefields (27).

Equation 1.71 is the basis for the calculation of the signal induced by the precession of the spins.After a !

2 pulse, we may assume that the magnetization M0 is along the Ox axis while the detectioncoil is along the Ox axis. We get the emf :

* = !-

Sample

+

+t(!"B1 ·

!"M0)dVs. (1.72)

In the case where the magnetization is precessing around Oz, and if B1 is homogeneous over the samplevolume, this integral simplifies to

* = !0[B1]xyM0Vs cos(!0t) (1.73)

where [B1]xy is the e!ective field produced per unit current flowing in the coil.


Noise It is also necessary to quantify the noise due to the detection system. When the detectionchain is well designed and no external electromagnetic noise exists, the noise featured in the signalis due to the thermal noise associated to the resistance of the coil (the radiation noise is generallynegligible in NMR experiments), hence, we have for the thermal noise :

VN =1

4kBT!fR, (1.74)

where T is the temperature of the coil, !f is the detection bandwidth and R is the electrical resistanceof the coil.

SNR Thus, we have for a given configuration and signal,

SNR =!0[B1]xyM0Vs*

4kBT!fR(1.75)

SNR ' [B1]xy*R

. (1.76)

[B1]xy being the field created per unit of current, the preceding equation can be reduced to one term{B1}xy, the field created per square root of power:

SNR ' {B1}xy. (1.77)

Hence, the quantity {B1}xy is the figure of merit of a coil, in the absence of external losses (dueto dissipative surrounding medium, for example) or additional noise (like electromagnetic noise pickedup by the antenna the coil forms).

1.2 Magnetic materials

In order to proceed with magnet design, it is necessary to bear in mind some elementary aspectsof the materials used. We shall here recall some of these aspects. The topic of magnetic materialsbeing a whole field of material science, it is not our purpose to cover the entire subject here. Inparticular, microscopic aspects of magnetic materials will not be mentioned as our main interest focuseson the megascopic (as opposed to macroscopic which would be on the scale of the magnetic domains),"technical" magnetic properties of materials. The curious reader may find useful references in (28),(29), (30) and (31).

1.2.1 Magnetostatics and materials

The elementary di!erential microscopic laws of magnetostatics are!") (!"B = µ0

!"k (1.78)

!") ·!"B = 0 (1.79)

where!"B is the magnetic field and

!"k is the current density distribution.

However, these equations can be used when the distribution of current density!"k is known every-

where. This is not the case in materials at the macroscopic scale, where the atomic electrons give riseto rapidly fluctuating magnetic moments (due to their orbital moment and their intrinsic magneticmoment). Hence, it is necessary to adopt an approximation to obtain a description of magnetic fieldsin materials. The derivation of the macroscopic Maxwell’s equations requires a proper averaging of the

1.2. MAGNETIC MATERIALS 31

microscopic equations, which will not be described here but can be found in chapter 6 of (32). Duringthis process, one identifies the ability of each atom to produce a magnetic moment !"mi, giving rise toan ensemble average macroscopic magnetization distribution

!"M , or magnetic moment density

!"M(!"r ) =

$

i

Ni < !"mi > (1.80)

The final result of this averaging yields a new macroscopic equation for!"B

!") (!"B = µ0(!"k +

!") (!"M) (1.81)

It is then convenient to introduce a new field!"H such that!") (!"H =

!"k (1.82)

as!") ·

!"k = 0 because of the conservation of charge and of the steady-state nature of magnetostatics.

This allows to write the well-known equation!"B = µ0(

!"H +

!"M) (1.83)

It is customary to call!"B the magnetic induction, expressed in Teslas,

!"H the magnetic field,

expressed in Amperes per meter, and!"M the magnetization of the material, expressed in Amperes per

meter. The magnetic polarization or intensity (in Teslas) might also be used :

I = µ0M (1.84)

As the di!erence resides only in the µ0 factor, the word magnetization may refer to magnetization orpolarization.

The magnetization is defined by a constitutive equation of the material considered. This consti-tutive equation can in some cases be linear (the material is isotropic) and the susceptibility of thematerial , can be defined as !"

M = ,!"H (1.85)

In that case, we can write !"B = µ0(, + 1)

!"H (1.86)

It is then convenient to define a material permeability,

µ =B

H(1.87)

= µ0(, + 1), (1.88)

and a material di!erential permeability

µr =µ

µ0= , + 1 (1.89)

These quantities can often be encountered in manufacturer’s data sheets. In some cases, the response ofthe material may be linear but not isotropic : it exhibits an angular dependency of the susceptibility.Most materials that are of interest in this work exhibit a uniaxial anisotropy : an "easy" axis ofmagnetization can be identified. It is in such cases necessary to define a susceptibility tensor ,̂ (33).


The knowledge of a permeability parallel to the easy axis µpar, and one perpendicular, µperp is su"cientfor uniaxial materials (34).

A usual consequence of the anisotropy of a material is that it demonstrates hysteresis : the materialmay retain some magnetization once the external field is turned o! and its general magnetic behaviordepends on its magnetic history. Such materials are of crucial importance as they can be used assources of magnetic field, even when they are subjected to external magnetic fields opposing theirmagnetization. These materials are obviously non linear and it is not possible to use a constantsusceptibility in equation 1.85. In that case, it is preferable to use the relative susceptibility

,r =dM

dH, (1.90)

assuming!"M and

!"H are colinear.

In addition, four important quantities shown in figure 1.4 help characterize the response of suchmaterials to an external magnetic field.

The saturation magnetization corresponds to the maximum magnetization achieved when in-creasing the external magnetic field. This corresponds to the situation where all magnetic moments inthe material are aligned with the external field.

The remanence of the material is its ability to maintain a magnetization once the external fieldis brought back to zero. One defines this way the remanent induction and remanent magnetization atzero field such that !"

BR = µ0!!"MR (1.91)

The general convention is to use the term "remanence" to refer to either the remanent induction ormagnetization when the external field is brought back to zero after reaching the saturation of thematerial (30).

The coercivity of a material is its capacity to maintain its magnetization in an opposing externalmagnetic field. The word coercivity is often associated to the opposite field strength HCB necessaryto bring back the field induction, µ0(

!"H +

!"M), to zero after the material has been saturated. The

intrinsic coercivity is the opposite magnetic field strength HCi necessary to cancel the magnetizationof the material once saturated. We always have HCB < HCi.

These quantities can be identified on the hysteresis curves of the material, as shown in figure 1.4.Such curves can be modeled for the purpose of numerical simulations and further details on this topicare given in appendix A. Another important quantity is the anisotropy field HA. This quantitymeasures the ability of the material to maintain its direction of magnetization in a transverse externalmagnetic field. We shall see very shortly how HA is determined.

All properties of magnetic materials are highly dependent on temperature as they are related tothe magnetic order of the material, which is competing with the thermal agitation (30).

1.2.2 Materials of interest

The desired properties of the material constituting an NMR magnet are high remanence (in orderto achieve the highest magnetic field possible) and high coercivity (to have flexibility for the designand minimum sensitivity to external fields). It is also desirable to have a high anisotropy field inorder to ensure the minimum angular sensitivity to external fields. Fortunately, HA is usually higherwhen the coercivity is higher. The desired material will hence belong to the "hard" family, likeferromagnets and ferrites. Remanence and coercivity can be summed up in one parameter, the max-imum energy product of the material BHMax. It corresponds to the maximum value of the productOpposing external field( Field induction.

1.2. MAGNETIC MATERIALS 33

1.0

0.5

-0.5

-1.0

2-2 4-4 0

H (MS units)

M (M

S uni

ts)

4

2

-2

-4

0 1 2 3-1-2-3

H (MS units)

B (!

0MS u

nits

)

Ms

Mr

HC i

Br

HC

Figure 1.4: General shape of magnetization curve of soft materials using a simple analytical description. M1

is plotted as the blue curve, M2 as the purple curve and M3 as the yellow one. k was chosen to be 3000 whileµ0MS was set to 1T.

There exists a variety of permanent magnets belonging to di!erent families depending on theirchemical formulation. Each family corresponds to new findings in material research, bringing higherperformances (higher BHMax). Figure 1.5 shows the di!erent families of materials with their dateof apparition and their BHMax. This curve exposes the breakthrough brought by the discovery ofsamarium-cobalt (SmCo) and neodymium-iron-boron (NdFeB) alloys at the beginning of the 80s. Thesematerials largely outperform aluminum-nickel-cobalt alloys (AlNiCo) and ferrites. Di!erent gradesallow manufacturers to o!er various ranges of remanence, coercivity, along with di!erent resistance tooxidization, di!erent temperature sensitivity, etc.

High performance rare-earth magnets are sintered materials. Their very involved fabrication processfeatures numerous steps. The production starts with powders of the alloy which are sintered into ablock of material while a strong external magnetic field is applied. This process creates the coercivityand anisotropy of the material. The resulting block is unmagnetized and can be machined to achievearbitrary shapes with tight tolerances. The parts are coated to protect the material from oxidationand finally magnetized. The many steps involved in the fabrication of these materials and the di"cultyto perfectly control the conditions of each of them make the repeatability of these materials not sogood. One can expect for the best quality of materials variations on the magnetization amplitude ofthe order of the percent or more, and on the orientation of the order of the degree. The materialswe are interested in are of the family of SmCo and NdFeB and we shall summarize their respectiveadvantages and drawbacks.

SmCo SmCo materials (modern Sm2Co17) o!er high coercivity (HCB up to 820 kA m#1, or µ0HCB =1.03 T, and HCi up to 1990 kA m#1, or µ0HCi = 2.5 T) but their remanence is fairly limited (Br < 1.15T). These materials are also the most resistant of the two families (SmCo and NdFeB). Their Curietemperature can reach 850ºC and their temperature sensitivity is about 300 ppm/K, three timesless temperature sensitive than NdFeB. Their very strong coercivity make them highly indicated forstructures with strong demagnetizing fields as the magnetization will not collapse quickly. However,their higher susceptibility in the demagnetization curve makes them more sensitive to demagnetizing


fields than NdFeB (in the sense of the variation of the working point, rather than the magnetizationcollapse). Some higher grades of SmCo (mostly based on SmCo5) display a smaller susceptibilitythan NdFeB and may be chosen in some cases, though their remanence is much lower. Materialsbased on Sm2Co17 provide a higher remanence but their susceptibility rapidly increases with thedemagnetizing field. The new grades of SmCo o!er improved resistance to corrosion and makes themparticularly indicated for humid environments. The strongest advantage of these materials is theirsmall temperature sensitivity.

Figure 1.5: The di!erent families of permanent magnets with their date of apparition and their BHMax (35).

NdFeB NdFeB materials o!er the highest energy product. They have the highest remanence avail-able (up to 1.5 T with reasonable coercivity) and also o!er the highest coercivity (HCB up to 1115 kA m#1,or µ0HCB = 1.4 T, and HCi up to 2265 kA m#1, or µ0HCi = 2.85 T). However, NdFeB is much moresensitive to temperature than SmCo. The Curie temperature reaches 370º maximum and the rema-nence varies with a coe"cient of the order of 1000 ppm/K. It is also much more sensitive to corrosion.Low-grade materials of this alloy feature an oxidization chain-reaction which completely destroys thematerial in a few days. More recent and higher quality grades have eliminated this issue so that ox-idization is contained in the surface layer of a magnet part but these materials remain sensitive tocorrosion very much like iron, and need to be coated to protect them from the atmosphere. NdFeBis hence the material of choice for high-performance, well-controlled environment applications. It ishence the material we will use in this work. In addition, NdFeB is lighter than SmCo (about 7.8 gcm#3 for NdFeB against about 8.4 g cm#3 for SmCo) and less brittle, which makes it easier to handle.In addition NdeFeB is 30 % to 40 % less expensive than SmCo.

1.3. THE DEVELOPMENT OF PERMANENT MAGNETS IN NMR 35

1.3 The development of permanent magnets in NMR

1.3.1 From low-field to high-field

Since the first description of a nuclear magnetic resonance experiment by Rabi et al. in 1938 (36),much work has been done in the field of NMR. The technique has been continuously improved, givingrise to the pulsed Fourier transform technique (37) and all the known NMR sequences used for NMRof liquids and solids (38). NMR spectroscopy can yield precious information on the chemical structureof materials, and is hence a tool of choice for chemists and biologists. It is nowadays a standard tech-nique in chemical analysis of materials, elucidation of protein structures, drug research and productioncontrol, etc.

Much later on, in 1973, Lauterbur and his team revealed how NMR could be used as well toobtain images of objects using in addition magnetic field gradients (39). Magnetic Resonance Imagingwas born. It was actually first referred to as Nuclear Magnetic Resonance Imaging (NMRI) but theword "nuclear" having a negative connotation in the public opinion, the first letter of the acronymwas dropped when the technique became popular in medicine. The first approach was to use theprojection-reconstruction (Radon transform) scheme used in X-ray tomography. MRI has evolved sincethen, using pulsed Fourier transform techniques and k-space spanning. MRI can yield tridimensionaldata of numerous kinds of subjects in a non-invasive way. It is hence a tool of choice for medicalimaging and other applications requiring non-destructive 3D visualization. It is today a critical toolin the diagnostic of some cancers, the study of human brain (through functional imaging, known asfMRI) and many other medical applications.

The major drawback of NMR is its lack of sensitivity, as stated in the previous section. This lack ofsensitivity makes di"cult the study of low-" nuclei in NMR spectroscopy by increasing the experimenttime. The low sensitivity is also a limitation of MRI resolution. The resolution of an image dependsindeed on the size of the voxel which defines the elementary volume of sample being detected in MRI.As a result, the resolution is mostly limited by our inability to detect the signal generated by smallvoxels. Hence, it seems highly beneficial to increase the external magnetic field in order to improvethe SNR. Furthermore, the chemical shift being proportional to the external field, the increase of thelatter provides improved resolution for NMR spectroscopy. Thus, the increase of the magnetic fieldfinds a double motivation.

The first experiment performed by Rabi and his team was done at a field of 6000 Gauss (about 25.6MHz) while Lauterbur’s experiment was carried at about 1.4 T (60 MHz). The magnetic field usedin NMR laboratories remained limited to 100MHz until the beginning of the 1960s as the saturationof iron is about 2.35 T (about 100 MHz for proton frequency). It is in 1964 that a very large gap infield strength was crossed with the first NMR spectroscopy experiment in a persistent superconductingmagnet at 200 MHz (40). Since then, the field strength of NMR spectrometers has never stoppedincreasing, reaching lately 1GHz proton frequency (about 23.5 T) for a machine installed in the Centrede RMN à très hauts champs in Lyon, France. At the same time, superconducting magnets have beenextensively used for MRI. The magnetic field strength being limited by governmental regulations to3 T for clinical use, the field strength race has not been as strong for MRI as for NMR. In addition,the bore of a whole body magnet being much larger than the bore necessary for a spectrometer, thetask of obtaining very high fields becomes much more di"cult due to the energy stored in the magnetand the limits of the superconducting wires (critical current, temperature and field). A few 7 and 9T scanners have been built for research purpose and an 11.75 T (500 MHz proton frequency) wholebody scanner is undergoing development (41). NMR spectroscopy research is also carried on at muchhigher fields (e.g. 30 T) using resistive or hybrid magnets (42).


1.3.2 Back to low-field

The era of high-field NMR has brought a wealth of resolution and variety of nuclei studied. However,very high fields present some major drawbacks. The machines resulting from these latest developmentsare extremely costly to build (the initial investment counts in many millions of dollars). These systemsare also very large and heavy (dimensions are of the order of a few meters) and require heavy main-tenance (cryogens for superconducting magnets, water and electrical power of several Megawatts forBitter magnets). Furthermore, with very high fields come new concerns in the NMR experiment. MRIwhich performs experiments on heterogeneous samples and subjects faces the problem of magnetic sus-ceptibility : the local abrupt changes of susceptibility create distortions in images and line-broadeningin spectra. Such e!ects remain unnoticed at lower fields but when reaching 7T and higher, these e!ectsstrongly compromise the resolution enhancement expected from the SNR improvement. The increase ofthe frequency also brings the wavelength closer to the dimensions of the NMR experiment and solutionshave to be found as near-field detector design is not suitable anymore (43). In NMR spectroscopy, thesuperconducting technology has brought high fields but imposed the use of saddle-shaped coils whichhave significantly lower detection e"ciency than solenoids, thus limiting the increase of SNR (24).

Furthermore, the extremely high cost of an NMR spectrometer or scanner limits the range ofapplications of this technique. In addition, the very heavy magnet (several metric tons) forbids anymobile application (though some mobile MRI with a magnet loaded on a truck can be found (44))while the need for cryogens makes arbitrary orientation una!ordable. To summarize, high-field NMRis constrained to the laboratory and asks its customer a high price. There exists however manyapplications where the subject or sample cannot be moved in the laboratory, or simply cannot fit inthe magnet and, finally, would benefit from NMR but cannot a!ord the price of high-field NMR. Onemajor example of such applications is the oil industry and well-logging. This is a very specific niche ofNMR, which has been evolving since 1952 when Varian filed the first patent for an apparatus capableof performing NMR while drilling (45) in order to discriminate water from oil. The successful use oflower field, lower cost NMR in well-logging is an example of the di!erence made by a tool of reducedpower (lower field) that can give data in a particular situation where the most powerful tool (high-field)would not give any as it is not applicable.

A non-exhaustive list of fields of study where low-field NMR has proven to be useful includefood industry, polymer studies, elastomer studies, cultural heritage, biomedicine, porous media andmoisture. A detailed review of low-field NMR and its application can be found in (46).

A first wide application of low-field transportable NMR is the measurement of porosity of materials.Already a central part of the analysis of NMR well-logging data (47), porosity measurements may be ofhigh interest in many other contexts, such as drill-core characterization (48; 49) where a portable single-sided tool comes in handy to measure such objects without machining them, or a tailored cylindricalmagnet is part of a fast and low-cost scanner. Furthermore, the use of a unilateral NMR surface sensorprovides su"cient data to characterize the geometrical structure of porous media (50).

Another use closely linked to porous media is the measure of moisture in materials. It is of highinterest in the field of cements and concretes. While high-field NMR has been of proven use to studyand validate new technologies of concrete curing (51), the study of pores structure of concrete can bedone using portable devices (52). Such studies are critical for the evaluation of the characteristics ofcements and concretes.

NMR has also found applications in food industry for twenty years. For example, it is a tool ofchoice for on-line fruit internal structure examination and fruit sorting (53). Portable NMR has alsoproven to be useful in diverse studies, such as food emulsions. The field gradient of unilateral magnetallows the study of di!usion in emulsions, providing discrimination of oil and water (54).

Cultural heritage is also a promising field of application for portable NMR. Most samples to be


studied (ancient artifacts, master’s paintings, historical monuments) usually cannot be taken apart,neither can a small sample be removed to be taken to an NMR spectrometer. The most suitablemethod of study of such objects is to bring the spectrometer to the subject. The possibility of usingunilateral low-field NMR for cultural heritage study has been demonstrated several times, for examplefor the study of paintings on wood (55).

The last field of study we will cite, biomedicine, is by no mean the least. Low-field NMR canprovide useful information at a lower cost, opening the door to routine NMR examination. Amongother examples, we can mention the ability to obtain well-resolved and relevant images of the kneewith 0.2T imaging devices (56). Portable, single-sided NMR can also be useful in the study of humanskin. Skin hydration has been studied this way in the context of moisturizer cream evaluation (57).

Finally, on a more prospective side, advances in low-field, lower-cost MRI could help tremendouslycancer detection. Studies have demonstrated the relevance of MRI for skin-tumors monitoring (58).Lower-cost unilateral MRI scanners could make a!ordable flexible skin examination in a routine fashion.Besides, while the suitability of MRI for early-stage breast-cancer detection has been proven true,its advantages on X-ray mammography are still debated (59; 60). A decreased cost of MRI breastexamination would make possible the economically-viable combination of both techniques for earlystage detection.

1.3.3 Magnets for low-field NMR

The development of strong and homogeneous permanent magnet structures is the result of e!orts intwo fields of research that are not directly connected : particle accelerator magnets and NMR magnets.The development of permanent magnets for accelerators started earlier than for NMR and providedsome basis for portable NMR, such as Halbach’s dipoles. The NMR requirements are however verydi!erent for field homogeneity, as well as for geometry and demand di!erent approaches.

An extensive work has been provided since the beginning of the 1990s to design and producemagnet prototypes to accomplish NMR experiments at low cost and in unusual conditions. One of themain advantages of these prototypes were their portability and, for some, their single-sided geometryallowing the measurement of objects bigger than the magnet. The proposed systems can be separatedinto two families. One of them contains all magnets which feature a "closed" geometry, meaning thatthe sample must fit inside the magnet. Such a configuration is best suited to produce reasonably highfields (up to 2T) with a compact system. We will call such magnets in situ magnets. The other familycontains all magnets designed to perform so-called stray-field NMR, outside of the magnet. In suchgeometries, the field is bound to be weaker (less than 0.5 T for reasonably sized systems) but there isno constrain on the size of the sample. Such systems are particularly suited for surface studies. Wewill call these systems ex situ magnets. Figure 1.6 gives a schematic of typical configurations for insitu and ex situ NMR.

The design of permanent magnets for NMR must satisfy several constraints : there must be enoughspace for the sample to be placed in the region of interest (RoI) of the magnet, the field must be ashigh as possible and as homogeneous as possible. We will now expose briefly the state of the art ofpermanent magnet design for NMR in order to identify progresses to be done.


Magnet

RoIBore hole Object RoI

Magnet

Object

a) b)

Figure 1.6: Typical configurations for low-field NMR. a) Schematic of an In situ magnet. b) Schematic of anEx situ magnet. The Region of Interest (RoI) is shown in red. While the object must be able to fit inside thein situ magnet, it can be larger than the ex situ one.

1.3.3.1 In situ magnets

As stated in the previous paragraph, in situ magnets o!er the most field e"ciency per unit of mass.This is due to the ability to confine the field in such geometries. Mainly, two e"cient schemes havebeen developped. They rely either on the use of a ferromagnetic yoke (usually made out of iron) (61)or on the Halbach’s scheme (62). Due to the field confinement, such structures are less sensitive toexternal perturbation (63).

Yoked magnets Such structures usually have the shape of a C or an H. The design of yoked magnetsgoes back to the design of electromagnets, formerly used in NMR and still in use in particle accelerators.The basic principle of these systems is to use a source of magnetic induction, such as a coil or apermanent magnet in conjunction with a medium to "confine" the created induction and carry it tothe point of interest. Iron and steel are commonly used for the so-called yoke part that acts as thismedium. Examples of yoked structures are shown on figure 1.7. The medium must have very highpermeability and high saturation magnetization in order to keep its confining properties up to highfield inductions. The field strength in the RoI is usually limited by the strength of the field source andthe magnetization saturation of the yoke. The saturation magnetization is generally around 2T, thuslimiting the field in the RoI to a similar value. The field strength is in fact limited to about two thirdsof the remanence of the permanent magnet used as a source (64). The homogeneity of such systemscan be controlled through the use of properly shaped pole pieces placed at the end of the yoke andfacing the RoI (65). The high permeability allows to assume the surface of the poles are equipotentialsof the magnetic scalar potential. Hence, the shape of the pole directly define the variations of the fieldin the RoI. Lately, a yoked magnet has been realized for portable NMR with a field of 1T in the RoIand a homogeneity of 0.25 ppm over a very small volume (21 nanoliters) (63).


B0B0

B0 B0

a) b)

c) d)

Figure 1.7: Examples of cross-section of iron-based magnets with translationnal symmetry. The high-permeability yoke provides a suitable confinement of the field to concentrate it in the gap. Proper shapingof the pole faces (surfaces of the yoke facing the gap) provides correction for homogeneity of the field in the gap.a) C-shaped magnet with coils as induction sources. b) Equivalent C-shaped magnet using permanent magnetsas field sources. c) H-shaped magnet based on currents d) Equivalent H-shaped magnet based on permanentmagnets.

Halbach dipole magnets and yokeless magnets In 1980, Klaus Halbach had already seen thenew possibilities o!ered by rare-earth materials and proposed a new way of assembling permanentmagnets to generate 2D multipoles of arbitrary order and strength (62). These structures are extremelye"cient for field confinement : they cancel the field outside of the structure while reinforcing it at thecenter. These so-called Halbach multipoles are simple infinitely long rings of permanent magnets witha magnetization orientation depending on the angular position so that

"2 = k"1 (1.92)

where "1 is the angular position in the ring and "2 is the angular orientation of the magnetization (seefigure 1.8 a). Of particular interest is the situation where k = 2 : the resulting structure is a perfectdipole with a completely homogeneous field in the bore. In addition, it is possible to show that thefield strength at the center is (62; 61)

B0 = M0 ln!

r2

r1

"(1.93)

where B0 is the field at the center, M0 is the magnetization amplitude, r1 is the inner radius and r2 isthe outer radius (see figure 1.8 b). Equation 1.93 gives a very important property of Halbach’s dipoles :their field strength in the bore hole is theoretically unlimited if one is prepared to use a lot of material.


However, in practice, a limitation occurs due to the coercivity of the material. There are indeed verystrong demagnetizing fields in the structure, and as the ratio r2

r1increases, some parts are more and

more demagnetized, so that there exists a limit to the strength of such dipoles. An extended work isbeing conducted in the field of accelerator magnets to increase the field strength of Halbach’s dipolesand deal with the demagnetization problem. Field strengths of 4T have been reached with specificallydesigned assemblies based on Halbach’s scheme (66–68). Yokeless structures (which are mostly derivedfrom Halbach’s structure) are more e"cient than yoked systems: they require less material to generatethe same field in their gap. This advantage increases with field strength (69). In addition, Halbachdipole structures have no stray field in theory.

!1

!2

M

x

y

r2

r1

B0

a) b)

Figure 1.8: a) 2D original Halbach structure. The magnetization is at an angle "2 with the y axis dependingon its angular position "1. b) Halbach dipole. The field at the center is completely homogeneous and itsstrength depends solely on the magnetization amplitude, the inner radius and the outer radius.

Halbach’s dipoles are highly homogeneous only in the case of a 2D structure with continuouslyvarying magnetization. Both of these conditions are not realized in practice for NMR magnets. A NMRmagnet, especially if it is meant to be portable, should not be very long, so that 3D characteristics of thestructure have to be taken into account. In addition, the angular variation of the magnetization mustbe discrete, due to the fabrication process of permanent magnets. Both of these imperfections impactthe field strength and the homogeneity of the magnet. In the case of NMR or MRI, the homogeneity ofthe field is a critical feature of the magnet. While field strength is desired for better SNR and improvedspectral resolution, homogeneity of the order of 10#4 (for relaxation measurements) to 10#7 (for NMRspectroscopy) over the sample volume is necessary to conduct standard experiments. The researchconducted in the domain of accelerator magnets is focused on field strength while homogeneity remainsa secondary issue (achieved homogeneities are of the order of 10#3 after shimming) and it has beenhence necessary to carry specific developments oriented toward homogeneity for NMR applications.

One problem posed by the Halbach structure is its assembly. There are many ways of discretizing themagnetization orientation, with various results on field strength, demagnetization and field uniformity.The tiling of permanent magnet pieces and its e!ect on homogeneity has been studied (70) and some


prismatic structures for parallelepiped geometries easier to build have been proposed (69; 71; 61).Ways of obtaining a homogeneous field have been proposed for such structures for MRI (72; 61) anda rather large MRI scanner for mice with a field of 1T and good homogeneity (10ppm in 3cm DSV)has been recently demonstrated (73). This system remains very compact compared to standard smallanimal MRI scanners.

All of the previously cited work on Halbach-like structures has not been concerned about highorders of homogeneity in order to keep the magnet structure small compared to the subject of study.No truly portable system has been introduced up to now with a high degree of homogeneity over alarge volume. There has been important work done in order to produce compact magnets for portableand low-cost NMR. A first Halbach structure dedicated to portable NMR was proposed in 2004 byRaich and Blümler, the NMR Mandhalas (74). This system is based on magnet bars and has provensuccessful from the point of view of cost, ease of realization and reasonable field strength (0.3T). Thehomogeneity was however poor at the time. This concept has been then extensively used to constructvariations of the initial system, in order to provide easy access to the region of interest (75) or increasethe field strength (76). The homogeneity of these systems remains however poor (tens of ppm atbest over small sample sizes (millimeter or sub-millimeter scale). A more recent work based on theMandhalas concept has been published, proposing a larger magnet (33cm diameter, 27cm length, 20cmbore diameter) with a field of 0.22T and homogeneity of 0.85ppm over 1cm3 (77). However, while theperformance is much better than previous Mandhalas systems, the sample is still very small comparedto the magnet volume.

It is important to realize that some NMR experiments do not require high homogeneity. Analysisof relaxation time can be done through CPMG sequences which refocus the dephasing due to theinhomogeneities. Hence, there are many experiences that can be conducted though the magnet isnot perfect. That is why Halbach’s dipoles have been extensively used for portable NMR, as one ofthe main application lies in the measurement of moisture and porosity (78; 49). In such cases, evensimple inhomogeneous Halbach systems are highly indicated as they tremendously reduce the cost ofthe system, compared to an homogeneous standard NMR spectrometer. Figure 1.9 illustrates some ofthe in situ prototypes cited in this section.


Figure 1.9: Views of previously published prototypes for permanent magnet based in situ NMR. a) Yokedmagnet achieving sub-ppm resolution on micrometric samples with a field of 1T (63).b) Example of C-shapedyoked magnet with pole pieces profiled in order to achieve a homogeneous field (65). c) Simple Halbach structurewith poor homogeneity and low field (about 90 mT) but easy access to the sample (75). d) NMR Mandhalas :Halbach magnet constituted of magnet bars or cubes (74). e) Halbach magnet fabricated with stacked rings ofmagnet rods (76). f) Parallelepipedic Halbach magnet with high homogeneity (10 ppm in 3 cm DSV) for miceMRI (73).


1.3.3.2 Ex situ magnets

While in situ magnets are e"cient at concentrating the field and achieving higher field strength (evenhigher than the remanence in the case of Halbach’s system), they still feature the limitation of regularNMR spectrometer : the sample must fit inside the bore hole. Ex situ magnets (also called single-sidedor unilateral) attempt to solve this issue by moving the experiment location outside of the magnet.This results in an "open" structure, which is much less e"cient in terms of field strength.

Another di"culty coming with unilateral magnets is their intrinsically inhomogenous field. Theplanar symmetry that can be used in closed magnets is absent in this situation and the spatial variationof the field is very fast. The obtention of a homogeneous field requires even more specific care than forclosed magnets, as we will see later on.

The first examples of "open" structures can be found in well-logging were the NMR system is placedin the drilling system and probes the surrounding rocks, outside of the magnet. Several systems havebeen used (79–82; 45) and remain quite inhomogenous (unsuitable for spectroscopy or imaging), theirpurpose being only to measure di!usion and relaxation times in rocks in order to assess the presence ofwater, oil and the rock’s porosity. NMR well-logging tools are still undergoing development as recentpatents have been filed (83–85) and design considerations are still discussed (86).

The concept of NMR "outside of the magnet" was later introduced to more common NMR appli-cations in the late 1980s and the beginning of the 1990s with the STRAFI (STRAy Field Imaging)techniques (87). The idea there is to use the extremely strong gradient o!ered by the stray field ofsuperconducting magnets (in excess of 50 T m#1 while standard gradient coils in a scanner hardlyachieve 30 mT m#1 (88)) in order to obtain high resolution NMR imaging (down to the micrometer).One of the main issue in such techniques is the SNR, which can be increased by using train of echoessuch as solid echoes or spin echoes. The very stable gradient of superconducting magnets can also beused for di!usion measurements. An extensive review of STRAFI can be found in (89). These workshelped showing how useful information could be obtained from NMR experiments performed in openspaces, despite a highly inhomogenous field.

The interest shown in STRAFI techniques probably triggered the development of permanent mag-nets for more regular NMR (as opposed to NMR well-logging). It really started with the demonstrationof the NMR MOUSE in 1996 (90; 91) by the team of Prof. Blümich. This very small hand-held sys-tem featured a simple yoked magnet generating a field parallel to its surface very close to it withan integrated detection coil. The field is reasonably high (about 0.5T) and the homogeneity verypoor. However, the compactness of this system and its ability to perform NMR analysis of surfacesof large objects made it a success as it found a wide range of applications (92; 54) and similar sys-tems built by the main NMR instrumentation companies (Bruker, Varian) have also been extensivelyused (93; 94; 55). The NMR-MOUSE found several declinations later on, such as a system based ona simple bar magnet (95) and a more involved design in order to improve single-sided detection (96).Inhomogeneous single-sided NMR has been explored further by Glover et al in order to accomplishSTRAFI on thin films and coating with optimized magnets for strong gradients (20 T m#1) (97; 98).

While STRAFI may be highly useful, another goal pursued by the single-sided NMR community isthe achievement of su"cient homogeneity to accomplish more standard NMR experiments and providea versatile tool with the most of the power of NMR. Modified versions of the NMR-MOUSE werereported and used with improved homogeneity in 2001 and 2003 (99; 100) and it is only in 2005and 2006 that significant progresses in the design of homogeneous magnets were done. The magnetwas broken down in several elements with adjustable gaps between them in order to control the fieldhomogeneity to some extent (101; 102). One of these systems was even equipped with a unilateraltransverse gradient system in order to perform 3D imaging (101). In the meanwhile, the concept thatinhomogeneities of the static field could be corrected by correlated inhomogeneities of the RF field


was introduced (8). If the orthogonality between the static and the RF field is achieved in the sametime as the correlation of the static field variations with the RF field ones, nutation echoes can beobtained while preserving the chemical shift, thus regaining resolution that is lost during a classicalspin echo. Based on the previous designs and that concept, a much higher performance system wasintroduced shortly after in order to achieve a 8 ppm resolution (103; 104), allowing spectroscopy oflarge chemical shift range nuclei (in that case, fluorine). A few radically di!erent designs were alsoproposed. We can cite the NMR-MOLE proposed in 2006 (105), which o!ered a 1.5% homogeneity overan extended volume (about 6000 mm3), and the surface-GARField (106) which achieved essentiallythe same functions as the NMR-MOUSE, with an attempt to obtain a constant field in planes parralelto the surface of the magnet. Another original system introduced was a single sided magnet basedon the stray field of a Halbach ring (107). All these systems remained however very inhomogeneous.Such inhomogeneities can be put in good use as shown by Rahmatallah et al in 2005 (108) in order tospatially encode or localize the NMR experiment.

It is only starting in 2007 that a sub-ppm single-sided magnet was proposed by Perlo et al (109).This system was based on the previous designs of the team (103; 104) and made use in addition ofa passive shimming system. This system was constituted of permanent magnet blocks that could bemoved with micrometric precision. A resolution of 0.25ppm could be achieved on a volume of 5x5x0.5mm with a magnet of size 28x28x12 cm. This prototype has achieved the ultimate homogeneityuntil now. A few other systems have been proposed more recently but most of them remain veryinhomogeneous systems (110).

Most of the previously cited systems are based on empirical designs, sometimes numerically opti-mized based on FEM softwares (98). It is worth noting that some other works have been based ontheoretical considerations in order to provide systematic methods for designing single-sided magnets.Marble et al proposed in 2005 a method of designing 2D magnets based on the scalar potential (111)and described a prototype of constant-gradient magnet based on this method (112). In addition oftreating the problem homogeneity in 2D, Marble treated on the problem of the maximum field achiev-able for a 2D magnet (113). Finally, Marble et al proposed some elements for the design of 3D magnetsbased on the scalar potential (114) but never published a highly homogeneous system based on thismethod.

Another recent inventive design proposed in 2009 by Paulsen et al might also be of interest tothe reader. This system o!ers the ability to move in space the sweet spot generated by the magnet,through the mechanical rotation of its magnet elements (115).

This short account of the state of the art of single-sided NMR is by no means exhaustive and thereader should refer to (46) for more details. Figure 1.10 illustrates some selected examples of prototypescited in this section.


a) b) c)

d) e) f )

Figure 1.10: Views of published prototypes for single-sided NMR. a) NMR-MOUSE with short dead-time (96).b) Optimized structure to induce a strong field gradient (20T m!1) for STRAFI (98) called GarField. c)NMR-MOLE generating a somehow homogeneous field in a sweet spot (105) d) Open tomograph based onNMR-MOUSE developments (101). e) Permanent magnet with an iron pole piece calculated based on scalarpotential theory (114). f) High resolution single-sided magnet for NMR spectroscopy, based on a mechanicallyadjustable and electromagnetic shimming system (109)

1.3.4 Motivation for this work

One of the major di"culties encountered in the design of portable magnets is the achievement of homo-geneity. All the systems stated above do not provide the homogeneity necessary to NMR spectroscopyor high-resolution MRI over a macroscopic volume (scale of centimeter). A few of the systems proposedachieved such homogeneity on very small sample size (5x5x0.5 mm in (109) or 21 nanoliter (63)) thuslimiting the signal to noise and imposing precise positioning of the sample. The goal of this work isto push further the performances of portable systems by increasing the ratio of the "sweetspot" dimensions to the magnet dimensions while maintaining a field as high as possible.We have worked on both families of magnets for di!erent purposes. Single-sided magnets are the maintopic of this work, with the demonstration of a single-sided magnet generating a uniform gradient overa wide volume with unprecedented penetration depth. Theoretical design and perspectives for sucha magnet generating a uniform field over a wide volume are also given. Unilateral detection systems(RF coils and gradient coils) are also studied and optimized but do not constitute our main focus here.In situ magnets are also studied for the generation of longitudinal and transverse fields. This provides


the ability to control the orientation of the field with respect to the magnet, leading us to demonstratea magnet generating a field pointing at the magic angle, which could be interesting for MAgic AngleTurning experiments.

Chapter 2

General approach to magnet design andfabrication

2.1 Preliminary remarks on field homogeneity

As it was discussed in the previous chapter, NMR is sensitive to the modulus of the magnetic field.This modulus must achieve sub-ppm uniformity in the case of proton NMR spectroscopy while therequirements on the field homogeneity of MRI vary with the strength of the gradients but usuallyrequire a few ppm over the volume of interest. This homogeneity must be achieved on the modulusof the field, that is to say on all three components of the field. In addition to its requirements onhomogeneity, NMR requires the highest magnetic field possible in order to increase its poor sensitivity.

We are now going to see how the problem of scalar homogeneity of the field (meaning that themodulus of the field must be homogeneous) can be simplified.

The field!"B0 at the center can be used as a reference and one can choose a vector basis such that

!"B0 = B0

!"uz. (2.1)

In the case of a homogeneous field, the field can be expressed in the vicinity of the center as :

!"B (x, y, z) = B0

!"uz +!"b , (2.2)

with!"b = bx

!"ux + by!"uy + bz

!"uz where222!"b

222 = b + B0. As a result, if one considers the modulus ofthe field,

B(x, y, z) =,

(B0 + bz)2 + b2x + b2

y,

or, B(x, y, z) = B0

3

1 + 2bz

B0+

b2

B20

. (2.3)

Expanding in Taylor series yields

B(x, y, z) = B0

!1 + 2

bz

B0+ O

!b2

B20

"", (2.4)

47

48 CHAPTER 2. GENERAL APPROACH TO MAGNET DESIGN AND FABRICATION

which shows that one can remain concerned only by one component of the field (the main one), as theother components a!ect the field only starting at the second order. As a result, the following consid-erations and example will deal only with one component of the field. In some cases, this assumptiondoes not apply and we will see how to deal with it in the specific case of a field gradient.

2.2 Homogeneity and Spherical Harmonic Expansions (SHE)

2.2.1 Theoretical considerations

The problem of homogeneity brings up the one of qualifying the variations of the field. In the end,one could think that the largest variation of the magnetic field throughout the sample accounts for thequality of the field in the Region of Interest (RoI). However, this might not be true as the inhomogeneitya!ecting the final NMR spectrum is weighted by the number of spin "seeing" that inhomogeneity. It ispossible to discuss what is the best so-called field quality factor for a long time, but the final criterionis the linewidth of the spectrum of a nucleus in a simple solution (classically, 1H in deuterated water).This linewidth can be computed from simulations and can theoretically be used as a criterion duringthe design of the magnet. However, spectrum simulation is a heavy numerical process and is notdesirable in an optimization task. It seems that the most straightforward criterion of optimization is infact the maximum field variation. But it appears quickly that this single number is not helping muchin the quest for a good magnet configuration, because it usually is a highly non-linear function of thediverse magnet parameters. As a result, a local optimum can be found but it can still be far from aglobal optimum and no control is exerted on the value of this local minimum.

The use of a spectral method seems more appropriate for the tailoring of the magnet field. Sucha method relies on the decomposition of the field component variations (a scalar function) onto somebasis. This allows to organize hierarchically and to qualify the variations. Hence, the optimizationprocess starts by setting the homogeneity target, which defines which terms of the decomposition (orspectrum) should be cancelled. Consequently, the definition of the magnet parameters relies on anoptimization with a known objective . Some initial considerations provide the insurance that theobjective can be achieved and that the final system will perform as initially specified. A method ofchoice in magnet design is the Spherical Harmonics Expansion (SHE).

In the case of NMR, the region of interest (RoI) is empty when there is no sample: the RoI is freeof any field sources. Hence, we have, from Maxwell’s equations :

!") (!"B =!"0 . (2.5)

As a result, one can write

!"B = !!")"!, (2.6)

where "! is a pseudo-scalar potential that verifies the Laplace equation :

!"! = 0 (2.7)In the case we are interested in, it is appropriate to represent the RoI as a sphere. The center

of this sphere will be called the origin. The Laplace equation is separable in the spherical coordinatesystem and one can obtain a unique decomposition of the potential on the spherical harmonics base,centered on the origin. The general solution for the potential can then be written (32):

"!(r, ',&) ="$

n=0

n$

m=#n

[Anmrn + Bnmr#(n+1)]Y mn (',&), (2.8)

2.2. HOMOGENEITY AND SPHERICAL HARMONIC EXPANSIONS (SHE) 49

where

Ynm =

32n + 1

4$

(n!m)!(n + m)!

Pmn (cos ')eim%. (2.9)

The Y mn s are the angular part of the spherical harmonics while the Pm

n are the associated Legendrepolynomials.

Useful properties of Legendre polynomials can be found in classic mathematics textbooks suchas (116). Important parity properties are that the Legendre polynomial of degree n has the sameparity as n, while the associated Legendre polynomial of degree n and order m has the same parity asn + m. The angular part of the spherical harmonics can be expressed in terms of associated Legendrepolynomials :

Y mn (', &) =

32n + 1

2(n!m)!(n + m)!

P |m|n (cos ')

2222222

1&2!

for m = 01&!

cos m& for m > 01&!

sin m& for m < 0

2222222(2.10)

The spherical harmonics and the associated Legendre polynomials are orthogonal, making eachterm of an expansion in spherical harmonics completely independent. The spherical harmonics andthe Legendre polynomials are defined here without the (!1)m Condon-Shortley phase factor. A shortsummary of useful identities and defining equations of Legendre polynomials and associated Legendrepolynomials can be found in appendix B.

The SHE has been extensively used in the fabrication of electromagnets and superconductingmagnets for NMR and MRI. Many analytical developments have been presented in the past (117)and this formalism has been used in permanent magnet design by Marble et al for the design of polepieces (114). The method proposed by Marble et al resembles the one existing in the design of iron-dominated magnets (see for example (65)): pole pieces are placed in the gap in order to adjust theproperties of the field in the RoI. The pole pieces method relies on the high permeability of somemagnetic materials such as iron. It can be indeed shown that, at the interface between two media (32)

(!"B2 !

!"B1) ·!"n = 0 (2.11)

!"n ( (!"H2 !

