CPMAS of quadrupolar S = 32 nuclei

17

Solid State Nuclear Magnetic Resonance, 1 (1992) II-32 Elsevier Science Publishers B.V., Amsterdam

CP/MAS

of quadrupolar

S = 3/2 nuclei

Alexander J. Vega Du Pont Central Research

and DeLseloprnent I, Experimental

Station, P.O. Bm 80356, Wilmington, DE 19880-0356, USA

(Received 28 September 1991: accepted 28 October 1991)

Abstract

The spin dynamics of Hartmann-Hahn cross-polarization from I = l/2 to quadrupolar S = 3/2 nuclei is investigated. A density-matrix model applicable to cases where the quadrupole frequency up is much larger than the rf amplitude yrs of the S spins, predicts the time development of the spin state of an isolated I, S spin pair in static situations and in three distinct cases of magic-angle-spinning speed vR. These cases are characterized as slow, intermediate, and fast, depending on the magnitude of the parameter LY= v &./ vnvR relative to the intermediate value of 0.4. The model predictions are supported by numerical simulations. The polarization transfer from I to S is efficient in the limits of slow and fast sample spinning. When CKK 1, the Hartmann-Hahn condition is shifted over once or twice vR. When the spinning rate is intermediate, poor spin-locking of the quadrupolar spins prevents the accumulation of a cross-polarization signal and, in addition, depletes the spin-locked I magnetization. Experimental CP/MAS data obtained in NaOH show that the concepts developed for isolated spin pairs are also applicable to cross-polarization in a strongly coupled multi-spin system. Keywords:

cross-polarization;

quadrupolar nuclei, NMR spectroscopy

Introduction Cross-polarization (CP) [ll has proven to be a valuable method for the enhancement of NMR signals of nuclei with low gyromagnetic ratio or long T, relaxation time [2]. It has found particularly wide application in conjuction with magicangle spinning (MAS) [31. Furthermore, the selective nature of CP is often utilized to observe only those nuclei that are dipole-coupled to protons. Despite these successes, CP/MAS has received relatively little attention in the area of half-integer quadrupolar nuclei, in surprising contrast to the growing interest in high-resolution solid-state NMR of those nuclei during the last decade. Recently however, several publications reported experimental evidence of satisfactory CP from

’ Contribution No. 5998. 0926-2040/92/$05.00

I = l/2 nuclei to the central-transition coherence of quadrupolar S = 3/2, 5/2, and 7/2 nuclei under MAS conditions. It was demonstrated for ‘H/ 27A1(S = 5/2) by Blackwell and Patton [4] and by Morris et al. [5,6]; for ‘H/ ‘iB (S = 3/2) by Woessner [7]; for ‘H/ 43Ca (S = 7/2) by Bryant et al. [8]; for ‘H/ “0 (S = 5/2) by Walter et al. [93; for ‘H/ ‘“Na (S = 3/2) by Harris and Nesbitt [lo]; and for ‘H/ 95Mo (S = 5/2) by Edwards and Ellis [ll]. Although much of this work was quantitative in nature, a systematic study of the necessary conditions for successful polarization transfer in rotating samples was not yet reported. An important practical question to be addressed is whether experimental circumstances exist under which CP does not work in principle. This is, for instance, relevant in experimental situations where a vanishing or weak CP/MAS signal could be used as evidence for the absence of protons in the vicinity of the quadrupolar nuclei. In this

0 1992 - Elsevier Science Publishers B.V. All rights reserved

18

A.J. Vega /Solid State Nucl. Magn. Reson. I (1992) 17-32 +90°

(a) I:

Y

S:

where we is the quadrupole

x

6J

I

X MY

(b) s:

Iy z

Fig. 1. (a) Cross-polarization pulse sequence for I = l/2, S = 3/2. (b) Single-resonance spin-lock experiment for S = 3/2.

paper we report theoretical and experimental investigations of Z = l/2 + S = 3/2 CP spin dynamics. The results show that in comparison with its static equivalent, CP/MAS often performs poorly, in particular when fast decay of the S magnetization interferes with the accumulation of the GP signal. The unusually rapid decay of spinlocked magnetization, which is observed in samples rotating at “intermediate” spinning speeds, is an inherent property of half-integer quadrupolar nuclei, as was shown in a recent investigation [12]. The concepts developed in that work and those put forth in Vega’s theory for static CP of half-integer quadrupolar nuclei [13] form the basis for the present discussions. All experiments of polarization transfer from protons to quadrupolar half-integer spins reported to date used the conventional CP pulse sequence shown in Fig. l(a). However, the appropriate matching of the amplitudes of the Z and S radiofrequency (r-f) fields wil and wis in this experiment is not necessarily prescribed by the conventional (I = S = l/2) Hartmann-Hahn condition [l] wil = wis, but rather by the more general rule that polarization transfer to a particular transition of the quadrupole-split spectrum of S requires that wi, be equal to the nutation frequency WnUtof that transition [4-11,131. Our interest is limited to the central l/2 -+ - l/2 transition of S, as this is usually the only portion of the spectrum that is not broadened beyond detection. Its nutation behavior depends on the relative magnitudes of wis and the first-order quadrupole splitting Q [14-161, Q(O> cp) = (1/2)w,(3

cos’8 - 1 + 77sin20 cos 29) (I)

p = 3ezqQ/2S(2S

frequency,

- 1)h

(2)

77 is the asymmetry parameter, and 13and cp are the polar angles of the Zeeman field with the electric field gradient principal axes, and e2sQ/h is the quadrupole coupling constant. When Q is very small, the nutation frequency is not affected by the quadrupole interaction, hence w,,~ = @is. In the other extreme, when Q is very large, the central transition behaves as an isolated two-level fictitious spin-l/2 system with o,,~ = (S + [13,14,17,18]. Thus in the two limiting l/2)% cases the matching conditions for the central transition are: @lI 0

=

WlS

,I=(s+1/2)w,,

(l&I (IQ1

-=-1s) BWIS>

(3) (4)

In the intermediate range, 1Q I = wIs, the Hartmann-Hahn condition is not well defined, in part because then the nutation spectrum of S consists of several frequencies [15,161. An extreme example of this is found in an g5Mo sample with a rather small or? where separate Hartmann-Hahn conditions for the nutation frequencies of the central and satellite transitions were identified experimentally [ll]. Another transition of interest is the triple-quantum coherence 3/2 + -3/2 of S = 3/2, for which the nutation frequency [18] and, hence, the matching condition [13] are given by wll = (2/3)&/~2

( 1Q I B uls)

(5)

