REVIEW OF SOME MAGNETIC PROPERTIES OF bcc SOLID 3He IN THE NEIGHBOURHOOD OF THE PHASE TRANSITION

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REVIEW OF SOME MAGNETIC PROPERTIES OF bcc SOLID 3He IN THE NEIGHBOURHOOD OF THE

PHASE TRANSITION

A. Landesman

To cite this version:

A. Landesman. REVIEW OF SOME MAGNETIC PROPERTIES OF bcc SOLID 3He IN THE

NEIGHBOURHOOD OF THE PHASE TRANSITION. Journal de Physique Colloques, 1978, 39

(C6), pp.C6-1305-C6-1313. �10.1051/jphyscol:19786563�. �jpa-00218053�

JOURNAL DE PHYSIQUE Colloque C6, suppldment au no 8, Tome 39, aolit 1978, page

C6-

1305

REVIEW OF SOME MAGNETIC PROPERTIES OF

bcc

SOLID

' ~ e

IN THE NEIGHBOURHOOD OF THE PHASE TRANSITION

A. Landesman

Service de Physique du SoZide e t de Rdsunance Magndtique, C.E.A., Omne des Merisiers, BP n02, F-91190 Gif-sup-Yvette, Frcmce

R6sum6.- On pr6sente les principaux faits experimentaux caract6ristique.s des propridtes thermodyna- miques magndtiques de l'hdlium trois solide cubique centre. On compare ensuite ces r6sultats 1 quelques modcles thdoriques proposes pour decrire ce systsme.

Abstract.- This review recalls basic surprising experimental facts about thermodynamic magnetic properties of bcc solid 3 ~ e and compares these results with some of the theoretical models proposed for this system.

1.

THE

NAIVE PICTURE.- The phase transition in bcc

%e was first observed in 1973 / I / on the melting curve at a temperature Tc z 1 mK but from experi- ments previously performed at temperatures of the order of 10Tc and above (NMR /2,3/, pressure

141)

some predictions were made on the system which are worth recalling.

Solid 3 ~ e is a lattice of atoms, presenting a large zero point motion, but still well-separated and obeying Fermi statistics. The prescriptions of these statistics can be accounted for by a spin Hamilto- nian, usually called exchange Hamiltonian,xex, ac- ting on nuclear spins which can be labelled accor- ding to the lattice sites

(I-).

The very general and

-3

useless form of this exchange Hamiltonian can be written as

where the index p runs on the N! possible permuta- tions of the N atoms of the crystal.

P

is the spin

P

operator describing the corresponding permutation of spin variables and Jp is a number (or, eventually, a phonon operator !).

The exchange frequency Jp is negative for an odd permutation and positive for an even permutation.

Naturally one would postulate that the overwhelmin- gly dominant term in equation

(I)

is a transposition between nearest neighyour atoms.

If

one neglects the effect of all other permutations, the spin Hamilto- nian is reduced to :

where Pij =

&

^{+ 2}

^_Ii . -

^{I j}

berg Nearest Neighbour Antiferromagnetic

151.

The antiferromagnetic tendancy is a priori likely : firstly nature is not very fond of ferromagnetic dielectrics; secondly the deviation of the suscep- tibility with respect to Curie law (i.e. the negati- ve sign of the Curie-Weiss constant 0 ) and the de- crease of pressure in presence of a magnetic field

/ 6 / both support antiferromagnetism (i.e. J<O). As

far as long range order is concerned, HNNA leads to antiferromagnetism with two simple cubic sublattices and a Nee1 temperature which is TN = 0.7 181 ZmK, according to theories based on high temperature ex- pansions and Pad6 approximants 171 or TN= 0.81 18

1

if one makes an experimental analogy with a well behaved electronic HNNA systems as for instance SmGaG Ill/.

One also notes that 3 ~ e is a very isotropic magnet.

The only anisotropy comes from the weak dipole-dipo- le magnetic interactions. But the dipolar energy of a cubic lattice of spins does not depend on the di- rection of the magnetization with respect to the lattice, so that an anisotropy shows up in a second order perturbation theory /8,9/. A very weak aniso- tropy means a very weak antiferromagnetic resonance and also that a very weak magnetic field Hs will produce a spin-flop phase with the magnetization of

the antiferromagnetic sublattices roughly perpendi- cular to the applied field. One has on the melting curve :

Hs

-

^{2 p ~= 1}0.65 gauss

3 a3 ( 3 )

p is the magnetic moment and a the nearest neighbour distance in the bcc lattice. This very weak anisotro- py is likely to remain whatever aspect of the

naive

The Hamiltonian (2) is usually called KNNA : Heisen-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786563

C6-

I

306

JOURNAL DE PHYSIQUE

p i c t u r e one i s l e a d t o abandon.

