Quantum Diffusion in Solid Helium

by

Andre Landesman

Commissariat a L'Energie AtomiqueService de Physique

du Solide et de Resonance MagnetiqueOrme des Merisiers

France - 91190 GIF SUR YVETTE

I I .

Introducti on

Vacancies in b.c.c. 3HeA. NMR Observation of VacanciesB. Various Channels for Vacancy MotionC. Discussion

Isotopic Impurities of 3He in h.c.p. 4He-3HeA. The Impuriton GasB. The Experimental Behavior of the Diffusion ConstantC. The Dense Fluid of ImpuritiesD. A Direct Determination of the 3He-4He Transition Frequency

References

Page105

106108109110

1101 111 12114

116

I . Introduction

We are interested in the motion of aparticle in a solid such that: j_) itsprobability density obeys a diffusionequation, and J_i_) the transfer fromsite to site is not a thermally acti-vated jump over some barrier but rathera quantum tunneling through that bar-rier. We shall call such a motion"quantum diffusion". Examples of it insolid helium will be shown. Obviouslyhelium is a good atom for such studiesbecause of its mass. Other particlesconvenient for quantum diffusion stud-ies include positive muon in an iron oraluminum lattice and solid mixtures of

"Presented at the 7th Waterloo,Summer School, June 8-13, 1981.

NMR

hydrogen species.A few facts about solid helium are

as follows. Helium behaves quantummechanically from the point of view ofenergetics and statistics. The firstone is more important. As shown inFigure 1, at moderate pressures, a liq-uid phase of helium exists to T = 0 K.The very existence of a liquid at O Kis a macroscopic quantum effect. Thequantum nature of this system is alsodemonstrated in the cohesive energy peratom, Eo, which is well known fromexperimental thermodynamics. We canwrite Eo as the sum of the averages ofthe kinetic energy (KE) and of thepotential energy (PE),

= KE + PE (1)

A very rough evaluation of the kineticenergy of helium in terms of Debye

Vol . it, No. 3 A 105

PRESSION (atmospheres)

HELIUM TROIS

28.9 atm.

3 / IWK

135.93 atm.

Liquide

- Tin = 0,32 K

J I I I I 1 I

HELIUM aUATRE

Solide hep

25- —

Tm = 0,78 KTEMPERATURE (K)

I I I I L

Solide cc.

Liquide

I 1 I L

200

10080

60

40

20

Figure 1. The pressure temperature phase diagram for both isotopes of helium.

temperature 8$ in a system of unitswhere kg = 1 is:

KE (9/16) (2)

We proceed to evaluate the potentialenergy for three rare gases, as shownon Table 1, by relating it to Eo and6n . The kinetic energy of helium iscomparable to |PE| while for a classi-cal system such as Xe, the kineticenergy is much smaller than the poten-tial energy (in absolute value).

It is also instructive to evaluatethe root mean square deviation o of anatom, again in the Debye approximation(which is a very crude approximationfor helium). As is seen in Table 1,this deviation is about ten times aslarge in helium than in the "extremeclassical" rare gas xenon.

Solid helium thus appears as a sys-tem with very large zero point motions.Not surprisingly atoms which are asbadly localized as they are in helium

are very likely to tunnel. In pure 3Hethe Fermi statistics of the atoms,which leads to the effective exchangeHami1tonian necessary to describe thevery low temperature magnetic proper-ties of pure solid 3He has to be con-sidered also (1,2,3). Presently we areonly interested in tunneling of dilutepoint defects. These are either thevacancies in pure 3He or the isotopicsubstitutiona1 impurities of 3He in*He-3He mixtures. The statistics arenot going to play any role.

