IMPURITIES AND VACANCIES IN SOLID HELIUM

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IMPURITIES AND VACANCIES IN SOLID HELIUM

A. Andreev

To cite this version:

A. Andreev. IMPURITIES AND VACANCIES IN SOLID HELIUM. Journal de Physique Colloques,

1978, 39 (C6), pp.C6-1257-C6-1263. �10.1051/jphyscol:19786554�. �jpa-00218043�

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JOURNAL DE PHYSIQUE

Colloque C6, supplbmenr au no 8, Tome 39, aotit 1978, page ~ 6 - 1 2 5 7

I M P U R I T I E S AND VACANCIES I N S O L I D H E L I U M

A.F. Andreev

I n s t i t u t e of Physical Problems, USSR Academy of Sciences, Vorobyevskoe Shosse 2 , 1 1 7 3 3 4 Moscow, U S 3

Rbsum6.-

I.

Diffusion quantique d'atomes 3 ~ e dans les cristaux de 4 ~ e . 2. Quasiparticules de 3 ~ e 5 une ou deux dimensions dans le volume de cristal tridimensionel d' 4 ~ e . 3. Lacunes dans le cristal d' 4 ~ e .

4.

Lacunes et proprii5t6s magngtiques des cristaux d' 3 ~ e .

Abstract.- 1. Quantum diffusion of 3 ~ e atoms in 4 ~ e crystals. 2. Two- and one-dimensional 'He quasi- within the volume of the three-dimensional 4 ~ e crystal. 3. Vacancies in 4 ~ e crystals.

4.

Vacancies

and magnetic properties of 3 ~ e crystals.

The main feature of helium crystals is the large amplitude of the zero-point motion. The straightforward consequence of this fact is the number of quantitative anomalies of solid helium which is a strongly anharmonic crystal even at zero temperature. However the more interesting thing is the qualitatively new effect of delocalization of atoms in the crystal. In ordinary crystals, atoms are lo- calized near definite equilibrium positions. They are individualized by belonging to definite lattice sites. In solid helium, there is a considerable probability of quantum tunneling of an atom to an adjacent lattice site which leads, as in liquid helium (a quantum liquid), to the picture of quantum-mecha- nically indistinguishable particles of the crystal

(a quantum crystal).

The tunneling process, in which two neighbouring particles exchange their positions, is evidently unobservable unless these particles differ in some respect. Depending on this. difference there are two types of phenomena connected with the delocalization effect. First, a direct exchange interaction of nuclear spins and nuclear magnetism appear in 3 ~ e crystals. Second, both in 3 ~ e and 'He crystals there are distinctive phenomena connected with the behaviour of impurities and point lattice defects (in particular, vacancies).

The aim of this talk is to review mainly the second type phenomena, though in He3 crystals both types of phenomena are closely connected.

I.

There is the following direct way of detecting the effect of delocalization of atoms in a crystal.

Let us consider a 'He crystal containing one 3 ~ e impurity. Even at absolute zero, this impurity atom can migrate in the crystal by the above-mentioned tunneling. Owing to the periodicity of the potential

in which the impurity atom moves, the good quantum number is not the coordinate, but quasi-momentum p. -+

The energy is some periodic function c(p) of the -+

quasimomentum. The situation here is quite analogous to the well-known case of electrons in a metal. The impurity atoms behave like quasiparticles that free- ley move through the crystal with constant velocity

/ 1 , 2 / . Their most important characteristics are the

width

A

of the energy band (or the tunneling frequen- cy J % A / H ) and velocity of motion v = ac/ap Q aJ, where a is the interatomic distance. Typical values for a 3 ~ e impurity in a 4 ~ e lattice are :

A

<

^{K, J}^<I ^{1 MHz,}

v <I

0.1 cm/s.

If the concentration of the impurities is small enough, they constitute a rarefied gas of quasiparticles. Thus the simple arguments given above imply an important conclusion on the nature of the diffusion of 3 ~ e impurities in 4 ~ e crystals /I/. Namely, a quantum diffusion must take place that has the same features as for diffusion of particles in a gas.