!"H1) =

!"K, (2.12)

where!"K is the surface current density and !"n is a unit vector normal to the surface pointing from

region 1 to region 2. For media satisfying equation 1.85, these boundary equations can be rewrittenfor the field induction as

!"B2 ·!"n =

!"B1 ·!"n (2.13)

!"B2 (!"n =

µ2

µ1

!"B1 (!"n . (2.14)

Hence, for the transition from a medium of high permeability to a medium like air (µ2 = 1), thetangential component is very small. High permeability materials like iron or mu-metal can achieverelative permeability between 10000 and 100000. It is hence usually assumed in the magnet designthat the ratio µ2

µ1is zero, so that the tangential component of the induction in air is zero and the

field induction is normal to the surface of the pole. As we saw that the field induction derives from ascalar potential in empty space, the surface of the pole is an equipotential of that scalar potential. Itis hence su"cient to calculate the appropriate equipotential shape that provides the desired SHE inthe RoI. However, the surface of the pole is not exactly an equipotential as the ratio of the tangentialcomponent to the normal component is of the order of 10#4 to 10#5. Besides, as mentioned in (114),the equipotential shape has to be truncated in order to fabricate the magnet and end-e!ects have a


significant impact on the field profile. Both of these imperfections induce deviations from the idealmagnet which become very visible when trying to achieve ppm or sub-ppm homogeneity. A shimmingstep is of course required to compensate these imperfections and obtain initial specifications. This hashowever not been published yet by Marble et al.

The pole pieces method is sound and seems likely to be able to provide high-performances. Ithowever highly relies on the quality of the pole piece and of the magnet used as the field source. Inaddition, soft ferromagnetic materials are di"cult to simulate with a high precision (better than 10#5),which is required for the type of homogeneity necessary for NMR. We will hence explore here anothermethod which does not require any high-permeability material but only high-quality magnet parts andlight shimming. We shall first make some additional considerations on the SHE of the potential andof the field and their relations with the magnetic sources distribution.

It is important to keep in mind that the potential exists only in empty areas of space. One candivide the space into two areas where the potential exists: inside the biggest sphere centered on theorigin that does not contain any source, and outside the smallest sphere centered on the origin thatcontains all the sources. Figure 2.1 summarizes these regions.

OInner region

Source region

Outer region

Figure 2.1: Schematic representation of the di!erent regions of space relevant to the spherical harmonicexpansion. The Laplace equation is only verified in areas free of sources, hence the expansion is possible onlyin the inner sphere and in the outer sphere.

When the sources are located outside of this sphere, we can write this expansion as :

"!(r, ',&) =1µ0

#Z0 +

"$

n=1

rn

%ZnPn(cos ') +

n$

m=1

(Xmn cos m& + Y m

n sin m&)Pmn (cos ')

&'. (2.15)

Where we call the Zn terms axial terms (also "Zonal harmonics" (117)) and the Xmn and Y m

n termsskewed terms (also "Tesseral harmonics" (117)). However, NMR is interested in the magnetic fieldand not in the potential. As each component of the field satisfies the Laplace equation, it is possibleto expand each component in the same way as the potential. However, the di!erent terms are not thesame. One can write the expansion of each component as follows :

2.2. HOMOGENEITY AND SPHERICAL HARMONIC EXPANSIONS (SHE) 51

Bx(r, ',&) = Zx0 +"$

n=1

rn

%ZxnPn(cos ') +

n$

m=1

(Xxmn cos m& + Y xm

n sin m&)Pmn (cos ')

&

By(r, ',&) = Zy0 +"$

n=1

rn

%ZynPn(cos ') +

n$

m=1

(Xymn cos m& + Y ym

n sin m&) Pmn (cos ')

&

Bz(r, ',&) = Zz0 +"$

n=1

rn

%ZznPn(cos ') +

n$

m=1

(Xzmn cos m& + Y zm

n sin m&) Pmn (cos')

&.

These equations show that the homogeneity requirements correspond to an expansion of the fieldfeaturing the degree 0 term only. This is not possible, but one can cancel as many terms as necessaryto achieve the required homogeneity over the desired volume. If we call a the radius of the largestsphere including no sources, for a given reference radius r0 < a, the Zn, Xm

n and Y mn terms have

a dependency in(

1r0

)n. Thus, the variations induced by a term of degree n scale as

(rr0

)nwith

r < r0. Consequently, one should cancel the k first degrees until4

ra

5k+1 is small enough. We shall nowsee how it is possible to cancel these terms through the design of the magnet. Symmetries can helptremendously this task. However, the magnet structure symmetries will reflect directly in the potentialsymmetries. It is hence necessary to consider first the potential expansion to determine the e!ect ofsymmetries and then convert it to field expansions.

One can establish in a straightforward way the following analytical relations between the potentialterms and the field terms after the relations given in (117) :

Xx1n = Zn+1 !

(n + 2)(n + 3)2

X2n+1 , n % 1

Xxmn =

12Xm#1

n+1 !(n + m + 1)(n + m + 2)

2Xm+1

n+1 , n % 2, 2 - m - n

Y x1n = !(n + 2)(n + 3)

2Y 2

n+1 , n % 1 (2.16)

Y xmn =

12Y m#1

n+1 ! (n + m + 1)(n + m + 2)2

Y m+1n+1 , n % 2, 2 - m - n

Zxn = !(n + 1)(n + 2)2

X1n+1 , n % 0

Xy1n = Zn+1 !

(n + 2)(n + 3)2

Y 2n+1 , n % 1

Xymn =

12Y m#1

n+1 ! (n + m + 1)(n + m + 2)2

Y m+1n+1 , n % 2, 2 - m - n

Y y1n = Zn+1 !

(n + 2)(n + 3)2

X2n+1 , n % 1 (2.17)

Y ymn =

12Xm#1

n+1 !(n + m + 1)(n + m + 2)

2Xm+1

n+1 , n % 2, 2 - m - n

Zyn = !(n + 1)(n + 2)2

Y 1n+1 , n % 0


Xzmn = !(n + m + 1)Xm

n+1 , n % 1, 1 - m - n

Y zmn = !(n + m + 1)Y m

n+1 , n % 1 1 - m - n (2.18)Zzn = !(n + 1)Zn+1 ,n % 0

These relations can also be found completely demonstrated in (31).From these equations and from the expansion of the potential, one can infer immediately that an

axisymmetric configuration is advantageous as it will cancel all the skewed terms in the potential andhence cancel many terms for any component of the field.

Depending on the desired direction of the field, one might be interested in one or the other com-ponent of the field. As the relations between the expansion of each component with the expansion ofthe potential is di!erent, one must treat di!erently each component.

In the case of Bz, it is straightforward to conclude that in order to obtain a homogeneous field, oneshould find a source distribution that creates a potential for which the expansion contains only the Z1

term. An n-fold rotational symmetry will remove the skewed terms up to order n!1, the first non-zeroskewed terms being then Xn

n and Y nn . Once skewed terms have been taken care of, one is left with

axial terms. Another helpful symmetry is then the planar antisymmetry with respect to xOy, whichwill leave only odd axial terms in the potential, hence leaving only even terms in the expansion of Bz.This symmetry cannot be achieved in the case of ex-situ systems, which allow to have sources onlyon one side of the RoI. It is then possible to arbitrarily cancel p degrees by arranging properly p + 1independent sources featuring the n-fold rotational symmetry (we need p + 1 independent variables).

The case of Bx and By can be treated at the same time. One can notice that the relation tothe potential is not as straightforward as for Bz. The e!ect of the n-fold rotational symmetry onthe transverse field expansions is not as obvious as for Bz, as some skewed terms of the field arecombinations of the potential’s axial terms and skewed terms. Nevertheless, axial symmetry cancelsall skewed terms in the potential so that we have only Xx1

n and Y y1n terms left in the field expansion.

However, we must remind that full axial symmetry (in terms of magnetic sources) can only induce alongitudinal field, so that here, Bx and By can only be null at the origin. The creation of a transversefield requires to give up the axial symmetry. In these conditions, a suitable arrangement is required tokeep the X1

1 or Y 11 term in the potential to obtain a degree zero term for Bx or By and cancel other

terms.Once the proper number of elements and the proper rotational symmetry are chosen, analytical

formulae for the remaining terms of the expansion provide the guides to the optimization of theproper dimensions of the magnet. The basic elements can be disks, rings, extruded polygons or evencylindrically symmetric arrangements of disks or cylinders, rings, extruded polygons. We shall explicitin the following section this analytical formulae. These formulae and their demonstration can be foundin the book in preparation by Guy Aubert (31). In everything that follows, the z axis is always therotational symmetry axis.

We have used here a cumbersome notation for the terms of the component expansions. From nowon, as we will consider only one component at a time, the terms of the relevant component will benoted Zn, Xm

n and Y mn and we will not refer anymore to the potential.

2.2.2 Design methodology

Based on the previous considerations, we can propose a systematic approach to the design of anaxisymmetric magnet achieving a specified homogeneity !B0 in a given sphere of diameter R0. Thefirst step is to obtain the degree n0 up to which field SHE terms must be cancelled. This can be done

2.3. ANALYTICAL BRICKS FOR AXISYMMETRIC MAGNETS 53

by assessing the order of magnitude of the radial term in the expansion. This term gives the amplitudeof the Legendre polynomial. The homogeneity condition can be written as

!r

R0

"n0

- !B0, (2.19)

hence,

n0 %ln !B0

ln rR0

, (2.20)

which defines the first non-zero axial term. For example, if we specify a homogeneity of 1 ppm over asphere of radius a quarter of the distance to the magnet (i.e. r

R0= 1

4), we have n0 % 10. Of course,this is a first crude estimation, as the geometry may cause higher degree terms to be smaller or greaterthan one. This estimation is usually conservative. In the case of axial symmetry for geometry andmagnetization, the field can only be longitudinal (along Oz) and the relationship between the Bz SHEand the scalar potential SHE is very straightforward (see equations 2.18). The number of segmentsin the structure must be greater than n0 + 1 in order to deal with the field skewed terms, while it isnecessary to use n0 + 1 independent axisymmetric elements in order to cancel axial terms in the fieldexpansion up to degree n (degree n is the first non-zero term). The lowest degree terms should betaken care of first, as they induce the most variations in the RoI (as the origin of the expansion is thecenter of the RoI).

When a di!erent orientation of the field is desired (for example, along Ox), the structure cannotfeature axial symmetry for the magnetization distribution, but we will see that geometries such as theHalbach dipoles keep a geometric axial symmetry while providing a transverse field with su"cientlyfew terms in the field expansion, so that these remaining terms can be cancelled by using a number ofindependent rings.

When possible, the xOy planar symmetry should be used in order to divide by two the number ofterms to be cancelled by positioning elements of appropriate dimensions.

The appropriate dimensions of the independent elements must be calculated in order to provide thecancellation of the desired axial terms of the potential. The cancellation of the terms can be used asconstraints to optimize the field strength or the field gradient strength for a given volume of material.This problem is highly non-linear when canceling more than two terms, hence it is necessary to havea good initial guess and to guide the optimization with analytical expressions. We will discuss thederivation of such formulas in the next section.

2.3 Analytical bricks for axisymmetric magnets

2.3.1 Dipoles

Dipoles are very important elements for magnet design as any magnetic part eventually "behaves" asa dipole at long distances. As a result, it is in most cases a good approximation to model first theelements of a design by dipoles and then move toward a specific geometry.

The magnetic field of a magnetic dipole can be written as (32)

!"B (x, y, z) =

µ0

4$

3(!"M ·!"u )!"u !!"M

R3(2.21)

with !"u being the unitary vector going from the dipole position P to the measurement point Q and Rbeing the distance from P to Q. The z component of the field can then be written as :


Bz(x, y, z) =µ0

4$

3(Mxux + Myuy + Mzuz)uz !Mz

R3(2.22)

It is possible to convert this expression to spherical coordinates but the direct calculation of theSHE of the field is not possible. The use of the scalar potential is however more successful.

P!0

"0

O y

z

x

!1

R0

"1

M1

Q

"

!r

R

Figure 2.2: Definition of the di!erent spherical coordinates relevant to the dipole.

Transforming to spherical coordinates, with!!"OP (R0, '0, &0),

!!"OQ(r, ',&), and

!"M(M1, '1, &1) as

shown on figure 2.2, we can obtain the following expressions for the SHE of the potential (see derivationin appendix C), originally shown by Aubert (31),

"! = ! M1

4$R20

6T 0

0 +"$

n=1

!r

R0

"n%T 0

nPn(cos ') +n$

m=1

(n!m)!(n + m)!

Tmn Pm

n (cos ')

&7. (2.23)

T 0n = (n + 1) cos '1Pn+1(cos '0) + sin '1P

1n+1(cos '0) cos(&1 ! &0)

Tmn = 2(n!m + 1) cos '1P

mn+1(cos '0) cos m(&! &0)

!(n!m + 1)(n!m + 2) sin '1Pm#1n+1 (cos '0) cos{m&! [(m! 1)&0 + &1]}

+ sin '1Pm+1n+1 (cos '0) cos{m&! [(m + 1)&0 ! &1]}

These equations, though large, o!er a completely analytical calculation of the SHE terms of thepotential up to an arbitrary degree and order.

Using equation 2.18, we can deduce analytical expressions of the SHE terms of Bz from the potential:

Bz =µ0M1

4$R30

"$

n=0

!r

r0

"n%HnPn(cos ') +

n$

m=1

(Imn cos m& + Jm

n sinm&)Wmn Pm

n (cos ')

&, (2.24)


with

Wmn =

(n!m! 1)!!(n + m! 1)!!

Hn = (n + 1)!

r0

R0

"n .(n + 2) cos '1Pn+2(cos '0) + sin '1P

1n+2(cos '0) cos(&1 ! &0)

/(2.25)

8888Imn

Jmn

8888 = (n!m + 1)(n!m)!!(n + m)!!

!r0

R0

"n

9::::::;

::::::<

2(n!m + 2) cos '1Pmn+2(cos '0)

8888cossin

8888m&0

!(n!m + 2)(n!m + 3) sin '1Pm#1n+2 (cos '0)

8888cossin

8888 [(m! 1)&0 + &1]

+ sin '1Pm+1n+2 (cos '0)

8888cossin

8888 [(m + 1)&0 ! &1]

=::::::>

::::::?

(2.26)

where r0 is a reference radius, {R0, '0, &0} are the spherical coordinates of the dipole position, {M1, '1, &1}are the spherical coordinates of the dipole magnetization. The Wm

n term allows a normalization ofthe associated Legendre polynomials so that the Im

n and Jmn terms have a physical significance (the

symbol !! depicts the double factorial, n!! = n(n ! 2)(n ! 4)..., 0!! = 1!! = 1, and 2!! = 2). Theseexpressions procure the ability of computing any SHE term of Bz at any degree and with arbitraryprecision. They are hence very useful to create a crude model of a magnet with an evaluation of theorder of magnitude of the SHE terms, and also, very importantly, they give a perfect test-bench forSHE terms measurements. In addition, they can prove very useful in the context of passive shimming,as we will see in section 3.1.1.4.

As the required field homogeneity is of the order of 10#6 to 10#8, approximating field sourcesby dipoles (so-called dipolar approximation) becomes rapidly inappropriate and it is necessary totake into account the geometry of the parts used in the design. It is possible to derive analyticalformulas of the z component of the field and of its SHE for some specific geometries with specificmagnetization orientation. As we adopt an axisymmetric geometry, two field orientation are of interest: the longitudinal one, along Oz and the transverse one, in the xOy plane, or, to make it simpler,along Ox.

2.3.2 Creation of a longitudinal field with rings

As we saw in section 2.2, structures featuring axisymmetric geometries are of high interest, as theysuppress a number of terms in the SHE of the potential and thus in the SHE of the field. Cylindricalrings are hence keys to homogeneous magnet structures. In many cases, it will not be possible tofabricate a magnet ring in a single piece and it will require to be segmented. A segmented geometryvery close to a cylindrical ring is a polygonal rim and we will hence also treat such geometries. Wegive here a few examples of elementary structures for which analytical expressions of the SHE of Bz

can be given. In what follows, Oz is always the axis of symmetry. The rings are defined by theirinner radius a1, their outer radius a2, their thickness b2 ! b1 where b1 is the z coordinate of the faceclosest to the RoI and b2 is the z coordinate of the furthest one. Polygonal rings feature in additiona number of segments N and an angular aperture of the segments 2&seg. When the polygonal ring issolid, consecutive segments touch each other so that &seg = !

N . Figure 2.3 provides a view of a ringwith the di!erent parameters defining it.

For such rings, we can identify two main types of magnetization schemes that are useful for thefabrication of an NMR magnet : a magnetization distribution in one direction parallel to the axis


a2a1

!seg

z

b1

b2

O

Figure 2.3: Definition of the geometry of polygonal rings with N = 12 segments. On the left, schematicdefining the di!erent radii and the segment aperture. On the right, 3D view of such a ring, b2 ! b1 being theheight of the ring.

or a radial magnetization distribution (inward or outward). Figure 2.4 shows schematically thesedistributions. These configurations are of interest when a field parallel to Oz is to be generated.

As described in section 2.2, axial symmetry provides a mean of cancelling skewed terms. Thepotential can be written in the form

"!(r, ') ="$

n=1

rnZ!!n Pn(cos '), (2.27)

and the field SHE terms can be derived using relation 2.18. We have changed notation here for thesake of simplicity. We now call potential axial terms Z%!

n and field terms Zn.We can infer from the parity of Pn(cos ') that a planar antisymmetry may cancel all even terms

in the potential expansion, thus leaving only even terms in the field expansion. The field expansion inthe general case of an axisymmetric structure can be written as

Bz(r, ') = Z0 +"$

n=1

rnZnPn(cos ') (2.28)

It might be necessary to adopt a discrete axial symmetry, like cylinders of polygonal cross-section.In that case, N segments ensure the first non-zero skewed term appears at degree N in the potentialSHE and at degree N !1 in the Bz SHE. Thus, it is possible to choose to use enough segments so thatskewed terms appear only at negligible degrees, so that equation 2.28 is still valid for all non-negligibleterms. The homogeneity can then be achieved by canceling the few remaining axial terms (that mighthave already been decimated by a planar antisymmetry).

It is possible to give analytical expressions of these axial terms. For the sake of readability, wewill not detail the derivation of these expressions. The details of these expressions and more largely ofproperties of axisymmetric distributions of currents and permanent magnets can be found in (31).

2.3.2.1 Cylindrical rings

Longitudinal magnetization The entire ring is magnetized in one direction parallel to the axis ofsymmetry with magnetization M . This ring is equivalent to the superposition of two solid cylinders


a) b) c) d)

Figure 2.4: Di!erent shapes of ring, segmented or not with the two types of magnetization distributions. a)Cylindrical ring with longitudinal magnetization. b) Segmented ring (N = 12) with longitudinal magnetization.c) Cylindrical ring with continuous outward radial magnetization. d) Segmented ring (N = 12) with segmentedoutward radial magnetization

of radius a1 and a2 and delimited by the same planes as the ring. Cylinder 2 is magnetized in thesame direction as the ring and cylinder 1 is magnetized in the opposite direction. Such cylinders areequivalent to solenoids of same radii and height with surface current density equal to M and !Mrespectively. The field generated on the axis by a solenoid of finite length is well known and can bewritten (29) as

Bzsol(r = 0, z) =µ0M

2

%b! z1

a2 + (b! z)2

&b2

b1

. (2.29)

The brackets [f(b)]b2b1 are a short notation for f(b2) ! f(b1). Hence, the field generated by the ring isgiven by

Bz(r = 0, z) =µ0M

2

%%b! z1

a2 + (b! z)2

&a2

a1

&b2

b1

. (2.30)

Thus,

Z0 =µ0M

2

%*b*

a2 + b2

+a2

a1

&b2

b1

=µ0M

2.[cos -]a2

a1

/b2b1

, (2.31)

with

a = c sin-,

b = c cos -, (2.32)

c =1

a2 + b2.

These definitions of - and c are going to be used throughout this work. A term of degree N + 1 canbe derived from the term of degree N thanks to the following relation (simple di!erentiation)

Zn = ! 1n

%*+Zn#1

+b

+a2

a1

&b2

b1

. (2.33)

After applying this relation, we find

Z1 = !µ0M

2

%*1c

sin-P 11 (cos -)

+a2

a1

&b2

b1

. (2.34)


However, if we look at the derivative of 1cn sin-P 1

n(cos -), we get

+

+b

!1cn

sin-P 1n(cos -)

"= ! n

cn+1cos - sin-P 1

n(cos -) ++-

+b

!1cn

(nP 1n+1 ! n cos -P 1

n(cos -)"

= ! n

cn+1sin-P 1

n+1 (2.35)

Thus, by induction, we find that

Zn'1 = !µ0M

2n

%*1cn

sin-P 1n(cos -)

+a2

a1

&b2

b1

. (2.36)

Radial magnetization The ring is magnetized radially inward or outward (see figure 2.4) : themagnetization points along ±!"ur everywhere in the ring. In the case of an outward magnetization,the cylinder is equivalent to two surface pole densities on the inner and outer cylinder with density#! = +M on the outer one (radius r2) and !M on the inner one (radius r1), and a volume pole density)! = !div

!"M = !M

a . The z component of the field on axis can be written as

Bz(r = 0, z) =µ0M

2

%%a1

a2 + (b! z)2! tanh#1 a1

a2 + (b! z)2

&a2

a1

&b2

b1

. (2.37)

The axial terms of the field expansion are not straightforward to establish. We will not give the detailsof the formulae and their derivation, which can be found in (31). All terms can be written in the form

Zn =µ0M

2.[F (a, b)]a2

a1

/b2b1

(2.38)

It is very di"cult to accurately and precisely fabricate such rings (because of the continuouslyvarying magnetization within the ring). It is however possible to fabricate cylindrical rings wherethe magnetization direction has a discrete variation (segmented) and this paragraph gives a goodapproximation of such rings. This approximation can be refined further during the magnet design.

2.3.2.2 Polygonal rings

Longitudinal magnetization The magnetization of all N segments points in the same directionparallel to Oz. As a result, each segment of the ring is equivalent to two trapezoïdal surfaces withfictitious poles surface density +#! = M at z = b2 and !M at z = b1. For a point on the axis, thefield takes the form (see (31) for the full expression and its derivation),

Bz(r = 0, z) = Nµ0M

2$

@[F (a, b,&seg, z]a2

a1

Ab2

b1. (2.39)

The degree zero follows from that relation and higher degree terms can be found using relation 2.33.The resulting expressions become rapidly extremely complex and a compact recursive form can bederived in order to compute directly an axial term of arbitrary degree. It is however not the point ofthis work and the reader should refer to (31) for this form. The equations given here are usable forthe magnet models discussed later in this work.


Radial magnetization The magnetization of the N segments points in the !"ur direction of eachsegment. Hence, for the outward magnetization, each segment is equivalent to an inner and an outersurface with surface pole density respectively #! = !M and +M , and also two side surfaces withsurface pole density #! = !M sin&seg . The signs would be opposite for all surfaces for an inwardmagnetization. Here again, the z component of the field for a point on the axis takes the form (see (31)for the full expression),

Bz(r = 0, z) = Nµ0M

2$

@[F (a, b,&seg, z]a2

a1

Ab2

b1, (2.40)

from which all terms can be derived using equation 2.33.

2.3.2.3 E!ciency of longitudinal and radial magnetization

We have identified two types of magnetization distributions in the preceding paragraphs. It is importantto determine which is the most e"cient for a given location in order to obtain the highest field strengthpossible. This assessment can be done in terms of the e"ciency of a dipole. As we saw previously,the contribution to the z component of the field at point O(0, 0, 0) from a dipole placed at pointP (r0, '0, &0) with dipolar moment

!"M1(M1, '1, &1) is given by

Bz(O) =µ0M1

4$R30

.2 cos '1P2(cos '0) + sin '1P

12 (cos '0) cos(&1 ! &0)

/. (2.41)

We can notice that for the longitudinal magnetization, we have sin '1 = 0 so that the absolutecontribution is proportional to |2P2(cos '0)| while for the radial magnetization, we have cos '1 = 0 andthe absolute contribution is proportional to |P 1

2 (cos '0)| cos(&1 ! &0). In addition, &1 is in that caseeither &0 or &0 + $. Thus, the cosine term is a constant (either 1 or !1). Thus, in both cases, theamplitude of the contribution of the dipole is controlled by '0. Hence we can give the following intervalon which the radial magnetization is the most e"cient,

*17! 3

4- | cot '0| -

*17 + 3

4. (2.42)

If the dipole is located at a distance r = a from the axis with altitude z = b, we have cot '0 = ba and

the interval where the radial magnetization is most e"cient is

0.28a - b - 1.78a. (2.43)

These considerations solve the problem of which magnetization distribution is to be used to generatea longitudinal field depending on the location of the element. A schematic shows the areas where eachmagnetization distribution is the most e"cient in order to generate Bz in O on figure 2.5. The figureis to scale.


BzO

b

a

Figure 2.5: Schematic of the areas of highest e"ciency of the longitudinal and radial magnetization distribu-tions. The space has a cylindrical symmetry around Oz and only the plane xOz is shown.

2.4 Generation of a transverse field with Halbach rings of finite length

In the design of NMR magnets, another interesting magnetization distribution for a ring is the Halbach’sdipole scheme. This structure is not truly axisymmetric as the magnetization distribution does not o!ersuch a symmetry. However, the geometry itself remains axisymmetric and that is why we are treatingit here. We consider a Halbach cylinder of finite height with a continuously varying magnetizationfollowing equation 1.92 with k = 2 (it is hence a dipole). A view of such a magnetization distributionhas already been shown in figure 1.8. Such a structure has no practical application but it is a verygood starting point for a segmented structure as we will see later on. The geometrical parameters arethe same as for the cylindrical ring of the preceding section (inner radius a1, outer radius a2, altitudeb1 and b2 with b2 > b1). The derivation of analytical expressions for the field components and for theSHE of the transverse field (the main component) is not straightforward. The most e"cient way is toderive the analytical expressions for the scalar potential, from which derive easily the ones of the fieldcomponents. The full derivation can be found in (31). We will state here the result of this derivation.The scalar potential can be written in cylindrical coordinates (), &, z) as

"! =M

2$) cos &

.[F1]a2

a1

/b2b1

!M

2$

cos &

)

%(b! z)

*a2 ! )2

r1K(k)! r1E(k) +

(a! ))(b! z)2

r1(a + ))#(., k)

+a2

a1

&b2

b1

+M

4cos &

)[sgn(a2 ! ))! sgn(a1 ! ))][(b2 ! z)|b2 ! z|! (b1 ! z)|b1 ! z|], (2.44)

where [F1]a2a1

can be found in (31), K(k), E(k) and #(., k) are respectively the complete elliptic integralsof first, second and third kind, with

k =2*a)

r1

. =4a)

(a + ))2

r1 =1

(a + ))2 + (b! z)2.

2.5. FIELD MODULUS PROFILE CONTROL 61

The strong outcome of this is that the variable & enters the expression of the potential only ascos &, so that the SHE of the potential adopts the following form (spherical coordinates (r, ',&))

"!(r, ',&) ="$

n=1

rnX"1n cos &P 1

n(cos '). (2.45)

Using the preceding equations for the potential, it is possible to find simple analytical expressions forthese terms (31)

We have for the x component of the field, using equation 2.16,

Zxn = !(n + 1)(n + 2)2

µ0X1n+1 (2.46)

Xx2n =

12µ0X

1n+1 (2.47)

Y y2n =

12µ0X

1n+1 (2.48)

Xz1n = !(n + 2)µ0X

1n+1 (2.49)

Thus, given the parity of P 1n(cos '), a planar symmetry cancels all even terms in the potential and

leaves only even terms in the field expansion. We need then p + 1 such Halbach cylinders with properdimensions to cancel p terms in the field expansion.

We have presented the spherical harmonic analysis theory and described a systematic methodologybased on that theory. The design of a homogeneous magnets is described as the choice of the properaxisymmetric structure with the appropriate number of independent elements. The dimensions ofthese elements can be computed through non-linear optimization based on the variety of analyticalexpressions for the scalar potential and relevant field components SHE we have given. The upcomingchapter presents some examples of structures useful to NMR and based on these considerations.

2.5 Field modulus profile control

We have until now considered only the problem of field homogeneity. It can be interesting in somecases to achieve di!erent field profiles, such as a very well-controlled and strong field gradient in agiven direction. We have seen that in the case of a homogeneous field, the modulus of the field canbe assimilated to the amplitude of the component parallel to the field at the center of the region ofhomogeneity (which we call here main component). This however does not hold in the case of fieldswith arbitrary variations and it becomes not su"cient anymore to only consider one component of thefield. This is due to the Gauss’s law for magnetism,

!") ·!"B = 0, (2.50)

which can be written for example in cartesian coordinates as

+Bx

+x+

+By

+y+

+Bz

+z= 0. (2.51)

As a result, a variation of a component of the field automatically induces proportional variations inthe other components. Hence, if the variations of the main component are large, variations of othercomponents might not be negligible even if they enter in second order. These additional gradientsare known as a problem in the context of NMR in very low field and gradient design, and have been


dubbed concomitant gradients (118; 119), also known as Maxwell terms. Despite the fact that wecannot consider only one component at a time, the SHE of the main component still proves useful.This is because all components derive from the same potential in free space. Thus, once one componentis known, all components are known.

If we turn again towards axially symmetric structures, only two components can exist in cylindricalcoordinates (), &, z), B& and Bz. We already know the form of the SHE for the potential and Bz (seeequations 2.27 and 2.28), using spherical coordinates (r, &, '). We can in addition derive the one ofB&. We have indeed

B& = !+"!

+), (2.52)

and+

+)= sin '

+

+r+

cos '

r

+

+'. (2.53)

If we take now only one term of degree n in the potential, we have

A =+

+)

(rnZ!!

n Pn(cos '))

= sin 'nrn#1Z!!n Pn(cos ') +

cos '

rrnZ!!

ndPn(cos ')

d'. (2.54)

In the meanwhile we have in appendix B

P 1n(x) = sin '

dPn(x)dx

. (2.55)

Consequently,A = rn#1Z!!

n

4n sin 'Pn(cos ')! cos 'P 1

n(cos ')5. (2.56)

We can now use the following relations (117)

(2n + 1) sin 'Pmn (cos ') = Pm+1

n+1 (cos ')! Pm+1n#1 (cos '), (2.57)

(2n + 1) cos 'Pmn (cos ') = (n!m + 1)Pm

n+1(cos ') + (n + m)Pmn#1(cos '), (2.58)

so that we can write+

+)

(rnZ!!

n Pn(cos '))

= !Z!!n rn#1P 1

n#1(cos '). (2.59)

Thus, as Zn = !(n + 1)Z!!n+1, we have

B& ="$

n=1

! 1n + 1

ZnrnP 1n(cos '), (2.60)

Bz = Z0 +"$

n=1

ZnrnPn(cos '). (2.61)

Looking now at the square of the modulus of the field, we have

222!"B

2222

= B2& + B2

z = R0 +"$

n=1

Rn(')rn, (2.62)

where Rn(') can be obtained based on the expansion of B& and Bz. As a result, the square of themodulus of the field can be expressed only based on the terms of the SHE of Bz. We can give the first

2.5. FIELD MODULUS PROFILE CONTROL 63

terms of this expansion,

R0 = Z20 (2.63)

R1 = 2Z0Z1 cos ' (2.64)

R2 =12

!Z0Z2 +

54Z2

1

"+

32

!14Z2

1 + Z0Z2

"cos 2' (2.65)

R3 =32

!Z1Z2 +

12Z0Z3

"cos ' +

12

!Z1Z2 +

52Z0Z3

"cos 3' (2.66)

The following terms become rapidly di"cult to handle. This expansion can be converted in thefollowing, more useful, one

222!"B

2222

="$

n=0

"$

m=0

Smn )mzn, (2.67)

where Smn can be obtained by transforming the Rn terms. We can give a few Sm

n as

Smn =

9::::;

::::<

n|m 0 2 4

0 Z20

14Z2

1#Z0Z214Z2

2#38Z1Z3+ 3

4Z0Z4

1 2Z0Z1 #3Z0Z334 (Z2Z3#Z1Z4+5Z0Z5)

2 Z21+2Z0Z2 # 3

2Z1Z3#6Z0Z434 (Z2Z4+15Z0Z6+ 3

2Z23 )

3 2(Z1Z2+Z0Z3) #(Z2Z3+4Z1Z4+10Z0Z554 (Z2Z5+3Z3Z4+3Z1Z6+21Z0Z7)

4 Z22+2(Z1Z3+Z0Z4) #( 3

4Z23+3Z2Z4+ 15

2 Z1Z5+15Z0Z6) 154 (Z2

4+Z2Z6+ 32Z3Z5+ 7

2Z1Z7+14Z0Z8)

=::::>

::::?

.

(2.68)All odd terms in ) are null.

In addition to the highly homogeneous field, another field profile useful to NMR is a uniformlinear variation of the field in one direction. Such a profile finds a particular use in the context ofSTRAFI (89; 97) and di!usion measurements (120; 121). As the resolution of the obtained 1D imageis inversely proportional to the gradient strength (122), it is desirable to achieve the strongest gradientpossible to obtain a better resolution. As stated in (123), this gradient should be preferably constant inthe volume of analysis. The generation of such a gradient with high uniformity is not a trivial process,as we are now confronted to a highly inhomogeneous field.

It appears clearly from equation 2.68 that, if a gradient is maintained (Z1 .= 0), it is not possibleto obtain uniformity of this gradient along z and ) at the same time for

222!"B

222. Indeed, canceling S21

requires that Z2 = 14

Z21

Z0, which forbids that S0

2 = 0 if the field is not zero. However, it is possible toobtain a desired homogeneity along one dimension. For example, homogeneity along ) up to degreefour (S2

0 = S40 = 0, the first non-zero term is at degree six) and cancelation of a few cross terms can

be achieved by setting

Z2 =14

Z21

Z0,

Z3 = 0, (2.69)

Z4 = ! 148

Z41

Z30

Z5 = 0.


2.6 Conclusions on the theory for magnet design

To conclude this discussion, the description in terms of SHE of the main component of the fieldcreated by an axisymmetric magnet is su"cient to describe the variation of the modulus in any case.While it is possible to obtain a homogeneous field in an arbitrary volume to an arbitrarytolerance, it is not possible to achieve a uniform strong gradient in an arbitrary volume.It is however possible to obtain its uniformity along one direction in the said volume. Wediscussed in details a method to obtain a desired homogeneity, based on the SHE of the potential andthe field components and derived analytical formulas for the SHE of building blocks. Symmetriesare powerful tools to simplify the task of the designer. We thus favor axisymmetric structures,which can only generate a longitudinal field. However, another interesting structure is the Halbachring, generating a strong transverse field, as it also naturally cancels a large number of terms in theSHE. We can also envision the combination of both types of structures to control the orientationof the field with respect to the axis of the structure. These developments have lead to four patentapplications (124–127).

2.7 Magnetic field measurements

Field measurements are a very critical step in the fabrication of a magnet, as they are the basis forthe field profile assessment, for the field stability assessment, and for the shimming of the magnet.The field profile is obtained through field mapping : it is necessary to obtain the field value in severallocations of the RoI. The field stability is a wider topic as, depending on the magnet configuration andnature, the field profile can evolve with time. It is hence necessary to assess the possibility of temporalvariations of the spatial variability. In contexts where only the field strength is expected to vary, onlyone measurement point taken at di!erent times is necessary while it is necessary to take a whole fieldmap at several times in other contexts (e.g. deuterium lock in high field superconducting systems).We shall here discuss the two measurement devices which are appropriate for three dimensional fieldmapping.

2.7.1 Magnetic field measurement devices

2.7.1.1 Hall probes

Theory Hall probes make use of the well-known Hall e!ect which is well-described in many textbooks,such as (128). We will give a short overview of the theory underlying the measurement of magneticfields with Hall devices. Discovered by E. Hall in 1879 (129), the Hall e!ect arises when a magneticfield

!"B is applied perpendicularly to a flow of electric charge carriers. In such conditions, the flowing

carriers undergo the Lorentz force, which can be written for a given carrier of charge q as

!"Fq = q(

!"E +!"vq (

!"B ). (2.70)

If the applied magnetic field is perpendicular to the flow (i.e. to !"vq ) and we forget for now about!"E ,

the resulting force on the carrier is orthogonal to its flow and proportional to the magnetic field.Let us now consider a long conducting plate of width w in which we feed a current. As the plate,

or strip, is long, we can neglect the extremities and concentrate on what happens in the middle ofthe plate. The carriers flow is induced by an electric field

!"E 0 collinear with the plate. If we apply

a magnetic field!"B perpendicular to that plate, the trajectory of each carrier is deflected due to Fq.

However, the carriers cannot escape the conductor and a charge build-up takes place on the sides of

2.7. MAGNETIC FIELD MEASUREMENTS 65

the plate (one side is overpopulated while the other side is depleted). Hence, a transverse electric fieldarises, the Hall electric field

!"E H , which compensates the deflection induced by the magnetic field. We

hence have for the total force applied on the carriers, assuming small magnetic fields and small Hallelectric field,

!"Fq = q(

!"E 0 +

!"E H + µq

!"E 0 (

!"B ) (2.71)

where µq is the mobility of the carriers and !"v q = µq!"E0. When the system reaches steady-state, the

Hall electric field compensates the e!ect of the magnetic field so that!"Fq = q

!"E0 and the carriers move

parallel to the plate again. The Hall electric field is in that case

!"E H = !µq

!"E 0 (

!"B. (2.72)

The presence of this electric field induces a Hall voltage VH across the plate such that, when themagnetic field is orthogonal to

!"E 0,

VH = µqE0Bw. (2.73)

Hence, for a long plate of given width of known material, with a known applied electric field, the voltageacross the strip is proportional to the perpendicular field and the strip can be used as a magnetic fieldmeasurement device.

These equations may be refined further to pull out additional characteristic quantities (128). Thecurrent density associated to a carrier can be written

!"Jq = qµq)q

!"E0 where )q is the carrier density.

We can define the Hall coe"cient RH asRH =

1q)q

, (2.74)

so that, in steady-state,!"E H = !RH

!"J (!"B (2.75)

If we introduce t the thickness of the strip, we have the current Iq = Jqwt and the Hall voltage is

VH =RH

tIB, (2.76)

when the magnetic field is perpendicular to the plate.In the case of a metal, there is only one type of carrier, the electron (charge !e) and the Hall

coe"cient can easily be calculated (e.g. in copper, RH / !5.12 10#11 m3 C#1).A particular type of materials is the semiconductors. In such materials, the carriers can be electrons

(charge !e) or/and holes (charge e). These materials are of great interest, for the mobility of thecarriers is usually much higher. As a result, the Hall coe"cient is also much higher. If we considern-type doped silicon with )n = ND = 4.5 1015 cm#3, we have RH / !1.39 10#3 m3 C#1, eight ordersof magnitude higher than copper ! That is why all modern Hall devices are made out of semiconductormaterials.

Much more details can be given on the behavior of Hall devices, taking into account semiconductorsproperties, considering di!erent geometries, etc. It is however not the purpose of this work, which onlyintend to use Hall devices as tools for magnetic measurement. The interested reader should refer toreference (128) for a detailed description of Hall devices.

Nonetheless, the description given above is a crude way of analyzing Hall plates. When precisionfield measurements are to be performed, it is necessary to consider some of the terms we neglectedpreviously. Equation 2.71 was indeed written assuming weak field so that the Hall electric field contri-bution to !"v q could be neglected in the force calculation. In addition this description does not take into


account the possible variations of the transport properties with the magnetic field. A more thoroughapproach based on kinetic equations yields the following result for the electric field (128)

!"E = rB

!"J !RH

!"J (!"B + PH(

!"J ·!"B )

!"B, (2.77)

where rB is the resistivity, RH is the Hall coe"cient, and PH is the planar Hall coe"cient. All thesecoe"cients feature non-linear variations with the magnetic field and the electric field. Equation 2.77shows that the resulting electric field can be separated into three vector terms,

!"E 1 = /1

!"J ,

!"E 2 =

/2!"J ( !"

B , and!"E 3 = /3(

!"J · !"B )

!"B .

!"E 1 is colinear with the current flow and corresponds to the

voltage drop across the plate.!"E 2 corresponds to the usual Hall e!ect while

!"E 3 is an additional

electric field collinear with!"B . This latter term is called the planar Hall e!ect and is an important

source of imprecision when measuring 3D fields. The measured transverse voltage arising is indeed theresult of both the Hall and the planar Hall electric fields. The planar Hall field is proportional to themagnetic field component collinear with the current and thus exists only when the magnetic field isnot orthogonal to the plate.

In practice, most magnetic field measurement probes are based on Hall plates which give a voltagereadout assumed proportional to the field component orthogonal to the plate. A calibration in tem-perature and field is necessary to provide accurate measurements of a field completely orthogonal tothe plate and relative precision of 10#3 to 10#4 can be achieved. Hall probes can measure very smallfields of the order of the earth field up to strong field of several teslas.

As the magnetic field is a vector quantity and a Hall plate measures only a quantity orthogonal toits surface, the full characterization of the magnetic field requires to develop a more involved systemcalled a three-axis Hall device. A very common way of obtaining such a device is to simply use threeindependent Hall plates set orthogonal to each other. Each plate gives a measurement of a component ofthe field. Such a system features however several flaws. The biggest one is that each plate is measuringa field component in a di!erent location, depending on the size of the device. The distance betweentwo plates can be as large as a few millimeters, which is unacceptable in precision field mapping. Theseposition di!erences could be corrected by a proper mapping scheme so that each plate would pass overthe same position. Proper data processing could in the end co-register the data sets from each plate.However, such a process is cumbersome and is subject to the uncertainties on the relative positionsof the plates and to the temporal variations of the field in between each measurement, as the fieldstrength may drift. In addition, the tolerance on the orthogonality of the sensors a!ects greatly theaccuracy of such measurements. Another way of measuring the three components of the field is touse integrated three-axes Hall devices (130; 131). Such devices allow the measurement of the threecomponents of the field in the exact same place, which can be highly localized thanks to the small sizeof the crystal used (few hundred micrometers). The three components are obtained through a propercombination of voltage readouts across the di!erent sides and corners of the device.

However, as soon as the field features a component collinear with the fed current, the planar Hallvoltage arises and perturbs the readout. As a result, measurements of magnetic fields of unknowndirection using three-axis probes are impaired by the planar field e!ect. While the noise performanceof most devices allow for 10#4 precision for a given location, the 3D mapping of a field varying inthree dimensions will see its precision lowered by the variations of the planar field e!ect combined withthe variation of the actual Hall e!ect. It is hence necessary to ensure either that the field orientationis constant in the region of measurement or that the device is insensitive to the planar field e!ect.The latter can actually be achieved by proper operation of the Hall device, using the spinning currentmethod (132; 133). In this case, the planar field e!ect can be reduced down to 10#4 relatively to theHall voltage and the precision reestablished for 3D mapping of a field of varying orientation so that an


absolute accuracy better than 10#3 can be achieved. Commercial devices based on these concepts areproposed by Senis with rated accuracy of 0.1%.

In addition to the planar field e!ect, Hall sensors are subjected to several non-linearities, includingcross-talk between the three axes. An involved calibration method as been proposed by Bergsma etal (134) to assess the whole 3D response of the device and attain precision and accuracy of the order of10#4 on all axes (135). This however requires great care during measurement and periodic calibration(every month).

It is interesting to note that in specific conditions, axial probes (only one field component is mea-sured) can achieve absolute accuracy of the order of ±100 ppm in very high field gradients (30 T m#1)after thorough calibration (136).

