tutorial on analytic theory for cross-polarization in
TRANSCRIPT
Tutorial on Analytic Theoryfor Cross-Polarization inSolid State NMRDAVID ROVNYAK
Department of Chemistry, Bucknell University, Moore Avenue, Lewisburg, PA 17837
ABSTRACT: This tutorial aims to be a self-contained and explicit analysis of the basic
cross polarization (CP) solid-state NMR experiment (SSNMR) in the isolated spin-pair
approximation using standard quantum mechanical arguments. The general result of
obtaining coherence transfer between two dipolar coupled spin-active nuclei while
applying radio frequency fields to both nuclei is described, with emphasis on the origin
of the well known Hartmann-Hahn matching conditions. No new theory is presented;
rather several common analytical methods in SSNMR are demonstrated in the context of
cross-polarization under static and magic-angle spinning (MAS) conditions. A background
in NMR and quantum mechanics is assumed, however this work attempts to minimize
the need for excursions into the literature. This article was written to aid a reader in advanc-
ing into more detailed descriptions of CP and dipolar recoupling in general. � 2008 Wiley
Periodicals, Inc. Concepts Magn Reson Part A 32A: 254–276, 2008.
KEY WORDS: cross polarization; solid state nuclear magnetic resonance; Hartmann-
Hahn matching condition; dipolar recoupling
INTRODUCTION
The use of double radio frequency (r.f.) irradiation
on a two spin system to transfer coherence among
the nuclei was presented by Hartmann and Hahn, (1).The technique of cross polarization (CP) has since
become immensely important in the practice of solid
state NMR. Perhaps the greatest value of CP is in
enhancing the signals of low gamma nuclei (13C or15N) that are dipolar coupled to proton spin baths. In
addition, the modern development of CP has led to
recent experiments that perform highly selective cou-
pling among nuclei (2, 3). A few examples of other
extensions include: amplitude modulated spin-lock-
ing pulses that achieve improvements in CP dynam-
ics, (4) Lee Goldburg decoupling which can be per-
formed simultaneously with the spin-locking step to
attenuate homonuclear proton couplings, (5) and
multiple-quantum CP which can be performed in
half-integer quadrupole systems (6–8).CP is arguably the gateway experiment into
SSNMR, particularly of biomolecules, and em-
Received 29 January 2008; revised 23 April 2008;
accepted 6 May 2008
Correspondence to: David Rovnyak; E-mail: [email protected]
Concepts inMagnetic Resonance Part A, Vol. 32A(4) 254–276 (2008)
Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/cmr.a.20115
� 2008 Wiley Periodicals, Inc.
254
bodies core concepts in dipolar coupling, magic
angle spinning, and common approximations used
throughout. Although CP encompasses a very large
family of experiments such as those noted above, it
is also a member of a ‘‘super family’’ of experi-
ments that perform dipolar recoupling among spins
(9–11).No new theory is presented. This article aims to
be a didactic, explicit development of the core results
for cross-polarization in static and magic-angle spin-
ning conditions. There are many treatments of CP in
texts and articles, (1, 11–17) as well as an excellent
set of papers on experimental aspects of CP/MAS
(18, 19). This article attempts to provide a tutorial of
the theory to complement prior reports that will aid
in learning solid-state NMR (SSNMR) theory or
serve as a refresher.
In this article, many assumptions will be ad-
opted to maintain progress. For example, homonu-
clear couplings are not explicitly treated, although
the Proton–Proton Interactions in Static CP section
previews their effects. Also, the thermodynamic
spin temperature description, while useful, is not
presented.
The pulse sequence to be considered is given in
Fig. 1. In this article, the capital letters I, S will be
used to represent an abundant, high-gamma nucleus
and a rare, low-gamma nucleus, respectively. Histori-
cally, the mnemonics are I ¼ insensitive (high abun-
dance, high sensitivity, often 1H), and S ¼ sensitive
(low abundance, low sensitivity, often 13C or 15N).
Briefly, the basic CP experiment offers:
1. Enhancement (Z) of the signal of a low-
gamma nucleus (S) by a factor on the order
of the ratio of the gyromagnetic ratios
Z / gIgS
:
2. Recycling of magnetization dependent upon
the T1 of the abundant, high-gamma nucleus.
For organic and biological solids, the 1H T1’sare much shorter (T1 ¼ 1–3 s) than those of13C (T1 ¼ 5–15 s).
3. Spectral simplification. With appropriate cy-
cling of phases of the initial I spin p/2 pulse
and the phase of the receiver detection of the
S spin (termed spin-temperature alternation in
the thermodynamic picture), CP results in S-
spin signals that originate only from dipole
coupled I-spins.
4. Multidimensional dispersion. CP can be used
as the mixing period in a 2D heteronuclear
correlation experiment.
The key to carrying out CP is to achieve what is
termed a Hartmann-Hahn match in which the r.f.
power levels applied to the I and S spins meet certain
criteria. There are important differences in how this
match is achieved in static and spinning samples.
Essentially, the principal aim of this tutorial is to
carefully obtain the matching conditions for success-
ful CP between an I–S spin pair.
CP IN A STATIC SOLID
A Hamiltonian in SSNMR
CP will be modeled by a heteronuclear two-spin sys-
tem consisting of Zeeman and dipolar interactions.
The chemical shift anisotropy is neglected. The Ham-
iltonian written in the lab frame is
HLAB ¼ HZI þ HZ
S þ HrfI þ Hrf
S þ HDII þ HD
SS þ HDIS
¼ �gIB0IZ � gSB0SZ � 2gIB1;I cos orf ;It� �
IX
� 2gSB1;S cos orf ;St� �
SX þ HDII þ HD
SS þ HDIS; ð1Þ
and it is already expressed in frequency units, which
was accomplished by dividing each side by � once.
Hamiltonian operators are denoted with a caret (^),
and nuclear spin angular momentum operators are
given in boldface type. Other symbols are:
Figure 1 Schematic diagram of the basic experiment
employing cross polarization. A p/2 pulse applied to the I
spin creates transverse I spin coherence which is trans-
ferred to the S spin during the double irradiation period,
the period that will be modeled here. Decoupling is
applied to the I-spin for line narrowing of the S-spin spec-
trum during acquisition.
ANALYTIC THEORY FOR CROSS POLARIZATION 255
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B0 ¼ static applied external field
gI ¼ gyromagnetic ratio for spin I
gS ¼ gyromagnetic ratio for spin S
orf ;I ¼ frequency of applied r:f: field on spin I
orf ;S ¼ frequency of applied r:f: field on spin S
B1;I ¼ magnitude of r:f: field applied to spin I
B1;S ¼ magnitude of r:f: field applied to spin S
HZI ; H
ZS ¼ Zeeman interaction Hamiltonians
HrfI ; H
rfS ¼ applied ratio frequency Hamiltonians
HDII ¼ homonuclear I�I dipole coupling
HDSS ¼ homonuclear S�S dipole coupling
HDIS ¼ heteronuclear I�S dipole coupling
Additional symbols are given in an appendix. The
factor of 2 in the magnitude of the r.f. terms is used
as a convenience. We will find that since the r.f. field
is linearly polarized, half of the magnitude remains
in the rotating frame.
The Dipolar Coupling Hamiltonian
The homonuclear dipole coupling terms, HDII and
HDSS, of course do not exist for an isolated heteronu-
clear two-spin system. They are included in Eq. [1]
to raise the question of whether the two-spin model
is acceptable for CP. Given that the S spin is taken to
be ‘‘rare,’’ as with natural abundance 13C (nat. abund.
�1%), the homonuclear S–S dipole coupling is very
safely neglected. The I spins usually are abundant,
and a significant I–I dipole coupling will exist, as in
the most common case of I ¼ 1H. The effect of HDII
can extend the breadth of the matching conditions
(Proton–Proton Interactions in Static CP section),
enables enhancements that exceed those observed in
the isolated two-spin approximations, influence the
rate of approach to thermal equilibrium, and allow
higher order matching conditions.
Previous work has explored analytical treatments
with HDII by treating the IS spin pair in the context of
an I-spin bath (13, 20). We mainly neglect HDII as
there is little other way to get a straightforward ana-
lytical result for CP, however we will briefly consider
its effects later. The two-spin model will be found to
provide a good description of CP. The dipole–dipole
Hamiltonian in the LAB frame often employs the
‘‘dipolar alphabet’’ notation (21–23),
HDIS ¼ m0gIgS
4pr3Aþ Bþ Cþ Dþ Eþ F� �
: [2]
The constant term m0/4p imbues Eq. [2] with SI
units but will be dropped for convenience, putting
Eq. [2] in cgs units. Equation [2] shows the high sen-
sitivity of the magnitude of the dipole coupling to
internuclear separation via r�3, where r is the inter-
nuclear I–S distance. The terms are
A ¼ 1� 3 cos2 b� �
IZSZð Þ;B ¼ 1
21� 3 cos2 b� �ðIZSZ � I
* � S*Þ
¼ �1
21� 3 cos2 b� �ðIþS� � I�SþÞ;
C ¼ �3
2sin b cos be�if IþSZ þ IZS
þð Þ;
D ¼ �3
2sinb cos beif I�SZ þ IZS
�ð Þ;
E ¼ �3
4sin2 be�2if IþSþð Þ;
F ¼ �3
4sin2 be2if I�S�ð Þ; ½3�
where b and f are spherical coordinates relating the
internuclear vector to the laboratory x, y, and z axes.Specifically b is the angle between the external mag-
netic field and the vector connecting the I and S spins
(often denoted y in many sources) and f is the angle
of rotation about the lab Z axis. Complete spherical
coordinates are: (r, b, f). The terms B� F are sup-
pressed for heteronuclear coupling, while C� F are
suppressed for homonuclear coupling. Two common
explanations are summarized here. Consider the ma-
trix representation of the Zeeman and dipolar terms:
HZI þ HZ
S þ HDIS ¼
þþj i þ�j i �þj i ��j i12ðo0;I þ o0;SÞ þ a c c e
d 12ðo0;I � o0;SÞ þ a b c
d b 12ð�o0;I þ o0;SÞ þ a c
f d d �12ðo0;I þ o0;SÞ þ a
26664
37775: ½4�
256 ROVNYAK
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
In Eq. [4] o0;I ¼ �gIB0 and o0;S ¼ �gSB0 are the
Zeeman frequencies of the I and S spins, and lower
case letters show which states are coupled by the cor-
responding parts of the dipolar alphabet in Eq. [3].
