Journalof Low TemperaturePhysics,VoL39, Nos.5/'6, 1980

Shear Viscosity of the B Phase of Superfluid 3He. II. s-p-d-Wave Approximation Jun'ichiro Hara Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois

Yoshimasa A. Ono*'t Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois

Katsuhiko Nagai~ Max-Plank-Institut fiir Physik und Astrophysik, Munich, West Germany

Kiyoshi Kawamura Department of Material Science, Hiroshima University, Hiroshima, Japan (Received N o v e m b e r 7, 1979)

The shear viscosity rl(T) of superfluid 3He-B is studied throughout the temperature region 0 ~< t ~< 1 (t = T/To). The Boltzmann equation for Bogoliubov-Valatin quasiparticles is solved by a variational method. In the calculation of the collision integral, the scattering amplitude is estimated in the s - p - d - w a v e approximation by taking into account the spin-dependent part as well as the spin-independentpart. The reduced shear viscosity (I = 7I(T)/~I(Tc) in this approximation is smaller than those in the s-wave and s-p-wave approximations at all temperatures. The numerical result for r in the s - p - d wave approximation is in good agreement with the recent experimental data of Parpia et al. and of Alvesalo et al. in the temperature range 0.5 <~t <~1.0. 1. INTRODUCTION A large n u m b e r of i n v e s t i g a t i o n s h a v e b e e n p e r f o r m e d o n t r a n s p o r t p h e n o m e n a in the A a n d B p h a s e s of s u p e r f l u i d 3He since t h e i r d i s c o v e r y in 1972. a-~9 *Supported in part by the Nishina Memorial Foundation and in part by the NSF under Grant No. DMR-76-24011. tPresent address: Department of Physics, Case Western Reserve University, Cleveland, Ohio. ~:Present address: Department of Physics, Yamaguchi University, Yamaguchi, Japan. 603 0022-2291/80/0600-0603503.00/0 9 1980 PlenumPuNishingCorporation

604

Jun'ichiro Hara e t al.

In a previous paper 19 (referred to henceforth as I), we have investigated the shear viscosity rl of superfluid 3He-B over its whole temperature range by the use of a variational method to solve the Boltzmann equation for Bogoliubov-Valatin quasiparticles. The transition probabilities for the scattering processes were computed by taking into account the coherence factors correctly, while for simplicity only the spin-independent s-wave part of the scattering amplitude was retained. Recently, Parpia et al. 2~ carried out measurements of the shear viscosity in 3He-B at various pressures in the temperature range 0.8 ~
Shear Viscosity ot the B Phase ot Superfluid 3He. !!

605

is found that the reduced shear viscosity ~ in the s-p-d-wave approximation is appreciably smaller than those in the s-wave and s-p-wave approximations. The temperature dependence of r~ in the s-p-d-wave approximation is in good agreement with the experimental plots of Parpia et al. as well as of Alvesalo et al. In Section 2, we derive the Boltzmann equation for BogoliubovValatin quasiparticles. A formula for the shear viscosity is derived in Section 3 in the single-relaxation-time approximation by making use of the variational method. In Section 4, the shear viscosity is calculated in the s-p-dwave approximation, and the results are compared with experimental data. A summary and discussion are given in the last section. 2. B O L T Z M A N N E Q U A T I O N In this section, we derive the Boltzmann equation for the BogoliubovValatin quasiparticles microscopically26 (Ref. 26 will be referred to henceforth as 1I) and discuss the general form of the collision term. We have shown in II that the deviation of the matrix Wigner distribution function from the local equilibrium distribution is given by the sum of the "particle" part 6F+1 and the "hole" part 6F_1. The matrix kinetic equation for 6F~ (v = +1) in the presence of a shear strain is given by 26

_l{u~+p,3(O)}af< { 8~ - ~1 ~0 s _ "~op X i/ , + h [ ~ O , 6F.]=L(p ) -~!~Pi~pi

(1)

where [A, B] = AB - BA is the commutator, s~ is the kinetic energy of the 3He quasiparticle measured from the Fermi energy, E is the excitation energy of the Bogoliubov-Valatin quasiparticle, and f < ( E ) = 1/[exp (/3E) + 1] is the Fermi distribution function (/3 = 1/kuT). The shear strain tensor X~j is given by

1/Ovni Ov,i 2 v,,) Xii = ~ Ori + arj -~Sij div

(2)

with the normal velocity v,,. The quantity ~o expresses the equilibrium energy matrix in particle-hole space: ~o =

@; (~) + ~p;

(~)

(3)

In the above equations, we have used a new set of Pauli matrices in particle-hole space given by 1 ' O

Ap\

,

1

0

--

Jun'ichiro Hara et ai.

