ACTAUNIVERSITATISUPSALIENSISUPPSALA2006

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 163

Neutron-Deuteron Scattering andThree-Body Interactions

PHILIPPE MERMOD

ISSN 1651-6214ISBN 91-554-6514-5urn:nbn:se:uu:diva-6739

To Kristina

List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I J. Klug, J. Blomgren, A. Ataç, B. Bergenwall, A. Hildebrand, C.Johansson, P. Mermod, L. Nilsson, S. Pomp, U. Tippawan, K.Elmgren, N. Olsson, O. Jonsson, A.V. Prokofiev, P.-U. Renberg,P. Nadel-Turonski, S. Dangtip, P. Phansuke, M. Österlund, C. LeBrun, J.F. Lecolley, F.R. Lecolley, M. Louvel, N. Marie-Noury,C. Schweitzer, Ph. Eudes, F. Haddad, C. Lebrun, A.J. Koning,and X. Ledoux, "Elastic neutron scattering at 96 MeV from 12Cand 208Pb", Phys. Rev. C 68, 064605 (2003).

II C. Johansson, J. Blomgren, A. Ataç, B. Bergenwall, S. Dangtip,K. Elmgren, A. Hildebrand, O. Jonsson, J. Klug, P. Mermod, P.Nadel-Turonski, L. Nilsson, N. Olsson, S. Pomp, A.V. Prokofiev,P.-U. Renberg, U. Tippawan, and M. Österlund, "Forward-angleneutron-proton scattering at 96 MeV", Phys. Rev. C 71, 024002(2005).

III P. Mermod, J. Blomgren, B. Bergenwall, A. Hildebrand, C.Johansson, J. Klug, L. Nilsson, N. Olsson, M. Österlund, S.Pomp, U. Tippawan, O. Jonsson, A. Prokofiev, P.-U. Renberg,P. Nadel-Turonski, Y. Maeda, H. Sakai, and A. Tamii, "Searchfor three-body force effects in neutron-deuteron scattering at 95MeV", Phys. Lett. B597, 243 (2004).

IV P. Mermod, J. Blomgren, A. Hildebrand, C. Johansson, J.Klug, M. Österlund, S. Pomp, U. Tippawan, B. Bergenwall, L.Nilsson, N. Olsson, O. Jonsson, A. Prokofiev, P.-U. Renberg, P.Nadel-Turonski, Y. Maeda, H. Sakai, and A. Tamii, "Evidenceof three-body force effects in neutron-deuteron scattering at 95MeV", Phys. Rev. C 72, 061002(R) (2005).

V P. Mermod, J. Blomgren, B. Bergenwall, C. Johansson, J. Klug,L. Nilsson, N. Olsson, A. Öhrn, M. Österlund, S. Pomp, U.Tippawan, P. Nadel-Turonski, O. Jonsson, A. Prokofiev, P.-U.

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Renberg, Y. Maeda, H. Sakai, A. Tamii, K. Amos, R. Crespo, andA. Moro, "95 MeV neutron scattering on hydrogen, deuterium,carbon and oxygen", submitted to Phys. Rev. C (2006).

Reprints were made with permission from the publishers.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Fundamental physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Nuclear applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The present work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Examples of three-body forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Emotional interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Nuclear interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Strong interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Electromagnetic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Gravitational interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Theory of nuclear interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1 Nucleon-nucleon potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 From nucleons to nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Many-body systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 The off-shell problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Three-nucleon forces and effective field theories . . . . . . . . . . . 14

4 Search for three-nucleon forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1 Three-nucleon force observables . . . . . . . . . . . . . . . . . . . . . . . 174.2 Neutron-deuteron scattering experiments at TSL . . . . . . . . . . . 194.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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1. Introduction

Like a coin, physics has two sides. Physics research is driven by curiosityabout nature (neophile side) as well as practical prospects (technophileside). This coin is in a state of constant flipping, since applications oftenmotivate−and finance−fundamental research in particular fields, and theorycan be used to predict quantities or phenomena which can lead to new orimproved technologies.

The present thesis is a good example of this duality. The neutron-deuteronscattering experiments on which it is based were designed to investigate fun-damental interactions between three nucleons, but at the same time the in-formation extracted from the experimental data may be useful for medicalapplications, and the work was carried out as part of a program on studies ofneutron-induced nuclear reactions of importance for incineration of nuclearwaste.

1.1 Fundamental physicsAll known interactions between physical objects can be interpreted in terms offour1 fundamental interactions: the electromagnetic interaction, the weak in-teraction, the strong interaction, and the gravitational interaction. Very often,the notion of force−which is what causes a change in an object−is mixed upwith the notion of interaction since a force is necessarily due to an interaction(with another object). In this thesis, we are interested about the nuclear force,which is a manifestation of the strong interaction described phenomenolog-ically as the effective interaction between nucleons (nucleons can be eitherprotons or neutrons).

Most theories of fundamental interactions involve hypothetically funda-mental objects called elementary particles. In this approach, in principle, atotal knowledge of the existence, properties and interactions of elementaryparticles allows to describe all physical phenomena: if the elementary parti-cles are really fundamental, then they are the building blocks of every largerstructure, and thus all structures must obey some complex (in practice, usu-

1 Traditionally, modern physicists have counted four interactions; however, electromagnetismand the weak interaction can be shown to be two aspects of a single electroweak interaction,and somewhat more speculatively, the electroweak interaction and the strong interaction can becombined using grand unified theories. How to combine the fourth interaction, gravity, with theother three is a topic of research in quantum gravity.

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ally too complex to be computed exactly by human means) combination of thefundamental laws governing elementary particle dynamics. In quantum fieldtheories, specific elementary particles (the gauge fields) act as mediators be-tween other elementary particles. This exchange mechanism is used as a basisfor the description of interactions between pairs of elementary particles.

Three-body (as well as four-body, five-body, etc.) interactions are oftenoverlooked in the description of physical systems, probably because, in na-ture, they tend to be insignificant compared to two-body (pairwise) interac-tions. As we will see in Chapter 2, quantum field theories of, e.g., electromag-netic and strong interactions predict new types of interactions which arise assoon as more than two objects are involved. These three-body interactions canbe viewed as the coupling between the exchange particles themselves whilethey are mediating the interaction between the real objects.

The present thesis primarily concerns experimental studies of three-bodynuclear forces in neutron-deuteron scattering at 95 MeV. The primary aim isto provide data in the three-nucleon system for testing the models of three-nucleon forces. The work is put in a broader perspective by considering ex-amples of three-body forces in other areas of physics.

1.2 Nuclear applicationsThere are a number of new applications under development where neutrons ofhigher energies than in the traditional applications (nuclear power and nuclearweapons) play a significant role. The most important are transmutation of nu-clear waste [1], medical treatment of tumors with fast neutrons [2], and themitigation of single-event effects in electronics [3]. These applications wouldbenefit from refinements of nuclear models, especially for neutron-inducednuclear reactions up to 1 GeV.

None of these applications require precise data on neutron-deuteron scat-tering for their further development in a short-time perspective. In a longerterm and a broader physics perspective, however, such data could be of largeimportance. It is an ultimate goal since long in nuclear physics to be able todescribe heavy nuclei from simple fundamental forces only. At present, thisgoal is far from being fulfilled. In nuclear energy applications, precise knowl-edge of heavy nuclei, like uranium and plutonium, is required, but for thesenuclei ab initio theories are not in a state where cross section predictions canmeet the requirements for technological applications. Advanced nuclear the-ory can be used to describe some particularly favorable cases, but for mostnuclei, specific models are constructed phenomenologically.

