Shear viscosity of superfluid 3He–A1 at low temperatures

Annals of Physics 309 (2004) 281–294 www.elsevier.com/locate/aop

Shear viscosity of superfluid 3He–A1 at low temperatures M.A. Shahzamanian and R. Afzali* Department of Physics, Faculty of Sciences, University of Isfahan, Isfahan 81744, Iran Received 25 June 2003

Abstract The shear viscosity tensor of the A1 -phase of superfluid 3 He is calculated at low temperatures and melting pressure, by using the Boltzmann equation approach. The two normal and superfluid components take part in elements of the shear viscosity tensor differently. The interaction between normal and Bogoliubov quasiparticles in the collision integrals is considered in the binary, decay, and coalescence processes. We show that the elements of the shear viscosities gxy , gxz , and gzz are proportional to ðT =Tc Þ
2 . The constant of proportionality is in nearly good agreement with the experimental results of Roobol et al. Ó 2003 Elsevier Inc. All rights reserved. PACS: 67.57.De; 66.20.+d; 67.65.+z Keywords: Shear viscosity; Superfluid 3 He–A1

1. Introduction There are three distinct stable superfluid phases of bulk 3 He, namely A, B, and A1 phases. In zero magnetic field, only the A and B phases are stable [1]. The A1 phase appears as a narrow wedge between the normal state and the A and B phases. In the presence of a magnetic field, the A and B phases are referred to as A2 and B2 phases [1]. It is widely believed from the nuclear magnetic resonance experimental data and

*

Corresponding author. Fax: +0098218711676. E-mail addresses: [email protected] (M.A. Shahzamanian), [email protected] (R. Afzali). 0003-4916/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2003.08.012

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theoretical work that the A phase of liquid 3 He is the Anderson–Brinkman–Morel state [2,3]. In this axial p-wave state, only ("") and (##) pairs exist, and in weak coupling limit may be totally decoupled. In the presence of a static magnetic field, the A phase of liquid 3 He splits into two phases, A1 and A2 , where in the A1 phase only a single spin population is paired and the A2 phase contains two independently paired spin populations [4]. Different theoretical researches have been so far been conducted on the shear viscosity of a superfluid 3 He, especially in zero magnetic field. The shear viscosity of isotropic superfluid state of 3 He has been discussed theoretically by Shumeiko [5]. Shumeiko, by taking the kinetic equation approach, showed that the shear viscosity of a superfluid s-wave state for temperatures close to the transition temperature Tc behaves like g ¼ gn ðTc Þ½1
cðD=KB T Þ, where gn ðTc Þ is the shear viscosity of the normal phase at the transition temperature Tc , KB is BoltzmannÕs constant, and c is a numerical constant. The shear viscosity of the A-phase, in zero magnetic field, has been calculated exactly for temperatures close to Tc by Pethick et al. [6] and Bhattacharyya et al. [7] in s–p approximation. Their result on the viscosity drops as ð1
T =Tc Þ1=2 and the exact coefficient of ð1
T =Tc Þ1=2 has been expressed as a function of normal state properties. The shear viscosity of the B-phase, in zero magnetic field, at low temperatures, along with the calculation of transition probabilities in s– p approximation, has already been investigated by Pethick et al. [8]. A microscopic calculation of the fourth-rank shear viscosity tensors of superfluid phases of 3 He, using Kubo formula approach, has been done by Shahzamanian [9]. The calculation of shear viscosity for A1 -phase, in non-zero magnetic field, has turned out to be more complicated than that for the other phases, due to the anisotropy of the gap. In high magnetic field, strong coupling effect must be considered for maximum gap parameter, and on normal fluid density components of the A1 and A2 phases of liquid 3 He [10]. The viscosity, in the presence of a magnetic field has been considered by Shahzamanian close to Tc in relaxation time approximation [10]. Shahzamanian and Afzali [11] calculated the shear viscosity of A1 -phase close to Tc , considering superfluid– normal interaction along with other interactions for binary, decay, and coalescence processes of quasiparticles, and we also moved beyond s–p approximation by the use of Pfitzner procedure. The shear viscosity of 3 He–B has been measured down to temperatures T =Tc ’ 0:4 at different pressures simultaneously by vibrating-wire and torsional-oscillator viscometers [12] and down to T =Tc ’ 0:3 by a vibrating-wire viscometer [13]. Experimentally, the shear viscosity of the A phase has been measured by Alvesalo et al. [14–16] in a vibrating string viscometer experiment and indirectly by Johnson et al. [17]. The shear viscosity, in the presence of low magnetic field, has been measured by Alvesalo et al. [16]. A number of viscosity measurements have been carried out in the A1 and A2 phases of superfluid 3 He at different high magnetic fields by Roobol et al. [18–20]. All the results show a sharp decrease proportional to the opening of the superfluid energy gap, D / ð1
T =Tc Þ1=2 near Tc1 and goes as T
2 for low temperatures. In this paper, we study the shear viscosity of A1 -phase of superfluid 3 He at low temperatures, using Boltzmann equation approach. We assume that magnetic field is sufficiently high so that all of quasiparticles with spin up go to superfluid state and all quasiparticles with spin down stay in normal state.