Because sample rotation causes some of the nuclear interactions to be time dependent, the behavior of the spins in CP/MAS is not necessarily the same as that just depicted for static samples. In fact, MAS introduces two main differences, the first of which arises from the periodicity of the Z-S dipolar interaction. This complication is also encountered when Z = S = l/2 in which case its effect on the Hartmann-Hahn condition is well understood [19-221: Optimum polarization transfer is then not achieved when ml1 and OI.S are equal, but when they differ by once or twice the rotation frequency wR. It is easy

A.J. Vega /Solid

19

State Nucl. Magn. Reson. I (1992) 17-32

to see that quadrupolar nuclei with very small quadrupole frequencies will follow the same sideband pattern for rf matching, i.e., w ,, = @IS + nwR,. n = 1, 2

(wp = 0,s)

(6)

However, an analogous generalization sideband rf-matching condition

to the

o,, = (S + 1/2)w,, ( my Z=-w1.s

1,

* nwR; n = 1,2 (7)

when the qudrupole interaction is strong, is not immediately obvious. The second deviation from conventional CP spin dynamics is due to the periodic time dependence of the quadrupole splitting, Q. Since the magic-angle condition implies that the time average of Q(t) vanishes, nearly every S nucleus in the sample experiences two or four sign changes of Q per rotor cycle [23]. This has a profound effect on the spin-locking characteristics of quadrupolar nuclei [12] and, as we will see, on the CP dynamics as well. In the theoretical sections of this paper we attempt to provide a framework for the understanding of the CP/MAS process by analyzing an elementary step of the process, i.e., polarization transfer from a single spin I = l/2 to a single quadrupolar spin S = 3/2. Our first approach is a simplified density-matrix model which is valid when we Z+ wrs and when the sample rotation is either very slow or very fast. A model applicable to intermediate spinning speeds is also proposed. The concepts developed in this theory are then tested by numerical simulations. The results of this analysis show that polarization transfer within a spin pair is inefficient when the spinning speed is intermediate. In view of the difficult nature of the theories of CP spin dynamics [24,25] and thermodynamics [26] in multi-spin systems, we did not attempt to extend the presently developed models to situations involving more than two spins. In stead, we present experimental data which demonstrate that the theoretical predictions obtained for an isolated spin pair are indeed relevant to CP/MAS in a strongly coupled multi-spin system.

Experimental The ‘H and 23Na NMR experiments were performed at 300 and 79.4 MHz on a Bruker CXP300 spectrometer. The amplitudes of the radiofrequency (r-f) fields were calibrated by determination of the lengths of 180” pulses applied to an aqueous NaClO, solution. All measurements were performed on a powder of NaOH (crushed pellets, vacuum dried at 75°C) contained in a Kel-F rotor. Cross-polarization and spin-lock data were obtained with the pulse sequences shown in Fig. l(a, b), employing simultaneous alternation of the phases of the first pulse and of the detection reference. The signal was detected after the completion of the x pulse and its intensity was measured as a function of the pulse duration, T-. The number of accumulations per FID was 2. The delay time between scans beginning with a proton 90” pulse was 10 min. For direct 2”Na excitation the delay time was 5 s.

Theory Operator and waue-function notation As in the theoretical treatment of MAS spinlocking [121, we find it convenient to work with a set of S = 3/2 spin operators which besides the conventional spin operators, ‘0 s,=

(l/2)

s?, = (l/2)

60

x

x

fi

0

0

2

0 fi

0

2

0

\ 0

0

43

’

0

_ ifi

ifi

0 0

0

0

-2i

0

0

2i

0

-ifi

\ 0

0

ifi

0

\

I

(8) S,=(1/2)x

‘3 0 0

0 1 o

\o

0

0 0 -1 0

0 0 0 -3 I

A.J. Vega /Solid State Nucl. Magn. Resort. I (19921 17-32

20

include the fictitious spin-l/2 operators ated with the central transition: lo

0

0

o\

;

7

:,

“D

I0

0

0

01

C,=(1/2)X

00

associ-

“0 0

loo

-;

u”

“0

00

0

1

0

0

R=lOOO X 0 0

0 0

0 1

1 0

! 1

00

I I

C,=(1/2)X

and finally a single-quantum-coherence associated with the satellite transitions,

S, = C, + 3TZ

(14)

S, = 2c, + (6/2)R,

(15) (1

S5!?(St-1)/3=

(0

T,= (l/2)

0

0

l\

0 x i 01

0 0

0 0

0 0I

(0

0

0

-i\

0

(100 T,=(1/2)X

;

;

;

i 0 0 0 -1 These matrices are representations of the operators in the basis set of the spin wave functions 13/2), I l/2), I -l/2), I -3/2). Among these fictitious spin-l/2 operators only C, and C, correspond to NMR-observable magnetization. In addition, we introduce the unit matrix,

(j=

(1

0

0

o\

0

I

0

0

01

1 0

0 01

I

the “unit submatrices” tral and triple-quantum

lo 0

0

o\

0 1

1 010 0 0 0 0 0

0 0 0 0

0

0

0

0

0

0

0

1

0

*=O i

0

0

; 01

(16) (17)

Obviously, any operator q commutes with any operator C&i, j =x, y, z, O), and the commutation relations among the fictitious spin-l/2 operators within the T and C manifolds follow the familiar rules of I = l/2. The spin state of an ensemble of S = 3/2 nuclei is commonly specified in the form of a density matrix. In an alternative view, we can often characterize the spin system as a collection of populated spin wave functions. This was found to be a particularly useful approach for the analysis of S = 3/2 spin-locking under adiabatically slow sample rotation [12]. The wave functions that are relevant in the context of this paper are, in addition to the standard wave functions 13/2), I l/2), I - l/2), I -3/2), the sum and difference combinations:

Ic+>={lW)+

I-W)}/fi

(18)

and associated with the centransitions.