The present work represents a step in the direction to establish a more fun-damental treatment in the realm of applied nuclear physics. It has been es-tablished that nuclear theory based only on the nucleon-nucleon (two-body)interaction fail to describe even such a basic quantity as the binding energy

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of A = 3 nuclei. With the introduction of three-nucleon forces, these prob-lems can be remedied, and moreover, it seems that a combination of two-and three-nucleon forces can provide a fairly good description all the wayup to A = 12. Thus, there is good hope that the quality of the description ofmany-body systems, like heavy nuclei, can progress significantly already byimproved understanding of three-nucleon forces.

In the experiments of the present thesis, in addition to the neutron-deuteronscattering data, we obtained also differential cross sections for neutron scatter-ing on carbon and oxygen at 95 MeV (see Paper V). Cancer therapy with fastneutrons may benefit directly from the carbon and oxygen data, since these el-ements are abundant in biological tissue and the evaluation of the cell damagedepends on the probability for the neutron to scatter inside the body.

1.3 The present workThe present work was part of the experimental effort in neutron scattering pur-sued since 1994 by the Department of Neutron Research at Uppsala Univer-sity. The laboratory work was undertaken at The Svedberg Laboratory (TSL)in Uppsala.

The papers (I-V) are interlinked and follow a consistent sequence. Paper Iby Klug et al. is about elastic neutron scattering on carbon and lead at 96 MeV,the first fully analyzed data obtained with the SCANDAL2 detector setup. InPaper II by Johansson et al., neutron-proton (np) data at 96 MeV were ob-tained with essentially the same technique specifically adapted to the purposeof np scattering.

When I joined the Uppsala neutron reaction research group in 2002, thedata for Papers I and II were still under analysis. I provided a modest con-tribution in multiple scattering corrections. At that time, both the MEDLEY[4] and the SCANDAL [5] detector setups had proven their capabilities andcould be used in a routine manner. This, and recent theoretical predictionsabout three-nucleon force effects in elastic neutron-deuteron (nd) scattering(see, e.g., Ref. [6]), motivated the nd scattering experiments presented in Pa-pers III, IV and V, in which I took active part at all stages. The MEDLEY nddata were analyzed first and published in summer 2004 (Paper III). Resultsof the SCANDAL experiments were obtained one year later (Paper IV) andconstituted both a cross-check and an improvement of the MEDLEY results.When analyzing the SCANDAL nd data, the work previously done for PapersI and II was very useful in the sense that the same detectors were read out andthe same type of corrections were applied. I have, however, almost completelyre-engineered the analysis routines, partly because I found it convenient to doit in my own way and partly because the special conditions of nd scattering

2SCANDAL stands for SCAttered Nucleon Detection AssembLy.

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demanded rather complex analysis procedures to obtain the required precision(details are given in Paper V). This allowed beneficial consistency checks ofthe routines since by-products of the nd experiments, i.e., elastic scattering oncarbon and on hydrogen, overlap with the results of Papers I and II, respec-tively.

An introduction to the thesis is given in Chapter 1. In Chapter 2, I dis-cuss three-body forces in a general context and I give examples of three-bodyforces outside the framework of nuclear physics. These personal contributionshave a philosophical nature distinct from the experimental work described inthe papers (in fact, except for three-nucleon forces, none of the three-bodyforce examples cited in this thesis are depicted in the literature). Relevantthree-nucleon interaction theory is outlined in Chapter 3. The nd scattering ex-periments and the search for three-nucleon force effects are outlined in Chap-ter 4. Finally, conclusions and an outlook are given in Chapter 5.

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2. Examples of three-body forces

In most cases, the complex interactions that arise in systems of three objectsor more are due to combinations of basic two-body interactions. In general,if the observed behavior of a system of more than two objects cannot be de-scribed by the two-body interactions between all possible pairs, the deviationis mainly due to a three-body force. The underlying mechanism, illustratedin Fig. 2.1, can be understood as follows: three-body forces are caused by aninteraction with the interaction.

Figure 2.1: The interaction between two bodies is represented as a dashed line. Onthe right, the interaction is affected by the presence of the additional body (wavy line).

2.1 Emotional interactionsAs an analogy, if we identify bodies with human beings and forces with feel-ings, jealousy is a good example of a three-body force: it is not felt as longas only two persons are acting, but it can show up if a third person enters thescene (see Fig. 2.2). In this case, the three-body force can be responsible forstrong effects upon the jealous individual: the jealousy can be expressed as anurge to fight the concurrent or to gain the beloved one by any means. But thesource of the jealousy has not much to do with the interaction with the con-current himself nor the beloved one, but rather arises from what is exchangedbetween them.

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Figure 2.2: Jealousy is a human example of a three-body force.

2.2 Nuclear interactionsThe nuclear force, or nucleon-nucleon (NN) interaction, is a particular aspectof the strong interaction which can be described very well phenomenologi-cally in the framework of pion-exchange models (see section 3.1). As we willdiscuss in detail in Chapters 3 and 4, there are very good reasons to believethat the description of a system made of three nucleons or more is not com-plete if three-body forces are not taken into account (and, in principle, alsofour-body forces, five-body forces, etc.).

Consistent models of three-nucleon (3N) interactions were proposed in the1960s [7, 8, 9]. The main motivation was to reproduce the triton binding en-ergy as accurately as possible. These early models contained already the basicsof 3N interactions, illustrated by the diagrams (b) and (c) in Fig. 2.3. Modelsof 3N interactions, including the most recent ones, will be further discussedin section 3.5.

The two-pion exchange 3N interaction represented in Fig. 2.3b is includedin almost all models of 3N interactions and is often believed to be the dom-inant contribution. This effect was stated in the pioneering 1957 article byFujita and Miyazawa [10] as follows: "Within a nuclear matter the potentialbetween two nucleons takes a different shape due to many body forces.". Moreexplicitly, due to the pion exchange between the first two nucleons, one of thenucleons is changed, e.g., it is excited to a N∗ or a ∆ resonance, thus modifyingthe interaction with the third nucleon via another pion exchange.

The 3N interaction shown in Fig. 2.3c, where a particle emitted by onenucleon interacts with the pion exchanged between the two other nucleons,is more intuitive. Since strong interactions conserve parity, the intermediate

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Figure 2.3: Possible interactions between three nucleons. In (a), the simple two-pionexchange does not correspond to a three-body interaction, since it can be reduced toa succession of two-body interactions. The three other diagrams represent three-bodyinteractions of different types. A typical two-pion-exchange 3N interaction is shownin (b), where one nucleon is excited between the absorption and the emission of thepions. In (c), one nucleon exchanges a scalar meson or a vector meson (double line)with a pion. In (d), there is a correlation between two nucleons while the pion is "inflight".

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Figure 2.4: Three-body interactions between quarks can arise due to gluon-gluoncoupling.

particle coupling to the pion cannot itself be a pion1. In a 1961 publication byLoiseau and Nogami [9], an interaction was proposed between the pion anda σ meson (spin 0 and even parity); in a 1965 article by Harrington [11], theparticle was described as a two-pion resonance (also a scalar); and in a 1974article by Yang [12], it was assumed to be a ρ meson (spin 1 and odd parity,or vector meson).

The type of 3N interaction illustrated in Fig. 2.3d has been proposed re-cently (2000) by Canton and Schadow [13] as a new contribution that mighthelp to reproduce experimental data which are sensitive to the spin structureof the 3N force (see section 4.1). It can be understood in the following way: apion is emitted by a nucleon, and a contact interaction (without pion exchange)occurs between the two other nucleons before the pion is absorbed.

2.3 Strong interactionsAt a more fundamental scale, strong interactions are the interactions betweenquarks and gluons as described by quantum chromodynamics (QCD). Due tothe fact that gluons, the mediators of the strong interaction, carry color charge,they can interact directly with each other. A gluon emitted by one quark cancouple with another quark or with the gluon emitted by another quark. In thelatter case, a three-body interaction can arise, as illustrated in Fig. 2.4.