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In this paper also we consider normal–superfluid interactions which come from the scattering between superfluid quasiparticles in the spin-up population, the socalled Bogoliubov quasiparticles and the normal fluid quasiparticles in the spindown population. In a normal Fermi liquid at low temperatures, the only important collision process is the binary scattering of quasiparticles, but in a superfluid the quasiparticle number is not conserved and other processes as well as binary processes can occur. So we take into account these processes such as decay processes in which one Bogoliubov quasiparticle decays into three and coalescence processes in which three Bogoliubov quasiparticles coalesce to produce one. Transition probabilities are explicitly extended in terms of spin indices. It has been cited elaborately how transition probabilities can be expressed in terms of singlet and triplet quasiparticle scattering amplitudes (see Section 2); and accordingly, Pfitzner procedure [21] has been used in the calculation of quasiparticle scattering amplitude. In Pfitzner procedure, necessary conditions are established to explicitly construct exchange-symmetric scattering amplitudes by adding higher angular momentum components. More clearly, Pfitzner introduces a general set of complete and orthonormalized eigenfunctions of the exchange operator on the Fermi sphere. Then, out of it, he manages to construct a general polynomial expansion of the quasiparticle scattering amplitude. By using this expansion, it is attempted to fit the quasiparticle scattering amplitude by including as many available experimental data (such as Landau parameters and transport times in the normal phase of 3 He) as possible and taking account of their respective uncertainties. This expansion of quasiparticle scattering amplitude fully contains the s–p approximation as a special case (see the sentences following Eq. (39)) but it should be borne in mind that this expansion is absolutely general. Therefore, there may be no need to say that s–p approximation is a simple approximation for quasiparticle scattering amplitude in terms of Landau parameters. Another study is also under way dealing with intermediate temperature regions, the complete solution of which requires numerical computations and will be published elsewhere. The paper is organized as follows. In Section 2, we obtain transition probabilities of different processes and discuss about them. In Section 3, by writing the collision integral, the Boltzmann equation is solved for A1 -phase. In Section 4, the shear viscosity has been calculated at low temperatures and in Section 5, we give some remarks and concluding results. 2. Transition probabilities To obtain transition probabilities, we start with the interaction term in the Hamiltonian, i.e., 1 X H¼ h3; 4jT j1; 2iay 4 ay 3 a1 a2 ; ð1Þ 4 1;2;3;4 where i ¼ 1; 2 ; 3, and 4 stands for both momentum (~ pi ) and spin ðri Þ variables.

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Suppose that only spin-up particles become superfluid. The creation and annihilation operators with spin down in H are left intact. By using Bogoliubov transformation, the quasiparticle creation a~yp;" and annihilation a~p;" operators may be replaced by the creation and annihilation operators a~yP ;r and a~P ;r in the superfluid. Bogoliubov transformation may be written as a~p;r ¼ u~p;rr0 a~p;r0
v~p;rr0 ay
p ~ 0;
p;r

ð2Þ

a~yp;r ¼ v~p;rr0 a~p;r0 þ u~p;rr0 ay
~p;r0 ;

ð3Þ

where the matrix elements u~p;rr0 and v~p;rr0 can be chosen for A1 -phase as [22]

   1=2  1=2 1 1 ð1 þ e~p =E~p ð1
e~p =E~p u~p;rr0 ¼ drr0 ; v~p;rr0 ¼ drr0 : 2 2

ð4Þ

2 1=2

In the A1 -phase, we may write E~p ¼ ðe~2p þ jD~p"" j Þ sgne~p , where e~p is the normalstate quasiparticle energy measured with respect to the chemical potential and D~p"" is the magnitude of the gap in the direction ~ p on the Fermi surface [22]. jD~p"" j is equal to DðT Þ sin hp , where DðT Þ is the maximum gap and hp is the angle between the quasiparticle momentum and gap axis ‘^ that is supposed to be in the direction of z-axis. For the non-unitary state of the A1 -phase, there are the following properties between u and v [22] u
~p;"" ¼ u~p;"" ;

v
~p;"" ¼
v~p;"" :