For dipole coupled heteronuclei the B� F terms are
nonsecular (i.e. noncommuting) with respect to IZand SZ, and are safely ignored since they are trun-
cated by the Zeeman Hamiltonian. In particular the Bterm (IZSZ �~I �~S ¼ �ðIXSX þ IYSYÞ) is noncom-
muting with IZ or SZ in the heteronuclear case. In a
frame rotating at the Larmor frequency (vide infra,
The (Doubly) Rotating Frame section), all nonsecular
terms are sinusoidally modulated by the Larmor fre-
quency and rapidly average to zero (see section 3.2.4
of Ernst for an expanded discussion (24)). For
a homonuclear spin pair, the B term is invariant to
the rotating frame transformation and must be
retained (since IZ ¼ Ið1ÞZ þ I
ð2ÞZ commutes with both
Ið1ÞZ I
ð2ÞZ and~Ið1Þ �~Ið2ÞÞ.
We can also show the dipolar truncation with a
perturbation approach. We see that B� F contributes
only off-diagonal elements in the matrix representa-
tion of the Hamiltonian (Eq. [4]), so that in first order
perturbation theory we must retain A for both homo-
and heteronuclear spin pairs. In the heteronuclear
case, we can apply second order nondegenerate per-
turbation theory to any of B� F to obtain energy
corrections that are directly proportional to the
square of the coupling and inversely proportional to
the Zeeman energy differences of the states that are
being mixed. We see from Eq. [4] that for B� F the
second order perturbation corrections will be inver-
sely dependent on one or a combination of the Zee-
man frequencies and so may be safely neglected.
In the case of homonuclear dipole coupling, we
may apply the above argument to the C� F terms
since they still couple states with nondegenerate Zee-
man energies. But the B term couples states that are
nearly degenerate (recognizing that two I spins will
still have different Larmor frequencies due to chemi-
cal shifts). The jþ 12;� 1
2i and j� 1
2;þ 1
2i (a.k.a. flip-
flop) states are so close in their Zeeman energies that
nondegenerate perturbation theory cannot be applied,
and indeed the energies of the flip-flop states will be
significantly changed by the B term, which must
therefore be kept (see Slichter (22) and Levitt (25)).It is convenient now to put
d ¼ gIgS�hr3
1� 3 cos2 y� �
; [5]
so that the laboratory frame dipole Hamiltonian
obtained from Eq. [2] is
HDIS ¼ dIZSZ: [6]
In the static case it will suffice to leave the orien-
tation dependent term d alone, but we will treat d ex-
plicitly for the case of magic-angle spinning.
The (Doubly) Rotating Frame
The next step is to write the Hamiltonian in frames
that rotate according to the applied r.f. frequencies.
This is a simplification strategy: by transforming the
Hamiltonian into a frame governed by the strongest
interaction, some terms may be truncated (i.e.
neglected). For rotating frame transformations (i.e.
when H0 below is proportional to IZ or SZ), it may
be shown that if H ¼ H0 þ H1 then
H� ¼ eiH0t H0 þ H1
� �e�iH0t � H0
¼ eiH0tH1e�iH0t ½7�
is the Hamiltonian which satisfies the time-dependent
Schrodinger equation in the rotating frame (see
standard texts, e.g. Levitt Ch. 9 (25), Ernst et al. Ch.2 (24), or Cavanagh et al. Ch. 2 (26)). In other words,
H� governs the time evolution of the spin system and
is the relevant Hamiltonian to keep track of when
working in rotating frames. The filled circle denotes
the use of a rotating frame. The rotating frame trans-
formation is most likely to lead to further insight
when H0 >> H1. We will allow for offsets by using
the frequencies of the applied r.f. fields to define the
rotating frame (i.e. orf,I and orf,S in H0), rather than
the Larmor frequencies of the I and S spins. To
obtain the necessary form for expressing the labora-
tory frame Hamiltonian of Eq. [1] in a frame rotating
according to the I-spin and S-spin r.f. frequencies,
add and subtract H0 ¼ orf ;IIZ þ orf ;SSZ to Eq. [1]:
H ¼ H0 þ H1 ¼ orf ;IIZ þ orf ;SSZ� �
þ"
�gIB0IZ � gSB0SZ � 2gIB1;I cosðorf ;ItÞIX�2gSB1;S cosðorf ;StÞSX þ dIZSZ
� �
� orf ;IIZ þ orf ;SSZ� �� ½8�
We are allowing for off-resonance pulses so we
have for the moment that orf,I � �gIB0 and orf,S ��gSB0. Then H1 consists of the r.f. power and dipole
coupling terms, but now includes presumably very
small resonant offset terms such as �gIB0 �orf,I.
Therefore Eq. [8] is consistent with the assumption
that H0 >> H1.
Before proceeding, we review how rotations are
handled in spin space. Derivations can be found, for
ANALYTIC THEORY FOR CROSS POLARIZATION 257
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
example, in Slichter Ch. 2 (22). A complete set of
examples showing the necessary operator sandwiches
is illustrated in Fig. 2, where the spin angular mo-
mentum operators are vectors and all angles are posi-
tive. Although Fig. 2 explicitly shows all cases, it
should be pointed out that the definition for the rota-
tion operator is RiðyÞ ¼ expð�iIjyÞ for j ¼ x, y, z.We now use these rules to express Eq. [8] in a
doubly rotating frame. It is important to note also
when performing sandwich operations such as Eq.
[7], that if H0; H1
� � ¼ 0, then H1 is unchanged.
Notice that the Larmor and dipole terms of H1 in Eq.
[8] commute with H0, and so they are unchanged
under the rotating frame transformation:
H� tð Þ¼eiH0tH1e�iH0t
¼
�gIB0IZ�gSB0SZþdIZSZð Þ�2gIB1;I cosðorf ;ItÞðIXcosorf ;It�IY sinorf ;ItÞ�2gSB1;S cosðorf ;StÞðSXcosorf ;St�SY sinorf ;StÞ� orf ;IIZþorf ;SSZ� �
26664
37775
¼ð�gIB0�orf ;IÞIZþð�gSB0�orf ;SÞSZþdIZSZ
�2gIB1;I cosðorf ;ItÞðIXcosorf ;It�IY sinorf ;ItÞ�2gSB1;S cosðorf ;StÞðSXcosorf ;St�SY sinorf ;StÞ
264
375:[9]
The final line in Eq. [9] is the complete rotating
frame Hamiltonian for a heteronuclear, dipole-coupled,
two-spin system under double radio-frequency irradia-
tion. However Eq. [9] is rarely seen or used—possibly
never—due to a common simplification to eliminate
the time-dependent terms, described next.
Average Hamiltonian in theRotating Frame
By taking an appropriate time average in the doubly
rotating frame, we will identify the parts of the Hamil-
tonian which remain time-independent after the rotat-
ing frame transformation. The use of average Hamilto-
nians is of great importance in understanding spin dy-
namics in SSNMR, but can become very complex
(27); hence, we only consider simple cases here. To
obtain an average Hamiltonian, H�, we must assume
that there exists a time t that simultaneously satisfies
the periodicity of both applied r.f. frequencies:
orf ;It ¼ n2p and orf ;St ¼ m2p; for n;m ¼ 1; 2; . . .½ �:[10]
This is reasonable since both r.f. frequencies are
in the MHz regime so that even several integer multi-
ples of the r.f. periods can safely be considered as
much shorter than the NMR time scale. The integral
is straightforward. First,
�H� ¼ 1
t
Zt0
H�ðtÞdt ¼ ð�gIB0 � orf ;IÞIZ
þ ð�gSB0 � orf ;SÞSZ þ dIZSZ
� 21
t
Zt0
ðgIB1;I cos2ðorf ;ItÞIX
þ gSB1;S cos2ðorf ;StÞSXÞdt: ½11�
Figure 2 Rotations in ‘‘spin space’’ refer to rotating the spin angular momentum operators, rep-
resented as vectors; initial positions of the SX vector are shown by the bold-face arrows. All
angles theta are positive.
258 ROVNYAK
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
The time-independent terms are clearly un-
changed, while integrals of the formRcosðtÞsinðtÞdt
are zero for periodic limits since the integrand is an
odd function. ThenZcos2ðatÞdt ¼ 1
2tþ 1
4asinð2atÞ [12]
is all that is needed to finish the integration:
�H� ¼ ð�gIB0 �orf ;IÞIZ þ ð�gSB0 �orf ;SÞSZ þ dIZSZ
� 2gIB1;IIX1
t1
2tþ 1
4orf ;Isinð2orf ;ItÞ
� �t0
� 2gSB1;SSX1
t1
2tþ 1
4orf ;Ssinð2orf ;StÞ
� �t0
¼ �gIB0 �orf ;I
� �IZ þ ð�gSB0 �orf ;SÞSZ
þ dIZSZ � gIB1;IIX � gSB1;SSX: ½13�
Here it was noticed that t is guaranteed to be a multi-
ple of 2p by Eq. [10]. If the irradiation is on-reso-
nance (e.g. (�gSB0 � orf,S) ¼ 0 and (�gIB0 � orf,I)
¼ 0), then Eq. [13] simplifies to the following aver-
age Hamiltonian in the rotating frame:
�H� ¼ o1;IIX þ o1;SSX þ dIZSZ; [14]
where o1,I ¼ �gIB1,I and o1,S ¼ �gSB1,S. This
expression is consistent with the vector picture often
used to describe the rotating frame in which vectors
corresponding to each applied transverse r.f. field are
stationary. Also the dipole coupling is not affected
by the rotating frame or average Hamiltonian trans-
formations.
The symbols o1,I ¼ �gIB1,I and o1,S ¼ �gSB1,S
reflect the power of the applied r.f. fields, not the fre-
quency. They are often termed the nutation rates for
the applied I and S spin r.f. fields, which can be
appreciated by example: if a given r.f. power pro-
duces a p/2 pulse in 5 ms, then a 20-ms pulse would
be a full 2p rotation and the nutation frequency is
1/20 ms ¼ 50 kHz.
A Special Tilted Frame
It is useful to relabel the axes, which is accomplished
in this case by applying a 908 rotation to each spin.
The r.f. terms will become IZ and SZ and the dipole
term will be transverse. This is illustrated for the I-
spin in Fig. 3. This is a special case of a more general
tilted frame that is described later. To further sim-
plify the notation we make the substitution
D ¼ 1
2d; [15]
where the factor of (1/2) will be convenient later.