We can use the ordinary formula D % vl of gas-kine- tic theory to calculate the diffusion coefficient, where 9. is the mean free path of the impurities 131.

Figure 1 shows the temperature dependence of the diffusion coefficient. THere are three characteristic temperature regions. In region

I

of low enough temperature, diffusion is limited

by

impurity-impurity scattering. The diffusion coefficient is independent of the temperature, and it is inversely proportional to the concentration of impurities. With increasing temperature, the interaction of impurities with phonons also begins to play an important role. This di- minishes the mean free path. In region 11, diffusion

is limited mainly by scattering of impurities by phonons, and it decreases with increasing temperature.

3 - 1.3

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786554

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JOURNAL DE PHYSIQUE

dent here.

Fig.

I

: Temperature dependence of the diffusion coefficient at different concentrations

X g > X2 > ^{X I}

One can show /1.3/ that the diffusion coefficient is inversely proportional to the ninth power of the temperature in the phonon region 11. However, for this reason, thermally activated diffusion, which rises exponentially with increasing temperature, must begin to play a more important role than quantum diffusion at a high enough temperature. In region 111, the thermally activated diffusion mecha- nism plays the major role. Since evidently diffusion is concentration-dependent only in region

I,

increasing concentration will narrow region 11, and finally maka it vanish. The dashed curves in figure 1 demonstrate the change in the character of the temperature dependence with larger concentrations,with X > X > X .

3 2 1

Figure 2 shows the experimental data on the temperature dependence of the diffusion coefficient of 3 ~ e impurities in hcp cryscals of 4 ~ e with molar volumes near 2 1 cm3 at different concentrations, as obtained by NMR /4,5/. The curve 1 corresponds to the concentration x = 6 x 141. Here we can see the temperature regions I and 11. In the phonon region I1 thediffusion coefficient is proportional to T-", where n =

9

^f1 , which is in a good agreement with the theory. The curve 2 corresponds to the much higher concentration x =

7.5

x 141. We can see the themperature regions I and I11 here. The intermediate phonon region 11 is absent. The curve

3

corresponds to the intermediate concentration

x =

5

x 1 5 1 . All three regions are clearly evi-

Fig. 2 : Temperature dependence of the diffusion coefficient of 3 ~ e impurities in 4 ~ e crystals at molar volumes near 21 cm3 at different concentrations : 1

-

⁶^x

^X

; 2 -0.75

X /4/

;

3 - 5 x 10-2 Z 151

Figure 3 shows the concentration dependence of the diffusion coefficient in the low-temperature region I /6,7/. The experimental data fit well the solid straight line corresponding to the law Dx = 1.2 x 10-~lcm~/s.

Thus we can consider it now firmly establi- shed that 3 ~ e impurity atoms in 4 ~ e crystals behave like delocalized quasi-particles that move freely through the crystal.

2. The peculiar situation arises when two 3 ~ e impurities are located at a small relative distance of the order of the interatomic distance. Although the corresponding probability is small for dilute solu- tions, there are phenomena (for example, spin-lattice relaxation) determined mainly by such pairs of 3 ~ e impurities.

The above-mentioned experiments make it pos- sible to determine a characteristic interaction energy

U

-I, 0.1

K

of two 3 ~ e impurities positioned at a distance

r

-I, a. Significantly, the interaction energy is much larger than the band width

A

2. 1oe4K of an isolated impurity. In this cas, a displacement of one of the impurities to a neighbouring lat-

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Fig. 3 : The concentration dependence of the diffu- sion coefficient of 3 ~ e impurities in an 4 ~ e crys- tal at a molar volume of 2 1 cm3 at T < 1.2 K :

1

-

data of / 6 / ; 2

-

data of / 7 /

tice site leads, generally speaking, to an energy change much greater than A. In these conditions,the probability of tunneling is proportional to (but not to A as for tunneling with conservation of ener- gy). Hence it is negligibly small. Equally small is the probability of simultaneous tunneling of both impurities with conservation of energy. Thus, the possibility of motion of the impurities practically disappears.