ð5Þ

Then, by using Eqs. (2) and (3) in Eq. (1), we have    1 X nh H¼ h3 " 4 " jT j1 " 2 "i u4"" ay4"
v4"" a
4" u3"" ay3"
v3"" a
3" 4~ ~ ~ ~ P1 ;P2 ;P3 ;P4   i h i  u1"" a1"
v1"" ay
1" u2"" a2"
v2"" ay
2" þ h3 # 4 # jT j1 # 2 #iay4# ay3# a1# a2# h     i þ h3 # 4 " jT j1 " 2 #i u4"" ay4"
v4"" a
4" ay3# u1"" a1"
v1"" ay
1" a2# h    i þ h3 " 4 # jT j1 " 2 #iay4# u3"" ay3"
v3"" a
3" u1"" a1"
v1"" ay
1" a2# h    i þ h3 # 4 " jT j1 # 2 "i u4"" ay4"
v4"" a
4" ay3# a1# u2"" a2"
v2"" ay
2" h    io þ h3 " 4 # jT j1 # 2 "iay4# u3"" ay3"
v3"" a
3" a1# u2"" a2"
v2"" ay
2" ; ð6Þ which contains the terms ay4 ay3 ay
2 a1 , ay4 ay3 a2 a1 , ay4 a1 a
3 a2 , ay4 ay3 ay
2 ay
1 , and a
4 a
3 a2 a1 . These terms decay one quasiparticle into three, convert two quasiparticles into two, coalescence three quasiparticles into one, create four quasiparticles from the condensate, and scatter four quasiparticles into the condensate, respectively. The last two processes are not allowed, because in each process the total energy should be conserved. The first bracket of Hamiltonian in Eq. (6) hence is related to the superfluid component interactions. The second bracket is related to normal component inter-

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actions, which is important and included in this study. The other terms are related to normal–superfluid interaction and vice versa. The transition probabilities for the processes may be defined, for example, as 2 jh~ p4 ; ";~ p3 ; " jH j~ p1 ; ";~ p2 ; "ij . By using WickÕs theorem, we obtain the following transition probabilities relating to the T-matrix (throughout this paper we put h  KB  1). For normal component interaction, we have
2

2

W22 ð##Þ ¼ 2pjh4 # 3 # jT j1 # 2 #ij  2pjTtI j ;

ð7Þ

where subscripts on W indicate binary processes in which two quasiparticles with spin down scatter to two quasiparticles with spin down and TtI is the triplet quasiparticle scattering amplitude. TtI is calculated by the Pfitzner procedure [21] (see Eq. (38)). For superfluid component interaction, by using relation (5), we have nh  2 2 2 2 2 2 2 2 W22 ð""Þ ¼ 2p jv1 j jv2 j jv3 j jv4 j þ ju1 j ju2 j ju3 j ju4 j þ 2u1 v1 u2 v2 u3 v3 u4 v4 i h 2 2 2 2 2  jh4 " 3 " jT j1 " 2 "ij
jv1 j jv4 j u2 v2 u3 v3 þ jv2 j jv3 j u1 v1 u4 v4  2 2 2 2  þ ju2 j ju3 j u1 v1 u4 v4 þ ju1 j ju4 j u2 v2 u3 v3  ðh4 " 3 " jT j1 " 2 "i i   h
1 " 3 " jT j
4 " 2 "i þ h4 " 3 " jT j1 " 2 "ih
1 " 3 " jT j
4 " 2 "i Þ h þ jv4 j2 jv2 j2 u1 v1 u3 v3 þ jv1 j2 jv3 j2 u2 v2 u4 v4 þ ju1 j2 ju3 j2 u2 v2 u4 v4  2 2  þ ju2 j ju4 j u1 v1 u3 v3  ðh4 " 3 " jT j1 " 2 "i h
2 " 3 " jT j
4 " 1 "i i  þ h4 " 3 " jT j1 " 2 "ih
2 " 3 " jT j
4 " 1 "i Þ h  2 2 2 2 2 2 2 2 þ jv1 j ju3 j ju2 j jv4 j þ jv2 j ju4 j ju1 j jv3 j þ 2u1 v1 u2 v2 u3 v3 u4 v4 i h 2 2 2 2 2  jh
1 " 3 " jT j
4 " 2 "ij
ju3 j jv4 j u1 v1 u2 v2 þ jv1 j ju2 j u3 v3 u4 v4  þ ju1 j2 jv2 j2 u3 v3 u4 v4 þ jv3 j2 ju4 j2 u1 v1 u2 v2  ðh
1 " 3 " jT j
4 " 2 "i i  h
2 " 3 " jT j
4 " 1 "i þ h
1 " 3jT j
4 " 2 "ih
2 " 3 " jT j
4 " 1 "i Þ h  2 2 2 2 2 2 2 2 þ ju1 j jv2 j ju3 j jv4 j þ jv1 j ju2 j jv3 j ju4 j þ 2u1 v1 u2 v2 u3 v3 u4 v4 io 2  jh
2 " 3 " jT j
4 " 1 "ij : ð8Þ The transition probability for decay process may be written as W13 ð""Þ ¼ jh~ p3 ";~ p4 ";
~ p2 " jH j~ p1 "ij2 or   W13 ð""Þ ¼ 2p v1 v2 u3 v4
u1 u2 v3 u4 h
1 " 3 " jT j
4 " 2 "i   þ v1 u2 v3 v4
u1 v2 u3 u4 h4 " 3 " jT j1 " 2 "i   2 þ v1 v2 v3 u4
u1 u2 u3 v4 h4 "
1 " jT j
3 " 2 "i  :

