THE NEW EXPERIMENTAL RESULTS ON MAGNETIC ORDER IN bcc SOLID 3He ARE WELL EXPLAINED BY PLANAR FOUR SPIN EXCHANGE

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THE NEW EXPERIMENTAL RESULTS ON

MAGNETIC ORDER IN bcc SOLID 3He ARE WELL EXPLAINED BY PLANAR FOUR SPIN EXCHANGE

M. Roger, J. Delrieu, J. Hetherington

To cite this version:

M. Roger, J. Delrieu, J. Hetherington. THE NEW EXPERIMENTAL RESULTS ON MAGNETIC ORDER IN bcc SOLID 3He ARE WELL EXPLAINED BY PLANAR FOUR SPIN EXCHANGE.

Journal de Physique Colloques, 1980, 41 (C7), pp.C7-241-C7-248. �10.1051/jphyscol:1980738�. �jpa-

00220176�

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JOURNAL DE PHYSIQUE ColZoque C7, suppZ&m~~?z$ n i l r.' 7, Tome 41, jui7,Zet 1980, page C7-241

THE NEW EXPERIMENTAL RESULTS ON MAGNETIC ORDER I N bcc S O L I D 3 ~ e ARE WELL EXPLAINED BY PLANAR FOUR S P I N EXCHANGE

M. Roger, J.M. D e l r i e u and J.H. ~ e t h e r i n ~ t o n *

DPh-G/PSRM

-

CEN Saclay, BP. 2, 91 190 Gif-sur-Yvette, France

*

Michigan S t a t e University, E u s t Lansing, N i . 48824, USA.

RBsumB

-

Un modble B deux parambtres : Bchange

a

t r o i s s p i n s J r e t 6chanpe p l a n 21 f l u a t r e s p i n s Kp ( l e s Bchanges B deux s p i n s s o n t nBglig6s) e s t e n accord ~ u a n t l t a t i f avec l e s f a i t s expBrimentaux s u i v a n t s :

1 ) En champ maenBtique f a i b l e H < 0

:O

s t r u c t u r e de l a phase ordonn6e dBduite des expBriences rbcen- t e s de rgsonance a n t i f e r r o m a g n 6 t i q u e d ' o s h e r o f f

;a

v a l e u r s de l a s u s c e p t i b i l i t B ; de l a frBquence d e rdsonance e t de l a v i t e s s e moyenne des ondes de s p i n ;

0

t r a n s i t i o n du premier o r d r e 1 1 mK.

2) En champ H > 4 kG :

0

prBsence d'une

?base

d ' a i m a n t a t i o n f' Btonnament f o r t e h 4 kG e t lentement va- r i a b l e (de 0.6 2i 0.7 f o i s 1 ' a i m a n t a t i o n de s a t u r a t i o n e n t r e 4 e t 72 kG ;

0

diagramme de phase inha- b i t u e l avec une temperature de t r a n s i t i o n du second o r d r e Tc2(H) c r o i s s a n t avec H .

3 ) Accord avec l e s dBveloppements de h a u t e s t e m s r a t u r e s de s u s c e p t i b i l i t d e t c h a l e u r s p c c i f i q u e . A b s t r a c t

-

A two parameter model ( t h r e e s p i n exchange J and p l a n a r f o u r s p i n exchanee Kp,(no t r a n s p o s i t i o n s whatsoever) a g r e e s quan t a t i v e l y w i t h t h e foflowing experimental r e s u l t s :

I ) A t weak magnetic f i e l d H< kG :&the s t r u c t u r e of t h e o r d e r e d phase deduced from t h e r e c e n t a n t i - ferromagnetic resonance experiments of Osheroff ;

0

t h e s u s c e p t i b i l i t y , t h e resonance frequency, and t h e s p i n waves mean v e l o c i t y ;

0

a f i r s t o r d e r t r a n s i t i o n a t T = 1 rnK.

2 ) A t H > 4 kG :

0

a phase w i t h unexpected high and slow v a r y i n g m a g n e t i z a t i o n : M i n c r e a s e s from

--

0.6 Ilo t o ²0.7 t'o between 4 and 70 kC- (Vo : s a t u r a t i o n magnetization) ;

0

an unusual phase diagram w i t h second o r d e r c r i t i c a l temperature Tc2(H) i n c r e a s i n g w i t h H.

3 ) High temperature s e r i e s expansion f i t s of t h e s u s c e p t i b i l i t y , s p e c i f i c h e a t above t h e phase t r a n s i t i o n .

The experiments, performed on bcc s o l i d 3 ~ e i n t h e c u b i c s y m e t r y and thus t h e d i p o l a r n a p n e t i c i n - l a s t decade, a t low t e m p e r a t u r e s , r e v e a l e d i n t e r e s - t e r a c t i o n s g i v e no a n i s o t r o p y a t f i r s t o r d e r t i n g unexpected n u c l e a r magnetic p r o p e r t i e s . w i t h i n t h e molecular f i e l d approximation (MFA)

.

