SPI N-POLARIZED QUANTUM 3He-4He SOLUTIONS

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SPI N-POLARIZED QUANTUM 3He-4He SOLUTIONS

E. Bashkin, A. Meyerovich

To cite this version:

E. Bashkin, A. Meyerovich. SPI N-POLARIZED QUANTUM 3He-4He SOLUTIONS. Journal de Physique Colloques, 1980, 41 (C7), pp.C7-61-C7-73. �10.1051/jphyscol:1980711�. �jpa-00220148�

JOURNAL DE PHYSIQUE CoZZoque C7, supple'ment au n o 7 , Tome 41, juiZZet 1980, page C 7 - 6 1

SPIN-POLARIZED QUANTUM 3 ~ e - 4 ~ e SOLUTIONS

E.P. Bashkin and A.E. Meyerovich

I n s t i t u t e for PhysicaZ Problems, Academy o f Sciences, Vorobyevskoe shosse, 2 , Moscow 117334, USSR.

R6sum6.- Nous consid6rons les phenomgnes inherents aux solutions quantiques de 3 ~ e - 4 ~ e et autres systSmes de Fermi 3 forte polarisation de spin. Nous Studions toutes les propri6t6s thermo, hydrodynamiques, cinStiques et de transport de ces solutions, ainsi que leurs transitions de phase. Nous suggcrons une th6orie simple qui n'implique pas dlhypothSses concernant le modele utilisg. Nous prgdisons plu- sieurs ph6nomSnes (ef f ets magn6tocin6tiquest superf luidit6 de 3 ~ e dans la solution, etc.) qu'il semble possible d'observer experimentalement 3 l'heure actuelle.

Abstract.- The phenomena inherent in the quantum solutions of 3 ~ e - 4 ~ e and other spin-polarized Fermi systems are considered. All the thermo and hydrodynamic, kinetic and transport processes and the phase transitions in solutions are inves- tigated. The suggested simple theory implies no model assumptions. Many phenomena predicted (magnetokinetic effects, the superfluidity of 3 ~ e in the solution, etc. )

seem to be within the reach for the experimental observation just at this moment.

1. I n t r o d u c t i o n

The p r o p e r t i e s of spin-polarized quantum Fermi systems can be d e s c r i b e d v e r y w e l l by t h e e q u a t i o n s of t h e u s u a l Landau t h e o r y of Fermi l i q u i d s . A s a re- s u l t main c h a r a c t e r i s t i c s of a system a r e expressed i n terms of a phenomeno- l o g i c a l Fermi-liquid f -f unc t i o n . Unfor- t u n a t e l y , t h e c o n s i s t e n t microscopic c a l c u l a t i o n of t h e f - f u n c t i o n f o r spin- p o l a r i z e d systems a s f o r o t h e r Fermi li- q u i d s i n most of the c a s e s is impossible.

Even the r e l a t i o n s between t h e v a l u e s of t h e Fermi-liquid f u n c t i o n f o r t h e same system a t d i f f e r e n t degrees of t h e spin p o l a r i z a t i o n a r e unknown. Such a r e l a t i o n i s e s t a b l i s h e d only i n t h e case of low-polarized Permi systems, when t h e f-function p r a c t i c a l l y i s not a f f e c t e d by the p o l a r i z a t i o n .

From t h i s p o i n t of view t h e v e r y i n t e r e s t i n g systems a r e low-density E ' e r m i l i q u i d s , f o r which a l l t h e p r o p e r t i e s can be e v a l c a t e d a t a l l degrees of t h e p o l a r i z a t i o n of a spin system. Many of r e s u l t s d e r i v e d f o r such o b j e c t s a r e also common f o r dense Fermi l i q u i d s . However t h e r e a r e p r a c t i c a l l y no examples of de- g e n e r a t e low-density Fermi ^liquids be- cause f o r r e a l gases t h e condensation be- g i n s before t h e e f £ e c t s of quantum aege- neracy become important. Maybe the o n l y e x c e p t i o n i s a degenerate Permi bas of

atoms d i s s o l v e d i n s u p e r f l u i d He 11.

F o r t h i s reason t h e i n v e s t i g a t i o n of spin-polari zed quantum s o l u t i o n s of 3t1 e- 4 ~ e i s of t h e p h y s i c a l i n t e r e s t no-t only

i n i t s e l f but provides v e r y important information about the n a t u r e and t h e o r i g i n of many phenomena common t o d i f -

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980711

JOURNAL DE P H Y S I Q U E

f e r e n t types of Eermi systems.

Below we s h a l l t r y t o give i n the s i m p l i e s t form the q u a l i t a t i v e descrip- t i o n of the main processes i n low-densi- t y spin-polarized Permi systems s i m i l a r t o ' ~ e ! - ' ~ e . More p r e c i s e r e s u l t s and the q u a n t i t a t i v e a n a l y s i s can be found i n the recent reviews /I / .

2. Uain p r o p e r t i e s of nonpolarized %e -

%e s o l u t i o n s

The b e - ^{He I1} mixture i s the Fermi l i q u i d of 3 ~ e p a r t i c l e s i n the superf l u i d Rose background of ' ~ e . The number of Bose-type e x c i t a t i o n s analogous t o pho- nons and rotons i n pure 4 ~ e vanishes ra- pidly a s the temperature decreases, and a t low temperatures most of thermodyna- mic and k i n e t i c c h a r a c t e r i s t i c s of t h e s o l u t i o n are determined by t h e impurity (Fermi) component exclusively.