I I Vacancies in b.c.c. 3He

In solid 3He Schottky type defectsof concentration xv can be thermallyexcited. The experimental proof oftheir existence is obtained by X-raydiffraction on crystals held at con-stant volume (h). As a function oftemperature T the variation Aa of theinteratomic distance is related to thevariation Axv of the concentration:

106 Bulletin of Magnetic Resonance

Table 1

Rare gas Cohesive Kinetic energy Potential energy Relative r.m.s. derivationenergy

9 §. = 1Eo KE - 75 UD fc a 2

[JL

3He(bcc) -O.k 11 -11.4

••Hedicp) - 5 . 6 15 -20 .6 28%

Xe(fcc) -1928 31 -1959 2.6$

All energies are in K per atom. Helium is considered at the minimum of the melting curve and lattice structure isgiven in parenthesis. The root mean square deviation 6 is evaluated in the Debye approximation as a function of theatomic mass, the Debye frequency fiR = 8D/n and the interatomic distance a.

oz

o

Aa/a + (X/3) Axv = 0 (3)

This is a well known method for study-

ing vacancies in metals (for example

Al). Experiments (h) lead to vacancy

concentrations at the melting tempera-

ture of the order of

melt = 5 x 10--

for molar24.86 cm3

v melt

volumes V between 20.3 andAlthough this concentration

is comparable to the concen-

tration at melting observed in somesolids (sodium or potassium), it isstill somewhat higher than a value com-patible with the analysis of the spe-cific heat experiments (5).

Generally speaking the equilibriumconcentration of thermally excitedvacancies is:

xv = exp(Sf -

In equation k, $> is the energy of for-mation and Sf is the entropy of for-mation. We assume Sf to be verysmall. Since the vibrational contribu-tion to Sf is probably negligible(note that T « 0Q) , this is equiva-lent to assuming that the ground stateof a vacancy in bcc 3He is non-degener-ate. From X-ray data (h) <£ can bedetermined. Sf is less well known.

A. NMR Observation of_ Vacancies

NMR is sensitive to vacancies

because they are mobile and lead to an

effective jump frequency l/rr for 3He

atoms:

1/Tr = Xv/T (5)

In equation 5 the quantity \/r is some

characteristic frequency for vacancy

motion. With NMR, only rr can be

measured. Several methods to be used

in different ranges of temperature were

developed (6) :

i. the effective spin diffusion

coefficient is

D = (1/6) aVrr (6)

whenever x v is large enough. This

coefficient is measured with the method

of spin echoes in an applied magnetic

field having a nonvanishing gradient.

ii. in the Zeeman-vacancy regime

for relaxation by modulation of the

dipole-dipole interactions (6,7). the

relaxation time Ti depends on the jump

time r In the high field weak1ision limit (Larmor frequencylarger than l/rr) one has,

1/Ti = (8/3) M2/(o,2rr)

col-w much

(7)

M2 being the NMR rigid lattice second

moment. The temperature range for this

mechanism is typically 0.8 to 2 K.

iii. at very low temperatures, spinrelaxation is in an exchange-vacancyregime. The bottleneck for the flow ofenergy occurs between the exchange sys-tem and the vacancies, which are ingood thermal contact with the phonons.In that "strong" collision regime theintrinsic exchange-1attice relaxationtime is:

T E L = 2(z-l)rr/z (8)

In equation 8, the coordination number

of the bcc lattice, z, equals 8. (This

equation assumes the effective exchange

Hamiltonian to be an Heisenberg tran-

sposition-only Hamiltonian, which is

not the case (1,3). The use of a more

realistic exchange Hamiltonian however

would only lead to a slight variation

of the proportionality factor between

T E L and rr.)

As discussed in reference (7), all

the experiments lead to the conclusion

that the jump frequency as studied by

NMR follows some Arrhenius relation:

'••• = T O " exp (-W/T) (9)

where the NMR activation energy W is

fairly well determined for the molar

volume V < 23-05 cm3. In fact for V =

20 cm3, where TE(_ has been studied to

0.1*5 K (8), the activation energy W is

108 Bulletin of Magnetic Resonance

over approximately ten decades.