I . A t f i e l d H < 4 kG a n u c l e a r a n ~ i f e r r o m a g n e t i c o r d e r n o t p r e d i c t e d by t h e Heisenberg model

The s u s c e p t i b i l i t y m e a s u r e ~ e n t s '

'

^'2'31 i n t h e ranpe 10

5

^T

5

30 mK ~ i v e a n e g a t i v e Curie Weiss temperature 8

=

2.9

+

0.3 mK. Thus idithin a n e a r e s t neighbour Heisenberg model (HNNA)

,

we would e x p e c t a second o r d e r t r a n s i t i o n t o a n u c l e a r magnetic l o n g ranp,e o r d e r a t T

--

2 mK (Hiph temperature s e r i e s e x p a n ~ i o n s ' ~ ] gi:e t h e o r e t i c a l l y 0.69 f3 ; experimen- t a l l y , u s u a l Heisenberp: antiferrornagnets give a va- l u e of t h e same o r d e r , a s example T

-

^0.8 ⁰ f o r t h e s p i n 1 /2 a n t i f e r r o m a g n e t s ~ G ~ G ' ~ ] ) .'Experimentally t h e t r a n s i t i o n o c c u r s a t a much lower temperature T

--

1 mK and i s f i r s t o r d e r . The o r d e r of t h i s t r a n s i t i o n was f i r s t s u p ~ e s t e d by t h e o b s e r v a t i o n of an a b r u p t drop of entropy a t 1 mK [ 6 y 7 1 and i s now confirmed w i t h c e r t i t u d e by r e c e n t experiments '91

.

The r e c e n t a n t i f e r r o m a g n e t i c resonance s t u d i e s of Osherof f e t a l L g l g i v e important i n f o r m a t i o n s on t h e s t r u c t u r e o f t h e low f i e l d ordered phase.

The l a r z e resonance frequency observed i n zero f i e l d f = 8 2 5 kHz excludes a l l phases p r e d i c t e d by a Heisenberg model w i t h f i r s t and second neiphbour

interactions!lo1. These phases have s u b l a t t i c e s with

Osheroff e t a 1 observe t h r e e resonances ( o r mul- t i p l e s of t h r e e ) a t l a r g e f i e l d which proves t h e presence of one ( o r s e v e r a l ) s i n p l e c r y s t a l w i t h domains havinp only t h r e e d i f f e r e n t s p i n o r i e n t a - t i o n s ; consequently t h e d i r e c t i o n of a n i s o t r o p y i s one of t h e t h r e e axes (OOl), (010) (100). Any o t h e r a x i s would l e a d t o more t h a n t h r e e dopains ( f o r i n s t a n c e (1 10) g i v e s s i x 6 o c a i n s ) . Among t h e s t r u c t u r e s which a r e d e s c r i b e d by t h e simple p l a n e w a E e q u a t i o n :

with t h e c o n d i t i o n S. + =

cSte

(cf method of V i l l a i n [ ' O 1 ) , O s h e r i f f e t a 1 f i n d o n l y one phase w i t h t h e r e q u i r e d symmetry and w i t h t h e c o r r e c t

z e r o f i e l d resonance frequency. I t s wave v e c t o r i s K = ( l o o ) , S t c o n s i s t of (100) p l a n e s of p a r a l l e l s p i n s a r r a n p e d i n t h e sequence UP-up-down-down-. (We r e f e r now t o t h i s phase by t h e n o t a t i o n "uudd", cf f i p - l c ) . I n z e r o f i e l d , t h e d i r e c t i o n of t h e s p i n s i s p e r p e n d i c u l a r t o t h e (100) a x i s and f r e e i n t h e plane (100l(It i s a "pla-

-+

n a r a n i s o t r o n y " ) . A small f i e l d H o r i e n t a t e s t h e s n i n s p e r n e n d i c u l a r t o i t s e l f , and rerroves t h e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980738

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JOURNAL DE P H Y S I Q U E

degeneracy. This is the simplest structure descri- bing the experiment but we point out that more

-f

complicated structures(S. being the superposition of more than one Fourier component K) with the 100 ani- sotropy direction give nearly the sane resonance frequency : for example antiferromagnetic states consisting of (100) planes of parallel spins arran- ged in the sequence n plane with spins up followed by n planes with spins down ~ i v e a relative change f -f2 of frequency with respect to the uudd plane

-

n

2 equal to 6% for n = 4 and 7.4Z for n=rn.

2. AT RELATIVELY LOW FIELD H 2 4 kG, A PHASE WITH UNUSUALLY HIGH MAGNETIZATION

The previous described antiferromagnetic phase is restricted to H <

4

kG. Another phase with unexpected behaviour is observed at H

2 4

kkG. There is a second order transition at a critical tem~erature T (H), increasing with H[778'11y123131. Its magne-

cl.

tization can be deduced from the dependence, with respect to H, of the limiting pressure P(H) in a Pomeranchuk cooling as measured by Adams et a1 [81 and Codfrin et a1[12'. If we neglect the entropy : S

--

0, the thermodynamical relations give :

F -ME :mapnetization andvolume

( ^=e; $'::

^{05 'solid}

d ~ L l t Vs-VL

R: magnetization

and volume

of liquid (2)

(This equation is similar to the Clapeyran-Clausius relation, the conjugate variables (T,S) being re- placed by (H,M)). FJe ne ME, thus M is propor-

tional to the slope fig. 3 we :eve plotted the curve AP = P(H)-P(0) obtained from [ 8 , 1 2 ] at low field and from[l2lat high field. The magnetization deduced from this curve is shown on insert 3a.

Ys varies from 0.6 Mo 4 kC to 0.7 Po at H around 60

-

^{70 kG} being the saturation magnetization). The extrapolation of this curve at H = O gives M (H=O)

--

^0.59^PI

.