1 1

c=o 0

0

(11)

00 _;

U=To+Co

o\ ;

-;

= To - C,

01

0

;

0

\o

co-

(10) \i

(13)

Some useful relationships are: (9)

0 01

and those associated with the triple-quantum herence:

operator

(12)

I t +_> = { 13/Z) + I -3/2)j/fi

(19)

Table 1 summarizes several operator forms of the density matrix and the corresponding relative occupancies of populated wave functions. As to the physical significance of negative populations in this Table, one should keep in mind that these number describe the “reduced density matrix”, i.e., the relatively small deviation of the actual density matrix from the state where all levels are

A.J. Vega /Solid

21

State Nucl. Magn. Reson. 1 (1992) 17-32

TABLE 1 Correspondence between wave-function populations Density matrix “

S = 3/2

Wave functions h I+ l/2);

C: C,

density

l-1/2)

/c+); Ic-) I + I /2); I - l/2) or: Ic+); Ic-) l + 3/2); I - 3/2)

c,, T: T, TO

It+>; It-> I +3/z); I- 3/2) or: It+); It-)

matrices

and

Populations 1:-l 1:-l 1:l 1:l 1: -1 1:-l 1:l I:1

a The operator notation is defined in eqns. (9)~(12). ’ The wave-function notation is defined in eqns. (18) and (19).

respect to the strong quadrupolar term [compare eqns. (13) and (16)]. The wlsSX term may thus be truncated and replaced by 2wisCX. This important simplification allows us to divide A? into two independent parts each of which operates exclusively in either the central-transition or the triple-quantum subspaces of spin S. We accomplish this formally by applying eqns. (14) and (16) and by considering 1, in eqn. (21) as being multiplied by the unit operator U of S, which in turn is the sum of C,, and T,, i.e., Z, = I, CT= Z.rTo + Z, C”

(22)

This leads to A?=&+Xr-

equally populated. We further point out that the “degeneracy” of populations of the occupied wave functions corresponding to C, or T,, allows us to choose any linear combination of these wave functions as the appropriate representation of level populations. Polarization

transfer within a static spin pair

In this section we discuss polarization transfer in a static isolated I = l/2, S = 3/2 spin pair, whereby we restrict ourselves to cases where the first-order quadrupole splitting Q of S is much larger than the rf amplitudes oil and wis applied to the two respective spins. The spins interact with a dipolar interaction of strength WD= (Y1YsV&)(3

cos2%, - I)

(20)

where the parameters have their usual meaning. oD is assumed to be much smaller than wil and wIs. Finally, we include a small frequency offset term 6 for the S spins. In the “doubly rotating frame”, rotating at the frequencies of the two applied rf fields, the Hamiltonian takes the form: Z=(Q,‘2)(S;-S(S+ + WISSX+ wJ,S;

1)/3) +w,,ZX + as,

(Q s- WI/> ~1s s+ ~0, 6)

(21)

It is clear that among the various spin operators in this equation only the R, contribution to the wIsS_ term [see eqn. (1511 is nonsecular with

(23)

Zc = - (Q/2)C,

+ wirl,C,, + 2wIsCx

+ wg ZZC, + 6CZ zT=

(Q/4To +

w,,Z,T,

(24) + 30,Z;Tz

+ 36T, (25)

The density matrix at time t = 0 of the CP period, immediately following the 90” pulse in Fig. l(a), can similarly be expressed as the sum of a C and a T contribution: P(O) = 1, = PC(O) + PT(0)

(26)

PC(O) = IXC,,

(27)

L+(O) = Z,T,

(28)

Since the S with those in p will evolve the respective

operators in C-subspace commute T-subspace, the two components of independently under the action of components of A?:

p(t) =&At)

f&t)

(d/dt)p,

Wdh

-1 = ibT3 &I = i[p,,

(29)

(30) (31)

While the separation into C- and T-subspaces is merely convenient in that it simplifies the description of CP dynamics in static samples, its full physical significance will become evident in the next section when we attempt to understand the CP process under MAS conditions. We first investigate the solution PC(t) of eqns. (24), (27) and (30). Since the -(Q/2>C, term

22

A..l. Vega /Solid

and the C, multiplier of Z, in the Hamiltonian (24) are inoperative as long as we stay within the C-subspace, it is apparent that this density-matrix problem is completely analogous to the CP case of two spins l/2. Following the standard description of Hartmann-Hahn matching for such a spin pair [1,19] we transform z= to the interaction representation of the two rf fields. This changes C, and Z, into time-dependent operators: C, -+ Cl cos(2w,,t)

+ Ci sin(2w,,t)

Z, + Zi cos( wirt) + Zi sin( Wilt)

(32) (33)

which substituted in eqn. (24) give rise to a collection of oscillating terms. Because of their vanishing time averages these terms have no appreciable effect on the time development of pc, unless the Hartmann-Hahn condition, wil = 2wis, is satisfied. Some of the dipolar terms are then independent of time and give rise to the average Hamiltonian, zc=

-(Q/2)C,+(wD/2)(Z;C;+ZJC;)

(34)

With the initial condition p=(O) = Z,C,, solution of the corresponding Liouville equation gives k(t)

coGb~/2)1 +(CA [1- CM%m] +(ZJC; - ZJC:) sin( w,t/2)

= (ZXC,/2)[1f

(35)

For later reference we also give the solution for a more general situation where the initial condition is pJ0) = aZ,C, + K,, i.e.,

+wwxGl+ CJ +K~-wwxcrC.J c+%m +(a -b)(Z;Ci -ZiCi) sin(o,t/2)

PcW = Ku

(36) The time development of the T-portion of the density matrix is entirely different in nature, because Xr [eqn. (25)] does not contain an rf term for the S spin, so that transformation to the rf interaction representation does not affect T,. Consequently, Hartmann-Hahn matching is not possible. The resulting time average of XT in the rf interaction representation is G@;.= (Q/2)To+3STz

(37)

State Nucl. Magn. Reson. I (1992) 17-32

Since this Hamiltonian have

commutes with p,(O), we

Z%(Z) = ZXZ,

(38)

At this point, we recall that the separation of p into pc and pr. relies on the validity of the assumption that the R, contribution to S, may be neglected (see above). There are, however, experimental circumstances where the nonsecular effects of S, can be appreciable. This is obviously the case when Q is not much larger than wis, but also when the rf radiation is very close to resonance (6 = 0). From previous discussions of this effect [12-14,181 one may infer that the T-portion of the Hamiltonian of a single S = 3/2 spin can then effectively be written as ~~=(Q/2)T,+o~~T,(Q,,ls;

where the third-order by

S=O)

(39)

rf coefficient of TX is given

0:“s = 2u;,/3Q2

(40)

In this form, the T-subhamiltonian contains an rf-radiation term which may be utilized for direct pulse excitation or for CP of the triple-quantum transition. Indeed, these two aspects of S = 3/2 spin dynamics were demonstrated in single-crystal samples by Vega and Naor [18] and by Vega [131, respectively. However, in the context of the current paper which deals with powder samples these effects are irrelevant, because w\2 varies widely with Q and thus with the orientations of the crystallites. More importantly, the TX term disappears whenever w$ < 16 I. In practice, this is nearly always the case as a result of the contribution of second-order quadrupole resonance shifts to 6 [12]. Therefore, we will continue to ignore the nonsecular part of S, as long as I Q I P+ wls. In summary, when the Hartmann-Hahn matching condition, wil = 2w,,, is satisfied and when the initial spin state is given by p(O) = Z,, the time dependence of the density matrix of a static isolated spin pair is given by p(t) = (Z,C,/2)[1$+ ( C/2)

cos(%t/2)1

[ 1 - CM wd/2)