The tilting is
HTrot ¼ ei
p2ðIYþSYÞ �H�e�i
p2ðIYþSYÞ
¼ o1;S sinp2SZþ cos
p2SX
þo1;I sin
p2IZþ cos
p2IX
þ ei
p2ðIYþSYÞ2DIZSZe�i
p2ðIYþSYÞ
¼ o1;SSZþo1;IIZþ 2Deip2IYIZe
�ip2IY
� � sinp2SX þ cos
p2SZ
¼ o1;SSZþo1;IIZþ 2D � sin
p2IX þ cos
p2IZ
ð�SXÞ
¼ o1;SSZþo1;IIZþ 2DIXSX: ½16�This is the tilted average Hamiltonian in the rotat-
ing frame and is often rewritten with raising and low-
ering operators:
HTrot ¼ o1;IIZ þo1;SSZ þ 2DIXSX
¼ o1;IIZ þo1;SSZ
þ 1
2D IþSþ þ IþS� þ I�Sþ þ I�S�ð Þ: ½17�
This tilting step is taken with the initial density
operator in mind. The initial condition for the density
Figure 3 Depiction of tilting transformation of the I-spin such that the r.f. term is relabeled as
IZ and is then ‘‘diagonal’’ in a Zeeman basis.
ANALYTIC THEORY FOR CROSS POLARIZATION 259
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
operator in the lab frame will be transverse I-spin
polarization due to the first 908 pulse. By tilting the
Hamiltonian, we will also need to tilt the initial den-
sity operator, rendering it longitudinal (i.e. propor-
tional to IZ), and this will lead to a picture of polar-
ization transfer in the tilted frame. Equation [17] is
often the starting point of many articles. In this arti-
cle the ‘‘bar’’ over the Hamiltonian is now dropped
since that initial averaging step is often regarded as a
straightforward truncation step and not worth extra
notation. We use the subscript ‘‘rot’’ to indicate the
average rotating frame Hamiltonian, and the super-
script T to remind us of the tilting. Authors may use
no special indication at all for Eq. [17], feeling that
the tilting and rotating frame are obvious from its
form. Equation [17] should be memorized.
Two Subspaces in the Tilted Hamiltonian
We wish to see more explicitly how the dipole coupling
in Eq. [17] can drive transitions that would lead to S-
spin signal. An important consequence of Eq. [17] can
be seen by writing out and diagonalizing the matrix rep-
resentation for HTrot. The matrix representation is
HTrot ¼o1;IIZþo1;SSZþ 1
2D IþSþ þ IþS� þ I�Sþ þ I�S�ð Þ
þþj i þ�j i �þj i ��j i
¼
1
2o1;Iþo1;S
� �0 0
1
2D
01
2o1;I�o1;S
� � 1
2D 0
01
2D �1
2o1;I�o1;S
� �0
1
2D 0 0 �1
2o1;Iþo1;S
� �
266666666664
377777777775; ½18�
where the bracket notation gives the I spin first, and
is simplified by using jþi ¼ j1/2i , and j�i ¼ j�1/2ifor the z-components of the spin operators in this
tilted, rotating frame. Recall that matrix representa-
tions of the spin operators are
SX ¼ 1
2
0 1
1 0
� �; SY ¼ 1
2
0 �i
i 0
� �; SZ ¼ 1
2
1 0
0 �1
� �;
Sþ ¼ 0 1
0 0
� �; S� ¼ 0 0
1 0
� �: ½19�
It is readily seen from Eq. [18] that the tilted
Hamiltonian is block diagonal. That is, the jþþi,j��i subspace is independent of the jþ�i,j�þi sub-space. Then define
J23Z ¼ 1
2
0 0 0 0
0 1 0 0
0 0 �1 0
0 0 0 0
0BBB@
1CCCA; J14Z ¼ 1
2
1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 �1
0BBB@
1CCCA;
J23X ¼ 1
2
0 0 0 0
0 0 1 0
0 1 0 0
0 0 0 0
0BBB@
1CCCA; J14X ¼ 1
2
0 0 0 1
0 0 0 0
0 0 0 0
1 0 0 0
0BBB@
1CCCA:
[20]
Then J23Z and J14Z can be associated with the differ-
ence and the sum, respectively, of the I and S diago-
nal terms,
J23Z ¼ 1
2IZ � SZð Þ;
J14Z ¼ 1
2IZ þ SZð Þ: ½21�
The 2–3 subspace is the ‘‘zero quantum’’ (ZQ)
subspace and describes the case in which the I- and
S-spins undergo opposite changes in spin states.
Informally these are often called ‘‘flip-flop’’ transi-
tions; this is a concerted event for two spins such
that the net change of the spin states is 0. There are
two possible flip–flop transitions for the I,S spin
system. If the I spin changes from the þ12to �1
2
state, then the S spin must undergo a change from
the �12to þ1
2state, and vice versa. The 1–4 subspace
is the ‘‘double quantum’’ (DQ) subspace and
describes the case in which I and S spins undergo a
concerted event so that the total change in spin
states is 62. Informally, these are often called
‘‘flip–flip’’ or ‘‘flop–flop’’ transitions. The two dou-
ble quantum transitions are when both spins change
from þ12to �1
2states, or when both spins change
from �12to þ1
2states.
260 ROVNYAK
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In the Zeeman basis used in Eq. [18], we have
IZ ¼ 1
2
1 0 0 0
0 1 0 0
0 0 �1 0
0 0 0 �1
0BBB@
1CCCA; SZ ¼ 1
2
1 0 0 0
0 �1 0 0
0 0 1 0
0 0 0 �1
0BBB@
1CCCA;
IXSX ¼ 1
4
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
0BBB@
1CCCA; IYSY ¼ 1
4
0 0 0 �1
0 0 1 0
0 1 0 0
�1 0 0 0
0BBB@
1CCCA:
[22]
We may now define
J23X ¼ IXSX þ IYSY;
J14X ¼ IXSX � IYSY: ½23�
The tilted Hamiltonian in this notation is
HTrot ¼ H23 þ H14;
where H23 ¼ o1;I � o1;S
� �J23Z þ DJ23X ;
H14 ¼ o1;I þ o1;S
� �J14Z þ DJ14X ; ½24�
and H23 and H14 commute since they are in inde-
pendent subspaces. The transverse terms in Eq. [24]
are proportional just to ‘‘D,’’ which is the reason for
introducing the factor of 12in Eq. [15]. Importantly,
Eq. [24] suggests the importance of the difference
and sum of the r.f. nutation frequencies and foreshad-
ows the Hartmann-Hahn matching conditions that
will be developed shortly. In review, Eqs. [20]–[23]
allow us to write the Hamiltonian with separate terms
that led to the zero quantum and double quantum
subspaces.
Diagonalizing the Tilted Hamiltonian
The Hamiltonians in the two subspaces given above
will be diagonalized. This is being done with an eye
towards simplifying the process of propagating the
density operator. The entire process is done ‘‘by
inspection’’ since diagonalizing each part of Eq. [24]
is accomplished by rotating a vector onto the Z axis.
The problem of diagonalizing Eq. [24] is to find out
what angle rotates the Hamiltonian onto the Z axis,
and write an expression for the magnitude of the vec-
tor. This is shown in Fig. 4 for the zero quantum sub-
space, where
sin y23 ¼ D
D2 þ o1;I � o1;S
� �2 1=2 ;
sin y14 ¼ D
D2 þ o1;I þ o1;S
� �2 1=2 ;
cos y23 ¼ o1;I � o1;S
D2 þ o1;I � o1;S
� �2 1=2 ;
cos y14 ¼ o1;I þ o1;S
D2 þ o1;I þ o1;S
� �2 1=2 ;
o23eff ¼ o1;I � o1;S
� �2 þ D2 1=2
;
o14eff ¼ o1;I þ o1;S
� �2 þ D2 1=2
; ½25�
determine y23 and y14 and the vector magnitudes o23eff
and o14eff . We now have
H23diag ¼ o1;I � o1;S
� �2 þ D2 1=2
J23Z ¼ o23effJ
23Z ;
H14diag ¼ o1;I þ o1;S
� �2 þ D2 1=2
J14Z ¼ o14effJ
14Z :
[26]
By simplifying the Hamiltonian in this way it
becomes much easier to predict the spin dynamics of
the two spin system without having to resort to com-
putational methods.
Predicting Spin Dynamics
The general approach for solving the dynamics of
the spin system under some Hamiltonian is to write
Figure 4 Diagram of the diagonalization of the tilted
Hamiltonian in the 2–3 subspace. Although in principle,
the expression above the arrow could be evaluated using
standard rules of spin operator rotations (i.e. Fig. 2), it is
easier to evaluate the diagonalization ‘‘visually’’ with trig-
onometric relations.
ANALYTIC THEORY FOR CROSS POLARIZATION 261
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out an initial density operator and carry out time
propagation with respect to an observable. We
review the steps to obtaining the Hamiltonian so
that we know how to write an appropriate initial
density operator:
1. Take lab frame Hamiltonian into a double
rotating frame and take the average;
2. tilt the Hamiltonian so the r.f. terms are longi-
tudinal;
3. identify two subspaces of the Hamiltonian and
diagonalize each.
The steps for deriving the spin behavior:
1. Take the initial density operator into the dou-
bly rotating frame: rLABð0Þ ! rrotð0Þ;2. apply the tilting operation as we did for the
Hamiltonian: rrotð0Þ ! rTrotð0Þ;3. apply the diagonalizing operations (i.e. y23,y14)
to the density operator: rTrotð0Þ����!y23;y14 ~rð0Þ;4. propagate: ~rðtÞ¼e�iðH23
diagþH14
diagÞt~rð0ÞeiðH23
diagþH14
diagÞt;
5. undo the diagonalization (i.e. �y23 and �y14);6. apply the operator for observable S spin sig-
nal (Sþ) to the density operator in either the
tilted rotating frame or the lab frame and
interpret the results.
To constrain the scope, an introductory level of fa-
miliarity with the density operator (r) and the prod-
uct operator method (28) is assumed. A good primer
on density matrix theory is given by Farrar (29, 30),while these topics are widely presented in standard
texts.
1. We wish to know what happens if we apply
double irradiation to a two spin system in which only
the I-spin is initially polarized. The equilibrium den-
sity operator in the lab frame is
rLABðeqÞ ¼ Z�1e�H�hkT � Z�1 1� H�h
kT
� �; [27]
where the expansion is justified using the common
high-temperature approximation (i.e. �hH ,, kBT),and Z is the partition function and is closely approxi-
mated by 2I þ 1 for a single spin I in the high-tem-
perature limit (24, 25). Since everything commutes
with 1 (identity operator/matrix), it will be invariant
to all subsequent operations and will not be carried
further. We could write a very general density opera-
tor for multiple I spins by inserting the Zeeman and
homonuclear dipolar interactions for just the I spin,
H ¼ �gIB0Iz þ HDII , where HD
II ¼ Aþ B, to use just
the secular dipolar interaction (17). We will write the
lab frame density operator for just the I spin
rLABðeqÞ �gIB0IZ�h
2kBT; [28]
assuming only the Zeeman interaction determines the
equilibrium populations, and where Z ¼ 2I þ 1 ¼ 2.