The v a r i o u s e x p e r i m e n t s w e l l d e s c r i b e d by t h e n a i v e p i c t u r e a r e summarized i n r e f e r e n c e / 3 / . I t i s use- l e s s t o review them h e r e . The p a r a m e t e r J of Eq.(2) f i t t i n g t h e s e e x p e r i m e n t s d e c r e a s e s v e r y r a p i d l y w i t h i n c r e a s i n g d e n s i t y and i s found i n r e f e r e n c e / 4 / :

I J I

⁼ 0.655 ( ~ / 2 4 ) ' ~ ' ' ~ m K ( 4 )

At v e r y low t e m p e r a t u r e and g o i n g up i n f i e l d , one e x p e c t s f o r a f i e l d Hs a f i r s t o r d e r t r a n s i t i o n from o r d i n a r y a n t i f e r r o m a g n e t i c phase t o a s p i n f l o p p h a s e and f o r a much h i g h e r f i e l d H c a second o r d e r t r a n s i t i o n from s p i n f l o p t o paramagnetism.

I n a mean f i e l d a p p r o x i m a t i o n f o r HNNA, one h a s H, =

-

⁼ 80 k g a u s s

?J

on m e l t i n g l i n e .

2 . THE EXPERIMENTAL FACTS.- We a r e o n l y i n t e r e s t e d h e r e i n m a g n e t i c thermodynamic a s p e c t s o f s o l i d 3 ~ e A l o t of d a t a c a n be found i n r e f e r e n c e s / 2 , 3 , 5 / o r

/ l o / .

We f i r s t r e v i e w h i g h - t e m p e r a t u r e d a t a , i . e . d a t a t o be u n d e r s t o o d i n t e r m s of a 1/T e x p a n s i o n of t h e p a r t i t i o n f u n c t i o n ; t h e n we r e v i e w low-tem- p e r a t u r e d a t a , i . e . w h a t e v e r knowledge we have on t h e t r a n s i t i o n i t s e l f , i t s n a t u r e , t h e p h a s e d i a - gram (H,T).

2. 1. High Temperature Data.

-

V a r i o u s thermodynamic q u a n t i t i e s a r e r e l a t e d t o t h e s p i n p a r t i t i o n func- t i o n of s o l i d 3 ~ e , which f o r a t e m p e r a t u r e T = I / B and a f i e l d H i s :

Z(T.H) = t r exp

{I-BWeX

⁺ BpHC

171

^{kg1} (5)}

i 1

We car. w r i t e i n a v e r y g e n e r a l f a s h i o n t h e l e a d i n g terms i n a 6 e x p a n s i o n f o r v a r i o u s i m p o r t a n t quan- t i t i e s g i v e n i n t a b l e

I.

The v a r i o u s c o e f f i c i e n t s a p p e a r i n g i n t a b l e I a r e c o n v e n i e n t l y w r i t t e n w i t h t h e n o t a t i o n o f r e f e r e n c e / 5 / . T h i s i s s u c h t h a t any J w i t h l i t e r a l s u b s c r i p t s i s some c o e f f i c i e n t of t h e e x p a n s i o n a f t h e p a r t i t i o n f u n c t i o n (5), s o d e f i n e d t h a t i t i s i d e n t i c a l t o J i f one r e p l a c e s i n e q u a t i o n (5) t h e exchange H a m i l t o n i a n

dQ

ex by t h e H e i s e n b e r g H a m i l t o n i a n ( 2 ) . We now r e v i e w some e x p e r i m e n t a l r e s u l t s which a r e i n c o n s i s t e n t w i t h HNNA. I f we a t l e a s t b e l i e v e i n t h e e x i s t e n c e o f a h i g h t e m p e r a t u r e e x p a n s i o n f o r t h e p a r t i t i o n f u n c - t i o n , we c a n use d i f f e r e n t e x p e r i m e n t s t o f i n d o u t d i f f e r e n t c o e f f i c i e n t s of t h i s expansion.

Note t h a t t h i s b e l i e f i s i n i t s e l f t h e a s s u m p t i o n t h a t t h e o v e r a l l p a r t i t i o n f u n c t i o n of 3 ~ e c a n b e f a c t o r i z e d i n t h e p r o d u c t of a l a t t i c e ( o r phonon) p a r t i t i o n f u n c t i o n and a s p i n f u n c t i o n . I f i n equa-

t i o n ( 2 ) . J i s a phonon o p e r a t o r w i t h l a r g e o f f d i a - g o n a l m a t r i x e l e m e n t s , t h i s assumption can be indeed c r i t i c i z e d / 2 7 / .

T a b l e I

High t e m p e r a t u r e e x p a n s i o n Zero f i e l d s p e c i f i c h e a t :

c ( T , o ) / N ~ ~ = 3 J:,

B~ -

3 J : ~ ~ 6 3 +

...

