Nuclear dipolar energy and relaxation in (NH4)2GeF6

JOURNAL OF MAGNETIC RESONANCE 31,377--386 (1978)

Nuclear Dipolar Energy and Relaxation in (NH4)2GeF 6 M . PUNKKINEN AND

L.

(~STERBERG

Wihuri Physical Laboratory and Department of Physical Sciences, University of Turku, SF-20500 Turku 50, Finland Received September 12, 1977; revised January 8, 1978 The nuclear dipolar energy and its relaxation in polycrystalline (NH4)2GeF 6 are studied at the proton and fluorine resonance frequencies of 33 and 38 M H z in the temperature range 4.2 to 300 K. The relaxation time Ttv is observed to have two minima at temperatures below 40 K. The most effective pulse sequence for transferring Zeeman energy into the dipolar reservoir in the presence of two kinds of nuclei is shown to be 90°-t-090 o with 0 > 45 ° and t of the order of the inverse linewidth. The initial rapid decrease of the dipolar energy is used for determining the time constant characterizing the approach of the dipolar reservoir toward internal equilibrium. INTRODUCTION

Experiments on the relaxation of the nuclear dipolar energy may reveal slow motions which cannot be detected by observing the Zeeman relaxation. Detailed information on motions is especially interesting for samples containing light symmetric molecular groups and having low barriers for molecular reorientations. This importance originates in the lack of a satisfactory theory for the spin-lattice relaxation and resonance absorption lineshape in terms of motional parameters. There is some evidence suggesting that the spin-lattice relaxation time of the nuclear dipolar energy, T1D , in (NH4)2SnCI 6 (1) and (NH4)2PbC16 (2) has two minima at temperatures below 40 K. These substances are both known to have rather large ammonium group tunneling frequencies (3, 4). Previous experimental data on the proton Zeeman relaxation in (NH4)2GeF 6 (5) lead to a rough estimate of 10 MHz for the tunneling splitting. Therefore, TIp can be expected to have two minima at low temperatures. This assumption is proved true by the present study on polycrystalline (NH4)2GeF 6. Hydrogen and fluorine nuclei have nearly equal magnetic moments. Hence the dipolar energy content of the secular time-independent parts of the proton-proton, fluorine-fluorine and proton-fluorine magnetic dipolar interactions, ~U~pp, ;U~F F and ~ v v , respectively, are comparable to each other. For a sample containing only one kind of magnetic nuclei the most effective transfer of the Zeeman energy into the dipolar energy reservoir is obtained by the pulse sequence 90°-t-45g0 o, where t is of the order of the inverse linewidth (6). When two kinds of magnetic nuclei are present, the maximum of the dipolar energy is shown to result from the pulse sequence 90°-t-09o o, with 0 > 45 °. If, for example, the Zeeman energy of protons in (NH4)2GeF 6 is transferred into the dipolar reservoir, the dipolar energy immediately after the 090° pulse resides in ~,~va and JF'~p F but not in JT~FF . It is only after a certain mixing time that 377

0022 2364/78/0313-0377502.00/0 Copyright@ 1978by AcademicPress, Inc. All rightsof reproductionin any form reserved. Printed in Great Britain

378

PUNKKINEN

AND

OSTERBERG

the energy reservoir reaches internal equilibrium and can be described by a spin temperature. The amount of the dipolar energy can be observed by applying another 090° pulse at a time interval t I from the first 090° pulse. The smaller the degree of mixing, the larger the amplitude of the dipolar signal. The mixing, or equilibration, process is studied by following the dipolar signal amplitude as a function of the interval t 1. To prevent relaxation from causing complications, the length of the pulse sequence is kept considerably shorter than Tlo. EXPERIMENTAL

The polycrystalline sample of (NH4)2GeF 6 was provided by Research Organic/Inorganic Chemical Corporation. It was sealed in a glass test tube. The temperature of the sample was controlled and measured by calibrated germanium 10