^{Adams et}

a1[I4] observe a broa;eninp, and ahifcof the resonance line almost independent of the

field between 4.3 and 29 kG. The undetermined shape of their sample and the weak accuracy on the temperature prevents a valuable analysis of their broadened NP!R line. Nevertheless the maximum shift

=

1.6 Gauss in the wing of the observed line corresponds fo the maximum possible

I!

demagnetizing field : 4~1I =

-

3.3 Gauss. This M

I

t1 0[8*121

.

This unex-

gives

- "

0.5 and agrees with

pected 'high magnetization at relatively low field

H

-- ⁴

kG ; the small change of Y (-- 15%) from H = 4 to 70 kG and the increase of the

critical temperatureTc2(H) with H are in com- plete contradiction with a Heisenbere model. The HNNA model gives an antiferromagnetic spin flop phase with two sublattices ;Tc2(H) decreasing mono-

4kB 6 tonicaly from 6 to zero at H

=

-

^{y U}

^--

72 kG (critical field of transition to the

se)

.

Its magnetization (at T = 0) : F = Vo

xlSRe

^is

-

linear with field and reaches the saturation at Hc

--

72 kG cf fig. 3. Thus the phase, observed in 3 ~ e has ferromagnetic tendancies but differs £roc a ferromagnetic phase which would give F = P f = C ste

S 0

(in the limit T + 0), at any H.

Some ferromagnetic tendancy is also observed in the behaviour of the susceptibility x(H+O) in the paramagnetic phase. Near the transition,

x

in- creases with respect to the extrapolated Curie Weiss law -At high temperature series expansion :

2131 has been fitted with

6

= -2.6 mK and B=-2.7mk ,

This behaviour is opposite with respect to that of a Heisenberg antiferromagnet where B = C 2 J 2 2 n (zn number of nth neighbours) is always positive.

3. THREE AND FOUR SPIN CYCLIC EXCHANGE : A THEORE- TICAL INTERPSETATION OF ALL EXPERIHENTAL RESULTS

A)

Microscopic origin of four spin exchange : simple geometrical arguments

The interaction potential of Lennard Jones between Helium atoms has a very strong repulsivehard core and a weak attractive part ; Helium atom are

thus well described by a hard sphere model (which has been successfully used to calculate the energy[151).Within this simple model the problem of exchange in Helium can be illustrated by the naive followinp; picture : suppose you stand in the sub- way at 6 p.m between "Concorde" and "Etoile" and you want to exchange place with your neighbour ; people are so squeezed that at least two or three persons have to move cyclicaly with you, "pair exchange" being practically impossible. In the geometrical exchange mecanism of n hard spheres, two parameters are essential :

1) the free space 6 available in "the box" sha- ped by the surrounding atoms in the exchange

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c o n f i g u r a t i o n ,

2) t h e t o t a l displacement d = (Xi 6 ):r

'I2,

of t h e p a r t i c l e s i n t h e 3 N dimensional space of configura- t i o n d u r i n g t h e exchange p r o c e s s . The exchange f r e - quency i n c r e a s e s w i t h 6 and d e c r e a s e s w i t h d . There i s a n optimum between t h e s e two c o n d i t i o n s which can f a v o r one type of exchange.

I n bcc He t h e f r e e space 3 6 i s much l a r ~ e r f o r t h r e e and f o u r s p i n exchange t h a n f o r p a i r exchange [ I 6 ' 17]. Although d i n c r e a s e s w i t h n, t h r e e and f o u r s p i n exchanges a r e l i k e l y t o be dominant. Higher o r d e r ( f i v e , s i x s p i n

...)

exchanges have a compara- b l e f r e e space 6 b u t a r e d i s f a v o u r e d by t h e i n c r e a - s i n g d. These c o n j e c t u r e s have been c o r r o b o r a t e d by q u a n t i t a t i v e c a l c u l a t i o n s [16'171 from which we conclude t h a t t h r e e and f o u r s p i n c y c l i c exchanges a r e most l i k e l y dominant w i t h r e s p e c t t o t r a n p o s i - t i o n and h i g h e r o r d e r exchanges. I n a bcc l a t t i c e , t h e r e a r e two kinds o f f o u r s p i n c y c l e s with f i r s t neighbours : one p l a n a r (P) and t h e o t h e r f o l d e d

(F) (cf r e f ' 7 1 ) . Our c a l c u l a t i o n s a r e too rough t o determine t h e h i e r a r c h y between them.

B) Yagnetic p r o p e r t i e s and phase diagrams w i t h i n t h e t h r e e and f o u r s p i n exchange model i n t h e H.F.A.

a ) H m i l t o n i a n :

Keeping o n l y t h r e e s p i n exchange J and f o u r t

s p i n exchanges : p l a n a r and f o l d e d KF, and n e g l e c t i n g p a i r t r a n s p o s i t i o n s , we w r i t e t h e Hamiltonian :

Q

^{i j k} i s t h e t h r e e p a r t i c l e c y c l i c permutation ope- ( T)

r a t o r ; t h e sum C i s t a k e n over t h e more compact t h r e e a t o z c y c l e s

Q

ijka i s t h e f o u r a r t i c l e cy-

(TP

^(P)

c l i c permutation o p e r a t o r ; t h e sums C

,

C a r e taken o v e r f o l d e d and p l a n a r c y c l e s . I t r e s u l t s from P a u l i P r i n c i p l e r l g l t h a t even, odd permutations a r e ferromagnetic, a n t i f e r r o m a g n e t i c r e s p e c t i v e l y ; t h u s ( c f r e f . f 9 ) , J t ,

$

and K a r e a l l n e g a t i v e .