B. Var ious Channels for Vacancy Mot ion

What is the physical mechanism forvacancy motion and what information can

the jump time rf asWe review four possi-(or channels) forIn each case we ask

is the vacancy charac-teristic frequency and what is the cor-responding NMR activation energy W ^ Rdeduced from equatio'ns k and 5? Onehas:

be deduced frommeasured by NMR?ble mechanismsvacancy motion,ourselves: What

WNMR = d(1nrr)/d(l/T) (10)

a) Classical diffusion: The motionof the vacancy is a thermally activatedjump over a barrier of height <?m:

exp(-$m/T)

WNMR

(Ha)

(12a)

b) "Incoherent" tunneling diffu-sion: The vacancy is coupled to astrain field and is somewhat similar toa small polaron. While the vacancy ishopping the phonon quantum numbers arechanged (9.10). The corresponding rateis phonon-enhanced and has a tempera-ture dependence given at very low tmep-eratures by:

1/T a T1 (lib)

This channel may have been observed forprotons in Ta or for positive muons inCu. In our case, the corresponding NMRactivation energy becomes temperaturedependent:

WNMR 7T (12b)

It should be remembered however thatexperimentally a power law with a highexponent is not always easy to distin-guish from the Arrhenius law.

c) "Coherent" tunnel diffusion: Ifa tunneling particle propagates, while

colliding with phonons, in such afashion that phonon quantum numbers areconserved in an elementary scattering,the characteristic frequency of theparticle is phonon hindered (11,12) andgiven by

/!2D)X(0D/T) ' (lie)

Here u>v is the tunneling frequency ofthe particle. Such a mechanism isobserved for 3He in 4H^ - 3He and prob-ably for positive muons in iron or alu-minum. The case of a tunneling vacancyin 3He is special because of the disor-der of the 3He spins in the paramag-netic state (13); strictly speaking thewave vector is not a good quantum num-ber for a vacancy which does not propa-gate. Collisions such as thosedescribed above lead to a diffusionwhich cannot be called coherent anymore for a vacancy. If these colli-sions are frequent enough, its charac-teristic frequency is neverthelessgiven by equation lie.

WNMR= <£> - 9 T (12c)

d) Pure tunneling motion: Thevacancy tunnels and its interactionwith phonons is negligible. If it tun-nels in 4He, it forms a band of states,the wave vector being a good quantumnumber. The diffusion of a tunnelingvacancy in *He is analogous to that ofan electron in a metal: it is coherent

the diffusion constant Dv i sandequal to an average group velocitytimes a mean free path. If a vacancytunnels in 3He, because of the disorderof the spins (13). its diffusion isincoherent and the diffusion

As shown in reference (14) for 3Hespins a vacancy-induced spin diffusionconstant D can be defined. By usingthe moment approximation to evaluatethe hydrodynamic limit to the behaviorof the spin-spin correlation function(14) , it is found to be

D = (l/6)Ao)va2xv ,

Vol . h, No. 3A 109

where A = 5-2. By usethe prefactor T Q inseen to be independentleading to

of equation 6,equation 9 isof temperature

frequencyThe result

byi s

tla).

us i ng

, = 2*0

the prefactor r

WNMR

C. Discussion

(12d)

In equations (12) the quantity onthe left side is an NMR observablewhile the right-side quantity is theenergy of formation obtained by X-rayscattering (it). The activation ener-gies obtained in the two experimentsare shown in Figure 2. The values arevery close. It should be noted howeverthat it is hard to evaluate the errorof the X-ray data. In view of theobserved agreement, the channel (d)model seems to be the most likely mech-anism.

20 22 23 V.

V (cmVmole)

25

Figure 2. Vacancy activation energy asa function of molar volume; X-ray dataare from reference {h) and NMR data arefrom reference (7).

We can then evaluate the tunneling

110

o*

(13)

for 20 cm3 < V <23.O5 cm3 (7). If apositive value for S f were used, thisfigure would be an upper limit only forthe tunneling frequency. If at thetemperature of the NMR experiments,only the bottom state of the vacancyband were populated, the vacancy con-centration would be effectively lessthan exp (-<J>/T) and the estimate (13)would then become a lower limit.