One would expect contributions such as the diagram in Fig. 2.4 to havesome kind of effect inside baryons. However, there are serious problems if wetry to perform calculations based on it. If all vertices are hard (high momen-tum transfer) then one can apply perturbation theory, but at this order (fourvertices), many other diagrams must be taken into account, thus complicat-ing the calculations [14]. In a realistic case, however, the vertices are morelikely to be soft (low momentum transfer) and then perturbation theory can-

1Pions are pseudoscalars with spin zero and odd parity, which forbids pion-pion vertices, butcontact interactions between real pions are still possible.

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Figure 2.5: Possible three-body interaction between electrons. The photons do notcouple to other photons, but high-order diagrams allow the formation of a virtualfermion loop.

not be used (not even for two-body interactions) due to a large coupling. Itis hard to imagine any kind of perturbative (and therefore calculable) processwhere three-quark forces would play a role. And in the case of baryons, it isin the completely non-perturbative regime where QCD must be replaced byan effective theory. Also, it is hard to think of any measurement that might besensitive to such a contribution [15].

Anyway, although this process is perhaps irrelevant in practical cases, it isinteresting from a philosophical point of view: it illustrates the fact that three-body forces should fundamentally exist in all quantum field theories wherethe exchange particles can couple to each other.

2.4 Electromagnetic interactionsThe fundamental theory of electromagnetic interactions is quantum electrody-namics (QED), where the exchange particles are photons. Photons do not carryelectric charge, and therefore they cannot couple to each other directly. How-ever, one can imagine a process through which a three-body force could alsoarise in electromagnetic interactions. Consider for example three electrons.One photon emitted by one electron can fluctuate into a fermion-antifermionpair, and another photon emitted by another electron can couple to one of thefermions. At first glance, one might think that the fermion-antifermion paircould annihilate into a photon which could couple to the third electron; how-ever, such a process−with two photons going into one photon−is forbiddensince electromagnetic interactions conserve charge conjugation (C-symmetry)and photons are C-odd [15]. But the process shown in Fig. 2.5, where two pho-tons are re-emitted and couple to two of the electrons, is thinkable. However,it involves eight vertices, which makes such contributions negligible since theelectromagnetic coupling constant is small (such a diagram appears only athigh orders in a perturbative expansion).

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2.5 Gravitational interactionsAre there three-body forces in gravitational interactions? The most successfultheory of gravity up to date is general relativity. General relativity is reallynon-linear and not even the two-body problem can be solved exactly [16]. Inlinearized general relativity, which is good enough for most purposes as longas gravity is not too strong, one considers only pairwise interactions betweenmasses. This is an approximation, which is not expected to give accurate re-sults in the vicinity of high energy densities.

Gravity, as any other interaction, can, at least to some extent, be viewed asthe exchange of particles of force, the gravitons. Interestingly, gravitons cancouple to themselves. When gravity is strong, this can lead to a process in athree-body system which is not captured by the pairwise interactions, wherea graviton leaves one of the bodies, splits into two, with one graviton goingto each of the other bodies [17] (the same kind of three-body force process asillustrated in Fig. 2.4 in the case of QCD). Thus, clearly, there are three-bodyforces in quantum theories of gravity, as in all non-linear theories with cubicvertices for the force particles.

The non-linear equations of general relativity are expected to automaticallytake everything into account. At large scales (typically in the neighborhood ofblack holes), one would expect the effects of three-body forces to play somerole. When applying general relativity to perform calculations in many-bodysystems, however, approximations must be introduced. Having worked in thefield of nuclear physics and seen the advantages of introducing a three-nucleonpotential in addition to the nucleon-nucleon interaction, I can imagine a simi-lar phenomenological approach to tackle large-scale many-body gravitationalproblems: it might be advantageous to use linearized general relativity andintroduce a three-body gravitational potential. The form of such a potentialcould possibly be based on a graviton-exchange theory. If high-precision ob-servations of, e.g., the dynamics of triple star systems can be made, they couldbe used to test the predictions with and without three-body forces.

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3. Theory of nuclear interactions

The ultimate goal of nuclear physics is to understand the properties of atomicnuclei in terms of the basic interaction between pairs of nucleons. This impliesthe assumption that neutrons and protons are the fundamental building blocksof nuclear matter, which is not true since they are themselves known to consistof interacting quarks. However, the perturbative techniques of QCD cannotbe used at the hadronic scale due to a too large coupling constant. We aretherefore limited to (semi-)phenomenological models for nuclear interactions.If we want one consistent description for all reactions, thus avoiding a jungleof particular phenomenological cases, our best hope is to rely on the nucleon-nucleon (NN) interaction and gradually attempt to describe systems of two,three, and many nucleons.

3.1 Nucleon-nucleon potentialsThe interaction between two nucleons can be described quantum mechanicallyby considering each nucleon as a wave packet exposed to the potential V of theother nucleon. The evolution of the system is governed by the time-dependentSchrödinger equation. The cross section for nucleon-nucleon scattering fromthe initial state |ψi > to the final state |ψ f > can be obtained from the scatter-ing amplitude, which is derived from the matrix element Ti f =< ψ f |V |ψi >.The usual technique to compute the matrix elements for a general collisionis to express the eigenstates of the time-independent Schrödinger equation asthe Lippmann-Schwinger equation [18].

The properties of the potential V have to be determined phenomenolog-ically. The general form of the potential must be invariant under rotations,reflections, and time reversal, and can be written as the sum of six indepen-dent terms accounting for the central and tensor components of the nuclearforce, the spin-spin interaction, and the spin-orbit, quadratic spin-orbit, andantisymmetric spin-orbit interactions.

In 1935, Yukawa proposed that the nuclear interaction arises from the ex-change of a massive scalar field [19], thus explaining its finite range. His the-oretical work predicted the existence of mesons (unknown at the time) and atthe same time provided a quantum field interpretation of the NN interactions,which can be used to derive the form of the NN potential. Shortly after the dis-covery of the pion in the late 1940s, the range of the nuclear force was conve-niently subdivided into three regions [20]: the phenomenological short-range

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core (strongly repulsive at short distance), the dynamical intermediate-rangedominated by two-pion exchange (which can be attractive), and the classicallong-range interaction mediated by one pion (which falls quickly to zero withdistance). In the 1960s, the models were refined by including heavy-mesonexchange (in particular, vector mesons) as well as correlations between pionsin two-pion exchange.

Modern potentials are still based on pion-exchange models. The compre-hensive NN models available in the 1980s [21, 22] gave a good overall de-scription of the data but still they were not sufficiently accurate for reliable abinitio calculations. In 1993, the Nijmegen group presented a detailed partial-wave analysis1 of NN scattering [23]. It was in this context that the world ppand np database (published before December 1992) was scanned very criti-cally to eliminate the data that deviated too significantly. In the mid-1990s,high-precision NN potentials were developed with parameters adjusted to fitthe large pp and np database surviving the selection of the Nijmegen group.The groups involved and the names of their potentials are, in chronologicalorder:

• Nijmegen group: Nijm1, Nijm2, and Reid93 potentials [24],• Argonne group: AV18 potential [25],• Bonn group: CD-Bonn potential [26, 27].

These potentials use about 45 parameters and fit the pre-scanned data basewith a reduced χ2 close to one, for energies between 0 and 350 MeV. How-ever, if the data published after December 1992 are considered, the χ2 are notperfect anymore [28]. Large sets of pp data were published between 1993 and1999, notably high-quality pp spin-correlation parameters from IUCF. It turnsout that the IUCF data contains new information that calls for an improvementof the NN parameters, without contradicting the old data [28]. When consider-ing np scattering differential cross sections, large sets of data were publishedafter December 1992, but they are not always consistent with each other norwith the old pre-scanned data. Systematic effects affecting the shape or the ab-solute scale of the np angular distribution are probably responsible for thesediscrepancies. In Paper II, we insist on the importance of high-quality np data,in particular for a precise determination of the strength of the fundamentalcoupling of the pion to the nucleon, the πNN coupling constant.