The rotating frame transformation is trivial:
rrotðeqÞ ¼ eiorf ;IIZtrLABðeqÞe�iorf ;IIZt ¼ gIB0�hIZ=2kBT;but it must be remembered from here on that all sub-
sequent operations must use Hamiltonians in the
rotating frame. We obtain the initial density operator
by applying an ideal 908 pulse to the I-spin using the
rotating frame r.f. Hamiltonian, Hrf ;I ¼ o1;IIY (see
Eq. [14]):
rrotð0Þ ¼ e�io1;rf tIYrrotðeqÞeio1;rf tIY
¼ gIB0�hIX=2kBT; ½29�
where we have put o1;rf t ¼ p2.
2. Next the tilting is
rTrotð0Þ ¼ gIB0�h=2kBTð Þeip2IYIXe�ip2IY
¼ gIB0�h=2kBTð ÞIZ¼ gIB0�h=2kBTð Þ J23Z þJ14Z
� �: ½30�
In order for the I-spin pulse that will be used dur-
ing the double irradiation period (e.g. HrfI / IX in
Eq. [1]) to be a spin-locking pulse, the preceding p/2pulse on the I-spin must be applied with a 908 phaseshift as we did in Eq. [29]. A good exercise is to con-
sider what would happen in Eq. [30] if the I spin p/2pulse were not 908 phase shifted from the I spin CP
pulse (i.e. allow the p/2 pulse to be applied as IX).
We set a0 ¼ gIB0�h/2kBT using notation similar to
Levitt et al. (13) to reduce clutter.
3. The remaining operations will be carried out
only in the zero quantum (ZQ) subspace (i.e., rT;23rot
(0) ¼ a0 J23Z ). Results in the double-quantum (DQ)
subspace can be written by analogy. The density opera-
tor for the ZQ subspace is now rotated into the diago-
nal frame by y23:
~r23ð0Þ ¼ eiy23J23
Y rT;23rot ð0Þe�iy23J23Y
¼ a0 cos y23J23Z � a0 sin y
23J23X : ½31�
4. The time evolution is based on the solution to
the Liouville-von Neumann equation for the density
operator at time t for a system subjected to a Hamil-
tonian H (24). The time evolution is rðtÞ ¼e�iHtrð0ÞeiHt, where we have assumed that the
262 ROVNYAK
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Hamiltonian is time-independent and self-commuting
over the time period, which is satisfied by Eq. [26].
We have been careful to express the density operator
and the Hamiltonian in identical frames, so
~r23ðtÞ ¼ e�io23effJ23Zta0 cosy23J23Z � siny23J23X� �
eio23effJ23Zt
¼ a0 cosy23J23Z � a0 siny
23
� ðcoso23eff tJ
23X þ sino23
eff tJ23Y
�: ½32�
5. Now rotate back out of the diagonal frame (i.e.
by �y23):
rT;23rot ðtÞ ¼ e�iy23J23Y
a0 cos y
23J23Z � a0 sin y23
� sino23eff tJ
23Y þ coso23
eff tJ23X
� �!eiy
23J23Y
¼ a0J23Z cos2 y23 þ sin2 y23 coso23
eff t� �
þ a0J23X sin y23 cos y23 � sin y23 cos y23 coso23
eff t� �
� a0J23Y sin y23 sino23
eff t� �
: ½33�
Thus, Eq. [33] describes the spin dynamics in the
tilted rotating frame for a static solid. The time-de-
pendent terms give rise to transient oscillations that
are orientation-dependent (20). In polycrystalline
(powder) samples the oscillations are dampened by
averaging over b (14), and also by interactions with
other spins and r.f. inhomogeneity (13). We will
return to the time-dependent terms shortly, but can
neglect them by examining the spin system after a
long time period (i.e. t � 0):
rT;23rot ðt >> 0Þ ¼ a0J23Z cos2 y23� �
þ a0J23X cos y23 sin y23� �
;
rT;14rot ðt >> 0Þ ¼ a0J14Z cos2 y14� �
þ a0J14X cos y14 sin y14� �
: ½34�
Equation [34] may be analyzed by considering
two limiting cases separately. One case allows cross-
polarization via flip–flop transitions in the spin pair
(i.e. jþ�i $ j�þi) and so this is a zero-quantum
matching condition since the I and S spins undergo
opposite changes in sign of the spin states. The sec-
ond case will be mentioned as an exercise where CP
is driven by flip–flip transitions (jþþi$j��i) and
so this limit is a double-quantum matching condition.
We will explicitly develop the zero-quantum case
which is obtained by noticing that the sum of the
nutation rates of the I and S spins can be chosen large
enough (ca. 50–100 kHz) such that
o1;I þ o1;S >> D: [35]
In this limit we have from Eq. [25] that y14 ? 0,
and Eq. [34] can be simplified to
rT;14rot ðt >> 0Þ ¼ a0J14Z ¼ 1
2a0 IZ þ SZð Þ [36]
Now rewrite the density operator in the ZQ sub-
space from Eq. [34] by substituting J23Z ¼ 12ðIZ � SzÞ,
J23X ¼ IXSX þ IYSY and the rotation identities from
Eq. [25], to give
rT;23rot ðt >> 0Þ ¼ 1
2a0 IZ � SZð Þ o2
D
D2 þ o2D
� �þ 1
2a0 IXSX þ IYSYð Þ DoD
D2 þ o2D
� � ; ½37�
with oD ¼ (o1,I � o1,S). Combining Eqs. [36] and
[37] gives the total density operator for long times as
rTrotðt>> 0Þ ¼ 1
2a0ðIZ þ SZÞ
þ 1
2a0ðIZ � SZÞ o2
D
D2 þo2D
� �þ 1
2a0 IXSX þ IYSYð Þ DoD
D2 þo2D
� � : ½38�
6. The special condition of Hartmann-Hahn
matching occurs when the nutation rates are equal;
then oD ¼ 0, and we create S spin polarization in the
tilted frame:
rTrot t >> 0;oD ¼ 0ð Þ ¼ 1
2a0 IZ þ SZð Þ: [39]
When D is small but nonzero, the third term of
Eq. [38] will lead to so-called dipolar order, which
will not be considered further. Notice that the þSZterm from the double-quantum subspace is independ-
ent of the I and S spin r.f. powers so long as the lim-
iting condition of Eq. [35] is maintained, but the
magnitude of the subtractive �SZ term due to time
evolution in the zero-quantum subspace is a sensitive
function of oD.
Recall that the initial density operator consisted of
equilibrium I spin polarization only in the lab frame;
the 908 pulse on the I spin was assumed to be perfect
so that the entire I spin polarization became trans-
ANALYTIC THEORY FOR CROSS POLARIZATION 263
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
verse in the lab frame, and then longitudinal in the
tilted frame (e.g. Eq. [30]). Then comparing Eqs.
[30] and [39], we see that half of the polarization
from the I spin is transferred to the S spin. Generally,
if O is some operator for an observable of interest,
we may calculate the expectation value of that
observable as hOi ¼ TraceðOrÞ. The lab frame
observable could be IX for the real part of the signal
or Iþ ¼ IX þ iIY for the complex signal. We have
two options to find the NMR signal. First, one could
take the observable into the rotating-tilted frame, and
then take the trace with Eq. [39]. Equivalently one
can transform the density operator back into the lab
frame, and take the trace of the result with IX or Iþ.The key is to be certain that O
_
, H, and r are
expressed in consistent frames. Also, we initially
neglected the identity operator 1 in Eq. [27] since it
was invariant to all transformations; furthermore
since all angular momentum operators are traceless,
then clearly hOtracelessi ¼ TrðO1Þ ¼ 0, and the iden-
tity operator in Eq. [27] makes no contribution to
the signal.
It is useful to rewrite Eq. [38] in a manner that
makes it easier to appreciate the relative influences
of D and oD on the density operator (and ignoring
the term for dipolar order):
rTrotðt >> 0Þ ¼ a0 1� D2
2 D2 þ o2D
� � !
IZ
þ a02
D2
D2 þ o2D
� �SZ: ð40Þ
It can be observed that this matching condition
has a Lorentzian line shape with respect to the mis-
match oD.
As noted above, to determine the observed time
domain signal in the lab frame we can take Sþ into
the rotating tilted frame and then find the expectation
value with respect to Eq. [40]. But for purposes of
illustration an efficient method is to begin with SX in
the lab frame, skip the rotating frame transformation,
and then tilt along Z as usual; by skipping the rotat-
ing frame we are just finding the value of the first
point of the signal, which is always directly propor-
tional to the peak area. Figure 5 is the graphical rep-
resentation of finding hSXi in this fashion and using
Eq. [40]. Figure 5 illustrates the Hartmann-Hahn
matching conditions for the two spin system for sev-
eral dipole couplings. It is seen that the tolerance to
r.f. mis-matching improves for larger dipole cou-
plings. The dipole coupling was not powder averaged
for the calculations, however this only slightly per-
turbs the matching spectra.
The time-dependence of the polarization transfer
is written by retaining the time-dependent polariza-
tion term, yielding
rT;23rot ðtÞ ¼ J23Z cos2 y23 þ sin2 y23 coso23eff t
� �;
rT;23rot ðtÞ ¼ J23Zo2D
D2 þ o2D
þ D2
D2 þ o2D
coso23eff t
� �;
rT;14rot ðtÞ þ rT;23rot ðtÞ ¼ 1
2IZ þ SZ� �þ 1
2
1
D2 þ o2D
� o2D þ D2 coso23
eff t� �
IZ � SZ� �
: ½41�After a little algebra, this leads to an expression for
the build-up of polarization:
rT;14rot ðtÞ þ rT;23rot ðtÞ ¼ 1
22� D2
D2 þo2D
1þ coso23eff t
� �� �
� IZ þ1
2
D2
D2 þo2D
1� coso23eff t
� �SZ;
STZ� � ¼ 1
2
D2
D2 þo2D
1� coso23eff t
� �: ½42�
If we satisfy the zero-quantum match, the frequency
of oscillation is o23eff ¼ D ¼ d/2 and can be a valuable
reporter on the dipole coupling (20), while similar
oscillations are important under magic-angle spin-
ning (31),We treated here the zero-quantum matching con-
dition in which the dipole coupling can drive tran-
Figure 5 Matching profile (a.k.a. ‘‘spectra’’) correspond-
ing to the expectation value of SX in the lab frame
(details in text) is computationally evaluated using Eq.
[40] to show the dependence of the CP efficiency on the
mismatch and the dipole coupling. Although not shown
here, each matching spectrum above could be powder
averaged for all possible orientations of the dipole tensor,
but this does not affect the matching spectra significantly.