There are, however, important breakes of this result, which are evident from the spin-lattice re- laxation data of Richards et al. /8/. The spin re- laxation time strongly depends on the possibility of the motion of neighbouring spins. The relaxation is the result of the spin flips due to pairwise di- pole-dipole interactions. ?he main relaxation mecha- nism is the dipolar field fluctuations arising from the 3 ~ e impurity motion. Figure 4 shows the experi- mental data /8/ on the dependence of the spin-latti- ce relaxation time T 1 on the NMR frequency for 3 ~ e impurities in hcp crystals of 4 ~ e . The most interes- ting feature of these data is the presence of reso- nance anomalies at frequencies near 1 . 5 and 3.0 MHz.

That clearly demonstrates the motion of neighboring impurities with characteristic frequency of the or- der of 1 MHz, that is of the order of the tunneling frequency of an isolated 3 ~ e impurity.

We shall see below that a pair of near-by impurities can really move in the crystal so that such a pair behaves as one distinctive quasiparticle.

Fig. 4 : The spin-lattice relaxation time for 3 ~ e impurity of three concentrations in an 4 ~ e crystal at a molar volume 21 cm3 and T = 0.53 K as a func- tion of frequency : 1

-

0.1

X

; 2

-

⁵x Z ; 3

-

2.5 x %

The distinguishing feature of these quasiparticles is that they move freely, but only over certain planes or along certain axes of the crystal lattice.

That is, they are two-or one-dimensional quasipar- ticles within the volume of the three-dimensional crystal 19-121. The complete classification of such quasiparticles in hcp-crystals was given by

Meierovich I l l / . Here we consider a number of typi- cal examples

.

Let the impurities be positioned at points A and B in a hexagonal plane of the hcp 4 ~ e crystal

(figure 5a). Here the tunneling of the first impuri- ty from the point A to the point A l is not accompa- nied by any change in the interaction energy, since the pairs of points AB and A B are equivalent in the

1

crystal. Then the second impurity can tunnel from the point B to the point B 1 , etc. One can easily understand that the system can move as a whole along a straight line parallel to the direction AAl by displacements of this type. Since the studied dis- placements to not alter the energy, the motion here is fully coherent. The pair of impurities behaves like a single one-dimensional quasiparticle that moves as a free particle, but only along the direc-

tion AA1. In more general case, the analogous one- dimensional quasiparticles may consist of two impu-

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rity atoms positioned at different liexagonal planes, other dissipative properties of helium crystals but in such a way that the point A in figure 5a is /3,10-121 one- or two-dimensionality of their motion the position of the first impurity atom, and the has not yet been directly confirmed experimentally.

point

B

is the projection of the second on the same

3. Let us come back to the temperature dependence of hexagonal plane.

the diffusion coefficient for 3 ~ e in hcp solid 'He, and consider the high temperature region I11 where the diffusion coefficient increases exponentially with the temperature. Two mechanisms of motion of

I

an impurity can occur here. First, there is the

classical thermallv activated diffusion in which the

Fig. 5 : a) A one-dimensional quasiparticle consisting of two impurity atoms ; b) A two-dimensional quasiparticle consisting of two impurity atoms ; c) A one-dimensional quasiparticle consisting of three impurity atoms

Figure 5b shows a configuration of two impurities that can move two-dimensionally. The point A is the position of the first impurity atom, and the point B, the projection of the second on the same hexagonal plane. Tunneling of the first impurity from A to Al or A does not alter the energy. The 2 same is true for tunneling of the second impurity in its own hexagonal plane to positions projected at A1 or A2. This type of a system can move over the entire hexagonal plane. The pair of impurities behaves libe a single two-dimensional quasiparticle.

Figure 5c shows an interesting example of a complex of three impurities that constitutes a one- dimensional quasi-particle/ll/. By a displacement each time of one of the impurities to a neighbouring lattice site, the initial configuration ABC can be transformed via the equivalent configurations AB C,

1 ABIC1, AB2C1 to the configuration A1B2C1, which dif-

fers from the original one by a translation along a straight line parallel to

AAl.

A dilute 3 ~ e - 4 ~ e solution contams such 3 ~ e 2 and 3 ~ e 3 "molecules" as well as isolated 3 ~ e atoms.