+ZXT, I

+ (Z,lCi - ZiCi) sin( w&/2)

(41)

A.J. Vega /Solid

23

%a& Nucl. Magn. Resort. I (1992) 17-32

where the primes in the notation of (ZjCi - ZiCi) indicate that these are operators in the rf-interaction representation, related to the doubly rotating frame by the transformations of eqns. (32) and (33). From eqn. (41) we calculate the expectation value of an observable A by the expression (A) = tr( PA)/2

(42)

which is arbitrarily normalized so that (I,) = 1 at t = 0. This gives for the x-magnetization of the central transition of S,

4

-2

-4

jc,-) 1

(C,> = [1 -cos(w&2)]/4 and for the x-magnetization

(43) of I,

( 1, > = ( I,C,, > + ( 1, T” >

= [ 1 + cos( q&q]

(44)

UJ,,)

= l/2

/4

(45)

(46)

The most striking feature of this result is that only half of the x-magnetization of spin I participates in the polarization-transfer process. In other words, only the portion of the I density matrix that is associated with the central transition of S is transferred to the S spin. The other half, which is associated with the triple-quantum transition, remains unaffected. When (C,) is at its maximum, the density matrix is p = C, + ZXT,,.With the dependence of the equilibrium density matrix on the gyromagnetic ratios taken into account, this corresponds to an enhancement factor of yI/ys over the signal intensity obtained by direct 45”-pulse excitation, i.e., exactly the same enhancement factor as in the I = S = l/2 case. MAS spin-locking of S = 3 /2 nuclei Before we extend the discussion of polarization transfer to CP/MAS conditions, we briefly summarize our recent observations concerning spin-locking of the magnetization of half-integer quadrupolar nuclei by cw rf irradiation under MAS conditions [12]. The arguments are based on the energy-level diagram of a spin S = 3/2 subject to the Hamiltonian Fs=(Q/2)[s,2--(S+1)/3]

-3

0

3

6

Q’w, s

where UJ,,>

-6

I

+w,,S,+SS, (47)

Fig. 2. Eigenvalues E and eigenstates of an S = 3/2 spin Hamiltonian .A?, consisting of a first-order quadrupole term (Q), an rf term toIs), and a frequency-offset term (6). The carrier frequency is slightly off resonance, S/o,, = 0.1. The eigenstates are indicated for the extreme cases of very large positive and negative Q/oIs ratios.

In Fig. 2 the eigenvalues of G!?Ysare plotted versus the quadrupole splitting Q. The figure also indicates the corresponding eigenstates of Zs when IQ1 >wis. See eqns. (18) and (29) for wave-function notation. The resonance offset 6 is assumed to be smaller than wrs, but larger than o&/Q2 at the high values of Q. The latter condition ensures that the wave functions I + 3/2) and I -3/2), rather than their linear combinations I t + > and 1t - >, are the proper eigenstates. For more details on the eigenvalues and eigenstates of GYs we refer to ref. 1121. A spin state is considered to be spin-locked if it can be described as a collection of popmated eigenstates of the Hamiltonian. Certain combinations of wave-function populations correspond with particular matrix forms of the density matrix, examples of which are given in Table 1. For instance, when the central transition of a spin S = 3/2 with large Q is excited via polarization transfer from a spin I = l/2, the resulting S-portion of the density matrix is of the form C, [see eqn. (401. Similarly, when the spins are excited by a phase-alternated 45” y-pulse, the density matrix is of the form C, [121. According to Table 1, the states I c + > and I c - > are then occupied with a 1: - 1 population ratio. Such a spin sate is

24

spin-locked in an rf field with phase X, as in the pulse sequence shown in Fig. l(b). Magic-angle sample spinning causes Q to oscillate back and forth between extrema of the order of Sue and to experience two or four zero-crossings per rotor cycle. What will happen to a spin-locked spin state during and after a zero-crossing depends on the size of the parameter CX,defined as a function of the relative magnitudes of wIs, we, and the sample rotation frequency wR, as follows [121:

When LYB 1, the passages from positive to negative Q, or vice uersa, are relatively slow and are characterized as adiabatic. The level populations are then transfered from the original eigenstates to the eigenstates that derive from them by continuity. For instance, if before the zero-crossing the density matrix is C, while Q is positive, then the system begins with populations of the wave functions I c + > and I c - ) denoted at the lower right of Fig. 2. Adiabatic passage to negative Q will then lead to populations of the ] +3/2) and I -3/2) states found at the lower left in the figure. In other words, the density matrix is transformed from C, to T,. Obviously, an original density matrix of the form T, is similarly transformed to C, by an adiabatic zerocrossing of Q. The same reasoning can be applied to density matrices that contain terms of the form C, or To, like those that were obtained by the CP process discussed above. These spin operators correspond to populations of Ic + >, I c - > or respectively (see Table 1). AcI +3/Z), I -3/Z), cordingly, adiabatic passages take C, to To and vice versa.

When the zero-crossings are fast such that (YK 1, the passages are categorized as sudden. They do not cause any change in the state of the spins. In the intermediate case LY= 1, the density matrix ends up in states that are generally not spin-locked. In powder samples this is experimentally observed as an irreversible loss of spin-locked signal. It is difficult to make a theoretical prediction of the exact value of LYfor which signal loss

A.J. Vega /Solid

State Nucl. Magn. Reson. I (1992) 17-32

is most pronounced, but experimental results of 23Na in NaClO, (vp = 380 kHz and 17= 0) have indicated that the strongest decay of spin-locked signal occurs when a is around 0.4 [12]. Obviously, the density matrix never disappears completely, because the portion represented by the unit operator U = C, + To is preserved under any conditions. Since the sum of C, and T, does not decay, we find that an intermediate passage causes only their difference (C, - To) to be transformed to unobservable coherences. In summary, we may categorize the S = 3/2 density-matrix transitions due to Q zero-crossings as follows: (YB 1 (adiabatic): CX++ TI ) G - To

(49)

a N 0.4 (intermediate):

C, + nsl, T, + ml, (G, - To) + nsl, (Co + T,,) --) (C, + To)

PO)

a -=K1 (sudden):

C, * C, , i-2 - T: , G,*G,,

To-T,,

(51)

where ml in eqn. (50) stands for combinations of spin states that, to a large extent, are not spinlocked. MXS polarization transfer of a spin pair

