PHYSICAL ORIGIN OF MULTIPLE THREE AND FOUR SPIN (OR ATOMS) EXCHANGE IN QUANTUM SOLIDS WITH SOME CONSEQUENCES

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PHYSICAL ORIGIN OF MULTIPLE THREE AND FOUR SPIN (OR ATOMS) EXCHANGE IN

QUANTUM SOLIDS WITH SOME CONSEQUENCES

J. Delrieu, M. Roger, J. Hetherington

To cite this version:

J. Delrieu, M. Roger, J. Hetherington. PHYSICAL ORIGIN OF MULTIPLE THREE AND FOUR SPIN (OR ATOMS) EXCHANGE IN QUANTUM SOLIDS WITH SOME CONSEQUENCES. Jour- nal de Physique Colloques, 1980, 41 (C7), pp.C7-231-C7-239. �10.1051/jphyscol:1980737�. �jpa- 00220175�

PHYSICAL ORIGIN OF MULTIPLE THREE AND FOUR SPIN (OR ATOMS) EXCHANGE IN QUANTUM SOLIDS WITH SOME CONSEQUENCES

J.M. Delrieu, M. Roger and J.H. Hetherington*

CEN-SacZay , Orme des Merisiers, B. P. N o 2, 91 190 Gif-sup-Yvette, France.

* Michigan State University, East Lansing, Michigan 48824, USA.

RBsumB. - Le mouvement de point zdro des solides quqntiques entraine une frequence non nulle d'C- change des atomes. Les calculs, utilisant les fonctions 1 2 corps de Jastrowdonnent essentiellercent des Bchanges B deux atomes, car ils ne peuvent pas d6crire correctement les corrBlations reelles dues 1 la gCometrie imposEe par les coeurs durs. Un modSle sinple de quatre sphPres dans une boite montre que lt&change cyclique B 4 atomes peut Ctre plus grand que 1'6change 1 2 atomes. Ce modPle conduit 1 une nouvelle approximation tenant conpte des correlations de coeur dur ; son application 1 1'3~e conduit B un &change 4 spins prgpondbrant dans la phase cubique centree, en accord avec les rdsultats expdrimentaux, B un Gchange 3 spins ferromagn6tique dans la phase hexagonale compacte de lt3He ou dans 13He absorb6 en surface et 1 des pernutations cycliques de 3 atomes dans ltH2 solide.

Abstract.

-

He leads to preponderant four-spin exchange in the b.c.c. phase as shown by the experimental re- sults and to a ferromagnetic three-spin exchange in h.c.p. 3He or adsorbed 3He or cyclic three-atom permutation in h.c.p. solid H2.

A two parameter model, assuming that the doni- cyclic permutations at a frequency in the range of nant contributions in the exchange hamiltonian are kilocycleswhich seems to explain several experimen-

cyclic four-spin exchange and three-spin exchange 3

tal results. For a small number of He aton in mi- explains all the experineqtal results for the ma- croscopic pores, we predict for n even (2 or 4) gnetically ordered phases in b.c.c. solid 3He. In antiferromagnetic properties and for n odd (3 for particular it predicts a low field ordered phase in example) on the contrary ferromagnetism.

agreement with the recent experiment of Osheroff et al.. However the early calculations of the ex- change rate predicted essentially pair exchange.

These earlier calculations used a variationnel wave- function (i.e. a product of one-particle gaussian and two-body Jastrow function) which is quite arbi- trary in the exchanging configuration (although it is quite good for the calculation of the ground state energy). Starting from simple physical and geometrical remarks we propose a new method to cal- culate exchange or more generally tunneling rate in many body systems with complex hard core correla-

tions, which cannot be described by two-body corre- lations functions. The simplest fornulation of this method gives a physical explanation of the prepon- derant four spin exchange in b.c.c. solid 3 ~ e and we propose some concequences :the prediction offer- romagnetic three spin exchange in h.c.p. solid 3 ~ e , in two dimensional triangular adsorbed 3 ~ e and a possible explanation of the ferromagnetisn observed in liquid 3 ~ e in contact with a solid surface. In solid h.c.p. H2 the same model predicts three-atom