ð9Þ

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Similarly the transition probability for the coalescence process may be written as W31 ð""Þ ¼ jh~ p4 " jH j~ p1 ";~ p2 ";
~ p3 "ij or

2

  W31 ð""Þ ¼ 2p v1 v2 u3 v4
u1 u2 v3 u4 h4 " 3 " jT j1 " 2 "i   þ v1 u2 v3 v4
u1 v2 u3 u4 h
1 " 3 " jT j
4 " 2 "i   2 þ u1 v v v4
v u2 u3 u4 h4 "
1 " jT j
3 " 2 "i  : 2 3

ð10Þ

1

For the superfluid–normal component interactions, one has n 2 2 2 2 2 2 W22 ð"#Þ ¼ 2p ju1 j ju3 j jh4 # 3 " jT j1 " 2 #ij þ jv1 j jv3 j jh4 #
1 " jT j
3 " 2 #ij
u1 v1 u3 v3 h4 # 3 " jT j1 " 2 #i h4 #
1 " jT j
3 " 2 #i
u1 v1 u3 v3 h4 # 3 " jT j1 " 2 #ih4 #
1 " jT j
3 " 2 #i 2

2

2

2



2

þ ju1 j ju4 j jh4 " 3 # jT j1 " 2 #ij þ jv1 j jv4 j jh
1 " 3 # jT j
4 " 2 #ij

2




u1 v1 u4 v4 h4 " 3 # jT j1 " 2 #i h
1 " 3 # jT j
4 " 2 #i
u1 v1 u4 v4 h4 " 3 # jT j1 " 2 #ih
1 " 3 # jT j
4 " 2 #i



o

ð11Þ

;

W22 ð#"Þ is obtained by replacing 2 with 1 and vice versa in Eq. (11). It is noted that decay of quasiparticle with spin up is impossible because in such a condition, two outgoing quasiparticles out of three must have spin down. This shows the creation of quasiparticles of normal state which is not allowed, since the number of quasiparticles in normal state is conserved. Then, we have the following result for decay of a quasiparticle with spin down: 2

W13 ð#"Þ ¼ 2pjh~ p3 ";~ p4 ";
~ p2 # jH j~ p1 #ij n 2 2 2 2 2 ¼ 2p ju3 j jv4 j jh
2 # 3 " jT j
4 " 1 #ij þ jv3 j ju4 j j  h
2 # 4 " jT j
3 " 1 #ij2 þ u3 v3 u4 v4 h
2 # 3 " jT j
4 " 1 #i  h4 "
2 # jT j
3 " 1 #i þ u3 v3 u4 v4 h
2 # 3 " jT j
4 " 1 #i o   h4 "
2 # jT j
3 " 1 #i

ð12Þ

The final quasiparticle in the coalescence process of the superfluid–normal interactions also must have spin down. Hence, we have 2

p1 # jH j
~ p2 #;~ p3 ";~ p4 "ij W31 ð#"Þ ¼ 2pjh~ n 2 2 2 2 2 ¼ 2p jv3 j ju4 j jh1 #
3 " jT j4 "
2 #ij þ jv4 j ju3 j 2

 jh
4 " 1 # jT j3 "
2 #ij þ v3 u3 u4 v4 h1 #
3 " jT j4 "
2 #i  h
4 " 1 # jT j3 "
2 #i þ v3 u3 u4 v4 h1 #
3 " jT j4 "
2 #i o   h
4 " 1 # jT j3 "
2 #i



ð13Þ

T-Matrix elements which appear in the transition probabilities may be expressed in terms of the scattering amplitudes for pairs of quasiparticles in singlet and