606

where Ap is the 2 • 2 matrix of the equilibrium energy gap for the BW state, A is its magnitude, and ~ is the unit vector parallel to p. Now we discuss the collision term I~(p). We employ an effective potential U of the form 27 U = 2 { V ; ( r i - r j ) + V~(ri-rj)et(i) " g(J)} i<]

=- 2

Y~

V(~)(ri-rj)o-.(i)(r,~(J)

i
(5)

where V (~ = V s, V (1) = V (2) = V (3) -= V a, o-, (a = 1, 2, 3) are Pauli matrices, and o-0 is the unit matrix. Since liquid 3He is nearly ferromagnetic in the normal state, the spin dependence of the interaction potential is very important. In terms of V" and V a we can express the scattering amplitudes for a pair of normal quasiparticles in the singlet and triplet states T, and Tt, 17"28respectively: Zs(Pl, P2, P3, P4) = [3 w a ( p l - P 4 ) -

W S ( p l - P 4 ) ] '[- [P4 <--)P3]

T,(pb P2, P3, P4) = [ V " ( p ~ - p a ) + V ' ( p t - p4)]- [p4~-->p3]

(6)

(7)

where V"~(p) are the Fourier transforms of VS'a(r). Using the above effective potential, we obtain the following collision term, which is an extension of Eq. (103) in II: I 1 /1\7< < > I.l(pl)= - dp2 dp3 dpa 1"2v3l/4 Y. ~pt~lE1 (2,7].)5 ~ h) fvlfM2fP3f 4

X [88q,~ '~. V('*)(p1-

p3) V(r

1- p3)

x ({r,A.3rr 8/~,} tr [A~r,A~,T~] -t-{TaAv3"I'B,Av',} tr [ ~ff'~'FaAv4 ?~ ] -{~A.~ro, A j

tr [A.~r. 6P~'o])

_ 1 )~ V(~)(pl_p3 ) V(m(pl_p4) 2 ,,,t3 X

({'r~A~s'r~A~2r~A.4r ~, 8P~,}

-- {~'a 8ff ~37"gA.2rc, A~a q'r A J

(8)

where {A, B} = A B + B A

.I is the anticommutator, ( ~ p = 8 ( p + p 2 - p 3 - p 4 ) ,

Shear Viscosity of the B Phase

of Superfluid 3He. II

607

6,.E = 6(vE + v2E2- u3E3 - v4E4), f~ = f<(vE) (f> = 1 - f < ) , A . = 1(1 + Vgo/E)

(9)

3F,, = aF.(p)/f>(E)f<(E)

(10)

and = ( O-a

r~

-~(r~)

(11)

where to-~ is the transpose of %. As in II, we find a solution of Eq. (1) of the form

8F~(p)=rt~;)~-~pi-~pj-56ijp"

X~j v

A.

roa 4 E 2 O;([~)

(12)

where we have assumed Ar/h >> 1 (r is a typical collison time) and also the particle-hole symmetry of the density of states. The relaxation times r(~) and rod obey the following equations:

Of
6

p~j +(W4+

0p~/

W C '[(2k2F 0~2 1 a~2 ii 5 34)~,~-~2) [pz,aTz/-~o,,p2.a---~2r(s%)j~

(13)

1 Of<

%a OE1 =-Idp2dp3dp4

~

~,,6+E,(~-hlh) 61

<

< > >

X [ W 1~r- W2C12C34 - W3(C24 q-- C13)]

(14)

where Cgj = v~vi'r~jA2/E,E/(i, ] ~ 1), C~i = uiy~iAR/Ellz), C*i = UiyliE*/Ei " [Yii = (P~ " Pi)], and the IV/are expressed in terms of T, and T, as w1

(2~-/h)~(g; + 3 T , 2)

W2 = (2~r/~)k[(T~ - T 2 ) + 4 T 2 , ( T 2 , - Tz,)] W3 = (2rr/h)~[(T, + T,)(T2s + T2,) +4TtT2,] W 4 =

(2~/h)~[(T~2 + 3 T 2 ) - 2(T22, + 3 T~,)]

W5 = (2"rr/h)~[(T3~ + T3,)(T2, - T2r)- 4 T3,T2,]

(15)

608

Jun'ichiro Hara et al.

with T2s,2t = Zs, t(pl, -P4, P3, -P2)

(16)

T3s,3t = Ts, t(Pl, -P3, - P2, P4)

(17)

The solution 8F~ in Eq. (12) is divided into two parts. The first part can be diag0nalized by the Bogoliubov-Valatin transformation as 0 <

1

0f <

U*A1.-Lf-~E61.U= ~ (1 + .03) ~ 6~

(18)

where

r = r(~)v,f/ E(p, ~/Opj -(1/3) ~Wk O~/Opk)Xq Equation (18) indicates that (Of
T,,t(02, r