Zero f i e l d e n t r o p y :

S(T,O)/Nkg = l o g 2

- 2 Jix ⁺

^{6 3}

⁺ . . .

2 Magnetic s u s c e p t i b i l i t y :

x

= ( N P ~

/

kgTV) { I + 4BJxzz ⁺ I ~ B ~ J ~ , , ~ +

. . ^{. I}

x-'

⁼ ( k B v / ~ p 2 ) ( T

-

8

+ R + ...

⁾

T Zero f i e l d p r e s s u r e : P(T.0) =

$

^{N B}

dt

^{J:, +}

^{. . .}

F i e l d dependent p r e s s u r e :

P(T,H)

-

P(T.0) = 2 li(0pH)' jxZz +

.. .

O t h e r n o t a t i o n s found i n q u o t e d l i t e r a t u r e : g 2 = 12 J ~ X ;

z3

= 12 J&, ; 8 = 4 Jxzz B ( 4 ~ , , , ) ~

-

12 J;,,,

Zero f i e l d p r e s s u r e measurements / 4 / a r e v e r y r e l i a - b l e ; t h e s e measurements performed between 15 and 100 mK f o r v a r i o u s m o l a r volumes, g i v e a v e r y t r u s t w o r -

t h y v a l u e f o r J,,, g i v e n by e q u a t i o n (4) where i t i s i d e n t i f i e d w i t h J. The r e a s o n why t h e s e measurements a r e s o a c c u r a t e i s p r e c i s e l y t h e l a r g e exponent of e q u a t i o n ( 4 ) , o f t e n c a l l e d a G r u n e i s e n c o n s t a n t and d e f i n e d a s :

yJ = d 1% JXX = 18.13 ( 6 )

d l o g V

The Jxx v a l u e deduced from t h e s e measurements i s i n r e a s o n a b l e agreement w i t h t h e v a l u e deduced from t h e T - ~ term of t h e s p e c i f i c h e a t measured above 4OmK / 1 2 , 1 3 / .

To have more i n f o r m a t i o n o n s p e c i f i c h e a t o r e n t r o p y one l o o k s f o r d a t a a t somewhat lower t e m p e r a t u r e , w i t h t h e hope of l e a r n i n g something on t h e n e x t term

i n t h e e x p a n s i o n , namely JXxx. We have d i r e c t s p e c i - f i c h e a t measurements performed a t La J o l l a , o f f t h e m e l t i n g c u r v e / 1 4 / and a l s o m e l t i n g c u r v e measure- ments performed a t C o r n e l l / I 5 1 and i n F l o r i d a / 1 6 / which, t h r o u g h t h e C l a u s i u s - C l a p e y r o n r e l a t i o n , y i e l d t h e e n t r o p y of s o l i d 3 ~ e on t h e m e l t i n g c u r v e . A l l t h e s e d a t a have b e e n t h o r o u g h l y d i s c u s s e d i n r e f e r e n c e 151. The q u a l i t a t i v e s t a t e m e n t made by b o t h C o r n e l l and La J o l l a g r o u p s i s :

JxXx> 0 o r

Z 3

^> 0 (7)

T h i s i s i n c o n t r a d i c t i o n w i t h HNNA (where JXxxSJ<O).

Q u a n t i t a t i v e l y t h e r e a r e p e n d i n g d i s a g r e e m e n t s on

absolute values; the T-' term measured above 15mK and given by equation (4) is never perfectly repro- duced in experiments at lower temperatures. Conse- quently the specific heat between

1

mK and 40

mK

is not very well known. There is still room for more measurements of that sort, to check for any devia-

tion of the specific heat from the leading T - ~ beha- viour at high temperature 127'1.

Regarding the susceptibility X, we already mentioned the antiferromagnetic sign of the deviation with respect to Curie law. In other words, one has

:

8

=

4

,,,J <

o (8)

This is the result of several experiments on the melting line /17,18,19,20/ where 8 is of the order of -3 mK, a figure the accuracy of which is still

questioned. One also has some results off the mel- ting line 121,221, which are not very accurate ei- ther so far, unfortunately. The volume dependence of the susceptibility is, thermodynamically, related to the effect of the field on pressure /6/

:

The leading term in a high temperature expansion of either side of equation (9) is N(B!.I)~ x. The pressu- re measurements in finite field /5/ are probably the av

more accurate to obtain that quantity. These measu- rements /5/ were done at several molar volumes off the melting line and the question of their quantita- tive interpretation with

HNNA

was raised immediately this interpretation is impossible 1231.

We may want to go further and investigate the second term in the expansion of the susceptibility, namely JxXzZ. More generally, for a well behaved antiferro- magnet, the molecular field expression for X, i.e.

the Curie-Weiss law, is at low temperature an upper limit to the real susceptibility as shown on figure

1.