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287

triplet states, Ts and Tt , respectively. By considering properties of Ts and Tt [23], we have h4 # 3 # jT j1 # 2 #i ¼ h4 " 3 " jT j1 " 2 "i  TtI ; h4 "
1 " jT j
3 " 2 "i  TtII ; h
1 " 3 " jT j
4 " 2 "i  TtIII ; h4 # 3 " jT j1 " 2 #i ¼ h4 " 3 # jT j1 # 2 "i  12ð
TsI þ TtI Þ; h4 " 3 # jT j1 " 2 #i ¼ h4 # 3 " jT j1 # 2 "i  12ðTsI þ TtI Þ; h4 #
1 " jT j
3 " 2 #i ¼ h4 "
1 # jT j
3 # 2 "i  12ð
TsII þ TtII Þ; h4 "
1 # jT j
3 " 2 #i ¼ h4 #
1 " jT j
3 # 2 "i  12ðTsII þ TtII Þ; h
1 # 3 " jT j
4 " 2 #i ¼ h
1 " 3 # jT j
4 # 2 "i  12ð
TsIII þ TtIII Þ; h
1 " 3 # jT j
4 " 2 #i ¼ h
1 # 3 " jT j
4 # 2 "i  12ðTsIII þ TtIII Þ;

ð14Þ

where TtI and TsI are given by [21] N ð0ÞTtI ;sI ðm; P Þ ¼

1 X k X k¼0

alk Xlk ðm; P Þ

ðl even; oddÞ

ð15Þ

l¼0

and N ð0Þ ¼ m pF =p2 is the density of states at the Fermi level. The coefficients with l even (odd) belong to the singlet (triplet) part of the quasiparticle scattering amplitude. Xlk appearing in Eq. (15) is ð2lþ1;0Þ

Xlk ðm; P Þ ¼ ðk þ 1Þ1=2 ð2l þ 1Þ1=2 ðP 2 =4
1Þl Pl ðmÞPk
l

ðP 2 =2
1Þ;

k ¼ 0; 1; . . . ; l ¼ 0; 1; . . . ; k;

ð16Þ

where the Pl ðmÞ and Pnða;bÞ ðxÞ are the Legendre polynomials and the Jacobi polynomials, respectively, and finally h ð17Þ P  2 cos ; m  cos u; 2 where h is the angle between the momenta of the incoming particles namely ~ p1 and ~ p2 , and u is the angle between the planes spanned by the momentum vectors of the incoming particles and the outgoing particles. Similar equations for TtII and TsII (TtIII and TsIII ) with replacement of cos h by cos hII (cos hIII ) and cos u by cos uII (cos uIII ) can be used. The following equations express hII , uII , hIII , and uIII in terms of h and u [8].



 h h h h cos u; cos hIII ¼
cos2
1
cos2 cos u; cos hII ¼
cos2 þ 1
cos2 2 2 2 2     1
cos2 h2 cos u þ 3 cos2 h2
1 1
cos2 h2 cos u
3 cos2 h2 þ 1    cos uII ¼  ; cos u ¼ : III h h
1 þ cos2 2 cos u þ 1 þ cos2 2 1
cos2 h2 cos u þ 1 þ cos2 h2

ð18Þ

At low temperatures, we have sin hpi ’ 0 (i ¼ 1; 2; 3; 4) [24]. Then we may write v ’ 0 and u ’ 1. It is noted that if we apply other approximations for v and u, the temperature dependence does not change, however, some small changes appear in the numerical coefficients of the transition probabilities. By substituting the values of v and u in the equations of the transition probabilitiesÕ (Eqs. (7)–(13)), we obtain

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M.A. Shahzamanian, R. Afzali / Annals of Physics 309 (2004) 281–294 2

W22 ð##Þ ¼ 2pjTtI j ; ð19Þ  p jTtI j2 þ jTtII j2 þ jTtIII j2 ; W22 ð""Þ ’ ð20Þ 4  p 2 2 j
TsI þ TtI j þ jTsI þ TtI j ; ð21Þ W22 ð"#Þ ¼ W22 ð#"Þ ’ 2 and the other transition probabilities have nearly zero value. It is noted that only binary processes are dominated at low temperatures, and this is also the case for calculating the thermal diffusion coefficient of the A-phase at low temperatures [24]. 3. Collision integrals The kinetic equation for the quasiparticle distribution function of superfluid, mp;r , may be written as [1] omp;r omp;r oEp;r omp;r oEp;r þ
¼ Iðmp;r Þ; ot o~ r o~ p o~ p o~ r

ð22Þ

where Iðmp;r Þ is the collision integral. Let us assume a disturbance of the form mr ðp; rÞ ¼ m0 ðpÞ þ dmr ðpÞ;