Consider an isolated I = l/2, S = 3/2 spin pair rotating at the magic angle and subjected to the CP pulse sequence [Fig. l(a)]. Assume further that wQ B wis. The spinning makes both Q and wD time dependent. If the spin pair belongs to a powder sample, we may assume without substantial loss of generality that at the time of the 90”~1 pulse the value of I Q I is much larger than wis. In the period between t = 0 and the first Q zero-crossing the density matrix will then develop from p = 1, to a combination of operators as given in eqn. (41), the only difference being that the product w,t in the goniometric expressions of eqn. (41) must be replaced by the time integral of o,(t). (This is so because zc as given by

A.J. Vega /Solid

State Nucl. Magn. Reson. 1 (1992) 17-32

eqn. (34) commutes with itself at different times, even when Q and wD are time dependent.) For simplicity we write for the density matrix just before the first zero-crossing PI-= (1 -P,JZ,C,

+I+,

(52) where pr represents the amount of transferred polarization. Subsequently, the density matrix is subjected to the first zero-crossing of Q. Since the form in which it emerges from that will depend on the magnitude of (Y, we discuss the various possibilities separately. First we consider the case of an adiabatic zero-crossing (a > 1). The spin-locked parts of the density matrix (ZXC,,, C,, Z,7’,) are then transformed according to the rules of eqn. (49). But the form into which the unlocked term develops during and after the zero-crossing depends strongly on the details of the dynamic process, so that destructive interference (in a powder sample) will eliminate its further contribution to the detectable signal. The relevant part of the density matrix immediately after the first zero-crossing is therefore

(53) The portions of the density matrix that were first involved in the polarization transfer are now frozen in the T-subspace, because Z,T, and T, commute with both G& and %r [eqns. (34) and (37)]. Concomitantly, a new polarization-transfer process starts from the Z,C, term, so that just before the second zero-crossing we get ~2-= (1 -P,)ZJ,+P,T,+

(1 -P~)Z,C”+P,C, (54)

The roles of the T- and C-subspaces are then reversed again, and so on. The generalized solution for p,(t) given in eqn. (36) shows that during any polarization-transfer period between consecutive zero-crossings the sum of the Z,C, and C, coefficients is conserved while their difference oscillates. These results show that, in contrast to the static case, the entire proton magnetization is involved in polarization transfer under adiabatic

25

MAS conditions. Furthermore, the cross-polarized S-spin coherence is created in the form of both C, and Tz. The adiabatic transfer from T, to C, coherence causes the excitation of additional observable S signal which does not come from CP. This signal originates from the Tz portion of the equilibrium density matrix p = S; [see eqn. (1411.Adiabatic sample spinning in the presence of a spinlocking rf pulse applied to the S spins transfers this coherence to an observable C, magnetization [12]. Since this process of “adiabatic magnetization” cia T, can always occur in addition to the Z-to-S polarization transfer, one must be aware of contamination of the CP signal with direct-polarization signal. Fortunately, spin-temperature alternation of the proton excitation in conjuction with phase alternation of the S-signal detection completely eliminates the adiabatic-magnetization contribution. As to the question of the sideband modulated Hartmann-Hahn condition in CP/MAS, we recall that exact Hartmann-Hahn matching (i.e., o,, = w,~) does not produce efficient CP in Z = S = l/2 samples with weak proton-proton interactions [19-221. The cause for this failure is that the cumulative polarization transfer is determined by the time integral of w,(t), which vanishes in MAS. This vanishing can be compensated by modification of the rf matching to one of the four sideband conditions given in eqn. (6) [19,20]. A recently introduced improved method for the restoration the polarization-transfer efficiency uses rotor-synchronized phase alternation of the rf fields of Z and S [21,221. Applying similar reasoning to CP/MAS of quadrupolar S = 3/2 nuclei, we note that a similar failure of Hartmann-Hahn matching (i.e., wII = 20,~) does not necessarily take place when the Q zero-crossings are adiabatic. This has to do with the on/off nature of the polarization-transfer process which implies that the integrations of wJt> are to be taken over limited fractions of the rotation cycle. Therefore, the cumulative effect of w,(t) does in general not vanish under exact Hartmann-Hahn matching. In any case, the entire discussion of the sideband pattern in the adiabatic case is mostly academic, because the condition (Y> 1 requires

26

A.J. Vega /Solid State Nucl. Magn. Resort. 1 (1992) 17-32

slow spinning speeds for which the sidebands tend to overlap. We now return to the density matrix pi_ [eqn. (52)] just before the first zero-crossing and consider what happens when the passages are sudden (cu -=x1). There will then be no change in any of the terms of p, including the portion that is not spin-locked. Hence, the polarization-transfer process will proceed uninterrupted. Consequently, no mixing of C- and T-type spin coherence will occur. The polarization will be transferred exlcusively to C,, while the ZJ, portion of the proton magnetization remains unperturbed, as in the static case. Furthermore, since o,(t) operates within the same spin subspace during the full rotor cycle, the Hartmann-Hahn condition must now be modified to the sideband pattern given by eqn. (7). The idealized behavior of the spins under adiabatic and sudden passages depicted thus far will only be observed when (Y is very much larger or much smaller than the intermediate value of 0.4, and when wiS is much smaller than we. However, in practical circumstances (Yis restricted by the following limits: lJis = vi*/2 = 20 to 50 (second-order broadening in kHz; viS > vi/v, field with Larmor frequency vL); ve > 300 = 2 to 15 kHz, limiting it to the range kHz; vR 0.03 < cr < 3. Hence, some degree of irreversible decay of various terms in the density matrix will often occur at the zero-crossings of Q, and the closer (Y comes to the intermediate value the more severely the signal decreases. There is also a gradual, rather ill-defined transition from sideband-type to centerband-type Hartmann-Hahn matching when (Yvaries from low to high values. To gain insight in the polarization-transfer dynamics at the midpoint of the transition range, we assume as an extreme case that the rules of eqn. (50) apply rigorously in the sense that the “ml” states are completely and irreversibly disappearing. We then find that the first zero-crossing transforms pl_ to pi+= (1 -P,/2)Z,(C,

Numerical simulations The approximations and predictions made in the preceding section were tested against the results of numerical integrations of the Liouville equation, d@)/dt

+ T,) = (I -P1/2)Z,

i.e., all accumulated S polarization is lost and the magnetization is reduced. This process repeats

= i[p(t),

z(t)]

(56)

where X(t) was the full Hamiltonian in the doubly rotating frame given by eqn. (21) without truncation of any nonsecular terms. The calculations were performed for a single ‘H/ 23Na spin pair in the following specific configuration. The quadrupole tensor was chosen to be axially symmetric with its unique axis of symmetry q perpendicular to the sample rotation axis R. The internuclear vector .rls was parallel to q and had a length of 1.5 A. At time t = 0, q and rlS were coplanar with R and the Zeeman field B,. Of course, R made the magic angle with B,. In this model, Q(t) and o,(t) oscillate in phase with each other at twice the rotation frequency: Q(t) = @o/2)

cos 2wQt

(57)

and

mg(t) = (?‘I?‘&~s) cos2oQt The

(58)

amplitude of the dipole coefficient was = 9.4 kHz. The operators p and 8 were represented by 8 x 8 matrices in the standard representation I mlmS > (m, = + l/2; m, = 3/2, . . . , - 3/2). The initial condition was p(O) = Z,. The density yIysh/rjs

(55) Z

itself at successive zero-crossings and leads to the total depletion of Z magnetization. In a powder sample, S magnetization will begin to build up with a normal CP rate at short contact times, but after about half a rotor cycle this process is increasingly frustrated by the zero-crossings of the quadrupolar nuclei. Each spin packet does resume the polarization transfer after every zero-crossing, but the steady-state magnetization achieved in this way for the total sample decays fairly rapidly because of the depletetion of the Z reservoir.

