Journ of Magn Res 72, 62-74 (1987)

JOURNAL OF MAGNETIC RESONANCE 71,62-14 (1987)

Two-Dimensional Solid-State Nutation NMR of Half-Integer Quadrupolar Nuclei

A. P. M. KENTGENS, J. J. M. LEMMENS, F. M. M. GEURTS, AND W. S. VEEMAN

Department of Molecular Spectroscopy, Faculty of Science, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands

Received May 13, 1986; revised July 11, 1986

The two-dimensional solid-state nut&ion NMR experiment for the determination of quadrupole parameters, as introduced by Samoson and Lippmaa is evaluated. A complete series of spectra (for spin Z = f, i, f, and p) resulting from density-matrix calculations is presented, and some experimental aspects of the method am discussed. Finally applications of the method to *‘Al (I = 3) in spodumene and to 45Sc (Z = i) in Sc2(SO& are shown. Q 1987 Academic Pres, Inc.

INTRODUCTION

The majority of elements’ in the periodic table have nuclei with half-integer quadrupole spins. It is therefore not surprising that there is interest in high-resolution NMR spectra of quadrupole nuclei in solids. Especially the recent studies of zeolites, clays, and ceramics have focused attention on obtaining structural information from NMR spectra of quadrupolar spins.

In comparison to nuclear spins with spin quantum number I = 1, the NMR spectra of quadrupolar spins contain two new features: ( 1) several transitions are possible and (2) the transition frequencies are determined not only by the interaction between the magnetic moment of the nucleus and the static external field but also by the interaction between the nuclear electric quadrupole moment and the electric field gradient of the surroundings of the particular nucleus. The chemical information which an NMR spectrum of a quadrupolar spin offers is not limited to the chemical-shift data, but such a spectrum can also provide the parameters that describe the quadrupolar interaction. These parameters depend on the local symmetry around the nucleus in con- sideration and thus give direct structural information.

Traditionally, quadrupole interaction parameters can be determined by NQR. There is usually a small magnetic field or none, and the nuclear spin levels are split mainly by the quadrupole interaction. The frequencies of the transitions between these levels provide the quadrupole parameters. The disadvantages of NQR are its low sensitivity when the quadrupole interaction is rather small (0- 10 MHz) and the wide frequency range one has to search for possible resonances. Both of these disadvantages can be overcome in principle, at least for small quadrupole interactions, in an NMR experiment.

0022-2364187 $3.00 Copyrishl8 1987 by Academic F’ms, Inc. AU rights of reprcduction in any form -ed.

62

NMR OF QUADRUF’OLAR NUCLEI 63

The description of the NMR spectra of quadrupolar spins depends strongly on the spin quantum number Z and the relative magnitude of the Zeeman and the quadrupole interactions. Here we limit our discussion to half-integer spins (I = 5, I, 5,:) and assume the quadrupole interaction to be small with respect to the Zeeman interaction, The (anisotropic) contribution of the quadrupole term to the NMR transition frequencies can then be calculated by perturbation theory.

In high magnetic fields all Zeeman transitions WZ, m’ shift in first order because of the quadrupole interaction except the f, - 1 transition which experiences a (much smaller) second-order shift. As a result of this, typical spectra of polycrystalline samples containing quadrupolar spins give characteristic powder patterns for the 1, - 1 transition whereas all other transitions are usually broadened beyond detection (I). Because the magic angle has no “magic” properties for the second-order quadrupolar interaction, MAS will not average this interaction, but yields powder patterns for the 1, - f transition which are approximately four times narrower than for static samples (2).

Although, in principle, it is possible to extract information about the electric field gradient from the spectra of static or spinning polycrystalline samples, in practice the powder patterns are often blurred by a spread in chemical shift and/or by the presence of more than one quadrupole interaction. To overcome these problems Samoson and Lippmaa (3) introduced a simple two-dimensional experiment which allows one to separate the quadrupole interaction from the chemical-shift interaction. This technique is based on a nutation experiment (4) where the evolution of the spin system in the presence of a radiofrequency field Br is studied in the rotating frame. This evolution yields a low-field (B, - O.OOl&) NMR spectrum, the nutation spectrum, with the sensitivity of the high-field spectrometer. In this respect it can directly be compared to the zero-field NMR technique developed by Pines and co-workers (5) where the sample is pneumatically shifted in and out of the magnet. The advantage of the nutation experiment is that, in contrast to the zero-field technique, there is no limitation on the T, spin-lattice relaxation time. A nutation spectrum, however, is more difficult to analyse than a zero-field spectrum.