(19)

T3s,3t = Ts, t(03, r

where 0~ and r are defined as follows: r is the angle between the planes spanned by (pl, P2) and (P3, P4) and 01 is the angle between Pl and P2 (see Fig. 1). The angles 02 and r are defined similarly as 01 and r by interchange of p2*--~-p4, and the angles 03 and &3 by interchange of P24"-~-P3. The integrals over the momenta P2, P3, and p4 are converted into integrals over the energy variables ~:2, ~c3, and ~4 and over the angles as 29 t" P / dp2 dp3 alp4 (~Pl~+/31 = ~ / d~2 d~3 d~4 (2qT)2(m*)3~+E 1 1'21'3v4d 1/2/)31.4 d--oo

x

4---~ cos ~0~

~

(20)

where m* is the effective mass at the Fermi surface and 4~ is the azimuthal

Shear Viscosity of the B Phase of Superfluid aHe. II

609

~xxx

L Fig. 1. Definition of the angle variables. (~1 is the angle between the planes spanned by (p~, P2) and (P3, P4); 01 is the angle between p~ and P2angle of P2 a r o u n d the axis Pl- After computing the angular integrals, we finally obtain from the B o l t z m a n n equation (13) Of< 3E1

--

i

d~2 dr dr

2

:~

E rr ( m ) ~2V3~'4 A4 X { [ 0l 1 q- 0~21"2/Y3P4 . . . .

3

(1)6 o13(p2134~2E4~-P3~1E~)IT(~l) ~

<

<

>

3,E,flf (Ej)f~f.,

A2

>

A2

q-(a4 q-OL5P3P4E3~E4)(E~) 2T(~2)}

(21)

where cq = (2 rr/h)~(T 2 + 3 T,2 ) 9 1 2 a2 = (2r Ts - Tt2 ) + 4 T2,(T2t - T2s)]3q2y34) o~3 = (27r/h)l([(Ts + Tt)(T2s + T2t) q-4TtT2t]'Y13)

(22)

a4 = (2 rr/h)l([(T 2 + 3 T~ ) - 2(T2s + 3 T~t)]89 3,22 - 1)) ces=(err/h) 89

T3t)(Tzs - T 2 t ) - 4 T3tT2t]'Y34~(3"y12 1. 2 - 1)) ,

and the brackets d e n o t e the angular average defined by ( e ) = I dOlddpl 2 - ~sirl01 ; ~ 4~r cos ~0~

A

(23)

610

Jun'ichiro Hara et aL

It is worth noting that the al are related to Einzel and W61fe's parameters 2z as oLz/al = 60, O/3/a1 = "Yo, oL4/al = --~-2, and a s / a 1 = -Yz. 3. V A R I A T I O N A L

SOLUTION

FOR SHEAR

VISCOSITY

In this section, the expression for the shear viscosity is derived in the single-relaxation-time approximation by using the variational method. 24 The qualitative behavior of the shear viscosity is discussed in the following two extreme cases: 6( = flA)<< l ( t = 1) and 6 >>l ( t = 0). The shear viscosity coefficient ~7 is expressed as 26

2 NF p4 ( ( ( ~ ) r ( s e ) ) )

(24)

~7= 15 (m*) 2 where PF is the Fermi momentum, NF( = pFm*/'rr2h 3) is the density of states at the Fermi surface for both spin projections, and <> denotes the energy average defined by <> = -

d

A

co .

(25)

It is difficult to solve the Boltzmann equation (21) directly and calculate the shear viscosity from Eq. (24) using its solution. Therefore, we calculate the shear viscosity variationally, introducing the single-relaxation-time approximation: ~-(~) = const = %. Although % is independent of energy, it does depend upon temperature. According to the variational principle, 24 the true shear viscosity 77 is bounded from below by "Or calculated variationally: (26)

r/>~n~-~-~ - (m,)2 \ \ k 2 ) fiT. where % is given by

g11.. 9

I\\E1] /c

A4

a2(/J2b'3/J4 E1E2E3E4 \Eli

/c

__0~3((b,2/]4 ~---~A + / ~ 2 /]3 A2 ~('1~2~

E2 4

~lE3] \-E-I] /c

~2 2 ~1 2 + 0~5(/,3 b,4 A2 ( r

el ]2;

(27)

Shear Viscosity of the B Phase of Superfluid aHe. !1

611

H e r e (A)~ is d e f i n e d b y (A)~ = - f oo dsCx d~:2 dr

d~4

oo

rr2(m*) 3 ( ~ h )

Z

6

v2 v3 v4

< > > 8+E,flf < (E~)f,..f,.af.,,A (28)

T h e n w e find

(m~2 i=1

r / ~ - 15 7r2h 3

(29)