3.2 From nucleons to nucleiWith high-precision NN potentials at hand, it becomes possible to try the abinitio approach and describe systems of more than two nucleons from the basicinteractions between nucleons. There are three major obstacles to overcome

1In a partial-wave analysis, the parametrization is not necessarily well founded on a theory, butrather chosen to optimize the fit to the data.

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before such models can become reliable: the difficulty to perform calculationsin many-body systems, the so-called "off-shell problem", and the need formany-body forces.

3.3 Many-body systemsDealing with the spin components of the NN force in systems of more thantwo nucleons is a formidable numerical problem. The quantum-mechanicalthree-body problem can be solved exactly by using an NN potential in theFaddeev equations [29]. However, before powerful computers were available,it was very difficult to obtain converged solutions, and approximations werenecessary. Consequently, thirty years ago, there was typically a 10−20% un-certainty (∼ 1 − 2 MeV) in the triton binding energy, while a precision of∼ 1% would be necessary to disentangle the physics [30].

Today, thanks to the ongoing advances in computational methods and re-sources, many few-nucleon properties can be reproduced accurately from abinitio calculations [31, 32]. Calculations in many-body systems are no longeran overwhelming obstacle.

3.4 The off-shell problemThe momentum dependence of the NN force cannot be obtained from NNscattering data only, because in NN scattering (as well as in the deuteron),the energy of the two nucleons is conserved, i.e., it is the same before andafter the interaction (they are "on the energy shell"). By contrast, in a many-body system, two nucleons may have different energies before and after theyinteract, i.e., their mutual interaction may be off the energy shell. Thus, thenuclear many-body problem does not have a unique solution. The off-shellNN interaction is empirically undetermined. Only theory can provide it. For-tunately, off-shell behavior can be derived from a relativistic meson-exchangefield theory, as has been done for the CD-Bonn potential [26]. The other poten-tials apply the so-called static/local approximation, which increases the tensorforce off-shell [28]. Therefore, among the potentials presented in section 3.1,it is likely that the CD-Bonn potential has the most realistic off-shell behavior.The triton binding energy calculated with the CD-Bonn potential is about 0.4MeV above the results obtained with the other potentials, and this discrepancyis probably due to different off-shell behavior [26].

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3.5 Three-nucleon forces and effective field theoriesThere is evidence that, in systems of more than two nucleons, three-nucleon(3N) forces (and maybe also four-nucleon forces, five-nucleon forces, etc.)play a significant role. Thus, we would like to include at least 3N forces in thedescription. In fact, in three-nucleon systems such as the triton and nucleon-deuteron scattering, it is possible to solve the Faddeev equations exactly withan additional 3N potential [6]. The challenge is to develop models of 3N in-teractions which are theoretically well-grounded and consistent with NN in-teractions.

Originally, 3N interactions were investigated to solve the discrepancy inthe binding energy of the triton, of typically 1 MeV between the experimentalvalue and the expectations from NN interactions. The models of 3N interac-tions were built by successively adding new terms to the interaction ampli-tude. The first models in the early 1960s were of Fujita-Miyazawa [10] type.Their essence was in the part of the 3N interaction corresponding to two-pion exchange between three nucleons with the intermediate excitation of a∆-resonance (see Fig. 2.3b on page 7). Heavier mesons (like the ρ) couldreplace the pions in two-pion exchange models, having a shorter range. Inter-actions of heavy bosons (such as σ , ρ , ω , and ππ correlations) with one ofthe exchanged pions, corresponding to Fig. 2.3c, may be added to contributeto the short-range 3N force [33]. The type of 3N interaction correspondingto Fig. 2.3d, where two nucleons interact while the pion is in flight, is a re-cent development [13] which is not yet included consistently with the otherdiagrams. It is believed to contribute to the spin dependence of the 3N force.

In the late 1970s, it was realized that models of two-pion exchange 3Ninteractions should be constructed using chiral constraints, i.e., consistentlywith the (approximate) chiral symmetry of QCD. Two physically equivalentapproaches were considered: one of them uses the so-called current algebra[34], and the other is based upon effective lagrangians in relativistic field the-ories [35]. In the field-theory approach, the lagrangians are not fundamentalobjects (they are "effective" or "phenomenological"), but they are quick andefficient tools for implementing chiral symmetry.

Here is a list of modern 3N potentials, with short descriptions on how theyare constructed.

• The Urbana 3N potential [36] does not consider chiral constraints and isbased on the two-pion exchange process (or ρ exchange) with an interme-diate ∆ (diagram 2.3b). An updated version is called Urbana IX [32].

• The Tucson-Melbourne 3N potential [34] uses current algebra for settingchiral constraints and is based on the two-pion exchange process (or ρexchange) (diagram 2.3b). An updated version is called TM99 [37].

• The Brazil 3N potential [35] uses an effective lagrangian to implement chi-ral symmetry. It includes the two-pion exchange (or ρ exchange) processwith an intermediate ∆ (diagram 2.3b) as well as processes mediated by

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Figure 3.1: Three-nucleon interactions at next-to-next-to-leading order in chiral per-turbation theory: two-pion exchange (left), one-pion exchange with NN contact (short-range) interaction (middle) and 3N contact interaction (right).

the interaction of a σ or ρ with a pion (diagram 2.3c). The more recentRuhrPot 3N potential [38] is constructed in a similar way.

• The Padova-Vancouver 3N potential [39] is generated by the exchange ofone pion in the presence of a NN correlation (diagram 2.3d). It does not yetcontain other processes.

An appropriate way to describe consistently both NN and 3N interactionsis provided by tailored effective field theories. This approach allows to unifythe physics of nucleons and nuclei in connection with QCD [40]. In particu-lar, chiral symmetry breaking can be analyzed in terms of chiral perturbationtheory (CHPT). The theory uses perturbation techniques, providing a straight-forward way to improve the results by going to higher orders in a perturba-tive expansion, and also keeping theoretical uncertainties under control. In thepast two decades, this framework has successfully been applied to a varietyof low-energy reactions in the meson and baryon sectors. The original ideaof applying CHPT to few-nucleon systems was formulated in the early 1990sby Weinberg [41, 42]. In few-nucleon systems, one has to deal with the non-perturbative nature of the nuclear force, which can be understood as contactinteractions (without meson exchange). According to Weinberg [42], a resum-mation of certain classes of diagrams can be achieved via solving Faddeev-like equations with the effective nuclear potential derived using the CHPTtechnique. The first model of this type (the Texas model) was proposed byOrdóñez and van Kolck [43, 44]. Subsequent developments by Epelbaum etal. allowed to perform a consistent analysis of nd scattering at next-to-next-to-leading order (NNLO, or third order) [45]. In parallel, nd phase shifts atlow energy were extracted at NNLO by Bedaque et al. [46].

In CHPT, 3N forces appear at NNLO in the perturbative expansion, as il-lustrated by the diagrams in Fig. 3.1. These diagrams require some furtherexplanation since they are not exactly of the same type as the 3N interactionsintroduced in section 2.2. In the CHPT diagrams, the NN interaction can beeither mediated by a pion (πN vertices are represented as black dots) or dueto a contact interaction which parametrizes the short-range physics (contactvertices are represented by crossings between the nucleon lines). Superposi-

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tions of these vertices (black squares) are responsible for the 3N interactions.It is also possible to introduce the 3N-force contributions due to intermediate∆ excitations (such as represented in Fig. 2.3b) in the CHPT representation.Such terms would arise already at NLO; however, the NNLO diagrams of Fig.3.1 actually recover a large part of the terms arising from ∆ excitations, sothat it is believed that the inclusion of ∆ intermediate states is not needed inthe context of CHPT provided that the expansion is performed at an appropri-ately large order [47]. Fourth-order (NNNLO) calculations have been maderecently in the two-nucleon system [48, 49], showing that the fourth order isnecessary and sufficient for a NN potential reliable up to 290 MeV, with anaccuracy comparable to the one of the high-precision phenomenological po-tentials presented in section 3.1. With such a solid theoretical basis, accuratecalculations in three-nucleon systems−which would naturally include bothNN and 3N interactions−should become feasible in the near future.