In this paper we discuss the calculation of these nutation spectra using the density matrix formalism, and present a number of simulated spectra for all half-integer quadrupole spins (I = $ to Z = 5) which can be used as fingerprints. In a previous paper on 27A1 nutation NMR of zeolites (6) it was shown how nutation NMR can be used to separate the signals from 27A1 with a large quadrupole interaction from the Al signals with a small quadrupole interaction. Here we want to show that by comparing an experimental spectrum to a set of simulated spectra one can determine the quadrupole parameters. This will be demonstrated for 27Al in spodumene and for 45Sc in S@SO4)3. Further, some experimental aspects of the method will be discussed.

THEORY

The pulse scheme of the experiment is outlined in Fig. 1. During the evolution period tl , an rf field is present and the system evolves under the secular Hamiltonian &“, in the rotating frame, assuming that the sample is static (i.e., no MAS):

9, =~o~+&“~+‘zQ=(n,-a)zz-n,zx+n~(3z~-z(z+ 1)) 111

64 KENTGENS ET AL.

b F, -projection

FIG. 1. (a) Pulse scheme of the experiment; a free induction decay is acquired during tz as a function of the pulse length tI . Subsequent Fourier transformation of the signals obtained yields a typical 2D powder pattern as shown in (d). This spectrum was calculated for Z = f with a ratio Q&l,+ = 0.45 and 9 = 0. The F2 projection of this pattern (c) gives the second-order quadrupole powder pattern, and the F, projection (b) is a characteristic nutation spectrum for the ratio Q,$&.

with

% = ;;;!;)(3 cos28 - 1 + 7j sin28 cos 2@) = Qo(3 cos2B - 1 + 7) sin28 cos 2@)

where Q,+ = y& and B and @ are the polar angles orienting the magnetic field &, in the principal axis system of the field gradient, and 1 represents the asymmetry parameter. Here dipolar interactions and the nonsecular part of the quadrupole Hamiltonian have been neglected. During t2 there is no rf field present and the FID of the f, - f transition is acquired (assuming that all other transitions are too broad to be detected). The system is now governed by the Hamiltonian S2:

z2=&“z+2Yf3+‘zQ PI where SF’= and Sacs are the Zeeman and the chemical-shift interaction and So is the quadrupolar interaction which only contributes in second order to the lineshape of

NMR OF QUADRUPOLAR NUCLEI 65

the central 4, - 4 transition (7). With the knowledge of these Hamiltonians we can calculate the signal s(t,, fz) using the density-matrix formalism. At time tl = 0 we start with the equilibrium density matrix u(O) in the high-temperature approximation. As one can see the dominating Zeeman interaction X’z is not present in 3?‘, . So if we irradiate close enough to resonance (zOf small) then %‘o and 9’ti are the most important terms of 2,. This means that the eigenfunctions of ~3?r depend on the ratio of Qo and Q,. From the Liouville-von Neumann equation we get at time tl

u(tl) = exp(-i~ltl/~)u(0)ex~i~,~,/~). [31

Starting in a basis of eigenfunctions of Z,, we get after (numerical) diagonalization of z,, using the orthogonal transformations T

u(tr) = Texp(-iE~l/ti)~+~(0)Texp(iE~I/~)~+. [41

Here Z? represents the diagonalized matrix of X1 giving the eigenvalues

Ej= C Z TpjzlpqTqj 151 P 4

and eigenfunctions lj) = C Tij14 m). [61

We assume that we only detect the coherence between the f and the - f states during t2, so only the element ~r(t~)~~~,-~,~ has to be evaluated. The signal of the central transition then can be calculated as a function of tl and t2:

- a(t1)1/2,-1,2exp(-i~2t2)

= C(R-*,2,,,2)i,jeXp(i~2iitl)eXp(-i~2t~) i,j

171

where Q2 represents the Larmor precession of the spins during t2 and Q2, = (Ei - Ej)/h is the transition frequency in the rotating frame between the states Ii) and /j). The coefficients (R-1/2,1/&

(R-1/2,1/& = Tl/2,iT-l/2,j C Tk,iTk,jdO)k,k 181 k

represent the contributions of the coherence between Ii) and lj) in the rotating frame during tl to the 4, - 1 coherence detected during t2.