I n accordance with the Landau

-

meranchuk theory / 2 / 3 ~ e impurity atom being introduced i n t o s u p e r f l u i d 4 ~ e be- comes a delocalized quasi-particle with a b i g & Rroglie wavelength. The energy spectrum of such quasi-particles with not very l a r g e momenta $ i s quadratic i n

where fo i s the binding energy, isnd M stands f o r the q u a s i - p a r t i c l e e f f e c t i v e mass.

F i f t e e n y e a r s ago the phenomenon of the f i n i t e s o l u b i l i t y of %e i n He I1 a t zero temperature was discovered /3/

( t h i s f a c t corresponds the negative s i g n to of 4 ^{i n} f$. ⁽¹⁾ a t low pressures).

It was found out t h a t %e and %e a t T=0 a r e miscible up t o t h e %e concentration

of 6.5%. That i s wby t h e system of

q u a s i - p a r t i c l e s ( 4 = ^-2.8 K, M = 2. 3m3, h3 i s t h e He atomic mass) being d i l u t e 3 always becomes degenerate a s the teapera- t u r e decreases. The degeneracy temperatuse To and the Fermi momentum po a r e small corresponding t o the low concentration of 3 ~ e i n the s o l u t i o n x:

Here N i s a number of ' ~ e atoms i n a unit 3

volume, i s an atomic radius. For t h e s e reasons the system of dissolved ' ~ e atoms behaves when T (< To a s a degenerate P e p m i gas of slow quasi-particle s 4 /k ^V1

* ^x: v3 << Y

The study of t h e i n t e r a c t i o n of qua- s i - p a r t i c l e s i n such a system p r e s e n t s no s p e c i a l d i f f i c u l t i e s . It i s w e l l known

'4 / that the s c a t t e r i n g amplitude of slow p a r t i c l e s can be e a s i l y evaluated by expanding i n a s e r i e s i n even powers of a small momentum f 2C ( L! = ^{0; 1} ; 2.. . -

i s a n o r b i t a l moment of c o l l i d i n g part- i c l e s ) . The f i r s t term of t h i s expansion corresponds t o the s-wave s c a t t e r i n g ( = 0). The s-wave cross-section

i n the case of slow p a r t i c l e s does not depend on momenta ( a i s the s-wave s c a t t e r i n g length).

The Fermi-liquid f u n c t i o n can be ex- pressed in terms of t h e two-particle s c a t t e r i n g amplitude by means of t h e per- t u r b a t ion theory which formally coincides

i n our case with the expansion i n powers of low d e n s i t y x ' / ~ . Por instance i n f i r s t - order p e r t u r b a t i o n theory ( t o the princi- p a l approximation i n the concentration) the f-function i s defined by t h e forward-

arrd

s c a t t e r i n g amplitude does not depend on t h e concentration of the s o l u t i o n

where 3, f ' a r e spins of Fermi-liquid e x c i t a t i o n s . As a r e s u l t one obtains t h e expression f o r t h e f-function i n the form of the s e r i e s i n powers of %e concentra- tion. This expression contains only one unknown parameter - the s-wave s c a t t e r i n g length a

.

The value of was determined /6/

from t h e comparison between the r e s u l t s of such c a l c u l a t i o n s and a l l a v a i l a b l e experimental d a t a on the p r o p e r t i e s of ' ~ e - ^$e solutions. It was found aut t h a t

a = - 1.5 0 ^A ( 5 ) The negative s i g n of the s c a t t e r i n g length ( 5 ) corresponds t o an a t t r a c t i v e i n t e r - a c t i o n between be q u a s i - p a r t i c l e s in the solution. This a t t r a c t i o n permits the propagation of s p i n waves i n s o l u t i - ons and enables the s u p e r f l u i d i t y of %e i n a s o l u t i o n t o be caused by the u s u a l BCS +Wave pairin@; of 3 ~ e q u a s i - p a r t i c l e s These p r o p e r t i e s of %Ie - ^'He s o l u t i o n s a r e aust opposite t o t h a t of pure b e .

This approach does not need any mo- d e l assumptions about the i n t e r a c t i o n

which are necessary f o r o t h e r d e s c r i p t i - ons such as well known /7/ .

3. Fermi-liauid i n t e r a c t i o n i n Be ? - 4 ~ e s o l u t i o n s

The nuclear spin system of he i n d i l u t e mixture can be polarized simply by means of the e x t e r n a l magnetic f i e l d . The degree of the p o l a r i z a t i o n i s defined by t h e r e l a t i o n between the magnitude of

the f = e l d 2 ~ ( 4/i ⁼ 0,OR W k O e i s

-

t h e %e nuclear magnetic moment ) , ^{t h e}

Fermi-degeneracy temperature To

(To = ^2.6 x 2'3 K) and the temperature T.

It i s obvious t h a t magnetic f i e l d s H < ^{200 K} Oe which a r e a v a i l a b l e a t t h i s moment can n o t appreciably p o l a r i z e t h e high concentrated s o l u t i o n s ( x > o,?%), and i n order t o obtain the high degree of the p o l a r i z a t i o n , may be i t w i l l be ne- cessary t o use the methods which a r e s u i t a b l e f o r p o l a r i z a t i o n of the pure

%e ( /8/ - ^{/lo/ ).} However t h e way of obtaining the p o l a r i z a t i o n of the s o l u t i- on i s not e s s e n t i a l f o r f u r t h e r results.