A.J. Vega /Solid

21

State Nucl. Magn. Reson. 1 (1992) 17-32

matrix at t > 0 was found by step-wise integration, ~[
+i[p(nAt),

c%‘(nAt)]At

(59)

and by application of an additional, shortly to be described, correction. This correction was necessary to overcome the accumulation of excessive numerical errors in the “nonsecular” matrix elements, i.e., the elements that connect centraltransition with triple-quantum S-states. These matrix elements were ignored in the simplified density matrix model presented above, but they do play an important role in the transition region When (Y is in the intermediate when IQ1 so,,. range, these matrix elements do not return to zero after the transition and they begin to oscillate with a frequency equal to Q(t). An essential point of our theory is that they are to be ignored after the completion of the zero-crossing, because destructive interference in a powder leads to their rapid decay, similar to that of the satellite coherences following a direct single-pulse excitation of S. In an exact calculation of a single spin pair their magnitude is, however, not automatically diminished. On the contrary, the finite value of QAt in the step-wise integration procedure causes a rapid increase of their amplitudes. This is exactly analogous to the artificial lengthening of the radius of a circular motion which is numerically simulated by step-wise tangential increments. If the angular frequency is Q, this radius increases by a factor l/cos

QAt = 1 + (QAt)2/2

(60)

at every step. To remedy this numerical problem, we multiplied the nonsecular density-matrix elements at every step with a factor 1 -h(QAt)’

(61)

whereby the intended correction is made by taking A = 0.5. However, in order to also simulate the observed decay of these matrix elements in powder samples, we actually chose A > 0.5. When simulated in this manner, the decay is faster for larger values of Q, as it should be in the case of

0.5

cMIvvvvvvvvvvvvvvvv\

@Jo)

0

J

W _____________._____.________.__.____

Fig. 3. Results of numerical integration of the density-matrix equation for a ‘H/ 23Na spin pair in a static CP experiment. The geometric model is described in the text. v,, = 80 kHz; vIs = 40 kHz; vp = 1.6 MHz; 6/2~ = 5 kHz; At = 5 ns; A = 1. Plotted are the expectation values of the indicated spin operators.

inhomogeneous broadening of Q. The secular matrix elements (those belonging to either the Tor the C-subspace) were not subjected to this correction. The first example of calculated CP dynamics, shown in Fig. 3, is for a static case. The numerical parameters are indicated in the figure caption. . . They satisfy the conditions wrS <
A.J. Vega /Solid

28

0

200 r (I&

400

0

200

400

7 w

Fig. 4. Results of numerical integration of the density-matrix equation for a ‘H/*‘Na spin pair in a CP/MAS experiment under adiabatic conditions (LY= 2.2). The geometric model is described in the text. v,, = 80 kHz; v,~ = 40 kHz; ve = 0.8 MHz; 6/27r = 5 kHz; I/~ = 0.9 kHz; At = 10 ns; A = 1.5. Plotted are the expectation values of the indicated spin operators. The vertical broken lines mark the zero-crossings of Q. The figures on the left and the right display identical data. The left side emphasizes that between zero-crossings the polarization transfer occurs in the C space only. The right side emphasizes that at the zero-crossings corresponding C and T operators exchange their expectation values.

An example of CP/MAS with adiabatic zerocrossings ((u = 2.2) is shown in Fig. 4 where the development of the various spin coherences is followed until after the second zero-crossing of Q. Most of the above-made theoretical predictions are demonstrated in this plot. For instance, between successive zero-crossings the polarization transfer takes place between (Z,C,) and (C,), while their sum (Z,C,) + (C,) remains constants (top-left figure). Although polarization transfer does not take place in the T-subspace, coherence mixing by the adiabatic passages causes the entire proton magnetization to be involved in the CP process (bottom left). At the zero-crossings we see a nearly perfect interchange of ( Z,C,> with (Z,Z’,> (top right) and of (C,) with (7”) (bottom right). For instance, the first zero-crossing resets (C,) and (Z,C,> to their original values of 0 and 0.5 and thus initiates a second polarization process similar to the first (top left). The CP/MAS behavior predicted for fast sample rotation is supported by the numerical results shown in Fig. 5, calculated for cy = 0.02. The plots on the left were simulated for centerband Hart-

State Nucl. Map.

Reson. 1 (19921 17-32

mann-Hahn matching (w !, = 2~,~). These experimental conditions do not cause any appreciable changes in the various expectation values, other than some fluctations in (Z,T,) which are not related to polarization transfer. However, when the Hartmann-Hahn matching is modified to the second-sideband condition (w,, = 2~,~ + 2w,), the cumulative polarization transfer is substantial, as is shown on the right. The spin dynamics has the same characteristics as the static CP case shown in Fig. 3, i.e., (Z,C,> is almost completely transfered to (C,) while ( Z,T,,) and (Tz> remain essentially unaffected. The second sideband is the appropriate rf-matching condition in this particular case, because wD oscillates with twice the frequency of sample rotation [see eqn.

c58)l. Figure 6 shows the numerical simulation of CP/MAS at a rotation speed in the intermediate range (a = 0.19) with centerband HartmannHahn matching on the left and second-sideband matching on the right. The precise choice of rf-matching condition is apparently of little significance in this situation. In either case, the C,

Fig. 5. Results of numerical integration of the density-matrix equation for a 1H/23Na spin pair in a CP/MAS experiment under fast-spinning conditions (a = 0.02). The geometric model is described in the text. vts = 20 kHz, vQ = 2 MHz; 6/2~ = 5 kHz; vR = 10 kHz; At = 5 ns; A = 0.5. Plotted are the expectation values of the indicated spin operators. The vertical broken lines mark the zero-crossings of Q. The plots on the left were calculated for exact Hartmann-Hahn matching, v,, = 2vls = 40 kHz. On the right the rf matching is modified to a sideband condition, v,, = 2~,~ t2v, = 60 kHz.