To summarize, we have during tl a system with eigenfunctions and eigenvalues which depend on the ratio of no and C&r (Fig. 2). The coherences between all these levels, with frequency a,,, develop during tl and give a certain contribution to the 1, -1 coherence detected during tz . Therefore, a two-dimensional Fourier transformation of the acquired signal will give a characteristic powder pattern (Fig. Id), whose projection on the F2 axis (Fig. lc) yields the normal powder lineshape due to the combined effect of chemical-shift anisotropy and quadrupole interaction. Projection onto the F, axis (Fig. 1 b) gives the nutation spectrum which depends on the quadrupole parameters e2qQ and 1, the spin quantum number Z and the rf field strength B, , and is independent

66 KENTGENS ET AL.

energy (MHz)

.04

.03

-.os

0 25 5 1.5

I Ok-----

:;‘::I:- -%?ii-xi

y6 (kHz) -

FIG. 2. Energy diagram of an isolated spin I = $ in the presence of a magnetic field B with quadrupole parameters e2qQ/h = 1 MHz and t) = 0. On the left is the low-field situation Zz < &“o and on the right is the situation &“= $ Zo, where X’o appears to be negligible.

of the chemical shift. As we shall see later, and as we reported before (6), the nutation spectra show more detail than MAS NMR spectra and therefore we concentrate on the nutation spectra instead of on the whole 2D powder pattern.

In the extreme cases l.%‘o[ 6 (%‘A and l%‘o[ + I%‘,.rj the nutation spectrum consists of a single line. In the first situation X’o may be neglected with respect to Xti and the nutation frequency is simply f&r. In the second situation straightforward perturbation theory shows that the nutation frequency becomes (I + $)fiti (6-84b).

In intermediate cases, I~??‘ol - IsA, the spectra are complicated and several peaks can occur because many transition frequencies t&in the rotating frame exist. In addition the nutation spectra will be powder patterns because of the anisotropic nature of the quadrupole interaction. Therefore for intermediate cases the experimentally obtained nutation spectra have to be compared to simulated spectra, calculated as described above.

Figures 3-6 each show a series of calculated nutation spectra for different ratio t&&.r, for the different spins Z = 5 to Z = 3. In all spectra the asymmetry parameter 71 and the resonance offset are assumed to be 0. These spectra were calculated for 7500 crystallite orientations, taking up to 5 minutes computer time on a mainframe (NAS 9060). The spectra appear to be very characteristic powder patterns so for these intermediate cases with !& known, one can determine no (no = e2gQ/h8Z(2Z - 1)). Because no depends on the nuclear quadrupole moment Q and the spin quantum number Z, and with Q,+ limited for experimental reasons, what range of electric field gradients eq can be determined without getting in the extreme situation with only one nutation frequency left will depend on the nucleus in question.

Figure 7 shows the effect of the change of the asymmetry parameter 17 for a spin Z = $ with G-J& = 0.6. The calculations are for 100 X 100 crystallite orientations taking up to 10 minutes computer time. The overall appearance of the powder pattern


I = 3/2

i, Orf 2 Orf

FIG. 3. Calculated nutation spectra for an isolated spin I = 3 as function of the ratio &/&, with 9 = 0. No resonance offset is taken into account. The scale factors of the spectra are 1, 4.7, 15.2, 14.9, 10.2, 7.3, 5.7, and 3.0 with respect to the G@, = 0 spectrum. In all spectra a Lorentzian line broadening of 2.5 kHz was applied.

remains the same but the intensity of some of the lines changes drastically. So it is also possible to get a good estimate of T) from the nutation spectra.’

So far it has been assumed that the sample is static, i.e., no magic-angle spinning applied. Magic-angle spinning would be desirable in view of the resolution along F2. However, numerical computer calculations of the simulated nutation spectra under MAS conditions becomes very time-consuming because the change of the orientation of the sample during tl and thus of &“r cannot be neglected. Experimentally it has been observed that the nutation spectra change with MAS. The discussion of these spectra and their interpretation will be postponed to a later publication.

