PHYSICAL ORIGIN OF A LARGE FOUR SPIN EXCHANGE IN b.c.c. SOLID 3He

(1)

HAL Id: jpa-00218049

https://hal.archives-ouvertes.fr/jpa-00218049

Submitted on 1 Jan 1978

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

PHYSICAL ORIGIN OF A LARGE FOUR SPIN EXCHANGE IN b.c.c. SOLID 3He

J. Delrieu, M. Roger

To cite this version:

J. Delrieu, M. Roger. PHYSICAL ORIGIN OF A LARGE FOUR SPIN EXCHANGE IN b.c.c. SOLID 3He. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-123-C6-125. �10.1051/jphyscol:1978656�.

�jpa-00218049�

(2)

JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 8, Tome 39, aoi2t 1978, page C6-123

PHYSICAL ORIGIN OF A

LARGE

FOUR

SPIN

EXCHANGE

I N

b.c.c. SOLID He 3

J.M. Delrieu and M. Roger

DPh-G/PSRM

-

CEN.SACLAY

-

B.P. N o 2

-

91190 GIF-sur-Yuette, France

RQsumd.- Une mQthode de calcul de l'dchange est prgsentde, en utilisant un potentiel effectif pour corriger la fonction d'onde variationnelle. L'Qtude gdomdtrique des configurations les plus probables, qui tient compte de la r6pulsion des coeurs durs, explique l'origine d'un Qchange 4 spins du mgme ordre que l'dchange 2 spins.

Abstract.- A method of calculating the exchange is presented, using an effective potential to correct the variationnal wave function. The geometrical study of the most likely configurations, taking account of hard core repulsion, explains the origin of four spin exchange of the same order as twospin exchange.

The experimental magnetic properties of b.c.c.

3 ~ e near its ordering temperature are interpreted with four spin exchange of the same order as two

spin exchange /1,2,3/. On the contrary the theore- tical evaluations of the spin exchanges 14 to 71 predict a small four spin exchange. All the exchange calculations take the variational /7/ wave function @ which minimize the energy E as a good approximation to the "home base function" J, defined by

Fig. 1 : Schematic drawing of exchange : D is the Herring /4,8/ "duct" connecting the two cavities C1 and Cp corres-

ponding to the exchanged configurations of 3 N par-

- $

^(ri-~i)2 ticles. C is the exchange surface midway between C,

@ = r e r f(ri-r.) ( 1 ) and C 1 .

i i< j _J

where f is the Jastrow function such that f is small when the two particles potential U is large. The relation /8,9/

sically the small value of @ inside the "duct" has, on the variational energy a negligeable effect of order J so that the value of $J on C can be quite wrong, even when it provides a very good variational energy E. Thus the use of formula (I) with 6 = @ is incorrect.

gives the exchange frequency, where P is the cor- responding permutation of particles and C is the exchange surface midway between the two exchanged configurations CI and C2. Following the terms of Thouless /9/, "a home base function" is the lowest energy solution of the ~chrgdin~er equation H$ = E$

in both "one cavityr' C1 (i.e a usual configuration of the atoms) and the "duct" D connecting the two exchanged configurations C1 and C2 (or "cavities"), which has been artificially closed at its end Anear the second cavity as shown on figure 1 . Unfortuna- tly the variationnal @ does not verify HJ, = E$ in '\e "duct". The analysis of Herring 181 on simple systems, shows that in general a variational 4 is very different from J, in the "exchange duct". Phy-

Apart from fondamental mathematical problems

1 5 1 , the perturbative expansion forJ given by Guyer

/5/ and Mullin / 6 / , which is equivalent to calculate

I )

, is also very'questionable ; @ is so different from the true J,, that it is not possible to obtain a good approximation using a small number of terms (practically limited to second order).

Because H@/@ > E everywhere inside the "duct", we must determinate the ratio g = J,/9 as we follow

the line along the center of the "duct", defined by the variable t. As a first simplifying approximation, we take for t, the straight line distancecon- necting the configuration to the center of the ca-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978656

(3)

vity (atoms on their sites). In order that J, = g$

verify as closely as possible HJ,

-

^EJ,in the "duct", g(t) is given by the minimum of

I = <

I v $ ~

+ (u-El$2> (3) with $ fixed. This gives

<+h(@g) + (u-E)J,~> = 0. Taking go(t) = g(t)(<+2> ) I h

It

with <$2>lto =

1

dySN 6(t(r)

-

to) as the mean value for fixed to, we obtain after arrangement asim- ple ~chrEdinger equation

Atgo + V(t)go = Ego with A Log

$>1

V(t) =

- -

₂

+ L A

Log<$2> +

<4271t 4 t

I t

The calculation of go gives J with the relation (I), if go is normalized : J =

- -

2 g2 g

m

J is given by the hopping frequency from one cavity to the other with the effective potential barrier V(t). We remark that the same equations are validif t is a vector, for example the coordinates of the

-

exchanging particles, so that a more realistic calculation is possible with the effective potential V($).