Hard core correlations and exchange in solid He 3 Quantum zero point motion prevents '~e from being solid at zero pressure (IC) ; it is only abo- ve 35 bars that He becomes a b.c.c. solid at zero 3 temperature. Physically the 3 ~ e atoms, interacting through a Lenrard Jones potential with an effective hard core diameter (4) 0 = 2.14 2 do not have enough space to move across each other as in a liquid. Ne- vertheless they can unfrequently exchange their place. A sinple analogy is interesting : in the un- derground or French "netro", around 6 p.m. when you are squeezed because there is too many people, you cannot move freely ; nevertheless you can exchange your place with your neighbours, and you can experi- mentally observe that this exchange is far more easier if 2 or 3 persons accept to permute cycli- cally with you than if only one accepts, the avai- lable space being smaller. In the underground in the "solid state" (nore precisely "glass state") three and four persons permutations are more easier than that of two persons. This kind of exchange mo-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980737

JOURNAL DE PHYSIQUE

Fig.1 : Geometrical hard c o r e e f f e c t s f o r t h e ex-

-

change of f o u r hard c o r e d i s k s i n a two d i - n e n s i o n a l box, of shape g i v e n by f i x i n g t h e

surrounding atoms a t t h e i r s i t e s ; t h e d i s - p o s i t i o n corresponds t o t h e p l a n e 110 of b . c . c . s o l i d He. T w o s i z e s o f t h e hard c o r e 3

d i a n e t e r a a r e shown : t h e dashed l i n e cor- responds t o t h e t r u e hard c o r e a of Lennard Jones p o t e n t i a l U ( a s shown i n Fig.3) ; t h e f u l l l i n e t o t h e c r i t i c a l s i z e a = b c where t h e l a s t p o s s i b l e t y p e of exchange i s sup- p r e s s e d by i n c r e a s i n g a ; h e r e i t i s f o u r - a t o n exchange.

a ) f o u r atom exchange i n t h e c r i t i c a l c o n f i - g u r a t i o n O=U on exchange s u r f a c e C where d i s k s a r e t a n g e n t t o each o t h e r . I t c o r r e s - p o n d s i n F i p . 2 t o t h e c a s e where t h e d u c t D b e - comes c l o s e d and exchange i s suppressed.

b ) three-atom exchange with o v e r l a p i n g d i s k s on C f o r U = U c , (with novenent of t h e f o u r t h d i s k ) .

c ) two-atom exchange w i t h w e r l a p i n g d i s k s on C f o r o=O (with movement of t h e two o t h e r a t o n s ) .

The d i f f e r e n c e Ocn-a=6 g i v e s 6 used i n ( 7 , 8 ) t o c a l c u l a t e t h e i n c r e a s e of k i n e t i c e n e r - gy V d u r i n g exchange. The d o t t e d l i n e g i v e s t h e e q u i l i b r i u m p o s i t i o n of t h e atoms and t h e f u l l l i n e t h e i r p o s i t i o n on t h e exchan- g e s u r f a c e C (which i s d e f i n e d i n Fig.2 i n c o n f i g u r a t i o n s p a c e ) .

being possible.