EXPERIMENTAL ASPECTS

Resonance ofiet. When the system is irradiated on resonance (Q = s2,) then the matrix of &“r in Eq. [2] becomes symmetrical with respect to both diagonals of the matrix. As a result of this

’ A complete set of simulated spectra with asymmetry parameter variation for all spins (I = $ to I = $) can be obtained on request to the authors.

I = 5/2

c, r

Qf 34f

FIG 4. As in Fig. 3, but for spin I = $. Scale factors: 1, 7.4, 8.2, 25.0, 25.1, 19.4, 15.7, and 6.9.

0.6

0.05

0

0 Qf Wf

FIG. 5. As in Fig. 3, but for spin I = $. Scale factors: 1, 6.7, 16.0, 22.0, 39.6, 32.6, 26.2, and 12.7.

68

-0 0 Orf %f

FIG. 6. As in Fig. 3, but for Z = f. Scale factors: 1, 12.1, 25.2, 19.8, 72.0, 61.6, 52.3, and 25.7.

FIG. 7. Nutation spectra for a spin Z = f as lknction of the asymmetry parameter t (&/Q, = 0.6). The overall appearance of the powder pattern remains the same, but the intensity of some lines decreases with increasing 9.

69

70 KENTGENS ET AL.

and W1/2,1,2h,j = -v-1/2,1/2)j,i i#j

(R-1/2, I&j = 0 i=j.

Substitution in Eq. [7] shows that the signal then becomes

[91

WI, f2) = 2 U- I,z,l,z)i,jsin QijtlexP(-ifizt2). iJ

[lOI

The amplitude of the signal detected during t2 is sine modulated which allows us to obtain pure absorption spectra (9). The presence of an off-resonance term in z1 lowers the symmetry of its matrix and the modulation now becomes a phase-modulation, which cannot be changed to amplitude-modulation by phase cycling. Thus there will be dispersive contributions to the lineshapes which can easily distort powder patterns.

Another effect of off-resonance irradiation is the appearance of a dispersive line at fir = 0. This is due to the magnetization component along the BeE field in the rotating frame which does not evolve during tl .

Both effects of off-resonance irradiation can even be observed for spins with Z = 5 for which the signal in such a case can be written as

S(tl, t2) - [A( 1 - cos QeEtl) + iB sin !&&l]exp(-iQ2t2) [Ill where A = Q,AQ/2(Q$ + AQ2) and B = Qd2w. When A0 = Q - Q0 # 0 then A # 0 and thus a constant and a cos fled1 term is introduced causing, respectively, the Q, = 0 signal and the phase modulation. To avoid complications it is clearly recommendable to irradiate on resonance, i.e., with the excitation frequency Q equal to the Larmor frequency yB0. That “on resonance” in this case does not mean at the frequency of the 1, - 4 transition is shown by the following magic-angle spinning experiment. Here the Larmor frequency lies outside of the quadrupole lineshape (2) and Fig. 8 shows the nutation spectra of NaN02 (e2qQ/h = 1.1 MHz, 9 = 0.1) as a function of the excitation frequency. In this case we are in the extreme situation no b Qti and MAS does not influence the nutation spectrum. It is clear that the line at Qi = 0 increases relative to the line at 2Qfi. The line at Bi = 0 is minimal when we irradiate outside the powder pattern near the Larmor frequency.

Line broadening. An important cause for line broadening in the F, dimension is the inhomogeneity of the rf magnetic field. The effect of rf inhomogeneity cannot simply be described with a Lorentzian broadening exp(-tr/rJ because a spread in !& will also cause a spread in the ratio Qo/s2ti, and will thus change the whole nutation spectrum. Furthermore the lineshapes due to rfinhomogeneity will not be Lorentzian but asymmetric (see Ref. (ZO), Fig. 3). Only in the extreme case Q, $ no, do we get a line at nutation frequency Qti with a linewidth directly determined by the rf inhomogeneity. In the other extreme case Qd 4 no there will be a line at (I + f)&, so the linewidth will be (I + 4)~ times the rf inhomogeneity.