2.7.1.2 NMR

Equation 1.2 gives the relation between the frequency of the NMR signal and the field stength. It isstraightforward to conclude that, if we know with high accuracy the magnetogyric ratio of a nucleus in avery pure sample (and we also know its chemical shift), we can use NMR as a very accurate and precisetool to measure magnetic field strengths as long as we have an accurate and precise spectrometer. It ispossible to achieve ppb (part per billion) precision on the field strength value if it is homogeneous overthe volume of the NMR probe. NMR is hence the tool of choice for high precision field measurements.It is important to bear in mind that this measurement is a scalar measurement : NMR yields themodulus of the field vector and hence does not give any information on the orientation of the field.Actually, it is necessary to know the direction of the field in order to perform NMR as the RF field tobe applied must be orthogonal to the static field (in fact, the NMR experiment selects the orthogonalcomponent of the RF field, so that we really need to have B1 0 B0 to obtain no signal). As a result,NMR methods can only be used for fixed position measurements or for mapping of constant orientationfields.

The quality of the magnetic field NMR measurement depends on many parameters. Assuming thespectrometer is not limiting accuracy and precision, the measurement is limited by the linewidth ofthe NMR signal. The precision on the field measurement is indeed directly linked to the resolutionwe have on the position of the NMR line and that resolution is usually given by the Full Widthat Half Maximum (FWHM) of the line. The linewidth can be impaired in first place by the fieldinhomogeneity, which thus a!ects tremendously the precision of the measurements. In addition, for agiven volume of sample (and hence a given number of spins), an increased field inhomogeneity (andhence increased frequency spread) results in a decreased signal strength, and hence decreases the SNR.As a result NMR field measurements can take place only in relatively homogeneous fields. The firstNMR magnetometer systems were based on the continuous wave method (137; 138) and some havealso been proposed based on the pulsed method (139). However, solutions exist when the field tobe measured is not so homogeneous. In first place, it is possible to greatly reduce the volume ofmeasurement by using microcoils. These devices have been introduced in 1994 by Wu et al (140) andcan feature dimensions as small as 100 µm. They are mostly used in the context of high resolutionstudies of limited-size samples (141) in liquids and also for solid-state NMR in the context of MagicAngle Coil Spinning (MACS) (142). Such coils allow to decrease the sample size while increasingthe sensitivity, hence maintaining a good SNR though decreasing tremendously the number of spinsinvolved in the experiment. These remarkable performances can thus be used for field measurementsin inhomogeneous fields: we get a weaker signal with narrower linewidth that we detect with a highersensitivity device, resulting in a much better SNR and linewidth. Such coils also o!er the abilityto miniaturize the field measurement device, which can be built into a probe that can be moved indi!erent locations to map field variations, even if the volume of interest is small.


In some cases, as we will see, even small samples of the order of 100 µm are still too large forhigh resolution measurements, due to the extreme field gradients. It is possible to use additionalproperly designed coils to compensate locally the inhomogeneities of the field over the sample in orderto reduce the linewidth, without modifying the field variations to be mapped. We will discuss such acompensation system in an upcoming section.

In the context of field stability measurements where the field profile is not expected to vary (forexample, in the case of electromagnets or superconductive magnets), the field measurement probe isnot to be moved and should only monitor subtle variations of the field strength at one location. Suchvariations are seen in the NMR spectrum as a shift of the signal frequency. We are here plainly limitedby the linewidth of the signal. However, as the field inhomogeneities are not supposed to change withtime, we can expect the linewidth of the signal to remain the same during the whole experiment. Hence,deconvolution methods can be used to assess frequency shifts smaller than the FWHM. The precisionthat can be obtained only depends on the SNR of the signal collected. We described an original methodof field stability measurement under low temperature conditions and very large inhomogeneities basedon microcoils and similar assumptions in (143).

We have considered until now the e!ect of field inhomogeneities on the spectral linewidth. Thiswill be the dominating issue for measurement precision in the rest of this work, but it is worthwhileto consider the case where the field inhomogeneities are not the limiting factor. In that case, thelimitation is the duration of the NMR signal, linked to the T2 and T1 relaxation times. It is hencenecessary to select an appropriate sample for which these relaxation times are long enough that we geta narrow enough linewidth. In addition, the T1 should not be too long, so that the NMR experimentcan be repeated often in order to gain time resolution in the context of field stability monitoring, andmapping speed in the context of field mapping. The latter is of great importance during the shimmingof the magnet as this procedure requires numerous field mappings and hence becomes quickly timeconsuming.

We have discussed two types of magnetic field measurement devices. It appears that, unless greatcare is taken, 3D Hall probes have a limited precision of the order of 0.1%. They are hence limited tocoarse measurements in the context of this work. It is necessary to use NMR in order to achieve thefine accuracy required to verify the expected homogeneity.

2.7.2 Retrieving SHE terms from field measurements or computations

2.7.2.1 Theory

The previous sections have made clear that the SHE of the field has become our reference frame for theevaluation of the field variations generated by a magnet in a RoI. We have given analytical formulasand design guidelines in order to build a magnet based on the expansion that we desire. We still have todiscuss the inverse process of determining the SHE of the field generated by a given magnet structure.We will rely for this operation on the following expression of the SHE of the field component (takenhere as Bz)

Bz(r, ',&) = Z0 +"$

n=1

!r

r0

"n%ZnPn(cos ') +

n$

m=1

(Xmn cos m& + Y m

n sinm&) Wmn Pm

n (cos ')

&, (2.78)

with r0 a reference radius and Wmn = (n#m#1)!!

(n+m#1)!! , a factor so that Wmn Pm

n (cos ') remains within [!1, 1],giving a more physical meaning to the values taken by the Xm

n and Y mn terms.

This section intends to cover the critical theoretical aspects of the extraction of the SHE termsfrom a set of measurements. A short description of a way to do it can be found in (117) and we willexpose here a more thorough study.


Field measurements can only be done in a finite number of discrete locations and the operationof retrieving the expansion terms from such measurements corresponds to a polynomial interpolation.This raises the issues of aliasing and appropriate measurement points choice. Various literature coveringthe topic of sampling and interpolation are available (144; 145). We will base our description of issuesconcerning the measurement of the expansion terms on the mathematics provided by this literature.

Aliasing In order to perform the right measurements and carry the right interpolation, it is importantto define the number of terms to retrieve and this cannot be dealt with without having aliasing in mind.If we sample a polynomial P in n points xi, we can define the polynomial # such that

#(x) = (x! x1)(x! x2)...(x! xi)...(x! xn) (2.79)

As a result, if we divide P by #, we obtain a quotient Q and a rest R :

P (x) = Q(x)#(x) + R(x) (2.80)

For the sample point xi, we obtainP (xi) = R(xi) (2.81)

and it is not possible to make the di!erence between P and R. We shall call # the sampling polynomial.For example, let the field variation V to be measured of the form

V (r, x) = Z0P0(x)+!

r

r0

"Z1P1(x)+

!r

r0

"2

Z2P2(x)+!

r

r0

"3

Z3P3(x)+!

r

r0

"4

Z4P4(x)+!

r

r0

"5

Z5P5(x)

with x = cos '. Let us sample that field variation in 3 points lying on the sphere of radius r < r0 andcorresponding to cos ' = 1

2 , 0,!12 . The sampling polynomial is hence defined as

#(x) = x

!x! 1

2

" !x +

12

".

One then finds out that R, as defined above, can be written as

R(r, x) =

%Z0 !

4996

!r

r0

"4

Z4

&P0(x) +

r

r0

%Z1 !

78

!r

r0

"2

Z3 +23128

!r

r0

"4

Z5

&P1(x) +

!r

r0

"2%Z2 !

8548

!r

r0

"2

Z4

&P2(x).

If we write the terms calculated from sampling (primed terms) in function of the real terms (unprimedterms), we obtain

Z %0 = Z0 !

4996

!r

r0

"4

Z4

Z %1 = Z1 !

78

!r

r0

"2

Z3 +23128

!r

r0

"4

Z5

Z %2 = Z2 !

8548

!r

r0

"2

Z4.

This demonstrates that for insu"cient, non-optimized sampling points, the higher degree termsof the expansion fold into the evaluated terms. This happens because we limit the interpolation to


three terms while the actual field variation features six of them. This is a situation we always facein magnetic field mapping, as we always have to set a finite number of measurement points while theexpansion potentially features an infinite number of terms. However, this expansion usually convergesand there exists an degree above which terms can be neglected and will not a!ect the extraction ofthe coe"cients. The number of necessary terms depends on the system producing the field. Onehence should assess first what is the spectral content likely to be found, based on the geometry andimperfection sources. The validity of the choice of the necessary number of terms to be retrieved shouldlater be checked by verifying the stability of the interpolation while varying the number of measuredterms.

Measurement points choice The first thing to do is to determine the number of points to bemeasured. This is a very well known problem : one needs n + 1 points to interpolate a polynomial ofdegree n. As the Legendre polynomials are a basis of the polynomial space, all theorem applying topolynomial interpolation can apply to the measurement of the SHE. Hence, once the number of termsto be retrieved has been decided, we know how many points should be measured.

The number of measurement points being chosen, it is necessary to decide where these pointsshould be. A property of solutions to Laplace equations is that if one knows the field on a closedsurface, it is possible to compute the field anywhere inside this surface. This is valid ifthis surface does not contain any sources, in order the Laplace equation for the potential to bevalid. Hence, it is possible to retrieve the inner spherical harmonic expansion from measurements on aclosed surface that does not contain any source. The sphere is obviously the most adapted geometry.It is also helpful to notice that the e!ect of aliasing depends in addition on the radius of measurement.The radius of measurement should contain the entire volume of interest (inaccuracies resulting fromthe measurement are amplified by extrapolation outside the measurement sphere). However, one mayneed to compute more terms (and use more points) to avoid the increase of the folding.

We still need to decide how to sample the surface of the chosen sphere. The problem of retrievingthe SHE terms boils down to solving a system of linear equations. We are bound to have errors inthe measurements and it is a worry to know how these errors will impact the result. This is estimatedby the condition number of the matrix linking the measurement points to the polynomials. For anequation of the type [A]X = Y , with appropriate norms defined on the set of X and the set of Y ,there exists a condition number / % 1 such that

0!X00X0 - /

0!Y 00Y 0 , (2.82)

where !X and !Y are respectively the error on X and Y . Hence, the errors on Y can be greatlyamplified if / is large and the problem is called ill-conditioned. For a square Hermitian matrix, wehave / = 'max

'minwhere the 0i are eigenvalues of the matrix A. In the case of a rectangular matrix, one

can write / = smaxsmin

where the si are singular values of the matrix A. Hence, / is large when someeigen- or singuar values are small compared to the others. For more details on condition numbers,the reader is invited to consult (146). The conclusion of this is that the construction of the matrixA defines the sensitivity -or immunity- to measurement errors of the process. We must thus choosecarefully the location of the measurement points.

A possible sampling scheme, which we here will refer to as the "Fourier scheme" is a combinationof an even distribution in & so that the m orders can be calculated through a simple Fourier transformand an even distribution in ' in order to get the n degrees through a simple fit. One thus needs atleast 2M + 1 points to retrieve all orders m up to M and at least N + 1 di!erent ' in order to retrievethe n degrees up to N . The even sampling in ' insures the lower degrees are the least a!ected by


aliasing, making it easier to obtain a reliable measurement (through increase of measurement points).Such a sampling scheme can be seen in figure 2.6. However, this scheme is not the most e"cient as itlargely overdetermines the problem. In the case of practical measurements, it is important to reducethe number of points to the minimum necessary in order to reduce the quantity of data to store andthe time necessary to perform the measurements. As we will see later, it is also important to take intoaccount measurement errors, whose e!ect can be reduced by over-determining the problem. There isthus a compromise to find, within which the Fourier scheme lies.

The definition of an optimum measurement grid (i.e. minimum number of measurement points andbest conditioning) is not a trivial problem and many methods can be used. Following the previousconsiderations, an intuitive way is to distribute the points evenly on the sphere (minimal maximumdistance between two points) and to perturb this distribution in order to minimize the condition numberof the resulting matrix. Another more systematic method follows from sampling theory. Elements tocalculate an appropriate sampling grid can be found in (145). However, reducing the number ofmeasurement points to a minimum makes the procedure weaker against noise. A trade-o! must thusbe made.

The aliasing problem must be treated on an individual case basis. The number of terms to beretrieved in order to obtain accuracy on the relevant terms depends directly on the harmonic contentof the field variations. Hence, it is necessary to make several maps of the magnet with an increasingnumber of terms retrieved until the relevant terms are stable.

!10!5

05

10x

!10!5

05

10y

!10

!5

0

5

10

z

Figure 2.6: 3D view of the measurement scheme for a SHE extraction. This measurement pattern is constitutedof 16 points in ' and 32 points in &.


2.7.2.2 Experimental errors in the extraction of the SHE terms

We have just covered the theoretical aspects of field SHE terms measurements. While we can insurethat aliasing does not impair the accuracy of the SHE terms retrieval, we cannot completely cancelout experimental imperfections during the measurements.

The measurements can be done in practice with any magnetic field sensor mounted on a positioningsystem, or with any magnetic field sensor array providing the appropriate measurement locations, asproposed by MetroLabs (147). We used in this work both Hall probes and home-made microcoil-basedNMR probes. The measurements were made with a single detector which is moved to the di!erent mappoints. The positioning system was made of three motorized translational stages (Newport LTA-HSmotors and M-443 stages). A LabView software was developed to synchronize motion and measure-ments both with the Gaussmeter (Lakeshore Model 360 with Lakeshore 3-axis probe) and the NMRspectrometer (Tecmag Apollo) in order to automate the measurement procedure.

We are hence confronted to the uncertainty of the field measurement in one location, !Bz, orfield noise, which is due to the detector itself, and to the positioning uncertainty, which is due tothe mechanical system providing the motion of the detector. That position uncertainty features arandom component and a systematic one. In the error assessments that follow, the unit "ppm" alwaysrepresents parts per million of the main field.

Field noise We call here "field noise", or field uncertainty, the precision associated to the measure-ment instrument (e.g. noise in the Hall probe signal, or linewidth in the NMR probe). The field noiseyields a measurement error essentially independent of the field profile measured (besides its influenceon the linewidth for the NMR probe). It can be expected to be a random, zero-mean, gaussian-like dis-tribution and a!ects thus mostly the precision of the measurement while the accuracy is left untouched(the average measurement is the correct one).

Such errors can be easily simulated by adding to each field measurement a pseudo-random numberbounded by the expected measurement error. These boundaries are given by the precision of theHall probe system or by the linewidth (most of the time FWHM) of the spectral line. We give intable 2.1 the average standard deviation on all terms for di!erent field measurement precision. Thecomputation retrieves all terms up to degree five and is performed using a measurement grid based oneight points in ' and fourteen points in &, so that aliasing does not a!ect significantly the measurementon the considered radii (we consider here a field profile created by a magnet similar to the single-sidedmagnets described in chapter 4). The deviations are expressed on the same radius (5 mm) but withdi!erent radii of measurement, showing the advantage of using a larger radius of measurement (as longas aliasing is not significant). These numbers show only the average absolute error for most terms.Some isolated terms may display a much larger spread (more than ten times larger).

!Bz (ppm) ±1 ±10 ±50 ±100 ±500Average error (ppm),r=6 mm 0.1 1 5 10 50Average error (ppm),r=10 mm 0.06 0.5 2 5 25

Table 2.1: Average absolute error for most SHE terms for di!erent radii of measurements and di!erent fieldmeasurement error !Bz. The measurement scheme makes aliasing negligible. A larger measurement radius maysignificantly help damping measurement error as here going from 6 mm to 10 mm divides by two the resultingerror on all terms retrieved. This average error is valid for most terms except for higher degree and order skewedterms (in this case, particularly X4

4 , Y 44 , X4

5 , Y 45 , X5

5 , and Y 55 are much more far o!). The detailed distribution

of errors is shown in table 2.2 for an uncertainty of ±30 ppm as an example.


n / m 0 1 2 3 4 50 2 - - - - -1 2 2 - - - -2 2 2 2 - - -3 2 2 2 5 - -4 2 2 5 35 155 -5 2 2 5 35 155 770

Table 2.2: Example of distribution of errors for !Bz = 30 ppm with a measurement radius of 6 mm. Valuesare expressed in ppm for a reference radius of 5 mm. Axial and skewed terms are treated together: m=0corresponds to Zn terms

Random position errors In the case of a single probe moved to the di!erent sampling points,such errors can be due to the random angular change of a translation stage, or the positioning errorof an actuator. The deviation from the correct position is di!erent each time the probe is positionedand the absolute deviation is bounded by a tolerance. That tolerance can be obtained based on thespecifications of the actuator, the translation stage, and the length of the arm holding the probe(translational stages induce small random angular errors, which are amplified by the length of thearm). In the same way as field-noise, random position errors have zero-mean. Thus, the average resultis the correct measurement (accuracy is not a!ected but precision is). The consequence on the precisionof the SHE terms extraction depends on the measured field profile (the measurement of a field profilewith small variations will be less sensitive to such errors than the one with large variations). Thus wecan expect that, for example, a high precision measurement of a magnet generating a field gradientwill be more challenging than for a magnet generating a homogeneous field. We can estimate thecontribution of such errors on a case-to-case basis, through simulations.

Axes orientation and centering errors When the probe is positioned successively in the di!erentsampling points, a positioning system has to be used, which we assume to be a 3-axes translationalsystem. In addition to the previously discussed errors, the accuracy of positioning is thus conditionedby the orientation and center location of the positioning system frame in regard of the magnet frame,and the orthogonality of the three axes. Such errors are usually made once and for all when settingup the measurement system, and are thus systematic. The consequences of such systematic errors arecritical for the measurement as they a!ect accuracy and are di"cult to compensate for afterwards. Wecan explicit further these errors.

The error on center location is simply introducing an o!set in the measurement coordinates rela-tively to the magnet position. In the context of SHE measurements, it simply corresponds to the shiftof the center of the measurement pattern (all measurement points following it). Were this error known,it could be corrected through standard translation of spherical harmonics techniques (148). We canwrite the measurement points coordinates in the magnet frame as a function of their coordinates inthe positioning system frame as

B

Cxyz

D

E =

B

Cx%

y%

z%

D

E +

B

Cx0

y0

z0

D

E , (2.83)

where (x0, y0, z0) are the coordinates of the positioning frame origin in the magnet frame.Following the notation of figure 2.7, we can define a "straight" tilted frame where all axes are still

orthogonal but the attitude is di!erent from the magnet frame. This attitude can be completely defined


by a rotation matrix with the Euler angles (with the usual convention of - a rotation around Oz, 1 arotation around the subsequently new Ox and " a rotation around the subsequently new Oz (149)),

B

Cx%

y%

z%

D

E = MEuler

B

Cxyz

D

E , (2.84)

with

MEuler =

B

Ccos " cos 1 ! cos - sin1 sin " cos " sin1 + cos - cos 1 sin " sin " sin-! sin " cos 1 ! cos - sin 1 cos " ! sin " sin1 + cos - cos 1 cos " cos " sin-

sin- sin1 ! sin- cos 1 cos -

D

E (2.85)

As MEuler is unitary, we can write B

Cxyz

D

E = MTEuler

B

Cx%

y%

z%

D

E . (2.86)

We can use one axis of this new frame as fixed reference (we choose Oz% in figure 2.7) and definethe orientation of the other axes relatively to the ideal orthogonal axes, as shown on figure 2.7. Itfollows the following relation between the ideal tilted frame and the actual skewed tilted frame,

B

Cx%

y%

z%

D

E = MSkew

B

Cx%%

y%%

z%%

D

E , (2.87)

where

MSkew =

B

Csin 'x cos &x ! sin 'y sin&y 0sin 'x sin&x sin 'y cos &y 0

cos 'x cos 'y 1

D

E . (2.88)

Thus, we have the following relation between the coordinates of a point in the magnet frame andthe skewed tilted frame, B

Cxyz

D

E = MTEulerMSkew

B

Cx%%

y%%

z%%

D

E +

B

Cx0

y0

z0

D

E , (2.89)

If the measurement frame is not skewed (MSkew = 1), the retrieved spherical harmonic terms arecorrect in the tilted frame and could be transformed to give the correct terms in the magnet framethrough well-known techniques of spherical harmonics rotation (148). However, the Euler angles arenot known most of the time (they result from imperfections of assembly) and it is hence di"cult todetermine the correct transform. Nevertheless, the retrieved terms are at least correct in a referenceframe. It is hence not a problem when characterizing a homogeneous field which features sphericalsymmetry. It might be an issue when characterizing other field profiles, such as a unidirectionalgradient as the tilt will obviously induce "leakage" of the expected gradient in the transverse axes. Insuch cases, if MEuler is not known, it is di"cult to devise how much of a contribution of a SHE termis due to the tilt of the measurement frame or to actual imperfections.

If the measurement frame is skewed, the measurement points are not taken in the correct positionsin any reference frame and the retrieved SHE terms are wrong in all reference frames. Hence, non-orthogonality of the measurement axes impairs the characterization of any field profile. Again, if thedi!erent skewing angles were known, it would be possible to correct the e!ect of MSkew but, most ofthe time, they are not.


x

y

z

x’

y’

z’

z’=z’’

x’’

y’’

x’

y’

!x

!y

"x

"y

Magnet reference frame Tilted frame Skewed tilted frame

MSkewMTEuler

Figure 2.7: Transformation from the magnet frame to the measurement frame.

We have seen the three main imperfections of the measurement system that introduce systematicerrors. We assume from now on that we cannot correct them as we have no knowledge of the imper-fections, save the fact they are there. The extent of the e!ect of these imperfections on the accuracyof the computation of the SHE terms depends obviously on the field profile. Profiles featuring stronggradients are obviously less tolerant to such errors than homogeneous fields.

While we do not have a precise knowledge of the errors introduced by the axis configuration, it isfeasible to estimate the tolerances on their orientation and thus assess the resulting error which givesus the accuracy of the measurement. This should be done on a case to case basis, through simulationsof the expected field profile.

Example: measuring a homogeneous field We seek to characterize the field generated by anaxisymmetric magnet designed to produce a region of homogeneous field (!Bz - 30 ppm in a sphereof 8 mm diameter). This magnet is discussed later on in section 4.1.2.1. We expect thus the field atthe center of this region to be along Oz and fairly homogeneous over the volume of interest (variationsof maximum 500 ppm due to imperfections). Due to the weak variations of Bz, the variations of Br

are small and we can write that B = Bz over the entire volume of interest. Thus, we can use a NMRmicro-coil to measure the field and apply the SHE extraction method described above to characterizethe field variations e"ciently (a Hall probe could be used but would require a larger measurementradius and more measurement points to retrieve the same number of terms with the same precision).In this context, based on the previous considerations and numerical simulations, we can assess the e!ectof geometrical errors on accuracy and precision of the SHE terms. Table 2.3 displays the standarddeviation, maximum error and mean error for the di!erent types of geometrical errors. These resultswere simulated using a scheme of 10 points in ' by 14 points in & to retrieve all terms up to degreefive with minimal aliasing from a field profile displaying no skewed terms and the axial terms shownin table 2.4. The radius of measurement is 4 mm. It appears that the measurement is most sensitiveto center position errors. We can remark however that the maximum absolute position error inducedby an axis tilted by 5º over 4 mm is only 0.34 mm while the o!-center error results in the translationof all measurement points by the same value (we consider here values larger than 0.34 mm).


Error ! on terms (ppm) Average !̄ Average #" Min !̄ Min #" Max !̄ Max #"

Tilt, 1º 10#4 - 10#6 - 0.01 -Tilt, 5º 0.05 - 10#4 - 1.3 -

Skewness, 1º 10#4 - 10#6 - 0.03 -Skewness, 5º 10#3 - 10#6 - 0.03 -

Center error, 0.5 mm 0.2 - 5 · 10#5 - 5 -Center error, 1 mm 0.5 - 5 · 10#4 - 10 -

Position noise, ±0.01 mm 0 0.01 0 0.005 0 0.02Position noise, ±0.05 mm 0 0.2 0 0.06 0 2

All, 1º, 1º, 1 mm, ±0.05 mm 0.5 0.05 10#3 5 · 10#2 10 0.5

Table 2.3: Deviation from actual SHE for di!erent error sources in the case of a homogeneous one-sidedmagnetic field. All numerical measurements are done with the same radius of measurement and same numberof points. "All" corresponds to all geometric errors together with tolerances values chosen to be realistic andgiven in this order, "Tilt", "Skewness", "Center error", "Position noise". It appears that the measurement ofsuch a homogeneous field is quite robust to these errors (considering that 1 mm o!-center is very large comparedto the radius of measurement).

Z0 (T) Z1 (T m#1)0.146 0

Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

Amplitude (ppm) 0 0 0 0 9.8 6.7 2.7 0.8

Table 2.4: Field, gradient values and contribution of the di!erent axial terms of the measured field profile, ina sphere of radius 4 mm.

2.7.3 Measurements in a strong field gradient

We desire here to characterize as precisely as possible a strong field gradient $Bz$z of about 3.3 T m#1

with a field at center along Oz and of value about 0.33 T generated by an imperfect rotationallysymmetric magnet (this magnet is discussed in section 4.1.2.2).

2.7.3.1 High precision measurements of B0 with NMR

Due to the need of spatial localization, it is necessary to use a small detector (sub-mm size). In addition,we seek here the highest accuracy and precision possible, thus making a NMR probe indicated. Wehence use micro-coils wound around capillaries of 700 µm outer diameter (500 µm inner diameter).However, even such a small coil yields a spectral width as large as 5000 ppm (the 3.3 T m#1 gradientinduces 10000 ppm mm#1). Hence, a bare micro-coil is not o!ering any more precision than a Hallprobe.

Nevertheless, it is possible to locally cancel the gradient with a small-sized compensation coilsystem centered on the NMR coil. Such a system should only induce a linear variation of the fieldwithout modifying the field at the center and without introducing additional inhomogeneities. Anindicated simple configuration is the Maxwell coil. This structure comprises two coaxial loops ofcurrent circulating in opposite direction. In terms of field SHE, the axial symmetry cancels all skewed-terms while the planar antisymmetry cancels all even axial terms (the field at the center is zero). Foridentical loops of radius a, a distance

*3a between them cancels Z3, so that we have Bz variations


almost only due to Z1 on a wide volume in the coil system (the first higher degree variation is at degreefive). The field on the axis can be written as (29)

Bz(0, z) =µ0Ia2

2

9:;

:<1

@a2 + (z !

&3a2 )2

A 32

! 1@a2 + (z +

&3a2 )2

A 32

=:>

:?, (2.90)

so that we can write Z1 as

Z1 = µ0I

F37

4849a2

(2.91)

while the first term introducing inhomogeneities is

Z5 = !µ0I

F37

101376016807a6

(2.92)

Thus, if we consider a system of reasonable size for the mapping of the single-sided magnet (a = 6mm),we would need a current of about 150 A to create a gradient of about 3.3 T m #1 balancing locally theone we want to measure. In order to use a more easily available power supply (less than 10 A), we needto build a coil featuring a large number of turns. Each loop will thus be replaced by a group of turns.The relatively small size of the coil and the preservation of the uniformity of the gradient at the centerrequires to take into account the geometrical extent of the ensembles of loops. The resulting stackingof loops is equivalent to a thick solenoid of finite length. The appropriate inner radius, outer radiusand length of the two solenoids, along with the distance separating them can be calculated analytically(based on the discussion on thick solenoids in (29), or as seen in (31)) or through numerical simulationsin order to cancel Z3 in the same fashion as the Maxwell coil. A micro-coil can be inserted betweenthe two coils, where it can detect the compensated field. As the field at the center of the Maxwell coilis zero, the field measured by the NMR probe is the one of the magnet.

We were able to reduce the linewidth of the signal from 6000 ppm to less than 70 ppm (accuracyof ±35 ppm on the field value at the center). We thus have made an e!ective tool to achieve highprecision measurements of the field in a strong gradient. A view of the coil model, along with theactual system is shown in figure 2.8. The compensation coil was wound with 0.7 mm diameter copperwire, with about 64 turns on each thick solenoid (eight turns per layer and eight layers). The desiredcompensation is achieved experimentally with a current of about 4 A fed by a stabilized power supply(TTi model EX354RT 300W). Spectra with and without compensation are shown on figure 2.9. NMRsignals are acquired with a Tecmag Apollo spectrometer using a CPMG sequence and adding 50 echoestogether in order to increase SNR. Exploitable signals can be obtained in two seconds (two scans) withwater samples doped with CuSO4.

The measurement precision of this system is however a!ected by the stability of the compensationsystem. It is hence necessary to assess the e!ect of its likely instability. We consider here that thecoils system is warmed up and that the loops are all glued together, so that no mechanical motion ispossible. We are left only with the power supply instabilities, which are rated to be lower than 1 mA.

Were the NMR coil perfectly positioned on the center of the compensation system (where the fieldinduced is zero), only the linewidth would be a!ected by variations of the compensation gradient. Ifthe compensation system is exactly adjusted, the variation of the linewidth is exactly the variation ofthe gradient. We have here about 0.825 T m#1 A#1, so that the maximum linewidth variation is about1.25 ppm over the micro-coil inner diameter (0.5 mm). We call this variation !width.

However, the fabrication tolerances lead us to assess that the micro-coil is bound to be located o!the center of the compensation system, with an uncertainty of 1 mm. In the worst case, the NMR


Figure 2.8: Views of the gradient-compensated field measurement probe. a) 3D Model of the coil. b) View ofthe coil assembly with a significant radiator for cooling, and holding system for the NMR micro-coil in betweenthe two coils. c) View of the micro-coil (circled in white). The outer diameter of the capillary is 0.7 mm, theinner diameter 0.5 mm. d) View of the whole system in place above the one-sided magnet.

!"####!$####!%#### !&#### # &#### %#### $#### "#####

&

%

$

"

!

"

#$%&'%()*+,-./0+12234

5326.0'7%+15

8984

!&'## !&### !'## # '## &### &'###

&

%

$

"

!

"

#$%&'%()*+,-./0+12234

5326.0'7%+15

8984

Figure 2.9: Typical spectra obtained at the center of the RoI of a 3.3 T m!1 gradient magnet. Left: compen-sation coil o!. Right: Compensation coil on. This shows clearly the improvement by more than one order ofmagnitude of the line width. The CPMG sequence features 50 echoes with 3.86 ms between each echo and a !

2pulse width of 2.5 µs. The spectra shown are obtained in 100 scans with a sample of water doped with CuSO4.This system provides an accuracy of ±35 ppm on the measurement of B0. The probe system is small enoughso that it can be moved around the center of the RoI in a radius of 1 cm.


coil is thus located 1 mm away from the center along Oz (where the gradient is strongest), so thatthe compensation coils o!set the field measured by the NMR coil. Given the nominal compensationgradient, we can expect an o!set of 3.3 mT (about 1 % of the field), or, per unit of current and interms of the magnet field, 2.5 ppm mA#1. Thus, we can expect the o!set variation to be no more than2.5 ppm of the magnet field. We call this variation !offset.

Hence, calling the uncertainty due to the linewidth itself !main (about ± 35 ppm), the totaluncertainty on the relative field value (from one point to another) is

,!2

main + !2width + !2

offset,which is about± 35.1 ppm. The relative error, or precision, on the field measurement is thus given bythe compensated linewidth, despite the power supply instabilities. However, the absolute uncertainty(i.e. the accuracy) is about 1 % because of the o!set induced by the position of the NMR coil. As weare mostly interested in measuring field variations, our limitation is the relative error.

2.7.3.2 Measuring Bz with NMR and extracting SHE terms

As we already stated in section 2.5, $Bz$z cannot exist alone, as

!") ·!"B = 0. (2.93)

In a rotationally symmetric system and in cylindrical coordinates (r, &, z), we have $B!

$% = 0 and thus,

1r

+(rBr)+r

= !+Bz

+z, (2.94)

which can be solved to obtain+Br

+r= !1

2+Bz

+z. (2.95)

Thus, a strong gradient $Bz$z implies a strong gradient $Br

$r . This calls for attention when we attemptto measure only one component of the field (e.g. SHE terms extraction) using NMR, which is sensitiveto the modulus of the field. We showed in section 2.1 that the transverse components enter only atsecond degree in the modulus variation for a field dominated by one component. However, in this case,as the variations of these components are very important, they a!ect significantly the modulus, eventhough entering at second order (e.g. if we consider a point at the elevation of the center of the RoI,located at a radius 5 mm from it, we have !Br = 8.25 mT. This corresponds to a variation of themodulus of about 300 ppm).

Nevertheless, we can use equation 2.95 to compensate NMR measurements and obtain a correctcomputation. In first approximation, we have indeed at a point of coordinate (r, &, z),

|!"B | = B =1

B2z + B2

r . (2.96)

As the sign of Bz does not change in the region of measurement, we can write

Bz =1

B2 !B2r . (2.97)

If we take measurements on Oz, we can compute the axial SHE terms of Bz without being a!ectedby $Br

$r . Thus, we can compute accurately $Bz$z on axis. Assuming this quantity does not vary much

over the measurement volume, we can compute Br in any measurement point with equation 2.95 andconvert into Bz the field measured by NMR. It is however necessary to assess the error made on the


value of Bz with such a method. We can distinguish the error !BBz due to uncertainties !B on B,and the error !BrBz due to inaccuracies !Br in Br.

!BBz =B

Bz!B, (2.98)

!BrBz =Br

Bz!Br. (2.99)

We have BBz1 1 and, for a radius r = 6 mm, Br

Bz1 0.05. !Br is given by the di!erence between the

actual Br and the one we predict based on the gradient on-axis, using Br = !12

$Bz$z r. In the case of a

perfectly axisymmetric magnet, this di!erence can be estimated to be less than 2.5 ppm of the nominalBz at a radius of 6 mm. Thus, we have !BrBz = 0.125 ppm. The total uncertainty on the calculatedBz being given by

1(!BBz)2 + (!BrBz)2, if we assume the uncertainty on B (the linewidth of the

signal) is greater than ± 1 ppm, we have !Bz 1 !B. Thus, for a perfect magnet, a SHE calculationbased on such measurements features the same precision as for a homogeneous magnet with the sameNMR signal linewidth.

However, as we will see later, the magnet to be characterized will feature imperfections and provewrong the symmetry assumption. This induces larger discrepancies between the actual Bz and thecalculated one, with a high dependence on the field profile. This can be explained by the possiblecompensation of variations of Bz by variations of Br in the modulus value seen by the NMR probe. Inthe context of this field gradient, we expect variations of Bz in & below a 1000 ppm (see section 4.1.3).Simulations show that for this extremity, the maximum error on Bz is about 100 ppm. The averagedeviation from the mean error is about 35 ppm. Of course, if the field profile has a better axialsymmetry (less variations in &), we gain accuracy and precision (less o!set and smaller standarddeviation). These errors are not random and their e!ect on SHE terms measurement is not equivalentto the e!ect of a field noise with a matching standard deviation.

Once we are capable of extracting Bz from the NMR measurements, we can perform the SHEterms measurements as described previously. However, we need to estimate the precision and accuracywe can expect. As we just mentioned, the imperfections of the field profile impact the accuracy ofBz measurements. X1

1 and Y 11 are the most a!ected terms with inaccuracies up to 80 ppm while

other terms are accurate to better than 10 ppm for profile imperfections of about 1000 ppm over thevolume of measurement. These errors are fundamental and occur even if the system measuring thefield modulus is perfect.

It is also important to assess the impact of the imperfections of that system. We are dealing herewith severe inhomogeneities due to the gradient, in addition to the magnet imperfections. This impactsgreatly the e!ect of geometrical errors, which become significant. Position noise now yields a standarddeviation of about 10 ppm on all terms for 10 µm position precision and 40 ppm for 50 µm precision.Angular errors have also a tremendous impact on the absolute accuracy of the measurement. However,these errors a!ect mostly the degree one terms (Z1, X1

1 , and Y 11 ). The absolute error made on a term

decreases with its degree, as(

rr0

)ndampens it. For example, a small tilt (0.1º) of the measurement

frame around Ox causes errors on X11 and Y 1

1 of the order of a hundred ppm while the accuracy ofother terms remains better than 10 ppm.

2.8. CONCLUSIONS ON MAGNETIC FIELD MEASUREMENTS 81

Error ! on terms (ppm) !̄ X11 Y 1

1 Average !̄ others Max ! othersTilt, 0.1º 100 2 10Tilt, 0.5º 500 5 20

Skewness, 0.1º 70 2 10Skewness, 0.5º 450 5 20

Center error, ± 0.5 mm 100 15 50Center error, ± 1 mm 200 30 100

All errors, 0.1º, 0.1º, ±0.5 mm 100 15 50

Table 2.5: Deviation from actual SHE for di!erent error sources in the case of a "pure" magnetic field gradient.All numerical measurements are done with the same radius of measurement (6 mm) and same number of points.A di!erence is made between degree one terms and other terms as they are impacted di!erently. "Other" termscorrespond to terms of degree strictly higher than zero and degree lower than four (average is over the set ofterms). As the geometry is asymmetric along Oz, the e!ect of de-centering is stronger for axial terms (Z3, Z4)when the measurement center is translated toward the magnet than when it is moved away from it.

2.8 Conclusions on magnetic field measurements

We discussed in details the mapping of the field generated by a magnet in order to retrieve SHE terms.This mapping is based on localized probes such as Hall probes or NMR probes, which are farsuperior if the field is su!ciently homogeneous in the volume of the NMR sample.

While the mapping and SHE terms extraction for a homogeneous (or almost) field is straightforward,it appears that the precise measurement of the SHE of Bz in a strong gradient is a verychallenging task. It requires a compensation system to make the NMR probe precise enough. Themeasurement system must be fabricated with great care and its positioning requires tight tolerances.Even so, a high inaccuracy is to be expected on degree one terms. However, we can expect areasonable precision (below 20 ppm) for all terms and excellent accuracy for terms other than X1

1 andY 1

1 . It is worth noting that in case of o!-center measurements, the inaccuracy on Z0 can be relativelyhigh (more than 500 ppm) compared to the perfect measurement (especially for de-centering alongOz).

We must stress here that most of these errors are related to the fact that the measurement frameis simply not where it is supposed to be (o!-center) and not oriented the way it should be (tilt). Thus,the measurements are not wrong, strictly speaking, and still provide a correct assessment of the fieldprofile, in the measurement frame. This error analysis merely tells us that, in practice, even a perfectmagnet would look imperfect to us, due to the measurement system.

The true goal of such measurements is to find out what must be done to adjust theimperfect magnet so the final user has the field profile necessary for his NMR experiment.Thus, these measurements are still useful to this purpose, as long as their precision issu!cient (see section 4.1.4). The final profile will be the desired one, only in a slightlydi"erent reference frame which can be found experimentally.

Chapter 3

In-situ magnets

One common family of magnets for portable NMR is the in situ one, where the RoI is located insidethe magnet and a bore hole provides access to it. We have seen in section 1.3.3.1 that it contains C, Has well as cylindrical dipole magnets. We intend here to revisit some very early designs that producea longitudinal field and the Halbach dipole in the light o the SHE as presented in chapter 2.

3.1 Longitudinal field - Aubert configuration

3.1.1 A simple and low-cost test-magnet

Based on an early design by G. Aubert in 1991 (150), generating a longitudinal field, we can fabricatea simpler magnet at a low cost and apply our di!erent tools to get the most out of it.

3.1.1.1 Theoretical description

This simple system features only two radially magnetized rings. As it is di"cult to obtain cylindricalrings with a uniform radial magnetization, it is necessary to segment the ring into discrete elements.We chose to use cubes as they are readily available for a low price. A schematic of the cube-basedstructure can be seen in figure 3.1. Twelve segments were used to maximize the density of materialaround a bore of 52 mm diameter. Two rows of cubes are placed in each ring to reinforce the field,requiring a total of 48 cubes. Each cube is 12x12x12 mm. The resulting assembly is about 66mm longand has a diameter of roughly 90mm, with a total weight of about 1kg.

Based on the previous considerations, we can assess the homogeneity that can be ultimately achievedby such a system. Starting with the segmentation into 12 pieces (i.e. 12-fold rotational symmetry) ofthe rings, we can expect that skewed terms will be cancelled up to the 11th degree (the first non-zerobeing the 12th). As we have two rings, one axial term can be cancelled by adjusting the inter-ring gap(Z2 is cancelled with a gap of 42.122 mm). The plane antisymmetry helps us by cancelling all oddterms. As a result, one might expect the homogeneity of this system to be dominated by the axial termof degree four (Z4). Given the dimensions of the magnet, this results in a homogeneity of about 15 ppmover a sphere of 3 mm in diameter. This is the best homogeneity achievable by this magnet. Achievinga better homogeneity would imply the addition of a shimming system of significant size compared to themagnet itself. Simulations of the magnetic cube arrangement of figure 2, performed using Radia (151)bear this out. A contour plot of the field in the xOz plane (figure 3.2) shows that the variations ofthe field are indeed dominated by the degree four and predicts the field at the center to be about 120mT. The magnets made of material designated N48 were purchased from Supermagnete.de withoutany requirements on the mechanical or magnetic tolerances. The two rings of cubes are supported by

83

84 CHAPTER 3. IN-SITU MAGNETS

machined aluminum mounts mounted on linear translation stages (Owis brand, model VT 65-Z-FGS,non-magnetic version) in order to adjust the gap between the two rings. Bracings maintain the gaponce the magnet is adjusted and removed from the linear stages. A photograph of the assembly isshown on figure 3.1.

Making an axial magnetic field is usually less e"cient than making a transverse one, using theconfiguration of Halbach. In our case, with the same number of cubes representing 630g of magneticmaterial, one could generate a "homogeneous" transverse magnetic field of 190 mT with the same boreradius, with a Halbach scheme. The homogeneity of this magnet would be 15ppm in 3 mm DSV. Wecould also use only axially magnetized blocs. This would lead to a worse e"ciency compared to theAubert or Halbach configurations. In our case the field generated by such a configuration is 90mT,with a homogeneity of 26 ppm in 3mm DSV.

It is straightforward to estimate the e!ect of position errors of the di!erent magnet parts fromthe theoretical calculations. These estimations indicate the positioning of the magnets must be veryprecise. For example, an error of 0.05 mm on the gap between the rings induces the rise of 15 moreppm of inhomogeneity on the same diameter, due to the increase of Z2.

Figure 3.1: Left: Basic layout of the magnet system. It consists of two rings of radially magnetized blocks.The combination of these rings creates a longitudinal field at the center of the system and confines the field inthe structure. The magnetization of each cube is shown as an arrow on the side. Right: Photograph of the finalimplementation of the cubes in their aluminum mounts set on translation stages for adjustments.

3.1.1.2 Part measurements

Good knowledge of the magnetic properties of each piece seems necessary in order to achieve highperformance. The most important quantity is the magnetization magnitude and orientation of thepart. It is extremely di"cult to retrieve the magnetization distribution of the magnet part but onecan measure with a good precision (but poor accuracy) the average magnetization of a part. This canbe done by assuming a dipolar approximation of the magnet part. Several methods can then be used,either based on Hall probes (128) or on flux coils (see for example section 5.2 in (152)). The dipolarapproximation is a very good one for any geometry as long as one measures the field far away enoughfrom the part. In the case of a cube, one should be located at least at a distance five times the size ofthe cube. The field of a dipole can be written as

3.1. LONGITUDINAL FIELD - AUBERT CONFIGURATION 85!"#$$%

&"#$$%

'()* '*)+ ()**)+'*),'*)-'*).'*)/'()*'(),'()-

!"#$$%

00$

Figure 3.2: Left: Contour plot of Bz in the XZ plane, as the theoretical magnet should produce. The variationsare dominated by degree four, and are limited to about 40 ppm over a sphere of 4 mm diameter. Levels are inppm. Right: Plot of the field variation in ppm along Ox.