Therefore f o r the sake of s i m p l i c i t y w e suppose f u r t h e r the s o l u t i o n t o be pols- r i z e d by the e x t e r n a l magnetic f i e l d H. +

Otherwise the magnetization of the solu- t i o n ( o r , t h a t i s j u s t the same, t h e po- l a r i z a t i o n degree)and not the f i e l d in- t e n s i t y 8 would be t h e b a s i c thermodyna- mic v a r i a b l e .

A uniform e x t e r n a l magnetic f i e l d does not a f f e c t the motion of an i s o l a t e d uncharged Fermi-parti c l e of ' ~ e , nor does it a f f e c t t o the n o n r e l a k i v i s t i c approximation t h e two-particle interac-

JOURNAL DE PHYSIQUE

tion. However when a inagnetic f i e l d i s applied the occupation numbers f o r the p a r t i c l e s with the d i f f e r e n t spin orien- t a t i o n change and consequently, the Femi-liquid e x c i t a t i o n energy which i s the f u n c t i o n a l of t h e d i s t r i b u t i o n f u n c t i o n a l s o change .

I n p r i n c i p l e i n the case of not too high magnetic f i e l d s a l l the conclusions of Sec. 2 concerning the s e m i - l i q u i d in- t e r a c t i o n i n unpolarized s o l u t i o n s a r e valid. Apparently, since f -function (43

does not depend on momenta of p a r t i c l e s , the expression f o r f-function i n a mag- n e t i c f i e l d t o the main approximation i n the concentration does not depend on t h e f i e l d and coincides with i t s value (42.

The f i e l d dependence of t h e f-function appears only i n higher orders of the p e r t u r b a t i o n theory. /11/

This conclusion i s not t r u e f o r t h e s o l u t i o n with the high degree of t h e po- l a r i z a t i o n . Owing t o P a u l i ' s exclusion p r i n c i p l e only p a r t i c l e s with oppositely d i r e c t e d spins i n t e r a c t i n tihe s-wave

s c a t t e r i n g f o r which the f-function i n the absence of t h e magnetic f i e l d has been considered above.

That i s why i n the highly polarized s o l u t i o n when p r a c t i c a l l y a l l the 3 ~ e q u a s i p a r t i c l e s p i n s have t h e same orien- t a t i o n , the s-wave s c a t t e r i n g becomes uneffective. I n t h i s case the Fermi-li- quid i n t e r m t i o n i s determined by the p-wave s c a t t e r i n g .

I n the spin-polarized s o l u t i o n t h e i n t e r a c t i o n only between p a r t i c l e s on the

Fermi surface with the r i d i u s ~ ~ = 2 ~ / ~ ~ ~

t a k e s place. The p-wave s c a t t e r i n g ampli- tude i n t h e centerof-mass frame i s pro- p o r t i o n a l t o the square of t h e r e l a t i v e momentum of c o l l i d i n g p a r t i c l e s p 4

.,? P 4 5(rgz j = 3 6 ' 2 ~ ~ ~ 1 ² $ =,lrd&]ws -

a P ⁽⁶⁾ P

where i s t h e angle of s c a t t e r i n g i n the center-of-mass frame and the constant

8 i s of the order of t h e gas-kinetic 3

volume of the atom lV 5 . Then t h e Permi-liquid f u n c t i o n .f(@ ⁽ 8 ^{i s t h e}

angle between the momenta of c o l l i d i n g p a r t i c l e s 3 ^{and pa} + ^{: p,} = ^pp = pF, p = 2pF s i n (8/2)) i s determined i n t h e f i r s t o r d e r of t h e p e r t u r b a t i o n theory by t h e antisymmetrized forward-scattering amplitude (6) ( = 0) / I ? , 12/

With the a i d of expressions f o r the f -function one can c a l c u l a t e very simply a l l the Fermi-liquid c h a r a c t e r i s - t i c s of the ~ e t - 3 4 ~ e solution. Unfortu- n a t e l y the numerical value of the b a s i c c h a r a c t e r i s t i c of the p-wave s c a t t e r i n g

1 i s unknown a t t h i s moment.

4. Thermo- and hydrodynamic p r o p e r t i e s of

f - ^{4 ~ e} s o l u t i o n s

I n a n i s o t r o p i c Fermi l i q u i d of spin-1/2 p a r t i c l e s , the Fermi surf aces of q u a s i p a r t i c l e s with d i f f e r e n t l y o r i - ented s p i n s c o n s t i t u t e two Fermi spheres,

3 3

whose r a d i i p, and p- (p+ +

pz=

determined by the degree of p o l a r i z a t i o n of the s p i n system. That i s why t h e change of thermodynamic c h a r a c t e r i s t i c s of the s o l u t i o n depending on the magnetic

p o l a r i z a t i o n i s b a s i c a l l y connected with the change of Fermi momenta p+ (H) under t h e a c t i o n of the f i e l d . Corresponding e f f e c t s could be observed a l s o i n an i d e a l gas of impurity e x c i t a t i o n s . The e f f e c t s connected wiVn t h e change of the i n t e r - a c t i o n under the p o l a r i z a t i o n in t h e t h e r modynamics a r e considerably l e s s impor- t a n t than i n t h e case of t r a n s p o r t pheno- mena.