~

,

,

,

60

80

I.U Z in e~ -J

O

1 0

20

40

O0

t~ (msl

Fro. 1. The spin-lattice relaxation of the dipolar energy in polycrystalline (NH4)2GeF6 at 77 K as observed by the pulse sequence 90°-13 #sec-609~oo-t~-609*0o. The straight line represents the leastsquares fit to the data for t~ > 2 msec. The spectrometerwas operated at the fluorine resonance of 38 MHz. resistors. The experiments were carried out with a Bruker pulsed nuclear spectrometer SXP 4-100 at the proton and fluorine resonance frequencies of 33.0 and 3 8.0 MHz. The spin-lattice relaxation of the nuclear dipolar energy was studied by the pulse o o sequence 90 o -t-6090o-t1-6090o. It gave the largest signal at both the proton (t = 17 #sec) and fluorine (t = 13 #sec) resonances. The signal amplitude appeared to be a rather nonexponential function of t I (Cf. Fig. 1). The initial rapid decrease was over in 2 msec independent of temperature for T > 40 K. Therefore, the decrease cannot be related to temperature-dependent or time-dependent interactions but is believed to be associated with the equilibration of the dipolar energy reservoir. The t I dependence of the dipolar signal for t I > 2 msec was exponential at temperatures above 40 K. The T m data in Fig. 2 were determined from the exponential part. The shape of the proton free-induction signal was observed to change between 30 and 40 K. The change is probably related to the N H 4 reorientations, which become slow in comparison with the linewidth. Simultaneously with the lineshape transition the exponential part of the relaxation curve disappeared. In addition, the initial decrease during the first 2 msec following the pulse sequence was observed to vary with

R E L A X A T I O N 1N A M M O N I U M F L U O G E R M A N A T E

379

temperature for T < 40 K. In these experiments the proton Zeeman energy was transferred partly into the dipolar energies ~ p p and JUmpr. Because the intraammonium dipolar interaction of the protons is the dominant energy reservoir below the linewidth-transition temperature (or ~¢g'bpp (intra) > JP~Pr, JY~rF), the equilibration of the dipolar reservoir is not believed to have much effect on the initial behavior of the dipolar signal. Therefore, the initial decrease for T < 30 K should characterize the relaxation rather than the equilibration. The experimental T m values for T < 40 K in Fig. 2 are the times t I in which the signal amplitude decreased to the eth part. The equilibration of the dipolar reservoir causes some uncertainty in the values, especially in the linewidth-transition region. J

r

o.:. 1000 0 t I

@

o

100

E ,,..,,

0

¢

10

¢=

J

0

[]

%

i

0

i

I

100

i

200

i

i

300

T(K)

FIG. 2. The T m of polycrystalline (NH4)zGeF 6 against temperature at the proton ( 0 , [3) and fluorine (O) resonances of 33 MHz. The data ([3) for T < 40 K are 1/e values.

To study the transfer of the Zeeman energy into the dipolar reservoir, the pulse sequence 90°-t-090o-3 msec-60%° was used. Here 0 means a variable pulse length. The time t 1 = 3 msec was chosen to ensure that the dipolar energy reservoir was in internal equilibrium. The results of one proton-resonance run at the temperature of liquid nitrogen are shown in Fig. 3. The role of the initial decrease of the dipolar energy was studied at 77 K by observing the signal amplitudes S(0.2)e and S(3.0)e resulting from the pulse sequences 9 0 ° - t 090°-0.2 msec-60goo and 90°-t-09o°-3 msec-60g0o , respectively. The latter amplitude was extrapolated back to tl = 0.2 msec through multiplication: S(0.2)c = exp (2.8 msec/Tm) S(3.0)e. If the dipolar reservoir were in internal equilibrium at the instant t I = 0.2 msec, the experimental amplitude S(0.2)e and the extrapolated amplitude S(0.2) c should be equal. Because they differ (cf. Fig. 4) the dipolar reservoir does not reach equilibrium until t~ > 2 msec.

380

PUNKKINEN AND OSTERBERG 80

"~

r

I

60

E

=-

40

L,.