!"B (x, y, z) =

µ0

4$

3(!"M ·!"u )!"u !!"M

R3(3.1)

with !"u being the unitary vector going from the dipole position O (set to be the origin) to the mea-surement point P(x,y,z) and R being the distance from O to P. When reducing to the z component ofthe field, we obtain :

Bz(x, y, z) =3µ0

4$

Mxux + Myuy + Mz(uz ! 13)

R3(3.2)

As a result, the knowledge of Bz at 3 arbitrary points of free space is su"cient to retrieve Mx, My

and Mz. However, due to the low precision of a Hall probe (about 10#4) and the uncertainties onpositioning and registration of the probe axes to the cube axes, it is necessary to use more than3 points. Unfortunately, it is sometimes not possible to perform the measurements at a distancesu"ciently remote to safely assume a dipolar behavior of the field generated by the cubes.

It is possible to refine the model used to retrieve the average magnetization when a 3D field mapof the field can be obtained. Using the same measurement scheme as in our moment measurement,we can use analytical formulas for the field generated by a parallelepiped (153). One can indeed writethe field generated by a parallelepiped of dimensions 2a, 2b and 2c, magnetized along the z direction(figure 3.3) as :


BZx (x, y, z, a, b, c) = Mz

4! ln

%y+b+

*(y+b)2+(x#a)2+(z#c)2

y#b+*

(y#b)2+(x#a)2+(z#c)2· y#b+

*(y#b)2+(x+a)2+(z#c)2

y+b+*

(y+b)2+(x+a)2+(z#c)2

·y#b+*

(y#b)2+(x#a)2+(z+c)2

y+b+*

(y+b)2+(x#a)2+(z+c)2· y+b+

*(y+b)2+(x+a)2+(z+c)2

y#b+*

(y#b)2+(x+a)2+(z+c)2

&(3.3)

BZy (x, y, z, a, b, c) = Mz

4! ln

%x+a+

*(y#b)2+(x+a)2+(z#c)2

x#a+*

(y#b)2+(x#a)2+(z#c)2· x#a+

*(y+b)2+(x#a)2+(z#c)2

x+a+*

(y+b)2+(x+a)2+(z#c)2

·x#a+*

(y#b)2+(x#a)2+(z+c)2

x+a+*

(y#b)2+(x+a)2+(z+c)2· x+a+

*(y+b)2+(x+a)2+(z+c)2

x#a+*

(y+b)2+(x#a)2+(z+c)2

&(3.4)

BZz (x, y, z, a, b, c) = Mz

4!

%arctan (x+a)(y+b)

(z#c)*

(y+b)2+(x+a)2+(z#c)2! arctan (x+a)(y+b)

(z+c)*

(y+b)2+(x+a)2+(z+c)2

+ arctan (x#a)(y#b)

(z#c)*

(y#b)2+(x#a)2+(z#c)2! arctan (x#a)(y#b)

(z+c)*

(y#b)2+(x#a)2+(z+c)2

! arctan (x+a)(y#b)

(z#c)*

(y#b)2+(x+a)2+(z#c)2+ arctan (x+a)(y#b)

(z+c)*

(y#b)2+(x+a)2+(z+c)2

! arctan (x#a)(y+b)

(z#c)*

(y+b)2+(x#a)2+(z#c)2+ arctan (x#a)(y+b)

(z+c)*

(y+b)2+(x#a)2+(z+c)2

&.(3.5)

The field generated by the same parallepiped magnetized along x or y satisfies the same equationsafter the proper transformation of coordinates and permutations of a, b and c. We shall define thefollowing notations to refer to the field generated by the di!erent magnetization coordinates:

PX( (x, y, z, a, b, c) =

1Mz

BZ( (!z, y, x, c, b, a)

P Y( (x, y, z, a, b, c) =

1Mz

BZ( (x,!z, y, a, c, b) (3.6)

PZ( (x, y, z, a, b, c) =

1Mz

BZ( (x, y, z, a, b, c),

where - can be x, y or z. Using this notation, we can express any component of the field generated bythe parallelepiped through the following equations :

Bx(x, y, z, a, b, c) = MxPXz (x, y, z, a, b, c) + MyP

Yx (x, y, z, a, b, c) + MzP

Zx (x, y, z, a, b, c)

By(x, y, z, a, b, c) = MxPXy (x, y, z, a, b, c) + MyP

Yz (x, y, z, a, b, c) + MzP

Zy (x, y, z, a, b, c) (3.7)

Bz(x, y, z, a, b, c) = !MxPXx (x, y, z, a, b, c)!MyP

Yy (x, y, z, a, b, c) + MzP

Zz (x, y, z, a, b, c).

This can be summarized in matrix formalism for any component - of the field as

B( =.2x,( 2y,( 2z,(

/B

CPX

z P Yx PZ

x

PXy P Y

z PZy

!PXx !P Y

y PZz

D

E

B

CMx

My

Mz

D

E , (3.8)

where 2),( = 1 if - = 1 and zero otherwise.It can be noticed that only one component of the field is required to retrieve the 3 components of

the magnetization if one obtains several measurements in di!erent locations. This is quite useful when

3.1. LONGITUDINAL FIELD - AUBERT CONFIGURATION 87

Figure 3.3: Definition of diverse geometric parameters for the measurement of the cube. The laboratory frameis the same as the cube’s frame. Dimensions of the cube are shown, along the di!erent angles associated to theattitude of the detector.

a Hall probe is used to perform the measurements as it is usually di"cult to have a good colocationbetween the x, y and z detector. Using only one component of the field solves the issue of measuringthe three components at the exact same location. We will now remain concerned with the z componentof the field. A convenient way of performing the measurement is to measure Bz in several points ofa plane centered on the cube, parallel to xOy and located as far away as possible from its center.Such a solution eases alignments and ensures better performance of the Hall detector by measuring thedominant component of the field. As mentioned in section 2.7.1, one must be careful to have negligibleorthogonal components in the field in order to avoid errors due to the planar field e!ect (128).

It is very important to take into account the possible misalignment of the cubes referential withthe translation referential and the detector referential. In our context, we assume the translationreferential is aligned with the cubes referential by construction. However, the position of the detectorand its orientation is not very well known. The detector can be anchored to the center of the cube inthe translation referential quite easily to a precision of about 0.1mm by eyesight and contact betweenthe probe and the magnet. However, the tilt of the detector and the refinement of its position relatedto the cube requires more involved calibration. If one defines the attitude of the detector as shown onfigure 3.3, equation 3.8 can be modified for Bz, as measured by the Hall detector:

BMeasz =

.cos &d sin 'd sin&d sin 'd cos 'd

/B

CPX

z P Yx PZ

x

PXy P Y

z PZy

!PXx !P Y

y PZz

D

E

B

CMx

My

Mz

D

E , (3.9)


We can calibrate the attitude angles through the measurement of the cube in a plane, as statedabove, with the cube successively rotated by a known angle $ around the Z axis. In our case, thegeometry of a cube makes easy and reliable the rotation by !

2 . One then obtains a set of measurementmatrices corresponding to each rotation angle. All of these measurements are related to each other bya simple rotation of angle $ such that the measured Bz verifies :

BMeasz =

.cos &d sin 'd sin&d sin 'd cos 'd

/B

CPX

z P Yx PZ

x

PXy P Y

z PZy

!PXx !P Y

y PZz

D

E

B

Ccos $ sin $ 0! sin $ cos $ 0

0 0 1

D

E

B

CMx

My

Mz

D

E .

(3.10)As the cube properties and the detector orientation remain unchanged from one rotation to the

other, it is possible to use these measurements to retrieve at once the magnetization of the cube, theattitude of the detector, and the center of the cube compared to the position of the detector (if theset of measurements is well conditioned). However, this requires several sets of measurements (oneper rotation) involving a number of points in order to overdetermine the system of equations. It canhowever be assumed that the attitude of the detector and the center of the cube are not going tochange when switching to the next cube to be measured. As a result, it is possible to measure the restof the set of cubes by using only one set of measurements for a given orientation of the cube. Thebest possible alignment should nevertheless be achieved before calibration. If the detector is too muchtilted or too much o!-center, the issue of planar field e!ect may be encountered.

The outcome of this measurement method performed on 68 cubes of side 12mm is shown onfigure 3.4 the di!erent components of the magnetization of our set of cubes in spherical coordinates forthe north and south pole. This shows the typical asymmetry between the two poles. We assessed therepeatability of the measurement to 0.5% to 1% for the magnetization, 0.5( for ' and 2( for &. Thedi!erence between the magnetization retrieved with the dipole assumption and the cube assumption isabout 1.5 to 2% on the magnetization amplitude, between !1( and 1( on ' and a few tens of degrees on& (the precision on the latter quantity is low in any case as ' is close to zero). It seems hence significantlymore accurate to use the cube assumption in this case. The & angle measurement distribution is verywide (it extends the full 2$ range) as there is a four-fold indetermination of orientation for a cube(due to its symmetry). Each cube can be rotated four times by 90º without apparent di!erence. If thecubes were perfect, this would result in four groups of & angles but as there exist an actual spread in&, these groups merge to give an apparent 2$ range. The small value of ' explains this wide spread in& (the precision on & is worse as ' approaches zero). However, & looses relevance as ' approaches zero.


! "! #! $! %! &! '! (!")$#")$$")$%")$&")$'")$(")$*")$+")%")%"")%#

(a)

! "! #! $! %! &! '! (!!

"

#

$

%

&

'

,-./01-2./3

453670859/:5-670859/

(c)

! "! #! $! %! &! '! (!!"##

!$%#

!$##

!%#

!

&!

"!!

"&!

#!!

,-./01-2./3

(e)

")$& ")$' ")$( ")$* ")$+ ")% ")%"!"#$%&'(*+"!"""#

4-2

./305;0<-./=

! !)& " ")& # #)& $ $)& % %)& &!

#

%

'

*

"!

"#

4-2

./305;0<-./=

! "! #! $! %! &! '! (! *! +!!

"

#

$

%

&

'

(

>?@2-670A1B9/0CD/B3//=E

4-2

./305;0<-./=

(b)

(d)

(f )

,-./04-2./3

FAB1/6@?A6@510C0GE

FAB1/6@?A6@510CGE

H1<9@1A6@510A1B9/0CD/B3//=E

!0A1B9/0CD/B3//=E

" 0A1B9/0CD/B3//=E

Figure 3.4: Measurements of the magnetization magnitude for an ensemble of 68 cubes. a) Magnetizationamplitude. b) Histogram of magnetization amplitude c) Magnetization inclination. d)Histogram of magnetiza-tion inclination e) Magnetization azimuth. f) Histogram of magnetization azimuth. The azimuth measurementis inaccurate and less relevant as the inclination is very small. The most relevant parameters are here theamplitude and the inclination which show significant spreads (few percents for the amplitude and few degreesfor the inclination)


3.1.1.3 Screening of parts and simulations based on measurements

The magnetization measurements can be used in first place to skim o! the worst pieces which deviatethe most from the average values of the set of magnets. The performances of a particular combinationof parts can be simulated based on the measurements. In order to perform fast estimations of acombination, we assumed the cubes were simple dipoles. Equations 2.24, 2.25, and 2.26 allow the quickevaluation of the homogeneity of the magnet in view of massive screening. Such an approximation isnot inappropriate in this case as this step of fabrication aims at reducing coarse imperfections.

One can hence estimate the SHE of the magnet simply based on the positions of the cubes and theirmagnetic measurements. It is possible to rapidly screen thousands of possible combinations in orderto refine the layout of the cubes. It is however impossible to span the whole set of combination. Theresult of such a screening yields a better solution with higher likelihood than when doing it randomly.However, it is not su"cient for the achievement of NMR-grade performances as the knowledge of themagnetization of a magnet is limited to about 1%.

Based on the measurements of the cubes, it is also possible to simulate a given combination withRadia, in order to assess the performance of the magnet. The resulting simulations including demag-netization are accurate with a field at center predicted within 0.2% of the experimental field. We werealso able to obtain qualitative and quantitative assessments of the field variations on a large scale. It ispossible to predict the coarse variations of the system, which will have to be corrected before startingfine adjustments based on NMR. We present on figure 3.5 the contour plot of Bz in the xOz planefor the simulated magnet and for the experimental measurements using a Hall probe. These resultsconfirm the possibility of estimating quantitatively the performances of the magnet assembly once thecubes have been carefully characterized.

!"#$"

!"#$"!%&'()

!%&'(

!$'#

!$'#

!"(&

*&"

+&$

%,,&"%(#

"%(#"'(("'((**'"

-& -" # " & (

-&

-"

#

"

&

(

-(-(

Z (mm)

X (mm)

!%+'+

!%+'+

!%"&%

!%"&%

!,#*!,#*

"*,

+'*

%'%%

"&&+%'%

%*%$$"&&

+*+"(

-& -" # " & (

-&

-"

#

"

&

(

-(-(

a) b)Z (mm)

Figure 3.5: Contour plots (in ppm) of Bz in the xOz plane for (a) a simulation of the original magnet withRadia based on the measurements of the cubes, (b) the experimental measurements of the original magnet.Simulations based on part measurements can give qualitative as well as quantitative information on the fieldvariations.


3.1.1.4 Characterization of the magnet and correction system

Magnet system measurement Based on the discussion of section 2.7.2, we can now have a lookat the measurement of this simple magnet structure. Theoretically, the first non-zero term is Z4. Thismeans that the relevant terms to assess the quality of the magnet are all terms up to and includingdegree 3. Higher degree terms are bound to be present but should be negligible in the volume ofinterest (as we will keep the radius small enough to obtain small variations of the degree 4). However,we need to perform the measurements on a larger radius for practical reasons (the radius of the volumeof interest is only 1.5 mm) and are hence subject to aliasing. As we saw we can reasonably simulate thefield, the necessary number of measurement points can be evaluated through simulation. Figure 3.6presents the resulting values of the Zn terms for di!erent number of sampling points on a sphere ofradius 5 mm. Similar results can be found with the Xm

n and Y mn but are not shown here. While 32

points are clearly insu"cient (Z2 is o! by more than 30 ppm), 48 points seem su"cient to achieve a 1ppm accuracy on all terms. However, using more than 64 points is not useful. The measurements inpractice were performed using a Hall probe.

!"#$%

!"$$%

!&$%

!'$%

!($%

!#$%

$%

#$%

($%

"% #% )%

!!"#$%&#'#&'()*+#%$#,""#

-./#

)#%*+,-./%

(&%*+,-./%

'(%*+,-./%

&$%*+,-./%

"$$%*+,-./%

Figure 3.6: Simulated measurement of the relevant Zn terms for the discussed magnet, with a measurementsphere of radius 5 mm. The stability of the calculated terms with the increase of measurement points indicatesspectral folding is not significantly a!ecting the terms of interest. Similar results can be obtained with the Xm

n

and Y mn but are not shown here as they do not add information.

Correction system Some high performance magnets have been proposed in the past (109; 77),based on intrinsically inhomogeneous structures (due to geometry) corrected by motion of significantadditional magnet blocks. Here, however, we start with a magnet theoretically homogeneous to anarbitrary degree. The inhomogeneities are hence mainly due to material imperfections and scarcelyto geometry. This results in a reduced need in shims. One can a!ord the use of very small shimpieces to correct for the imperfections of the magnet. Adding small pieces of ferromagnetic materialto the magnet in order to compensate its inhomogeneities is a standard procedure in MRI (154–160)and we adapted this method to the context of permanent magnets. It is indeed necessary here to usepermanent magnets, and not only iron, as the field strength involved is not su"cient to saturate it.In addition, permanent magnets have a predictable orientation and provide more freedom as they willkeep it in any direction they are oriented in such fields.

The correction of our simple example magnet can be done by using small magnet cubes placed at


appropriate locations. The full correction of the magnet, meaning up to the theoretical homogeneity, ne-cessitates the cancellation of fifteen terms (Z1, Z2, Z3, X1

1 , Y 11 , X1

2 , X22 , Y 1

2 , Y 22 , X1

3 , X23 , X3

3 , Y 13 , Y 2

3 , Y 33 ).

One needs p + 1 independent parameters to cancel p terms. As a result, we need at least 16 param-eters. A single cube features 3 translational degrees of freedom and 3 rotational degrees of freedom.We decided to set the orientation of each cube to the most e"cient orientation given their location,i.e. antiparallel to the field at the center of the magnet, leaving no rotational degrees of freedom. Forease of realization, we require to tie the possible positions of each cube to a plane. As a result, onlytwo translational degrees of freedom remain for each cube. It follows that one needs at least 8 cubesin order to cancel all undesired terms and achieve theoretical homogeneity. The symmetries of themagnet lead to place each cube in a quadrant of the xOy plane, symmetrically in regard of that sameplane. The areas where each shimming cube can be placed are shown on figure 3.7.

Figure 3.7: Photograph of the shimmed magnet (left). The small magnet cubes can be seen between the twomain rings. Top right: schematic showing the four planes where the location of the shims was constrained.Lower right: schematic showing a 3D view of the magnetic elements of the system. Our method allows thefabrication of a shimmed magnet with a field strength at center very close to the theoretical one (about 2%di!erence).

It is of course necessary to verify that the variables o!ered by the shimming system can spanthe relevant part of the expansion. We hence conducted a theoretical study of such a shimmingscheme. Based on the measurements of the cubes constituting the magnet, we simulated the magnetinhomogeneities using Radia, in order to include demagnetization. The measurement scheme wassimulated numerically and the extraction algorithm was used in order to obtain simulated SHE terms.Based on these simulations, we computed the appropriate position of the shimming cubes and addedthem to the Radia simulation. The shimmed magnet is shown on figure 3.7 and the field profilesbefore and after shimming are shown on figure 3.8. Figure 3.9 presents the values of the di!erentterms before and after shimming. The results are close to perfect and very encouraging for proceedingexperimentally.

The compensation of the magnet can also be done through modifications of its geometry. It is


!"#"!$%

!&'!('

!#%

!)

#'

)*

+%

$'

""*

"*%"+'

,# ," ' " #,#

,"

'

"

#

Z (mm)

X (mm)

!"%

!"%

!"#

!"#

!&

!&

!)

!)

!)!)

!'-#

!'-#

!'-#!'-#

)

)

)

)&

&

&

&"#

"#"#

"#

"+

"+"+

"+

#'

#'#'

#'

,# ," ' " #,#

,"

'

"

#

Z (mm)

a) b)Figure 3.8: Contour plots of Bz (in ppm) in the xOz plane for (a) a simulation of an unshimmed magnetwith Radia based on imperfect cubes (not the experimental magnet configuration), and (b) for a simulation ofthe same magnet shimmed with small cubes. The position of each shim cube has been calculated in order tominimize the di!erent terms in the spherical harmonics expansion. One can observe the dramatic improvementof the field homogeneity after shimming.

possible to have some play on the gap between the rings of the magnet and also on the orientation ofeach ring relative to the other. These geometry modifications are of course very sensitive and have agreat impact on the field variations at the center. The terms mainly concerned by such movements arethe di!erent gradients Z1, X1

1 , Y 11 and Z2.

We applied this shimming scheme using Hall probe measurements. We successfully obtained theSHE terms and computed the necessary positions of the shimming cubes (priorly characterized followingthe same procedure as for the larger cubes). This resulted in a great improvement of the homogeneity,as it can be seen on figure 3.10. The NMR linewidth was improved from 40 ppm (161) to 12 ppm(see figure 3.11) for a sample of 3 mm3. The sequence used was a simple !

2 -acquisition featuring 512points with a dwell time of 130 µs. The acquisition was done at 5.04 MHz in 128 scans (25.6 seconds)on a sample of water doped with CuSO4. This result is not as good as the 5 ppm the theory lets usexpect for the same sample size, but it can be explained by the crude positioning of the shims in thisprototype (precision on the location of the shims of about 0.5 to 1 mm). Actually, one may wonderwhy this procedure is that successful as we indeed have a very poor knowledge of the shimming cubes(at best 1% on the magnetization and 0.5º on orientation) and a poor positioning. This is only dueto the fact that percent-level errors in the shimming system are only resulting in errors that are afraction of the desired correction. Hence, achieving the initially desired field profile is only a matter ofiterations (and accuracy and precision of measurement).


‐120

‐100

‐80

‐60

‐40

‐20

0

20

1 2 3

ppm

n

Zn

12

3

‐350

‐300

‐250

‐200

‐150

‐100

‐50

0

50

1 2 3

n

ppm

m

Xnm

1

2

3

123

‐200

‐150

‐100

‐50

0

50

1 2 3

n

ppm

m

Ynm

1

2

3

a) b)

c)

‐0.2

0

0.2

0.4

0.6

0.8

1

1 2 3

ppm

n

Zn

123

‐2

‐1.5

‐1

‐0.5

0

0.5

1 2 3 npp

m

m

Xnm

1

2

3

123

‐0.15‐0.1

‐0.050

0.050.1

0.15

0.2

1 2 3n

ppm

m

Ynm

1

2

3

e) f )

d)

Figure 3.9: Contribution of the di!erent degrees to the field variations on a sphere of radius 5mm for asimulated magnet with perfect geometry using the measured cubes, before (left) and after (right) correction.Only terms theoretically cancelled are shown (terms of degree higher or equal to four are not relevant as we donot have control over them). a) Contribution of axial terms before correction. One can notice the cancellationof Z2 due to the appropriate gap adjustment. The gradient Z1 is the the dominant term and Z3 is in this casevery small. b) Contribution of axial term after correction. All contributions are below 1 ppm. c) and d) Xm

n

terms before and after correction. The maximum contribution amplitude is dramaticaly reduced from about350 ppm to 2 ppm. e) and f) Y m

n terms before and after correction.


!"#$#!"#$#

!"%&"

!"%&"

!'()!'()

%)'

#$)

"$""

%&&#"$"

")"**%&&

#)#%+

,& ,% ( % & +

,&

,%

(

%

&

+

,+,+

Z (mm)

X (mm)

'")"

!

!")"'

!#%(

!#%(

!'%+!'%+

!'%+

!")"

!")"

!")"

!")"

%+)

%+) +'$+'$

%+)%+)

"('%

"('%"&&+

"&&+

!")"

"*&"

+'$+'$

"*&"

,+ ,& ,% ( % & +,+

,&

,%

(

%

&

+

Z (mm)

X (mm)

a) b)Figure 3.10: Left: Hall probe measurement of the initial field variations in the xOz plane. Right: Hall probemeasurement of the corrected magnet field profile. A significant improvement can be noticed, as the profile ismuch closer from the theoretical one (see figure 3.2).

!"## !$## !%## # %## $## "###

%

$

"

!

"

#

$

%

%&'(')*#+"',&-./&,'0'%$'112

!$###!%)##!%### !)## # )## %### %)## $###!#*)

#

#*)

%

%*)

$

$*)!

321456789':3

;<

1H @ 5.1 MHzFWHM = 40 ppm

.=9>79?@A'BC5D6':112<.=9>79?@A'BC5D6':112<

Figure 3.11: 1H NMR spectrum of a 3 mm3 sample of water (doped with CuSO4) before (left) and after(right) shimming. The achieved full width at half maximum is about 12 ppm, displaying a major improvementon the original 40 ppm linewidth. The poor quality of the base of the peak might be related to the poor noiseconditions and also to fast field variations on the edge of the sample.

3.1.1.5 Conclusion

This small prototype opens the road for more involved magnets based on the same concept as wewill see in the next section. Moreover, it demonstrates the feasibility of shimming the magnet withsmall additional magnets with a large tolerance to magnetic properties and positioning (the latter isimportant from the practical point of view as the handling of such magnets in the presence of the mainmagnet is not easy).


3.1.2 A theoretical highly homogeneous magnet

Following the success of the previous low-cost magnet, we can study a more complex structure stillderived from the concept of G. Aubert (150).

Based on the equations and e"ciency considerations given in section 2.3.2, we can define an ar-rangement of rings to create a strong and homogeneous longitudinal field. We set the bore diameter to5 cm so that we have reasonable access to the center of the magnet. In order to keep the fabricationfeasible, we limit the number of elements to six, so that the first non-zero term is at degree 12 (the firstnon-zero term should be a skewed term at degree 11 but the planar symmetry forbids odd degrees).Hence, using equation 2.20, we can expect to achieve a homogeneity of 10 ppm in a sphere of radiusabout 10 mm. In addition, we restrict the dimensions of the magnet to 20 cm in diameter and 20 cmin length, to keep in the transportable range of weight.

The final arrangement is constituted of two radially magnetized antisymmetric rings (one inwardsand the other outward), and four antisymmetric longitudinally magnetized rings placed between theradially magnetized pieces. The antisymmetry insures the potential expansion features only odd terms,so that the field expansion features only even terms. The radially magnetized blocks have to besegmented, and the manufacturer we work with has assembly tools ready for 12-fold symmetry systems.This also guarantees a high order of axial symmetry. The longitudinally magnetized rings have to beof large dimensions and will also be segmented in twelve as it is not possible to fabricate pieces of thatsize in one block.

Along the design process, it appears that the four middle longitudinally magnetized rings requireto be very thin, beyond what is mechanically reliable. Reducing the angular aperture of the segmentsprovide the solution by allowing thicker rings for the same homogeneity. These segments o!er in turna good basis for an adjustment system in view of imperfection correction. All rings can be individuallyrotated around the axis by an arbitrary angle, so that we can alternate the middle rings, leaving morespace for mechanical mounts. The geometry and magnetic scheme of the theoretical magnet is shownin figure 3.12

The final magnet o!ers theoretically a field of 0.918 T with a remanence µ0M of 1.35 T and ahomogeneity of about 2 ppm in a diameter of 20 mm. The dominating term is at degree 12 with asmall contribution, which explains why the magnet performs better than anticipated when we estimatedthe degree of homogeneity necessary.

Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12

Amplitude (ppm) 0 0 0 0 0 0 0 0 0 0 0 1

Table 3.1: Contribution of the di!erent axial terms in the field variation of the final model in a sphere ofradius 5 mm. The field strength is 0.918 T for a remanence of 1.35 T. All contributions are given for a sphereof radius 10 mm.

The performances as calculated by numerical simulations in Radia are shown in figure 3.13. Theyof course verify the analytical calculations. All of these results are given for a perfect geometry andmagnetization. A study of the e!ects of imperfections, demagnetization and correction systems on thehomogeneity is necessary prior to fabrication and could be carried in the same fashion as what we showin section 4.1.3 for a unilateral magnet.


Figure 3.12: Top left: 3D view of the final structure with the middle segmented rings all aligned. Top right:3D view of the final structure with the middle segmented rings shifted in order to leave more space for individualmechanical mounts. The magnetic properties are scarcely changed (the amplitude of variation of the field isabout the same) from one configuration to the other. Bottom: schematic of the magnetization distributionthroughout the structure. Two rings are radially magnetized and the four middle ones are longitudinallymagnetized, opposite to the field at the center.


Figure 3.13: a) Field variations in ppm in the xOz plane. b) Field variations in Tesla along Oz. The verylarge region where the field is constant can be noticed. c) Field variations in ppm along Oz in a radius of 15 mmfrom the center. d) Field variations in ppm along Ox in a radius of 15 mm from the center.

3.1.3 Conclusion

The Aubert configuration is thus well-suited for the design of highly homogeneous magnets. Thesuccess of the previous small magnet and the demonstration of the shimming possibilities allowed usto start fabricating this more complex prototype with a good confidence.

3.2 Transverse field - Halbach magnets

Another interesting type of configuration is the Halbach structure (62), which generates a transversefield. We revisit here this 2D concept to achieve homogeneity in 3D, folowing the considerations ofchapter 2.

3.2. TRANSVERSE FIELD - HALBACH MAGNETS 99

3.2.1 Theoretical design : from analytical to numerical calculations

We can design a Halbach-type magnet generating a transverse field using the relations given in sec-tion 2.4. However, these equations provide the expressions of the SHE terms for a continuously varyingdistribution of magnetization within a hollow finite cylinder. As a result, we still need to make anadditional step in order to discretize the magnetization distribution. This can be done numericallyusing a Radia model of a cylinder with a very large number of segments approximating a continuousdistribution, with the dimensions obtained from optimization using the equations in section 2.4. Wecan then progressively decrease the number of segments and readjust the dimensions of the magnetbased on the extraction of the field SHE terms. We can obtain in the end either a polygonal ring or asegmented circular cylinder with the desired number of segments. It appears that the segmentation ofthe circular cylinder does not a!ect the homogeneity (as long as there are enough segments comparedto the degree of the first non-zero term). The transformation into a polygonal ring requires, however,some minor modifications (between 0.1 mm and 4 mm : the overall magnet size is not changed) of thethickness of the parts and the gaps between them. The relevant field SHE terms values are given intable 3.2 and figure 3.14 illustrates the above considerations by showing the field variations in the xOzplane at the center of the magnet for cylindrical and dodecagonal geometries.

Discussions with the manufacturer made appear to be less costly to fabricate a Halbach magnetwith cylindrical shape. Hence, the final magnet model retained has the cylindrical cross-section withtwelve segment of magnetization and is shown, along with its performance in figure 3.15.

Amplitude (ppm) Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8

Cylinder 0.0 0.1 0.0 0.0 0.0 0.6 0.0 0.0Dodecagonal ring 0.0 50.8 0.0 2.3 0.0 0.7 0.0 0.0

Corrected Dodecagonal ring 0.0 0.0 0.0 0.0 0.0 0.6 0.0 0.0

Table 3.2: Contribution of the di!erent axial terms in the field variation of the di!erent Halbach modelscomputed after Radia simulations. The cylinder model corresponds to a Halbach magnet with cylindricalgeometry, but segmented magnetization (12 segments). The dodecagon model is the same Halbach with thesame dimensions (diameters and heights) but with a dodecagonal cross-section, with segmented magnetizationcorresponding to each edge of the dodecagon. The corrected dodecagon model is the same magnet with modifiedheights in order to obtain the same performance as the cylinder model. The field strength is about 0.635 T withmaximum variation of 10!4 from one model to the other. All contributions are given for a sphere of radius 5 mm.The cylindrical model is not corrected for the segmentation of the magnetization distribution (the dimensionsare taken straight out of the optimization assuming a continuous distribution), explaining the slight Z2 term.The corrected version of the dodecagonal cylinder shows perfect homogeneity (up to the targeted degree).

3.2.2 Demagnetization considerations

While Halbach-like magnets generate a very strong field at their center, they also generate a strongdemagnetizing field. It can be computed analytically in the case of the cylindrical, continuouslyvarying magnetization structure (31), thus providing with the precious information of the maximumdemagnetizing field, which is not to exceed or even approach the coercivity of the material. If it wereto approach or exceed the coercivity of the material, the magnetization distribution within the magnetwould change dramatically and we would not achieve either the targeted field strength, and even lessthe targeted homogeneity. It is necessary to select a material of higher coercivity if it is the case.Once we have good confidence that the coercivity is not to be approached, we can proceed to see thee!ects of slight demagnetization within the magnet using Radia simulations (based on susceptibilityvalues ,) = 0.07 and ,$ = 0.17, see section 4.1.3 for more precisions on the simulations). As expected


Figure 3.14: a) View of the cylindrical model with twelve segments of magnetization. Blue and green partsdi!er only by geometry. b) Field variations in ppm in the xOz plane with a radius of 5 mm for the cylindricalmodel. c) View of the dodecagonal model. d) Field variations in ppm in the xOz plane for the uncorrectedversion (using dimensions obtained for a cylindrical, continuously varying magnet). The di!erent geometryinduces a significant degradation of the homogeneity at the center. e) Field variations in ppm in the xOz planefor the corrected version. We are back to the homogeneity of the cylindrical magnet.

(because the demagnetizing field is much stronger than in the unilateral models), the e!ect is quitedramatic, inducing several hundreds of ppm of inhomogeneity, mostly at degree two. However, thisinhomogeneity is the simple result of the unbalanced magnetization within the magnet. This canbe compensated in the first place by a proper selection of material. Increasing the coercivity of thematerial prevents the magnet to enter a non-linear catastrophic demagnetization regime, but doesnot change the susceptibility of the material by much. Hence, it seems more appropriate to adjustthe starting point of each segment, that is to say its remanence. Simulations show that a properdistribution of three remanences di!ering by a few percent may give us back a homogeneity of 10 to20 ppm from the 800 ppm expected out of an initial homogeneous remanence. Table 3.3 presents thevalues of the relevant field SHE terms after relaxation for a regular initial magnetization distributionand an optimized one while figure 3.16 indicates the scheme of remanence distribution. Of course,these results are obtained supposing that the remanence of each piece is very well controlled. Thiscannot yet be implemented at a reasonable cost but could be someday, as the fabrication processesimprove.

3.2. TRANSVERSE FIELD - HALBACH MAGNETS 101

Figure 3.15: a) 3D view of the magnet geometry. b) Field variations in ppm in the xOz plane. c)Fieldvariations in Tesla along Oz. The very large region where the field is constant can be noticed. d) Fieldvariations in ppm along Oz in a radius of 8 mm from the center. e) Field variations in ppm along Ox in a radiusof 8 mm from the center.


Amplitude (ppm) Z2 X22 X4

4 Z6

Initial 0 0 0 0.6Regular 55 405 0.2 0.6

Optimized 0.4 4 4 0.6

Table 3.3: Contributions of the relevant terms for an initial homogeneous structure, after demagnetization.All other terms have a contribution below 1 ppm and all contributions are given for a radius of 5 mm. The"initial" version corresponds to the theoretical Halbach, with homogeneous magnetization. The regular versioncorresponds to a demagnetized (or "relaxed") magnet based on the theoretical dimensions and using identicalinitial magnetization amplitude throughout the magnet. The optimized version corresponds to a "relaxed"magnet based on the theoretical dimensions but using an optimized distribution of initial remanence.

1.26

1.26

1.35 1.35

1.33

1.33

1.33

1.331.33

1.33

1.33

1.33

Figure 3.16: a) Optimized magnetization distribution within the twelve-segmented cylindrical Halbach.

3.2.3 Conclusion

It is thus possible to design arbitrarily homogeneous finite-length Halbach-type magnets. It is alsopossible to compensate the demagnetization occurring in such structures by a suitable initial distri-bution of magnetization. A prototype based on this model is being built in a collaboration with thecompany RS2D and the ESPCI (Paris).

3.3 Conclusions on in situ magnets - Magic-Angle Magnets

We have shown the possibility of designing arbitrarily homogeneous magnets generating either a longi-tudinal field or a transverse field. Both configurations have their individual merits but an even greaterpotential resides in the possibility of combining both structures 3.17, which have both an axisymmetricgeometry. This o!ers the ability to control the field orientation with respect to the structure axis, whilemaintaining the homogeneity. The control of the orientation is achieved through the determinationof the ratio between the two components, the angle with the axis given by ' = arctan Bx

Bz. The com-

bination in practice can be done by interleaving rings of both types or by creating hybrid rings withalternated segments.

It is for example possible to generate a homogeneous field pointing at the magic angle, adaptedfor Magic Angle Turning experiments. Such experiments can provide a resolution enhancement in softanisotropic materials such as live tissues (162). We have realized a first prototype of such a structure,which is still under study. The design of the magnet combines the two structures by using alternatedsegments. This makes possible the use of standard cubes. The structure is equivalent to two Aubertrings and two Halbach rings (see figure 3.18). The magnet has already been adjusted to generate a

3.3. CONCLUSIONS ON IN SITU MAGNETS - MAGIC-ANGLE MAGNETS 103

field of 0.22 T pointing at the magic angle within 0.1º in its center, while close to being dominated bythe fourth degree. A view of the prototype can be seen in figure 3.19, along with the field variationson-axis and the angle of the field with respect to the structure axis.

B!!

Bx

Bz

Figure 3.17: Control of the orientation of the field with respect to the axis of the structure. Aubert (left) andHalbach (right) structures can be combined together to generate a field at an arbitrary angle with the axis ofthe resulting structure (center). This combination can be done by stacking rings of both structure types or bycreating hybrid rings with alternated segments of both structure types.

- !"

"

!"

- !"

"

!"

- !"

"

!"x (mm) z (mm)

y (mm)

Figure 3.18: View of the simple magnet structure used. The system is based on low quality magnet cubes(side of 1 cm) made out of NdFeB, grade N48. The magnetization is indicated with the black arrows. TheAubert and Halbach structures are alternated and optimized together to obtain the homogeneity of the fieldwhile controlling its orientation.


Figure 3.19: Left: Photography of the magnet prototype generating a field pointing at the magic angle. Right:Variations of the two components of the field, on-axis, along with the angle variations. It is visible that themagnet is close to being dominated by the fourth degree.

Finally, we can establish a table of performances for selected past prototypes and the ones proposedin this work, showing that the realized magnets compare well with the state of the art and that themagnet models presented go well beyond (see 3.4).

Reference Resolution (ppm) Mass (kg) Sample size (mm) Field strength (T)Moresi and Magin (163) 41 4 3 o.d. ( 1 0.6Raich and Blümler (74) 700 11.4 18 ( 18 ( 30 0.3Jachmann et al. (164) 56 5 0.25 o.d. ( 2 0.5

Demas et al. (165) 17 2 0.32 o.d. ( 1 2Manz et al. (166) 50 0.6 1.5 o.d. ( 1.8 1

McDowell et al. (63) 0.25 0.6 0.3 o.d. ( 0.3 1Danieli et al. (77) 0.85 50 10 o.d. ( 10 0.22

Aubert (100 !) 12 2 1 i.d. ( 2.5 0.12Magic (100 !) 50 2 1 i.d. ( 2.5 0.22

Advanced Aubert 1 50 20 DSV 0.9Halbach 10 10 10 DSV 0.6

Table 3.4: Performances of selected previously proposed prototypes, along with the systems shown in thiswork. The already fabricated prototypes compare well with the state of the art while the last two models, stillto be constructed, go well beyond from the point of view of homogeneity and field strength for the same amountof material.

Chapter 4

An open NMR/MRI system: The magnet

4.1 Single-sided magnets in theory

The design of a magnet is an iterative process and the final product is a compromise between fieldstrength, homogeneity, cost and practical feasability. Initial considerations provide a starting pointin terms of dimensions but as the process steps further, additional information and more detailedcalculations alter the initial idea or prove wrong some conclusions given at an early step. We shall herewalk through the important steps of the conception of axisymmetric single-sided magnets and see thisprogression. Such magnets pose the challenge of constraining the location of magnetic sources to halfof the space while the other half must remain free.

4.1.1 Preliminary considerations

Following the results of the previous chapter, we are going to adopt a fully axisymmetric structure(geometry and magnetization), which requires the field to be longitudinal in the region of interest (alongthe axis of symmetry). The achievement of homogeneity will result in a geometry that derives from asolid cylinder and this will hence be a good starting point for size and field strength considerations.

The region of interest is located inside the free area. Obviously, a large penetration depth isdesirable and one should place this region of interest as far as possible from the magnet, while keepingthe diameter and height of the magnet system to a transportable size (final weight < 40kg), with afield of reasonable strength (Bz % 100mT ). Targeted applications are spectroscopy and imaging.

The density of NdFeB (the material of choice selected in section 1.2.2) being of 7.8 g cm#3, thetotal volume of material that can meet the maximum weight specification is about 5130 cm3. Themaximum size of the magnet is hence set while we still have to define the shape factor (radius-to-heightratio). We shall only use one type of magnetization distribution for simplicity, and we shall choose thelongitudinal one, as it can tremendously simplify the practical magnetization of the structure. In thecase of imaging, we can imagine using such a system for small animals routine preclinical imaging. Amouse is about 4 cm across and hence, a region of interest, or sweet spot located 2 cm away from thesurface of the magnet seems appropriate.

4.1.1.1 Size and field strength

Figure 4.1 and equation 2.43 give us some insight on the optimum shape factor of the magnet interms of field strength. It is obvious that a cylinder longitudinally magnetized is not the optimumsystem, as it will extend in the orange and even red regions were it is not the most e"cient, or evencounter-productive. However, the choice of this magnetization distribution simplifies tremendously

105

106 CHAPTER 4. AN OPEN NMR/MRI SYSTEM: THE MAGNET

the assembly of the magnet, as it can be done unmagnetized, with simply shaped parts. The systemis also salvageable in case of accidental demagnetization as it can be re-magnetized once assembled.This is not the case for, say, a radial magnetization distribution. One can also argue that despite thelongitudinal magnetization is not the most e"cient one in orange areas, it still does contribute to thefield, it is only when entering the red areas that it will be counter-productive, if the magnetization isnot inverted. In fact, we can expect that there is a maximum radius for a given height above which thefield resulting from the magnet in O will decrease. Figure 4.1 illustrates the compromise to be donewhen choosing the form factor of the cylinder.

Bz O b

a

r0 Magnet cylinder

Figure 4.1: Schematic of the areas of highest e"ciency of the longitudinal and radial magnetization distribu-tions in the half space. The center of the RoI is O while the region of space where magnetic material is allowedcorresponds to b > r0. The space adopts a cylindrical symmetry around Oz and only the plane xOz is shown.The section of the magnet cylinder is overlayed in gray.

These simple considerations are only showing us that the definition of the radius-to height ratio ofthe cylinder is not trivial and requires some care.

The objective way of defining the form factor of the cylinder is to use relation 2.31 and the volumeconstraint to search for the height giving the maximum field strength, given the distance r0 fromthe origin to the cylinder. This calculation is simple and, for a volume of 5130 cm3, provides theoptimal radius ropt / 8.09253 cm and optimal height hopt / 24.9344 cm giving the maximum fieldBmax / 0.358892µ0M where M is the magnetization of the material.

In some cases, constraints on the dimensions may apply. It is thus interesting to study in additionthe evolution of the field strength for various combinations of radius and height. Another interestingfunction to consider is the e"ciency of the structure (longitudinal field induction generated per unit ofmaterial volume at the center of the RoI) , which gives an idea of the economics of the magnet to bebuilt. Figure 4.2 shows the variation of the field generated in O in function of the radius and heightof the magnet and the e"ciency of the structure in the meanwhile.

It is interesting to note that the structure looses very quickly its e"ciency per unit of volume sothat there is rapidly not much gain in adding material to the magnet. In addition, the optimal shapegiven previously lies in a region where the field strength varies slowly with both radius and height. Wecan also notice that, for a given height, there exists a maximum radius above which the generated fieldstrength decreases (this had been foreseen when we discussed figure 4.1). We can give the equation for

4.1. SINGLE-SIDED MAGNETS IN THEORY 107

either the minimum height hmin of a cylinder for a given radius a as

hmin =

3a2(*

1 + 4a2 ! 3) +*

1 + 4a2 ! 12

! 1, (4.1)

or for the maximum radius ropt for a given height h as

rmax =

GHHI!

21

" 13

%(h + 1)2 +

!1

2

" 23

&, (4.2)

with1 = (h + 1)2 + (h + 1)4 +

1(h + 1)4[(h + 1)2 ! 1]2.

In these equations, the distance from O to the surface of the cylinder is taken as the unit for allgeometrical quantities (i.e. hmin, h, a, and ropt).

!"#!#"$!

!"

#!#"

!"

!#

!$

!%

!&

!'

!(

!

%&'()*+,-./012'3&4+56'*78914'57+,-./012'3&4+56'*78

:;;'<'&6<=+>&/+56'*+.;+?.250

&+,@.(+.;+-

./012'3&4+56'*78

!"

#!#"

$!

!

"

#!

#"!

!A$

)*%

)*#

!AB

#

%&'()*+,-./012'3&4+56'*78914'57+,-./012'3&4+56'*78

C'&24+,-./012'3&4+56'*78

Figure 4.2: Left : plot of the field generated by a cylinder for diverse radii and heights. The field quicklystagnates as the volume increases and an optimal radius can be found for a given height. Right : plot of thee"ciency per unit volume of material for a cylinder of various radii and heights. The e"ciency falls very quickly(the logarithm of the e"ciency is shown). The normalized units of length are referenced to the distance betweenthe surface of the magnet and the center of the RoI.