Although the molecular impurity quasiparticles must play an important role in

NMR,

in diffusion and

impurity atom migrates to an adjacent lattice site by overcoming some energy barrier. Second, there is diffusion caused by the presence of thermally activated vacancies. In the former case, the activation energy that characterizes the temperature dependence of diffusion equals the height of the energy barrier.

In the later case, it is the vacancy formation energy. The movement of the impurity here results from the following process. While moving in the crystal, a vacancy can occupy a lattice site nearest to the impurity atom. Then the vacancy can migrate to a site occupied by the impurity atom. finally it travels away from the site of the impurity. The process involves displacing the impurity.

In order to answer what is the nature of the thermally activated diffusion of the 3 ~ e impurities, we must compare the diffusion coefficients of the

3 ~ e atoms and of any other impurity in the very same crystal. If the activation energies match, then this will be a strong argument in favor of the vacancy induced diffusion. Apart from isotopic admix- tures, one can introduce in a controllable way and study only ions in helium crystals. We can easily calculate'the diffusion coefficient of the ions fran the experimentally measurable mobility in an exter- nal electric field 131.

Figure

6

shows the temperature dependence of the diffusion coefficient of positive ions in solid 4 ~ e at a molar volume 20.7 an3 as found by Keshishev 1131. The same figure shows the corresponding data of Grigor'ev et al. 1141 for the diffusion of 3 ~ e

at concentrations 0.75 x

lo-'

and 2.17 x

lo-'.

In the temperature region I11 where diffusion is thermally activated, not only the activation energies but also the absolute values of the diffusion coefficients coincide. The experimental data are descri- bed by the equation

D

=

6.6 x

exp(-9.5/T). This corresponds to the straight line drawn in figure 6..

The vacancy formation energy is E = 9.5 K.

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Fig. 6 : Temperature dependence of t h e d i f f u s i o n c o e f f i c i e n t of i m p u r i t e s i n an He c r y s t a l a t a mo- l a r volume of 20.7 cm3. 1

-

p o s i t i v e i o n s /13/ ; 2

-

0.75 % 3 ~ e ; 3 - 2 . 1 7 % 3 ~ e 1141

The vacancy induced d i f f u s i o n i s of a special i n t e r e s t t o us. The p o i n t i s those v a c a n c i e s a r e transformed i n t o d e l o c a l i z e d q u a s i p a r t i c l e s i n quantum c r y s t a l s . Although t h i s f a c t has n o t y e t been d i r e c t l y confirmed e x p e r i m e n t a l l y , owing t o t h e s u b s t a n t i a l experimental d i f f i c u l t i e s of observing v a c a n c i e s , we can h a r d l y doubt i t , because vacan- c i e s a r e c o n s i d e r a b l y more mobile than impurity q u a s i p a r t i c l e s . According t o t h e c a l c u l a t i o n s of Hetherington / 1 5 / , who f i r s t t r e a t e d t h i s problem, and t h o s e of o t h e r a u t h o r s /3/ t h e width of t h e vacancy energy band i s of t h e o r d e r of 1-10 K , i . e . of t h e o r d e r of t h e vacancy formation energy. I n t h i s r e s p e c t , i t i s i n t e r e s t i n g t o n o t e t h a t t h e r e i s a t h e o r e t i c a l p o s s i b i l i t y i n quantum c r y s t a l s f o r t h e e x i s t e n c e of t h e s o - c a l l e d zero-point va- c a n c i e s t h a t e x i s t i n a c r y s t a l a t a b s o l u t e z e r o l i k e zero-point v i b r a t i o n s / 1 , 1 6 , 1 7 / .

To show i t , l e t us t r a c e t h e change i n t h e energy spectrum of a vacancy a s s o c i a t e d w i t h an i n c r e a s e of t h e p r o b a b i l i t y of quantum tunneling.