[ I 9 1

.

I n terms of s p i n P a u l i m a t r i x oi

w i t h

Thus we w r i t e t h e e f f e c t i v e Hamiltonian :

w i t h e f f e c t i v e p a i r i n t e r a c t i o n s between nth neigh- (

: ) +

*

bours ( J n , 0.0.)

=

J UP t o n = 3 and f o u r t h orderterms.

From (5) we have :

bl Low f i e l d ordered phase

a )

One parameter models

I t i s f i r s t i n t e r e s t i n g t o d i s c u s s t h e p r e d i c - t i o n s of t h e s i m p l e s t models where one of t h e t h r e e parameters (Jt,K.,KF) only i s r e t a i n e d . 1 - T h r e e s p i n exchanpe a l o n e g i v e s an e f f e c t i v e two s p i n Hamiltonian w i t h ferromagnetic i n t e r a c t i o n s 2 -We t h e n t a k e only f o u r s p i n exchanges ( J _ = O ) Tn molecular f i e l d , t h e enerpy

?

of one i i o l a t e d f o u r s p i n c y c l e ( 1 , 2 , 3 , 4 ) : = < 'J fl _1234-+

+ e -'

_1234'

i s minimized when t h e f o u r s p i n v e c t o r s S , a r e

I

i n a same p l a n e and p e r p e n d i c u l a r t o each o t h e r :

= & i n $

n : ( 1 , 2 , 3 , 4 ) +(+,+,I,+) w i t h

E-i

^{a t}

T = O . I f

$

i s t h e dominant exchange ( t a k e

% -

^J

^--

o ) , t h e energy i s e a s i l y minimized a t t

T = 0 : i n t h e bcc l a t t i c e , t h e minimum configura- t i o n (+,+,4,+) can b e r e a l i z e d f o r a l l f o l d e d c y c l e s : we o b t a i n t h e s c a f [ 1 8 ' phase c f # i g . l a w i t h 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 orthogonal magnetization. This phase, proposed

i n t h e preceedinp; f o u r s p i n models [18,20,211 has f o u r s u b l a t t i c e s w i t h c u b i c symmetry and t h u s t h e magnetic d i p o l a r i n t e r a c t i o n s g i v e no a n i s o t r o p y a t f i r s t o r d e r ( w i t h i n t h e WA). I t i s t h u s incom- p a t i b l e w i t h t h e l a r g e a n t i f e r r o m a g n e t i c resonance frequency observed by O ~ h e r o f f ' ~ ] . The s i t u a t i o n i s more i n t r i c a t e w i t h dominant p l a n a r f o u r s p i n exchange

$

: i t i s n o t p o s s i b l e t o o b t a i n a l l p l a n a r c y c l e s w i t h t h e minimum c o n f i g u r a t i o n (+,+,+,+) t h e system i s somewhat " f r u s t r a t e d " . The r e n e r a l methods' lo' used t o minimize t h e energy w i t h q u a d r a t i c s p i n i n t e r a c t i o t s a r e t o o r e s t r i c t i v e :

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JOURNAL DE PHYSIQUE

they assume t h a t t h e ordered s t r u c t u r e i s d e s c r i b e d by ( 1 ) . Nith f o u r s p i n terms t h e s o l u t i o n c a n be more complex and be a s u p e r p o s i t i o n of more t h a n one F o u r i e r component. With a computer we used t h e f o l l o w i n g minimization procedure f o r a f i n i t e num- b e r of s p i n s with p e r i o d i c boundary c o n d i t i o n s a t z e r o temperature.

We s t a r t from randomly o r i e n t e d s p i n s ( N = O ) . For a s p i n c o n f i g u r a t i o n N, a f t e r c a l c u l a t i n g t h e a F on t h e s i t e i , t h e con- molecular f i e l d Hi =

- %

+

f i g u r a t i o n N+1 i s o b t a i n e d by t u r n i n g Si p a r a l l e l t o + H . . S t a t i o n a r y c o n f i g u r a t i o n s t h u s o b t a i n e d a r e o f t e n l o c a l minima of t h e e n e r e y . I n o r d e r t o e l i - minate t h e s e u n s t a b l e phases, we add some "thermal"

a g i t a t i o n . The c o n f i g u r a t i o n N + 1 i s o b t a i n e d w i t h

+ +

random f l u c t u a t i o n s

6

Si of t h e o r i e n t a t i o n of Si around H . s o t h a t we move through t h e c o n f i g u r a t i o n + space i n a f a s h i o n s i m i l a r t o t h e thermal a g i t a t i o n . When d e c r e a s i n g t h e temperature, i . e . t h e amplitude of random f l u c t u a t i o n s 6 + S;, we o b t a i n t h e configu- r a t i o n w i t h lowest energyZ221

.