-~ 6

20

,,,

0

o

o o

°

o

I-3 - 2 o o

'~

-40

--I

,,¢ z

- 60

(/I -

80

i

i

i

90

0

L

i

180

PULSE

i

i

270

360

LENGTH

(degrees

of

arc)

FIG. 3. The dipolar signal amplitude of polycrystalline (NH4)zGeF 6 at 77 K after the pulse sequence 9 0 ° - 1 7 #sec-090o-3 msec-60goo, as a function of the pulse length 0. The solid curve represents Eq. [6] with h = 59.1 and f = 46.1. The spectrometer was operated at the proton resonance of 38 MHz.

30

o I

o

• o

o

20 ©

6 © 0 0 i-

10

0

I

0

I

I

30 PULSE

I

60

I

I

90

20

LENGTH

( d e g r e e s of arc)

FIG. 4. The relative difference between the experimental (t~ = 0.2 msec) and extrapolated (4 = 3 msec) dipolar signal amplitudes of polycrystalline (NH4)zGeF~ at 77 K against the pulse length 0 in the pulse sequence 90°-t-Ogoo-tl-609~oo: Proton resonance (@); fluorine resonance (O)-

RELAXATION IN AMMONIUM FLUOGERMANATE

381

ANALYSIS OF THE T~D DATA The experimental results for T1D obtained by applying 90°-t-609°oo-ta-609°0 o pulse sequences at the proton and fluorine resonances agree with each other. This can be expected on theoretical grounds and has also been verified experimentally (7, 8). At temperatures above 200 K the relaxation time T m decreases with increasing temperature. This behavior cannot be related to the NH 4 reorientations because the proton-resonance line narrows, owing to the ammonium-group motion around 35 K, and therefore the corresponding contribution to 1~TaDshould decrease with increasing temperature. The reorientations of the GeF 6 groups start to shorten the fluorine Zeeman relaxation time in (ND4)2GeF 6 above 200 K (9). It seems logical, therefore, to assume that the GeF 6 motion is responsible for the shortening of T1D in (NH4)2GeF 6. Additional evidence comes from the magnitudes of the corresponding activation energies. The dipolar energy resides mainly in the intragroup part of ~"~FF and in ~7¢~vv and to a smaller extent in the intergroup part of ~ e v . The application of the slow-motion theory by Slichter and Ailion (10) gives the result TaD = k~"v. Here k is a constant close to unity and r F is the correlation time for the GeF 6 reorientations. When the correlation time is assumed to obey the Arrhenius law rv = rF0 exp (A/R T), the TaD data can be used to determine the activation energy A. The Tip values for T > 250 K are consistent with the activation energy 52 kJ/mole, which is inside the error limits of the previous value 56 _+ 7 k J/mole based on the fluorine Zeeman relaxation in (ND4)zGeF 6 (9). In the temperature range 40 to 170 K the dipolar energy resides in the same parts of the dipolar Hamiltonian as those mentioned above. However, its relaxation is most likely determined by the reorientations of the ammonium ions. This argument is consistent with interpretation of the proton, deuteron, and fluorine Zeeman relaxations in (NH4)2GeF 6 (5) and (ND4)~GeF 6 (9). Because the equilibrium orientations of the ammonium ions are not known~ a detailed evaluation of 1/TID from the general theory (11) is nearly impossible. The theory has been worked out, for example, for ammonium halides, where the magnetic moments of the other nuclei are much smaller than the proton moment (12). Although such derivations are not exactly true in the present case of two kinds of nuclei with comparable magnetic moments, their qualitative features should be valid. Therefore T1D should be inversely proportional to zv when covrv < 1 and covrv < 1 (or when T > 60 K). Here r v is the N H 4 correlation time and cov and cov are the proton and fluorine angular resonance frequencies. The application of the Arrhenius equation yields values for the activation energy which are in satisfactory agreement with the result 5.4 k J/mole of the previous Zeeman relaxation study (5). At T = 50 K there is a shoulder in the curve of TID against T. It arises from the fact that those terms of the dipolar Hamiltonian causing the Zeeman relaxation become ineffective in relaxing the dipolar energy below 50 K. Therefore TID is dominated by the time-dependent secular part of the dipolar interaction. Similar shoulders have been observed, for example, in the T1D curves of (NH4)zPbC16 (2) and NH4C1 and NH4Br (12), and they appear always at the same temperature with the minimum of the Zeeman relaxation time. A more detailed explanation of the effect is given in Refs. (2, 12). It should be noted that the relaxation is determined by the NH 4 reorientations in the entire temperature range 40 to 170 K. The shoulder around 50 K originates from the fact that the relaxing power of the various parts of the dipolar Hamiltonian vary differently with the frequency of the reorientations.