~/

w h e r e the 1 / 7 i a r e d e f i n e d b y

1

C{/( el ~ 2~

77 = 1

C
--72=

An

(1~1~2~

E1E2E3E4 \-Ell/ / c A2

1_

A2

2

(30)

"/73

r14

\s

k E t / I~

with C = -(77"/m ,)3fl 2 4 7"rh6[(((~t/Et)2})] -2

(31)

W e c a n e v a l u a t e the 1/'0i a n a l y t i c a l l y in t h e f o l l o w i n g t w o e x t r e m e cases: (i) 6 << l ( t - 1) a n d (ii) 6 >> l ( t - - - 0). T h e r e s u l t s a r e l i s t e d in T a b l e I u p to first o r d e r in 6 [case (i)] o r 6-1 [case (ii)]. TABLE I

Analytic Expressions for 1/~7, in the Limits t = 0 and t = 1

1/rh 1/7/2 1/r/a l/r/4

1/'q5

t~O

t=l

t 26 2 ~3 - (91 + 3 / 8 ) tz62 ~-Tr(1- 1/6) 1 ,. + 1/6) - /262 a~'tl t262 sS-rr(1/6) --/'262 3~r(1/6)

t 2 3~r I 2( l + ~5r r 6 )

0(32) 0(62) t 2 ~rr2(1 +~w6) 0(6 2)

Jun'ichiro Hara et al.

612

The shear viscosity at the transition t e m p e r a t u r e "oo(Tc) is obtained by substituting the values of 1/r/~ at t =- 1 into Eq. (29): 4~2 7 ,v (Tc)= 5 ~ : g 2 (N2oL 1~- NFo~4) -1

(

(32)

)

The exact theoretical value of the shear viscosity at T = T~, r/r calculated by H0jgaad Jensen et al.3~ 2 7

4flcPF

was

1

*hx(Tc) = 15rr2ti3(m,)2 S ~ a ~ YN

(33)

with 1 4a 2n+ 1 1 YN = -3+ ,= 1,3,5,. ~ .. ~2 n 2 ( n + l ) 2 n ( n + l ) - a

(34)

where a = - 2 o ~ 4 / o , 1. The value of a is about 1.5, as we shall show, and this a yields rh,(Tc)/rlex(Tc)~--0.97. It indicates that our trial function ~-(~:)= const = % is very close to the exact solution of the Boltzmann equation (21) at T = To, as was pointed out by Ono. 25 A t T = 0, our trial function r ( ~ : ) = % gives an exact solution of the Boltzmann equation (21), as was noted by Pethick et al. 14 Therefore ~70(0) is the correct shear viscosity. We obtain the following exact relation:

"o.(O) _ 8 (kBTc'~ 2 1 3~- \ A ( 0 ) ] (1 q- OL2/OLI-- ~Ot3/ 2 Oll) Y N

r/ex(Tc)

(35)

This expression was derived by Pethick et al. is in the s-p-wave approximation and then by W61fe and Einze123 for the general case. In the low-temperature limit, the reduced shear viscosity ~o = rlv(T)/'oATc) is expressed asymptotically as ~v = r}L,(0)[1 - C1(1/(~)]

(36)

where C1 = (9Oll -

3a2 - 2a3 + 5 a 4 - 3a5)/(3o~1 + 3 a 2 -

2o~3)

On the other hand, near T~, ~v is given by r}v = 1 - C2rr6

(37)

where C2 = (5Otl q- 2a4)/(16al + 16a4) In the intermediate t e m p e r a t u r e region 0 < t < 1, the 1/~/i are evaluated numerically by using Gauss' method for integration. We use the same

Shear Viscosity of the B Phase of Superfluid 3He. II

613

TABLE ii Numerical Results for 1/'0~ for Various Values of Reduced Temperature t = 6

1/,11

1/~2

--1/'03

1/'04

-1/'05

0.00000 0.03064 0.04333 0.05307 0.06129 0.1373 0.1945 0.2387 0.2762 0.3094 0.4384 0.6276 0.7784 0.9105 1.0315 1.1455 1.2546 1.3607 1.5676 2.0884 2.6664 3.3757 4.3435 5.8625 8.8185 11.7592 17.6388 35.2775 co

3.290 3.392 3.445 3.480 3.506 3.736 3.920 4.060 4.180 4.281 4.676 5.212 5.591 5.891 6.133 6.337 6.505 6.645 6.855 7.086 6.998 6.682 6.193 5.591 4.941 4.614 4.294 3.979 3.665

0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.007 0.011 0.015 0.042 0.115 0.207 0.310 0.421 0.539 0.661 0.786 1.040 1.665 2.220 2.669 2.994 3.206 3.349 3.415 3.487 3.571 3.665