A plot of the maximum radius can be found on figure 4.3. However, the cylinder shape providingthe maximum field for a given volume does not correspond to this maximum radius. When usingequation 4.2, the volume constraint (V = 5130 cm3) and the distance from the center of the RoI tothe magnet (2cm), we obtain the radius r / 9.85821 cm and the height h / 16.80241 cm that give afield 0.343412µ0M0, about 4.3% less than the optimum.


! " #! #" $! $" %!!

$

&

'

(

#!

#$

)*+,-./012/,3241567089+,:/.1;-,4<=

)*+,-./0109.,><1567089+,:/.1;-,4<=

Figure 4.3: Plot of the optimal radius in function of cylinder height. Height and radius are given in units ofthe distance from the origin to the cylinder. The optimal radius evolves slowly with the increase of the height.

We have hence given what is the optimal overall shape of our cylindrical magnet in terms of fieldstrength given our maximum weight. The loss in field strength is acceptable for reasonable alterationsof this optimum shape. It is now of importance to have a look at the e!ect of the form factor on thefield SHE and hence on the homogeneity of the field.

4.1.1.2 Size and homogeneity : from a cylinder to a tailored magnet

We will restrict the size of RoI of center O to 1 cm in diameter with a 10 ppm homogeneity, whichis enough to study small objects (the brain of a mouse, for example). Using relation 2.20, we canalready tell that the first non-zero term should be at degree nine or more. This estimation gives anorder of magnitude and does not take into account the coe"cients values due to the overall geometry.Equation 2.36 indeed tells us that the coe"cients scale as 1

cn , hence we can expect a severe dampingof the contribution of the higher degrees. We can use equation 2.36 to assess the value of the axialterms for a cylinder. The results for a cylinder of various dimensions with constant volume are shownin table 4.1.

Amplitude (ppm) Z1 Z2 Z3 Z4 Z5 Z6 Field Strength (µ0M)Max strength 76702 1647 102 7 0 0 0.358892

H=16.8 cm, R= 9.85 cm 60531 779 73 3 0 0 0.343412H=33.3 cm, R= 7cm 89171 2530 132 12 0 0 0.353089

Table 4.1: Amplitude of the axial terms of the field SHE on a sphere of radius 5 mm for various dimensionsof cylinder including the optimal one proposed in the preceding section. Degrees higher than three have smallenough contribution for the dimensions of highest field strength. While the field strength considerations recom-mend reducing the radius when reducing the height, it is detrimental to homogeneity as the contribution of alldegrees increase. In addition, at constant volume, the field strength does not vary much (less than 5% of themaximum field in what is shown) while the terms feature significant variations (20% to 50 % between line oneand line two).

It appears that terms of degree greater or equal to four are acceptable for the optimal dimensions(from field strength considerations), hence we would need only to cancel degrees one to three, requiring


four independent elements. However, as we will see, the cancellation of the lower degree terms mayresult in the increase of the higher degree terms (compared to the simple cylinder geometry) and wemay face the need of canceling more terms than we are anticipating from this consideration. Anotherconclusion arising from these results is that, as we could expect, the reduction of the radius of the cylin-der implies an overall increase of the contribution of the di!erent degrees while the field strength doesnot vary much from its maximum value. Hence, it might be desirable to use a cylinder radius greaterthan what the field strength considerations would recommend in order to reduce terms contributionsfrom the beginning.

Figure 4.4: Top : View of the concept of single-sided magnet. The system is constituted of a base (1) onwhich lie rings (2) of di!erent radii and positions. The RoI (3) in red resides at a fixed distance from thehighest magnet piece. The rings can be segmented into polygonal rings for fabrication purposes. All parts aremagnetized along Oz. Bottom : The definition of the number of "independent rings" is based on the numberof available variables. We need p + 1 variables to cancel p terms. The variables involved in the optimizationare the inner and outer radii of the magnetic material rings, so that we can count the air gaps as independentelements. We have five "independent" elements on the left and six on the right.

Once the dimensions of the magnet have been chosen, it is necessary to layout the di!erent inde-pendent elements. In order to maintain the symmetry and the field e"ciency, it seems appropriate touse concentric rings of alternate direction of magnetization (upward and downward), or, more simply,concentric rings of all same magnetization with rings of air (free space) in between (the rings might


be fully cylindrical or polygonal). The latter scheme is the easiest to assemble and magnetize in theend and we shall choose that one. A similar configuration is proposed in (167) but the calculationresults presented promise field strength below 0.05 T and limited homogeneity while using rather largemagnet rings (20 to 30 cm) and large distance between the magnet and the RoI (large R0, of the orderof 20 cm). We will try here to go further on the topic of homogeneity and field profile control, toachieve higher field strength with high homogeneity at shorter distances. To continue with the numberof rings to use, it is important to realize that the number of rings involved includes the air rings (whichcould also be rings of material magnetized in opposite direction if needed). Figure 4.4 clarifies thisaspect. Based on relation 2.36 or equation 2.39 and 2.33, it is then a matter of numerical optimizationto adjust the inner and outer radii of the di!erent rings in order to cancel the desired terms. In theprocess of this optimization, it appears that there is no need for the air gap in between the cylinders togo all the way down the magnet : we can set a large cylindrical base on which thinner rings can lie (thiscan be seen as a raw cylinder of magnetized material etched to a su"cient depth with an appropriatepattern of concentric rings). Furthermore, we have some freedom on the height and position of thedi!erent rings along the axis. All these considerations lead to a concept of single-sided magnet shownon Figure 4.4.

4.1.2 Theoretical design examples

We shall see here how the design evolves, based on the previous initial considerations, depending onthe homogeneity constraints. The analytical formulas provided in section 2.3 are used to define thegeometry of the magnet. The optimization is only done on the inner and outer radii of the di!erentrings, while the thickness of the di!erent elements and the outer radius of the magnet are adjustedmanually to make the optimization feasible and satisfy practical constraints. The precise modeling ofthe magnet is important as field variations are very quickly a!ected when we are located in the rangeof the ppm. Hence, all spacings required by mechanics should be included in the geometry, such as thethickness of a mount between two magnetic parts, or the thickness of the glue that must be used to fix inplace a magnetic part. Some of these spacings can be di"cult to model analytically and a good way toget round this di"culty is to go back and forth between the numerical and analytical simulations. Thenumerical simulation can be used to assess the e!ects of such alterations of the geometry and some partsthat are analytically modeled can be modified in order to balance these alterations. Hence, the design ofthe magnet is done based on both analytical and numerical calculations. All the structures we presentwere modeled in Radia to check the optimization results and analyze the e!ects of specific spacings.The field variation figures given in this section were obtained using Radia (151) and Mathematica.The 3D cartoons were generated with the free software Google SketchUp.


4.1.2.1 Homogeneous field

Following the previous considerations, we can go ahead and start numerical optimization in order tomaximize the field strength (Z0) while canceling axial terms Z1, Z2 and Z3. The cancellation of Z1

requires the removal of a significant amount of magnetic material close to the RoI, where it is moste"cient. As a result, the field strength is severely reduced and the final design achieves only about0.11µ0M0. The center of the RoI is located 18 mm above the upper surface of the magnet instead of20 mm in order to avoid reducing the field strength too much.

The cancellation of Z1 modifies considerably the geometry of the magnet, resulting in the increaseof higher degree terms, so that terms that were negligible in the previous discussion are not anymore.In fact, even the degree six features a contribution of about 30 ppm on a radius of 4 mm. This facthighlights the previous statement warning about the increase of higher degree terms when canceling thelower degree ones. Hence, while trying to maintain the geometry simple and achieving a homogeneitynot too far from the targeted one, we made a compromise by deciding to cancel all degrees up to degreefive (the first non-zero term is at degree 6). The resulting homogeneity is about 30 ppm in a sphere ofdiameter 1 cm. Two rings have been segmented with openings between the segments, in order to usethem later for imperfection compensation. The final model can be seen in figure 4.5, along with thevariations of the field in diverse directions and planes.

It appears from these results, that it would be necessary to cancel at least one more term (Z6, andprobably Z7 too) in order to achieve the initially targeted homogeneity. However, our desire to keepthe design simple for practical realization and cost-saving purpose discouraged us from going further.Nevertheless, would this model be satisfactory once built, subsequent systems (for commercial purposesfor example) will need at least an extra ring and a more involved and costly assembly system, whichcould be considered for series fabrication. The magnetic parameters are summarized in table 4.2.

Z0 (T) Z1 (T m#1)0.146 0

Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

Amplitude (ppm) 0 0 0 0 37 32 15 6

Table 4.2: Field, gradient values and contribution of the di!erent axial terms for the homogeneous field model,in a sphere of radius 5 mm. The remanence is set to 1.33 T.


Figure 4.5: a) View of the final model for a single-sided magnet generating a homogeneous field of about0.146 T with a remanence of 1.33 T, 18 mm away from its upper surface. The whole system is magnetizedvertically, according to the black and white arrows. b) Field variations in the xOz plane, the dominatingLegendre polynomial of degree 6 can be recognized. c) Overall field variations along Oz, the significant distancealong which the field is constant can be noticed. The field is given in units of µ0M d) Field variations along Ozover 1 cm around the center of the RoI. e) Field variations along Ox over 1 cm around the center of the RoIlocated 18 mm away from the magnet. All field variation plots were obtained using Radia to model the magnetafter the dimensions obtained from the optimization. Purple and blue parts are movable


4.1.2.2 Constant gradient

The creation of a homogeneous field required the removal of a significant quantity of material close tothe RoI, where the material is the most e"cient. As a result, the generated field is greatly reducedcompared to the maximum field attainable. Another type of magnet that can be useful to NMR, asdemonstrated by STRAFI experiments, is a constant-gradient magnet. We are going to see how thisrequirement modifies the design of the magnet.

We are here defining new specifications targeting the linearity and strength of the gradient, alongwith the field strength. Based on the results of previous section, we target to cancel degrees two to five.As Z1 does not need to be cancelled anymore, the final shape of the magnet is much closer to the oneof a cylinder. The final contribution of Z6 and Z7 are respectively only ±4 ppm and ±2 ppm over asphere of 1 cm diameter, which gives a gradient homogeneity of about 10 ppm in the specified volume.We have hence tremendously simplified the problem by removing the constraint of Z1 cancellation :the structure is simpler (only 5 independent elements) and we did not have to make any compromiseon the new specifications. The magnetic parameters are summarized in table 4.3.

Z0 (T) Z1 (T m#1)0.346 3.6

Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

Amplitude (ppm) 0 0 0 0 4 3 1 0

Table 4.3: Field, gradient values and contribution of the di!erent axial terms for the constant gradient modelin a sphere of radius 5 mm. The magnetization is set to 1.33 T.

The final model can be seen in figure 4.8, along with the field variations in diverse directionsand planes. Such a field profile may be of great interest for STRAFI imaging but also for di!usionmeasurements. The gradient produced here is smaller than the one encountered in the stray field ofa superconducting magnet but much more constant, over a very large volume. Its homogeneity couldbe compared to a MRI scanner gradient where the gradient strength is in general at least ten timessmaller. In addition, such a field distribution is advantageous for B0-B1 correlation experiment. Aswe will see later on, creating a correlated single-sided RF coil is equivalent to creating a single-sidedRF coil with constant gradient, which should not impair too much its sensitivity (as opposed to tryingto obtain a homogeneous single-sided RF coils, which results in a severe decrease in sensitivity). As aresult, this design seems very promising. Compared to the homogeneous design, it is much easier tofabricate, it produces a larger field and the gradient of Bz is homogeneous in a larger volume than isthe field in the other model.


Figure 4.6: a) View of the final model for a single-sided magnet generating a constant gradient field of about0.33 T and 3.3 T m!1 with a remanence of 1.33 T, 20 mm away from its upper surface. The whole structureis magnetized vertically, according to the black arrows. b) Bz deviation from a constant gradient in the xOzplane, the dominating Legendre polynomial of degree 6 can be recognized. c) Overall Bz variations along Oz,the significant distance along which the gradient is constant can be noticed (linear variation of the field). d)Bz deviation from a constant gradient along Oz over 1 cm around the center of the RoI. e) Bz variations alongOx over 1 cm around the center of the RoI located 20 mm away from the magnet. All Bz variation plots wereobtained using Radia to model the magnet after the dimensions obtained from the optimization. Green andblue parts in the magnet model are movable.


However, we have seen in section 2.5 that the presence of a strong gradient of the main componentof the field along Oz induces variations of the modulus of the field in ). It is not possible to cancel thedeviations from the ideal gradient along both dimensions (z and )). We have also seen in section 2.7.3.2that, in cylindrical coordinates (), &, z),

+Bz

+z= !1

2+B&

+). (4.3)

Thus, the significant gradient of Bz along Oz (about 3.3 T m#1) is accompanied by a rather stronggradient of Br in the xOy plane (about 1.65 T m#1), making the contribution to the modulus ofthe variations of Br no longer negligible in the volume of interest . This leads to a mostly quadraticvariation of the modulus as a function of r in cylindrical coordinates, with a maximum at the edge ofthe region of interest. As seen on figure 4.7, this maximum is about 350 ppm of the nominal field fora radius of 5 mm.

!"#

#"!

$!#$!# %##

%##

&'#

&'#

!"$'

!"$'

#"!

!#('

!#('

!#('

!$%$

!$%$

%##

)*+,,-

.*+,,-

!"# !$ " ( !"!"#

!%

!&

!'

!(

"

#

$

%

&

!"

!$##

"

(""

!"""

!(""

#"""

#(""

'"""

'(""

Dev

iatio

n fro

m id

eal g

radi

ent (

ppm

)

- ! - " " ! # $%%&'()*('*()"('"()+('+()

,-./0123 /140.3052 $66%&

- ! - " " ! 7 $%%&)'*''*)'"''")'+''+)'

8019/ /140.3052 $66%&

a)

b)

c)Figure 4.7: a) Contour plot of the deviation of the field modulus from the ideal gradient in the xOz plane.Contours are in ppm. The vertical contours indicate that the field gradient is very constant along Oz whileimportant field variations occur across that axis. b) Deviation of the modulus from ideal gradient along Oz inppm. c) Field modulus variations along Ox in ppm at z = 20 mm.

As seen in section 2.5, it can be interesting to use the theoretical Smn matrix to characterize the

variations of |B|2 (and thus, indirectly, those of |B|), rather than only looking at the SHE of Bz. TheSm

n matrix of this magnet configuration is


Smn = (µ0M)2

9::::::;

::::::<

n|m 0 2 40 0.0684014 0.0000479056 01 !0.00724151 0 02 !0.00704985 0 03 0 0 !6.6964( 10#8

4 0 !5.07014( 10#6 1.72467( 10#14

=::::::>

::::::?

. (4.4)

This shows clearly that, while many terms of the matrix are cancelled, the second degree term in ) ispresent and causes noticeable variations. The second degree in z appears naturally as we consider |B|2here.

As $Bz$z is substantially constant over the volume of interest (Bz varies by less than 10 ppm from

what would be predicted with a linear variation in z), $Br$r is substantially constant as well. Hence,

for a perfect magnet, surfaces of constant field values correspond to paraboloids of revolution parallelto each other in the RoI. Figure 4.8 shows the general aspect of these paraboloids. These paraboloidsfeature a curvature radius of about 30 µm, implying a 1D resolution of 30 µm over a 1cm x 1cm extentin the xOy plane, in a region extending over 1 cm along Oz.

Figure 4.8: Left: Surfaces of constant field in the RoI (1x1x1 cm) for equidistant field values. The surfacesappear as perfect planes because of the large extent along Oz. Right: Surfaces of constant field in a thin slice(1x1x0.02 cm). The curvature of about 30 µm due to "Br

"r is visible.

In addition, the knowledge of $Bz$z allows us to predict B0 everywhere in the RoI with an accuracy

similar to the deviation of Bz from its ideal gradient. In this case, we can predict B0 within 10 ppmof the nominal field. As the position is given by

z =f ! fObs

"Gz, (4.5)

the inhomogeneity-limited resolution !z is given by

!z =!B0

Gz. (4.6)

Thus, with the sole knowledge of B0 and $Bz$z at the center of the RoI, we can predict the elevation on

axis of the paraboloid corresponding to a given field value with a precision of ±0.5 µm. The curvature


of a paraboloid in the region of interest corresponds to 35 µm at the edge of the RoI. Consequently,in principle, we can achieve a 35 µm resolution for 1D profiling (this may be limited in practice byfactors inducing a faster damping of the FID, like T2, di!usion, etc). If we are provided in additionthe radial position, we can approximate the field by

B(r, z) =

3

[Bz(0, z0) + Gzz]2 +!

12Gzr

"2

, (4.7)

where Gz = $Bz$z

/z0

, with a maximum error of ± 1 µm for r - 5 mm. Consequently, in principle, wecan improve the resolution in z to 2 µm in the context of 3D imaging within the RoI.

It might be more interesting to achieve a constant field modulus for a given z in the RoI, whileaccepting variations of the gradient along Oz, yielding perfectly plane surfaces of constant field in theRoI. We have seen in section 2.5 that this can be achieved by setting the relevant Zn terms to theproper value, which depend on Z0 and Z1. A magnet achieving these constraints is not very di!erentfrom the magnet shown above. It can be achieved either by modifying the inner and outer radii ofthe top rings, or by shifting them radially and vertically. We calculated the appropriate dimensions ofthe upper rings, so that we can obtain this profile with a magnet of same diameter and height as theprevious one. The resulting theoretical Sm

n matrix is the following,

Smn = (µ0M)2

9::::::;

::::::<

n|m 0 2 40 0.0756757 0 01 !0.00937034 0 02 !0.00908028 3.50001( 10#8 !4.25478( 10#12

3 !8.99106( 10#6 1.31467( 10#7 !4.92443( 10#8

4 5.83549( 10#8 !3.18674( 10#6 5.55764( 10#15

=::::::>

::::::?

. (4.8)

These values indicate the very limited dependency on ), even in cross-terms, while the deviation fromthe ideal profile along Oz has been largely increased. However, in the context of imaging, toleranceson the gradient variation are fairly loose and gradient homogeneity of about 1% are standard in MRI(gradient homogeneity is defined here as Max

(B0#Gzz

B0(0)

)).

The performances of this magnet, in terms of field modulus are shown in figure 4.9. It appears itis possible to achieve micron resolution or better in a cube of 1 cm x 1 cm x 1 cm, as it can be seenin figure 4.10. This can be achieved with minor distortion (the gradient homogeneity is better than0.1 %). The same resolution can be achieved over cylindrical slices of diameter 2 cm x 2 cm with ashallower depth (about 2 mm).


!"!#

!"!#

!"!$

!"!$!"%!

!"%!

!"%#

!"%#

!"%$

!"%$

!"&!

!"&!

!"&#

!"&#

!"&$

!"&$

!"#!

!"#!!"##

!"##

!"#$

!"#$!"## !"##

'()**+

,()**+

!"# !$ ! $ %!!"#

!%

!&

!'

!(

!

&

'

&

%

%!

!

#)#$

!"%

#)"$

!"&

#)($

!"#

#)*$

#)'

#)'$

#)$

Dev

iatio

n fro

m id

eal g

radi

ent (

%)

- !" - # # !" $ %& ''"(!"()"(*"(+

,-./%0&12/03%.1%4&2567

- !" - # # !" 8 %& ''

- !+- !)- !"

- 9- :- +- )

;%0</ /03%.1%4& 5=='7

a)

b)

c)Figure 4.9: a) Contour plot of the deviation of the field modulus from the ideal gradient in the xOz plane.Contours are in percent. The very flat contours indicate the possibility of very high spatial resolution in Z,while the small deviation (< 0.1% over 1 cm) from ideal gradient promises low distortion over a wide range. b)Field modulus deviation from ideal gradient along Oz in percent. c) Field modulus variations along Ox in ppm.

z z

Figure 4.10: Left: Surfaces of constant field (equidistant values) at a height of 16 mm over a slice 0.2 mmdeep. Right: Surfaces of constant field (equidistant values) at a height of 26 mm over a slice 0.2 mm deep. Thecurvature of each surface is about or less than 1 µm in a volume 1 cm x 1 cm x 1 cm.


4.1.3 Imperfections

All the above considerations are based on the assumption that all magnetic parts are magnetized in ahomogeneous, rigid and identical way. Unfortunately, it is not the case when it comes to fabricatingthese parts. The very involved fabrication process does not guarantee a repeatability of the order ofthe ppm, rather of the order of the percent. We should expect magnetization amplitude variationsfrom one part to another of about 1%, but also an orientation variation of about 1(. Furthermore,a magnetic part of arbitrary shape is subjected to demagnetizing fields, due to the finite nature ofits geometry, while its magnetization features a relative susceptibility of the order of 0.07 (it is notinfinitely rigid) (34). The demagnetizing fields are not constant throughout the part and hence set adi!erent working point in di!erent locations of the part, resulting in an inhomogeneous magnetization.Neighboring parts also induce inhomogeneous demagnetizing fields which a!ect the local working point.In addition to this, the fabrication process itself introduces inhomogeneities of remanence within a part,which adds up with the demagnetizing field e!ect. To summarize, the actual magnetic parts are farfrom being perfect. In addition to these, we should also consider the e!ects of mechanical tolerances :the actual dimensions of the pieces and their positioning cannot be achieved to an arbitrary precisionand this will thus a!ect the final magnetic performances.

The e!ect of these variations can be estimated with numerical simulations, using finite elementsmethod (FEM), the method of moments (MoM) or boundary elements method (BEM). The inhomo-geneities and variations due to fabrication can be rendered through distributions of remanence eitherrandom or following some particular scheme, while the e!ect of demagnetizing fields can be computedthrough an iterative process using the field computation and the hysteresis curve of the material. Forthis purpose, equations such as relations A.16, A.17, A.18 and A.19 can prove very useful to facili-tate calculations. Several software packages are available and one particularly suitable for the typesof geometries we are considering is the one developed by the ESRF, called Radia (151; 168), whichbelongs to the BIM family. The BIM method allows the use of analytical formulas, yielding machine-precision results for perfect structure (homogeneous magnetization). All elements can be subdividedand be assigned a material behavior based on equations similar to the ones given in section A.0.3 andfree space does not need to be meshed (as it should be in a FEM calculation). The working point ofeach sub-element can then be found through an iterative process called "Relaxation". The resultingcalculated field in one point is precise (compared to experiment) to better than 2%, mostly due tounknowns in the material behavior. Hence, we can expect such simulations to give a good estimationof the order of magnitude of material imperfection e!ects on the field homogeneity. The e!ect ofmechanical tolerances can also be assessed using such numerical computation, but also simply basedon the analytical formulae used to design the magnet.

We will see in the example of the constant gradient magnet that these material imperfectionsand mechanical tolerances can be seen as perturbations of the perfect theoretical model, so that thismodel is an excellent starting point for a structure. That structure will need eventually some minoradjustments to obtain the targeted homogeneity. An e"cient way of adjusting the homogeneity ofthe magnet is to alter its geometry through displacements of some parts, as already demonstrated bythe team of Prof. Blümich (109; 77). Another way of compensating the imperfections is to add smallmagnetic parts which represent minor modifications of the structure but can compensate the minorimperfections (154–160; 169–171). It seems wise to anticipate a combination of both methods in orderto achieve our ambitious goal. As we have seen that the variations of B0 are governed by the variationsof Bz, most of this analysis is focused on Bz and its SHE.

In the case of our magnet concept, we plan to alter the geometry of the di!erent rings lying on topof the base, in order to apply the displacement technique. As a result, we employed the polygonal ringgeometry for these ring, using an angular aperture &seg < !

N , leaving free space between the segments


so that they can be held and moved around. We chose N = 12 in order to guarantee a high orderof axial symmetry while providing a number of variables for adjustments. This 12-fold symmetry hasbeen used extensively for all our magnet designs as the manufacturer we worked with had tools fordodecagons assembly.

The method of adding small parts does not require much preparation as we can always add astructure that holds the shimming magnets. A theoretical study based on simulations is howeverrequired to ensure that these adjustments provide enough degrees of freedom with enough amplitudeso that we can compensate all imperfections. The following section is going to show some examples oftheoretical magnet structures and evaluate the problem of imperfection compensation for one of them.

As we have seen that all variations of the field can be accounted for by the variations of Bz and itsSHE, we will concentrate on the impact of fabrication errors on the expansion of Bz.

4.1.3.1 Demagnetization

Demagnetizing field: analysis of a simple geometry The calculation of the working pointeverywhere throughout the magnet (called "Relaxation" in Radia) relies on the hysteresis data andthe calculation of the demagnetizing field. The latter can be done analytically or numerically. In theend, it is necessary to subdivide the magnet in sub-volumes of supposed homogeneous magnetizationin order to compute the magnetization in multiple locations of the magnet, leading to a numericalapproach for the whole process of "Relaxation". It is however instructive to bear in mind the analyticalbehavior of a simple geometry.

We consider here a cylindrical magnet of radius a centered on Oz, and delimited by two planesparallel to xOy at elevation z = b1 and z = b2. We will use cylindrical coordinates (), &, z) andalready note that all results are independent of & given the symmetry of the problem. In addition,due to the cylindrical symmetry, the vector potential

!"A has only one non-zero component, A% and

the field has only two non-zero component, B& and Bz. The magnetization is homogeneous and rigid,hence, the cylinder can be considered equivalent to a cylindrical sheet of radius a carrying an azimuthaldistribution of surface current density K% = M . The material is anisotropically magnetized along Ozwith a high anisotropy field, so that we can consider only the z component of the field. The expressionfor the vector potential in an arbitrary point of space for such a cylinder can be written as (29)

A% =µ0M

2$

.[G(), z; a, b)]a2

a1

/b2b1

(4.9)

G(), z; a, b) =2a(z ! b)

r1

*1k2

4E(k2)!K(k2)

5+ (

1m2

! 1)4#(m2, k2)!K(k2)

5+(4.10)

k2 =4a)

r1(4.11)

m2 =4a)

(a + ))2(4.12)

r1 =1

(a + ))2 + (z ! b)2 (4.13)

where K(m), E(m) and #(., m) are respectively the complete elliptic integral of the first, second andthird kind, as defined in (116). The magnetic field components can be deduced from these expressionsusing B& = !$A!

$z and Bz = 1&

$(&A!)$& . The use of the appropriate relationships between the complete

elliptic integrals and their derivatives helps simplifying the resulting expressions. Using!"B = µ0(

!"H +


!"M), we can write the following expressions for the demagnetizing field,

H&(), z) = M

J

K 12$)

%*a2 + r2 + (b! z)2

r1K(k2)! r1E(k2)

+a2

a1

&b2

b1

! 1

L

M (4.14)

Hz(), z) = M

J

K 12$

%*b! z

r1

*K(k2) +

a! )

a + )#(m2, k2)

++a2

a1

&b2

b1

! 1

L

M . (4.15)

H& diverges when ) = a and z = b. This arises directly from the fact that the magnetizationis assumed completely homogeneous in the volume of the magnet. This assumption is thus limitedin its physical relevance. The demagnetizing field variations of a cylinder centered on the origin ofdimensions comparable to the constant gradient magnet can be seen in figure 4.11. We have a = 10 cmand b = 13 cm. The maximum demagnetizing field along Oz is hence !0.593M , occurring at ) = 0and z = ±b and the minimum demagnetizing field occurs at ) = a and z = 0 with a value of !0.234M .We thus need to select a material that has a coercivity significantly larger than these values, so thatwe can consider the working point to be in the linear region of the demagnetization curve. The )component of the demagnetizing field is weak everywhere in the magnet, well below the anisotropyfield. This component diverges strongly in the corners where () = a, z = ±b). The perpendicularpermeability being small, we can expect the magnetization to remain vertical in most of the magnet,save the corners where a tilt might be noticeable due to the diverging H&.

! (cm)

Z (c

m)

! (cm)

Z (c

m)

a) b)

Figure 4.11: a) Contour plot of !Hz in the xOz plane, in multiples of the magnetization M . Hz opposes themagnetization everywhere in the magnet. b) Contour plot of H# in the xOz plane. H# is weak everywhere inthe magnet, except in the corners where it diverges.

E"ect of demagnetization on the constant-gradient magnet Demagnetization is often dis-carded as being very small due to the rigidity of the materials used. This is true in many applicationsof permanent magnets but the precision required by NMR makes it sensitive to the small variationsof magnetization occurring in the magnet. We discuss here the influence of demagnetization alone inorder to convince the reader of its importance.


As we start with a perfect magnet, the resulting demagnetized magnet must feature the samesymmetry, hence all skewed terms remain null. The expected distribution of magnetization in a magnetmade of a material identical everywhere is shown in figure 4.12. The remanence is 1 T vertically, thelongitudinal di!erential susceptibility is 0.07 and the transverse susceptibility is 0.17 (34). Artifactsof calculation can be noticed in the larger segments of the upper parts of the magnet. The "wrong"sub-volumes are rare and diluted in the number of "correct" sub-volumes, so that the end-resultstill provides an accurate estimate of the impact of demagnetization on the magnet. The maximumcomputed demagnetizing field is about 0.63 T, as could be expected from the geometry while themaximum vertical magnetization is 0.996 T and the minimum vertical magnetization is 0.956 T, thatis to say a variation of about 4% throughout the whole magnet. This is by no mean a small contribution.The magnitude of the e!ect of demagnetization on homogeneity is shown in figure 4.13 and we can seethat the most a!ected term is Z2 with a contribution about 150 ppm on a radius of 5 mm.

a) b) c)Figure 4.12: a) Overview of the magnetization distribution in the magnet model after relaxation. Yellowindicates stronger magnetization and black weaker magnetization. b) View of the magnetization distribution ina base slice. The colors are scaled (yellow indicates strongest magnetization within the slice and black weakestmagnetization). The magnetization variation follows the demagnetizing field variations found in section 4.1.3.1.c) View of the distribution within a segment of the upper part (one segment of each ring). The larger segmentfeatures a few calculation artifacts.

!"#$%

!"&$%

!"'$%

!"$$%

!($%

!#$%

!&$%

!'$%

$%

'$%

)'% )*% )&% )+% )#% ),% )(%

!"#$%&'()"#*+,,-.*

Figure 4.13: Contribution of the axial field SHE terms on a radius of 5 mm after relaxation of the perfectmagnet model.


4.1.3.2 Definition of imperfection models

Based on the numerical simulation code Radia, we can proceed to simulate imperfections in the magnetdue to geometry or magnetic properties. We will describe here the models used to assess the e!ects ofmagnetization distribution and geometry imperfection.

Magnetic properties variations We consider the possible imperfection in magnetization orienta-tion and amplitude for a part, and the possible inhomogeneities of these quantities within a piece. Wehence have a nominal magnetization vector with spherical coordinates (M0, '0, &0). The deviation fromthis ideal magnetization of the overall magnetization of a given part is given by (!M,!', !&), and thedeviation at a location within a part from the overall magnetization of the part is (2M, 2', 2&). For thepurpose of numerical calculation, we have to discretize the magnetization distribution. We hence havesub-volumes of constant magnetization with magnetization varying from sub-volume to sub-volume.We define the magnetization of a sub-element i from a magnet part p with its spherical coordinates(Mp,i, 'p,i, &p,i) with

Mp,i = M0 + !Mp + 2Mi (4.16)'p,i = '0 + !'p + 2'i (4.17)&p,i = M0 + !&p + 2&i. (4.18)

The spatial distribution of the di!erent deviations remains now to be defined.The description of the imperfections of a set of parts is a complex matter as part of the imperfections

feature some degree of correlation or some structure as they are due to biases in the manufacturingequipment. For example, the coil producing the magnetic field aligning the domains at the time ofsintering is bound to feature some inhomogeneities which will a!ect the local orientation of the easyaxis. Hence, for parts coming from the same location in that coil, the magnetization distributionwill be correlated, and a coherent inhomogeneity copying the field variations within each part is alsoinduced. However, the region of the coil a part comes from is not very well known after fabrication andwe hence lack the information in order to model such imperfection schemes. Generally, most of thesystematic imperfection schemes due to fabrication tools tend to be blurred by the random location ofthe final part in the initial bulk material produced. In addition, some imperfection factors are randomand isotropic, such as the e!ect of impurities contained in the initial powder.

To summarize, in practice, it is di"cult to determine the coherent non-random part of the im-perfections (unless a strong measurable bias exists) and we face a phenomenon that appears mostlyrandom and, to some extent, isotropic. Hence, we will base our imperfection assessment on randomdistributions (e.g. gaussian or uniform) of deviations from the nominal quantities, using the tolerancespecifications as boundaries or variance for these distributions.

One possible coherent imperfection may occur due to the process of fabrication. The di!erentparts are to be cut out of a larger piece of material dye-pressed from an initial batch. If di!erentbatches are necessary to produce the parts, there will be di!erent groups of parts with di!erent averagemagnetization, introducing a bias. As we will see, this can be especially harmful if those groups arethe di!erent rings.


Geometrical imperfections Geometrical imperfections refer to all imperfections related to theshape of each part and to the positioning of that part relative to the others. We need hence to settwo reference frames, one attached to the part itself, in order to define its shape imperfections, and alaboratory one which serves as the reference for the magnet assembly. It is in this latter frame thatthe positioning imperfections of each part are defined.

The characterization of the shape imperfections can adopt arbitrary degrees of precision and dif-ferent types of description. The fabrication of a part is made after drawings which give tolerances onheights, widths, angles, planarity, orthogonality, surface roughness, etc. Most of these tolerances giveonly a maximum deviation from a nominal value and do not give the complete distribution of dimen-sions (e.g. a supposedly planar surface may exhibit di!erent curvatures depending on the location onthe surface), as it would require way too many measurements. In the same fashion, we are limitedby the magnetic calculation tool as the geometry must remain simple enough to be calculated in areasonable amount of time. As a result, we are to model only imperfections of the parameters thatwere used in the optimization of the magnet (i.e. inner and outer radii, angular aperture, and height).The e!ects of variations of these parameters will dictate the fabrication tolerances on these quantities.We will be left with parallelism, planarity and surface roughness which should be defined so that themain parameters we just discussed do not deviate from nominal across the surface by more than theoverall tolerance on that parameter.

The positioning of each part can be summed up by six variables : the space coordinates of thecenter of the part and three angle of rotations around the three axes of the reference frame.

The variability of dimensions in the fabrication will be also described by uniform random distribu-tions for each variable (shape and positioning) with boundaries given by the tolerances.

In addition to the individual imperfections, it is necessary to take into account the possible me-chanical imperfection of the mounts, which may result in the change of elevation of a whole ring. Thisresults in a coherent error on the altitude of each part of a ring.

4.1.3.3 E"ects of imperfections

Once we have defined the imperfection models, we can use them to compute the performances ofnumerous randomly altered systems. The results can then be used to build some statistics on theconsequences for the field properties. We consider in what follows a magnet with a nominal remanenceof 1 T. The results would not be changed for a di!erent remanence, as all results are given in partsper million of the main field (proportional to the remanence).

Magnetization distributions We consider here the e!ect of magnetization imperfections from oneblock to another based on the tolerances provided by the manufacturer, that is to say 1% remanenceand 1º in orientation (1 on ' and 2$ radians on &, in our model). The input distribution is uniformwithin these bounds. Given the relatively large number of independent random variables, the resultingdistributions of deviation for the di!erent field SHE terms should approach a Gaussian distribution,centered around the theoretical value of each term. This can be used as a verification of the correctnessof the calculations and also to check the set of simulated magnets is large enough to assess the standarddeviation. The standard deviation depends on the tolerances and on the term studied. Given thesymmetry of the magnet, Xm

n terms and Y mn terms have the same standard deviation for a given

degree and a given order.We give in table 4.4 the standard deviation of dominant field SHE terms for two magnetization

tolerances. The tighter tolerance is typical of high quality, high cost parts while the second toleranceis typical of low-quality, low-cost parts. These estimations indicate it is necessary to invest in goodquality pieces in order to hope ending near the targeted performance. In addition, the individual


random imperfections seem to a!ect only lower degree terms (mostly degree one and two) while anaverage variation from one ring to another results in a stronger increase of higher degrees (three andfour), as seen in table 4.5. As a result, the di!erent rings should not be manufactured from singlematerial batches (unless all of them can be fabricated from the same batch), in order to reduce thisbias.

Z̄0 (T) !Z0 (T) Z̄1 (T m#1) !Z1(T m#1)Individual 1 % 1( 0.2615 0.0009 2.769 0.016Individual 5 % 5( 0.2611 0.0043 2.76 0.078

Amplitude (ppm) !Z2 !Z3 !Z4 !Z5 !Z6 !Z7 !Z8

Individual 1 % 1( 48.5 6.1 1.1 0.3 0.1 0.0 0.0Individual 5 % 5( 237.4 29.9 5.5 1.3 0.3 0.1 0.0

Amplitude (ppm) !X11 !X1

2 !X22 !X1

3 !X23 !X3

3 !X144 !X2

4 !X34 !X4

4

Individual 1 % 1( 394.0 61.2 33.9 8.5 6.9 2.8 1.6 1.4 0.8 0.3Individual 5 % 5( 1967.7 306.0 168.3 42.8 34.0 14.0 7.7 6.8 4.0 1.4

Table 4.4: Magnetization distribution. Standard deviation of the contribution of significant field SHE termsin a sphere of radius 5 mm for a set of 5000 simulated magnets. Two distributions are shown with 1% or 5%maximum amplitude variation and 1º or 5º maximum angular deviation. The average magnetization is 1 Tvertical. The inhomogeneous magnetization seems to a!ect all degrees in the same way (higher degrees do notincrease unexpectedly). Skewed terms are the most a!ected by individual parts errors, as it could be expected.Given the symmetry of the magnet, the standard deviations on the Xm

n terms are the same as for the Y mn terms.

Overall error on a ring does not a!ect skewed terms due to the symmetry. Ring errors are hence not shown inthe skewed terms table.

Z̄0 (T) !Z0 (T) Z̄1 (T m#1) !Z1(T m#1)Ring 1 % 1( 0.2615 0.0007 2.769 0.01Ring 5 % 5( 0.2611 0.003 2.76 0.05


Ring 1 % 1( 33.9 6.7 2.25 0.5 0.1 0.0 0.0Ring 5 % 5( 168.0 33.8 11.6 2.6 0.4 0.0 0.0


2 !X22 !X1

3 !X23 !X3

3 !X144 !X2

4 !X34 !X4

4

Ring 1 % 1( 140.6 29.7 0.0 6.3 0.0 0.0 2.3 0.0 0.8 0.0Ring 5 % 5( 659.1 137.3 0.0 30.5 0.0 0.0 11.1 0.0 0.0 0.0

Table 4.5: Magnetization distribution from one ring to another. Standard deviation of the contribution ofsignificant field SHE terms in a sphere of radius 5 mm for a set of 5000 simulated magnets. Two distributionsare shown with 1% or 5% maximum amplitude variation and 1º or 5º maximum angular deviation. The averagemagnetization is 1 T vertical. Magnetization error on a whole ring has a greater e!ect on higher degree axialterms (i.e. Z3, Z4, and Z5) than individual errors. Skewed terms are also a!ected due to the non-symmetricnature of a tilt of the magnetization on a whole ring. This e!ect is however lesser than for individual errors.


Geometry distributions In the same fashion as for the magnetization distribution, we used auniform distribution of deviations for inner radius, outer radius, height, location in 3D space, angularaperture of segments, and rotation around an axis parallel to Oz going through the center of each part.Again, given the large number of independent random variables, the final distribution of deviationfor each field SHE term should approach a Gaussian distribution centered on the theoretical valueof each term. This is again a verification element for the number of samples and the correctnessof the calculations. Histograms similar to the ones obtained for the magnetization imperfection canbe plotted but are not shown here as they do not bring further information. We will separate thetolerances related to the shape of the parts, fixed by fabrication, from the tolerances related to theirposition, which are set during the assembly and are correctable.

The e!ect of shape tolerance (namely inner radius, outer radius, and height) is shown in table 4.6for two standard fabrication tolerances (± 50 µm and ± 100 µm). It appears that the system is lesssensitive to shape tolerances than to magnetic tolerances (probably because mechanical tolerance aretighter than magnetic tolerances). It is still necessary to use the best fabrication tolerance achievableas the deviations arising for ±100 µm are not acceptable (taking a 3# confidence interval gives morethan 600 ppm peak to peak). However, these individual tolerances (on each part) result in the increaseof lower degrees only (mostly degree one, two and a little bit three)

The e!ect of positioning tolerances in conjunction with a fabrication tolerance of ± 50 µm is shownin table 4.7. While a positioning tolerance similar to the fabrication one does not change significantlythe deviations occurring in the di!erent terms, the ± 200 µm, ± 2( tolerance has a much greatere!ect, inducing deviations greater than for a fabrication tolerance of ± 100 µm perfectly positioned.It should be noted that the ± 200 µm and ±2( tolerance is the most likely to take place. Indeed, as wedesire some parts to be movable, the initial uncertainty on their positioning is larger than fabricationtolerances. However, this uncertainty is only related to the initial positioning, which can be refinedto find a proper configuration, as we will see in section 4.1.4. These individual imperfections a!ectmostly lower degrees (mostly degree one and two, and a little three). Global positioning in z of a ringis however more harmful to higher degrees.

As we said, it is also necessary to take into account vertical positioning errors of an entire ring.We can also carry a sensitivity calculation to random variations of elevation of the di!erent rings.Its results are shown in table 4.7 for tolerances of ± 50 µm and ± 100 µ (these are the same as forfabrication tolerances as a wrong vertical positioning of a whole ring is to be due to the mechanicalstructure). The e!ect of such errors is rather strong and these results advocate even more for tightestfabrication tolerances.


Z̄0 (T) !Z0(T ) Z̄1 (T m#1) !Z1(T m#1)±50µm 0.2615 0.0001 2.768 0.003±100µm 0.2615 0.0001 2.768 0.008


±50µm 17.1 5.5 1.6 0.4 0.1 0.0 0.0±100µm 38.3 11.5 3.4 0.8 0.1 0.0 0.0


2 !X22 !X1

3 !X23 !X3

3 !X144 !X2

4 !X34 !X4

4

±50µm 69.3 21.5 12.0 5.0 4.9 1.8 1.2 1.4 0.9 0.3±100µm 156.3 48.8 27.2 11.3 10.9 4.2 2.3 3.1 2.1 0.6

Table 4.6: Shape distribution. Standard deviation of the contribution of the di!erent field SHE terms in asphere of radius 5 mm for a set of 5000 simulated magnets. Two distributions are shown with ± 50 µm or ±100 µm tolerances on inner radius, outer radius and height of each part. The average magnetization is 1 Tvertical. Skewed terms are the most a!ected and, given the symmetry of the magnet, the standard deviationson the Xm

n terms are the same as for the Y mn terms.

Z̄0 (T) !Z0 (T) Z̄1 (T m#1) !Z1(T m#1)Individual ±50µm 0.2615 0.0001 2.769 0.005Individual ±200µm 0.2615 0.0004 2.769 0.014

Ring ±50µm 0.2615 0.00005 2.769 0.004Ring ±100µm 0.2615 0.0001 2.769 0.008


Individual ±50µm 31.8 10.0 2.8 0.7 0.1 0.0 0.0Individual ±200µm 108.5 33.9 9.3 2.3 0.5 0.1 0.0

Ring ±50µm 20.3 9.2 2.6 0.4 0.01 0.0 0.0Ring ±100µm 40.5 18.2 5.2 0.9 0.01 0.1 0.05


2 !X22 !X1

3 !X23 !X3

3 !X144 !X2

4 !X34 !X4

4

±50µm 82.0 29.4 14.5 8.6 5.9 2.5 2.3 1.8 1.2 0.4±200µm 190.8 85.2 34.3 6.9 29.0 13.3 3.2 1.2 8.2 4.3

Table 4.7: Position distribution. Standard deviation of the contribution of the di!erent field SHE terms in asphere of radius 5 mm for a set of 5000 simulated magnets. Two distributions are shown for individual partswith ± 50 µm, 0.5" or ± 200 µm, ±2" tolerances on 3D positioning and rotation around an axis parallel to Ozpassing through the center of the part. The results for vertical position error on the rings are also shown for twotolerances. The average magnetization is 1 T vertical. Skewed terms are the terms most a!ected by individualerrors, while axial terms are equally a!ected by individual errors or by ring position error.