The osmotic pressure and the second sound v e l o c i t y (the a c o u s t i c o s c i l l a t i o n s i n t h e system of impurity e x c i t a t i o n s ) are those thermodynamic q u a n t i t i e s which a r e t h e most s e n s i t i v e ones t o t h e change of the degree of p o l a r i z a t i o n . The change of such q u a n t i t i e s , which have no d i r e c t con- t r i b u t i o n from t h e pure ' ~ e and a r e propo*

t i o n a l t o some degree of t h e Permi momen- tum, t u m s out t o be highly considerable.

The osmotic pressure i n the d i l u t e mixture i s equal t o

4

where i s the chemical p o t e n t i a l of 3 ~ e i n the s o l u t i o n . Since t o t h e p r i n c i p a l approximation i n concentration t h e 3 ~ e chemical p o t e n t i a l i n t h e absence of t h e magnetic f i e l d i s equal t o

and i n t h e completely polarized s o l u t i o n

2 3 I 3

/$*

c+fi/h',

it i s c l e a r t h a t the osmotic pressure z / i n c r e a s e s from t h e value f l ~ (215) (e/ ₂

J t o t h e v a l u e f l z ( 2 / ~ ~ ( f i /

zr// 4 i.e. approximately by a f ao- t o r 22'3 ( t h e r e a r e more exact expressi- ons i n /I/).

The change of t h e second sound velo- c i t y i s analogous t o t h e magnetoosmotic e f f e c t mentioned above. To the main appro- ximation i n the concentration t h e speed

of the propagation of t h e density vibra- t i o n s i n t h e impurity e x c i t a t i o n gas i n the s o l u t i o n i s equal t o $z(4/Mj(F/ad

As a r e s u l t S2 increases approximately by a f a c t o r ~''3 from the value ^{$ 2} / f i M

t o the value 6 ⁿ e / d ? f l (more e x a c t l y see / I , 11, 12/). The r e l a t i v e change of t h e square of the second sound v e l o c i t y as a f u n c t i o n of t h e magnetic f i a l d i s shown i n Fig. 1. ( X = ~ ~ P / G ,

T, = ~ ' / z M f

Fig. 1

The p o l a r i z a t i o n of the s o l u t i o n a l s o l e a d s t o t h e e s s e n t i a l change of the equilibrium phase diagram of t h e system.

Namely t h e s o l u b i l i t y curve f o r b e i n 4 ~ e s h i f t s considerably /a/. With t h e a i d of t-he approximate values of

r 3

mentioned above it i s easy t o show t h a t the s o l u b i l i t y i n t h e degenerate f u l l y -

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polarized mixture decreases nearly by a f a c t o r 2. However it i s r a t h e r d i f f i - c u l t t o observe % h i s e f f e c t since f o r t h i s purpose the high-concentrated solu- t i o n s should be polarized.

Most of o t h e r thermo- and hydrodyna- mic phenomena i n t h e 3 ~ e f - ' ~ e d i l u t e mixture a r e l e s s s i g n i f i c a n t than dis- cussed above.

5. Magnetokinetic e f f e c t s

The influence of a spin p o l a r i z a t i - on on k i n e t i c p r o p e r t i e s of s o l u t i o n s i s much more pronounced than on thermodyna- mic ones. A l l the t r a n s p o r t phenomena a r e changed by the p o l a r i z a t i o n of fermion spins i n a r a t h e r dramatic manner. It i s due t o t h e rapid increase of t h e b e quasi-particle mean f r e e path and,conse- quently, the v a l u e s of k i n e t i c coef f i c i - e n t s such a s the thermal conductivity and v i s c o s i t y , a s a r e s u l t of t h e g o l a r i z a t i -

on.

The mean f r e e path of 'fie quasi-par- t i c l e s i n the degenerate s o l u t i o n i s in-

2 v e r s l y proportional t o I( ( T / T )

(here Ni i s t h e number of s c a t t e r i n g c e n t r e s i n a u n i t volume, stands f o r the s c a t t e r i n g cross-section, and the l a r g e f a c t o r (To/T) 2 i s due t o the non- e f f e c t i v e n e s s of c o l l i s i o n s i n the d e g e n ~ r a t e Fermi system). The s-Rave cross-sec-

2

t i o n G.v 70 (31, and i n the case of s - s c a t t e r i n g the mean f r e e paths

4

quasi-particles with opposite spins a r e

2 - I 2

proportional t o (N- ₊ To ) ( a ^{/ T )}

( N ~ a r e the numbers of p a r t i c l e s with d i f f e r e n t spin d i r e c t i o n s p e r u n i t vol-

ume, N+ N- = N3; a s a r e s u l t of P a u l i f s exclusion p r i n c i p l e only c o l l i s i o n s of f ermions with opposite spins a r e e s s e n t i - a l f o r the s-scattering). I n t h e absence of t h e p o l a r i z a t i o n N+ = N- and

I n high f i e l d s ~ i n e t i c c o e f f i c i e n t s i n c r e ase p r a c t i c a l l y without l i m i t s a s t h e number of p a r t i c l e s N- with s p i n s opgo- s i t e t o the f i e l d vanishes and inarea- ses. The l i m i t i n g value of [ i n the po- l a r i z e d s o l u t i o n (N- = 0) is d i c t a t e d by p-scattering w i t h t h e cross-secti on (6):

The growth of the v i s c o a i t y I ""

the thermal conductivity X c o e f f i c i e n t s a s a f u n c t i o n of an e x t e r n a l magnetic f i e l d = . Z ~ H / ? i s presented graf i c a l l y i n Fig. 2 (curve 1 - ^f (z),$'q

2 - ar/z)/alo) ^).

