FOUR SPIN EXCHANGE EXPLAINS SOME EXPERIMENTAL MAGNETIC PROPERTIES OF b.c.c. 3He

(1)

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FOUR SPIN EXCHANGE EXPLAINS SOME

EXPERIMENTAL MAGNETIC PROPERTIES OF

b.c.c. 3He

M. Roger

To cite this version:

(2)

JOURNAL DE PHYSIQUE

Colloque C6, suppldment au no 8, Tome 39, aotit 1978, page

C6-128

FOUR S P I N EXCHANGE E X P L A I N S SOME E X P E R I M E N T A L MAGNETIC P R O P E R T I E S O F

b.c.c. 3 ~ e

M. Roger

DPh-G/PSRM

-

CEfl.SacZay

-

B.P.

N o

2 -

91190

af-sur-Yvette, France

Rdsum6.- La prise en compte de termes d'dchange

1 quatre spins, du mSme ordre de grandeur que l'd-

change

1

deux spins, dans I'Hamiltonien, peut expliquer un certain nombre de propridtds magndtiques

de ~e~ c.c., en contradiction avec le modsle dtHeisenberg

:

i) la transition, apparemment du premier

ordre, I

T'

-

1

mK, ii) le signe ndgatif du terme en T - ~

de la chaleur spdcifique, iii) l'accrois-

sement de fa susceptibilit6 par rapport

3

la loi de Curie-Weiss au dessous de 10

mK.

Abstract.- Taking into account large four spin terms (of order of the pair exchange) in the Hamil-

tonian can explain some experimental magnetic properties of b.c.c. ~

e

which are inconsistent with

~

,

a Heisenberg model

:

i) the apparently first order transition at

~f

=

1 mK,

ii) the negative sign

of the

r3

term in the specific heat, iii) the increase of the susceptibility with respect to the

Curie-Weiss law below 10 mK.

Including pair exchange up to the third neigh-

bours and the two kinds of cyclic four spin exchange

with nearest neighbours

:

folded

(F) and planar

( P ) ,

we take the Hamiltonian /1,2/

:

where

yijka

=

The (P) four spin permutations contribute fo J 1 ,

J,

and Jj

;

we assume that J3 reduces only to the con-

tribution coming from

%

:

J3

=

%/2. The cyclic

permutation of four particles being odd, we assume

%

<

0 and

%

<

0 131. Taking into account the

Zeeman term

: -

y

Irl

tj

C

L ,

the partition function

i

Z is expanded /4,5/

:

B2

-

_{" +}

_...

_{+ y}

_{1 (BYIH)}

k i 2

N-' RnZ

=

NRn2

+

e,

_-

_{e3 24}

Theexpressions of

g 2 ,

g3,

0 . C

in terms of the ex-

change parameters are given in references /4,5/.We

R

deduce the specific heat

:

Cv(H

=

0)

=

(e",$2-e"3$3)

and the inverse susceptibility

B

x-'

(T)

= N-'

(T

-

0 +

;i;+

...)

with B

=

e2-c

High temperature experimental results give relations

between the four parameters of

Jbe.

Taking, according

2

to 16-91

:

ez

=

e",/0

=

1, J1

=

J,

/

10

1 are given in

*

terms of

<

=

$1

10 I

and

%

=

$1

10 I

by a second

order equation

;

we reject the unphysicial root such

as J1

>

0 or

IJ'

1

<

I.T21.

.In spite of large experi-

mental errors,

g3

is certainly positive. With 0

<

0,

e",

is negative when

I K ~

+

$1

2 l0lIl0 (cf. refe-

rence 151, Fig. 2). Thus, only large four spin terms

can explain the positive sign of e,. With a

Heisenberg model, B

= C

ZCL

J~

(za

:

number of a

th

a 2

a

neighbours, is positive and at low temperatures,

X

decreases with respect to the Curie-Weiss law. With

large four spin terms

1%

+

I

;t

10

1 /lo), B is ne-

gative 151, and

X

decreases with respect to the

Curie-Weiss law, in agreement with the experimental

results 19,101.

Using large cyclic folded four spin exchange,

Hetherington /11/ predicted at

T

=

1

mK a first

order transition between two particular antiferro-

magnetic orderings. With both kinds of four spin

exchange (folded an planar), we have investigated,

within the molecular field theory other possible

first order transitions between more general anti-

ferromagnetic orderings 141. Different phase dia-

grams are obtained in terms of

<

and

<

(with

ef

=

I).