264 ROVNYAK
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
sitions between the flip–flop states. A helpful pic-
ture of this will be shown in the next section as
well. We obtained this zero-quantum case by
assuming that the applied r.f. fields were suffi-
ciently large and of the same sign to justify the
assumption from Eq. [35] that o1,I þ o1,S � D. Topractice the above steps, consider how to obtain the
second case of double-quantum matching, begin-
ning by taking the r.f. field terms to have opposite
sign and developing an assumption analogous to
that in Eq. [35].
To wrap-up the static case, we will estimate the
enhancement of the S-spin polarization. Recall that
the initial density operator had the constant factor
gIB0�/2kBT. Thus, in the tilted frame the result of the
CP experiment for the S-spin polarization is half of
this. On the other hand, the equilibrium Zeeman S-
spin polarization would be gSB0�/2kBT, and so the
enhancement is:
Z ¼ gI2gS
: [43]
It is important to emphasize that the true enhance-
ment can vary quite a bit and often exceeds gI/2gS,particularly when multiple I spins are present.
Proton–Proton Interactions in Static CP
To wrap-up the static case, we have the opportunity
to use what we have covered to preview an intuitive
description for the role of I–I dipolar couplings given
by Marks and Vega (17). We will see in the static
case that one of the principal roles of a multi-I-spin
bath coupled to the S spin is to broaden the matching
conditions. This picture forms a very useful frame-
work that can be carried forward to treat the case of
sample spinning as well, but is outside the scope of
this article (17).Allow the S spin to be coupled to a set of N I
spins, which are also coupled among each other. All
S and I spins are spin ¼ 12particles. We first write the
Hamiltonian for all I–I dipolar couplings. The cou-
pling among two I spins must include both A and B
terms, as discussed in The Dipolar Coupling Hamil-
tonian section
HDII ¼
1
21� 3 cos2 b� �
Ið1ÞZ I
ð2ÞZ � I
*ð1Þ � I*ð2Þ ; [44]
which can be generalized to N I spins as
HDII ¼
1
2
XNi<j
1� 3 cos2 bij� �
IðiÞZ I
ðjÞZ � I
*ðiÞ � I*ðjÞ ;
[45]
where the meaning of the summation is over all pairs
(i,j) out of N spins such that i , j (to prevent double
counting). The angle b is defined as in Eq. [3] but
now refers to I–I internuclear vectors relative to the
static field.
Now Eq. [45] is invariant to both the rotating
frame and tilting operation since it commutes with IZand IY, so that the total Hamiltonian for the S(I)N sys-
tem in the tilted, double rotating frame is the sum of
Eqs. [45] and [17]:
HTrot ¼ o1IIZ þ o1SSZ þ
XNi�1
2DiIðiÞX SX
þ 1
21� 3 cos2 b� �XN
i<j
IðiÞZ I
ðjÞZ � I
*ðiÞ � I*ðjÞ� �
¼ o1IIZ þ o1SSZ þXNi�1
1
2D
� IðiÞþ Sþ þ I
ðiÞþ S� þ IðiÞ� Sþ þ IðiÞ� S�
þ 1
21� 3 cos2 b� �XN
i<j
IðiÞZ I
ðjÞZ � I
*ðiÞ � I*ðjÞ ;
½46�where the only other difference with Eq. [17] is that
the heteronuclear dipole coupling has been summed
over all of the N I spins. As usual, the heteronuclear
dipole term can couple zero-quantum or double-quan-
tum states. We will not perform further analytic manip-
ulations of Eq. [46] but will instead try to understand
its role in influencing energy levels of this system.
We will use a level diagram in the tilted rotated
frame (Fig. 6) to help interpret Eq. [46] (17). First,consider the isolated I–S spin pair in the tilted rotat-
ing frame, which follows from Eq. [17]. The energy
levels of the I–S spin pair are shown in Fig. 6(a). The
energies in this case are determined by the r.f. nuta-
tion powers, while the dipolar term is only shown
connecting flip–flop states, so that we are considering
the more common zero-quantum match (as usual, the
double-quantum case behaves in an analogous man-
ner). The key observation is that the exact zero quan-
tum match that we obtained (o1,I ¼ o1,S) renders the
flip–flop states degenerate in this frame so that they
can be coupled by the dipolar term. By rendering the
flip–flop transitions energy conserving, the Hart-
mann-Hahn match allows for the Boltzmann polar-
ization of the I spin to be redistributed among the I
and S spins through the dipole coupling that connects
these degenerate states.
ANALYTIC THEORY FOR CROSS POLARIZATION 265
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
We can represent N I spins in a coupled basis of
states represented by the quantum number M, such
that M spans –N/2, �(N þ 1)/2,. . ., (N þ 1)/2, N/2.
Recall that the homonuclear dipole coupling will per-
turb the spin energy levels of the I spin bath. If the I
spins were uncoupled, their energy levels would be
essentially degenerate as represented in the left hand
part of Fig. 6(b). Upon ‘‘turning on’’ the homonuclear
I spin coupling, a given M state will expand into a
manifold of energy levels. The width of this manifold
is approximately the average of the I–I dipole cou-
pling jdIIj where dII ¼ gIgI�hr3 ð1� 3cos2yÞ. Importantly,
the number of states in each manifold will vary as a
function of M, and this is not shown in the figure.
As depicted in the right part of Fig. 6(b), in the
event of a perfect match the manifolds for the flip–
flop transitions will overlap perfectly so that the I–I
homonuclear coupling does not affect the efficiency of
the perfect matching condition. However for a mis-
match that is small compared with the average magni-
tude of the I–I coupling, significant overlap of the flip–
flop states still exists so that significant flip–flop transi-
tions remain energy conserving, and CP will occur that
is much more effective than for the isolated I–S spin
pair.
In summary, a coupled I-spin bath will result in
greater tolerance to Hartmann-Hahn mismatch, signifi-
cantly broadening the practical matching conditions.
MAGIC-ANGLE SPINNING
The Time-Dependent Dipole Coupling
The Hamiltonian for the isolated IS spin pair will be
modified to account for rapidly spinning the sample
about a fixed axis that is tipped away from the direc-
tion of the static, external field by the magic angle,
which is defined by
1� 3 cos2 ym� � ¼ 0;
ym ¼ cos�1ffiffiffiffiffiffiffiffi1=3
p¼ 54:7356 . . . : ½47�
This is the angle for which any second-rank tensor
interaction will average to zero over one rotor period,
including the chemical-shift anisotropy (CSA), the
secular heteronuclear or homonuclear dipole cou-
pling, and the lowest order quadrupole coupling.
Only in the fast-spinning limit, when the rotation fre-
quency exceeds the magnitude of these interactions,
can such terms be removed from the Hamiltonian.
Figure 6 Energy level diagrams in the case of a static sample in the tilted rotating frame and
assuming the Zeeman basis functions shown in the kets. The matching conditions shown corre-
spond to the zero-quantum case. Energies are given on the figure and neglect the I–I coupling.
In (a) a perfect Hartmann-Hahn match for the isolated I–S pair is seen to render degenerate flip–
flop states, while the role of I–I couplings in (b) is seen to allow for degenerate flip–flop states
even when a perfect r.f. match is not present. The number of I spin states in a given manifold is
not accurately depicted and furthermore will depend on M.
266 ROVNYAK
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Otherwise we must add the additional time depend-
ence into the Hamiltonian and see how the spin dy-
namics are affected.
Starting with the Hamiltonian in the tilted, dou-
ble-rotating frame (i.e. Eq. [24]), we need to write a
time-dependent dipolar coupling,
H23 ¼ o1I � o1Sð ÞJ23Z þ DðtÞJ23X ¼ oDJ23Z þ DðtÞJ23X ;
H14 ¼ o1I þ o1Sð ÞJ14Z þ DðtÞJ14X ¼ o�J14Z þ DðtÞJ14X ;
[48]
where the symbols oD and oS represent the differ-
ence and sum of the r.f. nutation rates, respectively.
The static dipolar coupling used previously (e.g. Eqs.
[24] and [26]) was
DðstaticÞ ¼ 1
2d; d ¼ gIgS�h
r3ð1� 3 cos2 yÞ: [49]
To obtain the time dependence of the dipolar cou-
pling, D(t), for a spin system undergoing magic-angle
rotation, it is important to remember that in The
(Doubly) Rotating Frame section we averaged the
rotating frame Hamiltonian over a few Larmor peri-
ods. Introducing D(t) now assumes that the time de-
pendence of D(t) is very slow compared to the Lar-
mor precession so that D(t) appears static during a
few Larmor periods of each spin I and S. Larmor pre-
cession is at least 103 greater than the dipole fre-
quency so this is an excellent approximation.
A more convenient notation is helpful for express-
ing sample rotation. This section will derive D(t)using spherical-tensor notation and will verify the
equivalence to the A and B terms of the ‘‘dipolar
alphabet.’’ Since it is not practical to review irreduci-
ble spherical tensors here, some familiarity with this
terminology is assumed (see texts such as by Duer
(11), Mehring (23), or Spiess (32); also reviewed by
Eden previously in this journal (33)). One may jump
to the result for D(t) in Eqs. [57] and [59] and pro-
ceed to The MAS Rotating Frames section.