0.000 0.001 0.001 0.001 0.002 0.010 0.021 0.032 0.044 0.057 0.121 0.261 0.410 0.560 0.711 0.861 1.008 1.152 1.428 2.025 2.467 2.743 2.856 2.832 2.722 2.653 2.583 2.513 2.444

3.290 3.331 3.360 3.371 3.378 3.453 3.509 3.548 3.581 3.603 3.682 3.747 3.754 3.737 3.701 3.653 3.596 3.532 3.389 2.973 2.518 2.052 1.592 1.153 0.741 0.545 0.357 0.176 0.000

0.000 0.000 0.000 0.001 0.001 0.005 0.0ll 0.016 0.022 0.028 0.059 0.123 0.186 0.247 0.304 0.357 0.407 0.453 0.533 0.669 0.722 0.704 0.627 0.507 0.359 0.277 0.191 0.099 0.000

t 1.0000 0.9999 0.9998 0.9997 0.9996 0.998 0.996 0.994 0.992 0.99 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.15 0.10 0.05 0.00

T/Tc

e x p r e s s i o n f o r 6 o f t h e w e a k c o u p l i n g t h e o r y as e m p l o y e d 6 = [87r2/7~'(3)]~/z(1 ~" M T ) 6 -

t)l/Z/t,

i n I:

0 . 9 9 ~< t ~< 1.0

(38)

0 . 0 ~< t < 0 . 9 9

(39)

1

y zX(0) t

w h e r e y = 1 . 7 8 1 0 7 is t h e E u l e r n u m b e r a n d t h e v a l u e s o f A(T)/A(O) h a v e b e e n g i v e n b y M f i h l s c h e g e l . 31 T h e r e s u l t s c a l c u l a t e d f o r 1/~7i a r e t a b u l a t e d i n T a b l e II.* I n t h e t e m p e r a t u r e r a n g e s 0 ~ t <~ 0 . 1 5 a n d 0 . 9 9 9 6 ~< t ~< 1.0, we have checked that the numerical results agree very well with the analytic r e s u l t s g i v e n i n T a b l e I. *The values of 1/t2"0iare functions of only the parameter & If one uses other expressions for & one has to replace the values of t and 1/'0i in Table II by appropriate values corresponding to the value of the new 6.

614

]un'ichiro Hara et al.

4. SHEAR VISCOSITY IN THE

s-p-d-WAVE APPROXIMATION

In this section, we estimate the a~ in Eq. (29) in the s-p-d-wave approximation 32 and calculate the shear viscosity throughout the temperature range 0 ~
(40)

" V ....( p i - p j ) = V " " ( y q ) = ~ V"~'" PI('Yi]) 1

We also define St and Tl by

NFTt(O,, &,=O)= E T,P~(O,),

NFT,(O,, & , = O ) = E S~P,(O~) (41)

l

l

From Eqs. (6), (7), (40), and (41) we obtain the following expressions for the scattering amplitude in the s-p-d-wave approximation:

NFT,(Oi, qbi) = So+ S1 cos 0/+ $211(3 cos 2 0i - 3[) + 3(cos 0~- 1)Z(cos2 4 , - 1)]

(42)

NFT,(Oi, Oi) = [Tl(cos Oi- 1) + T2~(cos 3 2 01- ]-)] cos q~i

(43)

where

So=NF (3V~-Vg)+ S, (3V~'-- VT) l~0

SI = NF(3 V'; - V~ ), T , = N , z ( V ; + V~),

S2= NF(3 V; - V~ )

(44)

T 2 = N v ( V ~ + V~)

Because of the forward scattering sum rule 33

E Tt = 0

(45)

l

To does not appear in Eq. (43). Substituting the above expressions for T, and T, into Eq. (22), we find the following expression for the ai in terms of the five parameters So, $1, S z, Tt, and T2:

~i =--N2 hee,-- = ~ {D isms.•

SmT.~'

(46)

n, trl

The values of D s~s., i D ir..T., and D S~T~ ~ are listed in Table III. In the s-p-wave approximation, the above results for r are reduced to (W), (Wz), and (WD) in Eq. (82) in Ref. 22.