Fig.1 shows the same phenomen with a model of 4 hard core atoms (or disks in this figure) in a box of shape given by the positions in b.c.c. solid 3 ~ e of the surrounding atoms in the plane 110. This plane corresponds to the four spin planar exchange used to explain the

magnetic ordering properties of solid 3 ~ e . With in- creasing the hard core diameter a, above some cri- tical value ac, no exchange is possible, as shown in Fig.1 ; for a < a the first possible exchange

C'

is not pair exchange, but cyclic four-spin exchange as shown in Fig.l. Below some value oC3 < aC, three spin exchange becomes possible and one can have pair exchange only below Uc2 < aC3. A similar figu- re for an hexagonal lattice (7) or two dimensional triangular lattice shows that the first exchange appearing at ac corresponds to cyclic three-atons permutations and that pair exchange is even more inhibited. These simple geometrical remarks have important physical consequences : the parity of three and four-spin emut tat ions being respectively even and odd we expect ( 6 ) respectively ferromagne- tic and antiferromagnetic ordering i.e. opposite

Figure 1 shows clearly that the correlations during exchange processes cannot be described with one and two body correlations as it is assumed by taking the variational wavefunction @ used for the energy calculations (2,3) :

For this reason an accurate exchange calculation seems a formidable challenge ; nevertheless the study of Fig.l leads to a new method which includes these geometrical correlations in 4.

Exchange formula and approximations

The true wavefunction $ is a solution of the Schrgdinger equation with the Lennard Jones poten- tial U :

HJ, = - - IiL A$ + UJ, = EJ,

2m (2)

We know two qualitative properties of $ : I ) J , = O when two hard cores touch each other

The configuration where each particle is near its lattice site has a high probability I)$*,because the corresponding point in the 3N dimensional space of N atoms is far from the hard core boundary con- ditions where $ = 0. Follovling the terminology of Thouless ( 6 ) we call cavities these configurations with large available space. There are N! such ca- vities corresponding to all possible permutationsof atons. Exchange of atoms occurs through pathway or

"duct" connecting the cavities.

As in the example of Fig.], due to the hard core geometrical effects, the opening of theseducts is small, and thus $ is small. In configuration space exchange is a tunneling process between cavi- ties through small ducts. If these ducts are closed the ground state is N! fold degenerate. The basic assumption of the exchange model is that the opening of the small ducts has a negligible effect on in the interior of the cavities, only the relative phases of $ between different cavities is modified when the N! degeneracy is removed. In this case, a hopping hamiltonien between cavities in configura- tion space with fixed tunneling frequencies gives the energies of these N! states ; the exchange hamiltonian is obtained by expressing in terms of spin operators the permutations corresponding tothe hopping operators fron one cavity to another. In

this approximation, to calculate the exchange fre- quency between two cavities tie can isolate these two cavities independently of the others, as shown schematically in Fig.2 ; the exchange frequency is the difference of energy between the exact even ground state $ and first odd excited state so- lutions of the SchrGdinger equation (2) ; fron these equations, after partial integration we obtain :

where the hyper volume y o £ surface Cv in princi- ple arbitrary is choosen such that the exchange surface C is the 3B-1 dimensions plane Z midway

v

between C 1 and C2, where $ is mini~un. This mini- mizesthe errors due to approximations for i,bl and

Following Herring (') we note that if we know

JOURNAL DE PHYSIQUE

s:

a) Schematic drawing of exchange for a given permutation: two cavities C andC represent

1 2

the system in each exchanging configuration where the N atoms are near their lattice sites with free space for zero point motion;

exchange occurs through a "duct" D connec- ting the two cavities C1 and C2 ; the duct section is smaller than the cavities becau- se the free space for zero point motion is reduced by hard core during exchange. C is the exchange surface midway between C1 and C2. The line &(t) is the fall line or gra- dient line (or M.P.E.P.) along the center of the section of the duct where the real groundstate wavefunction $o is maximum. The

"home base function" of Herring is the exact solution in C, and D with the duct closed at surface A near C2 ($=O).

b) Values of the wavefunctions along the line a t ) at the center of the duct and cavi- ties :

$o the true symnetrical groundstate wave- function ;

$, the antisymnetric first excited wave- function ;

$ is the "hone base function" defined by Herrine :

$ = 0 on A at the end of the duct and 'JJ0=*+P$, $,=$-P$.

The true $ decrease exponentially on 1.

@ Jastrow wavefunction uniquely function J

of the distance from the wall ; it is too l a r ~ e in D, because it is constant on C.