First, the second rank portion of an NMR interac-
tion (subscript j ¼ chemical shift anisotropy, dipole–
dipole, quadrupole, spin-rotation, etc.) may be writ-
ten in its principal axes system (PAS) as
HPASðjÞ ¼X2m¼�2
ð�1ÞmrPAS2m ðjÞT2m: [50]
where the T2m terms represent the spin operator
degrees of freedom, while the spatial dependence of
the interaction is given in the PAS frame by the
rPAS2m (j) terms. The spatial terms will depend on the
magnitude and the symmetry properties of the partic-
ular interaction. To express an interaction in a differ-
ent coordinate systems, we manipulate the space
terms only, writing
HLABðjÞ ¼X2m¼�2
ð�1ÞmRLAB2m ðjÞT2;�m;
where RLAB2m ðjÞ ¼
X2m0¼�2
Dð2Þm0mða; b; gÞr2m0 ðjÞ: ½51�
The angles a, b, g are termed Euler angles and are
a convention for applying arbitrary frame rotations to
tensors, and the Dð2Þm0m (a,b,g) represent Wigner rota-
tion elements that perform the desired rotation (34,35). The dipolar coupling is completely described in
the principal axes system by only one component
(i.e. r00ðdipoleÞ ¼ 0; r20ðdipoleÞ ¼ffiffi32
qd; r2mðdipole;
m 6¼ 0Þ ¼ 0Þ). The other components vanish in the
PAS frame since the principal axis of the dipole ten-
sor is aligned with the internuclear bond vector,
requiring that the tensor be axially symmetric about
this vector. We will only expand the secular part of
Eqs. [50] and [51], corresponding to m ¼ 0. For the
dipole coupling d ¼ �2gIgS�hr3 , giving
RD;LAB20 TD
20 ¼X2m0¼�2
Dð2Þm00ða; b; gÞr2m0
1ffiffiffi6
p ð3IZSZ � I* � S*Þ
¼ dð2Þ00 bð Þr20
1ffiffiffi6
p ð3IZSZ � I* � S*Þ
¼ 1
21� 3 cos2 b� � 2gIgS�h
r3
� �1
2
� �3IZSZ � I
* � S*
¼ 1
21� 3 cos2 b� � gIgS�h
r3
� �3IZSZ � I
* � S*
¼ Aþ B; ½52�
where T20 ¼ 1=ffiffiffi6
p ðIZSZ � I � SÞ, and the reduced
Wigner element is dð2Þ00 ðbÞ ¼ 1
2ð3cos2b� 1Þ. We see
that Eq. [52] is in agreement with the A and B‘‘dipolar alphabet’’ expansion in Eq. [6]. By includ-
ing m ¼ 61, 62 the C� F terms are also generated,
which is suggested as an exercise. The complete
form of the Wigner elements is DðjÞm0m ¼ e�im0ad
ðjÞm0m
(b)e�img, however notice that Eq. [52] uses only the
case m0 ¼ 0, and m ¼ 0. Now we wish to find
ANALYTIC THEORY FOR CROSS POLARIZATION 267
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HD;MAS ¼X2m¼�2
ð�1ÞmRMAS2m ðtÞT2;�m;
RD;MAS2m ðtÞ ¼
X2n¼�2
RD;LAB2n Dð2Þ
nmðortþ f; yMA; 0Þ; ½53�
where yMA % 54.740, and D2nm(ort þ f, yMA,0) is
the Wigner rotation corresponding to the spinning
with frequency or, and f is the initial phase angle
of the rotation. We have already performed the
rotating frame and tilting transformation on the spin
coordinates (i.e. T20) when we treated the static
case. We only need to know what the space tensor
looks like in the MAS frame. We continue further
with just the secular term:
RD;MAS20 ðtÞ ¼
X2n¼�2
RD;LAB2n D
ð2Þn0 ðortþ f; yMA; 0Þ
¼X2n¼�2
RD;LAB2n d
ð2Þn0 ðyMAÞe�in or tþfð Þ
¼ RD;LAB22 d
ð2Þ20 ðyMAÞe�i2 or tþfð Þ
þ RD;LAB21 d
ð2Þ10 ðyMAÞe�i or tþfð Þ
þ RD;LAB2;�1 d
ð2Þ�1;0ðyMAÞei or tþfð Þ
þ RD;LAB2;�2 d
ð2Þ�2;0ðyMAÞei2 or tþfð Þ
¼ dð2Þ02 bð Þr20
dð2Þ20 ðyMAÞe�i2 or tþfð Þ
þ dð2Þ01 bð Þr20
dð2Þ10 ðyMAÞe�i or tþfð Þ
þ dð2Þ0;�1 bð Þr20
dð2Þ�1;0ðyMAÞei or tþfð Þ
þ dð2Þ0;�2 bð Þr20
dð2Þ�2;0ðyMAÞei2 or tþfð Þ;
½54�where we explicitly substituted the lab frame space
tensors in the last step. The dðjÞm0m() are the reduced
Wigner rotation elements, which can be found in many
texts (11, 32, 36). Several identities of the reduced
Wigner elements are useful for simplifying Eq. [54]:
d202ðÞ ¼ d20;�2ðÞ; d220ðÞ ¼ d2�2;0ðÞ;d201ðÞ ¼ �d20;�1ðÞ; d210ðÞ ¼ �d2�1;0ðÞ: ½55�
Then
RD;MAS20 ðtÞ¼r20d
ð2Þ02 bð Þdð2Þ20 ðyMAÞ e�i2 or tþfð Þ þei2 or tþfð Þ
þr20d
ð2Þ01 bð Þdð2Þ10 ðyMAÞ e�i or tþfð Þ þei or tþfð Þ
¼r20d
ð2Þ02 bð Þdð2Þ20 ðyMAÞ2cos 2ortþ2fð Þ
þr20dð2Þ01 bð Þdð2Þ10 ðyMAÞ2cos ortþfð Þ: ½56�
In deriving D(t) we must multiply by 1/2 since
that was included in Eq. [15] to simplify the appear-
ance of the subsequent expressions. Thus we have
D tð Þ ¼ g1 b; yMAð Þ cos ortþ fð Þþ g2 b; yMAð Þ cos 2ortþ 2fð Þ; ð57Þ
which employs a common notation:
g1 b; yMAð Þ ¼ffiffiffi6
p gIgS�hr3
d201 bð Þd210 yMAð Þ;
g2 b; yMAð Þ ¼ffiffiffi6
p gIgS�hr3
d202 bð Þd220 yMAð Þ: ½58�
Equation [57] is another common starting point in
SSNMR literature. The initial phase can be set arbi-
trarily to zero, to give
D tð Þ ¼ g1 b; yMAð Þ cos ortð Þ þ g2 b; yMAð Þ cos 2ortð Þ[59]
.
Interestingly, Eq. [59] shows that D(t) has compo-
nents that are modulated not only at the rotor fre-
quency but also at twice the rotor frequency as well.
We will see the effect of both modulation frequencies
when obtaining the matching conditions.
The MAS Rotating Frames
In this section, the Hamiltonian is written in a frame
rotating according to the MAS frequency, diagonal-
ized, and then used to evolve the density operator.
The inclusion of MAS requires an extra step in sim-
plifying the Hamiltonian and thus an extra step in
transforming the density operator before it can be
evolved. We will work in the double quantum sub-
space and begin with Eq. [48]:
H14 ¼ o�J14Z þ DðtÞJ14X : [60]
Based on Eq. [59] there are several choices for a
second rotating frame, namely 6or, and 62or. We
choose þor for the rotating reference frequency and
add and subtract orJ14z in Eq. [60]:
H14 ¼ H0 þ H1 ¼ orJ14Z þ o� �orð ÞJ14Z þDðtÞJ14X
� �;
[61]
which is written in the same form as Eq. [7] to obtain
a Hamiltonian that describes spin dynamics in the
rotating frame
268 ROVNYAK
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H14;or ¼ eiH0tH1e�iH0t
¼ eiorJ14Z t o� � orð ÞJ14Z þDðtÞJ14X� �
e�iorJ14Z t
¼ o� � orð ÞJ14Z þDðtÞ� cosortJ
14X � sinortJ
14Y
� �: ½62�
As before, average the Hamiltonian over one pe-
riod of the motion, tr ¼ 2p/or:
�H14;or ¼ 1
tr
Ztr0
½ðo� � orÞJ14Z
þ g1 cos ortþ fð Þ cosortJ14X � sinortJ
14Y Þ�dt: ½63��
One may verify that the g2 term does not survive
the integration over periodic limits by inspecting the
following identities:
cos 2xþ 2fð Þcos xð Þ¼ 1
2cos 3xþ 2fð Þþ 1
2cos xþ 2fð Þ;
cos 2xþ 2fð Þ sin xð Þ¼ 1
2sin 3xþ 2fð Þ þ 1
2sin xþ 2fð Þ: ½64�
If we had chosen one of the 62or frames, the g2term would survive and the g1 term would be aver-
aged to zero. This averaging step incurs a loss of
generality—we need to repeat this recipe for each of
the MAS-determined frames. These are easily
obtained by analogy to the case shown explicitly here
for þor. We complete the integral of Eq. [63]:
�H14;or ¼ o��orð ÞJ14Z
þg1tr
Ztr0
cos ortþfð Þ cosortJ14X � sinortJ
14Y
� �� �¼ o��orð ÞJ14Z
þg1tr
Ztr0
�1
2cos 2ortþfð Þþ 1
2cosf
�J14X dt
�g1tr
Ztr0
1
2sin 2ortþfð Þ
�� 1
2sinf
�J14Y dt
¼ o��orð ÞJ14Z þ 1
2g1 cosfJ14X þ1
2g1 sinfJ14Y :
½65�
The ZQ subpsace is handled identically, so
�H14;or ¼ o� � orð ÞJ14Z þ 1
2g1 cosfJ14X þ sinfJ14Y� �
;
�H23;or ¼ oD � orð ÞJ23Z þ 1
2g1 cosfJ23X þ sinfJ23Y� �
:
[66]
Again, Eq. [66] is just the case for þor and is only
one of four total cases. For diagonalization, we will
again work in just the DQ subspace and generalize
the results to the ZQ subspace. To simplify the nota-
tion the following substitutions will be made:
�H14;or ) �H14;
�H23;or ) �H23: ½67�
An immediate observation from Eq. [66] is that
two steps are needed to perform the diagonalization.
The first will be the removal of the J14Y term by rotat-
ing the Hamiltonian into the XZ plane. The second
step then tilts the Hamiltonian along the Z axis to
eliminate the J14X term. By inspection of Fig. 7, the
result of the first rotation about the z axis is
�H14T ¼ eifJ
14Z �H14e�ifJ14Z ¼ o� � orð ÞJ14Z þ g1
2J14X ;
[68]
which is identical to the case that would be obtained
if f¼ 0 in D(t). By reading the magnitude of the
Hamiltonian vector from Fig. 7, the diagonalized
Hamiltonian for the 1–4 subspace is
�H14TT ¼ eicJ
14Y �H14
T e�icJ14Y
¼ o� � orð Þ2 þ g214
� �1=2J14Z ; ð69Þ
where the c angle is indicated in Fig. 8. The TT sub-
script indicates that both f and c tilting operations
have been performed. The same procedure in the 2–3
subspace yields
�H23TT ¼ oD � orð Þ2 þ g21
�4
h i1=2J23Z : [70]
The c angles for either subspace can be read off
of Fig. 8:
Figure 7 Diagram of the first step in diagonalizing Eq.
[66], shown here for the 1–4 subspace.
ANALYTIC THEORY FOR CROSS POLARIZATION 269
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
c14 ¼ tan�1 g1=2
o� � or
� �; c23 ¼ tan�1 g1=2
oD � or
� �:
[71]
Introducing the symbols oTTD ¼ [(oD � or)
2 þ g21/4]1/2 and oTT
� ¼ [(oS � or)2 þ g21/4]
1/2, we may
rewrite Eqs. [69] and [70] as
�H23TT ¼ oTT
D J23Z ;
�H14TT ¼ oTT
� J14Z : ½72�
CP-MAS Spin Dynamics
We have performed a number of operations to obtain
a Hamiltonian in Eq. [72] which accounts for the
MAS rate, or, and is diagonal in a chosen frame.