615

Shear Viscosity of the B Phase of Superfluid aHe. !1

TABLE 111

Values of D s.,s., Direr., and DIsn,r. i D&~ D~s~ D~ Ds~s~ Ds~s~ Ds2s2 D~7 ~ D~7 ~ D~ D~ D~ Ds~n Ds~ D&~ D&~

i=1 1/2 -1/6 -3/10 7/30 37/210 1/2 8/5 24/35 24/35 0 0 0 0 0 0

i=2 7/30 -9/70 -23/210 107/630 271/2310 7967/30030 104/105 472/1155 2088/5005 -32/105 -32/105 32/315 32/1155 32/231 2336/15015

i=3

i=4

i=5

1/6 1/30 -13/210 -29/210 -1/210 31/330 -4/7 4/21 36/77 -8/15 -8/35 0 0 0 32/385

-1/10 -5/42 -3/14 1/70 67/330 243,/770 -24/35 -24/55 -2568/5005 0 0 0 0 0 0

-23/210 -1/70 1/210 179/2310 -17/2310 -769/30030 20/231 -1004/3003 -60/143 32/105 64/385 8/165 8/5005 72/1001 -248/5005

Now we estimate the values of the parameters T~ and &(l <~2). These are related to the Landau parameters F I 'a by S t = A ~ - 3 A 7 and Tt = s+aa At ~ t , with A~ "a = F;'~/[1 + F;'~

+ 1)]

(47)

The parameters F~, Fg, F~, and F~ can be estimated from the experimental data on the molar volume, the specific heat, the magnetic susceptibility, the velocity of first sound, and the velocity of zero sound of normal liquid 3He at low temperatures. The remaining two parameters A~ and A~ are determined as follows. The values of the above four parameters and the truncated forward scattering sum rule up to l = 2 A~+A~+A~l +A'[+A~ +A~ = 0

(48)

enable us to write the right-hand side of Eq. (33) in term of A~. Putting the left-hand side of Eq. (33) equal to the experimental data for the shear viscosity in normal 3He,34 we can get the value of A~. We take the values of F~, F~), and F~ at 21 bar from the review by Wheatley 34 and take F~ = 0.924 at 21 bar estimated by Nettleton3S; then we obtain two sets of real roots for A~ and A~, which are referred to as solution A and solution B. On the other hand, if we take F~ = 0 . 6 9 at 21 bar determined by Dobbs, 36 Eqs. (33) and (48) have only imaginary roots. Since this is inconsistent with our theoretical program to determine the values of

616

Jun'ichiro Hara et al.

TABLE IV The Values of A ~'" and 4 i =N~hal/rr at 21 bar in the s-p-d-Wave Approximation ~

A~ A~ A~ Ao A~' A~

A

B

A

0.9835 2.4197 0.7799 -2.7843 -0.3732 -1.0256

0.9835 2.4197 0.7799 -2.7843 -1.2033 -0.1955

dl 42 43 44 &s

B

32.2 13.5 -0.5 -24.1 -2.7

33.1 9.9 4.1 -24.9 -4.1

~In columns A and B, the sets of solutions A and B for A~ and A~ are tabulated, and the kl estimated from those sets are given. The values A~, A~, and A~ are taken from Ref. 34, and the value A~ from Ref. 35.

1.0

,

,

,

,

,

,

,

,

,

0.8

J

0.6

0.4

~

....... /

f~l//1

0.2

0.0

I

0.0

'

I

0.2

I

I

0.4

I

I

0.6

I

I

0.8

I

1.0

t=T/Tc Fig. 2. Overall behavior of the reduced shear viscosity v = "%(T)/~v (To) in three approximations for the scattering amplitude, as a function of the reduced temperature t = T I T c. The solid curve denoted A is the calculated shear viscosity in the s-p-d-wave approximation with solution A and the solid curve denoted B, is that in the s-p-d-wave approximation with solution B. The dashed line is the result in the s-p-wave approximation and the dash-dot line is that in the s-wave approximation.

Shear Viscosity o[ the B Phase of Superfluid 3He. II

617

A~ and A~, we do not adopt these solutions in the following calculation. The sets of solutions A and B are tabulated in Table IV. Using these values of A~'a, we evaluate di from Eq. (46), with the results also given in this table. The behavior of the reduced shear viscosity ~ versus t is shown in Fig. 2 in three approximations for the scattering amplitude: the s-wave approximation, the s-p-wave approximation, and the s - p - d - w a v e approximation (A and B). There are considerable differences among the results for r~v(t) in the three approximations. This indicates that large-/components of the scattering amplitude play a significant role in the shear viscosity. In Fig. 3, the theoretical results are compared with the torsion pendulum data obtained by Parpia e t al. 2~ and Alvesalo e t al. 21 in the temperature range 0.5 ~< t ~< 1.0. The agreement between theory and experiments is fairly good. Although the values of A~ and A~ in solutions A and B are quite different, the difference in the shear viscosity becomes very small, in the temperature range where the experiments have been performed. Therefore at this point we have no way of knowing which solution is correct. 1.0

0.8

0.6 J

J

0.4

0.2

__._A.<__-- - - - - - ' - 0.0

I

0.5

i

0.6

I

I

0.7

I

i

0.8 t = T/Tc

i

I

0.9

I

.0

Fig. 3. Reduced shear viscosity ~o = ~%(T)/rlo(Tc) as a function of the reduced temperature t = T! T c in the region 0.5 ~
618

Jun'ichiro Hara et al.