382

PUNKKINEN AND <3STERBERG

Around 35 K the N H 4 reorientations become at least partly slow in comparison with the linewidth. Therefore, the shape of the free-induction signal changes and T1D has a minimum. At still lower temperatures the intragroup part of dgC'~ppmust also be taken into consideration. Consequently, a shorter interval t is needed in the pulse sequence to maximize the dipolar signal. The nonexponentiality of the relaxation below 40 K is probably a result of many contributions, including the anisotropy of the time-dependent and time-independent parts of ~ p p , the equilibration of the dipolar reservoir, and possibly also tunneling. The effect of the N H 4 tunneling on Tt~ of (NH+)zPbC16 was considered in Ref. (2). The intraammonium secular Hamiltonian ~,~¢'~pp(intra) can be divided into three kinds of terms or A, E, and T terms (13), where the symbols A, E, and T refer to the irreducible representations of the tetrahedral point group. The A-type term is totally symmetric and it is not affected by the N H 4 reorientations or tunneling. Actually this term vanishes for a perfect proton tetrahedron. The E-type terms are modulated by the 120 ° reorientations about the threefold axes at the rate 3k3 = 1/r' and the T-type terms are modulated by both the 120 and 180 ° reorientations at the rate k 2 + k 3 = 1/r. In analogy to (NH4)2PbC16 the proton linewidth transition in (NH+)EGeF 6 around 35 K is believed to arise from the fact that the time variation of the E-type terms becomes slow compared with the linewidth. The E-type terms also contribute to the dipolar energy reservoir below 35 K and, as mentioned above, constitute the major part of it. The Ttype terms do not contribute to the dipolar energy if the 180 ° reorientations are fast or the tunneling splittings between the T-species levels of the librational ground state are large in comparison with the linewidth (2, 13, 14). When the slow-motion theory (10) is applied on the E-type terms one obtains the approximate result TlO -- z'. This equation should be valid for 30 K > T > 15 K. (We have introduced three different correlation times for the N H 4 reorientations: zp, r', and r. If k 3 > k z the times r' and r are roughly equal and one can write rp ~_ ~' ~_ r. Otherwise if k 3 < k 2 the quantity rp should be chosen equal to the correlation time of that motion which dominates the relaxation.) Another interesting fact is the decrease of T~o with temperature for T < 10 K. It shows that the correlation time of some other motion is lengthening toward the inverse linewidth of the proton or the fluorine resonance curve. TtD is likely to have the corresponding minimum at some temperature below 4.2 K. This minimum should be accompanied by changes in the lineshape and the second moment. The NH+ groups in (NH4)2GeF 6 are located at threefold symmetry axes (15). So it is possible that there are nonequivalent axes for reorientations giving rise to the secondary minimum claimed. For example, the reorientations might occur more frequently about one of the threefold axes than about the others, thus producing a decrease of T~D with temperatures below 10 K. However, even in a perfectly tetrahedral crystalline environment there are the 120 and 180 ° reorientations of the N H 4 group, which may take place at different rates. It was proposed above that the 120 ° reorientations cause the T~D minimum at 35 K. Thus the 180 ° reorientations and consequently the time variation of the T-type terms of ~U~pp(intra) may still be fast in comparison with the proton linewidth at 10 K. When the corresponding correlation time r becomes longer the product Acoz approaches unity, making T~o decrease.

