TRANSPORT COEFFICIENTS OF SUPERFLUID 3He-B

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TRANSPORT COEFFICIENTS OF SUPERFLUID

3He-B

P. Wölfle, D. Einzel

To cite this version:

JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-1

TRANSPORT COEFFICIENTS OF SUPERFLUID

He"B

P. Wolfle and D. Einzel

Institut fUv Theoretische Physik, Teehnische Universitat Miinohen,

8046 Gavahing, F.R. Germany

Résumé.- Nous présentons un traitement théorique des phénomènes de transport pour l'état BW et don-nons des résultats numériques pour la 1ère et 2ëme viscosité, la conductivité thermique et la rela-xation spin-réseau dans toute la gamme de température. Ces résultats sont comparés à l'expérience. Abstract.- A transport theory of the BW state is described and numerical results for the shear and bulk viscosities, thermal conductivity, and spin relaxation rate in the whole temperature range are presented and compared with experiment.

Dissipative processes in a superfluid are asso-ciated with the thermal excitations, or normal com-ponent. In a paircorrelated Fermi liquid such as

superfluid 3He the dominant thermal excitations (at

not too low temperatures) are the Bogoliubov quasi-particles, characterized by momentum k, spin projec-tion a and energy E° = (£? + A2)1 / 2. Here B, = - U

it K K • n 34

Zm is the normal quasiparticle energy and A(T) is the energy gap, which is isotropic in the BW state. The transport of mass, momentum, energy, spin may be ex-pressed in terms of a distribution function of qua-siparticles in momentum space, V£ (r,t), describing the state of the quasiparticle system subject to an external driving force. V, obeys a kinetic equation describing the changes in the distribution by the streaming in phase space and by collisions,

h»\

+

Vk-V\ - l v

?

A =

T*±* Z

V

6

V

k

o

O )

where Sv, = Vfc - v° and oVfc = 6vfe -(3vk/3E)6Ekare

the deviations from global and local equilibrium, respectively, and v = [exp(gE ) + [ ]- 1. A derivation

of this equation from the microscopic theory, in-cluding explicit expressions for T, and C,, , is gi-ven in reference /1/.

SHEAR VISCOSITY.- The coefficient of shear viscosi-ty ri describes the response of the momentum current to a transverse velocity field v (r), i.e.

n

n = ]?k

(v

E

)6v

]/Iv^(?)n

The calculation of Sv from eq. (1) involves the solution of an integral equation. Rather than solving the exact problem approximately, as done in the well-known variational approach, we approximate the in-scattering part of the collision operator by

C E ^ . / T ^ , ) (2)

yielding the correct 1=2 moment of the exact col-lision integral. X is an angular average of the

2

quasiparticle scattering amplitude.

The transport equation (1) and (2) is readi-ly solved in terms of integrals over energy. We ha-ve evaluated this expression numerically using va-lues of X obtained from normal state viscosity da-ta together with values of the q.p. lifetime on the Fermi surface in the normal state deduced from sound absorption measurements and employing the so-called s-p approximation for the q.p. scattering amplitude for estimating some further cross sections appea-ring in the expression for 1/T. .

In figure 1 the theoretical result using pa-rameter values appropriate for melting pressure is compared with data of the Helsinki group 13/.

•li 1 1 1 1 1 1 1 1 1

Tic

ooo AlvesaLo et al. this work 06 I

\ °

Fig. 1 : Reduced shear viscosity versus reduced temperature.

In this calculation we used an interpolation formu-la for A(T) including strong coupling corrections near T but not for T-K). A comparison of the theory

with the recent Cornell data /4/ at intermediate

pressure is shown in figure 2.

1 ' I ' I I ' I J I

...

-

Fig. 2

Relative change of the shear viscosity

squared versus reduced temperature.

The leading correction term near Tc, shown as a

straight line 151, is seen to approach the data on-

ly very close to T

.

The data show a systematic de-

viation towards lower

at lower temperatures. This

is probably due to the fact that the scattering of

quasiparticles by the walls in the narrow slab geo-

metry employed in the Cornell experiment renders

the mean free path finite at low T, causing the

viscosity to vanish for T+Q.

THERMAL CONDUCTIVITY.- We have further calculated

the coefficient of diffusive thermal conductivity

(reference /2/).

is found to be roughly propor-

tional to I/T in the whole temperature range. Con-

trary to what one might naively expect, KT is found

to increase by

with increasing temperature

before tending to the T=O limiting value, which is

lower than the value at T

.

SECOND VISCOSITY.- Among the three coefficients of

second viscosity present in the hydrodynamics of an

isotropic superfluid the two appearihg in the

stress tensor are negligibly small. Only the coef-

ficient

5

in the acceleration equation of the su-

3

perf

luid,

+

vs

3~

-

S3div 0,

:

,

(

-

\g

is relevant. We have calculated

t3

with the result

121

4

5,

N~ T~(O,T~>/A(T>,

second sound should be well-defined below about

100

Hz at T

0.8 Tc.

INTRINSIC SPIN RELAXATION.- We have also calculated

the intrinsic spin relaxation time

introduced

phenomenologically by Leggett and Takagi /6/

describing relaxation of the quasiparticle spin S

g

towards the equilibrium polarization for given to-

tal spin S, S

:

(x:lxO)~,

where

X0

is the q.p.

4

partial spin susceptibility without Fermi liquid

corrections.

-rLT

is found to tend to TN(0) for T+Tc, in

agreement with the exact result and tends to infi-

nity for T+O as

3,

wherer is the p.q. lifetime at

the Fermi surface.

T~~

has been measured by Webb et al. 171

from the frequency decrease of the wall-pinned rin-

ging mode in 3 ~ e - ~ .

A comparison of theory with

these data, using a value of

(0)

0.31 us

r n ~ ~

as obtained from sound attenuation measurements,

is shown in figure

Fig. 3

Relaxation parameter

a

of the wall-pinned

magnetization ringing mode. Data circles

Webb et

a1.171. Solid line

theory.

where Nf

mwkf/r2 is the density of states, and

is the normal q.p. lifetime.

5

governs the at-

3

References

/ I

/

Einze1,D. and ~ E l f l e , ~ .

,

J. Low Temp. Phys.

to be published

/ 2 /

wglfle,P. and Einzel,D., J. Low Temp. Phys.

to be published

/ 3 /

Alvesalo,T.A., Collan,H.K., Loponen,M.T.,

Lounasmaa,O.V. and Veuro,M.C., J. Low Temp.

Phys.

19 (1975)

/ 4 /

Parpia,J.M., Sandiford,D.J., Berthold,J.E.

and Reppy,F.D., to be published

/ 5 /

Bhattacharyya,P., Pethick,C.J., and Smith,H.,

Phys. Rev.

B15

/ 6 /

Leggett,A.J. and Takagi,S., Phys. Rev. Lett.

34 (1975) 1424

-

/ 7 /

Webb,R.A., Sager,R.W

Wheatley,J.C., Phys.