The opposite is true for solid 3 ~ e as shown in several experiments /19,20,24,25,26/ the last one being off the melting line. Figure 2 shows that for T c < T < 5 m K

By use of Table I one could express unequality

(10)

by stating that

B

f

(4 Jx,,)2 - ¹²

^J& < 0 (1 1)

For

HNNA,

this quantity would be B

E

4~~

=

02/4

> 0.

reference /2,6/ reports for a molar volume fairly

,

close to the melting line

8 =

- ^2.6

^mK

^{8'14 = I .7mK2}

and B

= (-

2.7

f

0.5) mIC2. Even if the error bar on B is optimistic, it is reasonable to think that its sign is indeed negative and unequality

(10)

seems also well established. For a temperature just above

Tc one has a susceptibility equal to approximately twice the Curie-Weiss value.

I I I I I

1 2 3 4 5 L

Tc Fig.

1 :

The deviation of the susceptibility with respect to Curie-Weiss law in the paramagnetic region. The upper curve is an experimental one taken from reference 1261. The lower one is taken from reference 1551.

2.2. Low temperature data.- The observation of the phase transition at

1 mK is either a sharp decrease

of the entropy deduced by Clausius-Clapeyron rela-

tion from melting line measurements /1,16/ or a cusp-like maximum of the susceptibility measured off the melting line 1261 (Figure 2).

The order of magnitude of Tc

I 1

mK is hard to understand with

HNNA.

One has Tc = -0.38, which is rather low.

There is also a qualitative disagreement, illustra- ted by figure

3

which shows the drop of entropy

around the transition temperature for both solid 3 ~ e and Samarium Gallium garnet which has a second order transition to an antiferromagnetic phase. By compa- rison one is led to claim that the transition in 3 ~ e is first order. However no hysteresis of any kind is ever reported. Besides the entropy of 3 ~ e falls significantly below its maximum R In 2 well above Tc, a pretransitional drop uncommon in anti- ferromagnetism. This point is dramatically shown by figure 7 of reference /5/

:

taking at 1/T expansion with a ten term series for the

HNNA

entropy, it is impossible to reproduce the experimental curve of the entropy of 3 ~ e , with any value of J, even for T _> ^2Tc.

Of course, since no neutron diffraction experiment

has ever be performed on '~e, one has no direct

JOURNAL DE PHYSIQUE

proof that in the low temperature phase there is a long range order of the spins

!

Fig.

2

:

The inverse susceptibility

of

3 ~ e as a function of temperature 1281. The two solid curves are a Curie-Weiss straight line and the experimental curve taken from reference 1261. The black dots are the approximation to the susceptibility given by the equation

The dashed curve is the estimated susceptibility for H

=

9 kG with the discontinuity calculated as is explained in the article.

The susceptibility measured in La Jolla 1261 shows a sharp maximum, at a temperature reported as 1.25

mK

rather than I

mK,

although this discrepancy might be a thermometric problem. Also reported there, is the decrease of Tc with decreasing molar volumes.

The important property relevant to the low tempera- ture phase is the decrease of the magnetization below the critical temperature (see figure 2). At Tc the magnetization drops rapidly to 40% of its maximum value and becomes temperature independent at the lowest temperature investigated.

The other important experiment to give information on the low temperature phase (s) is the Florida measurement of the melting line in applied fields / 16,331 up to 12 kG. The field dependence of the

melting pressure Pm is related to the magnetization M of the solid

vL and vs being the molar volumes of the liquid and the solid. Equation (12) is written assuming the magnetization of the liquid as negligible with res- pect to that of the solid.

Fig. 3

:

The entropy versus temperature in reduced units. Both curves are experimental

:

the solid one is 'He, taken from reference 111; the dashed one is the garnet SmGaG, taken from reference/ll/.

In low field, if the susceptibility x

=

is assu- med to be field independent, equation (12) can be integrated as

The analysis of Florida measurements 1161 with equation (13) at

3

kG does reproduce the general behaviour of La Jolla measurements 1281. Note that equation (13) provides an absolute measurement of X.

These pressure measurements also give useful and unique information on the (H,T) phase diagram on the melting curve. For H

<

4.1 kG, the transition at T

=

Tc appears as

a

rapid drop in entropy (figu- re 3) for a given pressure, the temperature Tc slowly decreasing for increasing field, as on figu- re 4, always with a drop in entropy suggesting

a

first order transition. For H

>

4.1 kG, figure 4 of reference 1161 shows that the behaviour of the sys- tem is very different

:

rather than a discontinuity

*

in entropy, one has a very rapid change in the slope

. If this change is indeed a real discontinui-

= m

ty, one is lead to believe into a second order tran-

sition of the solid along the melting line, such

that its temperature Tt increases with increasing

field, as shown on figure 4. The associated discon-

tinuity of specific heat is given by :

where Pt is the melting pressure at the transition and St the entropy at the transition.