5. S U M M A R Y A N D D I S C U S S I O N Using the variational method in the single-relaxation-time approximation, we have calculated the shear viscosity in the whole t e m p e r a t u r e range in the s - p - d - w a v e approximation. We found that the t e m p e r a t u r e dependence of the reduced shear viscosity in the s - p - d - w a v e approximation differs very much from those calculated in the s-wave and s-p-wave approximations. This indicates that the large-/ components of the scattering amplitude are very important in the calculation of transport coefficients. The reduced shear viscosity in the s - p - d - w a v e approximation is in good agreement with the experimental plots of Parpia et al. 2~ and of Alvesalo et al. 2j in the t e m p e r a t u r e range 0.5<~t <- 1.0, in spite of our insufficient knowledge of the scattering amplitude. Further precise experiments to obtain higher order Landau parameters (especially A~ and A~) are desirable. As Fig. 3 shows, our result for the reduced shear viscosity is always smaller than that of W61fle and Einzel at all temperatures and shows a better agreement with the experimental data of Alvesalo et al. In order to examine the validity of their approximation, we have made a variational calculation of ~ using the same parameters as W61fle and Einze123; i.e., a 2 / a l = 0.29, 1.0

,

,

,

,

,

,

0-8

0-6

0.4

0-2

O'O

I

0.s4

i

I

I

0.a8

I

0.92 t

I

o.96

I

1.0

= T / Tc

Fig. 4. Comparison of the variational result with that of W61fleand Einzel.23The solid line represents the variational result using the same parameters as WSlfle and Einzel. The dashed line is the result by WSlfle and Einzel.

Shear Viscosity of the B Phase of Superfluid 3He. II

Of3/~' 1 =

619

0.12, a 4 / a l = - 0 . 7 , a s / a l = O . O , and 6:l.761tA a n hC[ 1]- -I~ (/3 2 ( C1 ~ - t \ t

, '/z

(49)

As Fig. 4 shows, when the same parameters are used, the variational method gives a result which agrees with that of W61fle and Einzel. In this paper, we have neglected size effects in the narrow slab geometry of the experiments. We discuss this effect briefly. The relaxation time rt for the Bogoliubov-Valatin quasiparticles is determined from the following equation: fl 2

1

-<<(F1)>>; ~-~ >c4-of2
2

A4

\El/

A2

--Of3<(/)2b'4

/~

A2

E2Eg-}-b'3F1E~)(~l~2~\E/c1]

(50)

This equation is obtained from Eq. (27) by retaining only the out-scattering

! 0 -I

10-2

~....- B

tm(cm)

A... ~

10-3

10-4 0.2

0.4

0.6 t = T/Tc

0.8

.0

Fig. 5. Behavior of the mean free path of the BogoliubovValatin quasiparticles as a function of the reduced temperature t = Curve A: O71= 32.2, 62 = 13.5, o73= - 0 . 5 . Curve B: O7l =33.1, O72= 9.9, O73=4.1.

T/Tc.

620

Jun'ichiro Hara et al.

terms. According to Einzel and W61fle, 22 the mean free path of the Bogoliubov-Valatin quasiparticles is given by

lm(t)=[f_~d~v~(-~)2f<(E)/I_~d,f<(E)]l/2rt

(51)

The behavior of the mean free path versus t is plotted in Fig. 5. As Fig. 5 shows, the mean free path is longer than the size of the sample container (9.5 • 10 -3 cm) 2~ for reduced temperature t lower than 0.3, which is consistent with the estimates by O n o 37 and by Einzel and W61fle. 22 Therefore, in this temperature region, a transport theory which takes only quasiparticle collisions into account is no longer valid, and other collisions mechanism (such as collisions with the walls of container) should be taken into account. On the other hand, in the temperature region 0.5 ~
ACKNOWLEDGMENTS One of the authors (YAO) thanks Prof. Chris Pethick, Prof. Gordon Baym, and Prof. David Pines for comments and for making his stay at Urbana a very enjoyable one. One of the authors (KN) is indebted to Prof. P. W61fle and to D. Einzel for discussion of the approximation. The numerical calculations were performed with the FACOM M-190 computer at the Computer Center of Kyushu University.