Fig. 2

The corresponding c a l c u l a t i o n s took in- t o account only the s-scattering and supposed the degeneracy of a l l t h e Fermi component of the s o l u t i o n including the

i m p u r i t i e s with s p i n s opposite t o the f i e l d . For t h i s reason N- tended t o zero when 3[ -1, and k i n e t i c coeff icientS 2 ^,

and &' became inf i n i - t e hen -1 a s Pig. 2 shows. ~ c t u a l l y / l 3 / a t nonzero temperature N- # 0, and i n high f i e l d s

2 3 I k i n e t i c c o e f f i c i e n t s grow slower than it i s shown i n Fig. 2:

More d e t a i l e d d e s c r i p t i o n of t r a n s p o r t phenomena i n almost coapletely polarized

s o l u t i o n s can be found i n ~ e f s . / l , 13/

.

As a r e s u l t the values of k i n e t i c c o e f f i c i e n t s r e s t r i c t e d i n high f i e l d s only by p-scattering (10) /12/ $re many times l a r g e r (xw4l3 * 1) than t h e i r values i n the abscence of the f i e l d .

The main experimental d i f f i c u l t y f o r observing the magnetokinetic e f f e c t s i s t o achieve considerable degree of the po-

3 _ern.

l a r i z a t i o n of the He nuclear s p i n sy-t A f i e l d H ~ 7 5 kOe can p o l a r i z e a t a suf- f i c i e n t l y low temperature (2pH

-

a s o l u t i o n with a %e concentration a d l o 4 ^(TF d 8mK). I n these conditions the %e atom mean f r e e path grows more

t h a n 105 times and runs up t o t e n s of c- timetres. It r e s u l t s i n the g i a n t growth of o t h e r k i n e t i c c o e f f i c i e n t s (e.g. the hydrodynamic v i s c o s i t y increases t o a va- l u e exceeding s e v e r a l c e n t i p o i s e s which i s l a r g e r t h a n the v i s c o s i t y of the water).

I n such circumstances one can enco- unter some nonlocal e f f e c t s inherent i n the Knudsen regime, such a s the tempera- t u r e jump o r the radiometric e f f e c t . Ca- p i l l a r s about s e v e r a l centimetres i n di-

ametres become superleaks permeable only t o the s u p e r f l u i d component of the l i q u i d . Such a growth of t h e mean f r e e path pre- s e n t s a l s o some s p e c i a l d i f f i c u l b i e s f o r measuring of the k i n e t i c c h a r a c t e r i s t i c s . As the r e s t of k i n e t i c c o e f f i c i e n t s do, the r e l a x a t i o n time 7 a l s o increa- ses. This phenomenon reduces the region of existance of hydrodynamic o s c i l l a t i - ons ~)37<<1.

The observation of magnetokinetic e f f e c t s i n more concentrated s o l u t i o n s implies t h e p o l a r i z a t i o n of a spin system r a t h e r by some a l t e r n a t i v e methods t h a n by the d i r e c t rnagnetizati on i n an exter- n a l magnetic f i e l d .

6, Spin waves i n the ' ~ e 4 - ^{4 ~ e} s o l u t i o n I n t h e quantum - ' ~ e s o l u t i o n the d i f f e r e n t high-f requency modes

m 7 5 > 1 inherent both i n tne Bose sys- tem ( t h e high-frequency f i r s t sound) and i n the case of t h e Permi-liquid ( t h e sym- metric s p i n waves) can propagate. Since the 3 H e impurity atoms i n the s o l u t i o n experience t h e e f f e c t i v e a l tract ion

( a < ^{0) and} the Fermi-liquid i n t e r a c t i o n i s .small, the existence of weakly damped asymmetric s p i n waves with azimuth num- b e r s m # 0 and the modes of zero-sound types t u r n s out t o be impossible /6/.

The influence of the e x t e r n a l mag- n e t i c f i e l d ( t h e p o l a r i z a t i o n of the 3~re

nuclear s p i n g s t e m ) on the propagation of high-frequency o s c i l l a t i o n s i n t h e so- l u t i o n i s very appreciable. Since, due t o the magnetokinetic e f f e c t , t h e charac=- r i s t i c r e l a x a t i o n time increases consiue-

JOURNAL DE PHYSIQUE

rably, the region mT>,l, i n which t h e c o l l i s i o n damping of high-f requency mo- des i s small, a l s o noticeably increases.

It means t h a t i n a high polarized s o l u t i - on the condition CQrbb f can be va- l i d even f o r inf ra-acoustic frequencies.

High-frequency o s c i l l a t i o n s of the s o l u t i o n a r e described by the c o l l i s i o n - l e s s k i n e t i c equation, the c o n t i n u i t y equation and the superf luid-motion equa- t i o n . The s p i n p o l a r i z a t i o n of 3 ~ e i n the s o l u t i o n .changes quali.t;atively a l l the p a t t e r n of the spin-wave propagation.

Instead of one three-times degene- r a t e d branch of s p i n o s c i l l a t i o n s , two brances of t r a n s v e r s e s p i n waves with the gap in the energy spectrum ( i n the presence of the e x t e r n a l magnetic f i e l d ) and one l o n g i t u d i n a l branch of s p i n waves coupled with high frequency v i b r a t i o n s of the He s u p e r f l u i d background a r i s e 4 /I?/, The d i s p e r s i o n law of t r a n s v e r s e s p i n waves(osci1lations of t h e magneti- zation i n the plane which i s perpendi- c u l a r t o the magnetic-polaryzation vec- t o r d i r e c t i o n ) i s quadratic i n small wave v e c t o r s + k J-/J 9

~ 3 ~- & 2 ~

w = % + l ( 3 6 r ) ^-7-

d ⁹

i 4 - A

a s i t - i s inherent i n t h e systems with the f errornagnet symmetry. The d i s p e r s i o n laws of s p i n waves i n t h e absence of a magnetic f i e l d ($ = 0) and trans- verse s p i n waves i n an e x t e r n a l magne- t i c f i e l d fl # 0 a r e shown i n Fig. 3.