Fig. 4 : The phase diagram of 3 ~ e as deduced from melting pressure measurements in presence of a magnetic field /16,33/. The solid line is presumed

to be a first order transition and the dashed line is presumed to be a second order transition. The inset shows an isotherm magnetization curve for T=1.4mK, as estimated in reference 1281 from the same experiments.

For a line of second-order phase transition points in the (H,T) plane, there is a corresponding dis- continuity of the susceptibility

AX = x(T<Tt)

-

x(T>Tt)

so that going down in temperature one has a sharp increase of susceptibility (or a positive "kink" in the isotherm magnetization curve)/28/. There is thm an important qualitative difference between the low field low susceptibility phase (I) for H<4.1 kG and the high field high susceptibility phase (11) for H>4.1 kG. These conclusions are drawn from mea- surements performed on the melting line, with solid-

liquid mixture present in the apparatus : surface effects might play a part in the game.

3. THE THEORETICAL EFFORT.- Several models are pro- posed to account for the experimental facts. Rather than a detailed review, what follows is more

a

con- sumer's report on these models, listed according to their ingredients.

3.1. Lattice-Dynamical Models.- If the zero field transition at Tc 2

1

mK is indeed first order, we can first think of the exchange magnetostriction 1291. The exchange

J

given by equation (4)

is

a ra- pidly varying function of the interatomic spacing.

A molecular field approximation 1301 yields a first order transition at constant pressure when the com- pressibility

K

of 3 ~ e obeys the unequality

However such a simple explanation (also leading to a change of interatomic spacing) does not hold : unequality (16) is not obeyed and one is off by a factor 60. Even the fact that being on the melting line differs from constant pressure conditions is not likely to make this explanation better.

A more sound idea in the same direction is that there ought to be, between phonons and the exchange process, a coupling i.e. equation (2) is written with

J

being an operator in phonon space. The spin- phonon partition function is then

where #,is the phonon Hamiltonian and where the trace is on both spin and phonon variables. The high temperature expansion of the partition functim ( 17) is a complicated renormalization problem and there is so far no detailed comparison of such an approach with experiments /27,31,32/. Recently an attempt was made to consider a deformation assisted pairing which can indeed make the onset of long ran- ge order a first order transition and account for the high susceptibility just above Tc 1341.

3.2. Multiple-Exchange models.- Now we forget the lattice effects, the phonons and take for granted that there is a spin Hamiltonian such as

(I)

whate- ver objections might be raised 1351. High tempera- ture expansions such as the ones given in table

I

also exist. The summation in equation ( 1 ) now goes beyond the nearest neighbour transpositions and the generalization can be made in several direc- tions :

a) One considers transpositions between spins which are not nearest neighbours to one another, with

JOURNAL DE PHYSIQUE

n e g a t i v e exchange i n t e r a c t i o n . T h i s l e a d s t o an a n t i f e r r o m a g n e t i c Heisenberg Hamiltonian w i t h a range longer than i n HNNA.

b) One c o n s i d e r s two c l a s s e s of permutations : both t r a n s p o s i t i o n s and c y c l i c permutations of t h r e e par- t i c l e s 1231. The l a t t e r have p o s i t i v e exchange inte- r a c t i o n s b u t t h e o v e r a l l s p i n Hamiltonian i s s t i l l a Heisenberg one because of t h e i d e n t i t y

Pijk

'

^{P. P.. =}

^-

(P. .+P +P

)

+ 2 i I.. (_IkAlj)

1 13 13 j k k i - 1

(18) The r e s u l t a n t Heisenberg Hamiltonian can have nega- t i v e n e a r e s t neighbour exchange and p o s i t i v e n e x t n e a r e s t neighbour exchange; t h i s explained t h e p r e s s u r e measurements i n f i n i t e f i e l d / 3 , 5 / b u t s e v e r a l f a c t s remain unexplained, a s t h e f i r s t o r d e r of the low f i e l d t r a n s i t i o n .

C) Going up i n s o p h i s t i c a t i o n i n e q u a t i o n ( I ) , one i n c l u d e s c y c l i c permutations of f o u r p a r t i c l e s 127.

36,371. The s p i n o p e r a t o r f o r such a permutation i s I391 :

'ijkk= '5% ' i j k

- - I

-

+

- ( I . . I + I . . I . + I I + I . I . ) 8 2 -1 -8 ^{-1 -J} - j m - k -k -1

+ 2

Ii.IR

( I j . I k + I i . I k ) + 2

Ii.Ij

( I i . I k +

Ik.Zd

-

2

:;._Ik _Ik.;,.