Recycle delay. Another important experimental aspect in the nutation experiment is that one has to ensure that the recycle delay or relaxation delay is long enough for the system to return to equilibrium before the next pulse arrives. If the recycle delay is short with respect to T, then the build up of magnetization along the z axis is incomplete. This will distort the pure sine amplitude modulation of the FID


NaN02

a b 1 F1 %f 0

C

, 5 *%f 0

FIG. 8. MA!3 spectra of NaN02 together with their nut&ion spectra as a function of the excitation frequency fi. (a) Off-resonance irradiation; in the nutation spectrum we see a large line at R, = 0. (b) Irradiation close to the Larmor frequency (outside the powder pattern) gives the best nutation spectrum. (c) With the carrier frequency equal to the average 4, - f transition frequency (in the middle of the powder pattern) the line at ti, = 0 in the nutation spectrum increases again.

(Eq. [ lo]). Fourier transformation of a distorted sin Q I t wave will give a line at frequency Q, plus a number of harmonics at 2%) 3!4, . . . . The number and amplitude of these harmonics depends on the distortion of the sine wave, and they can easily be mistaken for components with a large quadrupole frequency. Figure 9 shows this effect

- F2

FIG. 9. 2D nutation spectrum of ‘Li in Lick where the recycle delay (0.25 s) is short with respect to T, . Li has a small no in Lick so only one line at t& is expected. Because of the short recycle delay, however, there are several harmonics of this frequency present.

72 KENTGENS ET AL,

for the ‘Li nutation spectrum of LiCl. Here Li has a very small quadrupole interaction, but due to the short repetition rate we see not only a line at 52, = Qfi but also at Q, = 2&, 3&, and 4Q+ In fact this experiment is proposed as a method to determine Q,.r in solution, for nuclei with long T, and low natural abundance (II).

EXPERIMENTAL

NaN02, LiCl, and SCZ(SO& were all commercially available chemicals. NaN02 and spodumene spectra were recorded on a Bruker CXP-300 with, respectively, 64 and 128 tl increments of 2 ~.ls. A standard Bruker probe with an rf field of 36 kHz was employed. LiCl and Sc#O.& spectra were recorded on a Bruker WM-500 (with 256 t, increments of 2 and 1.5 ps, respectively). Here a specially constructed probe equipped with a 6 X 12 mm solenoid perpendicular to BO, operating with an rf field strength up to 70 kHz, was used.

SPODUMENE

J‘ _ ~-_- - / L-.. .c d.--L .-,., --r--r- L +------- .~_

, I I 0 Yf 150

- KHz

FIG. 10. (a) 27AI spectrum of the central transition of spodumene recorded at 78.2 MHz. (b) F, projection of the 2D nutation spectrum of spodumene recorded on a Bruker CXP-300 (128 t, increments of 2 ps and Q, = 36 kHz). (c) Simulated spodumene spectrum using the quadrupole parameters e*qQlh = 2.95 MHz and 7 = 0.94 (12) and a Lorentzian broadening of 2.5 kHz.


Sc*(SO&

SC 1=7/2

200 khz

-F 2

FIG. 11. 2D nutation spectrum of 45Sc in SC&SO& recorded on a Bruker WM 500, together with its F, projection (256 tl increments of I .5 ps with a 70 kHz rf field).

RESULTS

To demonstrate the effectiveness of the nutation experiment, Fig. 10a displays the NMR spectrum of 27Al in powdered spodumene recorded on a Bruker CXP-300 at 78.2 MHz. The spectrum consists of one featureless line 5 kHz wide. It is clear that no accurate quadrupole interaction parameters can be extracted from this spectrum. In addition, MAS does not solve the problem; again we see a rather featureless line, 1.5 kHz wide. In Ref. (2~) it is shown that this MAS spectrum can be reproduced theoretically (using a large line broadening) with the known quadrupole parameters (e*qQ/h = 2.95 MHz, 1 = 0.94 (Z2)), but it will be 1 c ear that it is almost impossible to do so without any preknowledge of the quadrupole parameters. The result of the 2D nutation experiment, however, appears to be a well structured pattern. From the F1 projection of this pattern (Fig. lob) we can estimate the magnitude of the quadrupole parameters e*qQ and rl using our set of calculated spectra. It appears that flo/Q,.r - 1 with an rf field strength of 36 kHz; this means that e*qQ/h - 3 + 0.5 MHz, and that I) must be between 0.8 and 1.