With only p l a n a r f o u r s p i n exchange (J = K F = O ) , t h e s t r u c t u r e t h u s

t

o b t a i n e d h a s t h e maximum nunber (two o u t of t h r e e ) of p l a n a r c y c l e s " s a t i s f i e d " w i t h t h e lowest energy c o n f i g u r a t i o n ( f

,+,

J , + ) , t h e o t h e r being " f r u s t r a - ted" w i t h c o n f i g u r a t i o n ( I , + , + ,+) and t h e energy

= +

-

This phase (cf f i g . Ib) h a s ferromagne- t i c l i n e s along t h e d i r e c t i o n Ox p a r a l l e l t o one of t h e t h r e e axes ( l o o ) , (OlO), (001). P e r p e n d i c u l a r t o Ox, t h e r e a r e two p l a n a r simple s q u a r e i n t e r p e n e - t r a t i n g a n t i f e r r o m a g n e t i c s u b l a t t i c e s w i t h orthogo- n a l o r i e n t a t i o n (''ssq a£" p h a s e ) . This phase having more t h a n one F o u r i e r component i s n o t d e s c r i b e d by ( 1 ) . I n c o n t r a s t t o t h e "scaf" phase, t h i s phase has a l a r g e d i p o l a r a n i s o t r o p y . The a n i s o t r o p y a x i s b e i n g one of t h e t h e r e a x i s ( l o o ) , (010)

,

(001) r i t would g i v e t h r e e d i f f e r e n t domains i n agreement with O s h e r o f f ' s r e s u l t s . However i t s symmetry d i f - f e r s from t h a t of t h e "uudd" phase, t h e o r d e r parameter b e i n g a t r i e d r e given by two orthogonal d i r e c - t o r s ( d l , d 2 ) corresponding t o t h e two antiferroma- g n e t i c s u b l a t t i c e s . The d i p o l a r a n i s o t r o p y energy i s

by t h e s i g n of C =-

-

C 0 2 0 . 5 i s such t h a t d 1 and d a r e p a r a l l e l t o (010) and (001). The second

a 2 ~ _

d e r i v a t i v e s

---f

with r e s p e c t t o t h e r o t a t i o n s of

ae

( d l , d 2 ) around (010) and (001) g i v e two degenerate zero f i e l d l o n g i t u d i n a l a n t i f e r r o m a g n e t i c resonan- c e s R2

a ' ~ ~

and t h e second d e r i v a t i v e

7

w i t h r e s p e c t t o t h e

a~

r o t a t i o n of (d d ) around t h e a n i s o t r o p y axe 1' 2

Ox// (100) g i v e s a n o t h e r resonance Ql

(xI

and

xl1

a r e t h e s u s c e p t i b i l i t i e s f o r H respec- t i v e l y p e r p e n d i c u l a r and p a r a l l e l t o t h e p l a n e (d d ) ) . The z e r o p o i n t motion renormalize C1 by

1' 2

a f a c t o r around .85 and C by -.95(cf 5 6). Thus t h e r a t i o (C1-2Co)/2C1

=

1121.5 i s small and g i v e s Q1

=

5Q2.

The r e n o r m a l i z a t i o n f a c t o r cominp from s p i n waves i s e s t i m a t e d roughly ( c f 5

6)

t o 0.85. Thus t h e h i z h frequency : Q '

==

0.75 Ql

--

1067 kHz, (when

1

t a k i n g t h e e x p e r i n e n t a l v a l u e of X[31) i s too l a r - I

g e r g l . I n a mapnetic f i e l d , t h e d e g e n e r a t e fre?uency R2 would s p l i t i n t o two modes, g i v i n p a spectrum more comvlicated t h a n t h a t of Osheroff

191 .

The conclusion i s t h e n t h a t t h e experimental r e s u l t s of O s h e r o f f r g l cannot be explained w i t h i n a one parameter model.

p =

-

N

v

i s t h e number d e n s i t y .

C = 2.16 and C, = 5.31 were c a l c u l a t e d by numerical summation. The e q u i l i b r i u m c o n f i g u r a t i o n , determined

(6)

( a )

s c a f (b) s sq af'

(c)

_-

u u d d

Fig.1 : The bcc l a t t i c e i s s e p a r e d i n two simple- cubic l a t t i c e s ( f u l l and dashed l i n e s ) . The s c a f ( a ) and p f ( d ) phases have no d i p o l a r a n i s o t r o p y a t f i r s t o r d e r . T h e s s q a f phase (b) minimizes i t s d i p o l a r energy w i t h i t s s p i n d i r e c t e d a l o n g t h e (018) o r (001) a x i s p e r p e n d i c u l a r t o t h e f e r r o m a g n e t i c l i n e s . The a n i s o t r o p y of t h e uudd phase i s p l a n a r . The d i - r e c t i o n of t h e s p i n i s p e r p e n d i c u l a r t o t h e 100 a x i s and f r e e i n t h e 100 ferromagnetic planes(shadded a r e a s )

B)

Two parameter model ( J

t ' K ~ )

Neglecting two s p i n and f o l d e d f o u r s p i n exchanges, we can account f o r a l l experimental d a t a with a two parameter model i n c l u d i n g t r i p l e exchange J and pZanar f o u r s p i n exchange K p . For t h e energy

t

minimization, another kind of " f r t l s t r a t i o n " i s in- troduced by J t < 0 which " l i k e s " ferromagnetic t h r e e s p i n c y c l e s (f,+,+). The computer method e x l a i n e d above g i v e s t h e "s s q a f " phase f o r

! ~~1

^<

lTl.

EP

For

1

^.Jt

1

^>

/>I

we o b t a i n t h e "uudd" phase Fro- , ^#

posed by Osheroff !. I n t h i s phase, a l l p l a n a r f o u r s p i n c y c l e s have t h e same c o n f i g u r a t i o n : ( + , + , 3 , L ) w i t h

(&

⁼

- l.