+ non h e r m i t i a n terms (19) The o p e r a t o r (19) permutes t h e c o o r d i n a t e s of s p i n s l o c a t e d a t f o u r d i f f e r e n t s i t e s ( i , j , k , R). A s p i n Hamiltonian t a k i n g i n t o account such f o u r par- t i c l e exchanges w i l l n o t look any more a s a Heisen- berg Hamiltonian. The e f f e c t of week four-spin i n t e r a c t i o n s was r e c e n t l y d i s c u s s e d with r e f e r e n c e t o a n o t h e r system / 3 8 / .

To b u i l d a s p i n Hamiltonian w i t h more and more ex- change parameters i s not a v e r y s a t i s f a c t o r y game i f one makes o n e s e l f c o n t e n t w i t h simply i n c r e a s i n g t h e number of a d j u s t a b l e parameters. Some P h y s i c a l comments s. ^t i n o r d e r . As i s almost obvious, m u l t i - p l e exchange i s b e n e f i c i a l only i f i t s amplitude i s a t l e a s t comparable t o t r a n s p o s i t i o n exchange. I n t h e frame of a Bubbard model t h e h i g h e r t h e o r d e r of exchange, t h e weaker i s t h e amplitude. A simple argument w i t h o v e r l a p p i n g b e l l shaped s i n g l e par- t i c l e wave f u n c t i o n s l e a d s t o t h e same conclusion.

On t h e o t h e r hand one may argue t h a t t h e e x i s t i n g microscopic c a l c u l a t i o n s of exchange a r e n o t t h a t f i r m t h a t t h e i r conclusions ought t o be taken a s Gospel t r u t h . An exchange i n t e r a c t i o n corresponding t o t h e c y c l i c permutation of a f i n i t e number of s p i n s i s a s s o c i a t e d t o t h e t u n n e l i n g r o t a t i o n f r e - quency of t h e corresponding polyhedron. S t e r i c argu-

ments might convince t h a t t h e e a s i e r r o t a t i o n i s that of e i t h e r a t r i a n g l e o r a t e t r a e d r o n r a t h e r than t h a t of a dumb b e l l 1401. Of c o u r s e , t h e problem i s t o prove t h a t f o u r p a r t i c l e exchange i s important b u t t h a t h i g h e r o r d e r exchange remains week, o t h e r w i s e t h e whole scheme does n o t make sense any more. Gene- r a l l y speaking, i t seems t h a t f o u r p a r t i c l e exchange i s indeed a t e n a b l e model 15,441 and i s t h e one ex- p l a i n i n g t h e l a r g e r number of experiments, a s is shown below.

d) Let us f i r s t c o n s i d e r an extreme s i t u a t i o n : the only s i z a b l e exchange corresponds t o t h e permutation of f o u r atoms making a f o l d e d r i n g (F) s o t h a t both p a i r s of o p p o s i t e c o r n e r s of t h e t e t r a e d r o n a r e se- cond neighbours 1371. I f t h e corresponding i n t e r a c - t i o n i s KF (< 0 ) , ttie exchange Hamiltonian looks a s f o l l o w s , a f t e r summation :

nn nnn

gex

= -6k&

1 Ii.Ij -

4kBYp

1

i < j i < j

F

i < j z k < l q j k % (20)

YijkL

⁼ ( t i . ~ j ) c ~ k . ~ e ) + ( ~ i . f a ) ( ~ j . ~ k ) - ( ~ i - t k ) ( ~ . ~ ) -3

-

This Hamiltonian n e c e s s a r i l y c o n t a i n s both two s p i n terms and f o u r s p i n terms. The high temperature ex- pansion of t h e corresponding p a r t i t i o n f u n c t i o n i s of course made only f o r f i r s t terms 141,421 : 8=18%

J$, = 14.6 K;. One f i n d s a p o s i t i v e s i g n f o r

~ 2 , ~

⁼

-

189.75 K; which i s a s u c c e s s and one f i n d s B = 0 which i s unexplained.

The phase diagram has been i n v e s t i g a t e d i n t h e frame of molecular f i e l d approximation /37,41,43/. What i s done i s t o guess t h e long range ordered s t r u c t u r e g i v i n g t h e lowest f r e e energy and t o c a l c u l a t e t h i s f r e e energy i n t h e molecular f i e l d approximation.

Another t h i n g t o have i n mind i s t h e almost n e g l i g i - b l e a n i s o t r o p y of 3 ~ e . So t h a t when a f i e l d i s ap- p l i e d , i t s d i r e c t i o n i s t h e o n l y p r i v i l e g e d o r i e n t a -

t i o n of t h e system. A 1 1 p o s s i b l e s t r u c t u r e s i n a f i e l d l a r g e r t h a n approximately a Gauss a r e assumed t o be s p i n - f l o p s t r u c t u r e s , i . e . s t r u c t u r e s w i t h a non z e r o m a g n e t i z a t i o n a l o n g t h e a p p l i e d f i e l d . The important t h i n g about Hamiltonian (20) i s t h a t

t h i s procedure l e a d s t o a f i r s t o r d e r t r a n s i t i o n a t Tc

- ^-

^{6.9 KF} from a paramagnetic phaSe t o a long range ordered phase w i t h f o u r s u b l a t t i c e s , l a b e l l e d