I n t h e c l a s s i c a l l i m i t a vacancy

i s

l o c a l i z e d , and p o s s e s s e s a c e r t a i n energy E > 0. The presence of a small b u t f i n i t e t u n n e l i n g l e a d s t o t h e appearan- ce of a band of f i n i t e width where t h e middle of t h e band c o i n c i d e s w i t h Eo. As t h e t u n n e l i n g proba- b i l i t y i n c r e a s e s , t h e w i d t h of t h e band i n c r e a s e s ,

and t h e r e f o r e t h e minimum energy ^E ( t h e bottom of t h e band) i s reduced. Thus, i n a s t r o n g l y quantum c r y s t a l l i k e helium, a s i t u a t i o n i s p o s s i b l e i n which t h e energy & becomes n e g a t i v e . This means

t h a t t h e c r y s t a l i n i t s ground s t a t e has t o c o n t a i n a number of v a c a n c i e s .

The vacancy induced d i f f u s i o n d a t a mentioned above show t h a t t h e number of v a c a n c i e s i n 4 ~ e c r y s t a l s d e c r e a s e s e x p o n e n t i a l l y when t h e tempera- t u r e d e c r e a s e s . The minimum energy ^E i s p o s i t i v e , s o t h a t t h e zero-point v a c a n c i e s a r e l i k e l y t o be a b s e n t .

I n 3 ~ e c r y s t a l s , t h e quantum e f f e c t s a r e more d i s t i n c t due t o t h e s m a l l e r atom mass. But i n

t h i s c a s e t h e r e a r e a l s o experimental d a t a on X-ray s c a t t e r i n g 1181 showing t h a t t h e number of vacan- c i e s d e c r e a s e s more or l e s s e x p o n e n t i a l l y w i t h t h e lowering of temperature. However, i n t h e c a s e of 3 ~ e c r y s t a l s , from t h a t i t does not follow t h a t t h e r e a r e no v a c a n c i e s a t z e r o temperature.

4. The p o i n t i s t h a t t h e behaviour of v a c a n c i e s i n ' ~ e c r y s t a l s has s u b s t a n t i a l s i n g u l a r i t i e s connec- t e d w i t h t h e presence of n u c l e a r s p i n s i n t h e 3 ~ e atoms. I f a l l t h e n u c l e a r s p i n s a r e p a r a l l e l , then t h e c r y s t a l i s i d e a l l y p e r i o d i c and, owing t o t h e quantum-tunneling e f f e c t , t h e vacancy i s t r a n s f o r - med i n t o a q u a s i p a r t i c l e w i t h energy band width A ^2,10 K. A c t u a l l y , however, t h e n u c l e a r s p i n s a r e d i s o r d e r e d down t o temperatures of t h e o r d e r of t h e exchange i n t e r a c t i o n J ^2, K , s o t h a t t h e c r y s - t a l i s not p e r i o d i c . The vacancy energy spectrum problem i n t h i s c a s e i s mathematically e q u i v a l e n t t o Hubburd's model f o r e l e c t r o n s . The r e s u l t s obtained f o r t h i s model by Nagaoka /19/ show t h a t t h e energy of t h e vacancy i n t h e spin-disordered s t a t e of a b c c - c r y s t a l i s p r a c t i c a l l y always h i g h e r t h a n t h e minimum energy E corresponding to t h e bottom of t h e band i n f e r r o m a g n e t i c s t a t e by an amount of t h e o r d e r of 6. The sam'e i s t r u e a l s o f o r t h e s t a t e w i t h a n t i f e r r o m a g n e t i c a l l y o r d e r e d s p i n s . Thus, t h e ferromagqetic p o l a r i z a t i o n of t h e n u c l e a r s p i n s can be accompanied by an a p p r e c i a b l e d e c r e a s e of t h e vacancy energy.

The r e s u l t a n t s i t u a t i o n i s analogous i n many r e s p e c t s t o t h e known problem of t h e behavior of t h e e l e c t r o n i n l i q u i d helium. I t i s e a s i l y s e e n t h a t e a c h vacancy should produce i n t h e c r y s t a l a macroscopic r e g i o n i n which t h e n u c l e a r s p i n s a r e f u l l y p o l a r i z e d / 2 0 , 2 1 / .