All contributions combined Based on the previous results, we decide to use the best commercialmagnetic tolerance available (± 0.5% and ± 0.5º), along with tight fabrication tolerances (± 50 µm)and expected looser tolerances on radial positioning and Oz rotation of each part. Vertical positioningof rings is also taken into account. We show in table 4.8 the standard deviation for relevant terms whenall errors are combined. We show for comparison in table 4.9 the expected contribution average andstandard deviation of each term for a simple cylinder of similar size made out of twelve segments withsimilar tolerances. It appears clearly that the analytical design makes a di!erence, despite the fabri-cation imperfections. These imperfections induce indeed deviations that are smaller than the intrinsicinhomogeneity of a solid cylinder, despite the optimized structure is more sensitive to imperfections(larger standard deviation on axial terms) than the cylinder. The cylinder is however more sensitivein terms of transverse gradients (X1

1 and Y 11 ), most likely because of the smaller number of parts: the

larger number of parts in the optimized magnet may average out some of the imperfections.

Z̄0 (T) !Z0 (T) Z̄1 (T m#1) !Z1(T m#1)Value 0.2615 0.001 2.769 0.03

!Z2 !Z3 !Z4 !Z5 !Z6 !Z7 !Z8

Amplitude (ppm) 126.0 36.7 10.2 2.5 0.5 0.1 0.0

!X11 !X1

2 !X22 !X1

3 !X23 !X3

3 !X144 !X2

4 !X34 !X4

4

Amplitude (ppm) 454.2 105.5 49.9 30.1 15.2 7.6 8.4 4.6 3.3 1.2

Table 4.8: All errors together. Standard deviation of the contribution of the di!erent field SHE terms in asphere of radius 5 mm for a set of 5000 simulated magnets.

Z̄0 (T) !Z0 (T) Z̄1 (T m#1) !Z1(T m#1)Amplitude (ppm) 0.3187 0.001 3.98 0.03


Intrinsic 682.3 88.3 !3.5 -0.1 0.0 0.0 0.0Standard Deviation 101.6.0 17.1 3.1 0.6 0.1 0.0 0.0


2 !X22 !X1

3 !X23 !X3

3 !X144 !X2

4 !X34 !X4

4

Standard Deviation 710.3 156.7 90.7 30.7 26.8 11.3 6.4 6.7 4.2 1.4

Table 4.9: E!ect on the SHE terms of all errors together, for a cylinder of comparable dimensions made outof twelve joining segments.

We are thus to expect significant imperfections leading to inhomogeneity of the field profile, whichshould still be much better than for a simple magnet cylinder. We have chosen to keep some parts of themagnet free to move in order to adjust the geometry and compensate these errors. This decision resultsin larger tolerances on the position of these parts and thus degrades the initial profile. Improving thesetolerances is di"cult as the mechanics involved in the motion of the parts are subjected to importantforces and cannot use much space to hold the parts. For example, the larger segments undergo forces of120 N towards the magnet base, making the sliding operation di"cult, while longitudinal and transverseforces can reach a few tens of newtons depending on the displacements. In addition, there are serious


practical concerns for the final clamping of the parts, once adjusting is over (perfect machining of thesole of each magnet mount is here a key to success). It is thus a real question wether or not it is abetter strategy to fabricate the whole magnet with factory tolerances and to only shim with additionalsmall parts, as seen in section 3.1.1.

4.1.4 Corrections

The preceding section has demonstrated that the imperfections bound to occur during the fabricationof a magnet impair the magnetic performances in an unacceptable way. It is hence a requirement toadjust the magnet in order to compensate these imperfections. The analytical design provides an initialstructure that is very close to the desired structure (the one providing the desired performances) andhence facilitates its adjustment. We present in this section the theoretical details of the method weare going to use to compensate the fabricated magnets by moving parts. We will rely for this purposeon the example of the constant-gradient magnet.

4.1.4.1 General description

The target of the compensation process is to cancel the field SHE terms due to the magnet imperfectionswhile we intended to zero them. As we have seen in section 2.7.2.1, we can obtain the SHE terms of thefield generated by an arbitrary magnet, provided we can measure (or compute) the field in a su"cientnumber of points. We can build a compensation approach based on these measurements, similarly tothe one introduced by Perlo et al and Danieli et al (109; 77).

We have left space around some of the ring segments in order to move them. The e!ect of the motionof these parts can be assessed in terms of field SHE terms through successive measurements before andafter an elementary motion. Assuming that the starting point is close enough to the optimum, thesystem can be considered as behaving linearly over the range of correction and a correction matrix canbe built. We have then an equation of the following type to solve

MCompVMotion = !VSHE0 , (4.19)

where VSHE0 = [Z1, ...Zi, ...Zn, X11 , ...Xp

i , ...Xmn , Y 1

1 , ...Y pi , ...Y m

n ] includes the field SHE terms for theinitial configuration, MComp is the correction matrix that includes the e!ect on each field SHE termof each elementary motion, and VMotion is the vector of required motion for each parts, in terms ofelementary motion. The solution of this equation is found through linear least-square solving. Therewill be a reasonable solution to that equation only if MComp has rank greater than the number of termsto be corrected and the target variation of the SHE terms !VSHE0 is in the range of the correctionmatrix.

It is important to estimate the sensitivity of the correction system to measurement errors. Theseerrors are involved at two levels: the initial field characterization (VSHE0) and the construction of thecorrection matrix. As the least-square inversion provides a set of displacements that cancels the initialset of variation contributions, we can expect that the residual contributions are of the order of themeasurement error. In addition, each row or column of the matrix corresponds to the measured SHEterm variations induced by the elementary motion of a part. Thus the correction matrix is itself flawedby measurement errors and is relevant only if the variations recorded are larger than the measurementprecision. As we are facing a linear least squares problem, the issue of the conditioning has to betaken into account. As long as the computed VMotion is reasonable (all displacements smaller than,say, 5 mm) and cancels out the unwanted contributions, its sensitivity to the measurement error onVSHE0 has no relevance (the solution of the least-square problem has no relevance to us, only its e!ect).However, the sensitivity of the solution to errors in the measurement of the correction is important.


The condition number as discussed in section 2.7.2.1 provides us an estimation of this sensitivity. Aslong as the least square process is successful (i.e. the residual is small), the error on VMotion is roughlyproportional to / (146). The size of / suggests a high dynamic-range in the eigenvalue or singularvalue spectrum of the matrix. This is closely related to the correction system a!ecting e"cientlysome terms and scarcely modifying some others. The condition number is indeed also a measure ofhow close to being rank-deficient the matrix is (146). In turn, the correction of the least sensitiveterms requires large displacements of parts which induce large variations of the sensitive terms andamplifies errors introduced in the correction matrix during its construction. In addition, the largerdisplacements induced may venture some parts outside of their linear safe-area and introduce moreerrors in the correction. Thus, the condition number / can be used to estimate the e"ciency of thecorrection system, along with its sensitivity to measurement errors.

We can also conclude from this discussion that the measurement error has a double importance inthe correction system and special care must be taken to achieve the best measurement accuracy andprecision possible. The increase of the elementary displacement size results, to some extent, in theincrease of the amplitude of variation of the SHE terms, thus increasing the "SNR" on the correctionmatrix. This is however limited by the linearity of the variations of the SHE terms. We may introduceeven greater errors in the correction matrix if these variations exhibit strong non-linearities on long-range displacements. The choice of elementary displacement is thus important too. The correctionmatrix "SNR" may also be improved by measuring it several times and averaging it. This however mayincrease tremendously the measurement duration. Furthermore, the adjustment process may requireseveral iterations due to the errors introduced by measurements and non-linearities. After completionof an iteration, the correction matrix may have to be re-built to take into account the new behavior ofthe system and complete another adjustment. If carried on appropriately, the process must convergeto residual SHE terms of size dependent on the measurement errors.

We used the above-mentioned numerical simulations of imperfect magnets to test numericallycorrection schemes in the context of the constant-gradient magnet described in section 4.1.2.2.

4.1.4.2 E!ciency of the considered correction system

The study of imperfections is an important tool in the elaboration of a correction scheme in the sensethat it shows us what parameters are the most sensitive and thus more e!ective for corrections. Wehave shown in section 4.1.3.3 that motions of the di!erent parts a!ect mostly lower degree terms (oneand two). We expect thus that the correction matrix will have similar e"ciency.

Following the discussion of the previous section, we can approach this problem in a more globalway by calculating the condition number of the correction matrix. In order to take into account thevariability of the magnet (or uncertainty on its exact initial state), we calculated the correction matrixof a large number of simulated random magnet samples. We calculated from this set of magnets anaverage condition number of the correction system, along with its standard deviation. Table 4.10presents the average condition number computed as defined in section 2.7.2.1 for di!erent numbersof corrected degrees. These computations are done for corrections involving only radial motions ofthe free parts and for both radial and vertical motions. It appears clearly that correction matricestreating degrees up to two are well-condtionned and adjustment system is e"cient, while degree threehas already degraded performances for radial motion and higher degrees have poor sensitivity to thecorrection system.

This can be verified by looking at the content of the average correction matrix, as seen graphically infigure 4.14. Degree three and four have indeed a poor sensitivity to the adjustment system. Figure 4.14gives some insight on the way the correction system behaves. The part numbers are sorted by ring andordered in positive trigonometric sense. Hence, we can verify that on average, all parts of a given ring


Up to degree 2 3 4/̄ radial 6.5 1140 61640#* radial 0.03 330 -

/̄ radial/vertical 4.8 27.7 61730#* radial/vertical 0.02 0.2 -

Table 4.10: Condition number of the correction matrix treating terms of degree respectively up to two, threeand four. The elementary displacement is 0.05 mm. The addition of vertical displacements greatly improvesthe conditioning (and thus the e"ciency) of the matrix up to degree three. Correcting degree four remains verydi"cult. No standard deviation is given for degree four as the variability is extreme (maximum / can reachmore than 106 while / is lower-bounded by 1)

have the same e!ect on axial terms while a periodicity equal to the rank appears for skewed terms. Asexpected, a 90º phase shift (three parts) appears between X and Y terms.

It appears already from this discussion that we will not be able to compensate all degree threeand four terms su"ciently with this correction system. Luckily, we have seen in section 4.1.3 thatimperfections of degree four and above introduce little field inhomogeneities (few ppm) in the RoI. Itis hence not an issue. We however need to be prepared to deal with degree three in a more e"cientway. It seems thus interesting to add the ability to change the elevation of each part, as it improvessignificantly the conditioning of the matrix (see table 4.10). If a good precision of field measurementcan be achieved, and the terms of degree three are not too large, we can conceive to correct them. Assuggested by the condition number study, degree four seems out of reach and would need additionalshimming if the expected small residuals are not tolerable. This discussion lets us also foresee practicaldi"culties during measurement, as the small variations of the less sensitive degrees may remain buriedin the measurement precision.

Another important aspect of these results is that, despite the variability of the magnet due toimperfections, the correction matrix at the initial point is very stable up to and including degree 2.Thus, it should be possible to correct a number of terms (first, second degree, and also Z3 to someextent) with good accuracy, based solely on a correction matrix obtained through simulations. Thiscan save a lot of time in the correction process as experimental construction of this matrix requiresnumerous measurements.


Figure 4.14: Average simulated correction matrix with elementary displacement of 0.05 mm and measurementradius of 5 mm. The standard deviation is also shown. These standard deviation values correspond to 1% to5% of the change rate for e"cient matrix elements.


4.1.4.3 Linearity and correction simulations

Another concern is the reliability of the linear hypothesis. Our correction concept relies on the as-sumption that the system behaves linearly (or not too far from it) in the range of the correction vectorwe compute (VMotion). It is thus interesting to know the extent of the linear behavior of the system.Can we actually use the same correction matrix from one iteration to another and spare the work ofre-building it ? What is the threshold distance after which it is necessary to build a new correctionmatrix. The previous section already gave us the hint that the behavior is fairly linear, as within thetolerance of the magnet fabrication, the correction matrix remains the same.

We can verify this through simulations of imperfect magnets and computation of the e!ect ofdisplacements of selected parts. Figure 4.15 presents selected results from such simulations.

Z2 Z4

X21 X3

2

- ! - " " !

- !#- "$- "#

- $

$"#"$

- ! - " " !

#%&'

#%&(

"%##

"%#!

"%#)

"%#'

- ! - " " !

- )

- !

!

)

Normalized unit Normalized unit

Normalized unit

Normalized unit

Displacement (mm)

Displacement (mm)

Displacement (mm)

Displacement (mm)

- ! - " " !

#%*#%(#%&"%#"%""%!

Figure 4.15: Simulated variations of selected terms with the radial motion of one piece of the outer ring forten di!erent imperfect magnets. Vertical axes are normalized to the value at zero displacement. We can observetwo types of behaviors for the di!erent terms. The part motion is either e"cient and the variations induced arefairly linear over millimetric ranges (Z2, X2

3 ), or the terms are insensitive to the part motion and the variationsdisplay non linearities, mostly quadratic variation (X1

2 , Z4). Such curves can also be observed with other parts,with di!erent behavior of the SHE terms. The absolute variation associated to the motion of a part variesvery little from one magnetization distribution to another. The apparent large di!erence between the curves isrelated to the starting point, which can be close to zero or very large.

It appears that the linearity assumption is valid on a su"cient range (1 mm) for most of the targetedterms. An important consequence of this is that the correction matrix may have to be constructed onlyonce, or less often if the required displacement are large (1 mm or more). In addition, the adjustmentprocedure should converge very quickly. Some terms like X1

2 may seem problematic because of their


low sensitivity and their non-linearity. However, we must remember that we are only considering themotion of a single part here. As seen in figure 4.14, other parts may be more e"cient as skewed termsfeature a & dependency.

We can finally simulate the complete correction of simulated sample magnets in order to test theselected correction matrices, verify the stability of the correction matrix from an iteration to another,and observe the dependence of convergence speed and residual terms on measurement precision. Suchsimulations show that the selected adjustment scheme can correct up to degree three with verticalshimming and iterative radial motion (see figure 4.17).

Such a correction leads to a field profile very close to the one desired, as shown in figure 4.16. Wehave seen in section 2.7 that our SHE terms are flawed with systematic errors impacting accuracy.The accuracy errors we have identified, due to tilts, o!-center positioning, and skewness imply for thecorrection procedure that the final field profile will be the one desired, but only in the reference frame ofthe measurement system. We thus need to make sure that this reference frame corresponds to the one ofthe subsequent NMR experiments, when the adjusted magnet is used. When considering the gradientdesign with extreme gradient homogeneity along Oz (cancellation of terms), it is possible to achievethe cancellation of all terms up to three within the measurement precision (save the measurementinaccuracy). In the case of the design providing homogeneity in planes parallel to xOy, it is possible toapproach the required combination of terms (see section 2.5) and the final curvature of constant |B0|surfaces within a volume of 1 cm diameter and 1 cm depth is better than 5 µm (see figure 4.18 for fieldprofiles). The maximum radial displacement is about 2 mm and the proper adjustment is achievedwithin three iterations.

!"##!$%%&'

%&'

"(%

"(%

)'# )'#

*""

*""

%')+%')+

%&'

%#$)

%#$)

%#$)

%+#("(%

!"#$$%

&"#$$%

!%' !+ ' + %'!%'!*!)!&!#'#&'(%'

!+''

'

+''

%'''

%+''

#'''

#+''

"'''

!%*$*!%"#)

!(+&

!%*#

"*$

$)%

%+""

"*$

#)((

"#&$

!"#$$%

&"#$$%

!%' !+ ' + %'!%'!*!)!&!#'#&'(%'

!"'''!#'''!%''''%'''#'''"'''&'''+''')'''('''

-!" -# # !"$ %&&'#"!""!#"(""

)*+,-.,/0 12/& 3*21*4. 52-6,*0.%33&'

-!" -# # !" 7 %&&'(""8""9"":""!"""!(""

;,*<6 =-2,-.,/0 %33&'

-!" - # # !"$%&&'#"!""!#"(""(#"

)*+,-.,/0 12/& 3*21*4. 52-6,*0. %33&'

-!" - # # !" 7 %&&'#""!"""!#""("""

;,*<6 =-2,-.,/0%33&'

Gra

dien

t dev

iatio

n (p

pm)

Gra

dien

t dev

iatio

n (p

pm)

Befo

re a

djus

tmen

tsAf

ter a

djus

tmen

ts

Figure 4.16: Design with a uniform gradient along Oz: simulated variations of B0 before (top) and after(bottom) adjustment. The contour plots to the left show the variations in ppm of B0 in the xOzplane, the plotin the middle presents the variations in ppm along Ox and the plot on the right the deviation from the idealgradient along Oz.


Before adjustments After adjustments

Figure 4.17: Design with a uniform gradient along Oz: relevant SHE term values in the measurement framebefore (left) and after (right) adjustment. The residual amplitude of all terms is within the precision of mea-surement. This demonstrates the capability of the system to control precisely a limited number of terms.


Gra

dien

t dev

iatio

n (%

)G

radi

ent d

evia

tion

(%)

Befo

re a

djus

tmen

tsAf

ter a

djus

tmen

ts

!"!#!"!$!"%!!"%&

!"!#

!"%'

!"!$

!"(( !"($

!"%!

!"()

!"%&

!"##

!"%'

!"#*

!"((

!"&%

!"($

!"+$

,-.//0

1-.//0

!"# !$ ! + %!!"#!%!&!'!(!(&$'%!

!

!"%

!"(

!"#

!"&

!"+

!"$

!#)((

!#)"'

!#)#&!#)#$

!"!#

!"!#

!"!#

!"%%

!"%%

!"%%

!"(!

!"(!

!"('

!"('!"('

!"#$

!"#$

!"&+

!"+#

!"+#

!"*! !"*'

!"$%

!")+

,-.//0

1-.//0

!"# !$ ! + %!!"#!%!&!'!(!(&$'%!

!#)(

!

!"(

!"&

!"$

!"'

%

- !" - # # !" $%&&'"()"(*"(+"(,!("!()

-./012034 563& 7.65.82 961:0.42;%<'

-!" - # # !" = %&&'

-)#""-)"""-!#""-!"""- #""

>0.?: @16012034%77&'

-!" - # # !" $%&&'"(!"()"(A"(*"(#"(+

-./012034 563& 7.65.82 961:0.42;%<'

-!" - # # !" =%&&'#"!""!#")"")#"A""

>0.?: @16012034%77&'

Figure 4.18: Design with a uniform gradient in the xOy plane: variations of B0 before (top) and after (bottom)adjustment. The contour plots to the left show the variations of B0 (%) in the xOzplane, the plot in the middlepresents the variations in ppm along Ox and the plot on the right the deviation from the ideal gradient alongOz.

After estimating the possible imperfections of the fabricated magnet, we have built up some confi-dence in our ability to adjust it and recover the desired field profile. These results have been obtainedin the case of a constant-gradient design but would be very similar for the homogeneous field de-sign. This process has also helped us develop a more detailed knowledge of the e!ect of the di!erentgeometry parameters. This theoretical study is an essential step before proceeding with the actualfabrication and adjustment of the magnet. Another important conclusion of this discussion is that themain corrections can be carried on based on a simulated correction matrix and the measurement ofthe magnet.

4.2 Fabrication of an open tomograph

4.2.1 Fabrication, assembly and magnetization

Due to diverse reasons, the selected design for the gradient magnet is the one o!ering a uniform gradientof Bz. The gradient magnet was fabricated according to our design by Vacuumschmeltze (VAC), acompany specialized in the production of magnetic materials and based in Germany. We chose touse NdFeB, grade 633HR (172) for its good compromise between high remanence (Br = 1.35 T) andsu"cient coercivity (HCi = 1275 A m#1). We performed the final assembly in our laboratory. The partswere initially de-magnetized, facilitating the assembly. The assembled magnet before magnetizationcan be seen in figure 4.19. The mechanical plans were drawn with the help of M. Bougeard and S.Cazaux.

Once all parts in place, the magnet system was taken to the Laboratoire National des ChampsMagnétiques Intenses in Grenoble, where we put the system in a resistive magnet with a large borehole (28 cm) and generating a relatively high field (>6 T over the volume of the permanent magnet).This field was su"cient to saturate the complete magnet system (it is recommended to achieve at least

4.2. FABRICATION OF AN OPEN TOMOGRAPH 137

4 T everywhere in the material). The magnetization system can be seen in figure 4.20.

Figure 4.19: Left: View of the assembled magnet before magnetization. Right: View of the magnet with itscover on before magnetization.

Figure 4.20: Left: Side view of the resistive magnet used for the magnetization. Right: Top view of thepermanent magnet in place in the resistive magnet.


The permanent magnet was then brought back to our laboratory and installed in an assemblysystem designed by us for adjustments (figure 4.21). This system comprises a set of 24 aluminum armscapable of independent radial motions with micrometrical precision. The movable magnets are gluedin brass mounts that can be connected to and de-connected from the arms. The mounts feature oblongholes, so that they can be bolted to the magnet structure once the adjustments are complete. Thebrass mounts simply slide on the top surface of the aluminum magnet structure, as seen in figure 4.21.The forces exerted on the di!erent parts of the system were calculated for various positions (simulatingthe approach of the magnetic segments on the top of the magnet) and were found to be compatiblewith our assembly approach.

Figure 4.21: Top: View of the magnet system in place in the adjustment bench. The two poles support ameasurement platform. Each arm (only the arms handling the outer ring magnets are shown here) is supportedby micrometric stages (Newport 460A series) Bottom left: View of the magnet connected to the adjustmentbench with the 24 arms. The NMR gradient-compensated field measurement probe can be seen approachingthe region of measurement. Bottom right: View of the assembly/disassembly process. The arm systems aresupported by carriages allowing the removal of the magnet parts, despite the important forces.


4.2.2 Magnet characterization and adjustment

4.2.2.1 Initial measurements

A first set of measurements of the field was performed with a Hall probe to verify the coarse variationsof the field. As seen on figure 4.22, the results were very promising, as they matched very well thesimulations on large scales. It is however necessary to measure fine variations (NMR-based and SHEterms extraction) on the distances considered during the design to really validate our theoretical work.

!"# !$# !%# # %# $# "##&$

#&$$

#&$'

#&$(

#&$)

#&"

#&"$

#&"'

#&"(

#&")

!"#$$%

& '"#(%

!"# !$# !%# # %# $# "##&$

#&$$

#&$'

#&$(

#&$)

#&"

#&"$

#&"'

#&"(

#&")

!"#$$%

*+,+%#+--*+,+%.+--

*+,+$#+--

*+,+$.+--

*+,+"#--

*+,+".--

*+,+'#+--

*+,+'.--

*+,+.#+--*+,+..+--

0 5 10 15 25 25 30 35 40 45 500.20

0.22

0.24

0.26

0.28

0.30

0.32

0.34

0.36

0.38

0.40

“Distance” from surface (mm)

Bz Component (Tesla)

OptimizedRegion

Figure 4.22: Top left: Theoretical field curves along Ox for di!erent altitudes Z. Top right: Experimentalfield curves along Ox for the same altitudes. Bottom: Field variation along Oz. The field variation is highlylinear after 5 mm up to more than 50 mm. The region where the field variations are optimized (minimumvariations parallel to xOy) is highlighted in yellow between the dashed lines.

4.2.2.2 NMR measurements, adjustments and final performance

Based on the probe and the considerations described in section 2.7.3, we developed a high precisionfield mapping device based on a NMR probe. We were thus able to realize field maps with a resolutionbetter than 50 ppm and experimental SHE terms measurements with a precision of the order of 10 ppmon the most relevant terms using a reasonable number of points (8 along ' and 14 along &), so thata complete map could be obtained in less than ten minutes. We saw in section 2.7 that while manyterms remain very reliable despite measurement noise, some terms of degree and order above threemay become very untrustworthy. However, section 4.1.3.3 showed us that, in this case, these degreesare unlikely to contribute to the field variations in the volume of interest as imperfections do not a!ect


them significantly. We can thus overlook them. The experimental measurements (see figure 4.23)give values not too far from the range expected from the simulations presented in section 4.1.3 (mostterms are within 3#). However, Z3, Z4 and Z5 are all in the high end of the expected distribution,with Z5 largely above (more than 3#). These anomalous values are likely due to the combination ofimperfections of the magnet and imperfections in the measurement system (the center of measurementwas probably too close to the magnet).

!"

#"$%"

$

Figure 4.23: Experimental values of the di!erent SHE terms for an unadjusted configuration of the magnet.All values are in the range of the expected imperfections. Overestimation of higher degree terms due to noisecan be noticed (especially for X4

5 and Y 45 ).

The linear dependence of the most relevant terms on the position of one part was also verifiedexperimentally and showed a good match with simulations, as seen in table 4.11 and figure 4.24.

Change rate (ppm/mm) Z2 Z3 X11 X1

2 X22 Y 1

1 Y 12 Y 2

2

Theoretical 41.8 -5.6 -196.2 4.4 -31 -50.2 1.2 -17Experimental 49.4 -9.4 -184.8 0.2 -30.2 -39 0.1 -13.2

Table 4.11: Computed and experimental change rate of the first SHE terms for the motion of one selectedmagnet piece. Good agreement (few percents) between experimental and theoretical is obtained on sensitiveterms. Discrepancies may partially be explained by the di!erent initial point used in the simulation and in theexperiment (it is not possible to know exactly what is the initial state of the magnet). Experimental values ofless sensitive terms are unreliable because of measurement noise.


!" !#$% !#$& !#$' !#$( # #$( #$' #$& #$% "!(##

!")#

!"##

!)#

#

)#

"##

")#

(##

!"#$%&'()(*+,-)).

/(0),1&0"&+"2*,-$$)

.

!,3(!,4"

"

!,4("

!,4((

!" !#$% !#$& !#$' !#$( # #$( #$' #$& #$% "!(##

!")#

!"##

!)#

#

)#

"##

")#

(##

!"#$%&'()(*+,-)).

/(0),1&0"&+"2*,-$$)

.!,3(!,4"

"

!,4("

!,4((

Figure 4.24: Left: Calculated variations of a few terms for the displacement of one selected magnet piece.Right: Experimental variations of the same terms for the displacement of the same piece. Sensitive terms displaya linear behavior (the Z2 measurement seems impaired by noise). The measurement noise however precludesany conclusion regarding X1

2 for this particular part motion.

These results show that the calculated matrix terms are su"ciently accurate to perform correctionsof the magnet. We were able to achieve an acceptable correction (see figure 4.25) within four iterationsbased solely on the simulated correction matrix. The quality of the correction is limited by higherdegree axial terms: Z3 did not decrease su"ciently while Z4 and Z5 increased significantly. Theseterms are also probably overestimated because of o!-centering of the measurement system.

!"

#"$%"

$

Figure 4.25: Experimental values of the SHE terms after correction of the magnet. The correction targetedthe cancellation of terms up to and including degree two, while maintaining higher degrees as low as possible.Top: Axial terms. Z0 and Z1 are not shown as these are the quantities we wanted to maintain. Bottom: Skewedterms. Most of these terms contribute less than 10 ppm to the variations.


Based on the experimental SHE terms, we can reconstruct the field profile anywhere in the mea-surement sphere. Our measurement system allows us to also completely map the field, so that wecan compare reconstructed profiles with experimental variations and verify that the whole SHE mea-surement procedure is correct and that we retrieve enough terms. Figure 4.26 presents reconstructedprofiles and experimental ones, showing an excellent match within measurement precision (better than±10 ppm at center and about ±50 ppm on the edge of the measurement volume).

!"#$

!"#$

%&#'

%&#'

()*#)

()*#)

(%"#"

(%"#"

&*&#&

&*&#&

!'*#+

!'*#*

!*%#'

!*%#'

!'*#+!'*#+

,-.//0

1-.//0

!" ' *!"

!#

!$

!%

!&

'

(

&

!

)

*

'

*'

(''

(*'

&''

&*'

!''

!*'

)''

)*'

*''

22/

&('#*

&('#*

&('#*

()*#%

()*#%

()*#%)'(

"(#)

"(#)

(+#"(+#"

&$*#(

&$*#(&$*

#(

&$*#(

!!%#$

!!%#$

!!%#$

)')#&

!!%#$)')#&

)')#&)+"#"

,-.//0

3-.//0

!" ' *!"

!#

!$

!%

!&

'

(

&

!

)

*

'

(''

&''

!''

)''

*''

22/

%#&

%#&

+!#)

+!#)

(($#+

(($#+

($(#%

($(#%

&&+#(

&&+#(

&"'#!

&"'#!

!!)#*

!!)#*

!!)#*

,-.//0

1-.//0

!" ' *!"

!#

!$

!%

!&

'

(

&

!

)

*

'

*'

(''

(*'

&''

&*'

!''

!*'

)''

)*'

22/

&!%#%&!%#%

&!%#%

("'#'

("'#'

("'#'

(&'#'

(&'#'

+'#'

+'#'

&%%#%

&%%#%&%%#%

&%%#%

!*%#%

!*%#%

!*%#%)(%#%

)(%#%!*%#%

)$%#"

)(%#%

,-.//0

3-.//0

!" ' *!"

!#

!$

!%

!&

'

(

&

!

)

*

'

(''

&''

!''

)''

*''

22/

Figure 4.26: Left: Reconstructed profiles from experimental SHE terms (top is the xOz plane passing throughthe center and bottom is the xOy plane containing the center). Right: Experimental field profile in the sameplane. The experiment verifies the reconstruction within measurement noise and gives us even more confidencein the SHE term measurements.

We have here proved that the uniformity of $Bz$z is achievable. However, concomitant components

make this uniformity irrelevant at the field level we are considering. Therefore, the theory exposed insection 2.5 can be applied. Despite the fabricated magnet is not meant to achieve uniformity of B0

in transverse planes, we can alter it by large displacements of the mobile pieces and partially achievea more suitable field profile. However, due to mechanical limitations, extensive correction could bedone only up to degree two, while reducing as much as possible the degree three. Simulations haveshown that the required compensation of Z2 with radial displacements induces a significant increaseof Z3, which can be compensated to some extent by vertical motion of the inner ring. This howeverincreases Z4 and Z5, so that we must expect uncontrolled values of Z3, Z4, and Z5 between 40 ppm and100 ppm. This results mostly in increased cross-terms in the Sm

n matrix, thus degrading field profiles


in transverse planes away from the center.In addition, such a field profile imposes an even greater challenge for the field measurement, as the

field gradient is varying by 0.1% over the measurement volume. To maintain precision, the compen-sation system should thus be adjusted throughout the SHE term measurement to match the varyinggradient. However, in practice, this increases greatly the time needed to perform one field map. Inaddition, as the NMR probe is not perfectly centered in the compensation coil, maintaining precisionrequires to take into account the field shift induced by the variation of the compensation strength.To save time and for the sake of simplicity, we chose to loose some precision and maintain the samecompensation during a field map.

We were able to bring Z2 within 5 ppm of its required value to meet the condition given inequation 2.70 while skewed terms up to degree two were reduced to less than 20 ppm. However, higherdegree terms could not be controlled and Z3 increased to 100 ppm. The final field is 0.329612 T (about14.03 MHz) and the gradient is 3.5016 T m#1 (corresponding to 149.1 kHz mm#1, or 10623 ppm mm#1).The experimental values of the di!erent SHE terms are shown in figure 4.27. We can also verify thatthe measured field profile in the entire volume corresponds to the one reconstructed from the SHEterms, as seen in figure 4.28. This comparison provides confidence in the SHE coe"cients to use themfor field calculation in NMR signal simulations.

!"

#"$%"

$

Figure 4.27: Experimental deviation of the SHE terms from their target values. The correction here aims atachieving a uniform field modulus in transverse planes. Top: deviation of axial terms from their target value.Z2 is within 5 ppm of the condition necessary to achieve cancellation of S2

0 (see equation 2.70). Bottom: skewedterms. All skewed terms must be cancelled and we maintained all terms up to degree two below 10 ppm.


!"#$

!"#$

%&'#'

%&'#'

%(%#&

%(%#&

!)*#*

!)*#*

*$$#"

*$$#"+*)#'

+*)#'

$!&#%

$!&#%

"&!#+"'+#"

"&!#+

)""#(

,-.//0

1-.//0

!" & $!"

!#

!$

!%

!&

&

%

!

*

+

$

&

%&&

!&&

*&&

+&&

$&&

"&&

)&&

22/

!&&'()!*$(%!'*("

!#"(*

!#"(*

!%%(%

!%%(%

%#+

!$#%

,-.//0

3-.//0

!" & $!"

!#

!$

!%

!&

&

%

!

*

+

$

!%++

!&"+

!&++

!"+

&

22/

!*)(*!)%(,

!''('

!"+(#

!$#(%

!&)(+

!&(*

%+#*

,-.//0

3-.//0

!" & $!"

!#

!$

!%

!&

&

%

!

*

+

$

!&#+

!&%+

!&++

!)+

!'+

!#+

!%+

&

22/

*%#(

*%#(

%%*#)

%%*#)

%($#$

%($#$

!))#+

!))#+

*$(#!

*$(#!

++%#%

++%#%

$!!#(

$!!#(

"&+#)

"&+#) "'"#"

"'"#""'"#"

,-.//0

1-.//0

!" & $!"

!#

!$

!%

!&

&

%

!

*

+

$

&

%&&

!&&

*&&

+&&

$&&

"&&

)&&

22/

Figure 4.28: Left: Reconstructed field profile from measured SHE terms. Right: Experimental profile. Top:Field deviation from an ideal gradient in the xOz plane. Bottom: Field variations in the xOy plane (planeat center of the RoI). The reconstructed profiles match very well the experimental ones, given the precision ofmeasurement.

We have thus been able, to some extent, to adjust the magnet in order to obtain the field profilewe desired. The final performance (field homogeneity in planes parallel to the surface of the magnet)was mainly limited by the improper design of the magnet, leading to an important correction of degreetwo and uncontrolled increase of higher degrees. The field profile for which the magnet was initiallydesigned could be partially achieved. However, the contribution of higher degrees was significantlyhigher than expected from the simulations. This issue was not investigated further by lack of time.We can however suppose that the tolerance in the elevation of the rings is larger than expected. Inaddition, the correction of lower degrees is not up to what could be ultimately achieved because ofsmall perturbations of the magnet during clamping of the magnets and disassembly of the arms at thefinal stage of the adjustment. This final operation still needs to be refined and improved to o!er amore reliable extraction of the magnet from the assembly system.


4.2.3 Field stability

Most NMR magnets have a field stability rating, so-called "drift", related to the slow decay of thecurrent circulating in the coil. It is usually provided in terms of ppm per hour (ppm/h) or evenppb/h. These magnets are thus extremely stable. Permanent magnets are however highly sensitive totemperature (about 1000 ppm/K for NdFeB (35)) and the temporal field stability concern is dominatedby matters of temperature stability. In our case, the magnet system and the room is not temperature-stabilized. We are thus subjected to temperature variations dependent on the environment, which canspan several degrees Celsius over a day. Such conditions, though not as di"cult as for field experimentsare much harsher than what could be expected in a reasonable laboratory environment. We used twoPt-100 temperature probes (IST, Pt100 sensors) mounted on the top and on the side of the magnetto monitor its temperature at all time, using a thermistor converter (Pico Technology, model Pt104)linked to a computer.

The stability of the field profile cannot be easily predicted, as it relies on the homogeneity ofthe temperature variation in the magnet. Indeed, if the relative distribution of temperature withinthe magnet structure remains the same, the field profile must remain stable. However, a varyingdistribution of temperature will induce a degradation of that profile. Nevertheless, we can hope, fromthe good thermal conductivity of NdFeB (5-15 W.m#1.K#1) and the compactness of the system, thattemperature variations will propagate quickly through the magnet and no strong temperature gradientswill set in the structure.

We thus monitored the temperature, field strength, and field SHE terms over long periods of time(several hours). The evolution of the field strength and SHE terms with temperature shows that themagnet responds well to slow temperature variations. The field variations map very well the temper-ature variations with a temperature coe"cient of about 900 ppm/K (see figure 4.29). In addition,the field profile is extremely stable and no actual trend of the main SHE terms can be observed (allfluctuations are within the measurement noise), as shown in figure 4.30. These results provide confi-dence that the magnet can be adjusted once and for all and subsequently used for NMR experiments.However, these results prove the stability only under mild temperature variations. Harsher local tem-perature variations are likely to a!ect the field profile on short time scales. Our experiments suggestsuch a behavior as the field strength curve features ripples corresponding to the periodic motion of thecompensated NMR probe (the compensation coil features about 400 Amp turns), which provides heatto the magnet locally.

It is also important to note that temperature variations are of concern during field mapping, asa typical SHE term measurement can take up to ten minutes, during which the magnet externaltemperature can vary by 0.02ºC to 0.2ºC (implying 18 ppm to 180 ppm field variations). This issue issolved by taking several reference measurements (always at the same location) at di!erent times duringthe procedure. Temperature variations of the magnet structure are slow and mostly linear over the timeperiod of a field map and few reference points are necessary to achieve an accurate representation ofthe temperature-related field variations during the measurement. This can then be used to correct themeasurements and obtain the e!ective spatial variations of the field, free of the temperature influence.Typical field variations during a SHE term measurement are shown in figure 4.31, along with thee!ective temperature variation.


! "!!! #!!!! #"!!! $!!!! $"!!! %!!!!!"###

!$%##

!$###

!%##

!

&'()*+,-'./+0112

3

!

!4"

#

#4"

$

5'2(+0,3

5(21(67/86(+,-'./+093

! "!!! #!!!! #"!!! $!!!! $"!!! %!!!!!$$##

!$###

!&##

!'##

!(##

!)##

5'2(+0,3

5(214+:;(..'<'(=/+0112>93

!

!4?

#*'

#4$

$*)

$

5(21(67/86(+,-'./+093

Figure 4.29: Left: Temperature and field strength variation over 8.5 hours. Right: Temperature and temper-ature coe"cient variations over the same time period. The left graph shows that the field variations follow thetemperature variations very well, with no apparent lag (even at the "bend" highlighted by the dashed line).Such a behavior is indicating that the temperature of the magnet remains mostly homogeneous. These mea-surements were taken with field maps intended for SHE terms calculations. The ripples visible in the field curveand temperature coe"cient curve are due to the variation of the proximity of the field probe to the magnet.This probe dissipates heat due to the large number of ampere-turns (about 400) and slightly warms-up theupper part of the magnet when in close proximity. The period of the ripples corresponds to one field map. Theinaccurate temperature coe"cient during the first 10000 seconds is due to the small temperature shift whichamplifies errors on the field shift measurement.

! "!!! #!!!! #"!!! $!!!! $"!!! %!!!!!"#

!$%

!$#

!%

!

"

#!

#"

$!

&'()*+,(-,.-/01*233)

4

5#5$5%565"57

&-)'*214! "!!! #!!!! #"!!! $!!!! $"!!! %!!!!

!&#

!"#

!$#

!

#!

$!

%!

&'()*+,(-,.-/01*233)

4

8##

8$#

8$$

8%#

8%$

8%%

86#

86$

&-)'*214

Figure 4.30: Variations of the di!erent SHE terms with time over more than eight hours. Y mn terms are not

shown as they have a behavior similar to the Xmn . The fluctuations seem to remain within the measurement

noise: the field profile is very stable over that period of time.

4.3. CONCLUSIONS ON SINGLE-SIDED MAGNETS 147

! "!! #!! $!! %!! &!! '!! (!!!"#

!$#

!%#

!&#

!'#

!(#

!)#

!*#

!+#

!

)*+,-./0*12.3445

6

#'7&'

*&,'$

#'7'

#'7'#

#'7'%

#'7''

*&,&$

#'7(

#'7(#

#'7(%

8*5+.3/6

8+54+9:2;9+.3<=6

Figure 4.31: Typical temperature variations during a field mapping intended for SHE terms calculation. Thefield variations monitored through the reference points are shown (nine points are used), while the temperatureis sampled every second. The wiggles in the temperature curve are only measurement noise.

4.3 Conclusions on single-sided magnets

We have presented theoretical designs for single-sided magnets capable of high performances in terms offield profile control. A detailed study of the possible imperfections of such magnets showed that, withreasonably tight fabrication tolerances, such magnets can outperform simple bar magnets without anyfurther adjustment. We also detailed a possible approach for the shimming of such magnets, based onthe displacement of some components of the magnet. Here again, detailed numerical analysis showedthe feasibility of the concept. We thus proceeded with the fabrication of a prototype generatinga strong field gradient and characterized it following the techniques described in chapter 2. Theexperiment proved that the correction of the magnet can be done on the terms expectedto be sensitive to the correction system, solely based on a simulated correction matrixand few experimental field maps. However, the strong gradient is accompanied by concomitantgradients that cannot be neglected in view of the medium field strength. It is thus preferable to usethe framework exposed in section 2.5. While this prototype was not designed according to thisconcept, it was still possible to adjust it to partially verify the conditions necessary toobtain a constant modulus in transverse planes. This shows how versatile the correction systemcan be. In addition, we verified experimentally that temperature induces a severe field drift, as it wasexpected, but also showed that the field profile is very stable despite temperature variations.

We can also compare this prototype to the current state of the art (see table 4.12) and observethat it compares well with prototypes proposed in the past, despite the di!erent issues that still needto be fixed.

blahblahblahblahblahblahblah


Reference Res (µm) Mass (kg) RoI (mm) Dist. (mm) G (T m #1) B0 (mT)Chang (110) 200 2 5 ( 5 ( 1 2 9.9 280

McDonald (106) 2000 20 10 o.d. ( 0.1 50 3.25 75Marble (112) 150 14 10 ( 10 ( 2 6 0.3 100Perlo (102) 10 ? 10 o.d. ( 0.1 10 20 400This work 20 40 10 o.d. ( 8 20 3.3 330

Table 4.12: Performances of selected previously proposed prototypes of single-sided magnet with stronggradient, along with the prototype discussed in this work. The resolution stated for this work is explainedand discussed in the next chapter. It appears it compares well with the state of the art, despite the diverseissues encountered which can be solved in a near future.

blahblahblahblahblahblahblah

Chapter 5

An open NMR/MRI system: Coils andNMR experiments

5.1 Single-sided coils

The magnet is a central component in a NMR system but is not su"cient. Detection and gradientcoils are integral part of such a system and require as much care in their design in order to obtaingood sensitivity or good imaging performances. A good magnet will never compensate for the poorperformances of the RF coil, so that the final NMR system can end up being mediocre, even thoughit is based on a high performance magnet. As we will see, the design of RF coils is an entire field ofresearch, as well as the design of gradient coils. Our goal in this work was to develop a system andreach the point were we could perform a few actual NMR experiments. We thus had a limited time todevote to the RF aspects. In this context, we concentrated on the development of an original designwith su"cient performance.

In a single-sided context, we limit the whole NMR system to a half-space. We thus need, in additionto a single-sided magnet, single sided coils for excitation/detection and for pulsed gradients. The highversatility of printed circuit boards techniques makes feasible arbitrary surface distributions of currentswhile saving space. We thus focus here on such distribution of currents that o!er optimal performancesfor some chosen parameters.