29

A.J. Vega /Solid State Nucl. Magn. Reson. I (1992) 17-32

/‘,

‘7

0

5

0

10

r (ms) 0

200

400

r w

0

200

400

r w

Fig. 6. Results of numerical integration of the density-matrix equation for a ‘H/ ‘“Na spin pair in a CP/MAS experiment under intermediate-spinning conditions ((u = 0.19). The geometric model is described in the text. v,,s = 30 kHz, v. = 1.2 MHz; 6/2~ = 5 kHz; vn = 4 kHz; At = 10 ns; A = 1.5. Plotted are the expectation values of the indicated spin operators. The vertical broken lines mark the zero-crossings of Q. The plots on the left were calculated for exact Hartmann-Hahn matching, v,, = 2vrs = 60 kHz. On the right the rf matching was modified to a sideband condition, v,, = 2v,s +2v, = 68 kHz.

magnetization passes through a relatively low maximum around T = rR. Its subsequent decay is linked to the rapid depletion of the two proton magnetization components (Z,C,> and (ZJ’,). The various elements of the spin-dynamics process are clearly illustrated in the figure: Between the zero-crossings, small smounts of spin polarization are transferred from ( Z,C, > to (C, >, while (Z,T,) makes its characteristic oscillations around a constant vaiue (compare Fig. 3). As predicted by eqn. (551, (C,) is irreversibly reduced at the zero-crossings, albeit not as thoroughly as implied by our extreme assumption of complete diappearante of the “nsl” states. On the other hand, we do see evidence that (Z.,C,> and (f,T,) equilibrate at each zero-crossing.

Experimental results The quadrupole tensor for 23Na in NaOH is axially symmetric and has a quadrupole frequency ue = 1.8 MHz, as was recently reported by Dee et al. [271 Its static single-resonance T,, relaxation

Fig. 7. Single-resonance z3Na spin-locking in NaOH with r,s = 40 kHz in a static experiment and in MAS eperiments Lith vn = 2 kHz ((u = 0.44) and in = 4 kHz ((u = 0.22).

.

.

time m a spin-lock field of vIs = 40 kHz is much longer than 10 ms, as is shown in the top trace in Fig. 7. This figure also shows the much faster decays at spinning speeds of 2 and 4 kHz which correspond to (Y values of 0.44 and 0.22, respectively. In Fig. 8 the same MAS-spin-lock data are replotted on an expanded time scale which is normalized with respect to the rotor period. This plot shows that the observed signal reduction is, indeed, stronger when (Yis closest to 0.4. Figure 8 also compares the experimental data with a theoretical curve that was calculated for a powder sample of axially symmetric quadrupolar nuclei whereby the signal of every individual spin packet was assumed to vanish abruptly at its first Q

100

‘;;

% 50 .-6-J cn 0 0

0.5

1 5/ZR

1.5

2

Fig. 8. Single-resonance 23Na spin-locking in NaOH with v rs = 40 kHz under MAS at vn = 2 kHz ((Y= 0.44) and vR = 4 kHz (cu= 0.22). These are the same data as in Fig. 7. but plotted with 7 normalized to the rotation period TV. The theoretical curve represents the signal decay calculated under the assumption that the observable magnetization of every spin packet disappears completely at the first zero-crossing of its Q.

A.J. Vega /Solid State Nucl. Magn. Reson. I (1992) 17-32

30

zero-crossing in an MAS experiment. The (Y= 0.44 decay rate is seen to approach the theoretical limit of signal reduction very closely. This is strong experimental support for the validity of the assumption that the central-transition coherences vanish irreversibly at intermediate-rate zero-crossings. Cross-polarized 23Na signals of NaOH were obtained by Hartmann-Hahn matching with vis = 40 kHz and ull = 80 161~. Static and MAS signal intensities were determined from the first points of the free-induction decays. Fouriertransform lineshapes cannot be used for this purpose, because the dependence of CP dynamics on the crystal orientations distorts the lineshapes. The signal intensitites are plotted in Fig. 9 as percent of the intensity obtained after direct 45”pulse excitation. Since the maximum theoretical signal enhancement- of an isolated spin pair is equal to the ratio yl/ys, as was pointed out in the discussion following eqn. (34), thermodynamic equilibration of the ‘H and 23Na spin baths having equal numbers of spins, could optimally lead to a signal enhancement of half that size [26], i.e., 189%. The experimental static CP signal reaches almost 130% and thus represents nearly 70% this theoretical limit. Spin-temperature equilibration between (Z,C,> and (C,) further implies that (I,C,) will be reduced to half its equilibrium size. Consequently, optimum CP could reduce the total residual proton magnetization, (Z,C,) + (IJ,), to 75%. If the CP process is only 70% effective, as it is according to the

0

5

10

z (ms) Fig. 9. ‘H/23Na CP results obtained in NaOH under the indicated static and MAS conditions. Y,, = 80 kHz; vIS = 40 kHz. The =Na signal intensity is plotted as percent of the signal intensity obtained by direct 45”-pulse excitation.

$00 a-

3

El

‘Z 50 I 0 0

5 10 T 0-W Fig. 10. Residual ‘H signal intensities following ‘H/ Z3Na CP in NaOH under the indicated static and MAS conditions. v ,, = 80 kHz; v,~ = 40 kHz.

23Na data, the corresponding residual proton magnetization should be 82%. This prediction is fairly well reproduced by the observed static-CP proton signal intensities shown in Fig. 10. Unfortunately, detection problems in the 300 MHz channel caused some slow fluctuations to occur in the data, but the initial drop to N 80% and the absence of a substantial subsequent signal reduction are in excellent agreement with the model that predicts exclusive CP from one half of the proton magnetization. The efficiency of CP/MAS is much lower than that of its static analogue, as shown by the 23Na data in Fig. 9. It is obvious that this is caused by the destruction of S magnetization at the zerocrossings, because the effect is most pronounced at the spinning speed of 2 kHz ((Y= 0.44) for which the 23Na spin-locked signal reduction was seen to approach its theoretical limit. Initially, the 2 kHz CP signal increases at the regular rate but it begins to decay after it reaches a maximum during the second half of the first rotation period (T = 0.3-0.5 ms>, in accordance with the theoretical predictions. Finally, Fig. 10 shows that the decay of CP signal under MAS conditions is linked to a simultaneous depletion of the proton magnetization. This experimental example illustrates that the concepts developed for the understanding of polarization transfer in an isolated spin pair are, at least in part, applicable to a system with a strongly coupled proton bath. In particular, it demonstrates that the separation of the spin system into

A.J. Vega /Solid

31

State Nucl. Magn. Reson. I (1992) 17-32

C- and T-subspaces appears to retain its validity in a multi-s, multi-l spin system. This is a somewhat surprising discovery, because there are many more than two subspaces in the presence of more than one S spin. For instance, two S spins necessitate the consideration of four subspaces: CC, CT, TC, 7”T; and the number of combinations increases exponentially with the number of S spins. We did not attempt to describe this large spin system in a theoretical model, but even if it turned out that the many subspaces were developing independently under Hartmann-Hahn matching, one could speculate that combined heteronuclear and homonuclear dipole interactions provide a mechanism for equilibration of the expectation values associated with all these subspaces. This would lead to a participation of the total proton magnetization in the CP process, even in the static case. Nevertheless, the experimental results obtained for NaOH seem to indicate that the dipolar interactions do not generate an appreciable deviation from the simplistic dual-space picture that describes the isolated spin pair.