Fig. 3

When the d e n s i t i e s of s t a t e s f o r differerrt- l y spin-oriented quas i p a r t i c l e s a r e no%

equal t o each o t h e r , the o s c i l l a t i o n s of the l o n g i t u d i n a l component of t h e magne- t i z a t i o n t u r n out t o be coupled with t h e o s c i l l a t i o n s of the impurity e x c i t a t i o n s d e n s i t y (and i n consequence with t h e v i b r a t i o n s of the He s u p e r f l u i d back- 4 ground; it i s due t o the i n t e r a c t i o n be-

tween t h e %e q u a s i p a r t i c l e s and the He11 background),

Then the dependence of the propaga- t i o n speed of the high-f requency first sound on the degree of s p i n p o l a r i z a t i o n a r i s e s only i n r a t h e r h i & orders i n t h e concentrations of 3 ~ e ( W x4'3). I n t h e case of t h e s p i n wave t h e r e i s t h e very i n t e r e s t i n g e f f e c t namely the supression of t h e l o n g i t u d i n a l s p i n mode by a s p i n p o l a r i z a t i o n .

The cause of t h i s phenomenon i s t h a t i n t h e nonpolarized s o l u t i o n the

propagation v e l o c i t y of the correspond- ing s p i n mode i s hardly larger t h a n the

Fermi v e l o c i t y and i s exponentially c l o s e t o it. /6/. When one begins t o PO- l a r i z e the ~ 0 l u t i o n t h e v e l o c i t y of t h e

l o n g i t u d i n a l s p i n wave UH begins t o decrease ( t h e t h i c k curve i n Fig. 4 ) .

Fig. 4

A t the same time t h e Fermi-surfaces f o r

%e q u a s i p a r t i c l e a with t h e d i f f e r e n t spin o r i e h t a t i o n s move a p a r t leading t o t h e in- crease of % and the decrease of ( t h e t h i n l i n e s i n Fig. 4 ) . Yhen ^a c e r t a i n c r i t i c a l v a l u e of the magnetic f i e l d H, i s reached ( i f t h e polaryzation has been obtained by means of the e x t e r n a l magne- t i c f i e l d ) the value of t h e spin-wave speed uH becomes equal t o tile Fermi ve- l o c i t y U,, ( H ~ ) = < ^{( H ~ )} . ^{Then the}

s p i n wave experiences a s t r o n g Landau damping due t o decay of the magnon i n t o a p a r t i c l e and a hole, and the appearing imaginary p a r t of the propagation veloci- t y of o s c i l l a t i o n s t u r n s out t o be of the same order a s the r e a l p a r t . The calcula- t i o n of the c r i t i c a l f i e l d H c ( t h e c r i t i - c a l degree of p o l a r i z a t i o n ) l e a d s t o the

following numerical expression /?I/.

The s i m i l a r e f f e c t - the supression

of the t r a n s v e r s e zero-sound, t h e veloci- t y of which i s a l s o very c l o s e t o the

Fermi v e l o c i t y , can be observed i n a s p i ~ polarized l i q u i d normal b e too. With

t h e a i d of experimental d a t a on ' ~ e one can estimate %he value of the c r i t i c a l f i e l d which t u r n s out t o be of t h e order of Hc a 4 kOe . ^/14/ I n c o n t r a s t t o t h e symmetric s p i n wave i n the d i l u t e mixtu- r e i n t h i s case even i n the f i e l d H>H, t h e t r a n s v e r s e zero-sound experiences a weak damping t o the extent t h a t t h e meanin@; of H - ^Hc i s small. I t i s connecti ed with the n e c e s s i t y of holding of t h e o r b i t a l moment conservation law during the absorbtion of transverse zero-sound e x c i t a t i o n s by quasipart i c l e s of a Fermi- l i q u i d .

I n the spin-polarized s o l u t i o n t h e propagation of s p i n waves t u r n s out t o be possible a l s o i n the Boltzmann region /14/. If the s p i n p o l a r i z a t i o n of b e i n the mixture i s created by means of t h e e x t e r n a l magnetic f i e l d , the magnon

spectrum i n the v i c i n i t y of the NMR- frequency uo i s j u s t a s follows

( k v f 7 ^2371~21 k2

W s Uo +2- > 3 , , = 7 ²

CJo 3 4

I n the short-wave region t h e pro- pagation of s p i n waves i n the nonaegene- r a t e s o l u t i o n i s impossible because of t h e s t r o n g c o l l i s i o n l e s s damping. Thus ti;e p o l a r i z a t i o n of the s p i n system makes the propagation of weakly-damped transverse s p i n mosed p o s s i b l e i n a wide temperature range. The increase of t h e temperature causes t h e narrowing the of

JOURNAL DE PHYSIQUE

wave-vectors range i n which the weakly damped rnagnons e x i s t . The observation of s p i n waves i n t h e Boltzmall region i s pos- s i b l e i n r a t h e r high magnetic f i e l d s .