It has to be pointed that the analy-sis (5) of the specific heat measure-ments leads to a tunneling frequencywhich is about ten times larger thanthe above estimate (13). At the pres-ent time is is not easy to explain thereason for this discrepancy. A morecomplete analysis of spin relaxationthrough narrow band vacancies isneeded, as well as a thorough investi-gation of the effect of strain fields.

Note that the channel (a) seems toapply to a vacancy in the hep 3He,where the inequality equation 12a holds(7).

111. I sotopi c impur i t i es o_f_ 3He j_nh.c.£. M±e - 3He

Due to the zero-point motion dis-cussed in the introduction, an isotopicimpurity can exchange site with a hostatom in a transposition. In fact, thisexchange might well happen with twohost atoms, in a three-body cyclic per-mutation, as argued in the case of anortho impurity in the h.c.p. solidhydrogen (15)- We do not have to dis-cuss the microscopic physics of theimpurity exchange since no qualitativeeffect is different forof a 3He impurity withtwo host atoms. It is3He atom is transferreding lattice site with a phenomenologi-cal tunneling frequency J; J will betypically of the order of 10~s K.

One isolated 3He atom in the crystalpropagates freely. Its eigenstates,analogous to the vacancy waves in *He,

Bulletin of Magnetic Resonance

a permutationei ther one orassumed that ato a neighbor-

called impuritons (16,17), form a bandof states with a width l6hJ. The rootmean square velocity of an impuritonover the band is

v = v'12Ja (UO

Two 3He atoms separated by a vectorR:k are two point defects, each onesurrounded by a strain field. Thesetwo strain fields interact and thiselastic interaction decreases asymptot-ically with Rjk"3- In an infinitecrystal the interaction is non vanish-ing if the sound velocity is aniso-tropic or if the local strain aroundone impurity is anisotropic. In ourcase, the asymptotic form for theinteraction could be written as

E j k3 3= (a/Rjk)MVo(l-3cos

2f)/2+..}.(15)

J_n equation 15» f is the angle betweenR:k and the trigonal axis of thecrystal and V o is an energy typicallyone thousand times larger than J.

A. The Impur i ton Gas

In low concentration these impuritiesmay be thought of as a gas, similar tothe case of electrons in a metal.

1. Collision between Impurities.

The scattering of two narrow bandparticles through a potential, such asgiven by equation 15, is a difficultproblem. A dimensionality argument(17) leads to a cross section,

o = irb2 , (16)

where b is some effective interactionradius related to the mean square value•hUo of the anisotropic elastic inter-act i on (15) :

b = a(U o/l£>J)m . (17)

Note that equation 16 can be written as

a a (F) -2'3 ,

which agrees with the well known fact

Vol. 4, No.

that the cross section for collision ofgaseous molecules having dipolarmoments depends on T~1/3.

The mean free path for impuritons is

/ = (l/cr)aVx/2 ,

where x is the concentration of 3Heatoms. Kinetic theory leads to a dif-fusion constant for impurity atoms

D = (1/3)"/ = (V.65JaVx) (J/Uo)2" .(18)

It should be noted that D is measur-able by NMR (see Section M-A-i) andalso that it can be much larger thanJa 2. If D is much larger than Ja 2, thediffusion is coherent, as opposed, forinstance, to an incoherent ordinaryspi n d i ffus i on.

If the impuritons are considered toform a gas, the density of this gasmust be sufficiently low so that onlytwo-particle collisions are occuring.In other words: / » b, which leads to

x « J/U, (19)

In section Ill-C the situation of 3Heconcentration larger than the limit setby equation 19 is considered.