@ Jastrow and Gaussian wavefunction which is quite arbitrary inside the duct, being fixed by the shape of the cavity C,.

c) Structure of the effective potential V de- fined by 9 and equal to the energy of lo- calisation of the system in a g'iven sec- tion of the duct, i.e. in the 3N-1 dimen- sional subspace orthogonal to the tangent to the line &(t) at fixed t, the curbili- gne coordinate along the line d(t).

to a good approximation for small J << OD with res- pect to zero point motion Debye frequency OD.

@,= @ + P@ and ₌ @ - P@

where P is the permutation operator from C 1 to C2 ; in terms of @ we obtain :

Approximations and effective potential V

The true @ being very difficult to calculate (even with four hard spheres in a box like in Fig.l), we must use approximations ; the precision on J is directly of the same order as that on @ on C inside the duct D.

First we can use the wavefunction $J of formu- la 1 with A = 0, i.e. with Jastrow functions only as used for the energy of liquid ; in this case A = 0 the solidification ( 4 ) with $J is obtained at

a density quite larger than the experimental one ; we can understand the physic of this result :

$J being a function only of the distance between hard cores is in Fig.2 schematized as a function only of the distance from the walls ; the size of the duct near its niddle on C being nearly constant, QJ is also nearly constant. On the contrary the true $ decreases nearly exponentially because it must satisfy the SchrGdinger equation ; if we sepa- rate the curviligne coordinate t along the duct and the other 3N-1 variables noted I defining the posi- tion inside the duct section we can write :

We can define an effective potential V function of the position inside the duct D :

which is of the order of the energy of the locali- sation of the system inside the duct at fixed t, because the first tern of (6) measures the local curvature of @, which is fixed roughly by the hard core limiting conditions. The duct being much smaller than the cavity this energy V is much lar- ger than E and the true @ verifies :

0'

Because V _>> E this has a simple approximate W.K.B.

solution along the line &(t) at the center of, the duct, as function of t :

Near its hiddle C by symmetry the duct has a nearly constant section, giving constant V and thus J, has an exponential decrease which cannot be described by the nearly constant OJ. Because in the cavity C

1 and at the begining of the duct @ and QJ are of the same order, the fast decrease of @ shows that the Jastrow function

eJ

This remark that $ overestimate strongly the fre- J

quencies of any movement, gives the physical origin of the solidification with Jastrow functions $

J at a density quite larger than the experimental one.

For this reason a valuable description of so- lid He is obtained only with 3 A # 0 in fornula (1);

the frequency of atoms movements are strongly de- creased by the Gaussian functions. Unfortunately only one parameter A for describing the complex hard core geometrical correlations is strictly in- sufficient to obtain even the nature and order of magnitude of the exchange frequencies (293), for

example the formula 8 shows that @ does not decrea- ses like a gaussian near C but nore like an expo- nential.

These remarks show how one could proceed to estimate @ along the duct ; simply we need to cor- rect @J so that the SchrGdinger equation (1) is ap- proxinativelysatisfied at least along: the line&(t) at the center of the duct, i.e. the equations 7 and 8 are verified. I7e can use different methods to es- timate V defined by equation (6) with the unknown true @. As shown schematically in Fig.2, V isof the order of the localisation energy of the system in- side the duct at fixed t ; thus in (6) we can re- place @ by the variationnal QJ or $, because V use essentially the transversal curvature @ inside the duct which is correctly approximated by @ or $.

J 61e can use also a variationnal estimate using @

J or Qo to calculate by numerical integration the energy V at fixed t (*). The resulting effective

76

C 7 - 2 3 6 JOURNAL DE PHYSIQUE

potential has an approxinatively sinusoidal shape as shown on Fig.2 and not a parabolic shape corres- ponding to the gaussian @.