"scaf P" i n r e f e r e n c e / 4 1 / . Of course t h e very e x i s - tence of a long range o r d e r a t low temperature i s an assumption. The p h y s i c a l o r i g i n f o r t h e f i r s t o r d e r t r a n s i t i o n , a q u a l i t a t i v e d i f f e r e n c e w i t h HNNA, i s a

l a r g e term of t h e f o u r t h power i n o r d e r parameter i n the molecular f i e l d approximation t o the exchange energy i n t h e ordered phase. Besides t h a t the Hamil- t o n i a n (20) does not make a very good job. It i s d i f f i c u l t t o have a second o r d e r t r a n s i t i o n f o r any magnetic f i e l d and t h e change i n t h e o r d e r of t h e

t r a n s i t i o n f o r H>4.1 kG i s unexplained.

e ) A b e t t e r job i s done by a model w i t h more t h a n j u s t one p h y s i c a l parameter. I n t h e molecular f i e l d approximation, phase diagrams were i n v e s t i g a t e d with a s many a s f o u r parameters : t r a n s p o s i t i o n exchange t r i p l e exchange, f o u r - p a r t i c l e exchange on a f o l d e d r i n g (F) and f o u r - p a r t i c l e exchange on a p l a n a r d i a - mond ( P ) . The p l a n a r diamond i s shaped s o t h a t t h e ends of one diagonal a r e second neighbours and end of t h e o t h e r d i a g o n a l a r e t h i r d neighbours. D e t a i l s on t h e s e phase diagrams a r e i n r e f e r e n c e s /37,41.43/.

One reproduces t h e d i f f e r e n c e of o r d e r of t h e high f i e l d and of t h e low f i e l d t r a n s i t i o n , t h e s u s c e p t i - b i l i t i e s of phases I and 11. One a l s o f i n d s t h e posi- t i v e s i g n of Jxxx and t h e n e g a t i v e s i g n of B.

It remains t o be checked t h a t the magnetic s t r u c t u r e s a r e s t a b l e with r e s p e c t t o s p i n wave e x c i t a t i o n s , how the phase diagram behaves i n a random phase ap- proximation, which i s being done 1451. It i s a l s o of i n t e r e s t t o f i n d o u t t h e f r e q u e n c i e s of t h e a n t i f e r - romagnetic resonances which could be observed a s a f u n c t i o n of t h e a p p l i e d f i e l d 1431. I n f a c t even a f e r r o m a g n e t i c phase appears i n t h a t model 1411 f o r some s e t of parameters and some v a l u e s of t h e f i e l d . 3.3. Ground-State Vacancies Models.- S e v e r a l theore- t i c a l models a r e based on t h e p o s t u l a t e t h a t a t zero temperature i n s o l i d helium, t h e number of atoms i s s m a l l e r than t h e number of l a t t i c e s i t e s 1461, l e a - d i n g t o a f i n i t e c o n c e n t r a t i o n of s o - c a l l e d "ground s t a t e vacancies". Even i f no d i r e c t evidence f o r the- s e was e v e r p u b l i s h e d , one cannot r e a l l y exclude t h e p o s s i b i l i t y . These v a c a n c i e s would behave as deloca- l i z e d q u a s i - p a r t i c l e s obeying Fermi s t a t i s t i c s . a) A vacancy t u n n e l i n g i n a s p i n system has energy s t a t e s forming a band, t h e s h a p e o f w h i c h depends signi- f i c a n t l y o n t h e s p i n arrangement 1471. I f t h e tunneling m a t r i x element between n e a r e s t neighbours i s t

,

t h e bandwidth i s 2etwhere .z=8 i s t h e number of n e a r e s t neighbours. I f t h e arrangement i s a n t i f e r r o m a g n e t i c o r paramagnetic, the'bu1k"bandwidth 2wzt i s d e c r e a s e d (w<l), although a few vacancy s t a t e s remain a v a i l a - b l e i n band t a i l s e x t e n d i n g out t o t h e f u l l bandwidth, b) Since t h e bandwidth i s i n c r e a s e d i n t h e f e r r o s a - g n e t i c s t a t e , t h i s s t a t e i s t h e one where t h e vacan- c i e s w i l l have t h e lowest energy. But i f t h e s p i n s a r e coupled by an a n t i f e r r o m a g n e t i c Heisenberg i n t e -

r a c t i o n given by e q u a t i o n ( 4 ) , t h e r e s u l t a n t orde- r i n g w i l l be a compromise between both e f f e c t s 1481 and one can have a t r a n s i t i o n t o a vacancy induced f e r r o m a g n e t i c phase i f the c o n c e n t r a t i o n of vacancies i s l a r g e r than a c r i t i c a l v a l u e xc 0.1 J / t . The t u n n e l i n g frequency t i s known 1491 f o r thermally a c t i v a t e d v a c a n c i e s f o r molar volumes s m a l l e r than 23 cm3 : i t i s t = 40 mK and h a r d l y volume dependent.