7. Superf l u i d i t y of 3 ~ e ? i n 3 ~ e - ^%e

s o l u t i o n s

As it was mentioned i n Sec. 2, t h e negative sigh of the s - s c a t t e r i n g l e n g t h

a (5) i n d i c a t e s t h a t q u a s i - p a r t i c l e s i n t e r a c t i o n i n s o l u t i o n s i s a t t r a c t i v e , and t h a t the b e s u p e r f l u i d t r a n s i t i o n i n %e - ^{4 ~ e} s o l u t i o n s must be caused by the u s u a l BCS s-pairing of ferffiions.

Since the He atoms i n a s o l u t i o n form 3 a d i l u t e Fermi gas of quasi-particles with a quadratic energy spsctrum, the h e superf l u i d t r a m i t i o n temperature can be evaluated p r e c i s e l y using the BCS theory /15/ . The t r a n s i t i o n tempe- r a t u r e curves versus the 3 ~ e concentra- t i o n x f o r t h r e e d i f f e r e n t pressures a r e presented i n F i O

Fig. 5

The phase t r a n s i t i o n temperature proved t o be high enough t o hold out s t r o n g

hopes of discovering of the %e superflu- i d i t y i n 3 ~ e - ^%e s o l u t i o n s i n t h e not d i s t a n t f u t u r e .

The 3 ~ e s u p e r f l u i d t r a n s i t i o n i n

%e f - ^{4 ~ e} has some p e c u l i a r properties.

The degree of the p o l a r i z a t i o n d i c t a t e s the type of s u p e r f l u i d phase. P r o p e r t i e s of these e s s e n t i a l l y d i f f e r e n t phases /16/ appear t o be a s i n t e r e s t i n g a s the

p r o p e r t i e s of superf l u i d pure h e . I n n o t very high f i e l d s the super- f l u i d i t y of 3 ~ e i n s o l u t i o n s is provided a s i n t h e zero f i e l d case by s-wave p a i r i n g of 3 ~ e quasi-particles. But i n a p a r t i a l l y p o l a r i z e d s o l u t i o n Fermi m a menta of the coupling p a r t i c l e s p+ and p- a r e not equal t o eaah other, This f a c t prevents from t h e formation of BCS p a i r s with t h e zero momentum. A s a re- s u l t the t r a n s i t i o n temperature becomes lower with the growth of the polariza- t i o n , and i n some range of s p i n system p o l a r i z a t i o n s the p a i r i n g with t h e non- zero momentum i s e f f i c i e n t . The 3 ~ e su- p e r f l u i d t r a n s i t i o n temperature i n kt?-

%e s o l u t i o n s versus t h e e x t e r n a l magne- t i c f i e l d s t r e n g t h (i.e. t h e degree of the p o l a r i z a t i o n ) i s presented g r a f i c a l - l y i n Fig. 6.

Here TCH i s t h e t r a n s i t i o n temperature of a p o l a r i z e d s o l u t i o n . Together w i t h t h e d a t a of Fig. 5 on t h e t r a n s i t i o n tem- p e r a t u r e TCO i n t h e absence of a p o l a r i -

z a t i o n , t h i s curve d e s c r i b e s t h e depen- dence of t h e t r a n s i t i o n ter1:perature on t h e c o n c e n t r a t i o n of t h e s o l u t i o n , t h e p r e s s u r e and t h e degree of the p o l a r i z a - t i o n .

I n f i e l d s H HL ( t h e p o i n t L i n Fig. 6 corresponds t o I t, HL = 1.06 Tco, TL = 0.56 Tco) t h e p a i r s a r e formed w i t h t h e zero momentum, and t h e super- f l u i d phase is of t h e o r d i n a r y BCS type.

For H

.

-b +

momentum Q (H) t a k e s p l a c e (Q i s about

hi'& _, wher@ & - ftp,/bq0

i s a coherence l e n g t h ) , and t h e appearing s u p e r f l u i d phase i s s p a t i a l l y inhomogene- ous w i t h t h e h e t e r o g e n e i t y of t h e o r d e r of Y ... ~ / Q ( R J .

The study of t h e inhomogeneous su- p e r f l u i d phase i s of the pzbincipal i n t e - r e s t . I n t h e case of superconductors t h e analogous inhomogeneous phase /I?, I S / ( t h e F u l d e - F e r r e l l s t a t e ) was n o t observed because of t h e i n f l u e n c e of t h e e l e c t r o n diamagnetism, t h e c o n s i d e r a b l e spin-or- b i t i n t e r a c t i o n and t h e presence of im- p u r i t i e s . A l l t h e s e d i f f i c u l t i e s a r e a b s e n t i n t h e case of b e t - ^{4 ~ e solu-}

t i o n s , and t h e inhomogeneous s u p e r f l u i d phase seems t o be a c c e s s i b l e t o t h e ob- s e r v a t i o n . I n accordance w i t h t h e d a t a of Ref. / 1 6 / t h e i n t e n s i t y of t h e magne- t i c f i e l d Ht does not exceed s e v e r a l k i l o o e r s t e d s , and t h e t r a n s i t i o n tempera-

t u r e f o r t h e s o l u t i o n s w i t h h i g h concen- t r a t i o n s of He 3 i s h i g h enough (TL 3 la) t o make an experimental s t u d y of t h i s phase p o s s i b l e .