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4. Search for three-nucleon forces

As we have seen in the previous chapter, the theory of nuclear interactionspredicts three-nucleon forces. It can be shown in the framework of CHPT (seesection 3.5) that the NN interactions are more important that the 3N ones,which are more important than the four-nucleon interactions, and so on [42].Still, the absolute strength of the 3N-force components cannot be predicted bytheory: the unknown parameters must be determined experimentally.

There are indications that 3N-force effects should be small, but not neg-ligible. For instance, in the most simple three-nucleon system, the triton, itappears that the binding energy cannot be reproduced accurately by calcu-lations with NN interactions alone. Relativistic Faddeev calculations with theCD-Bonn potential−which should have the correct off-shell behavior, see sec-tion 3.4−underbind the triton by 3.5% [26]. The inclusion of a 3N potentialin the calculations does account for this discrepancy [50].

Now we must face the following problem: 3N forces are small comparedwith NN forces and thus they are difficult to observe, but they still give acontribution comparable to, e.g., off-shell behavior effects, relativistic effects,or Coulomb effects. In an ab initio model of nuclei and nuclear reactions, allthese effects must be correctly implemented. To disentangle them and studytheir details experimentally, we have to find experimental observables whichare especially sensitive to each particular phenomenon.

4.1 Three-nucleon force observablesThree-nucleon forces can be best identified in three-nucleon systems. Systemsof two neutrons and one proton, i.e., the triton and neutron-deuteron (nd) scat-tering states, are preferred since they are free of Coulomb interactions.

In three-nucleon bound systems, 3H and 3He, as we have discussed, thebinding energy is a very good 3N-force indicator, mainly because its exper-imental value is known to a very good accuracy. That is the reason why, inmost theories, parameters related to the 3N interactions are fixed from the tri-ton binding energy. It is, however, interesting to note that other bound-systemobservables can be sensitive to 3N forces. For example, the description of the3H and 3He charge form factors is also improved when 3N forces are consid-ered [51].

In the low-energy limit, the nd and pd scattering lengths (or strengths ofthe interaction) show a sensitivity to 3N forces. In addition to the triton bind-

17

ing energy, the doublet nd scattering length can advantageously be used tofix the two parameters related to the 3N interactions at NNLO in tailored ef-fective field theory approaches [45, 46]. A recent high-accuracy measurementof the coherent nd scattering length, which is almost equivalent to the dou-blet nd scattering length, allows to set tight constraints on model descriptionsbased on modern NN and 3N potentials [52]. It is observed that most models,although their description is improved when including 3N forces, fail to repro-duce exactly the world-average experimental value of the nd scattering length.This has lead to the suggestion that additional diagrams may be needed in 3Npotentials, where the additional parameters would be set to fit the nd scatteringlength in addition to the triton binding energy [53].

Spin observables in nd and pd scattering are interesting in the sense thatthey can be used to study the spin structure of 3N interactions. In particular,it has been observed since a long time [29] that the nucleon and deuteronvector analyzing powers in elastic nucleon-deuteron scattering below 30 MeVcannot be explained by means of NN calculations, and that an inclusion ofexisting 3N potentials does not remove this discrepancy [54]. In principle, itcould be solved by a refinement of the 3N-interaction terms in CHPT [45].Alternatively, a new class of diagrams was proposed recently [13] (see Fig.2.3d on page 7) which are expected to affect the vector analyzing powerswhen implemented into a 3N potential.

Another rich source of information is the nucleon-deuteron breakup reac-tion. In this case, pd is a much more practical choice than nd, since the com-plete kinematics of the reaction can be obtained by detecting the two outgoingprotons. Effects of 3N forces in pd breakup can be studied in the intermediateenergy range (65−250 MeV) if a sufficiently large fraction of the full phasespace is measured [55, 56]. Recent pd breakup experimental results at 130MeV show a clear preference for the predictions in which 3N forces are in-cluded [57]. Such an approach is further encouraged by the recent advancesin including Coulomb interactions in the calculations [58].

Finally, 3N forces are expected to affect significantly the shape of the angu-lar distribution in elastic nd and pd scattering at intermediate energies [6, 59].The elastic nucleon-deuteron scattering differential cross section looks liketwo hills with a valley in between (see the drawing on the title page of thisthesis). This is due to the basic NN scattering processes. However, if 3N forcesare present, their contribution should scatter the incident nucleon in a muchmore isotropic way, since all three nucleons participate at once in the interac-tion, potentially absorbing an arbitrary fraction of the transverse momentum.Thus, the 3N forces are overwhelmed by NN forces in the regions of the hills,but they are on the same footing in the region of the valley. As a result, itturns out that 3N forces are expected to fill in the valley. While this effect ismost clearly visible in the shape of the nucleon-deuteron angular distribution,it has also been observed that calculations without 3N forces underestimatethe nd total cross section above 100 MeV [60].

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The effect in the minimum of the differential cross section has been seenin numerous elastic pd scattering experiments at intermediate energies (seethe references [23-29,32-34] of Paper V). However, the pd data sets do notalways agree with each other. For instance, at 135 MeV, the theoretical pre-dictions which include 3N forces agree very well with three japanese sets ofexperimental data [61], while they underestimate significantly the Groningen(Netherlands) data [62]. The Groningen data indicate a stronger effect thanexpected. To explain this, a possible candidate would be relativistic effects(which were not treated in the calculations), but a relativistic treatment per-formed without 3N forces indicates that such effects are not significant at thisenergy [61]. At 250 MeV, where a similar discrepancy is observed between theexperimental data and the predictions with 3N forces for both pd and nd scat-tering, it is not excluded that relativistic effects are responsible for the excess[63, 64]. The nd data by Palmieri at 152 MeV agree well with non-relativisticcalculations with 3N forces [65], thus supporting the picture in which, below200 MeV, relativistic effects are small and an inclusion of a 3N potential inthe calculations is necessary and sufficient to reproduce the differential crosssection. The nd data at 95 MeV presented in Papers III, IV and V as well asin the next sections are fully consistent with this picture. These results have adouble advantage. Firstly, elastic nd scattering is unaffected by the Coulombinteraction, and thus a comparison between nd and pd data, which has neverbeen done before around 100 MeV, allows to verify the theoretical indicationsthat Coulomb effects in pd scattering are not significant in the minimum ofthe angular distribution [66]. And secondly, at this energy, relativistic effectsare expected to be negligible.

4.2 Neutron-deuteron scattering experiments at TSLThe present elastic nd scattering experiments were performed at The SvedbergLaboratory (TSL) in Uppsala, using the neutron beam line simultaneouslywith the MEDLEY and SCANDAL setups (see Fig. 4.1). The TSL cyclotronproduced a 98 MeV proton beam which was directed toward a lithium-7 target,producing quasi-monoenergetic neutrons with a peak energy of 95 MeV atzero degrees. The charged particles were removed by deflecting magnets andthe remaining neutron beam was shaped by a set of collimators.