ð23Þ

0

where, in general, both m ðpÞ and dmr ðpÞ are functions of the variable r. We define the function wr ðpÞ by 1 dmr ðpÞ ¼
m0 ðpÞð1
m0 ðpÞÞwr ðpÞ; ð24Þ T p ~ uÞ=T þ 1
1 and u is the slightly inhomogeneous velocity where m0 ðpÞ ¼ ½expðEp0
~ in the liquid. By substituting (23) in (22), keeping the terms which contribute to the ~ ~ shear viscosity, and supposing ~ u and r u are zero at the point considered, to first order in dmr ðpÞ we have for the initial quasiparticle with spin up [1] 

1 om0 oEp;" oui ouk 2 ~
pi
dij r  ~ ð25Þ u ¼ Iðdm" ðpÞÞ; þ 2 oEp;" opk ork ori 3 where Iðdm" ðpÞÞ is nearly equal to I22 ð"#Þ. Eq. (25) for initial quasiparticle with spin down is given by changing the distribution function of superfluid state, m0 ðpÞ, to the distribution function of normal state, n0 ðpÞ, and Iðdm" ðpÞÞ to Iðdm# ðpÞÞ, Iðdm# ðpÞÞ is equal to I22 ð##Þ þ I22 ð#"Þ. Now by using (23) and keeping the terms to first order in wr ðpÞ and using Eqs. (19)–(21), the linearized collision terms in the Boltzmann equation may be written as: ðm Þ3 T I22 ð"#Þ ¼ 64p5

Z

sin h dh du du2 cos h2

Z

þ1

dx dyðwðtÞ þ wðx þ y
tÞ


1 2


wðxÞ
wðyÞÞ½j
TSI þ TtI j m0 ðtÞn0 ðx þ y
tÞð1
m0 ðxÞÞð1
n0 ðyÞÞ 2

þ jTSI þ TtI j m0 ðtÞn0 ðx þ y
tÞð1
n0 ðxÞÞð1
m0 ðyÞÞ;

ð26Þ

M.A. Shahzamanian, R. Afzali / Annals of Physics 309 (2004) 281–294 3

ðm Þ T I22 ð##Þ ¼ 16p5

Z

sin h 2 jTt j dh du du2 cos h2 I

Z

289

þ1

dx dyn0 ðtÞn0 ðx þ y
tÞ
1

 ð1
n0 ðxÞÞð1
n0 ðyÞÞðwðtÞ þ wðx þ y
tÞ
wðxÞ
wðyÞÞ;

ð27Þ

where u2 is the azimuthal coordinate of ~ p2 relative to ~ p1 , t  E1 =T , x  E3 =T , and y  E4 =T . To solve the linearized Boltzmann equation, it is suitable to define qðtÞ as 
oE1 oui ouk 2 ~ wðtÞ ¼ pi qðtÞ
dij r  ~ u : ð28Þ þ op1k ork ori 3 After expressing in Eq. (28) in terms of a series of spherical harP2 the bracket jmj U p ðcos HÞ eimU [25] and substituting (28) in the collision monics, i.e., m¼
2 m 2 integrals, the integration on u2 can be done conveniently. By noting some symmetries with respect to variables in the collision integrals [25] and performing integration on y, we get

Z Z 3 2 0 ðm Þ T 2 X sin h dh du jmj imU
om I22 ð"#Þ ¼ Um p2 ðcos HÞ e dxKðt; xÞ 32p4 m¼
2 oE1 cos h2 2

2

 ðj
TsI þ TtI j þ jTsI þ TtI j Þ½qðtÞ þ qð
xÞp2 ðcos hÞ
qðxÞ  fp2 ðcos h13 Þ þ p2 ðcos h14 Þg;

ð29Þ

p1 and ~ pi , and h13 and h14 are related to h and u with where h1i is the angle between ~ the following relations: cos h13 ¼ cos2

h h þ sin2 cos u; 2 2

cos h14 ¼ cos2

h h
sin2 cos u; 2 2

ð30Þ

and Kðt; xÞ is Kðt; xÞ ¼

e
t þ 1 x
t : e
x þ 1 eðx
tÞ
1

ð31Þ

By substituting Eq. (29) in Eq. (25) and considering Kðt; xÞ ¼ Kð
t;
xÞ, we have [25] Z dxKðt; xÞfqsr ðtÞ
k2sr qsr ðxÞg ¼ Br ; ð32Þ Z

dxKðt; xÞfqar ðtÞ
k2sr qar ðxÞg ¼ OðTBr Þ;