2. Impuriton-phonon Scattering

At finite temperatures phonons arescattered off the impurities. This iswhy a pure crystal has a larger thermalconductivity than an impure one. Theproblem can be analy2ed exactly (12),only the final result is given here.When impuritons collide with phononsmore often tha,n with other impuritons,the 3He diffusion constant becomes:

D = 1.3 X 10-< (J2a2/S2flD) (0D/T) ' , (20)

where S is some couplican be deduced fromthermal conductivity4He - 3He. Equationthe same as the resultnote, however, thatcoherent diffusionEquation 20 was estabdue ing a Kubo type

ng constant whichthe analysis ofmeasurements in20 i s essential 1yof equati on lie;v*e sti 1 1 have afor 3He atoms.1 i shed by i ntro-memory function

111

which can be integrated exactly and isnot based on a moment approximation.

B. The Exper imentai Behavior of theDiffusion Constant

When measured by the method of spinechoes in an inhomogeneous magneticfield, the diffusion constant 0 (for agiven molar volume) has a typicalbehavior shown on Figure . 3» for threedifferent 3He concentrations. Threedifferent regimes can be defined.

T-

Figure 3- Temperature dependence ofthe diffusion constant at differentconcentrations in 4He - 3He (17).

In the regime I, the mechanism for Dis not a 3He-4He transposition but goesthrough thermally excited vacancies,more or less like in the section II

above. Vacancies in 4He are numerousenough in that regime to show up when Tis higher than about 1 .i* K.

In the intermediate regime II, D isa decreasing function of T and dependsslightly on concentration when it islow enough. Here one observes theimpuriton-phonon scattering discussedin Section ll-A-2. This is illustratedin Figures k and 5- The latter shows aplot of 0"1 versus T' which confirmsrather well the power law of equation20. For the corresponding molar volumeof V = 21 cm3 the Debye temperature is

The coupling constant S 2

deduced from the thermal con-data. This enables us tothe 3He-4He exchange fre-

6U = 26 K.= 0.17 isduct ivi tyevaluatequency:

J = 5 X 10" (21)

In the regime III, D is a decreasingfunction of x, independent of T. Thisregime is established well below 1 K.Now the diffusion at low temperatures,such that impuriton-phonon scatteringis negligible, should be discussed athigher 3He concentration, such that xis high enough to violate the gas cri-terion given by equation 19-

C. The Dense Fluid of Impur i t i es

When x is large enough, an impurityreally interacts with the elasticfields of many impurities simultane-ously and tunnels in the mean elasticfield gradient of all the impurities ofthe crystal. This can be calculated(18) by the method of Anderson and Mar-genau, which was used also for the NMRlinewidth in a dilute system (AbragamChapter IV) as well as for the relaxa-tion time of ortho hydrogen impuritiesin parahydrogen (19)- The result is

112 Bulletin of Magnetic Resonance

1.0 1.5 2.0

Figure h. Temperature dependence of the diffusion constant in 4He - 3He for dif-ferent concentrations and molar volumes (21):

Curve

1

2

3

5

6

V(cm3)

20.95

20.95

20.95

20.7

20.7

20.7

6 X 10-5

5.2 x 10-4

1 .2 X 10-3

2.5 x 10-3

7.5 x 10-3

2.17 x 10-2

Vol. k, No. 3/4 113

3

2

103

-7

,''

11 13

Figure 5« Temperature dependence ofthe diffusion constant exhibiting theimpuriton-phonon collisions (22)(regime II: V = 21 cmJ = 6 X TO"5) .

Figure 6. Concentration dependence ofthe diffusion constant in the low temp-erature plateau (23) (regime I: circlesare for V = 20.95 cm3

V = 20.7 cm3) .squares are for

D = 0.1<7Ja2f J(22)

Note that the average elastic fieldgradient is indeed proportional toUox

4'3. The main difference between adense fluid and a gas is the differencein power laws. The data in the regimeIII are given in Figure 6. In thisfigure D vs x is plotted for two dif-ferent molar volumes. It is impossibleto conclude whether the power law is -1or - k/J), however.