The integrals in V(t) are similar to that gi- ving the energy and thus the most reliable methodis the Metropolis Monte-Carlo Method

11

0-1 21. A hard

e

spheres solid with diameter u = 2.14 A is a goodap- proximation to solid helium 112-131. Using this same method, at the melting density p = 0.25 of a hard spheres solid, we have calculated V2(t) and V,(t) along the "ducts" of J2 the two spin first neighbourg exchange, and of %,the folded four spin cyclic exchange. As a preliminary result, we find that V, is slightly smaller than V, in the "duct".

The value of g on tC being roughly proportional to

exp

-

^{J:P (6) we must take account of

the length d of the "ducts" i.e. the total displacement of the atoms in order to compare J2 and

$.

The center of the duct on C is defined by the maxi- mum of $, on C ; thus the spheres must be as far apart as possible. The geometrical configuration on C are shown : for J2 in figure 2.a and Figure 2.b

Fig. 2 : a) _:Critical configuration on C of two exchanging spheres, with fixed neighbours in a hard sphere b.c.c. solid at the melting density. d isthe length of the "duct" i.e. the total displacement of all atoms _;b) : Same as a) but allowing for the mo- vement of two neighbouring spheres ; c) same as b) but with the critical configuration of four spin cyclic planar exchange

complex). In figure 2.a, with all neighbour atomsat their sites, the free space 6 available for the two spheres is half the 6 available in figure 2.b and figure 2.c, where the four spheres are as far apart as possible with other neighbours at their sites-On the other side the length d of the exchange "duct"

is smaller in figure 2.a, than in figure 2.b or 2.c.

These configurations being the most typical configurations in the Monte-Carlo integration, and thuswith relation (6) go(tC) % exp{-dl&} J, and K are of the

P same order of magnitude.

The physical origin of large four spin excharr ge in b.c.c 3 ~ e is the following : two exchanging atoms, due to hard core repulsion, find place by large displacements of their neighbours, so that the width and length of the "duct" for J2 are similar to that for K (or $) ., 5 or 6 cyclic spin exchange, on

P

the contrary, is smaller because the length d ofthe '"duct" becomes proportionnal to the number of particles, with about the same place 6.

and for K the planar four spin cyclic exchange in P

figure 2.c. (The 3 dimensional figure for

%

^ismore

(4)

References

/ I / Hetherington, M.J. and Willard, F.D.C., Phys. Rev.

Letters

35

^A(1975) 1442

/ 2 / Roger, M., Delrieu, J.M. and Landesman, A., Phys.

Lett.

62

^A(1977) 449

Roger, M. and Delrieu, J.M., Phys. Lett.

63

^A

(1977) 309

/4/ Mc Mahan, A . K . , J. Low Temp. Phys.

8

(1972) 159 and 115

/5/ Guyer, R., Phys. Rev. A s (1974) 1785

Its expansion is incorrect, because it uses the inverse of the antisymmetrizer operator A, which does not exist

/6/ Mullin, W . , Phys. Rev. B

12

(1975) 3718

/7/ Mc Mahan, A.K. and Wilkins, J.W., Phys. Rev. Lett.

35 (1975) 376

-

181 Herring, C., In Magnetism Vol. I1 B, Chap. I edited by Rado C.T. and Suhl H. (1966) Academic Press London

/ 9 / Thouless, D.J., Proc. Phys. Soc.

86

(1965) 893 / l o / Mc Millan, W.L., Phys. Rev.

138

A (1965) 442 1111 Hansen, J.P. and Pilloch, E.L., Phys. Rev.

5

A

(1972) 2651

1121 Hansen, J.P., Levesque, D. and Schiff, D., Phys.

Rev.

2

A (1971) 776

1131 Kalos, M.H., Levesque, D. and Verlet, L., Phys.

Rev.

2

^A(1974) 2178