Let t h e c r y s t a l temperature T s a t i s f y t h e

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^JOURNAL^DE^PHYSIQUE

c o n d i t i o n J << T << A. I n t h i s c a s e t h e p r i n c i p a l r o l e i s p l a y e d by t h e v a c a n c y e n e r g y r e g i o n n e a r t h e b o t t o m o f t h e band, where t h e e n e r g y s p e c t r u m i s q u a d r a t i c , ~ ( p ) = co + p 2 / 2 ~ . Here p i s t h e quasimomentum, M i s t h e e f f e c t i v e mass o f t h e v a c a n c y and i s of t h e o r d e r of H 2 / A a 2 , and a i s t h e i n t e r - a t o m i c d i s t a n c e . We assume t h a t a l l t h e n u c l e a r s p i n s a r e p o l a r i z e d i n s i d e a s p h e r e of r a d i u s R and d i s o r d e r e d o u t s i d e t h i s s p h e r e . Then t h e Hamiltonim of t h e vacancy w i t h i n t h e s p h e r e i s e q u a l t o ~ ( p ) , and a t t h e boundary of t h e s p h e r e t h e wave f u n c t i o n s h o u l d b e e q u a l t o z e r o , b e c a u s e i n t h e d i s o r d e r e d s t a t e t h e vacancy h a s a l a r g e e x c e s s e n e r g y . The e n e r g y of t h e ground s t a t e o f t h e vacancy i s i n

t h i s c a s e e q u a l t o

The r a d i u s R of t h e f e r r o m a g n e t i c s p h e r e s h o u l d b e d e t e r m i n e d from t h e c o n d i t i o n t h a t t h e f r e e e n e r g y o f t h e s y s t e m F = E

-

TS b e minimal (S i s t h e c r y s - t a l e n t r o p y change due t o t h e o r d e r i n g of t h e s p i n s i n t h e volume of t h e s p h e r e ) . It i s c l e a r t h a t S = Nln2 where N i s t h e number of 3 ~ e atoms i n t h e volume of t h e s p h e r e . The f r e e e n e r g y i s e q u a l t o

where n i s t h e number, of 3 ~ e atoms p e r u n i t volume.

From t h e c o n d i t i o n t h a t t h i s e x p r e s s i o n b e a m i n i - mum, we o b t a i n t h e r a d i u s o f t h e s p h e r e

The number of p a r t i c l e s N i n t h e volume of t h e s p h e r e i s of t h e o r d e r of ( A / T ) ~ / ~ , which i s much l a r g e r th?- u n i t y . The m a g n e t i c moment

M

of t h e f e r r o m a g n e t i c r e g i o n i s t h u s e q u a l t o

where p i s t h e m a g n e t i c moment o f t h e 3 ~ e n u c l e u s . The t e m p e r a t u r e dependence of t h e e q u i l i b r i u m num- b e r o f v a c a n c i e s N v i n t h e c r y s t a l i s d e t e r m i n e d by t h e e x p r e s s i o n exp(-F/T), where F i s o b t a i n e d by s u b s t i t u t i n g (3) i n f o r m u l a ( 2 ) . We have

The d i s c u s s e d e f f e c t s h o u l d t h e r e f o r e m a n i f e s t it-

s e l f e x p e r i m e n t a l l y i n a l l phenomena d e t e r m i n e d by v a c a n c i e s ( m o b i l i t i e s of t h e c h a r g e s and i m p u r i t i e s , h e a t c a p a c i t y , d i r e c t measurement of t h e number of v a c a n c i e s w i t h t h e a i d of X-rays). The l a r g e f e r r o - m a g n e t i c moment of t h e v a c a n c y s h o u l d , o f c o u r s e , i n f l u e n c e a l s o t h e m a g n e t i c p r o p e r t i e s o f 3 ~ e c r y s - t a l s . M e i e r o v i c h 1221 h a s shown t h a t t h e a p p e a r a n c e of f e r r o m a g n e t i c r e g i o n s i s t h e c a u s e o f t h e singu- l a r i t i e s i n t h e h i g h - f r e q u e n c y s u s c e p t i b i l i t y which a r e c o n n e c t e d w i t h f e r r o m a g n e t i c r e s o n a n c e .