Let us assume t h a t t h e t u n n e l i n g frequency h a s t h i s v a l u e even f o r molar volumes c l o s e t o t h e minimum of the m e l t i n g curve and i s a l s o r e p r e s e n t a t i v e of ground s t a t e v a c a n c i e s . Then t h e c r i t i c a l concentra- t i o n xc t u r n s o u t t o be a s l a r g e a s 0.2%. One should s t i l l p o i n t o u t t h a t t h e t r a n s i t i o n t o vacancy indu- ced ferromagnetism, because the e x i s t e n c e of t h e band t a i l s , i s found t o be f i r s t o r d e r i n molecular f i e l d approximation 1481.

c) When t h e ground s t a t e v a c a n c i e s have a temperatu- r e independent c o n c e n t r a t i o n x l e s s than x c , they s t i l l p l a y a r o l e f o r the magnetic p r o p e r t i e s i f they form s p i n p o l a r o n s 1521. When surrounded by a s p h e r i c a l ferromagnetic r e g i o n ( t h e p o l a r o n ) , t h e vacancy lowers i t s k i n e t i c energy, w h i l e exchange energy i s r a i s e d and entropy lowered. The s i z e of t h e p o l a r o n i s found by minimizing t h e f r e e . energy : i t c o n t a i n s about 30 atoms a t T = 0 K, d e c r e a s e s w i t h i n c r e a s i n g temperature and d i s a p p e a r s a t 30 mK, a l l t h e s e numbers being o n l y f u n c t i o n s of J and t . I n f a c t a s p i n p o l a r o n i s s t a b l e o n l y i f t h e r a t i o t / J i s l a r g e r t h a n some c r i t i c a l v a l u e , which i s probably t h e c a s e . Spin p o l a r o n s do n o t e x p l a i n a f i r s t o r d e r t r a n s i t i o n . But when p o l a r o n s a r e f o r - med, t h e y have an e f f e c t i v e magnetic moment of 30p and w i t h x = 2.3 x

lo-'

one e x p l a i n s t h e d i f f e r e n c e between t h e observed s u s c e p t i b i l i t y and t h e Curie- Weiss law, a s i n f i g u r e 1 . A h a r d l y l a r g e r concen- t r a t i o n of v a c a n c i e s can a l s o account f o r t h e spe- c i f i c h e a t observed i n t h e 50 mK

-

1 mK temperature range, i . e . p r e c i s e l y t h e r e g i o n where s p i n p o l a r o n s a r e formed 1531. I n f a c t i n presence of v a c a n c i e s coupled t o s p i n s , t h e p a r t i t i o n f u n c t i o n c o n t a i n s b o t h d e g r e e s of freedom and t h e expansions of t a b l e I a r e n o t v a l i d any more.

I t was noted t h a t t h e maximum of s u s c e p t i b i l i t y il- l u s t r a t e d by f i g u r e 2 has t h e c h a r a c t e r of a cusp, which could b e t h e s i g n a t u r e of t h e f r e e z i n g below T c of s p i n p o l a r o n s t o form a s p i n - g l a s s 1541, t h e n a t u r e of t h e i n t e r a c t i o n s between p o l a r o n s remains obscure.

d) More g e n e r a l l y , we could c o n s i d e r t h e system a s having two o r d e r parameters, t h e number of v a c a n c i e s

JOURNAL DE PHYSIQUE

and a magnetic polarization characteristic of the structure under consideration 1501. It was recently proposed /51/ that the low field phase I is antifer- romagnetic, while the high field transition observed in Florida /16/ is a transition from paramagnetism with no vacancies to ferromagnetism, with a discon-

tinuity of the number of vacancies which is not clam- ped any more. This accounts for the difference of susceptibilities of phases

I

and I1 1.281. Unfortuna- tely the low field transition would remain second order and the high field one would be first order while the opposite seems to be the experimental situation. Also the ferromagnetic phase stable for H > 2 kG at 0 K would have a very high concentration of vacancies (0.6%)

.

4. CONCLUSION.- Many questions remain open about the magnetic properties of 3 ~ e at very low temperature.

We tried to list some of them. Several theoretical models are proposed and probably more experiments are needed in order to make a selection. For the moment, it seems that we are studying a system with antiferromagnetic tendency without having neutron diffraction at our disposal : this was the situation for electronic magnetism thirty years ago and the present situation for 3 ~ e is somewhat frustating.

Here is a challenge : do we take bets for this mil- lenium or the next one, that we have neutron diffrac- tion results for 'He ?

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