RELAXATION IN AMMONIUM FLUOGERMANATE

383

TRANSFER OF THE ZEEMAN ENERGY INTO THE DIPOLAR RESERVOIR A sample containing two kinds of magnetic nuclei is considered. In the present study these nuclei are protons and fluorines, denoted by p,p',p", ... and f,f',f", ..., respectively. The derivation proceeds along the same lines as the treatment for identical nuclei by Goldman (16). The two-pulse sequence 90°-t-090 o is applied on the spin system at the proton resonance, the first pulse along the x azis and the second along the y axis of the rotating frame. When the state of the spin system is considered some time later, which is long enough for reaching internal equilibrium but short in comparison with the spin-lattice relaxation times T a and TaD, the density matrix will be f Here crF is the inverse Zeeman spin temperature of the fluorines, fl is the inverse dipolar spin temperature, and ~JUD = ~'DPP + ~DPF + ~"DrF lS the secular time-independent dipolar Hamiltonian. The term/~ is obtained from the expression r

fl

~

flLh Tr(~D2 ~ Tr

i

¢

!

"

{ dU~exp \(-iO ~p Ipy) exp(-i~Yg'~t/h)

Equation [1] is similar to Eq. (2.53) of Ref. (16) with the exceptions that JY~ contains the three contributions mentioned above and that there is also the t e r m 03F~flfz corresponding to the nonresonant nuclei. It can be shown that the nonresonant nuclei have no effect through this term. The evaluation of Eq. [1] is facilitated by the identity

exp(iO ~p lpY)(~'DPP+2~PF+~'U~rr)exp( -it? ~ Ip'y J~'~DFF--~ ' + COS 0S'~;~F' -- sin 0

~

apfIfzIpx + ½(3 cos z 0--

1)~Dp P'

P,f

~pp,,(Ip+Ip,,++ I~Ip,,_)

+ sin 2 0 ~ p
-

sin 0 cos 0 Z

~pp,,[Ip~(Ip,,++ Ip,,_) + (Ip++ I~)Ip,,z],

p
where, for example,

apf

=

YP?vh--~2(1 y3pf

3 cos 2 0~).

The above equation and the fact that traces are independent of the cyclic permutation of operators lead to the expression

flLh°gv

(--sin0

/ / - Tr ('~'~2)

~ a,f Tr[Ipxlf~ exp(-i~t/h ) ~Ip,y exp(i~t/h)] p,f

t

p'

¢-

~app,Trl[Ipz(Ip, ++ Ip,_) + (Ip++ [ x exp(--i~/"Dt/h)~p,, Ip,,,,exp(iJS~t/h)]}. sin

0cos 0

p
[2]

384

P U N K K I N E N AND (JSTERBERG

Because ,~DPP, ~UDPF, nd ~DFF all commute with each other one has

Tr{ IpxIiz eXp(--i~'Dt/h)

~ Ip,y exp(i~'~'Dt/h)l p~

= Tr ( [fz exp (i~p

v t/h) Ipx exp (--i~pF t/h)

× exp (-iJF~rpt/h)

Z Ip,~ exp (i,~'Dppt/h)} p~

=---Tr{Ifzsin(~%s,[s,zt/h)Ivy x exp (--i~'Dppt/h)

Z Ify exp(i~'Dret/h)}.

[3]

p~

Similarly,

~ ~app,,Tr { [Ipz(Ip,,++ Ip,, ) + (Ip++ I~,)I¢,~] exp (-i~'~ t/h) ~ Ip,, exp(i~"ot/h) } pH

pr

f ,P'~

k

f

x exp (--iS~'~rp t/h) Ip,,y exp (i~<'~ppt/h) }

=--XTr{cos(~ap,~Ii~t/h)}Tr{i[Ip,,~'~pp] t/h) Iv,, exp (i~'Dr pt/h)}

x exp ( - - i ~ p v = h X Tr cos p~

%,Js~t/h ~Tr{exp(iS-/';prt/h)Ip, exp(-i~;'~';ppt/h)Iv,,I.