Only one of these diagrams agrees with the

experimental results /12,13/. It is sinilar to

Hetherington's diagram and occurs with

1

5

1 <<

1

5

1 =

9. With J1

=

-

0.545

mK ;

J

=

-

0.14

ILK,

%

=

0.355

mK,

J 3

=

$

= 0

(similar

to the parameters of Hetherington), the scaf lphase

(3)

(two simple c u b i c a n t i f e r r o m a g n e t i c l a t t i c e s w i t h o r t h o g o n a l m a g n e t i z a t i o n ) i s s t a b l e a t T = 0 K ; we observe a t T' = 1 mK a f i r s t o r d e r t r a n s i t i o n b e t - ween t h e scaf

1

and t h e normal a n t i f e r r o m a g n e t i c phase "naf", i n agreement w i t h t h e experimental r e - s u l t s . A second o r d e r t r a n s i t i o n o c c u r s a t T~ = 1.75mK between t h e naf and t h e paramagnetic phase. It h a s n e v e r been e x p e r i m e n t a l l y 0 b s e r v e d ; b u t we know t h a t t h e c r i t i c a l temperatures p r e d i c t e d b y t h e molecular f i e l d t h e o r y a r e always l a r g e r t h a n t h e experimen- t a l v a l u e s . I n o r d e r t o c o r r e c t t h i s v a l u e , we have a p p l i e d t h e Random Phase Approximation on t h e Ha- m i l t o n i a n ( 1 ) . The f o u r s p i n t e r n g i v e a f a c t o r

<sZP3,

t h a t we can n e g l e c t w i t h r e s p e c t t o <S b

n e a r t h e paramagnetic t r a n s i t i o n . The c r i t i c a l tem- p e r a t u r e i s g i v e n by t h e same r e l a t i o n a s f o r a Heisenberg Hamiltonian, w i t h t h e same v a l u e of J , , J2, J 3 . According t o 1141 we o b t a i n w i t h J1 =

-

0.545 mK and J 2 =

-

0.14 mK,

T: RPA

-

1.2 mK

-

Tc. The s p i n wave Hamiltonian f o r t h e scaf

1

phase has been r e c e n t l y c a l c u l a t e d 1151. There a r e t h r e e doubly degenerated a c o u s t i c modes and one doubly degenerated o p t i c a l mode. The mean v a l u e of t h e t o t a l u n i f o r m m a g n e t i z a t i o n i n t h e op- t i c a l mode a t k

-

= 0 i s zero f o r s y m e t r y r e a s o n so we p o i n t o u t t h a t t h i s mode cannot be seen expe- r i m e n t a l l y by conventional n u c l e a r magnetic r e s o - nance a s suggested i n 1151. With t h e a i d of a com- p u t e r , we observe t h a t t h e f o u r e n e r g i e s of t h e s p i n wave modes a s a f u n c t i o n of t h e wave v e c t o r a r e always p o s i t i v e w i t h o u r s e t o f parameters.Thus t h e scaf

1

phase i s s t a b l e . The mean s p i n d e v i a t i o n

1

a t T = 0 i s

- -

<s?>

=

6.6 x The ground s t a t e 2

enezgy E'

-

1.696 mK i s about a f a c t o r two lower t h a n t h e molecular f i e l d energy Eo 2

-

0.742 mK and

t h u s t h e f i r s t o r d e r t r a n s i t i o n between 'the s c a f a n d t h e paramagnetic phase o c c u r s a t a h i g h e r tempera- t u r e around 2 mK. More a c c u r a t e c a l c u l a t i o n s seem n e c e s s a r y , w i t h approximations b e t t e r t h a n t h e s p i n wave t h e o r y a t f i n i t e temperature. N e v e r t h e l e s s , t h i s shows t h a t by r e a d j u s t i n g t h e exchange para- m e t e r s we can o b t a i n a good agreement w i t h t h e experimental phase diagram /12,13/ ; i . e . a t zero f i e l d a d i r e c t f i r s t o r d e r t r a n s i t i o n between t h e s c a f

1

and t h e paramagnetic phases a t T = I mK and

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