The MEDLEY setup is a vacuum chamber of about 70 cm diameter con-taining up to eight detector telescopes each equipped with two silicon detec-tors and one CsI detector. Two fission-based neutron monitors are situatedbetween the MEDLEY chamber exit foil and the SCANDAL setup. Each ofthe two SCANDAL arms can be seen as a set of twelve detector telescopesand can be used either to detect protons and deuterons directly or to detectneutrons by converting them to protons inside a converter scintillator detector.More information about the TSL neutron beam facility and the MEDLEY and

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Figure 4.1: Overview of the Uppsala neutron beam facility before year 2004. The pro-ton beam coming from the left stroke a lithium target and charged particles were de-flected by magnets. The resulting neutron beam was collimated and available throughthe MEDLEY and SCANDAL setups.

Figure 4.2: Picture of the MEDLEY setup during the present experiment. The detectortelescopes were placed in the forward hemisphere of the vacuum chamber. Each ofthem was equipped with two silicon detectors and one CsI detector.

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Figure 4.3: Drawing of the SCANDAL experiment in deuteron detection mode. Seventargets (alternated CD2, CH2 and C) were used simultaneously. The setup was usedone arm at a time, and recoil protons or deuterons were detected directly by one plasticscintillator, two drift chambers, another plastic scintillator and an array of twelve CsIcrystals. A typical event is indicated.

Figure 4.4: Drawing of the SCANDAL experiment in neutron detection mode. Thetarget was a cylinder containing water, heavy water or graphite. Scattered neutronswere detected by letting them pass through a veto plastic scintillator to reject chargedparticles, convert them into protons inside two thicker plastic scintillators, and trackingthe protons with the same set of detectors as in Fig. 4.3. A typical event is indicated.

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SCANDAL setups−which are shown in Figs. 4.2, 4.3 and 4.4 to illustrate thethree present experiments−can be found in Ref. [4] (MEDLEY) and in Ref.[5] (SCANDAL) .

The goal of the experiments was to measure of the full elastic nd scatteringangular distribution. The simplest approach is to detect the recoil deuteronsfrom deuterated polyethylene (CD2) and graphite (C) target foils (the back-ground from carbon must be subtracted). The larger the deuteron scattering an-gle, the lower its energy, and therefore there is an angular limit beyond whichthe deuterons cannot be detected anymore (they are stopped in the materialbetween their production and the detectors). Thus, one has to compromise be-tween having a thick target which allows good statistics but a limited angularcoverage and having a thin target which allows a large angular coverage butpoorer statistics. With the MEDLEY setup, which is in vacuum and has a highsensitivity to low-energy particles, it was reasonable to choose a relativelythin CD2 target (0.280 mm thickness). On the other hand, the SCANDALsetup is comparatively large and massive since it is primarily designed to de-tect scattered neutrons. In the case where the recoil deuterons were detected,it was judicious to choose a relatively thick CD2 target (1.060 mm thickness),allowing good statistics in the limited angular range covered−which, fortu-nately, corresponds to the cross-section minimum, where 3N force effects areexpected to be most clearly visible. This experiment is illustrated in Fig. 4.3,where seven targets (alternated CD2, C and CH2) were used simultaneouslyinside a multi-target device. The neutron scattering forward angular range,which was not covered with SCANDAL in deuteron mode, could be obtainedby using SCANDAL in neutron detection mode. In this last experiment, illus-trated in Fig. 4.4, we used water and heavy water contained in aluminum cansas targets, and detected the scattered neutrons by converting them to protonsinside one of the converter scintillators and tracking the protons through thedetectors.

Thus, we used a convenient combination of three complementary exper-iments. With the first experiment (deuteron detection with MEDLEY), weobtained a full angular coverage−although quite sparse−with statistics suf-ficient to distinguish between presence or absence of 3N forces in the mini-mum. With the second experiment (deuteron detection with SCANDAL), weobtained high-accuracy data in the minimum, allowing to discriminate be-tween the different models. With the third experiment (neutron detection withSCANDAL), we covered the forward angular range, thus allowing a consis-tency check of the shape of the whole angular distribution (versus the MED-LEY data). Although the effect we are looking for is really an effect in theshape of the angular distribution, the absolute normalization of the data is cru-cial, especially for the second experiment which covers only the minimum,where an error in the normalization could potentially fake or disguise the ef-fect of a three-body force. For this reason, we took special care to designthe experiments such as to allow an unambiguous internal normalization, by

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measuring a known cross section at the same time as we measured nd scat-tering. For the two experiments in deuteron detection mode, the solution ofchoice was to use CH2 targets and normalize the data to np scattering. For theSCANDAL experiment in neutron mode, we could also measure np scatter-ing but with poor accuracy. It proved to be a better technique to use a graphitetarget and normalize the data to the total elastic scattering cross section oncarbon.

4.3 Data analysisThe data analysis for the three experiments described in the previous sectionrepresents a major fraction of my working time as a PhD student, but I willonly give a brief overview here, since all details are given in Paper V.

The SCANDAL nd experiment where scattered neutrons were detected re-sembles the np experiment presented in Paper II. It was the most difficult toanalyze, and the least significant as far as 3N forces are concerned, since itdid not cover the cross-section minimum. Angular bins were defined (one foreach CsI), by using the DCH tracking information and selecting the trajecto-ries which intersected the crystals inside well-defined areas. The scintillatorand CsI detectors were calibrated by detecting recoil protons from np scatter-ing and associating the recorded pulse heights with the expected proton ener-gies. Protons were identified in two-dimensional plots of the energy depositedin the CsI versus the energy deposited in the trigger scintillators. Signals fromthe scintillators allowed to identify in which detector the conversion occurred.The proton conversion angles and neutron scattering angles were obtained bytracing back the proton trajectories to one point in the converter scintillatorplane. A maximum conversion angle of 10◦ was required in order to separatekinematically events converted in hydrogen from events converted in carbon.The neutron scattering energies were calculated by knowing the proton ener-gies and conversion angles, and assuming np scattering inside the converter.The remaining steps to identify the elastic nd scattering events (including thesubtraction of the oxygen and deuteron breakup backgrounds) were done bymanipulating the neutron energy spectra obtained with the different targets.Elastic scattering on carbon (C target) was also identified in the spectra, whichis important for the absolute normalization of the data; and, as a bonus, elas-tic scattering on oxygen (H2O and D2O targets) was also extracted from thespectra, as well as inelastic scattering to the low-lying collective excited statesfor both carbon and oxygen. Finally, corrections were applied, e.g., for deadtime, multiple scattering inside the target, and the angular dependence of theconversion efficiency. For nd scattering, the systematic uncertainties per pointwere 10% to 20%, and were dominated by the uncertainties in the oxygen andbreakup background subtraction procedures. In the regions were they over-lapped in angle, the data obtained with the two SCANDAL arms and the two

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converters of each arm were combined, reducing both the statistical and sys-tematic uncertainties per point since these four sets of data were analyzedseparately.

The analysis for the MEDLEY nd experiment and the analysis for theSCANDAL nd experiment where recoil deuterons were detected were sim-ilar. The principle is rather simple: essentially, all one has to do is to identifywhich events are induced by a 95 MeV neutron elastically scattered by a targetdeuteron and count them. Deuterons were identified by two-dimensional ∆Eversus E plots; 95 MeV neutrons were selected by time-of-flight techniques;and the elastic events could be identified in the energy spectra. Backgrounddeuterons from carbon were taken care of by subtracting the spectra obtainedwith the graphite targets. With SCANDAL, it was important to take the ef-fects of the multi-target device inefficiencies into account since they wereangle-dependent and varied from one target plane to another: in particular,it could affect the relative normalization of the carbon spectra in the carbonbackground subtraction. For absolute normalization purposes, np data wereobtained in the same way as for nd scattering. With SCANDAL, there wasone more complication related to the accidental fact that the CH2 target foilwas larger than the beam. This problem was circumvented by using the DCHtracking information to trace back the events to the CH2 target plane, whichallowed to determine its effective area (the beam cross section area) with anaccuracy of 2.25%. Corrections were applied for effects which could slightlyaffect the np and nd angular distributions, like the CsI detection efficiencyand the contamination from low-energy neutrons in the neutron beam spec-trum. In the region of the nd cross-section minimum, the relative uncertaintyper point was comparable for MEDLEY and SCANDAL (about ±4% statis-tical uncertainty and about ±5% systematic uncertainty). For both MEDLEYand SCANDAL, these uncertainties could be reduced by combining the sets ofdata taken on both sides of the beam; the SCANDAL measurement allowed toreach a better precision for testing 3N forces than the MEDLEY measurementbecause it was concentrated in the cross-section minimum, providing twice asmuch data points in this angular range.