s-p-d-wave

REFERENCES 1. T. A. Alvesalo, Yu. D. Anufriyev, H. K. Collan, O. V. Lounasmaa, and P. Wennerst6m, Phys. Rev. Len. 30, 962 (1973). 2. T. A. Alvesalo, H. K. Collan, M. T. Loponen, and M. C. Veuro, Phys. Rev. Lett. 32, 981 (1974). 3. T. A. Alvesalo, H. K. Collan, M. T. Loponen, O. V. Lounasmaa, and M. C. Veuro, J. Low Temp. Phys. 19, 1 (1975). 4. T. J. Greytak, R. T. Johnson, D. N. Paulson, and J. C. Wheatley, ~'hys. Rev. Left. 31, 452 (1973). 5. R.T. Johnson, R. L. Kleinberg, R. A. Webb, and J. C. Wheatley, J. Low Temp. Phys. 18, 501 (1975). 6. C. W. Kiewiet, P. C. Main, and H. E. Hall, in Proc. 14th Int. Conf. Low Temp. Phys. (North-Holland, Amsterdam, 1975), Vol. 5, p. 413. 7. P.C. Main, C. W. Kiewiet, W. T. Band, J. R. Hook, D. J. Sandiford, and H. E. Hall, J. Phys. C 9, L397 (1976). 8. R. W. Guernsey, Jr., R. J. McCoy, M. Steinback, and J. K. Lyden, Phys. Lett. A 58, 26 (1976). 9. J. Seiden, Compt. Rend. 276B, 905 (1973); 277B, 115 (1973); 278B, 1009 (1974); 279B, 151 (1974); 279B, 487 (1974).

Shear Viscosity of the B Phase of Superfluid 3He. 11

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

621

T. Soda and K. Fujiki, Progr. Theor. Phys. 52, 1405 (1974); 53, 1218 (1975) (Erratum). O. Vails and A. Houghton, Phys. Lett. 50A 211 (1974). M. A. Shahzamanian, J. Low Temp. Phys. 21, 589 (1975). B. T. Geilikman and V. R. Chechetkin, Zh. Eksp. Teor. Fiz. 69, 286 (1975) [Soy. Phys. JETP 42, 148 (1976)]. C. J. Pethick, H. Smith, and P. Bhattacharyya, Phys. Rev. Lett. 34, 643 (1975). C. J. Pethick, H. Smith, and P. Bhattacharyya, J. Low Temp. Phys. 23, 225 (1976). P. W61fle, Phys. Rev. B 14, 89 (1976). P. Bhattacharyya, C. J. Pethick, and H. Smith, Phys. Rev. B 15, 3367 (1977). C. J. Pethick, H. Smith, and P. Bhattacharyya, Phys. Rev B 15, 3384 (1977). Y. A. Ono, J. Hara, K. Nagai, and K. Kawamura, J. Low Temp. Phys. 27, 513 (1977). J. M. Parpia, D. J. Sandiford, J. E. Berthold, and J. D. Reppy, Phys. Rev. Lett. 40, 565 (1978); J. Phys. (Suppl. 8) 39, C6-35 (1978); J. D. Reppy, in Proceedings ULTHakond International Symposium. (The Physical Society of Japan, Tokyo, 1978), p. 89. T. A. Alvesalo, C. N. Archie, A. J. Albrecht, J. D. Reppy, and R. C. Richardson, J. Phys. (Suppl. 8) 39, C6-41 (1978). D. Einzel and P. W61fle, Z Low Temp. Phys. 32, 19 (1978). P. W61fle and D. Einzel, J. Low Temp. Phys. 32, 39 (1978). J. M. Ziman, Electrons and Phonons (Clarendon Press, Oxford, 1960), Chapter 7, p. 275. Y. A. Ono, Progr. Theor. Phys. 60, 1 (1978). J. Hara and K. Nagai, J. Low Temp. Phys. 34, 351 (1979). P. W61fle, Rep. Prog. Phys. 42, 269 (1979). Y. A. Ono, Ph.D. Thesis (University of Tokyo, 1976), unpublished. D. Pines and P. Nozi~res, The Theory o[ Quantum Liquids (Benjamin, New York, 1966),

VoL 1, Chapter 1. 30. H. HCjgaad Jensen, H. Smith, and J. W. Wilkins, Phys. Lett. 27A, 532 (1968); Phys. Rev. 185, 323 (1969). 31. B. Miihlschlegel, Z. Phys. 155, 313 (1959). 32. P. W/51fle and D. Einzel, presented at LT15, Grenoble (1978) and to be published. 33. A. A. Abrikosov and I. M. Khalatnikov, Rep. Prog. Phys. 22, 329 (1959). 34. J. C. Wheatley, Rev. Mod. Phys. 47, 415 (1975). 35. R. E. Nettleton, J. Phys. C 11, L725 (1978). 36. E. R. Dobbs, Proceedings ULTHakondlnternationalSymposium (The Physical Society of Japan, Tokyo, 1978), p. 300. 37. Y. A. Ono, Progr. Theor. Phys. 58, 1068 (1977).