With @I at the place of I) the estimate of V

The second expression is valid when V is calculated along the line &(t) at the center of the duct which is a fall line or gradient line of $, so that V 6 = 0. In this estimate (V-E) is directly the

I

local measure of the error in the SchrSdinger equa- tion made with the variational $. A valid estimate of )I is obtained by modifying @I so that the

SchrGdinger equation is verified at least along the line &(t) at the center of the duct using equation

where $(& (t)) is the value of @ on the line & (t) at given t and r (t) is the position variable in

I

the section of the duct D, i.e. the 3N-1 dimensio- nal subspace orthogonal to the tangent ,L(t) for fixed t (in Fig.2 it is schenatized by the normal to the tangent). The formula, (4) with the estimate (10) of I), gives the exchange frequency J ; it is appro~imativel~ equal to the tunneling frequency of afictitious quantum one dimecsional systen ~ 5 t h the 3Fie mass in the effective potential V(t) , ^{as shown}

in ref.(8) (it is exactly equal with a suitable de- finition of V(t) when k(t) is a straight line).

Simple physical estimate of the exchange effective potential V(t)

For a physical estimate of V(t) i.e. of the energy of atoms in the exchanging configuration we can use the method of London (I0) : the potential energy is the value of the true potential U forthe given configuration and the kinetic energy is the kinetic energy of atoms in rigid spherical poten- tials of radius Ri which would fit inside theshell of neighbors located at their position in the con- figuration (although it is a very crude approxima- tion, London (I0) takes the equilibrium configura- tion i.e. the lattice sites and find a kinetic energy for hard spheres within 15 % of the exact value ( 4 ) ) ) . His formula gives :

and R. is the size of the maximun sphere which can be fitted among the neighboring lattice positions.

The equation (11) can be replaced by the more exact formula ( 9 ) , but it gives the essential phy- sical picture of exchange :

where L is the effective length of the exchange path from C1 to C in configuration space.

2

(where x h s

-

We have several physical effects: in the ef- fective potential V we have first the increase of kinetic energy given by the decrease of 6 . in the exchange configuration and secondly the change of potential U due to the decrease of the interatomic distance during exchange, as shown in Fig.3. In solid 38e at equilibrium, potential and kinetic energy are of the same order and thus it is essen- tial to take both into account in order to obtain even the correct order of magnitude. The Fig.3 shows that four-atom exchange without movement of surrounding atons corresponds to the distance where the Lennard Jones potential U(r) is minimum ; on the contrary two atons exchange without novement of the surrounding corresponds to a much smaller 6 i.e. V(r) is quite larger, both with potential and energy terms. Thus two-atom exchange needs the mo- vement of the surrounding atoms in order to prevent hard core overlap and thus the number n of moving atons for two-aton exchange is of the same order as for four-atom exchange. Thus the exchange length

L ". JG is of the sane order for these two types

of exchange.

More precisely we need to determine the line (t) of exchange in configuration space : it seems natural to define it as the path of maximun probability for tunneling from C1 to C2 which is estimated as the path giving the naxinum exchange frequency, i.e. that where : S = J~~(V-E~)'/~ is mininum. This path is the most probable excape path (M.P.E.P.) for this tunneling process (7,8,9);

atoms moving from configurations C1 to C2 with the potential -V(r) ; thus the M.P.E.P. lined(t) can be determined by numerical integration of the clas- sical movement of the atoms under the potential V(t) which connects the configuration C 1 to C2. A more detailed analysis of the minimum of S shows that the movement of surrounding atoms during the exchange process decreases exponentially with the distance r from the exchanging atoms : physically, in the product A . L , if we minimize V only, we ex- pect a long range elastic deformation like -, 1 but