This is an ideal frame in which to propagate a density
operator. Therefore we have to write the initial den-
sity operator in the same frame as the Hamiltonian of
Eq. [72]. As before we start with transverse polariza-
tion on the I spins. In the tilted, rotating frame this is
Eq. [30] again:
rTrotð0Þ ¼ IZ ¼ J23Z þ J14Z : [73]
We can see that Eq. [73] is invariant to the MAS
rotating frame transformation, that is eiorIZtIZe�iorIZt
¼ IZ. We next need to repeat the rotations by f and
c. The first f rotation has no effect, but later when
returning out of this double tilted frame the f rota-
tion still needs to be ‘‘undone.’’ Thus we have
r14TT 0ð Þ ¼ eic14J14Y J14Z e�ic14J14Y ¼ cosc14J14Z � sinc14J14X ;
r23TT 0ð Þ ¼ cosc23J23Z � sinc23J23X ;
[74]
where the TT mnemonic indicates that the density
operator has been tilted by both f and c. The densityoperator is now evolved. In the 2–3 subspace,
r23TT tð Þ ¼ e�ioTTD J23Z t cosc23J23Z � sinc23J23X
� �eio
TTD J23Z t
¼ cosc23J23Z � sinc23 cos oTTD t
� �J23X
�þ sin oTT
D t� �
J23Y : ½75�
Next Eq. [75] must be untilted by f and c. We
have to undo both f and c rotations since Eq. [75] is
not invariant under rotation about J23z any more:
r23ðtÞ ¼ e�ifJ23Z e�icJ23
Y r23TTðtÞeicJ23Z eifJ
23Y
¼ e�ifJ23Z
cosc23 cosc23J23Z þ sinc23J23X� �
� sinc23 cosoTTD t cosc23J23X � sinc23J23Z� �þ sinoTT
D tJ23Y� �
" #eifJ
23Z
¼
cos2 c23J23Z þ cosc23 sinc23 cosfJ23X þ sinfJ23Y� �
þ sin2 c23 cosoTTD tJ23Z
� sinc23 cosc23 cosoTTD t cosfJ23X þ sinfJ23Y� �
� sinc23 sinoTTD t cosfJ23Y � sinfJ23X� �
266664
377775: ½76�
Figure 8 The second of two tilts required to diagonalize
the Hamiltonian H14
in Eq. [66].
As before, we want to first obtain the long-time
behavior to demonstrate the basic polarization transfer,
so we discard the time-dependent parts and add the
subspaces back together. The dynamics of the double-
quantum subspace can be obtained by analogy to Eqs.
[74]–[76]. The net behavior in both subspaces is
270 ROVNYAK
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
rðt >> 0Þ ¼ cos2 c23J23Z
þ cosc23 sinc23 cosfJ23X þ sinfJ23Y� �þ cos2 c14J14Z
þ cosc14 sinc14 cosfJ14X þ sinfJ14Y� � ½77�
We can obtain the sines and cosines according to
Figs. 7 and 8. We also substitute the J operators with
the I and S spin operators to better see the polariza-
tion transfer. Expanding only terms containing IZand SZ from Eq. [77], we have:
rðt >> 0Þ ¼ cos2 c23J23Z þ cos2 c14J14Z
¼ 1
2cos2 c23 IZ � SZð Þ þ 1
2cos2 c14 IZ þ SZð Þ
¼ 1
2cos2 c14 � cos2 c23� �
SZ
þ 1
2cos2 c14 þ cos2 c23� �
IZ
¼ 1
2
o� � orð Þ2
o� � orð Þ2þ g21
4
� oD � orð Þ2
oD � orð Þ2þ g21
4
!SZ
þ 1
2
o� � orð Þ2
o� � orð Þ2þ g21
4
þ oD � orð Þ2
oD � orð Þ2þ g21
4
!IZ
[78]
If we were to include JX and JY we would find
that it is possible to create dipolar order,
rðt >> 0Þ / ðIXSX þ IYSYÞ, for r.f. mis-matching
conditions.
Strong r.f. fields. If the difference in applied r.f.
power levels matches the spinning frequency (typ-
ical conditions meeting this criteria are oD ¼oI � oS ¼ 50 kHz � 40 kHz ¼ or ¼ 10 kHz),
and if the r.f. fields are strong (oS � g1), then
Eq. [78] simplifies to:
rðt >> 0Þ ffi 1
2SZ þ 1
2IZ: [79]
Thus the polarization in the tilted frame is
hSZi ¼ TrðSZrðt >> 0ÞÞ ¼ 12. Half of the initial polar-
ization is transferred from the I-spin to the S-spin for
Hartmann-Hahn CP. Again, as in the static case, the
true enhancement may vary significantly depending
on homonuclear couplings and the experimental
design. By applying the strong r.f. field approxima-
tion (i.e. oS � or . g1/2, so c14 ? 0 in Fig. 8) to
the density operator prior to the untilting, and retain-
ing only longitudinal terms, one gets
rTT 0;c14 ! 0� � ¼ J14Z þ cosc23J23Z
rTT 0ð Þ ¼ J14Z þ oD � orð Þ2
oD � orð Þ2 þ g21
4
J23Z
rTT 0ð Þ ¼ 1
2IZ þ SZð Þ þ oD � orð Þ2
oD � orð Þ2 þ g21
4
1
2IZ � SZð Þ:
rTT 0ð Þ ¼ 1� 1
2
g21
4
oD � orð Þ2þ g21
4
!IZ
þ 1
2
g21
4
oD � orð Þ2þ g21
4
!SZ ½80�
This follows Eq. [28] in Levitt et al., for example
(13). This case exploits the difference or zero-quan-
tum subspace. By inspection of Eq. [76] and making
the analogy to the static case, it is found that the time
dependence for the polarization build-up is
STZ� �ðtÞ ¼ 1
2
g21
4
oD � orð Þ2þ g21
4
!1� cosoTT
D t� �
SZ
[81]
Weak r.f. fields and/or fast MAS. What happens
when the sum of the applied r.f. powers is equal to
the MAS rate? (e.g., suppose the MAS rate is 10 kHz
and the applied r.f. frequencies are oS ¼ o1,I þ o1,S
¼ 5 kHz þ 5 kHz). In this case, Eq. [78] reduces to
rðt >> 0Þ ¼ 1
2� o2
r
o2r þ g2
1
4
!SZ þ 1
2
o2r
o2r þ g2
1
4
!IZ
[82]
And if we allow or . g1,g2, we obtain
hSZi ¼ � 12. This situation therefore takes c23 ? 0
and operates in the sum or double-quantum subspace.
As an overview, this procedure can be repeated
for the remaining �or, 62or frames with the
expected additional matching conditions. The so-
called ‘‘zero quantum’’ (Eqs. [79] and [80]) and
‘‘double quantum’’ (Eq. [82]) matching conditions
are easily distinguished by the relative signs of the
enhancements.
CP-MAS Matching Profiles
Equation [78] can be numerically evaluated. The
results obtained in this way must be combined with
evaluations of the density matrices in each of the
other subspaces. Yet these can be written down from
ANALYTIC THEORY FOR CROSS POLARIZATION 271
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
inspection of Eq. [78], although some care should be
taken to keep g1’s and g2’s straight and to note the
needed sign changes. A sample calculation for high
power CP-MAS is shown in Fig. 9, which displays a
typical zero quantum matching spectrum.
The transfer predicted by Fig. 9 occurs only for
the zero-quantum conditions, that is, when the differ-
ence in applied fields is a multiple of the spinning
rate. If, on the other hand, the I-spin r.f. power is
very low, then the double-quantum matching condi-
tions may be introduced since it becomes possible for
the sum of the applied powers to equal the MAS rate,
and this is illustrated in Fig. 10, where the DQ trans-
fer is negative. Several CP-MAS matching spectra
are shown in Fig. 10 to illustrate how both ZQ and
DQ transfer conditions depend upon the I-spin r.f.
power level. One condition exists with o1I ¼ 5 kHz,
for which one ZQ (5 � 15 ¼ 10) and one DQ (5 þ15 ¼ 20) condition cancel exactly, which only occurs
when g1 ¼ g2.In practice, g1 and g2 may differ and one exercise
would be to predict how the matching profiles will be
altered when g1 and g2 are not equal. Additionally,
Figs. 9 and10 correspond to a single arbitrary crystal
orientation and could be averaged over all possible
powder orientations of the internuclear dipole vector
as well. Thus Figs. 9 and 10 are intentionally ideal-
ized to illustrate that coherence (which is polarization
in the tilted frame) transfer is obtained at the
expected matching conditions.
RESONANCE OFFSETS
A General Tilted Frame
The introduction of resonance offsets is a good
extension of these results. Frequency offsets are
among the most likely experimental parameters to be
encountered in setting up CP experiments after
r.f. power levels and MAS rates are determined.
This treatment can be very satisfying because discov-
ering the Hartmann-Hahn matching conditions for
CP-MAS with resonance offsets can lead to condi-
tions for spectrally selective CP-MAS polarization
transfer (2).An important change in the treatment is that the
tilting operation, shown before in Fig. 3, is general-
ized to deal with resonance offsets. Another impor-
tant difference is that we will use a new interaction
frame to quickly derive Hartmann-Hahn matching
conditions, skipping the procedure of diagonalizing
the two subspaces and propagating a density operator
in each subspace.
For continuity we will examine the dynamics for
CP under MAS first. We restore the resonance offsets
to the radio-frequency portion of Eq. [14] and insert
the time-dependent dipolar coupling. This gives
Figure 9 Plot of the zero-quantum matching conditions,
generated by evaluating Eq. [78] with a computer pro-
gram: o1I ¼ 50 kHz, or ¼ 10 kHz, and o1S ¼ [0,100]
kHz. Equation [78] must be evaluated in each frame
(6or, and 62or) and then the profiles of all four frames
added together to give the plot. Also, for convenience we
take g1 ¼ g2 ¼ 2000 Hz; although in reality they are
likely not equal. The double quantum matching conditions
can never be obtained here since the sum of the r.f. fields
never matches the MAS rate.
Figure 10 CPMAS matching spectra calculated for con-
ditions in which both ZQ and DQ matching can occur. As
mentioned in the text, g1 and g2 are taken to be equal and
no powder averaging of g1 and g2 has been performed.