I o r d a n s k i i h a s shown 1231 t h a t t h e o r d e r i n g o f s p i n s n e a r v a c a n c i e s g r e a t l y i n f l u e n c e s t h e i r m o b i l i t y .

It i s i m p o r t a n t t o n o t e t h a t t h e t e m p e r a t u r e dependence of t h e e q u i l i b r i u m number of v a c a n c i e s i n t h e h i g h t e m p e r a t u r e r e g i o n i s d e t e r m i n e d m a i n l y by t h e second t e r m i n e q u a t i o n ( 5 ) . T h i s t e r m i s a l w a y s n e g a t i v e , i . e . t h e number of v a c a n c i e s de- c r e a s e s f a s t w i t h t h e l o w e r i n g o f t e m p e r a t u r e , t h e s i g n o f co b e i n g u n i m p o r t a n t . T h e r e f o r e , t h e assump- t i o n E < 0 w i l l n o t c o n t r a d i c t t o t h e h i g h tempe- r a t u r e d a t a . Under t h i s a s s u m p t i o n , t h e 3 ~ e c r y s t a l w i t h p o l a r i z e d s p i n s i s u n s t a b l e d u e t o s p o n t a n e o u s c r e a t i o n o f v a c a n c i e s . T h i s i n s t a b i l i t y c a n i n f a c t b e o b s e r v e d i f t h e s p i n s a r e p o l a r i z e d by a n e x t e r - n a l m a g n e t i c f i e l d . A t s u f f i c i e n t l y h i g h m a g n e t i c f i e l d , t h e p h a s e t r a n s i t i o n of t h e I k i n d should o c c u r i n a p e c u l i a r s t a t e w i t h t h e f e r r o m a g n e t i c o r d e r i n g and t h e z e r o - p o i n t v a c a n c i e s . The l a t t e r b e i n g f e r m i o n s i n 3 ~ e , occupy a l l t h e s t a t e s w i t h n e g a t i v e e n e r g y , i . e . w i t h p < (-2M co)112. The i d e a of t h e p o s s i b i l i t y of t h e v a c a n c y induced ferroma- g n e t was advanced by S o k o l o f f and Widom 1241. The c o r r e s p o n d i n g H-T p h a s e d i a g r a m was found by Marchenko, M e i e r o v i c h , and t h e a u t h o r / 2 5 / . I t i s shown i n f i g u r e 7. Here VF i s t h e v a c a n c y ferroma- g n e t i c p h a s e , P i s t h e p a r a m a g n e t i c p h a s e , A i s t h e a n t i f e r r o m a g n e t i c p h a s e . The s o l i d c u r v e c o r r e s p o n d s t o t h e p h a s e t r a n s i t i o n o f t h e I k i n d and t h e dashed one t o t h e I1 k i n d . The c i r c l e s c o r r e s p o n d t o t h e d a t a on t h e m a g n e t i c f i e l d dependence of t h e orde- r i n g t e m p e r a t u r e o f s o l i d 3 ~ e by Kunrmer e t a l . /261.

The unknown p a r a m e t e r s o f t h e t h e o r y were f i t t e d t o t h e s e d a t a . The c o n c e n t r a t i o n o f t h e z e r o - p o i n t v a c a n c i e s i n VF-phase was found t o b e 6 ^x

(M/m)3/5, where m i s t h e mass o f t h e ?He atom, i . e . i t i s n o t l e s s t h a n 0.6 %.

It s h o u l d b e emphasized t h a t a t p r e s e n t t h e - r e a r e o t h e r p o s s i b i l i t i e s /27/ t o e x p l a i n t h e ano- m a l i e s i n t h e m a g n e t i c f i e l d dependence of t h e o r -

d e r i n g t e m p e r a t u r e . Only f u r t h e r e x p e r i m e n t s c a n

(8)

Fig. 7

:

H-T phase diagram of solid 3 ~ e at melting curve

answer the very interesting question on the exis- tence of the zero-point vacancies in solid 3 ~ e .

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