[4]

In the derivation of Eqs. [2] to [41 all the vanishing terms were dropped. By combining these expressions one obtains fl - T~ r ( ~ 2

sin 0 Z

%fTr Ii~ sin Xap1,If,zt/h

f

x Tr[ Ipyexp(-i~f;eet/h)

f'

P'
-hsinOcosOTr[cos(~apsIf~t/h)] x ~ Tr Ip, exp (igf'Dpet/h)

~ Iv,, exp (--i~e~;prt/h) . p'
[5]

R E L A X A T I O N IN A M M O N I U M F L U O G E R M A N A T E

385

Broekaert and Jeener (17) considered a similar problem and concluded that the inverse spin temperature after the two-pulse sequence mentioned above is

fl = h(t) sin 0 + f(t) sin 0 cos 0.

[6]

Equation [51 agrees with their result. In principle, h(O and f(t) can be calculated from Eq. [5]. In practice, however, such a calculation meets the same difficulties as the solution of the lineshape problem (18). It is worth noting that when ,~'DPF vanishes or apl = 0, Eq. [5] reduces to fl =

T r ( ' - ~ z) sin 0cos 0 - - T r

dt

Ipy exp (tPFDp P /h)l Z Ip,y e x p ( - i f f ; p p t/h) ,

p,

which is the familiar result for identical spins (6, 16). The experimental data of Fig. 3 agree qualitatively with Eqs. [5] and [6]. The maximum of the dipolar energy is seen to result from the pulse sequence 90°-t-090 o, with 0 approximately equal to 60 ° at both the proton and fluorine resonances. The solid curve in Fig. 3 represents Eq. [61 with h and f determined by the least-squares fit to the data for 0 between 0 and 180 °. The deviation of the solid curve from the experimental points for 0 > 180 ° is believed to arise from inhomogeneities in the static and rf magnetic fields. A P P R O A C H TO I N T E R N A L E Q U I L I B R I U M

When the rf pulses resonate with the protons the dipolar energy just after the sequence 90°-t-09o o resides in ¢?¢~Pv and ~ p p . The energy contents of these terms correspond to the first and second terms of Eq. [5], respectively. In internal equilibrium a considerable amount of the dipolar energy is also stored in ~ v v . Hence the energy distribution cannot correspond to internal equilibrium nor can the concept of the dipolar spin temperature be used. When the amount of the dipolar energy is observed by applying another proton-resonance pulse at an interval t I from the two-pulse sequence, the dipolar signal results from the energy stored in ~)PF and ~Zf~pp according to a derivation similar to that above. Therefore, when t 1 is so short that there is practically no transfer of the dipolar energy, to ~ ' DFF~ the dipolar signal should be larger than that for a long t~ corresponding to a nonvanishing energy transfer. The results in Fig. 4 are calculated from the signals for two different tl values. The interval t~ = 200 #sec corresponds roughly to the shortest time permissible to avoid oscillations in the signal amplitude [6]. The other time tj = 3 msec was chosen to make the subsequent relaxation exponential. The exponentiality is believed to mean that the dipolar reservoir is in internal equilibrium. The largest percentage difference between the observed and extrapolated signals, S(0.2) e and S(0.2)e, is obtained at 0 ~_ 45 ° for the proton resonance (cf. Fig. 4). The fluorine resonance data do not show such a clear maximum. At each resonance the difference decreases with increasing 0 for 0 > 45 °. This behavior can be qualitatively understood. For a relatively short 090° pulse at the proton resonance the energy contents of both ~ ' ~ p p and ~ p r are larger than the energy content in internal equilibrium and hence there must be energy transfer from both to ~ F F ' Because it is reasonable to assume that the transfer ,2/~p e --, ~"~vF is slower (probably a four-spin process) than ~"~pF --' ~W~,FF (a two-spin process) the