ð33Þ

where qsr ðtÞ and qar ðtÞ are symmetric and antisymmetric parts of qðtÞ, respectively, and subscript r may be indicated by " or #. At low temperatures, Eq. (32) is dominated and one can ignore Eq. (33) with respect to it [25]. In Eq. (32), k2sr and Br are

, Z Z sin h dh du 3 sin h dh du 2 2 W22 1
ð1
cos hÞ sin u W22 ; ð34Þ k2sr ¼ h 4 cos 2 cos h2

290

M.A. Shahzamanian, R. Afzali / Annals of Physics 309 (2004) 281–294

16p5 Br ¼ 3 2 m T

"Z

sin h dh du W22 cos h2

#
1 ;

ð35Þ

where, for r ¼" and #, the function W22 stands for 12W22 ð"#Þ and 14W22 ð##Þ þ 12W22 ð#"Þ, respectively. For obtaining k2sr and Br , one must indicate the functional relations of TtI , TsI , TtII and TtIII on h and u. Furthermore, we note that h is small for superfluid case and its maximum value is pT =Dð0Þ [24] where maximum gap, Dð0Þ, due to strong coupling effects is equal to 1:77Tc [26]. Then, we have





 2 h 4 h 6 h N ð0ÞTsI ¼ 2:47 þ 6:61 cos þ 17:69 cos
11:2 cos 2 2 2


 h h 2:9
0:96 7 cos2
1 ;  ð3 cos2 u
1Þ sin4 ð36Þ 2 2


 2 h 2 h 4 h cos u
3:3 þ 2:28 cos
5:82 cos N ð0ÞTtI ¼ sin 2 2 2  h ð37Þ
0:74 sin4 ð5 cos2 u
3Þ ; 2 h  u u i ; ð38Þ N ð0ÞTtII ¼
4:78 þ h2 0:54 cos2
3:05 sin2 2 2 h  u u i : ð39Þ N ð0ÞTtIII ¼ 4:78 þ h2 0:54 sin2
3:05 cos2 2 2 It is noted that in obtaining the above equations we truncate the sum in Eq. (15) for k ¼ 3 at melting pressure. Of course, it should be mentioned that the s–p approximation with the Landau sum rule fulfilled is equivalent to truncate the k sum in Eq. (15) at k ¼ 1 [21]. Finally, we obtain k2s# ’ k2s" ’ 0:75; B2s" ’

p5 N ð0Þ2 ; m3 T 2 186:53

ð40Þ B2s# ’

p5 N ð0Þ2 : m3 T 2 195:81

ð41Þ

Following the Sykes et al. [25] procedure, the following result is obtained from Eq. (32) Z þ1 dm0
2B q2sr ðtÞ ¼ 2 cðk2sr Þ; dt ð42Þ p dt ð1
k2sr Þ
1 where cðkÞ is [25] 1 ð1
k2sr Þ X ð4n þ 3Þ ðn þ 1Þð2n þ 1Þfðn þ 1Þð2n þ 1Þ
k2sr g 4 n¼0 o n k2s
1 1 1 ¼ r c þ ‘n2 þ wd ðs1 Þ þ wd ðs2 Þ ; 2k2sr 2 2

cðk2sr Þ ¼

ð43Þ

M.A. Shahzamanian, R. Afzali / Annals of Physics 309 (2004) 281–294

and c ¼ 0:577 . . . is EulerÕs constant, wd is a digamma function and 3 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8k2sr þ 1; s2 
8k2sr þ 1; s1  þ 4 4 4 4

291

ð44Þ

and we have cðk2s" Þ ’ cðk2s# Þ ’ 0:77. Now, we will proceed to calculate the shear viscosity in the following section by using Eq. (42) and the calculated values of Br , k2sr , and cðk2sr Þ. 4. Viscosity The shear viscosity in general is a fourth-rank tensor, which is defined by the relation 

X oui ouk 2 ~ plm ¼
glmik
dij r  ~ u ; ð45Þ þ ork ori 3 ik where plm , the momentum flux tensor, is Z oE plm ¼ dsp pl dmr ðpÞ: opm

ð46Þ

When Eq. (28) is substituted in Eq. (24) and then in Eq. (46) and compared with (45), we get

Z  Z Z 4pF5 om0 on0 glmik ¼
dXp p^l p^m p^i p^k dt q ðtÞ þ q ðtÞ : dt ð47Þ 3 ot 2s" ot 2s# ð2pÞ m By using (42) in Eq. (47), we have ! Z pF5 B" B# dXp p^l p^m p^i p^k glmik ¼ 5  cðk2s" Þ þ cðk2s# Þ ; pm 1
k2s" 1
k2s#