5.1.1 Design theory

The design method we chose relies on the stream function method. This method is widely used todesign shim and gradient coils (173–179) and has been surprisingly more scarcely used for the designof RF coils (180). We will re-formulate here the formalism of stream function for the optimizationof coils relative to the power dissipated in the coil. We have indeed seen in section 1.1.6.2 that thesensitivity of the coil is proportional to the ratio B1&

P. Thus, minimizing the power dissipated for a

given B1 provides the best sensitivity attainable for a coil. In the context of gradient coils, it canalso be desirable to be able to generate a strong gradient for a limited input current, thus requiring tominimize the power dissipated for a given field gradient. It can also be desirable to attain short risetimes for gradient coil, which corresponds to the minimization of the inductance, or equivalently thestored energy. Such a problem can also be treated with stream function methods. We will concentratehere on power dissipation minimization.

149

150 CHAPTER 5. AN OPEN NMR/MRI SYSTEM: COILS AND NMR EXPERIMENTS

We consider here a surface distribution of current!"k . In a static situation, we can write

!")S ·!"k = 0, (5.1)

!"k ·!"n = 0, (5.2)

where the divergence is taken on the surface S occupied by the current distribution, while !"n is theunit vector orthogonal to the plane tangent to the contour of S at the point of evaluation. Equation 5.1implies that their exists a vector

!"F such that

!") (!"F =!"k (5.3)

We will restrict ourselves to surfaces corresponding to a constant coordinate in the chosen coordinatesystem. We choose the Cartesian system in what follows for simplicity but the results can be usedwith other orthogonal coordinate systems (such as cylindrical or spherical). We consider the surfacez = const so that we have necessarily

!"k (x, y) = kx(x, y)!"ux + ky(x, y)!"uy, (5.4)

!"F = F (x, y)!"uz, (5.5)

kx =+F

+y, (5.6)

ky = !+F

+x. (5.7)

F (x, y) is called the stream function and its iso-contours are nothing else but the current lines. Thus, ifwe determine the flux function corresponding to the desired current distribution, we can realize it in astraightforward fashion by etching selected iso-contours of the stream function in a conducting surface.In the limit where the tracks formed by these iso-contours constrain the current lines to following theiso-contours, injecting a current in each track allows to reproduce the desired current distribution. Inaddition, if the iso-contours used as tracks boundaries correspond to evenly space values of the streamfunction, we can use the same current in all tracks.

We now need to define this stream function. This can be achieved through a variational approachbased on the constraint of the problem. We intend to minimize the power dissipated through the coil,which we can express as

P =1#e

--

S

4k2

x + k2y

5dx dy, (5.8)

where # is the conductivity of the conducting material and e is the thickness of the conductor (smallcompared to the skin depth). We may have constraints on any physical quantity Q which can be forexample a component of the field or a field gradient. This constrain can be written in the general form

Q0 =--

S(Qx(x, y)kx + Qy(x, y)ky) dx dy, (5.9)

and we may have additional constraints such as the homogeneity of the quantity Q, which can bewritten in the form

C0 =--

S(Cx(x, y)kx + Cy(x, y)ky) dx dy. (5.10)

5.1. SINGLE-SIDED COILS 151

We can introduce a stream function F (x, y) that we set equal to zero at the boundaries of thesurface and rewrite P . If we consider the k2

x term alone, we can write

--

Sk2

x dx dy =--

S

!+F

+y

"2

dx dy (5.11)

=- 6*

F+F

+y

+

boundary

!-

F+2F

+y2dy

7dx (5.12)

= !--

SF

+2F

+y2dx dy. (5.13)

We can write a similar derivation for k2y so that we have

P = ! 1#e

--F!F dx dy. (5.14)

P will adopt this form whatever the coordinate system is used, as long as the current distributionoccupies a surface defined by qi = cst where qi is one of the three coordinates. With similar derivation,it is possible to write Q0 in terms of F,

Q0 =--

SF

!+Qy

+x! +Qx

+y

"dx dy. (5.15)

Let!"Q be a vector such that

!"Q = Qx

!"ux + Qy!"uy, (5.16)

we can writeQ0 =

--

SF

(!")S (!"Q

)

z. (5.17)

We now use the Lagrange multipliers to minimize P under the constraint Q0. We need to solve

+

+F

!--

SF

@!SF ! 0

(!")S (!"Q

)

z

Adx dy

"= 0, (5.18)

which is equivalent to --

S!SF ! 0

(!")S (!"Q

)

zdx dy = 0, (5.19)

which is verified when we have identically

!SF ! 0(!")S (

!"Q

)

z= 0. (5.20)

We can derive the same relation for constraints of type C and all of this remains valid in any orthogonalcoordinate system. Thus, the creation of the quantity G0 for a minimum power dissipation by a currentdistribution on the surface q3 = cst is achieved as soon as the stream function satisfies

!F = 0(!")S (

!"Q

)

3+ µ

(!")S (!"C

)

3+ ... (5.21)

Thus, the problem boils down to solving a Poisson equation in two dimensions. Based on this result,we can find analytical solutions of the stream function for various planar surface coils.


5.1.2 RF coils

We intend here to produce a coil suitable for single-sided detection in conjunction with one of thesingle-sided permanent magnet discussed previously. These magnets generate a field along Oz whichis the B0 of the NMR experiment. We thus need a coil system generating a field B1 in a plane parallelto xOy at the center of the region of interest of the magnet. We can decide that the field will beoriented along Ox. We require the current distribution to be located in the plane xOy, right on top ofthe magnet.

Considering the range of frequencies at stake in NMR, the conductor thickness (25 µm to 70 µmfor Printed Circuit Boards, or PCB) is of the order of the skin depth (about 25 µm for copper at14 MHz) and the distance to the region of interest (of the order of 1 cm), so that we can considerthe current distribution close to a surface distribution. In addition, the size of the region of interest(1 cm) and of the whole detection system (coil and distance from coil to region of interest is of theorder of 1 to 10 cm) is very small compared to the wavelength (of the order of 10 m) so that we canuse the quasi-static approximation (the e!ect of retarded potentials can be neglected). As a result, wecan use the previous results to design an RF coil, as if it were a static coil. We can do it only becausethe working frequency is low enough. It would obviously be a di!erent story, were we working at highfields (43).

5.1.2.1 Optimal sensitivity in one point

We intend here to minimize the power dissipated for a given B1. Such an optimization yields a coilof optimum e"ciency (B1

P ), which is equivalent to an optimum sensitivity ( B1&P

). This has a doubleimportance as it will increase the excitation e"ciency and the detection sensitivity.

Given the symmetry of the problem, it appears suitable to use cylindrical coordinates for the currentdistribution. In addition, we require the current distribution to be located in the plane xOy. We canthus express the current distribution

!"k (r, ') as

!"k (r, ') = kr(r, ')!"u r + k+(r, ')!"u +, (5.22)

and we have

kr =1r

+F

+'(5.23)

k+ = !+F

+r(5.24)

We can express the field generated along Ox at a point located on Oz with z = z0 as

B1 =µ0z0

4$

--

S

kr sin ' + k+ cos '4r2 + z2

0

5 32

r dr d', (5.25)

while the dissipated power isP =

1#e

--

S

4k2

r + k2+

5r dr d'. (5.26)

Putting aside the multiplicative constant for now, the expression of B1 gives us the vector!"Q defined

previously,

Qr =z0 sin '

4r2 + z2

0

5 32

, (5.27)

Q+ =z0 cos '

4r2 + z2

0

5 32

, (5.28)


thus, we have (!")S (!"Q

)

z=

1r

!+(rQ+)

+r! +Qr

+'

"= ! 3rz0 cos '

4r2 + z2

0

5 52

, (5.29)

and the equation to solve is

!SF = !30z0r cos '

4r2 + z2

0

5 52

. (5.30)

The Laplace equation is separable in the cylindrical coordinate system and the form of the secondmember leads us to look for a solution of the form

F (r, ') =

#K1r +

K2

r! 0z0

r1

r2 + z20

'cos ' (5.31)

Boundary conditions induced by the spatial constraints on the current distribution (e.g. inner radius,outer radius) allow the calculation of K1 and K2. We can for example constrain the current distributionto a disk of radius r0 so that the stream function becomes

F (r, ') = 0r

B

C 1(z0 +

1r2 + z2

0

) 1r2 + z2

0

! 1(z0 +

1r20 + z2

0

) 1r20 + z2

0

D

E cos ' (5.32)

We thus have a current distribution symmetric in regard of Oy where F = 0, with currents circulatingin opposite direction on each side. As the contours of F are the current lines of the ideal currentdistribution, the design of the coil only takes to trace contours of the stream function for equidistantvalues (i.e. trace F (r, ') = F0

k with F0 the maximum of F ). These contours can be used as insulatingareas delimiting tracks in which the same current I flows. We show in figure 5.1 an example of coilwith minimum power dissipation for a given B1 (the generated field only depends on the current fedin the coil). The point of interest is located at normalized coordinate z0 = 1 and the coil is delimitedby a circle of radius r0 = 3.

We can then calculate the expressions of B1 in z0 and of P by integration over the whole disk andobtain

B1 = !0µ0K

16z20

, (5.33)

P = 02 $K

4#ez20

, (5.34)

with

K =(R0 ! z0)4(R2

0 + 4R0z0 + z20)

r20R

40

R0 =,

r20 + z2

0 .

We can thus write the relative sensitivity of the coil at the point of calculation, as defined in 1.1.6.22222B1*P

2222 =µ0*

#e

8*

$z0

*K, (5.35)

The SNR associated to such coils is given by

SNR =µ0!0M0VS

*#e

8z0*

4$kBT!f

*K. (5.36)


-! - " -# $ # " !-!

-"

-#

$

#

"

!

X (Normalized units)

Y (N

orm

aliz

ed u

nits

)

Figure 5.1: Example of optimum e"ciency planar coil design. The generated field is along Ox and theminimum power dissipation for the generated field is achieved on the Oz axis at elevation z0 = 1. The coil isdelimited by a circle of radius r0 = 3. The lines drawn correspond to contours of the stream function as definedin equation 5.32 for equidistant values. Currents in the blue region flow in opposite direction to the ones in thered region.

Plotting the sensitivity (we will call it later the intrinsic sensitivity to distinguish from the actualsensitivity related to the environment noise) of the coil as a function of r0

z0for di!erent z0 helps assessing

the benefit of increasing the radius of the coil for a given distance to the point of interest and observethe decrease in sensitivity with z0. As it can be seen on figure 5.2, the gain of sensitivity becomes slowto very slow after a critical radius, so that it becomes pointless to increase the size of the coil beyondthat radius.

It is important to note that the sensitivity of the coil at z = z0 varies as 1z0

, placing a fundamentallimitation on the increase of the coil range. Another curve of interest is the variation of B1 as afunction of z. Figure 5.2 shows such variations for di!erent coils of di!erent parameters z0 and r0. Aninteresting behavior to note is that, if we take two coils with same parameter z0 but di!erent r0, thelarger coil is indeed more e"cient for z % z0 but there exists z1 < z0 for which the smaller coil is moree"cient. It is important to define what the depth of observation is expected, as using the proper coilmay provide more than twice the sensitivity than another one. For example, in figure 5.2 on the right,coil 1 is more than twice as e"cient as coil 6 at z = 0.1 while coil 6 is about 20 times more e"cientthan coil 1 at z = 2.

We have now all elements to design a coil of highest e"ciency at one point on Oz. It might also bedesirable to achieve homogeneity in addition to e"ciency. This can be achieved analytically by addingconstraints in the Lagrange multipliers, like setting $B1

$z = 0, for example. We will not discuss thishere and proceed directly to practical aspects of the coils we just studied theoretically.


! " # $ % &! &" &#!

!'&

!'"

!'(

!'#

!')

!'$

!'*

!'%

!'+

&

,!-.!

/0,1234.56785984:4;4:<

.!=!')

.!=&

.!="

.!=(

! !') & &') " "') (!

!'&

!'"

!'(

!'#

!')

!'$

!'*

!'%

!'+

&

.7>2,?4:,2,<7359@:A7B94:C

D &7>90,1234.567B94:C

E0437&F7.!=&G7,!=&

E0437"F7.!=&G7,!=#

E0437(F7.!=&G7,!=&!

E0437#F7.!="G7,!="

E0437)F7.!="G7,!=%

E0437$F7.!="G7,!="!

Figure 5.2: Left: Coil sensitivity (SNR) at z = z0, as a function of the coil radius for a given z0 (distanceto point of interest). After a given radius, the increase in sensitivity is very slow. In addition, the maximumsensitivity varies in 1

z0. Right: Variation of the generated field with z for several coils of di!erent parameters.

The green dashed lines indicate the two z0 used. It appears clearly there are large di!erences depending on theparameters of the coil and the depth of observation. The detection coil should be built according to the depthat which the user needs to observe the signal.

5.1.2.2 Practical realization and experimental study

Practical realization Figure 5.1 gives the theoretical profile of the tracks of the desired coil. Weneed to perturb this layout in order to connect the tracks to each other. We thus choose a regionof lowest |

!"k | (small contribution to the field) for these connections and modify the pattern, baring

in mind the current must circulate in opposite sense from one side to the other. In addition, we candiscard the external track, which does not contribute much to the field. The modified pattern can beused as a mask for the fabrication of a printed circuit board (PCB). A view of such a mask is shownon figure 5.3, along with a picture of a resulting circuit. We used standard PCB with a 70 µm copperlayer, which is about three times the skin depth. As stated earlier, the thickness is still of the order ofthe skin depth, so that the variation of the current density throughout the depth of the circuit duringRF pulses will not be great. Thus, the B1 e"ciency should not be greatly degraded from the DCbehavior, and neither the resistance of the coil.

Field profile We can assess the experimental field variation along Oz. We built several coils withdi!erent r0 and z0 and measured the field generated for a given DC power with a Hall probe. Com-parison of the experimental sensitivity of the di!erent coils can be made through the normalization toa power of 1 W, while comparison with theoretical computations (continuous current density distribu-tion) can be made through normalization to the maximum field strength obtained. Figure 5.4 showsthe results of these calculations and measurements. The field measurements were performed by a Hallprobe (Lakeshore 3-axis probe with Gaussmeter model 460) mounted on our automated positioningsystem (position precision about 10 µm). A rather large quantitative discrepancy appears (a littlemore than a factor two) between the theory and the experiment. This loss of sensitivity might bepartially explained by the removal of the external track, as the theoretical calculation is performed byintegrating over the whole disk. Solving this issue is a work still in progress.


Figure 5.3: Left: view of a mask used for the fabrication of planar coils. Right: View of the fabricated coil.The parameters were chosen such that z0 = 1 and R0 = 3, with centimeters as units. The mask colors areinverted as the photoresist used is positive. The two elements are connected through the center points and twocurrent leads have been added to ease connections.

! " #! #" $! $" %! %"

!

#

$

%

&

" '(#!!"

)*!+(,!-.)$+(#-(/0)*!+(,!-.)&+(#-(/0)*!+(,!-.)&+(#1"-(/0)*!+(,!-.)"+(#1"-(/0)*!+(,!-.)#!+($-(/0

,()00-! " #! #" $! $" %! %"

!

!1$

!1&

!12

!13

#

#1$ '(#! !#

,()00-

4 #56#5$ ()7(8!$%&-

Figure 5.4: Left: Theoretical relative variations of B1 (DC) as a function of Z for di!erent coil parameters.Right: Experimental relative variations of B1 (DC) for the same coils, as measured by a Hall probe.

In addition, it is important to characterize the lateral variations of B1, in order to qualify theextent of the excited and detected region during the NMR experiment. Such variations can also becomputed and verified experimentally, as shown on figure 5.5 for two coils of parameters respectively(z0, r0) = (1, 4) and (1, 2) cm. As expected, B1 decreases faster laterally for a smaller coil. In addition,the measured variations match the simulations very well qualitatively and quantitatively, save themisalignment of the measurement window.

5.1.2.3 NMR detection performances

RF performance We were able to obtain !2 pulses with duration of 4 µs (that is to say, about

63 kHz in terms of Larmor frequency and 1.5 mT in terms of B1 seen by the nuclei), using RMS powerof roughly 140 W at 14 MHz, about 5 mm away from the coil. As the coil is linearly polarized, wemust apply a factor 2 to the e!ective B1 (24) to compute B1&

Pto about 2.48 ( 10#4 T W# 1

2 . This is


!""

!""

!"#

!"#

!$%

!$%

!$$

!$$

!&'

!&'

!&(

!&(

!&&

!&&

!&&

!)

!)

!*

!*

!"#$$%

&"#$$%

!% !* !$ # $ * %

!%

!*

!$

#

$

*

%

%

!%#!("!*%!*#

!""

!""

!$%

!$%!$%

!$#

!$#

!$#!&"

!&"

!)

!)

!"#$$%&"#$$%

!% !* !$ # $ * %

!%

!*

!$

#

$

*

%

!)#

!%#

!(#

!*#

!"#

!$#

!&#

!"#!"$

!%&

!%'

!%(!()

!()

!()

!("!("

!("

!#

!#

!#

!$

!$

!"#$$%

&"#$$%

!' ' '!'!$!"!%!('(%"$'

!*(!'$!$)

!$)

!$+

!$+!"$

!"$

!"$!%)

!%)

!%)!%+

!%+

!%+!("

!("

!)

!)

!"#$$%

&"#$$%

!' ' '!'!$!"!%!('(%"$'

!)+

!*+

!'+

!$+

!"+

!%+

!(+

%

Figure 5.5: Top left: Simulated lateral variations of B1 for (z0, r0) = (1, 4) cm and Z = z0. Topright: Experimental variations of B1 of the fabricated coil. Bottom left: Simulated lateral variations of B1

for (z0, r0) = (1, 2) cm and Z = z0. Bottom right: Experimental variations of B1 of the fabricated coil. Con-tours are in percent variation from the maximum B1 in the considered plane. Such plots can be realized fordi!erent Z, showing that the profile displays little changes over about 7 mm in depth for (z0, r0) = (1, 4) cmand about 5 mm in depth for (z0, r0) = (1, 2) cm.

in good agreement with the DC measurements, considering the uncertainties on these estimations andthe possible losses due to the increase in the coil resistance related to the skin depth.

We can compute the best-case SNR of the coil, provided its design parameters, approximating B1

to be homogeneous over the sample volume, and using formula 1.75 and equation 1.65 for the thermalequilibrium magnetization. The B1 homogeneity assumption is not unfounded as the detected vol-ume extent along Oz (where the B1 inhomogeneity is greatest) is limited by the detection bandwidth,because of the strong B0 gradient. Indeed, with a quality factor of 20, the maximum detection band-width is about 700 kHz, corresponding to about 5 mm. In practice, the working detection bandwidthis smaller than 100kHz, corresponding to about 700 µm.

At a temperature of 300 K and in a field of 0.33 T, the polarization of protons in water is about1.06 ( 10#3 A m#1. As a result, the best-case SNR per unit volume and per square root of Hz that


we may expect is about 2.84( 1010 Hz12 m#3. Considering a standard working bandwidth of 100 kHz

and a sample 1 µm thick and 10 mm wide, we can expect a SNR of 0.009. Signal averaging may helpas it may improve the SNR by a factor

*N for N accumulations, provided the noise is incoherent.

Achieving a SNR of 1 would require about 250 scans of 50 accumulated echos (using a CPMG andassuming grossly that no decay is observed in the echos). Considering a fast-relaxing sample such asolive oil (recycle time of 0.3 s), it would take about 75 seconds to reach a SNR of one and about twohours to reach a SNR of ten. This is neither unreasonable nor comfortable. Slices 10 µm thick in turnwould however yield a SNR of 10 within 90 seconds. Nevertheless, we observed a poorer experimentalSNR. We reached a SNR of about 10 within 5 minutes in conditions similar to the ones assumed in thisestimation. Though some of our assumptions lead to over-estimating the SNR, it does not account forsuch a di!erence. This loss of sensitivity indicates that other sources of noise are present.

Another important aspect of RF coils for NMR is their dead time. It is the time after the RF pulseis complete during which the voltage across the coil keeps oscillating due to mechanical oscillationsinduced by the pulse. These oscillations overwhelm the receiver, making the NMR signal unobservableduring that time. It is thus important to have short dead times, especially for NMR experiments ininhomogeneous fields, where the signal decays quickly. We observed in practice dead times shorterthan 20 µs. This can be related to the low quality factor Q of these coils (96), between 20 and 30(Q = "L

R , where ! is the signal frequency and L is the coil inductance). As a probe quality factor isusually dominated by the coil Q, a low value of this quantity implies a large bandwidth for the probe,which is advantageous in the context of large field gradients.

We did not extend further the study of these coils, but it is clear that much remains to be done inthe future to fully understand their behavior and to compare them to other types of coils.

5.2 NMR experiments with an open tomograph

Following the fabrication of the single-sided magnet and coils, we performed a few crude NMR ex-periments to demonstrate and start exploring the capability of the system. As the field is highlyinhomogeneous, the NMR signal is broadband and decays quickly. This decay is mostly related to thesample extent along Oz (about 7 µs for a sample 1 mm thick). This can be an issue when the RF coilhas a dead time longer than the signal duration. This problem can be solved by using Hahn echos,which enable "reviving" the signal an arbitrary time %1 after the last RF pulse (as long as %1 is shorterthan T2). We will proceed in what follows exclusively with echos, as they can also be used in CPMGtrains to improve SNR.

Reducing the extent of the detected sample (either physically or by reducing the bandwidth of theexcitation pulse (123)) allows to increase the signal duration, while decreasing the number of spinsparticipating to the signal.

Another problem accompanying NMR experiments in strong gradients is di!usion. As stated insection 1.1.4, di!usion in a gradient shortens the apparent T2, thus reducing the number of echos thatcan be collected. As a result, in order to achieve the best SNR, it is necessary to use low di!usionsamples. In the same time, this aspect can also be used to measure di!usion constants (123).

5.2. NMR EXPERIMENTS WITH AN OPEN TOMOGRAPH 159

5.2.1 Relaxation times and di!usion measurements

As we saw in section 1.1.1, the most elementary parameters of the NMR phenomenon to appear areits relaxation times. Such measurements can be used to discriminate various materials like oil andwater (181) or estimate the alteration of an object (94). While the strong static gradient forbids high-resolution NMR spectroscopy, the broad signals obtained are su"cient to measure these parameters,as no spectral information is required.

5.2.1.1 T1 measurements

A very simple inversion-recovery sequence (see (1) for example) was applied to demonstrate the capa-bility of the system to measure T1 relaxation times. A $ pulse is first applied, followed by a !

2 pulseafter a "recovery" time which is varied sequentially. The sequence was modified to increase SNR forthis system operating in an inhomogeneous field. We applied a CPMG sequence instead of the single!2 pulse, with a fixed delay of 70 µs between pulses and added together 1000 echoes. In addition, aphase cycle was used to compensate possible pulse imperfections. The initial $ pulse and the !

2 pulsewere in phase, alternating along x, !x, y, !y and the CPMG $ pulse where at 90º of the !

2 . Thisallowed to acquire signals with su"cient SNR within 4 scans for olive oil and 20 scans for distilledwater (di!usion is greater in water, thus damping the CPMG train faster than for oil).

Ox Oy

!1 !1 !1

Oy

!1 !1

Oy

!1 !1

Oy

t

!!!Ox

!"#$%&#"'

Figure 5.6: Sequence used to measure T1. A $ pulse is initially applied along Ox to invert the polarization.After a variable time %recovery, a !

2 along Ox is applied, followed by a CPMG train. The echos of the CPMGtrain are added together to improve SNR. For short %recovery, the system has not relaxed and the !

2 pulse flips thepolarization in the xOy, yielding a signal with opposite phase. As %recovery is increased, the inverted polarizationalong Oz decreases as it relaxes, and so does the opposite phase signal recorded during the CPMG train. Aftercanceling itself, non-inverted polarization along Oz increases, along with in-phase signal. We ensure this waywe record the full longitudinal relaxation curve.

! " # $ % &!!

!'"

!'#

!'$

!'%

&

()*+,-./

0,1,0!

234+5)*+6789:+;5+..)<6

(&,=,&'%#,.,>,!'!$

! !'? & &'? "!

!'"

!'#

!'$

!'%

&

()*+,-./

010

!

234+5)*+6789:+;5+..)<6

(&,=,!'&%,.,>,!'!@

A).7)99+B,C87+5D9)E+,<)9

Figure 5.7: Left: recovery curve of distilled water. The experimental points fit very well on a the theoreticalrelaxation law (equation 1.11) with a T1 of 1.84 s. Right: Recovery curve of olive oil with a much shorter T1 ata value of 0.18 s.


5.2.2 1-D profiling

As aforementioned, such a strong gradient allows a high localization of the NMR experiment and thuso!ers an excellent 1D resolution. The experimental results of section 4.2.2.2 indicated that a resolutionbetter than 20 µm is achievable in a cylinder of 1 cm diameter and 8 mm height. This must now beverified by NMR performance.

5.2.2.1 Point Spread Function and NMR simulations

A critical parameter of an imaging system is the resolution it can provide, that is to say the minimumdistance between two features that can be distinguished by the system. When the imaging systemcan be considered as linear, its resolution can be characterized by its Point Spread Function (PSF).Originating from optical imaging, it characterizes the transfer function of imaging system (consideredas a linear system). Following this definition, the PSF is the response of the system to a spatial impulse(a plane for 1D imaging, a line for 2D and a point for 3D), which corresponds a spatial spread of thesignal power. The resolution of the imaging system directly depends on the width of the PSF, astwo impulse separated by less than the width of the PSF will appear undistinguishable from a singlelarger feature. The definition of the width of the PSF and the corresponding resolution is not absoluteand di!erent criteria exist, depending on the PSF shape (e.g. the Rayleigh criterion for di!raction-limited optical systems with circular aperture). A commonly found definition is the Full Width at HalfMaximum (FWHM), which we will use here. The PSF merely gives the ultimate resolving capacity ofthe instrument. The resolution actually obtained depends also on the SNR of the image.

Formally, we can define the PSF as the function such that

F (!"r ) =-

f(!"r ) · PSF (!"r !!"r1)d!"r1 , (5.37)

where f is the input spatial distribution of signal power and F is the output spatial distribution. Asthe signal power must be conserved, we have

-PSF (!"r )d!"r = 1. (5.38)

In addition, if the input distribution is an impulse 2(!"r ), we have

F (!"r ) = PSF (!"r ). (5.39)

Thus, a practical way of estimating the PSF experimentally is to image a very small object, muchsmaller than the expected PSF. Another possibility is to use a very sharp edge. We have indeed (in1D for simplicity)

2(x) =d

dx(Heavyside(x)) . (5.40)

So that, if we have F (x) the image of a step function, we get

dF (x)dx

= PSF (x). (5.41)

Based on the density matrix formalism presented in section 1.1.2, we can perform simulations ofthe NMR signal for various sequences. These can be carried out by defining volume units (voxels)containing zero, one or more spins. These voxels mesh a desired measurement volume with a desiredspatial resolution. Assigning a static field B0(x, y, z) and B1(x, y, z) to each voxel, and assuming no


interaction between voxels, we can compute the response of each voxel to a given sequence and sumall their responses to obtain the overall response.

We can use the experimental SHE terms to reconstruct the static field in any point of the mea-surement sphere with excellent accuracy. In addition, we have seen that the spatial variations of theRF field can be reliably calculated. Thus, we can also compute a set of SHE terms of the RF field andreconstruct it straightforwardly anywhere in the volume. This provides an e"cient tool to generatethe field maps necessary for the mesh used in the NMR simulation. Thus, the NMR simulations canbe performed based on realistic field variations, giving estimations of expected position and relativeamplitude of the spectrum features.

Such simulations can also be used to calculate the PSF of the system by considering a slice extendedonly in the xOy plane. We show in figure 5.8 the computed PSF at di!erent heights, based on themeasurements of the magnet. The FHWM of the PSF remains below 20 µm on the whole range coveredand its minimum is about 2 µm at the center of the RoI. It degrades quickly when going above thecenter of the RoI while it remains fairly stable when shifting toward the magnet. We can suspect aresolution better than 10 µm over more than 8 mm depth may be achieved by shifting the center ofthe RoI towards the magnet.

!"# !$# !%# !&# # &# %# $# "##

#'#&

#'#%

#'#$

#'#"

#'#(

#'#)

!"#$%&

'()%*+,!-."*%/+,01.-

*+!",--*+!%,--*+#,--*+%,--*+",--

Figure 5.8: PSF of the system at di!erent elevation z (measured from the center of the RoI). The FWHMremains below 20 µm over the 8 mm covered. It is worth noting that the PSF does not degrade as fast whengoing toward the magnet than when going away from it. This was expected from the magnet measurements.The strong asymmetry of the PSF above the center can be related to the quadratic field profiles found in thisregion.

5.2.2.2 Experimental PSF and layer profiles

Due to the small SNR expected for samples of thickness smaller than 10 µm, we chose to use a glass-oilinterface to measure the PSF. Using oil instead of water helps getting higher SNR as the di!usioncoe"cient is smaller. This permits a longer apparent T2 and allows us to collect more echos with aCPMG train. The sample was a layer of olive oil contained in between two microscope glass slides.The following PSF profiles were obtained based on spectra collected with 1000 scans of 50 accumulatedechos with a delay of 2.6 ms between each echo. The RF coil used was the 2 cm radius surface coilmentioned earlier. Given the spatial variations of B1 associated to this coil, we can expect a goodspatial selection of the excited region within 1 cm2. This was verified by observing the absence of


increase of the PSF FWHM when changing the sample extent from about 1 cm2 to about 2 cm2.A di"culty coming along the targeted high resolution (about 10 µm) is that the result is extremelysensitive to the orientation of the sample. If the sample layer is tilted by only 0.5º compared to thexOy plane, the sharp edge 10 mm wide is seen by the system as a smooth edge about 90 µm deep. Itis thus crucial to achieve an excellent alignment of the sample.

We show in figure 5.9 an exemple of measured PSF slightly below the center of the RoI. We startfrom the spectrum (absolute value) of the oil sample. We then di!erentiate the spectrum, giving anon-normalized PSF. As the sample is very thin, we can observe the response to the rising and fallingedges (respectively positive and negative peak). We must stress here that the oil layer used for theedge measurement must be larger than the PSF. Were it not the case, the line shape of the layer wouldbe dominated by the PSF, and we would thus be computing the derivative of the PSF instead of thePSF itself. The symmetry of the two peaks linked respectively to the rising and falling edge is a goodindication that this condition is verified. It is the case in figure 5.9 and the observed FWHM is about15 µm.

!"## !$# !%# !&# !'# # '# &#

!(

#

(

"#

!"#$%&'()*+!

,-,.

/*+01.

!'## !"(# !"## !(# # (# "## "(# '##!"

!#)(

#

#)(

"

")(

!"#$%&'()*+!

,-,.

/*+01.

!*# !'# !"# # "# '# *##

(

"#

"(

!1#$%&'()*+!

,-,.

23)4')567*89%:&*+;</.

Figure 5.9: Measured PSF. Top left: a spectrum of the oil sample with a sharp edge is recorded. The width ofthe layer decreases with height, explaining the asymmetric line shape. The signal was zero-padded to twice itsduration. Bottom left: di!erentiated spectrum. The non-normalized PSF appears for both the rising and thefalling edge. The abscissae scale is inverted compared to the spectrum scale as the gradient is negative. Right:Zoom on the PSF of the system. The width is about 15 µm for both the rising and the falling edge.

An advantage of this magnet is that the resolution is maintained over a wide range of elevations.We recorded the PSF of the system 2 mm and 4 mm away from the previous one and observed thatthe FWHM degraded to 20 µm (see figure 5.10), demonstrating the capability for 1D profiling over awide range with high resolution. The larger PSF at center and the absence of significant degradationwhen going to higher z indicates that the line width is dominated here by the alignment of the sample.

As the system o!ers a high resolution over several millimeters and the RF coil delivers enough B1

to perform broadband pulses (2 µs pulses are possible, corresponding to a spectral width of 500 kHzwell within the bandwidth of the tuned circuit of Q = 20, translating to a 3.6 mm slice thickness), it is


!"## !$# # $#

!%

!&

!'

#

'

&

%

!"#$%&

'%()*+,-."#'

/0/&

# '# &# %# (# "## "'#

!&

!'

#

'

&

!"#$%&

'%()*+,-."#'

/0/&

Figure 5.10: Left: Measured PSF 2 mm above the the position used in figure 5.9, with the same sample.The FWHM is slightly degraded to 18 µm. Right: Measured PSF 4 mm above the position used in figure 5.9,with the same sample. The FWHM is slightly degraded to 20 µm. This demonstrates it is possible to achievehigh-resolution 1D profiling over a wide volume.

possible to record 1D images in one experiment without moving the sample or changing the observationfrequency. We show in figure 5.11 the image of a sandwich of microscope glass slides interleaved by oillayers. Six oil layers are present, and each glass slide is about 150 µm deep, leading to a total thicknessof about 0.75 mm. The changes in peak shapes are not due to the magnet but are actual sample shapevariations. This was confirmed by moving the sandwich by a few 1

10 of millimeters and observing nochange in the NMR profiles of the layers. The increase in width might be explained by the decrease inweight above each layer as z rises. The surface tension would indeed cause the liquid to push up theabove glass slide, increasing the layer thickness. The amplitude variation remains unexplained for themoment, as the frequency o!set cannot explain such dramatic amplitude variations.

!"## !$## !%## # %## $## "###

!

%

"

$

#

"

$

%&'()*

+),-./012&'+

343*

Figure 5.11: Image of a sandwich glass-oil containing six layers of oil. The image thickness from the first oillayer to the last is about 800 µm (112.5 kHz), matching the depth estimation based on the glass slide thickness(150 µm).

We have thus shown the capability of these single-sided system for two main applications. Thedemonstrated imaging performances are besides on par with what was expected from the field profiling.


This coherence validates the relevance of the method used to characterize the magnet. We havenevertheless not explored all the possibilities o!ered by this system, such as T2 measurements, di!usionconstant measurements, application to useful samples (such as for example imaging of cultural heritageobjects), and 3D imaging using x and y flat gradients designed based on the method described here.

Chapter 6

Conclusions and perspectives

We presented an original consistent and systematic method for the design, fabrication and test of per-manent magnet-based systems for NMR. We applied it to design several magnet models and fabricatetwo prototypes. While the current experimental performances of these prototypes call for improve-ment, the results are very encouraging and su"cient to already conduct useful NMR experiments.These experiments validate the results obtained during the magnet characterization and confirm therelevance of the method for NMR magnets.

Extensive analytical developments based on the expansion of the magnetic potential and of thefield components in spherical harmonics were shown, providing a basis for the exact design of magnetsystems with desired profiles. In the light of this formalism, the importance of axial symmetry washighlighted and we focused our work on configurations featuring such a symmetry. We also providedsome theoretical results for finite length Halbach dipole cylinders, which are of particular importancebecause of the high magnetic field they can create.

Highly homogeneous field profiles can be achieved by considering only the main component ofthe field, making the use of SHE straightforward. However, it is sometimes desirable to obtain astrong uniform field gradient in a given direction. This situation requires more care as the transversecomponent variations, also called concomitant gradients, may impact significantly the field modulus.We exposed a framework for this case, based on the SHE of the main component (it is su"cient toknow one component to know all of them).

In addition to the aspects of homogeneity necessary to NMR, the economics of the fabricatedmagnet are important. The cost of a magnet is directly linked to the amount of magnetic materialused. It is thus important to make the most of each cubic centimeter of material and we addressedpartly this issue by giving some considerations on the e"ciency of a magnetic part depending on itslocation and orientation compared to the center of the RoI in the case of an axially symmetric magnet.

In addition to design tools, a measurement methodology has been studied, in order to providehigh precision field measurements and mapping. It is possible to retrieve the SHE terms from suchmeasurements and we exposed a detailed analysis of the precision and accuracy that can be expected.A method to measure the SHE terms of the main component in a strong field gradient with goodprecision and reasonable accuracy has also been exposed.

Based on the analytical results, we developed the design of a highly homogeneous in situ axiallysymmetric magnet (1 ppm in 2 cm DSV) and of a simple very homogeneous (1 ppm in 1 cm DSV)Halbach. The planar symmetry cannot be used in the context of ex situ magnets where half of the spacemust remain free of magnetic sources. Their design is thus much more challenging. We developed inaddition a design for a very homogeneous single-sided magnet (10 ppm in 8 mm DSV) located 18 mmaway from the magnet. The cancellation of the first degree costing a lot in field strength, we also

165

166 CHAPTER 6. CONCLUSIONS AND PERSPECTIVES

proposed a model of magnet generating a constant gradient along the axis of the magnet with higherfield strength. This model o!ers a field strength almost three times greater than the homogeneousdesign. Uniformity of the field across the axis and constance of the gradient along Oz have both beenexplored in theory and the e!ect of fabrication imperfections, along with possible corrections throughgeometry alterations are detailed. This study shows that despite the variability of the magnet, thecorrection of the main terms can be carried out using a correction matrix built out of simulations,saving significant measurement time. We supported our analytical work with models created with thecode Radia which was found to be extremely appropriate for the calculation of these structure.

Following our theoretical considerations, we realized two magnet prototypes to demonstrate thevalidity of these developments. A first low-cost in situ magnet generating a longitudinal field of120 mT (about 5 MHz proton frequency) was proposed, achieving a 12 ppm homogeneity over avolume of about 2 mm long by 1 mm diameter after shimming. The shimming method in this caseconsisted in using several additional small magnets that could be approximated by dipoles to computetheir appropriate positions, compensating the magnet imperfections. The fabrication of this magnethas also been the occasion of laying out a methodology for cube measurement and sorting, and todemonstrate the possibility of reproducing accurately the magnet imperfections with Radia, based oncubes measurement and simple parameters. Thus, we can use simulations to identify imperfections inthe parameters over which we have control (tilt of a ring, for example).

We also built a single-sided prototype generating a constant-gradient and based on the theoreticaldesigns we proposed. However, we made the unfortunate design choice of a gradient constant alongOz, which is of small interest for low-field NMR (because the gradient is large compared to field). Wewere able to partially achieve the said profile through corrections of the magnet geometry based ona calculated correction matrix. Some parameters that were not well-controlled like the centering ofthe measurement system and some fabrication tolerance are most likely responsible for the unexpectedgreater contribution of higher degree terms. We then transformed this profile to a more useful gradientwith constant field across the axis by another geometry correction. This correction however did notfulfill the theoretical performances of the proposed model because of the limitation imposed by therange of the adjustable part and the lack of control of higher degree terms due to the required largemodification of the magnet. Final homogeneity in the central plane is about 60 ppm and degrades asthe observation plane moves away from the center of the RoI, up to 200 ppm 5 mm away from thecenter. The magnet will be updated in the future by replacing the mobile magnet parts by others oneswith an appropriate geometry, while keeping most of the magnet unchanged. The situation could alsobe improved by applying the shimming method of additional magnet blocks. This actually raises a realquestion for the fabrication strategy. Keeping mobile parts increases tolerances on parts positioningand thus increases the initial imperfections. These parts have in addition a limited impact on thedi!erent SHE terms and cannot fight unexpected imperfections. Adding small magnet parts afterfabrication is most likely more versatile. Nevertheless, space constraints may also limit the impactof these additional parts and the final shimming method might be to use both movable parts andadditional parts.

In addition to the magnet generating the external field, the NMR experiment requires an excitationand detection system. We thus also developed a novel planar RF coil generating a suitable excitationfield with high e"ciency. The method used for its design can also be used for gradient coils, andRF homogeneity control. However, many experimental aspects of this type of RF coils remain to bestudied in details. This includes the noise behavior, the interaction with the main magnet, extensivecomparison with other types of coils. The exploration of shielding techniques (182) could also be asource of improvement for these coils.

Using our magnet prototype and excitation-detection coils, we were able to perform a few NMR

167

experiments including accurate measurements of relaxation times and high resolution tomography(10 µm) in a large volume (10 mm diameter by 6 mm height) at a remote distance of the magnet(about 2 cm) and the coil (about 4 mm). We have thus gone through all the steps of the fabricationof a single-sided NMR system for high-performance tomography.

While the gradient uniformity performances could still be improved, much remains to be done toexplore all the possibilities of NMR experiments already o!ered by this system. In particular, itsuse with real applications remains to be done to observe its performances in conditions not as wellcontrolled as presented in this work.

The control of the field profile of such a strong gradient is very di"cult due to the di"culty tomeasure it with precision. The very encouraging results obtained with this first magnet prototype letthink that the fabrication of a homogeneous prototype based on the theoretical model we give wouldbe very successful as the measurement conditions would be considerably eased. As the field profile ofthe magnet and the field profile of the RF coil can be well-controlled with the methods we propose,it seems possible to achieve a very good field correlation in order to perform high resolution NMRspectroscopy based on nutation echo, taking advantage of the field strength this magnet provides.

The merits of permanent magnets are attractive to many fields where NMR was scarcely used.While it has been shown that NMR can provide useful information for rheology studies, apparatususually used for this purpose are of small size and much less costly than a typical high-field NMRspectrometer (183). Permanent magnets could bring the benefits of NMR to rheology by an appropriatestructure fitting on existing rheology apparatus.

Another important advantage of permanent magnet is their lack of current leads, making easy theirtransportation. This can be also put to use in the context of Magic Angle Turning (MAT). MAT hasbeen proposed to improve in vivo resolution for imaging and localized spectroscopy. While the firstexperiment involved the turning of the subject (162), turning the field seems much more desirable.The method described in this work is suitable for the design of a magnet generating a strong field atthe magic angle which could be turned at su"cient speed (10Hz).

One major drawback of these single-sided magnets is their low field strength, which reduces tremen-dously the SNR of the NMR experiment. However, numerous methods to increase the polarization(hyperpolarization) exist. While some of these methods are very specific to a nucleus (184), someseem more flexible, like Dynamic Nuclear Polarization (DNP). DNP allows the transfer of the highpolarization of the orbital electrons to the nucleus. This requires to irradiate the sample at the elec-tron paramagnetic resonance (EPR) frequency, which is reasonable at the magnet field we consider inour magnets (in the case of the constant gradient, it corresponds to about 9.24 GHz). While DNPis currently a growing field of research at high fields, according to (185), the field generated by ourgradient magnet (0.33 T) is exactly at the optimum for maximum polarization gain. Thus, there aremost likely very good prospects in the conjugation of DNP with permanent magnets and it seems thelogical next step in the development of these systems.

Another possibility of significant improvement resides in the detection coil. Cryogenic surface coilhave already been proposed in the past, o!ering very high quality factor and thus high sensitivity (186).While this limits the detection bandwidth, we can suppose that most applications requiring micronresolution do not have a very large extent and thus do not require a very large bandwidth.

This brings us to the topic of miniaturization. NMR detection of very small objects and minia-turized NMR systems (microcoils (142), miniaturized arrays (187), microfluidics and biosensors (188))have been growing in the past years. Magnets such as the ones proposed in this work, significantlyreduced in size could benefit to such experiments (it is possible to scale permanent magnet structureat will. The region of homogeneity scales with the dimensions of the magnet while the field strengthis not modified.)

168 CHAPTER 6. CONCLUSIONS AND PERSPECTIVES

In a farther future, we can envision a wide development of such magnet systems for routine medicalMRI. With proper engineering, and after improvement of the detection sensitivity, compact NMRsystems could be produced in large quantities at low cost and made a!ordable to small medical o"ces.

Appendices

169

Appendix A

Magnetic materials

The magnetic properties of materials depend on their constituents and their crystalline and chemicalstructure. There exists di!erent types of magnetic materials for which the magnetic properties origi-nate from di!erent phenomenon. One can distinguish diamagnetism, paramagnetism, ferromagnetism,antiferromagnetism, ferrimagnetism and helimagnetism.