Conclusions We have investigated some theoretical aspects of Hartmann-Hahn polarization transfer from a single I = l/2 nucleus to a single quadupolar S = 3/2 nucleus, assuming that wp B wis. The various concepts that were developed in a simplified density-matrix model could be verified by numerical simulations. Experimental results demonstrated that a strongly coupled multi-spin system follows similar spin-dynamics principles. We anticipate that most of the following conclusions, reached for S = 3/2, wil1 also be valid for quadrupolar nuclei with higher half-integer spin. In static CP, the Hartmann-Hahn condition is [or oil= (S + 1/2)w,, for any half*II = 2wis integer S]. The theoretical enhancement of signal intensity obtained in the thermodynamic-equilibrium CP limit, as compared to the signal intensity following a direct-excitation 45” pulse, is the same as that for spin-l/2 cross-polarization. In contrast to the latter, only one half of the I

magnetization participates in the polarization transfer whereas the other half remains unaffected. In CP/MAS the nature of the spin dynamics is determined by the magnitude of the parameter (Y= o&/wQoR. When a B 1 (slow spinning), the Hartmann-Hahn conditions is again wi, = 2~,~, and the signal enhancement is comparable with the static CP. However, the entire I magnetization participates in the polarization-transfer process. In the other limit when (YK 1 (fast spinning), the CP mechanism is also efficient, but Hartmann-Hahn matching must satisty any of the sideband conditions, wi, = 2wis + nw,,, (n = 1, 2). In contrast to the slow-spinning case, only half of the I magnetization participates under fast-spinning conditions. When (Yis in the vicinity of 0.4 (intermediate spinning rate), the CP signal begins to build up during the first half of the first rotor period, but the magnetizations of both S and I decay rapidly soon after the completion of the first rotor period. This occurs whether Hartmann-Hahn matching satisfies the centerband condition or a sideband condition. In practical signal-enhancement applications one should attempt to avoid intermediate spinning rates. This may be achieved by going to very slow or to very fast spinning. The latter is probably the more successful approach in many situations (e.g., v,, 5 60 kHz, v,~ I 30 kHz, vQ 2 800 kHz, vR 2 10 kHz). Another possible solution [28] is the use of dynamic-angle spinning (DAS) [29] whereby the sample rotation axis is placed parallel to the Zeeman field during the CP period after which it is switched to other angles for signal detection. The CP dynamics is then like in static samples. On the other hand, for indirect detection of S through observation of the proton signal decay, intermediate MAS rates are to be preferred.

References 1 S.R. Hartmann and E.L. Hahn, Phys. ReL,., 128 (1962) 2042. 2 A. Pines, M.G. Gibby and J.S. Waugh, J. Chem. Phys., 59 (1973) 569.

A.1 Vega /Solid State Nucl. Magn. Reson. I (1992) 17-32

32 3 J. Schaefer and E.O. Stejskal, J. Am. Chem. Sot., 98 (1976)

4 5 6 7

1031. C.S. Blackwell and R.L. Patton, J. Phys. Chem.., 88 (1984) 6135. H.D. Morris and P.D. Ellis, J. Am. Chem. Sot., 111 (1989) 6044. H.D. Morris, S. Bank and P.D. Ellis, J. Phys. Chem., 94 (1990) 3121. D.E. Woessner, 2. Phys. Chem. (Miinchen) N.F., 152 (1987)

51. 8 R.G. Bryant, S. Ganapathy

and S.D. Kennedy, J. Magn. Reson., 72 (1987) 376. 9 T.H. Walter, G.L. Turner and E. Oldfield, J. Magn. Reson., 76 (1988) 106. 10 R.K. Harris and G.J. Nesbitt, J. Magn. Reson., 78 (1988) 245.

11 J.C. Edwards and P.D. Ellis, Magn. Reson Chem., 28 (1990) s59. 12 A.J. Vega, J. Magn. Reson., 96 (1992) 50. 13 S. Vega, Phys. Reu. A, 23 (1981) 3152. 14 A. Wokaun and R.R. Ernst, J. Chem. Phys., 67 (1977)

1752. 15 A. Samoson and E. Lippmaa, Phys. Rev. B, 28 (1983) 6567. 16 A.P.M. Kentgens, J.J.M. Lemmens, F.M.M. Geurts and W.S. Veeman, J. Magn. Reson., 71 (1987) 62.

17 A. Abragam, Principles of Nuclear Magnetism, Clarendon, Oxford, 1961, p. 36. 18 S. Vega and Y. Naor, J. Chem. Phys., 75 (1981) 7.5. 19 E.O. Stejskal, J. Schaefer and J.S. Waugh, I Mugn. Reson., 28 (1977) 105. 20 A.J. Vega, J. Am. Chem. Sot., 110 (1988) 1049.

21 X. Wu and K.W. Zilm, private communication. 22 T.M. Barbara, A. Brooke, H.D.W. Hill and E.H. Williams, 32nd Rocky Mountain Conference, Denver, 1990, Abstract No. 227. 23 M.M. Maricq and J.S. Waugh, J. Chem. Phys., 70 (1979) 3300. 24 D.E. Demco, J. Tegenfeldt and J.S. Waugh. Phys. Rw. B,

11 (1975) 4133. 25 X. Wu and K.W. Zilm, J. Magn. Reson., 93 (1991) 265. 26 M.H. Levitt, D. Suter and R.R. Ernst, J. Chem. Phys.. 84 (1986) 4243. 27 SF. Dee, G.E. Maciel and J.J. Fitzgerald,

J. Am. Chem. Sot., 112 (1990) 9069. 28 K.T. Mueller and A. Pines, private communication. 29 K.T. Mueller, B.Q. Sun, G.C. Chingas. J.W. Zwanziger, T. Terao and A. Pines, J. Magn. Reson., 86 (1990) 470.