Another example is the 45Sc (I = $) nutation spectrum of Scz(SO& (Fig. 11). The spectrum of the central transition measured at 12 1.5 MHz is 2.6 kHz wide and shows no structure, MAS narrows the spectrum to 600 Hz. The MAS spectrum recorded at 43.8 MHz does show some structure from which it becomes clear that there must be sites with different quadrupole parameters but the same chemical shift. The 2D nutation spectrum is well structured (Fig. 11). Comparing its projection to our set of calculated spectra reveals that the major constituent has a e*qQ/h - 2 f. 0.5 MHz with a low

74 KENTGENS ET AL.

asymmetry parameter (7 between 0.1 and 0.4). P. P. Man recently also showed an intermediate case for Mn in KMn04 (8~).

CONCLUSIONS

The 2D nutation method appears to be very useful for the determination of quadrupole interaction parameters of half-integer quadrupolar nuclei in a field gradient, especially when these parameters cannot be determined from MAS experiments. The method combines the sensitivity of high-field measurements with the information one gets from experiments at low or zero field. The spectra can easily be simulated with a straightforward density matrix calculation. To get high sensitivity and maximum resolution in the F2 dimension, which is advantageous if nuclei with different chemical shifts are present, one preferably performs the experiment in the highest available magnetic field. Because there is no obvious need for MAS the method will also be very suited for high-temperature studies (e.g., of zeolites).

ACKNOWLEDGEMENTS

Mr. P. van Dael and Dr. C. Haasnoot, of the SON hf-NMR facility at the University of Nijmegen, are thanked for their support with the experiments on the 500 MHz spectrometer. We thank Mr. J. W. M. van OS and Mr. P. van Dijk for their technical assistance and Prof. dr. E. de Boer for critically reading the manuscript. Dr. B. de Jong kindly supplied the spodumene sample. This work was carried out under the auspices of the Netherlands Foundation of Chemical Research (SON) and with the aid of the Netherlands Organization for the Advancement of Pure Research (ZWO).

REFERENCES

1. (a) K. NARITA, J.-I. UMEDA, AND H. KUSUMOTO, J. Chem. Phys. 44,2719 (1966); (b) J. B. BAUGHER, P. C. TAYLOR, T. OYA, AND P. J. BRAY, J. Chem. Phys. 50,4914 (1969).

2. (a) E. KIJNDLA, A. SAMOSON, AND E. LIPPMAA, Chem. Phys. Left. 83,229 (198 1); (6) H.-J. BEHRENS AND B. SCHNABEL, Physica B 114, 185 (1982); (c) A. SAMOSON, E. KUNDLA, AND E. LIPPMAA, J. Mugn. Reson. 49,350 (1982).

3. (a) A. SAMOS~N AND E. LIPPMAA, Chem. Phys. Lett. 100,205 (1983); (b) A. SAMOSON AND E. LIPPMAA, Phys. Rev. B 28,6567 (1983).

4. C. S. YANNONI AND R. D. KENDRICK, .I. Chem. Phys. 74,747 (1981). 5. D. B. ZAX, A. BIELECKI, K. W. ZILM, A. PINES, AND D. P. WEITEKAMP, .I. Chem. Phys. 83, 4877

(1985), and references therein. 6. F. M. M. GEURTS, A. P. M. KENTGENS, AND W. S. VEEMAN, Chem. Phys. Lett. 120,2 (1985). 7. A. ABRAGAM, “Principles of Nuclear Magnetism,” Oxford Univ. Press, Oxford, 196 1. 8. (a) A. T~ONIKER, P. P. MAN, H. THEVENEAU AND P. PAPON, Solid State Commun. 55, 929 (1985);

(b) P. P. MAN, H. THEVENEAU, AND P. PAPON, J. Mugn. Reson. 64, 27 I (1985); (c) P. P. MAN, J. Mugn. Reson. 67,78 (1985).

9. A. BAX, “Two Dimensional Nuclear Magnetic Resonance in Liquids,” Reidel, Dordrecht, 1982. 10. D. HORNE, R. D. KENDRICK, AND C. S. YANNONI, J. Mugn. Reson. 52,299 (1983). II. J. R. WEISENER AND H. GUNTHER, J. Mugn. Reson. 62, 158 (1985). 12. H. E. F%TCH, N. G. GRANNA, AND G. N. VOLKOFF, Can. J. Phys. 31,837 (1953).

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Journ of Magn Res 72, 62-74 (1987)