4.4 ResultsThe results for np scattering are shown in Fig. 4.5. We should not be espe-cially impressed by the remarkable agreement with the high-accuracy Rahmet al. data [67] since the absolute scale of our data (with proton detection) wasadjusted to agree with the Rahm et al. data. It is comforting, however, that theshape of our np angular distribution matches well both the previous data andthe Nijmegen partial-wave analysis PWA93 [23] (high-precision NN poten-tial predictions are not shown in the figure but they give very similar curves).Since the nd data were measured with the same technique and in essentially

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the same conditions as the np data, we are confident that the shape of the ndangular distribution will be free of unexpected systematic errors.

The nd final results are shown in Fig. 4.6. We observe a good agreementbetween the three experiments in the regions where they overlap. The data arein overall agreement with the expectations both in shape and absolute scale.In the minimum region, including a two-pion-exchange 3N potential in thecalculations causes a 30% effect which fills the minimum (the dashed anddotted curves correspond to inclusions of the Tucson-Melbourne TM99 [37]and Urbana IX [32] 3N potentials, respectively, to be compared with the solidcurve where 3N forces are not included). Our data actually fill the minimumas expected from theories of 3N interactions.

In Fig. 4.7, by plotting the ratio between the nd data and the np data inthe minimum region, the effects of 3N forces are definitely confirmed. In-deed, normalization errors (and some other sources of systematic errors) arecancelled out for this ratio. Thus, it is difficult to imagine what kind of ex-perimental error might significantly affect this result. As a possible differencein detector behavior, I can identify the CsI efficiency and the multi-target effi-ciency, which have been taken into account. Other possible effects would be aninaccurate relative number of irradiated target nuclei (but this was determinedwith an accuracy better than 2.3%, see section 4.3), or the contamination ofprotons from deuteron breakup in nd scattering (but those were rejected bya particle identification cut). Relativistic effects, which were not included inthe calculations, are expected to be negligible at this energy [68], and the rel-ative differences in these effects for np and nd scattering must be even lessimportant. Thus, the clear excess observed in the minimum must be due toeither a three-body force or a totally unknown and unsuspected effect. Amongthe calculations using various NN and 3N potentials shown in Fig. 4.7, whichare all considerably closer to the data when 3N forces are included, the bestdescription (χ2 = 1.2) is obtained with the combination AV18 + Urbana IX.

The present nd data have the advantage to be free of Coulomb force ef-fects. The situation is different in elastic pd scattering, where, for studying3N forces, one would like to be sure that Coulomb effects are negligible in theminimum region. The present nd data give us an opportunity to validate thisassumption by providing the first comparison between elastic nd and pd scat-tering around 100 MeV (see Fig. 4.8). Moreover, a comparison in the forwardangular range, where the Coulomb effects are expected to be strongest, wouldallow to test the models which attempt to include these effects in the calcula-tions. Up do date, we cannot perform a very conclusive comparison becausethe only pd data available, by Chamberlain and Stern [69], do not have therequired precision. New pd data at this energy are expected to be publishedsoon and will allow a more detailed comparison [70].

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Figure 4.5: The np differential cross section at 95 MeV. The black dots, black squaresand open squares are results from the present experiments with MEDLEY, SCAN-DAL with proton detection, and SCANDAL with neutron detection, respectively. Thedata were normalized to the Rahm et al. data [67], shown as small triangles. The opentriangles are the previous SCANDAL data in neutron mode from Paper II. The solidline is the Nijmegen partial-wave analysis PWA93 [23] (see section 3.1).

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Figure 4.6: The nd differential cross section at 95 MeV. The MEDLEY data (blackdots) and SCANDAL data with deuteron detection (black squares) were normalizedto the np differential cross section using the Rahm et al. np data [67] as reference (seeFig. 4.5). The SCANDAL data with neutron detection (open squares) were normalizedto the total elastic 12C(n,n) cross section (see Paper V for details). The solid curve wasobtained with Faddeev calculations with the Argonne AV18 NN potential [25] without3N forces, and the dashed and dotted curves were obtained with the same calculationsusing an additional 3N potential: the Tucson-Melbourne TM99 3N potential [37] andthe Urbana IX 3N potential [32], respectively. The gray band corresponds to CHPTcalculations at NNLO [45].

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Figure 4.7: The ratio of the nd to the np differential cross sections at 95 MeV inthe region of the cross-section minimum, as a function of the proton/deuteron detec-tion angle. The dots and the squares are the MEDLEY and SCANDAL results (withdeuteron or proton detection), respectively. The thin, middle-thick and thick curveswere obtained with the AV18 [25], Nijm2 [24] and CD-Bonn [27] NN potentials, re-spectively. The dashed curves were obtained by including the TM99 [37] 3N potentialin the nd calculations, and the dotted curve was obtained by including the UrbanaIX [32] 3N potential. Note that the curves obtained with the AV18 and Nijm2 poten-tials are almost indistinguishable from each other; this is probably because they havesimilar off-shell behavior (see section 3.4).

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Figure 4.8: Comparison between the nd and pd differential cross sections at 95 MeV.The dots are the complete set of the present nd data, and the triangles are pd data byChamberlain and Stern [69]. With the present accuracy of the pd data, Coulomb forceeffects cannot be observed.

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5. Conclusions and outlook

Given that the interactions between nucleons are sufficiently well understood,through the Faddeev formalism it is possible to predict the properties of three-nucleon systems. For example, one can calculate the shape of the angular dis-tribution for 95 MeV neutrons elastically scattered from deuterons, taking intoaccount all possible exchanges between the nucleons taken individually. Byactually measuring this angular distribution we found that such a descriptiondoes not reproduce exactly the experimental data: it predicts a deeper mini-mum than the data suggests. This is simply because three-body interactionshave to be taken into account. This effect was foreseen by theory [6] beforeany measurement existed at 95 MeV−it was actually what motivated the ex-periments in the first place. After introducing a three-nucleon potential in theFaddeev equations, the theoretical predictions agree very well with our data.

An excellent characterization of the physics of few-nucleon systems is ob-tained by a combination of modern NN and 3N potentials. Although far frombeing unique, the present case has the merit of providing a good illustration ofthis success: the elastic nd scattering differential cross section at 95 MeV isaccurately described by calculations relying on NN and 3N potentials whoseparameters are fixed solely by NN data and the triton binding energy. Chiralperturbation theory can also give reasonably good predictions, and provides aconsistent framework for the description of few-nucleon systems.