D . k_ D i rect Determi nat i on of the 3He -4He Transpos i tion Frequency

An experiment in which the tunnelingfrequency J could be observed with nointerference of the much larger elastici nteraction U, is needed. This

provided by the possibility of a cohe-rent propagation of a two dimensionalquasiparticle formed by a cluster oftwo nearest neighbor 3He atoms (17,20).Suppose this cluster or this (3He)2molecule, remains in the basal plane ofthe hep crystal. All vectors joiningnearest neighbor sites are crystal 1o-graphically equivalent. Consequentlythey correspond to identical elasticinteractions. If the "molecule" movesin that plane and in such a manner thatthe interatomic distance is conserved,the elastic interaction remains con-stant and one has a free privilegedtunneling. Of course, this is possibleonly if the concentration is low enoughthat the other impurities can be neg-lected. For such a motion, the tunnel-ing Hamiltonian can be diagonalized.The density of states for these twodimensional excitations is shown inF i gure 7-

111* Bulletin of Magnetic Resonance

(E/J14>

Figure ~]. Density ofspin triplet (3He) 2quasiparticIe confinedplanes is plotted asenergy (2k).

states for thetwo-d imensi onal

to hexagonala function of

These excitations modulate theintramolecular dipole-dipole interac-tion leading to a relaxation mechanismwhich is dominent when the Larmor fre-quency is related to the energy differ-ence between the bands of Figure ~]. Asa result, T1 versus fielat a given concentration,shown in £igure 8. Thesefrequencies which do notthey are more pronouncedcentrations.

This contribution to the relaxation/ate has been calculated theoretically(20) and the result is illustrated byFigure 9> which shows relaxation maximaoccuring at 03 ~ 2.5J and a> = 5J • Byfitting this result to the data of Fig-ure 8, at V•.= 21 cm3, the following Jis obtained:

exper iment,shows dips

d i ps occur atdepend on x;at lower con-

Figure 8. The spin lattice relaxationtime Ti in 4He - 3He as a function offrequency for three concentrations (25)(squares: x = 10~3; circles: x = 5 x

10-*; triangles: x = 2.5 X 10"4; V = 21

cm T = 0.53 K)

J ^ k X 10' (23)

which is fairly close to the estimate(21).

One might question however whetherthe tunneling frequency deduced fromthe experiments shown in Figure 8 isindeed the same as the one which deter-mines the diffusion. It could beargued that the phenomenon determiningthe experiments of Figure 8, is not a3He-4He transposition but a more com-plicated permutation involving two 3Heatoms and one 4He atom. We assume thatthese various tunneling frequencies arevery similar, so that we can indeed usethe result of equation 23 to analyzethe data of Figure 6 quantitatively asf ollows.

For V = 20.95 cm3, we consider the

Vol. 4, No. 3A 115

1/T,M2.5

o/J

Figure 9. Spectral density histogramin arbitrary units for spin-latticerelaxation due to coherent tunneling of(3He)2 molecules in 4He (20).

lower concentration studied (x10"4) for which the measured 0cmVs. With equation 18 in theton gas regime we find

- 2.5 xis 10-7

impur i-

J/U, 1.5 X 10"4 = 0.6x

(dense fluid ofand with equation 22impurities) we find

J/Uo = 6 X 10"4 = 2.4x.

Apparently this concentration x=2-5 X10~4 is approximately the transitionconcentration between the impuriton gasregime and the dense fluid regime.Indeed, for x > 5 X 10"4 the data ofFigure 6 for V = 20.95 cm3 fit the

J/Un * 4equation 22 fairly well withX 10-4 or fiUo = 8 X 10"

2 K.Other interesting aspects about

quantum diffusion in 4He - 3He systemsare the values of the relaxation x.Also several theoretical problemsregarding quantum diffusion remain to

be solved. The solution of these prob-lems is relevant not only to the solidhelium but also to other situationswhere quantum diffusion is observed.

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