4 One h a l f of t h e t h r e e s p i n c y c l e s a r e ferromagnetic w i t h

Y '

= <

e

⁺

^; ^@ ^;

^>⁼ and t h e

i i k ., 2

o t h e r have t h e " f r u s t r a t e d " c o n f i g u r a t i o n ($43) with

'

⁼

- 2

1

.

Within t h e V F A , t h e f r e e energy of t h i s phase i s :

p = <2Sz) i s t h e p o l a r i z a t i o n and S(p) t h e entropy The r a t i o of t h e c o e f f i c i e n t s of t h e p and 4 p L term i s l a r g e - : f o r example

I J ~ I -151

K - 3 $ h J t - 2 . 2 . Thus we o b t a i n a t Tcla s h a r p f i r s t o r d e r t r a n s i t i o n between t h e uudd and paramagnetic phase, with p

-

¹up t o t h e t r a n s i t i o n . A good approximation t o T i s given by

cl

t h e r e l a t i o n :

4 uudd 3

N - ~ =-T F ~Rn2=N ~ F ~ ~(p = 1) = 25

+

- K

cl t 2 P

The Curie Weiss c o n s t a n t i s

With J t = - 0 . 1 mK and Kp = -0.355 IEK we o b t a i n 9 = 2.79 mK _and'I= 1.06 mK.

c 1

The i n v e r s e p e r p e n d i c u l a r s u s c e p t i b i l i t y - - : X-l = -c-'8(-4.J +3K )

= - c - ~

1.91 6, i s i n good

t P

agreement with t h e experimental r e s u l t C 3 ] . The d i - p o l a r a n i s o t r o p y and t h e a n t i f e r r o m a g n e t i c resonance frequency were c a l c u l a t e d by Osheroff e t a l . Ve p o i n t o u t two c o r r e c t i o n s t h a t must b e a p p l i e d t o t h e v a l u e c a l c u l a t e d w i t h i n t h e !"FA : f i r s t

9'

due t o t h e z e r o p o i n t s p i n waves d e v i a t i o n s estima- t e d t o

--

0.85 by Osheroff e t a l " ] , secondly t h e atomic l a t t i c e z e r o p o i n t motion o r phonons renor- m a l i z a t i o n f a c t o r I)

,lz3'

which,by h n t e Car10 i n t e - g r a t i o n of t h e v a r i a t i o n a l wave function,we have e s t i m a t e d i n t h i s c a s e t o be around 0.8 t o 0.85 (Osheroff e t al"] d i d n o t t a k e i n t o account t h i s second f a c t o r ) . We t h u s e s t i m a t e t h e c o r r e c t i n g f a c t o r t o $RN = $s@F'

=

0.75 ⁵0.1 and f i n d f

--

800 kHz i n p e r f e c t agreement w i t h t h e e x p e r i - mental r e s u l t .

This two parameter f i t a g r e e s a l s o w i t h high temperature d a t a ; i t gives f o r t h e high tenpera- t u r e s e r i e s expansion of t h e s p e c i f i c h e a t : C = .251(;

B2 - g3

B 3 ) , 5

=

7.15 mK 2

,

i n good

v 2

3 . 1251 agreement w i t h [ 2 4 y 2 5 1 and

--

0 . 8 (mK) ( r e f

3

g i v e s

r3

^>0 i n c o n t r a s t t o t h e Heisenberp model ; i t s a b s o l u t e v a l u e i s n o t determined w i t h accuracy.

The c o e f f i c i e n t of t h e t h i r d term i n t h e i n v e r s e s u s c e p t i b i l i t y expansion of t h e paramagnetic phase i s ( c f E q . 3) e q u a l t o B = -0.5 PK 2

.

^{I t}r i v e s q u a l i - t a t i v e l y t h e observed i n c r e a s e of

x

with r e s p e c t t o t h e Curie-Weiss law (cf 1 2 ) . I t s magnitude i s small c o m ~ a r e d t o t h e experimental f i t L 3 ] , b u t i t i s p o s s i b l e t h a t a t very low temperatures t h i s f i t g i v e s , r a t h e r than e x a c t l y B, t h e c o n t r i b u t i o n of

s e v e r a l h i p h e r power tern? i n t h e

-

T 1 s e r i e ~ e x ~ a n s i o n .

el H i g h f i e l d phase

The g e n e r a l method of ~ i l l a i n ' l O ] can be a p p l i e d t o f i n d t h e o r d e r e d phase i f we assume i t appears from t h e ~ a r a m a g n e t i c phase through a second o r d e r t r a n s i t i o n , a s i s i t w e l l e s t a b l i s h e d now. I n t h e paramagnetic phase, a l l s p i n s a r e

(7)

JOURNAL DE PHYSIQUE

-+ -+

e q u i v a l e n t S. = So ( w i t h i n t h e IIFA). An ordered phase appears a t Tc through a second o r d e r t r a n s i - t i o n i f t h e s u p e r p o s i t i o n of a small component 6

xi

p e r p e n d i c u l a r t o S lowers t h e energy. The

-+

l i n e a r i z a t i o n of t h e I.*F e q u a t i o n s w i t h r e s p e c t t o

6s.

