tutorial on analytic theory for cross-polarization in

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

Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a

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


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



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


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.



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


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.



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


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-



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


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.



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


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



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


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.



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


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


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


4

!IZ

þ 1

2

g21

4


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


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



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


�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.



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


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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pp 1–190.

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


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