272 ROVNYAK
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
�H� ¼ HRF þ 2DðtÞIZSZ¼ OIIZ þ OSSZ þ o1;IIX þ o1;SSX þ 2DðtÞIZSZ;
[83]
where the resonance offset terms are the difference
between the Larmor frequency and the frequency of
the applied r.f. field (orf,I or orf,S):
OI ¼ �gIB0 � orf ;I;
OS ¼ �gSB0 � orf ;S: ½84�
The purpose of the tilt is to render the r.f. and off-
set terms diagonal, to make the new Z axis parallel to
HRF. This type of rotation is now familiar, however
we illustrate this in Fig. 11 because it is important to
understand that this tilt only diagonalizes the r.f.
terms, while the effect of this tilt on the dipole term
will initially appear more complicated. This will lead
to an intuitive shortcut for deriving matching condi-
tions. The tilt is expressed as
HT ¼ eiyIIY eiySSY OIIZ þ OSSZ þ o1;IIX þ o1;SSX�
þ 2DðtÞIZSZÞe�iySSY e�iyIIY ;
where yI ¼ tan�1 o1;I
OI
� �; yS ¼ tan�1 o1;S
OS
� �; ½85�
giving
HT ¼ O2I þ o2
1;I
1=2IZ þ O2
S þ o21;S
1=2SZ
þ eiyIIY eiySSY 2DðtÞIZSZð Þe�iySSY e�iyIIY
¼ oI;eff IZ þ oS;effSZ þ 2DðtÞ cos yIIZ � sin yIIXð Þ� cos ySSZ � sin ySSXð Þ
¼ oI;effIZ þ oS;effSZ þ 2DðtÞ
cos yI cos ySIZSZþ sin yI sin ySIXSX� sin yI cos ySIXSZ� cos yI sin ySIZSX
26664
37775:
[86]
It is common to express this using raising and
lowering operators:
HT¼oI;effIZþoS;effSZ
þDðtÞ
cosyI cosySIZSZþsinyI sinyS12 IþSþþIþS�þI�SþþI�S�ð Þ�sinyI cosySI6SZ�cosyI sinySIZS6
26664
37775;
[87]
where we used the identities:
2IXSX¼1
2IþSþþIþS�þI�SþþI�S�ð Þ;
2IXSZ¼IþSZþI�SZ¼I6SZ: ½88�
A convenient short-hand for D(t) is
DðtÞ ¼X2n¼�2
Rlab2n d
ð2Þn0 yMAð Þe�in or tþfð Þ
¼X2n¼�2
oD;ne�inor t [89]
with oD,n defined as
oD;61 ¼ Rlab2;61d
ð2Þ61;0ðyMAÞeif;
oD;62 ¼ Rlab2;62d
ð2Þ62;0ðyMAÞe2if: ½90�
For a review, compare Eqs. [89] and [90] to Eq.
[54] to verify that they are consistent.
A New Interaction Frame
Previously, we diagonalized the two subspaces of the
Hamiltonian in order to propagate a density operator
and obtain analytic expressions for the Hartmann-
Hahn matching conditions. This was a convenient
and efficient way to take advantage of the simplicity
of the tilted Hamiltonian in the absence of resonance
offsets (i.e. see Eqs. [17] and [24]). This method no
longer serves us due to the complexity of the tilted
Hamiltonian when resonance offsets are present, i.e.
Eq. [87].
We transform the Hamiltonian of Eq. [87] into a
new interaction frame based on the two effective r.f.
nutation frequencies that were defined in Eq. [86]:
Figure 11 A general tilting operation for the rotating
frame Hamiltonian.
ANALYTIC THEORY FOR CROSS POLARIZATION 273
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
HrfT ¼ ei oI;eff IZþoS;effSZð ÞtDðtÞ
�
cos yI cos ySIZSZ
þ sin yI sin yS1
2IþSþ þ IþS� þ I�Sþ þ I�S�ð Þ
� sin yI cos ySI6SZ� cos yI sin ySIZS6
266664
377775
� e�i oI;eff IZþoS;effSZð Þt: ½91�
The following identity will greatly simplify Eq. [91]:
eiaIZIþe�iaIZ ¼ eiaIZ IX þ iIYð Þe�iaIZ
¼ cos aIX � sin aIY þ i cos aIY þ sin aIXð Þ¼ eiaIþ: ½92�
We see that it is desirable to obtain a Hamiltonian
in which the r.f. terms are diagonal and the coupling
terms can be expressed as raising and lowering oper-
ators. Then Eq. [91] evaluates to
HrfT ¼ ei oI;effIZþoS;effSZð Þt X2
n¼�2
oD;ne�nor t
:::
:::
:::
264
375e�i oI;eff IZþoS;effSZð Þt
¼ DðtÞ cos yI cos ySIZSZ þX2n¼�2
oD;n
�sin yI sin yS
1
2
ei �eff�norð ÞtIþSþ þ ei Deff�norð ÞtIþS�þei �Deff�norð ÞtI�Sþ þ ei ��eff�norð ÞtI�S�
!
� sin yI cos ySI6SZei 6oI;eff�norð Þt � cos yI sin ySIZS6e
i 6oS;eff�norð Þt
2664
3775; ½93�
with Seff ¼ oI,eff þ oS,eff and Deff ¼ oI,eff � oS,eff.
Suppose we ask the question: what is the average of
Eq. [93] over one rotor period? It is apparent from
Eq. [59] that D(t) must average to 0 over one rotor
period. Importantly, appropriate choices of r.f. field
strengths and resonance offsets can cause other parts
of Eq. [93] to become time-independent. In other
words, we are interested in conditions where some
part of Eq. [93] will survive an integral over one
rotor period so that a dipole coupling exists that can
recouple the I–S spin states.
Example
Suppose we propose a zero-quantum condition in
which the difference in effective r.f. fields matches a
multiple of the MAS rate, such as Deff ¼ or. If we
neglect all terms that remain time dependent after
this substitution, we have
1
tr
Ztr0
HrfT ðtÞdt ¼
1
tr
Ztr0
1
2ðoD;1 sin yI sin ySIþI�
þ oD;�1 sin yI sin ySI�IþÞdt; ½94�
which integrates to
�HT Deff ¼orð Þ¼ 1
2ðoD;1 sinyI sinySIþS�þoD;�1 sinyI sinySI�SþÞ
¼ 1
2sinyI sinySðoD;1IþS�þoD;�1I�SþÞ
¼ 1
2d1;effðIþS�þ I�SþÞ ½95�
In Eq. [95] we wrote the effective dipolar cou-
pling element by taking the initial rotor phase to be 0
for convenience (i.e. f ¼ 0), giving
d1;eff ¼ Rlab2;1d
ð201;0 yMAð Þ sin yI sin yS: [96]
So Eq. [95] represents a dipole coupling in the zero-
quantum subspace that can drive flip–flop transitions.
The double quantum condition, where Seff ¼ nor, rep-
resents a dipole coupling in the double-quantum sub-
space that can drive flip–flip (or flop–flop) transitions:
�HT �eff ¼orð Þ¼ 1
2ðoD;1 sinyI sinySIþSþ þoD;�1 sinyI sinySI�S�Þ
¼ 1
2sinyI sinySðoD;1IþSþ þoD;�1I�S�Þ
¼ 1
2d1;effðIþSþ þ I�S�Þ ½97�
274 ROVNYAK
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Therefore without propagating or even forming a
density operator, it is already clear that a time inde-
pendent dipolar term will be created with appropri-
ately chosen resonance offsets and nutation powers
that can drive the CP effect. We also see that the
matching is explicitly determined by the effective
fields, which are strong functions of the resonance
offsets. In the absence of homonuclear couplings,
which can broaden matching conditions consider-
ably, and assuming small r.f. field strengths, Eq. [93]
indicates that the matching conditions will be very
narrow and will recouple only certain frequency
regions of I- and S-spin spectra (2). Instead of con-
sidering an 1H-13C spin pair, where homonuclear 1H
couplings could cause unwanted broadening of the
matching conditions, this approach could be used to
good effect in 13C-15N spin pairs, and this experiment
has been dubbed ‘‘spectrally induced filtering in com-
bination with CP’’ or ‘‘SPECIFIC CP’’ (2).Additional analysis of this and other topics in CP is
beyond the scope of the article, but the reader should
be better prepared to tackle them independently.
CONCLUSION
Analytical theory has been explicitly reviewed in the
form of a tutorial for two-spin, heteronuclear CP dy-
namics in static and MAS samples. In the static case,
the role of I–I couplings in broadening matching con-
ditions was shown. This introduction to CP provides
good opportunities to present operations that are com-
mon in SSNMR theory, such as graphical (by inspec-
tion) diagonalization, rotations of spin operators and
space tensors, the use of sum- and difference-fre-
quency subspaces, and the utility of tilted and interac-
tion frames. It is hoped this introduction will be a use-
ful reference for tackling more difficult problems in
CP and in modern SSNMR theory in general.
ACKNOWLEDGMENTS
Notes and valuable discussions from Henry Spindler
and Dr. Phil Costa are gratefully acknowledged, as
are invaluable discussions with Dr. Vladimir Ladiz-
hansky. This tutorial evolved over many years and
benefited from discussions with numerous members
of the Griffin lab, from feedback received when pre-
senting this tutorial in lecture format in more recent
years, and from critical readings from Dr. Kristo-
pher Ooms and Prof. Robbie Iuliucci that led to
many improvements. Very helpful comments from
peer reviewers are gratefully acknowledged.
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APPENDIX: ADDITIONAL SYMBOLS
IX; IY; IZ spin angular momentum
operators
I* ¼ ðIX; IY; IZÞ total angular momentum
vector
I* � S* ¼ IXSX
þ IYSY þ IZSZ scalar (a.k.a. dot) product
of two angular momenta
Ri rotation operator applied
about i = (x, y, z) axisH Hamiltonian operator
r density operator
m0 permittivity of free space
kB Boltzmann constant
�h Planck’s constant divided
by 2p (i.e.¼ h/2p)
BIOGRAPHY
David Rovnyak received his B.S. from
the University of Richmond in 1993
where he eagerly observed NMR work
by Prof. R. Dominey (UR) and Prof. J.
N. Scarsdale (VCU). He did Ph.D. stud-
ies in the group of Robert Griffin at
M.I.T. on quadrupolar NMR methods and
applications in solids, then did post-doc-
toral work in the group of Gerhard Wag-
ner at the Harvard Medical School on
biomolecular NMR methods in liquids. Currently, Dr. Rovnyak is
an Assistant Professor at Bucknell University and is pursuing a
combined liquids/solids NMR research program with interests in
zinc coordination, bile salt micelles, metalloprotein structure, and
new NMR methodology.
276 ROVNYAK
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a