-+ g i v e s :

The s t a b l e s t r u c t u r e i s t h e one l e a d i n g t o t h e h i g h e s t c r i t i c a l temperature. By t h i s method we f i n d a t h i g h f i e l d K =

-

27T (1,0,0) which corresponds (cf f i g . Id) t o a phase w i t h two simple c u b i c l a t - t i c e s A and A ' h a v i n g t h e r e s p e c t i v e m a g n e t i z a t i o n s

-t -t

H and V symnetric w i t h r e s p e c t t o t h e magnetic A + A '

f i e l d H :

(10) This r e s u l t i s g e n e r a l w i t h Jt < 0 ,

$

and Kp ne- g a t i v e . I n t h i s phase, f o l d e d and p l a n a r f o u r s p i n c y c l e s a r e e q u i v a l e n t w i t h t h e same c o n f i g u r a t i o n

('?,f,t,?).

The c r i t i c a l f i e l d of t r a n s i t i o n a t -1

t h a n t h e v a l u e given by t h e same phase w i t h i n t h e HNNA model ( f a c t o r 2 f o r a = 0 . 5 ) .

I n c o n t r a s t t o t h e HNNA model, t h e c r i t i c a l temperature Tc2(N) i n c r e a s e s w i t h t h e f i e l d up t o a l i m i t HL (cf diagram of f i g . 2 ) , t h e n d e c r e a s e s and tends t o z e r o a t Hc.

The p r o p e r t i e s of t h i s phase a t r e l a t i v e l y low f i e l d a r e s u r p r i s i n g . With J s t r i c t l y p o s i t i v e , i n

t

c o n t r a s t t o t h e Heisenberg model, t h e a n g l e

,d

( r e f . 1 0 ) determined by minimizing t h e energy does n o t tend t o z e r o a t H = O . I n t h e l i m i t H = O we o b t a i n a spontaneous magnetization :

rJT

^(I(^r^0.53^;¹ w i t h t h e two parameter

=

Kp

f i t J =-0.1 mK ; %=-0.355 mK). The c r i t i c a l f i e l d t

H ( l i m i t N = 1 F ) b e i n g v e r y h i g h H

=

157 kG 11 v a r i e s very slowly with t h e f i e l d . This behaviour i s t h a t of t h e experimental r e s u l t s d e s c r i b e d i n 5 2. Thus t h e syrmnetry of t h i s phase i n t h e l i m i t H + 0 i s t h e same a s t h a t of a "weak ferromagnetic"

phase b u t w i t h an important d i f f e r e n c e , t h e spontaneous m a g n e t i z a t i o n a t H -+ 0 i s n o t "weak"

-

We c a l l t h i s phase a "pseudo f e r r o m a g n e t i c one"

( p f ) . H i t h increasinp: T, t h i s phase g i v e s a t some c r i t i c a l temperature Ti1 a f i r s t o r d e r t r a n s i t i o n t o t h e normal a n t i f e r r o m a g n e t i c phase ( n a f ) having t h e same symmetry a s t h e "pf" phase i n a magnetic

f i e l d b u t w i t h

$

⁺~ / 2 and ?? + 0 a t H + 0. The corresponding f i r s t t r a n s i t i o n l i n e s ends a t a c r i - t i c a l p o i n t . Crossing t h i s l i n e , we have a discon- t i n u i t y of p o l a r i z a t i o n p and a d g l e

4.

The whole phase diapram w i t h i n t h e two parameter f i t J =-0.1 mK,

t

$

= -0.355 I ~ K i s shown on f i g . 2 and compared w i t h b o t h t h e diagram p r e d i c t e d by t h e HNNA model (with t h e same 8 ) and w i t h t h e

experimental r e s u l t s .

The c r i t i c a l f i e l d Hc, of t r a n s i t i o n between t h e "uudd" and t h e "pf" phase i s Hcl= 12 kG, somewhat h i g h compared t o t h e experimental v a l u e H c , r 4 k c . But f l u c t u a t i o n s call a p p r e c i a b l y change t h i s molecular f i e l d v a l u e . A t h i r d phase appears i n a small domain i t i s a n h e l i c a l phase d e s c r i b e d by (1) , w i t h wave v e c t o r K = Z (O,O,KZ) ;

K = a r c o s

( ^- 3:212;f).

We presume t h i s i s a n a r t i f a c t of t h e mean f i e l d approximation.

The c r i t i c a l temperature of t r a n s i t i o n of t h e naf phase i n c r e a s e s up t o H

=

120 kG corresponding

F i g . 2 : Phase diagram o b t a i n e d w i t h i n t h e molecular f i e l d approximation w i t h o n l y t h r e e s p i n exchange.3 and p l a n a r f o u r s p i n exchange Kp. ( A l l o t h e r exchan- t ges a r e n e g l e c t e d ) . F u l l and d o t t e d l i n e s i n d i c a t e r e s p e c t i v e l y f i r s t and second o r d e r t r a n s i t i o n . C i r - c l e s a r e experimental p o i ~ t s f r 0 m [ ~ * l l l , 3 : f i r s t or- d e r t r a n s i t i o n , e second o r d e r ' . t r a r s i t i o n .

- C r o s s e s a r e rough v a l u e s of e second or- d e r c r i t i c a l temperature o b t a i n e d byfT2 ]. The dashed d o t t e d l i n e i s t h e phase d i a p r a n w i t h i n t h e HNNA model w i t h 8 = - 2 . 8 mK.