The f o r m a t i o n of p a i r s w i t h t h e non-zero momentum means that t h e energy gap A becomes a f u n c t i o n of co o r d i n a t e s

A ( 2 ) . 3 The e x a c t type of t h e c o o r d i n a t e

- 3

dependence A ( 2 ) i s n o t known y e t , The

3

most probable a r e t h e cubic d (z/= 2A0 *

o r t h e laminated A ( ~ = z B ~ C ~ S ( @ Z / ~ )

s t r u c t u r e s . I f t h e phase occurs t o be laminated t h a n t h e He superf l u i d t r a n - 3 s i t i o n i n t h e f i e l d s H > ^{HI, (} /fn = 1.28 Tco, TM = 0.13 Tco; see Fig. 6) i s of t h e second o r d e r and i s of t h e f i r s t o r d e r when H < H < HMO

L

The inhomogeneous s t r u c t u r e of t h e l i q u i d l e a d s t o s e v e r a l i n t e r e s t i n g con- sequences. Though t h e d e n s i t y of %e a t a s remains c o n s t a n t , t a e e q u i l i b r i u m magne- t i c moment of t h e s o l u t i o n t u r n s o u t t o be p e r i o d i c i n t h e space. The v e l o a i t y of quasi-par t i c l e s i n some d i r e c t i o n s may be c l o s e o r equal t o zero. It r e s u l t s i n a sharp a n i s o t r o p y of k i n e t i c pheno- aena and i n a slow temperature decrease of t h e s p e c i f i c heat. The hydrodynamic p r o p e r t i e s of such phases a r e a l s o unusu- a l . For example, t h e s u p e r f l u i d d e n s i t i e s ( t h e r e a r e f o u r s u p e r f l u i d d e n s i t i e s i n b e - ^{4 ~ e} s o l u t i o n s w i t h both %Ie and 'Fie

condensates ; t h e corresponding r e f e r e n c e s can be found i n Ref. /I / ) a r e a n i s o t r o - p i c , s p a t i a l l y o s c i l l a t i n g and may change s i g n s .

JOURNAL DE PHYSIQUE

S-wave p a i r i n g become s p o s i t i v e l y impossible when P ^{H > 1.33 T o (see}

Pig. 6). I n such f i e l d s t h e 3 ~ e super- f l u i d t r a n s i t i o n can be caused only by the p a i r i n g w i t h higher o r b i t a l moments.

I f the p-wave s c a t t e r i n g cocresponds t o the a t t r a c t i o n , the p r o p e r t i e s of the

superf l u i d phase of %e i n %e f - ^{4 ~ e}

s o l u t i o n s a r e s i m i l a r t o t h a t of %e - ^A

i n a high (compared w i t h t h e energy gap) magnetie f i e l s . But such a s u p e r f l u i d t r a n s i t i o n occurs a t the temperature be- low 10-5 K and i s of a l i t t l e experimen- t a l i n t e r e s t now.

More concise d e s c r i p t i o n of s u p e r f l u i d phases of %e t - %e s o l u t i o n s can

be found i n Ref. /16/ and i n the second review / I /

.

8. Conclusions

Recently t h e study of spin-polarized quantum systems became p r a c t i c a l l y a new f i e l d of t h e physics of the condensed matter. Many of r e s u l t s discussed above may be s u c c e s s f u l l y used f o r the wide range of p o l a r i z e d systems, such a s vacancy quantum s o l i d s , e l e c t r o n l i q u i d s , pure ?He, e t c . ( s e e ~ e f ./I/ ) The consi- derable progress i n the experimental in- v e s t i g a t i o n of polarized quantum systems

and i n the low t e m ~ e r a t u r e technique holds out a hope of f u t u r e experimental achievements i n the ?He+- %e s o l u t i o n s study and of t h e discovery such i n t r i g u - ing phenomena as the s u p e r f l u i d i t y i n s o l u t i o n s and magnetokinetic e f f e c t s .

R e f e r e n c e s

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Nauk, B, 279, 1980

E.P.Bashkin, andAE M e y e r ~ v i c h ~ Adv.

i n Phys. 1980.

/2/' L.D.Landau and 1.Ya Pomeranchuk.

Doklady Akademii Nauk USSR, 2, 669, 1S)lr8

/3/ D.O.Edwards, D.P.Brewer, P.Seligmann, Bd.Skertic, and M.Yaqub, Phys. Rev.

Lett., 2, 773, 1965.

/ ^4/ L.D.Landau and E .M.Ligshit z. Kvanto- vaya Mekhanika, M. , Nauka, 1974.

(Quantum Mechanics)

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'I 956.

/6/ E.P.Hashkin. Sov. Phys. JETP Lett., 25, 1, 1977 Pis1ma Zh. Eksp. Teor.

Piz., 3 , 3 , 1977 ; Sov. Phys. JEW, 46, 972, 1977 Zh.Eksp.l!eor.Fiz.,

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2, 1849, 1977

/7/ J.Bardeen, G.Baym, and D.Pines. Phys.

Rev. L e t t . , 'J' ,372, 1966; Phys.Rev.

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K), 257, 1979-

/ 9 / B.Castaing and P.Nozieres. 3. de Phys.

40, 2579 1979-

-

. ^.

/lo/. M.Chapellier, G.Frossatti and F.B.Ras- mussen. Phys. Rev. L e t t 42, 904, 1979

/11/ E.P.Bashkin and A.E.Meyerovich. Sov.

Phys. JETP, 3, 992, 1978 Zh.Eksp0 Teor. Biz. z, 1904, 1978