r 3 in this case L is too large ; the deformation of the surrounding atoms is limited by the correspon- ding increase of the exchange length L, the optimum between fi and L being an exponential decrease with r. The length L being proportional to 6, for this reason only a finite number of atoms have a large movement during exchange. Figure 1 shows that this number increases with the density ; roughly the number of moving atoms corresponds to the number of atoms which must move in order to give sufficiently space to the exchanging atoms. Near melting in b.c.c. solid He the movement of the surrounding, 3 shell is small for four-atoms exchange because a straight line for the M.P.E.P. &t) is nearly the optimum one ; each atom in real space moves direc- tly from its old to its new position without tou- ching the three other exchanging atoms. On the con- trary two atom exchange needs a large deformation, the line d(t) is far from a straight line, and the effective number of moving atoms is larger than for four-atoms exchange. Three-atoms exchange gives essentially the same order for /T L as four atoms exchange in b.c.c. solid 3 ~ e .

On the contrary in h.c.p. solid He or bidi- 3 mensional adsorbed solid He the triangular struc- 3 ture leads to the result (7) that three-aton ex- change is favored with respect to four-atom exchan- ge, two-atom exchange remaining fundamentally smaller. In particular at large density (7) the hierarchy is given simply by the increase of the volume 6'f occupied by the exchanging atoms (7) ,

because both V and L increase w i t h 6 y . In solid 3 ~ e two-atom exchange is defavored with respect to three and four atom-exchange because the distortion of surrounding atoms is large during exchange. In particular the exchange frequency J is a function

solid

For solid '~e and other hard core quantum so- lids, preponderant four-aton e::change for the b.c.c.

phase and three-aton exchange for h.c.p. phase have izportant experimental consequences. In the b.c.c.

phase, the existence of four-spin exchange leads to both antiferromagnetic (in low field with negative Curie Feiss tenperature and a first order transi- tion) and pseudo ferrocagnetic properties (in finite field) which are in agreenent") with the experinen- tal results but which cannot be explained with two spin exchanze by itself. In the h.c.p. phase cyclic three spin exchange being an even permutations gives f erronap,netism(6) : under even permutations-identical fernions with parallel spins behave like bosons, so that the rounds state is the sage as witli bosons, with ferronagnetic alipfienent of spins. 'n inportant experinental test would be to neasure thecurie Keiss constant 6 for h.c.p. 3 ~ e which is estimated : 0.975 mK at the lo57er pressure;this could be obtained by a- diabatic demaenetization of solid 3 ~ e .

The sane prediction of a positive 0 is made for adsorbed 3 ~ e on various substrates, as long as the arrangement of atoms is roughly triangular, i.e. for the first and second larger. Very likely, this remains true for an irregular arrangement like a glass. The experiments on adsorbed He remain to 3 be made, but it is observed that liquid 'He in con- tact with various substrateshas ferronagnetic pro- perties with positive O ' 14) , which are supl~rns- sed by roughly 2 layers of 4 ~ e (I3). A simple way to explain this result is to remark that in these two first layers, with approxinatively triangular arrangement of atoms, the geometrical hard core effects favor three-spin exchange, explaining the observed ferromagnetism. The theory of a ferrona- gnetic layer (I5) of liquid 3 ~ e near a solid sur- face is another possibility, which in principle does not depend on the nature of the substrat

(hard solid or solid He) 4 ; a discrimination could be made by accuraten!easurenent of the number of 4 ~ e

layers required to suppress ferromagnetism ; the result could be function of the nature of the substrat. In particular if it has very small mi- croscopic pores, as function of the number even or odd of He atoms in these pores (which are a phy- 3 sical realization of the box of Fig.1 with fixed

JOURNAL DE PHYSIQUE

seems to explain the motionally narrowed N.M.R. re- ^I sonance lines observed for H3 impurities in solid

h.c.p. parahydrogen (16). On the contrary the per-

walls) we expect respectively antiferromagnetic or in Fig.] cannot be suppressed in the liquid phase ferromagnetic properties. It would be interesting to and thus are oversimplified with Jastrow functions.

study the magnetism of 3 ~ e atoms in well defined ho- mogeneous pores as function of the number of atoms : for a given pore (or box) size and shape, with two atoms we expect antiferromagnetism, with threeatoms

mutation movement of orthohydrogen impurities is severely reduced by their quadrupolar interactions,