=

3.8 mK

H -

I

,

/ 1

I X

2 -

I ^I

I I

,

I

1 2 3 ^I^I

,

v ^Pf

^I^I/ I

1 -

^H

t

^1:

^H 1 0 0

...

The l ' m i t i g p r e s s u r e i n a Pomeranchuk c e l l mea- r e d by "12' i s p r o p o r t i o n a l t o d e energy

a t

T.0 :

+ 4? C C C t . 4 . f .

+ + + + +.+ +

-

_{o g}

u u d d

o

/y

i ^para

:

T (mK) O 0

In I

1

^I

(8)

I n t e g r a t i n g r e l a t i o n (2) we o b t a i n : H

AP(H) = P(H)

-

P(o) =

(

v ( H ) ~ H = E(H)-E(O)

a

9.; _{=AE (H)}

Fig.2 compares t h e experimental curve AP(H) o b t a i n e d from [ 8 ' 12' and t h e t h e o r e t i c a l curve AE(H,T = 0 ) . Both curves have t h e same shapes. The main d i f f e - r e n c e comes from t h e too high t h e o r e t i c a l f i e l d of t r a n s i t i o n g i v i n g a s h i f t between t h e two c u r v e s . I n s e r t 2a compares t h e experimental and t h e o r e t i c a l m a g n e t i z a t i o n c u r v e .

F i g . 3 : The experimental e q u i l i b r i u m p r e s s u r e A P = P ( H ) - P ( 0 ) i n a Pomeranchuk c e l l (dashed l i n e ) i s compared w i t h t h e t h e o r e t i c a l curve ( f u l l l i n e ) Experimental p o i n t s m a g n e t i z a t i o n ?1/1Io deduced f rom AP.

d l Spin wave spectrwn of t h e low f i e l d ordered phase IJe have c a l c u l a t e d t h e s p i n wave spectrum of t h e "uudd" phase. There a r e one doubly d e g e n e r a t e o p t i c a l mode Q+ and one doubly d e g e n e r a t e a c o u s t i c mode R- given by : (The a n i s o t r o p y i s n e g l e c t e d )

2 2 2 2 2

R,

= w -Y +rl -v

2 2 2

' I 2

t 2 [ ~ 2 ~ 2 + ~ 2 v 2 - 2 w y ~ ~ o s akx-ri

v

s i n akxl with :

w = -8J -1 2K +4(2Jt-K (cosak +cosak )

t P

2

^Y

-4Kpcos ak cosak Y y = +8 J cos akx

t

v = 2 4 ( J _t-K _P) c o s

"7 ^-

^cos^-akz ₂

q = 24Jt cos cos akz

We t e s t e d t h e s t a b i l i t y of t h e uudd phase w i t h r e s - p e c t t o t h e s p i n waves. With t h e a i d of a computer, we found w ^f> 0 f o r 10 000 p o i n t s t a k e n a t random i n t h e B r i l l o u i n zone.

A t low wave v e c t o r ka << 1, t h e a c o u s t i c mode i s

By i n t e g r a t i n g over a n g l e s we f i n d t h e "mean s p i n wave v e l o c i t y " e x p e r i m e n t a l l y deduced by Osheroff

from m e l t i n p p r e s s u r e [ 2 6 1 . The t h e o r e t i c a l v a l u e i s C

-

^6.7 m/s w i t h i n our two parameter model ; Osheroff g i v e s 8 . 4

+

^{0.4 m/s.}

4 ) CONCLUSION

With o n l y two parameters : dominant p l a n a r f o u r s p i n exchange Kp = -0.355 mK and s m a l l e r t h r e e s p i n exchange J =-0.1 mK t

--

0.3 Kp, we f i t p r a c t i c a l l y a l l . experimental r e s u l t s a t low tempe- r a t u r e on bcc 3 ~ e . A b e t t e r aereement w i t h t h e hiph temperature d a t a ( i n p a r t i c u l a r w i t h B [ ~ ] ) i s o b t a i n e d w i t h t h r e e parameters :

But t h e unwanted f e a t u r e s of t h e uhase diapram cannot be e l i m i n a t e d t h i s way, however and s o we d e c l i n e t o f i n e tune a t t h e expense of a n e x t r a parameter. From p r e l i m i n a r y r e s u l t s of kfonte-Carlo s i m u l a t i o n s w i t h c l a s s i c a l s p i n s , we t h i n k t h a t t h e f i r s t o r d e r t r a n s i t i o n c a l c u l a t e d a t 1 mK i n t h e mean f i e l d approximation happens a t a s l i g h t l y lower temperature,with our parameters. (T r 0.8 mK).

Thus w i t h i n a more a c c u r a t e approximation t h a n t h e FIFA, t h e parameter shouldbe s l i g h t l y c o r r e c t e d

t o f i t de phase diagram. This could g i v e a lower f i e l d of t r a n s i t i o n between t h e uudd and naf phase and a s l i g h t l y h i p h e r "mean s p i n wave v e l o c i t y "

i n b e t t e r agreement w i t h t h e experimental r e s u l t s .

(9)

JOURNAL DE PHYSIQUE

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t

(4) o f t h e p r e s e n t p a p e r . The f i r s t f o r m u l a of page L 141 of t h i s r e f e r e n c e must b e cor- r e c t e d f o l l o w i n g

22

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We thank D. E s t e v e f o r s u g g e s t i n g u s t h i s me- t h o d

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26

-