With a good understanding of the nuclear forces in three-body systems, aswell as impressive advances in computer resources and computational tech-niques, it becomes gradually possible to unify the physics of nucleons andnuclei by trying to describe four-nucleon systems, and then five-nucleon sys-tems, etc. The alpha particle binding energy is reproduced very accuratelybased on combined NN and 3N forces [71], suggesting that the role of four-nucleon forces is not significant. If four-nucleon forces are, indeed, negligiblein nuclei, then it is reasonable to assume that forces of higher order (five-nucleon, etc.) are also negligible. In the near future, we will probably seemajor advances in this field of research: the behavior and properties of lightisotopes could in principle be described by a fully consistent ab initio theoryof nuclear interactions built upon the underlying two- and three-body interac-tions. In fact, recent models (in particular, the so-called ab initio no-core shellmodel (NCSM), see Refs. [72, 73] and references therein) are now able to givepredictions for the binding energies and nuclear structures of light isotopes upto carbon. An inclusion of a 3N potential generally improves the description,

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by increasing the binding energies and rearranging the level ordering and levelspacing of the low-lying excitation spectra.

Such an approach cannot be sustainable without constant refinements ofthe structure of the NN and 3N potentials, which imply stringent comparisonswith experimental data. For instance, spin observables such as analyzing pow-ers call for a better understanding of the spin structure of the 3N forces. Ac-curate measurements of the scattering lengths, deuteron break-up, and elasticscattering observables in the nucleon-deuteron system provide a good testingground for the 3N potentials. Concerning specifically elastic nd scattering, thedata are scarce in the intermediate energy region and the present work repre-sents an important contribution. As 3N forces are expected to be more accen-tuated with increasing energy and progress is being made in estimating therelativistic contributions, it would be natural to extend the present measure-ment up to higher energies. There has been a recent proposal for nd scatteringexperiments at TSL at 135 MeV, this energy being especially attractive asthe pd scattering experiments performed at 135 MeV by two different groups[61, 62] differ by about 30%. Elastic nd scattering data might help to resolvethis discrepancy. Above this energy (the maximum energy for the TSL neu-tron beam is 180 MeV), some upgrade of the CsI detectors is necessary, andsuch an upgrade is already in progress for MEDLEY.

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6. Acknowledgements

I am extremely grateful to my supervisor Jan Blomgren for his positive spiritand dynamic guidance.

Special thanks to Stephan Pomp and Udomrat Tippawan for precious con-tributions with the MEDLEY setup, and to Leif Nilsson for meticulously go-ing through my articles. Thanks to Jan Källne and Nils Olsson for their interestand good advices.

I am very thankful to all the other people who have been activelyparticipating in my work during the last four years, including BelBergenwall, Anders Hjalmarsson, Cecilia Johansson, Joakim Klug, PawelNadel-Turonski, Pär Olsson, Angelica Öhrn, and Michael Österlund. Thanksto Susanne Söderberg for administration support, and to all INF colleaguesfor a pleasant working environment, among others, Peter Andersson, ErikAndersson Sundén, Maria Back, Jean-Christophe Bourselier, Sean Conroy,Göran Ericsson, Anna Flodin, Maria Gatu Johnson, Luca Giacomelli,Wolfgang Glasser, Joel Gustafsson, Moinul Habib, Masateru Hayashi, CarlHellesen, Mikael Höök, John Loberg, Philip Magnusson, Christine Marklund,Emanuel Ronchi, Nils Sandberg, Henrik Sjöstrand, Janne Wallenius, LovisaWallin, Matthias Weiszflog, Gustav Wikström and Martin Wisell.

If I had not been part of the AIM graduate research program, I could prob-ably not have made an average of 1.25 transcontinental trips per year: manythanks to Erkki Brändas and Bo Thidé for their excellent leadership, as well asMarcus Dahlfors, Emma Hedlund, Roger Karlsson, Mattias Lantz, Joa Ljung-vall, Otasowie Osifo, Fredrik Robelius, Christofer Willman, and other AIMstudents for good companionship.

I wish to thank Alexander Prokofiev and Per-Ulf Renberg for their remark-able work with the neutron beam monitoring, Olle Jonsson for the 7Li targethandling, Anatoly Kolozhvari for making the data acquisition system work,as well as the other members of the TSL technical staff who were operatingthe cyclotron. I also want to thank Ib Koersner and Teresa Kupsc for computersupport.

I thank very much Yukie Maeda, Hideyuki Sakai and Atsushi Tamii forcollaborating so closely with us and having participated in some of our exper-imental runs. I have also appreciated the precious collaboration with KichijiHatanaka and Nasser Kalantar-Nayestanaki.

Fruitful discussions with Ulf Danielsson and Ruth Dürrer (gravity), andJohn Field and Gunnar Ingelman (QCD), were crucial for broadening my un-

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derstanding of three-body forces beyond my expertise area. I am also verygrateful to Kenneth Amos, Raquel Crespo, Evgeni Epelbaum, Walter Glöckle,Hiroyuki Kamada, Arjan Koning, Antonio Moro, and Henryk Witała for theirenthusiastic supply of theoretical calculations.

Finally, I want to express my deep gratitude to my parents, my brothersAlexandre and Kevin, and my wife Kristina for their love and support.

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7. Summary in Swedish

Avhandlingens titel översatt till svenska är Neutron-deutronspridning ochtrekropparväxelverkan.

Den handlar framför allt om tre experiment där vi mäter differentiellatvärsnitt för elastisk neutron-deutronspridning vid energin 95 MeV.Experimenten utfördes vid The Svedberglaboratoriet i Uppsala meddetektorutrustningarna MEDLEY och SCANDAL. För att kunna täcka helavinkelfördelningen med en bra precision har vi använt en kombinationav två olika tekniker, där man detekterar antingen rekyldeutroner direkt(med både MEDLEY och SCANDAL) eller så detekterar man spriddaneutroner som man konverterar till protoner i en plastscintillator (medSCANDAL). Absolutnormaliseringen bestämdes relativt till ett tvärsnittsom är välkänt och som mättes samtidigt som neutron-deutronspridning, ivårt fall neutron-protonspridning och elastisk neutronspridning på kol. Denexperimentella metodiken och diskussion av resultaten redovisas i artiklarpublicerade i internationella tidskrifter.

Neutron-deutronsystemet består av tre nukleoner som växelverkar baragenom stark växelverkan, vilket underlätter studiet av trekropparkrafter.Växelverkan mellan två nukleoner kan påverka kraften på en tredje nukleon.Detta innebär att krafterna mellan tre nukleoner inte kan beskrivas exakt somsumman av de individuella nukleon-nukleonväxelverkningarna, utan manbehöver också att ta hänsyn till trekropparkrafter. Detta är inte unikt för starkväxelverkan, utan det finns andra exampel inom fysik där trekropparkrafterkan uppstå.

Egenskaper hos trenukleonsystemet kan beräknas genom användning aven nukleon-nukleonpotential (för att beskriva nukleon-nukleonväxelverkan)samt en trenukleonpotential (för att ta hänsyn till trekropparkrafter) i de såkallade Faddeevekvationerna. Sådana beräkningar visar att trekropparkrafterökar det differentiella tvärsnittet för elastisk nukleon-deutronspridning medungefär 30% i det vinkelområde där tvärsnittet är minst.

Teoretiska förutsägelser som inkluderar trekropparkrafter stämmer bra öv-erens med våra experimentella data. Eftersom det är svårt att tänka sig någonannan effekt som skulle kunna modifiera vinkelfördelningen på samma sätti minimumområdet, kan vi tolka resultatet så att vi faktiskt ser effekter avtrekropparkrafter.

Detaljerade undersökningar av trekropparväxelverkan mellan nukleonerkan vara av avgörande betydelse för en fundamental beskrivning av

35

atomkärnor. Att kunna beskriva nukleoner och kärnor i en och samma modellär ett viktigt mål i kärnfysik. Då slipper man använda fenomenologiskamodeller som bara kan tillämpas i speciella fall. Nyligen har det blivit möjligtatt beskriva lätta kärnor upp till A=12 (kol) ganska väl med beräkningar sombaseras på nukleon-nukleon och tre-nukleonpotentialer. Genom att utvecklapotentialerna kan den här metodiken sannolikt så småningom ge bättre ochbättre resultat.

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