2

1 1 ^-1--..* _{.. . . .}

3 4 5 6

I 1 I -

-

ferromagnetic properties (with a decreased exchange frequency, the free space being smaller) and with four atoms very peculiar properties : if the shape

20-

of the box is cubic, four-aton exchange is prepon-

derant, and we expect antiferromagnetic properties 15- as in b.c.c. solid 3 ~ e ; if the shape of the box is

triangular or pyramidal three-atom exchange is pre- ponderant and we expect ferromagnetic properties as

3 10-

for liquid He in contact with a surface.

Multiple atom-exchange or pernutation is not limited to solid He 3 : three-atom permutation for the description of movement of impurities of 3 ~ e in

5 -

solid 4 ~ e could have some consequences : in parti- cular the models of movement of two or three impu-

0

-

rities bound states must be modified (I8). Even in

which are much larger than the permutations frequen- cies ⁽¹⁷⁾

solid H which is much more localized than helium, 2

the nethod of calculation of exchange presented

here, gives a frequency of cyclic three-Hz molecule - 5 -

permutation quitelareer (10 4 ) than that calculated previously with gaussian function ; the predicted

-1 0-

permutation frequency in the range of the kilocycle

U(r) K

one pair

.

K

1 2

I

The method outlined in this paper for calcula- ting the exchange frequency could be applied to any tunneling mechanism in quantum systems with hard core correlations with the condition that its fre- quency is small in comparison to the zero point mo- tion frequency. In particular this shows that in liquid (in equilibrium with solid) the quantumwave- function

+

Fie.3 : Variations of potential (full line) and kinetic energy (dotted line) within the London approximation (fornula 1 1 ) as func- tion of the distance r between atoms, or radius R of the maximum sphere which canbe fitted among the neighboring lattice posi- tions, for the different type of n-atom exchange in the plane 110 in 3 ~ e , without movement of surrounding atoms : n=2,3,4,

this conference

(2) Mc. Elahan A.K., J. Low Temp. Phys. (1972) 159 and 115.

(3) Mullin W., Phys. Rev. B12 (1975) 3718.

( 4 ) Kalos M.H., Levesque D. and Verlet L., Phys.

Rev. (1974) 2178.

(5) Herring C., In magnetism Vo 11 B chap.1 edited by Rado C.I. and Suhl H. (1966) Academic Press London.

(6) Thouless D.J., Proc. Phys. Soc. 86 (1965) 893.

(7) Delrieu J.M. and Roger M. and Hetherington J.H.

To be published in J. Low temp. Phys.

(8) Delrieu J.M. and M. Roger, xvth international conference on low temperature physics (L.T.15), Colloque C6 123 39 (1978) Journal de Physique

(1 978).

D.8 3345 and 3366 (1973).

(10) London F., in Superf luids (John l7iley and Sons N.Y. 1954) Vo I1 Sec. 56 pp.29-31.

(11) Bozler H.11. et al., Journal de Physique Collo- que (L.T.15) 283 (1978).

(12) Sato K. and Sagawara T., Journal de Physique, L.T.15 Colloque C6 281 (197.2).

(13) Godfrin H. et al., Journal de Physique L.T.15 Colloque % 287 (1978).

(14) Ahnonen A.I. et al., Journal de Physique Collo- que L.T.15 % 285 (19781, J. Phys. 5 1665 ( 1 976)

.

(15) Beal Monod N.T. and Ifills D.L., Journal de P h p sique L.T.15 Colloque C6 290 (1973).

(16) Schweizer R., Ilashburn S. and !{eyer H. , ^Phys

.

Rev. Lett. 60 1035 (1978) and 2 913 (1978).

(17) Sullivan N., Delrieu J.M., to be published.

(18) Andreev A.F., Journal de Physique L.T.15 Collo- que 1257 (1 978).