Diamagnets Such materials have a small and negative susceptibility (, 1 !10#5). Diamagnetismis a general e!ect that occurs in all materials. The diamagnetic susceptibility is due to a Lenz’s lawalike e!ect : the orbiting electrons will see their orbit modified in order to oppose the magnetic fieldvariations. A more detailed description of diamagnetism theory (Langevin’s theory) can be foundin (29). This e!ect is very weak and is mostly noticed in diamagnetic materials, which are made outof elements without permanent electronic magnetic moment (because all electrons are paired). Thee!ect of diamagnetism may also be observed in weak paramagnetic materials. Examples of diamagneticmaterials are copper, mercury, gold, bismuth and most organic compounds. Such materials are usuallyconsidered as non-magnetic, as they hardly acquire a magnetic moment. Superconductors are a specialexample of diamagnets with a susceptibility , = !1. They are hence very strong diamagnets.

Paramagnets Paramagnetic materials exhibit a positive susceptibility usually very small to moder-ate (, 1 10#3 ! 10#5). The elements constituting these materials usually have unpaired electrons sothat a net electronic magnetic moment arises (some exceptions with an even number of electrons exist).The paramagnetic susceptibility can be derived from quantum mechanics considerations, resulting inthe Brillouin function (29). Following these considerations, it appears that , is highly dependent onthe temperature in most paramagnets (where the magnetic moments of the participating electronsfollow the Boltzmann statistics). The Curie-Weiss law gives a simple expression of this dependency

, =C

T ! TC(A.1)

where T is the temperature in kelvin, TC is the Curie temperature and C is the Curie constant. Theseresults are in very good agreement with experimental measurements. In metals, the paramagnetismdue to free electrons following the Fermi statistics is independent of temperature. Typical examples ofparamagnets are aluminum, platinum and manganese.

Ferromagnets The ferromagnetic susceptibility is positive and large to very large (, 1 50!10000).Ferromagnets usually exhibit hysteresis, anisotropy, and most important have non-zero remanence.

171

172 APPENDIX A. MAGNETIC MATERIALS

Ferromagnetic materials have elementary constituents with a net atomic magnetic moment. Ferro-magnetism is the result of the existence of a magnetic order in the material. Magnetic domainsconstituted of many aligned magnetic moments exist even when the material is demagnetized. Thebulk magnetization is the result of the alignment of the di!erent domains (one usually speaks aboutthe growth of the domains favored by the external magnetic field).

Ferromagnets are highly sensitive to temperature. There exists a critical temperature, the Curietemperature above which the material becomes paramagnetic and all properties such as coercivity andremanence vanish. This can be explained in short by the thermal agitation energy overcoming themagnetic order energy, making the domains disappear. A modified Curie law, the Curie-Weiss law canbe derived in order to characterize the variation of susceptibility with temperature :

, =C

T ! Tc(A.2)

where Tc is the Curie temperature.Typical examples of ferromagnets are iron, cobalt, nickel, several rare earth metals like neodymium

and samarium and their alloys.

Antiferromagnets Some paramagnetic materials see their susceptibility reach a maximum at atemperature TN called the Néel temperature. Under this temperature, the structure of the materialis suddenly modified and it becomes antiferromagnetic. The structure under the Néel point is orderedand constituted of two sub-lattices of identical magnetic moments of opposed direction. The resultingmagnetization within a domain is zero, and hence the bulk magnetization is zero. As a result, they areof little interest as sources of magnetic field. However, these substances are of high interest in materialscience to help understand magnetism in materials. Typical examples are chromium below 37ºC andmanganese below 100K.

Ferrimagnets Ferrimagnets, or ferrites are very commonly found and the first magnetic materialever found by mankind, magnetite (Fe3O4) is a ferrite. They are widely used for electronic inductors,electrical transformers, electromagnets. Ferrites are very similar to antiferromagnets, their di!erencebeing that the two sub-lattices are di!erent. The net magnetization within a domain is hence non-zeroand the bulk magnetization can be non-zero.

Other types of magnetism in materials exist, such as helimagnetism, but these have no relevanceto this work.

A.0.3 Magnetic materials modeling

When delivering a material, manufacturers usually provide the experimental magnetization curve orhysteresis cycle (to saturation) of the material and may also give the anisotropy field. Such data aresampled experimentally and in the course of simulations, it may be more convenient to use simpleanalytical models of the response of materials to an external field, which give the relevant behaviour ofthe material with a good approximation, before using experimental data in a final stage of the design.

As we just saw, the response of a material to an external field is conditioned by its constitutiveequation, which can be written in a general way

!"M = f(

!"H ). (A.3)

Before going any further in the modeling of this constitutive equation, we shall first distinguish thetwo main families of materials :

173

• "Soft" materials exhibit no to very little hysteresis and are generally isotropic. A typical exampleis soft iron, which is used in electromagnet, electric transformer, and more generally as a magneticcore.

• "Hard" materials which exhibit moderate to strong hysteresis and demonstrate usually stronganisotropies (generally unidirectional). Typical examples are all permanent magnet materialssuch as ferrite, NdFeB, SmCo, AlNiCo, etc.

A.0.3.1 "Soft" materials

The response of these materials to an external field can be described by a simple magnetization curvewith a variable susceptibility. It is useful in such cases to use the di!erential susceptibility dM

dH . Wewill here remain concern with regimes of "high" external magnetic fields, that is to say above 1 mT,so that we can neglect the initial part of the curve, governed by the Rayleigh law (30). Several formscan be used, depending on the shape of the magnetization curve (31)

M1 =kHF

1 +(k H

MS

)2(A.4)

M2 =kH

1 +222k H

MS

222(A.5)

M3 = MS tanh!

kH

MS

", (A.6)

the parameters k and MS being used to adjust the curve to experimental data. Figure A.1 shows thegeneral shape of the curve for each equation.

1.0

0.8

0.6

0.4

0.2

0.00.0005 0.0010 0.0015 0.0020 0.0025 0.00300.0000

!0H (T)

M (u

nits

of M

S)

Figure A.1: General shape of magnetization curve of soft materials using simple analytical description. M1

is plotted as the blue curve, M2 as the purple curve and M3 as the yellow one. k was chosen to be 3000 whileµ0MS was set to 1T.


A.0.3.2 "Hard" materials

Hard materials usually also feature uniaxial anisotropy and we will cover here only such materials. Inthe uniaxial situation, the response of the material will change greatly depending on the orientation ofthe external field. It is necessary to consider the case where the field is parallel to the easy axis, andthe one where it is orthogonal.

Parallel case The external magnetic field is parallel to the magnetization and it is possible tomeasure both of them algebraically along the same axis. The external field favors the growth ofmagnetic domains in the same direction. The Hysteresis e!ects is due to a phenomenon of anchoringof the domain walls due to the crystalline structure and its defects. Hysteresis make the behavior of amaterial dependent on its magnetic history.

The data provided by the manufacturer correspond usually to a part of a cycle of hysteresis whendecreasing the external field H from a very large positive value to a very large negative value. Thematerial is hence saturated at each extremity of the curve (we will call it the saturated hysteresis cycle).It is however the only case where the behavior of the material is completely known. If the materialis subjected to an opposite magnetic field which is then brought back to zero, the magnetization willincrease again but will not follow the saturation hysteresis curve because of irreversible losses. Theevolution of the magnetization is then following a new hysteresis cycle inside the saturated cycle. Wesuppose in what follows that any state of the material is reached by decreasing the external field froma state where the material was saturated, so that we remain on the curve of the saturated hysteresiscycle.

We can then use three simple mathematical forms to describe the saturated hysteresis cycle. Theseare similar as the ones used for soft materials but include an additional parameters, the intrinsiccoercivity HCi. We impose that the magnetization must cancel for H = !HCi in the upper part ofthe cycle (H going from positive to negative) with +1 and for H = HCi in the lower part of the cycle(H going from negative to positive) with !1.

M1 = MS

k(

HHCi

± 1)

F1 + k2

(H

HCi± 1

)2(A.7)

M2 = MS

k(

HHCi

± 1)

1 + k222 HHCi

± 1222

(A.8)

M3 = MS tanh*k

!H

HCi± 1

"+. (A.9)

The parameter k provides the control over the "squareness" of the cycle. Figure A.2 shows the generalshape of such curves for the magnetization while figure A.3 presents the variations of the field inductionB during that cycle. The second quadrant (H < 0 and M > 0) is usually the part interesting the userof the material and most manufacturer only provide this quadrant.

Orthogonal case The crystalline structure of the material imposes constraints on the magneticdomains orientation. These constraints are contained in a term of anisotropy energy EAn(',&). Whenan external field is applied at an arbitrary angle, a competition arises between this anisotropy energyand the field energy EH , so that the total energyETot can be written

ETot = EAn(',&) + EH , (A.10)

175

1.0

0.5

-0.5

-1.0

2-2 4-4 0

!0H (T)

M (M

S uni

ts)

Figure A.2: General shape of saturated hysteresis cycles for the magnetization using simple analytical de-scription. M1 is plotted as the red curve, M2 as the green curve and M3 as the blue one. k was chosen to be 5while µ0MS and µ0HCi were set respectively to 1T and 2T .

4

2

-2

-4

0 1 2 3-1-2-3

!0H (T)

B (!

0MS u

nits

)

Figure A.3: General shape of saturated hysteresis cycles for the field induction using simple analytical de-scription. The variations of B = µ0(H + M) are shown for M1 as the red curve, M2 as the green curve and M3

as the blue one. k was chosen to be 5 while µ0MS and µ0HCi were set respectively to 1T and 2T .

where EH = !µ0!"MS ·

!"H and

!"MS is the magnetization of the domain considered. It shall be considered

equal to the saturation magnetization.


In the case of uniaxial anisotropy, the lowest order term of EAn can be written as

EAn = K sin2 ', (A.11)

where ' is the angle of the magnetization with respect to the easy axis and K is a positive anisotropyconstant. We have then for an applied field orthogonal to the easy axis

ETot = K sin2 ' ! µ0MSH$ sin ', (A.12)

as the angle between!"MS and

!"H$ is !

2 . Hence, the minimum of energy is achieved for ' verifying

sin ' =MSH$

2K. (A.13)

One can then define a quantity homogeneous to a field, the anisotropy field HA such that

HA =2K

MS. (A.14)

HA is a fictitious quantity and has no physical existence. Modern materials such as NdFeB or SmCoexhibit anisotropy fields greater than 20T. As a result, ' is most of the time very small and it is possibleto neglect the orthogonal component of the magnetization.

Arbitrary orientation If it is not possible to neglect the orthogonal component of the magnetiza-tion, the preceding equations (A.16,A.17,A.18)can be modified by taking into account the decrease ofthe longitudinal magnetization and the growth of the transverse magnetization. We can consider thetwo components of the field, H) parallel to the easy axis and H$ orthogonal to it. The magnetization!"M hence has two components M) and M$. Replacing MS in A.7,A.8 andA.9 by

MS cos ' = MS

3

1!!

H$HA

"2

, (A.15)

we obtain

M1) = MS

3

1!!

H$HA

"2 k(

HHCi

± 1)

F1 + k2

(H

HCi± 1

)2(A.16)

M2) = MS

3

1!!

H$HA

"2 k(

HHCi

± 1)

1 + k222 HHCi

± 1222

(A.17)

M3) = MS

3

1!!

H$HA

"2

tanh*k

!H

HCi± 1

"+. (A.18)

andM123$ = MS

H$HA

(A.19)

so that a full model of the hysteresis cycle can be obtained based on the 4 quantities k, MS , HCi andHA.

Appendix B

Legendre functions and associatedLegendre functions

This section summarizes all the identities of Legendre functions used throughout this work and is byno mean a full description of Legendre functions and their properties. More detailed introductions canbe found for example in (116; 26; 32).

The Legendre polynomials can be defined by the Rodrigues’ formula

Pn(x) =1

2nn!

!d

dx

"n

(x2 ! 1)n. (B.1)

The associated Legendre polynomials can be derived from the Legendre polynomials through thisidentity

Pmn (x) = (1! x2)

m2

dm

dxmPn(x), (B.2)

where we have ignored the (!1)m factor that can occur in some definitions.In this work, we are concerned mostly about functions of the type Pm

n (cos ') with 0 < ' < $, sothat

*1! cos2 ' = sin '. These functions feature a number of functional relations, starting with the

di!erentiation in '. A few of these identities reported in the literature are given here ((116; 26; 117)

sin 'dPm

n (cos ')d'

= n cos 'Pmn (cos ')! (n + m)Pm

n#1(cos ') (B.3)

sin 'dPn(cos ')

d'= !(n + 1) cos 'Pn(cos ') + (n + 1)Pn+1(cos ') (B.4)

dPmn (cos ')

d'= !1

24Pm+1

n (cos ')! (n + m)(n!m + 1)Pm#1n (cos ')

5(B.5)

sin 'dPm

n (cos ')d'

= n cos 'Pmn (cos ')! (n + m)Pm

n#1(cos ') (B.6)

sin 'dPm

n (cos ')d'

= ! sin 'Pm+1n (cos ') + m cos 'Pm

n (cos ') (B.7)

sin 'dPm

n (cos ')d'

= (n + m)(n!m + 1) sin 'Pm#1n (cos ') + m cos 'Pm

n (cos ') (B.8)

sin 'dPm

n (cos ')d'

= (n + m! 1)Pmn+1(cos ')! n cos 'Pm

n (cos ') (B.9)

177

178 APPENDIX B. LEGENDRE FUNCTIONS AND ASSOCIATED LEGENDRE FUNCTIONS

A few other identities allow to step up or down orders and degrees (116; 26; 117):

(2n + 1) cos 'Pmn (cos ') = (n + m)Pm

n#1(cos ') + (n!m + 1)Pmn+1(cos ') (B.10)

2m cos 'Pmn (cos ') = ! sin '

4Pm+1

n (cos ') + (n + m)(n!m + 1)Pm#1n (cos ')

5(B.11)

(2n + 1) sin 'Pmn (cos ') = Pm+1

n+1 (cos ')! Pm+1n#1 (cos ') (B.12)

(2n + 1) sin 'Pmn (cos ') = (n + m)(n + m! 1)Pm#1

n#1

!(n!m + 1)(n!m + 2)Pm#1n+1 (cos ') (B.13)

Pm+1n#1 (cos ') = cos 'Pm+1

n (cos ')! (n!m) sin 'Pmn (cos ') (B.14)

Pm+1n#1 (cos ') =

2m

sin 'Pm

n (cos ')! (n + m)(n + m! 1)Pm#1n#1 (cos '). (B.15)

Appendix C

Derivation of SHE for dipoles

The scalar potential due to a dipole can be written as

&2 =!"M ·!"u4$R2

(C.1)

= ! 14$

!"M ·!")P

1R

(C.2)

where the subscript on ) indicates derivatives are taken in P. Hence, the SHE of the potential of adipole derives directly from the SHE of 1

R . We shall now concentrate on the SHE of 1R .

P!0

"0

O y

z

x

!1

R0

"1

M1

Q

"

!r

R

Figure C.1: Definition of the di!erent spherical coordinates relevant to the dipole.

Transforming to spherical coordinates, with!!"OP (R0, '0, &0),

!!"OQ(r, ',&), and

!"M(M1, '1, &1) as

shown on figure C.1, we can write

R =.(r sin ' cos &!R0 sin '0 cos &0)2 + (r sin ' sin&!R0 sin '0 sin&0)2 + (r cos ' !R0 cos '0)2

/ 12

=.r2 + R2

0 ! 2rR0(cos ' cos '0 + cos(&! &0) sin ' sin '0)/ 1

2

179

180 APPENDIX C. DERIVATION OF SHE FOR DIPOLES

hence, if r < R0,

1R

=1

R0

1F1 +

(r

R0

)2+!2 r

R0[cos ' cos '0 + cos(&! &0) sin ' sin '0]

(C.3)

=1

R0

"$

n=0

!r

R0

"n

Pn(cos ' cos '0 + cos(&! &0) sin ' sin '0). (C.4)

The addition theorem of spherical harmonics (26) states that

Pn(cos ") = Pn(cos '1)Pn(cos '2) + 2n$

m=1

(n!m)!(n + m)!

Pmn (cos '1)Pm

n (cos '2) cos m(&1 ! &2) (C.5)

where cos " = cos '1 cos '2 + sin '1 sin '2 cos(&1 ! &2).As a result, we can write

1R

=1

R0

"$

n=0

!r

R0

"n%Pn(cos '0)Pn(cos ')

+n$

m=1

2(n!m)!(n + m)!

Pmn (cos '0)Pm

n (cos ') cos m(&! &0)

&(C.6)

This expression can be plugged in equation C.2. We will first express the di!erent cartesiancomponents of the gradient in order to ease the calculation of the scalar product. We can use thefollowing identities for this purpose:

+

+x0= sin '0 cos &0

+

+R0+

cos '0 cos &0

R0

+

+'0! sin&0

R0 sin '0

+

+&0(C.7)

+

+y0= sin '0 sin&0

+

+R0+

cos '0 sin&0

R0

+

+'0+

cos &0

R0 sin '0

+

+&0(C.8)

+

+z0= cos '0

+

+R0! sin '0

R0

+

+'0. (C.9)

We will have to make extensive use of the relations given in appendix B to complete the di!erenti-ations and re-arrangement of terms. We can also separate axial and skewed parts as the di!erentiationis done on the dipole coordinates. The easiest terms are the axial ones in the z component. We haveindeed for a single order n, using some of the previous identities

+

+z0

!rn

Rn+10

Pn(cos '0)"

=1

R20

!r

R0

"n 4! (n + 1) cos '0Pn(cos '0)

+(n + 1) cos '0Pn(cos '0)! (n + 1)Pn+1(cos '0)5

= ! 1R2

0

!r

R0

"n

(n + 1)Pn+1(cos '0). (C.10)

We can then tackle the axial terms in the x and y components. For example, we have for x,

181

+

+x0

!rn

Rn+10

Pn(cos '0)"

=1

R20

!r

R0

"n

cos &04!(n + 1) sin '0Pn(cos '0)! cos '0P

1n(cos '0)

5

=1

R20

!r

R0

"n

cos &01

2n + 14! (n + 1)P 1

n+1(cos '0) + (n + 1)P 1n#1(cos '0)

!(n + 1)P 1n#1 ! nP 1

n+1(cos '0)5

= ! 1R2

0

!r

R0

"n

cos &0P1n+1(cos '0) (C.11)

In the same fashion, we get for y

+

+y0

!rn

Rn+10

Pn(cos '0)"

= ! 1R2

0

!r

R0

"n

sin &04P 1

n+1(cos '0)5. (C.12)

We now turn to skewed terms in z:

+

+z0

!rn

Rn+10

Pmn (cos '0)

"=

1R2

0

!r

R0

"n (! (n + 1) cos '0P

mn (cos '0)

!n cos '0Pmn (cos '0) + (n + m)Pm

n#1(cos '0))

= ! 1R2

0

!r

R0

"n

(n!m + 1)Pmn+1(cos '0) (C.13)

We are finally left with skewed terms in the x and y components. We have for the x component

+

+x0

!rn

Rn+10

Pmn (cos '0) cos m(&! &0)

"= A

=!

sin '0+

+R0+

cos '0

R0

+

+'0

"·

!rn

Rn+10

Pmn (cos '0)

"cos &0 cos m(&! &0)

! m

R20 sin '0

!r

R0

"n

Pmn (cos '0) sinm(&! &0) sin&0.

The terms in &0 can be transformed into two factors,

cos &0 cos m(&! &0) =12

(cos(m&! (m + 1)&0) + cos(m&! (m! 1)&0))

sin&0 sinm(&! &0) =12

(cos(m&! (m + 1)&0)! cos(m&! (m! 1)&0))

We have thus

A =1

2R20

!r

R0

"n

(- cos(m&! (m + 1)&0) + 1 cos(m&! (m! 1)&0)) .


The term in cos(m&! (m + 1)&0) becomes

- = !(n + 1) sin '0Pmn (cos '0)! cos '0P

m+1n

+mcos2 '0

sin '0Pm

n (cos '0)!m

sin '0Pm

n (cos '0)

= !(n + m + 1) sin '0Pmn (sin '0)! cos '0P

m+1n (cos '0)

=1

2n + 1

(! (n + m + 1)Pm+1

n+1 (cos '0) + (n + m + 1)Pm+1n#1 (cos '0)

!(n + m + 1)Pm+1n#1 (cos '0)! (n!m)Pm

n+1(cos '0))

= !Pm+1n+1 (cos '0), (C.14)

and the term in cos(m&! (m + 1)&0) is

1 = !(n + 1) sin '0Pmn (cos '0)! cos '0P

m+1n

+mcos2 '0

sin '0Pm

n (cos '0) +m

sin '0Pm

n (cos '0)

= !(n + m + 1) sin '0Pmn (cos '0)! cos '0P

m+1n (cos '0) +

2m

sin '0Pm

n (cos '0)

= !Pm+1n+1 (cos '0) + Pm+1

n#1 (cos '0) + (n + m)(n + m! 1)Pm#1n#1 (cos '0)

= !Pm+1n+1 (cos '0) + Pm+1

n#1 (cos '0) + (2n + 1) sin '0Pmn (cos '0)

+(n!m + 1)(n!m + 2)Pm#1n+1 (cos '0)

= (n!m + 1)(n!m + 2)Pm#1n+1 (cos '0). (C.15)

Therefore, we have,

A =1

2R20

!r

R0

"n ((n!m + 1)(n!m + 2)Pm#1

n+1 cos(m&! (m! 1)&0)

!Pm+1n+1 (cos '0) cos(m&! (m + 1)&0)

). (C.16)

In the same fashion, we can proceed with the y component. This time, we will have terms insin(m&! (m+1)&0) and sin(m&! (m!1)&0). The terms in '0 are the same, so that we obtain, setting

B =+

+x0

!rn

Rn+10

Pmn (cos '0) cos m(&! &0)

", (C.17)

B =1

2R20

!r

R0

"n ((n!m + 1)(n!m + 2)Pm#1

n+1 sin(m&! (m! 1)&0)

!Pm+1n+1 (cos '0) sin(m&! (m + 1)&0)

). (C.18)

We now have all terms for each component of the gradient. We have in addition

M1x = M1 sin '1 cos &1 (C.19)M1y = M1 sin '1 sin &1 (C.20)M1z = M1 cos '1. (C.21)

183

The scalar product is now trivial and we only have to regroup terms in &0 with terms in &1. Infine we get the complete expansion for the potential:

"! = ! M1

4$R20

6T 0

0 +"$

n=1

!r

R0

"n%T 0

nPn(cos ') +n$

m=1

(n!m)!(n + m)!

Tmn Pm

n (cos ')

&7. (C.22)

T 0n = (n + 1) cos '1Pn+1(cos '0) + sin '1P

1n+1(cos '0) cos(&1 ! &0)

Tmn = 2(n!m + 1) cos '1P

mn+1(cos '0) cos m(&! &0)

!(n!m + 1)(n!m + 2) sin '1Pm#1n+1 (cos '0) cos{m&! [(m! 1)&0 + &1]}

+ sin '1Pm+1n+1 (cos '0) cos{m&! [(m + 1)&0 ! &1]}

It is then straightforward to derive the expressions for the di!erent components of the field, followingequations 2.16, 2.17, and 2.18.

Bibliography

[1] Levitt, M. H., Spin Dynamics, Wiley.

[2] Abragam, A., Principle of Nuclear Magnetism, Oxford Science Publications, 1994.

[3] Goldman, M., Quantum Description of High-Resolution NMR in liquids, Oxford UniversityPress, 1988.

[4] Messiah, A., Quantum Mechanics, Dover Publications.

[5] Hahn, E. L., Physical Review 80 (1950) 580.

[6] Carr, H. and Purcell, E. M., Physical Review 94 (1954) 630.

[7] Meiboom, S. and Gill, D., The Review of Scientific Instruments 29 (1958) 688.

[8] Meriles, C. A., Sakellariou, D., Heise, H., Moulé, A. J., and Pines, A., Science 293 (2001) 82.

[9] Jerschow, A., Chemical Physics Letters 296 (1998) 466.

[10] Ardelean, I., Scharfenecker, A., and Kimmich, R., Journal of Magnetic Resonance 144 (2000)45.

[11] Ardelean, I., Kimmich, R., and Klemm, A., Journal of Magnetic Resonance 146 (2000) 43.

[12] Sakellariou, D., Meriles, C. A., Moulé, A., and Pines, A., Chemical Physics Letters (2002) 25.

[13] Meriles, C. A., Sakellariou, D., and Pines, A., Journal of Magnetic Resonance 164 (2003) 177.

[14] Torrey, H. C., Physical Review 104 (1956) 563.

[15] Augustine, M. P., TonThat, D. M., and Clarke, J., Solid State Nuclear Magnetic Resonance 11(1998) 139.

[16] McDermott, R., Trabesinger, A. H., Mück, M., Hahn, E. L., and Pines, A., Science 295 (2002)22472249.

[17] McDermott, R. et al., PNAS 101 (2004) 7857.

[18] Pannetier, M., Fermon, C., Go!, G. L., Simola, J., and Kerr, E., Science 304 (2004) 1648.

[19] Pannetier, M. et al., IEEE Transactions on Applied Superconductivity 15 (2005) 892.

[20] Sidles, J., Applied Physics Letters 58 (1991) 2854.

[21] Rugar, D., Budaklan, R., Mamin, H. J., and Chui, B. W., Nature 430 (2004) 329.

185

186 BIBLIOGRAPHY

[22] Degen, C. L., Poggio, M., Mamin, H. J., Rettner, C. T., and Rugar, D., PNAS 106 (2009) 1313.

[23] Savukov, I. M., Lee, S.-K., and Romalis, M. V., Nature 442 (2006) 1021.

[24] Hoult, D. and Richards, R. E., Journal of Magnetic Resonance 24 (1976) 71.

[25] Hoult, D. I., Biomedical Magnetic Resonance Technology, Adam Hilger, 1989.

[26] Arfken, G. B. and Weber, H. J., Mathematical Methods for Physicists, Elsevier Academic Press,sixth edition, 2005.

[27] Insko, E. K., Elliott, M. A., Schotland, J. C., and Leigh, J. S., Journal of Magnetic Resonance131 (1998) 111.

[28] Coey, J. M. D., Rare-Earth Iron Permanent Magnets, Oxford Science Publications, 1996.

[29] Durand, É., Magnétostatique, Masson, 1968.

[30] Jiles, D., Introduction to Magnetism and Magnetic Materials, CRC Taylor and Francis, 1998.

[31] Aubert, G., Magnétostatique, To be published.

[32] Jackson, J. D., Classical Electrodynamics, Wiley, third edition.

[33] Stratton, J. A., Electromagnetic Theory, McGraw-Hill, 1941.

[34] Katter, M., IEEE Transactions on Magnetics 41 (2005) 3853.

[35] Vacuumschmelze, Rare-Earth Permanent Magnets VACODYM - VACOMAX, Product docu-mentation, 2010.

[36] Rabi, I. I., Zacharias, J. R., Millman, S., and Kusch, P., Physical Review 53 (1938) 318.

[37] Ernst, R. and Anderson, W. A., Review of Scientific Instruments 37 (1966) 93.

[38] Ernst, R. R., Bodenhausen, G., and Wokaun, A., Principles of Nuclear Magnetic Resonance inOne and Two Dimensions, Oxford Science Publications, 1994.

[39] Lauterbur, P. C., Nature 242 (1973).

[40] Nelson, F. A. and Weaver, H. E., Science 146 (1964) 223.

[41] IEEE Transactions on Applied Superconductivity, The Whole Body 11.7 T MRI Magnet forIseult/INUMAC Project, volume 18, 2008.

[42] Pauvert, O. et al., Inorg. Chem. 48 (2009) 8709.

[43] Brunner, D. O., Zanche, N. D., Fröhlich, J., Paska, J., and Pruessmann, K. P., Nature 457(2009) 994.

[44] http://www.mobileleasing.com/ (2010).

[45] Varian, R. H., Apparatus and Method for Identifying Substances, US Patent 3,395,337, 1968.

[46] Blümich, B., Perlo, J., and Casanova, F., Progress in Nuclear Magnetic Resonance Spectroscopy52 (2008) 197.

BIBLIOGRAPHY 187

[47] Kleinberg, R. L., Concepts in Magnetic Resonance 13 (2001) 404.

[48] Blümich, B. et al., Journal of Geophysics and Engineering 1 (2004) 177.

[49] Anferova, S. et al., Magnetic Resonance Imaging 25 (2007) 474.

[50] Marko, A., Wolter, B., and Arnold, W., Journal of magnetic resonance 185 (2007) 19.

[51] Friedemann, K., Stallmach, F., and Kärger, J., Cement and Concrete Research 36 (2006) 817.

[52] Karakosta, E. et al., Industrial and Engineering Chemistry Research 49 (2010) 613.

[53] Kim, S.-M., Chen, P., McCarthy, M. J., and Zion, B., J. agric. engng res 74 (1999) 293.

[54] Pedersen, H., Ablett, S., Martin, D., Mallett, M., and Engelsen, S., Journal of Magnetic Reso-nance 165 (2003) 49.

[55] Senni, L., Casieri, C., Bovino, A., Gaetani, M., and Luca, F. D., Wood Science and Technology43 (2009) 167.

[56] Riel, K.-A., Reinisch, M., Kersting-Sommerho!, B., Hof, N., and Merl, T., Knee Surgery, SportsTraumatology, Arthroscopy 7 (1999) 37.

[57] Dias, M., Hadgraft, J., Glover, P., and McDonald, P., Journal of Physics D: Applied Physics 36(2003) 364.

[58] Rajeswari, M. R. et al., Laboratory Investigation 83 (2003) 1279.

[59] Sardanelli, F. et al., American Journal of Radiology 183 (2004) 1149.

[60] Kriege, M. and et al., The New England journal of medicine 351 (2004) 427.

[61] Abele, M. G., Structures of Permanent Magnets. Generation of Uniform Fields, Wiley, 1993.

[62] Halbach, K., Nuclear Instruments and Methods (1980) 1.

[63] McDowell, A. and Fukushima, E., Applied Magnetic Resonance 35 (2008) 185.

[64] Leupold, H., Potenziani, E., and Abele, M., Journal of Applied Physics 64 (1988) 5994.

[65] Abele, M. G., Tsui, W., and Rusinek, H., Journal of Applied Physics 99 (2006).

[66] Bloch, F., Cugat, O., Meunier, G., and Toussaint, J., IEEE Transactions on Magnetics 34 (1998)2465.

[67] Kumada, M. et al., IEEE Transactions on Applied Superconductivity 12 (2002) 129.

[68] Li, C. and Devine, M., IEEE Transactions on Magnetics 45 (2009) 4380.

[69] Abele, M. and Leupold, H., Journal of Applied Physics 64 (1988) 5988.

[70] Ravaud, R. and Lemarquand, G., Progress In Electromagnetics Research 94 (2009) 327.

[71] Abele, M., Chandra, R., Rusinek, H., Leupold, H., and Potenziani, E., IEEE Transactions onMagnetics 25 (1989) 3904.

188 BIBLIOGRAPHY

[72] Jensen, J. H., IEEE Transactions on Magnetics 35 (1999) 4465.

[73] Shirai, T., Haishi, T., Utsuzawa, S., Matsuda, Y., and Kose, K., Magnetic Resonance in MedicalSciences 4 (2005) 137.

[74] Raich, H. and Blümler, P., Concepts in Magnetic Resonance 23B (2004) 16.

[75] Hills, B., Wright, K., and Gillies, D., Journal of Magnetic Resonance 175 (2005) 336.

[76] Zhang, X. et al., Design, construction and NMR testing of a 1 tesla Halbach Permanent Magnetfor Magnetic Resonance, in Proceedings of the COMSOL Users Conference Boston, 2005.

[77] Danieli, E., Mauler, J., Perlo, J., Blümich, B., and Casanova, F., Journal of Magnetic Resonance198 (2009) 80.

[78] Anferova, S. et al., Concepts in Magnetic Resonance Part B 23B (2004) 26.

[79] Givens, W. W., US Patent 4,714,881 (1987).

[80] Hanley, P., WIPO patent WO 94/18577 (1994).

[81] Herzog, G., Method and Apparatus for Well Logging Utilizing Resonance, US Patent 3,617,867,filed July 17, 1953, continuation-in-part of application No 238,754, July 26, 1951, 1971.

[82] Jackson, J. A. and Cooper, R. K., US Patent 4,350,955 (1982).

[83] Locatelli, M., Cattine, V., and Jeandey, C., US Patent 6,657,433 B1 (2003).

[84] Prammer, M. G. et al., US Patent Application 10/837,084 (2005).

[85] Reiderman, A., US Patent application 11/756,863 (2007).

[86] Goswami, J. C., Sezginer, A., and Luong, B., IEEE Transactions on Antennas and Propagation48 (2000) 1393.

[87] Randall, E. W., Encyclopedia of Magnetic Resonance .

[88] Benson, T. and McDonald, P., Journal of Magnetic Resonance Series B 109 (1995) 314.

[89] McDonald, P. and Newling, B., Reports on Progress in Physics 61 (1998) 1441.

[90] Eidmann, G., Savelsberg, R., Blümler, P., and Blümich, B., Journal of Magnetic Resonance A122 (1996) 104.

[91] Blümich, B. et al., Magnetic Resonance Imaging 16 (1998) 479.

[92] Krüger, M., Schwarz, A., and Blümich, B., Magnetic Resonance Imaging 25 (2007) 215.

[93] Capitani, D., Brilli, F., Mannima, L., Proietti, N., and Loreto, F., Plant Physiology 149 (2009)1638.

[94] Casieri, C., Bubici, S., Viola, I., and De-Luca, F., Solid State Nuclear Magnetic Resonance 26(2004) 65.

[95] Blümich, B. et al., Concepts in Magnetic Resonance 15 (2002) 255.

BIBLIOGRAPHY 189

[96] Anferova, S. et al., Concepts in Magnetic Resonance 14 (2002) 15.

[97] Glover, P., McDonald, P. J., and Newling, B., Journal of Magnetic Resonance 126 (1997) 207.

[98] Glover, P., Aptaker, P., Bowler, J., Ciampi, E., and McDonald, P., Journal of Magnetic Reso-nance 139 (1999) 90.

[99] Prado, P. J., Magnetic Resonance Imaging 19 (2001) 505.

[100] Prado, P. J., Magnetic Resonance Imaging 21 (2003) 397.

[101] Perlo, J., Casanova, F., and Blümich, B., Journal of Magnetic Resonance 166 (2004) 228.


[103] Perlo, J. et al., Science 308 (2005) 1279.


[105] Manz, B. et al., Journal of Magnetic Resonance 183 (2006) 25.

[106] McDonald, P., Aptaker, P., Mitchell, J., and Mulheron, M., Journal of Magnetic Resonance 185(2007) 1.

[107] Chang, W.-H., Chen, J.-H., and Hwang, L.-P., Magnetic Resonance Imaging 24 (2006) 1095.

[108] Rahmatallah, S. et al., Journal of Magnetic Resonance 173 (2005) 23.

[109] Perlo, J., Casanova, F., and Blümich, B., Science 315 (2007) 1110.

[110] Chang, W.-H., Chen, J.-H., and Hwang, L.-P., Magnetic Resonance Imaging 28 (2010) 129.

[111] Marble, A. E., Mastikhin, I. V., Colpitts, B. G., and Balcom, B. J., Journal of MagneticResonance 174 (2005) 78.

[112] Marble, A. E., Mastikhin, I. V., Colpitts, B. G., and Balcom, B. J., Journal of MagneticResonance 183 (2006) 228.

[113] Marble, A. E., IEEE Transactions on Magnetics 44 (2008) 576.

[114] Marble, A. E., Mastikhin, I. V., Colpitts, B. G., and Balcom, B. J., IEEE Transactions onMagnetics 43 (2007) 1903.

[115] Paulsen, J. L., Bouchard, L. S., Graziani, D., Blümich, B., and Pines, A., PNAS 105 (2008)20601.

[116] Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover Publications,1965.

[117] Roméo, F. and Hoult, D. I., Magnetic Resonance in Medicine 1 (1984) 44.

[118] Meriles, C. A., Sakellariou, D., Trabesinger, A. H., Demas, V., and Pines, A., PNAS 102 (2005)1842.

[119] Meriles, C. A., Sakellariou, D., and Trabesinger, A. H., Journal of Magnetic Resonance 182(2006) 106.

190 BIBLIOGRAPHY

[120] Kimmich, R. and Fischer, E., Journal of Magnetic Resonance Series A 106 (1994) 229.

[121] Hürlimann, M. D., Journal of Magnetic Resonance 148 (2001) 367.

[122] Callaghan, P. T., Principles of Nuclear Magnetic Resonance Microscopy, Oxford Science Publi-cations, 2001.

[123] Kimmich, R., NMR, Tomography, Di!usometry, Relaxometry, Springer, 1997.

[124] Hugon, C., Aubert, G., and Sakellariou, D., Application 0955892 to the french patent o"ce(INPI) (2009).




[128] Popovic, R. S., Hall E!ect Devices, Institute of Physics, 2004.

[129] Hall, E. H., American Journal of Mathematics 2 (1879) 287.

[130] Schott, C. and Popovic, R. S., Proceedings of the transducers, Sendai, Japan 7-10 June (1999).

[131] Schott, C., Waser, J.-M., and Popovic, R. S., Sensors and Actuators A 82 (2000) 167.

[132] Kejik, P., Schurig, E., Bergsma, F., and Popovic, R. S., The 13th International Conference onSolid-State Sensors, Actuators and Microsystems, Seoul, Korea, June 5-9 (2005).

[133] Popovic, D. R. et al., IEEE Transactions on Instrumentation and Measurement 56 (2007) 1396.

[134] Bergsma, F., Proceedings of the 13th International Magnetic Measurement Workshop, May19-22, stanford, California (2003).

[135] Aleksa, M. et al., Journal of Instrumentation 3 (2008) P04003.

[136] Schott, C., Popovic, R. S., Alberti, S., and Tran, M. Q., Review of Scientific Instruments 70(1999) 2703.

[137] Maki, A. H. and Volpicelli, R. J., Review of Scientific Instruments 36 (1965) 325.

[138] Boero, G., de Raad Iseli, C., Besse, P., and Popovic, R., Sensors and Actuators A 67 (1998) 18.

[139] Prigl, R., Haeberlen, U., Jungmann, K., zu Putlitz, G., and von Walter, P., Nuclear Instrumentsand Methods in Physics Research A 374 (1996) 118.

[140] Wu, N., Peck, T. L., Webb, A. G., Magin, R. L., and Sweedler, J. V., Journal of the AmericanChemistry Society 116 (1994) 7929.

[141] Olson, D. L., Peck, T. L., Webb, A. G., Magin, R. L., and Sweedler, J. V., Science 270 (1995).

[142] Sakellariou, D., Go!, G. L., and Jacquinot, J.-F., Nature 447 (2007) 694.

BIBLIOGRAPHY 191

[143] Hugon, C., Jacquinot, J.-F., and Sakellariou, D., Journal of Magnetic Resonance 202 (2010) 1.

[144] Hamming, R. W., Numerical Methods for Scientists and Engineers, Dover Publications, 1987.

[145] Higgins, J. R., Sampling Theory in Fourier and Signal Analysis. Foundations, Oxford SciencePublications, 1996.

[146] Trefethen, L. N. and Bau, D., Numerical Linear Algebra, SIAM, 1997.

[147] Metrolab Instruments S.A., 110, ch. Du Pont-du-Centenaire, CH1228 GENEVA, SWITZER-LAND, MFC-3045 Magnetic Field Camera, 2004.

[148] Steinborn, E. O. and Ruedenberg, K., Advances in Quantum Chemistry 7 (1973) 1.

[149] Goldstein, H., Poole, C., and Safko, J., Classical Mechanics, Addison Wesley, third edition, 2002.

[150] Aubert, G., US Patent 5,014,032 (1991).

[151] Elleaume, P., Chubar, O., and Chavanne, J., Proc. of the PAC97 Conference (1997) 3509.

[152] Onuki, H. and Elleaume, P., Undulators, wigglers and their applications, CRC Press, 2003.

[153] McCaig, M. and Clegg, A. G., Permanent Magnets in Theory and Practice, Pentech Press,second edition, 1987.

[154] Hoult, D. and Lee, D., Review of Scientific Instruments 56 (1985) 131.

[155] Belov, A. et al., IEEE Transactions on Applied Superconductivity 5 (1995) 679.






[161] Hugon, C., Aguiar, P. M., Aubert, G., and Sakellariou, D., Comptes Rendu de Chimie del’Académie des Sciences 13 (2010) 388.

[162] Wind, R., Hu, J., and Rommereim, D., Magnetic Resonance in Medicine 46 (2001) 213.

[163] Moresi, G. and Magin, R., Concepts in Magnetic Resonance Part B 19B (2003) 35.

[164] Jachmann, R. C. et al., Review of Scientific Instruments 78 (2007) 1.

[165] Demas, V. et al., Journal of Magnetic Resonance 189 (2007) 121.

[166] Manz, B., Benecke, M., and Volke, F., Journal of Magnetic Resonance (2008) 131.

[167] Fukushima, E. and Jackson, J. A., US Patent 6,828,892 B1 (2004).

[168] Chavanne, J., Chubar, O., Elleaume, P., and Van-Vaerenbergh, P., Proceedings of EPAC 2000,Vienna, Austria (2000) 2316.

192 BIBLIOGRAPHY

[169] Zijl, P. C. M. V., Sukumar, S., Johnson, M. O., Webb, P., and Hurd, R. E., Journal of MagneticResonance, Series A 111 (1994) 203 .

[170] Ren, Z., Xie, D., and Li, H., Progress In Electromagnetics Research M 6 (2009) 23.

[171] Hugon, C., D’Amico, F., Aubert, G., and Sakellariou, D., Journal of Magnetic Resonance 205(2010) 75.

[172] Vacuumschmelze, Rare Earth Permanent Magnets. VACODYM-VACOMAX, Technical report,2009.

[173] Schenck, J. F., Hussain, M. A., and Edelstein, W. A., US Patent 4,646,024 (1987).

[174] Edelstein, W. A. and Schenck, F., US Patent 4,840,700 (1989).

[175] Martens, M. A. et al., Review of Scientific Instruments 62 (1991) 2639.

[176] Turner, R., Magnetic Resonance Imaging 11 (1993) 903.

[177] Tomasi, D., Magnetic Resonance in Medicine 45 (2001) 505.

[178] Liu, W., Tang, X., and Zu, D., Concepts in Magnetic Resonance Part B 37B (2009) 29.

[179] Demyanenko, A. V. et al., Journal of Magnetic Resonance 200 (2009) 38.

[180] Shvartsman, S. M. et al., Magnetic Resonance in Medicine 45 (2001) 147.

[181] Song, Y. et al., Journal of Magnetic Resonance 154 (2002) 261.

[182] Suits, B. and Garroway, A., Journal of Applied Physics 94 (2003) 4170.

[183] Barroso, V. C., Raich, H., Blümler, P., and Wilhelm, M., Journal of Physics: Conference Series149 (2009).

[184] Abragam, A. and Goldman, M., Reports on Progress in Physics 41 (1978) 395.

[185] Wind, R. A. and Ardenkjaer-Larsen, J.-H., Journal of Magnetic Resonance 141 (1999) 347.

[186] Ginefri, J.-C., Darrasse, L., and Crozat, P., Magnetic Resonance in Medicine 45 (2001) 376.

[187] Anders, J., Chiaramonte, G., SanGiorgio, P., and Boero, G., Journal of Magnetic Resonance201 (2009) 239.

[188] Lee, H., Sun, E., Ham, D., and Weissleder, R., Nature Medicine 14 (2008) 869.

permanent magnets for nmr and mri, by cedric hugon

Documents