386

PUNKKINEN AND (JSTERBERG

approach toward internal equilibrium is slow and the percentage difference large. When 0 exceeds 45 ° the factor sin 0 makes the energy content of ~'~vr the largest. Therefore, it is mainly the fast transfer ~DVV ~ ~7"¢'DFEwhich is needed for reaching internal equilibrium, and thus the percentage difference is small. The rapid decrease of the dipolar signal was always complete in 2 msec independently of temperature for T > 40 K. This means a characteristic time of roughly 0.5 msec describing the approach toward internal equilibrium. Equilibration times of the same magnitude were recently observed for a LiF single crystal (17). Such values may seem somewhat long compared with the inverse linewidth 1~Ace of the relevant absorption curves. However, one should bear in mind that the transfer of the dipolar energy from ~ ) v v to ~W~vv is a four-spin process provided the Zeeman energy is conserved. Consequently, it corresponds to the second order in the perturbation expansion and thus explains qualitatively the long time constant. One consequence of such a long equilibration time is that the experimental TID values of the order of 1 msec or shorter may not be reliable because both the approach toward internal equilibrium and the spin-lattice relaxation occur at comparable speeds. Note added in proof. It was erroneously claimed in the derivation of Eq. [3] that ~,W~vv,~Td'~vz and f f ~ r v all commute with each other. Though ~W~vv and ~W~Fv really commute, they do not commute with ~W~pF. Therefore Eqs. [3-5] are only approximately valid. ACKNOWLEDGMENTS We thank Professor V. Hovi for the facilities of the Wihuri Physical Laboratory. The help of Mr. E. Ylinen, M.Sc., and Mr. J. Heinilii during the experiments and in the analysis of the results is gratefully acknowledged. REFERENCES 1. Z. T. LALOWICZ,C. A. McDoWELL, AND P. RAGHUNATHAN,in "Proceedings of the XIXth Congress Ampere, Heidelberg, 1976," (H. Brunner, K. H. Hausser, and D. Sehweitzer, Eds.), p. 574. 2. M. PUNKKINEN,J. E. TUOHI, AND E. E. YLINEN,Phys. Lett. A 54, 133 (1975). 3. M. PRAGER,W. PRESS, B. ALEEELD,AND A. HOLLER,J. Chem. Phys. 67, 5126 (1977). 4. M. PUNKNINEN,J. E. TUOHI, AND E. E. YLINEN, Chem. Phys. Lett. 36, 393 (1975). 5. E. E. YLINEN, J. E. TUOHI, AND L. K. E. NIEMEL/i, Chem. Phys. Lett. 24, 447 (1974). 6. J. JEENERAND P. BROEKAERT,Phys. Rev. 157, 232 (1967). 7. M. E. ZIaABOTINSKII,A. E. MEEED, AND M. I. RODAK, Soy. Phys. JETP 34, 1020 (1972). 8. H. T- STOKESAND D. C. AILION, in "Proceedings of the XIXth Congress Ampere, Heidelberg, 1976" (H. Brunner, K. H. Hausser, and D. Schweitzer, Eds.), p. 433. 9. J. E_ Tuom, E. E. YLINEN, AND L. K. E. NIEMEL~,,Chem. Phys. Lett. 28, 35 (1974). i0. C. P. SLICHTERAND D. C. AILION,Phys. Rev. A 135, 1099 (1964). 11. J_ JEENER, in "Advances in Magnetic Resonance" (J. S. Waugh, Ed.), Vol. 3 p. 206, Academic Press, New York, 1968. 12. M. PUNKKINEN, in "Proceedings of the XVIIth Congress Ampere" (V. Hovi, Ed.), p. 219, North-Holland, Amsterdam, 1973; J. Nonmetals 2, 79 (1974). 13. M. PUNKKINEN,J. E. TUOHI, AND E, E. YLINEN, or. Magn. Reson. 22, 527 (1976). 14. Z. T. LALOWlCZ,C. A. McDoWELL, AND P. RAGHUNATHAN;J. Chem. Phys. 68, 852 (1978). 15. J- L. HOARD AND W. B. VINCENT, or. Am. Chem. Soe. 61, 2849 (1939). 16. M. GOLDMAN, "Spin Temperature and Nuclear Magnetic Resonance in Solids," p. 36, Oxford Univ. Press, New York/London, 1970_ I7. P. BROEKAERTAND J. JEENER,Phy& Rev. a 15, 4168 (1977). 18. M. LEE, D. TSE, W. I. GOLDBURG,AND 1. J. LOWE,Phys. Rev. 158, 246 (1967).