ð48Þ

where we employ the fact that at low temperature all momenta lie on the Fermi surface. After substituting Br , k2sr , and cðk2sr Þ and taking the angular integrations, we have 2

1 N ð0Þ pF5 1 gxy# ¼ gxz# ¼ gzz# ’ 13:17  10
3 2 ; 4 3 T m

ð49Þ

2

gxy" ’

4 N ð0Þ pF5
4 T 4:28  10 ; m4 Tc6

gxz" ’

N ð0Þ pF5 T2 8:16  10
4 4 ; 4 m Tc

ð51Þ

gzz" ’

N ð0Þ2 pF5 1 10:36  10
4 2 : 4 T m c

ð52Þ

ð50Þ

2

292

M.A. Shahzamanian, R. Afzali / Annals of Physics 309 (2004) 281–294

By nothing that gxy  gxy# þ gxy" , gxz  gxz# þ gxz" , and gzz  gzz# þ gzz" , finally we have 2

1 N ð0Þ pF5 1 gxy ¼ gxz ¼ gzz ’ 13:17  10
3 2 : 4 3 T m

ð53Þ

Now we proceed to express Eq. (53) in terms of the shear viscosity at Tc , gðTc Þ. gðTc Þ is ð1=5Þqðm =mÞv2F s0 with s0 being a characteristic relaxation time and given by 8p4 =m3 Tc2 hWN i and density of the liquid, q, being mpF3 =3p2 . gðTc Þ depends on hWN i, m , vF , Tc , and s0 . In the following, we calculate the constant prefactor by using different values of these quantities and different methods. Case 1. If we use the Pfitzner procedure, after some algebra we have Z  130:56 dX 2p  2 2 3jT hWN i ¼ j j ; þ jT ’ t s 4p cos h2 8 N ð0Þ2

ð54Þ

then we have


2 1 T gðTc Þ: gxy ¼ gxz ¼ gzz ’ 0:33 3 Tc

ð55Þ

Case 2. If we use s–p approximation for calculating singlet and triplet quasiparticle scattering amplitude and the numerical values of Landau parameters in [1] ([27]) for all steps of calculating g, then we get the prefactor in Eq. (55) as equal to 0.15 (0.18) Case 3. If we use the values of gðTc Þ and Tc2 from [19], vF from [28], and m =m from [29], we get


2 T gxz ’ 0:34 gðTc Þ: ð56Þ Tc It is amusing that if we take m =m from [28], the above prefactor will be 0.47. Case 4. If we use the values of s0 Tc2 and vF from [28] and m =m from [29], we get


2 T gxz ’ 0:44 gðTc Þ: ð57Þ Tc It is amusing that if we take m =m from [28] the above prefactor will be 0.47.

5. Conclusion By supposing that Bogoliubov quasiparticles with spin up are formed in the superfluid component in A1 -phase, we calculate the transition probabilities for the cases where both the normal and Bogoliubov quasiparticles are presented in decay, coalescence, and binary processes, beyond s–p approximation. We use the Pfitzner [21] procedure for obtaining singlet and triplet quasiparticle scattering amplitude.

M.A. Shahzamanian, R. Afzali / Annals of Physics 309 (2004) 281–294

293

Then we calculate the components of shear viscosity tensor of the A1 -phase of superfluid 3 He at low temperatures and high magnetic field at melting pressure. Roobol et al. [20] results on the shear viscosity of the A1 -phase of superfluid 3 He in a static magnetic field up to 15 T at low temperatures indicate g ¼
2 0:48gðTc ÞðT =Tc Þ . They mention that in measuring the values of the viscosity, g, mainly the values of the component gxz with a few percent of the values of the component gzz will be contributed. If we take the contribution of gzz in g between 5 and 10%, we obtain the prefactor between 0.44 and 0.40, respectively. This result is in fairly good agreement with our result which is obtained in case 4. In this case, we benefit from the Pfitzner procedure plus the values of the quantities s0 Tc2 and vF from [28] and m =m from [29]. In the following, it would be suitable to present a quantitative comparison between results obtained from the Pfitzner procedure and s–p approximation. While the calculated prefactor for case 4 through s–p approximation is about 0.18, and considering the fact that the calculated figure has an error margin of a 100% and even more in comparison with the experimental data, the calculated prefactor for case 4 through Pfitzner procedure is 0.44 having a few percent margin of error. Hence, Pfitzner procedure gives better results than s–p approximation. It is noted that this is also the case for the results near Tc [11] where Pfitzner procedure was implemented for the first time to calculate one of the transport coefficients of superfluid 3 He. Moreover, implementation of this procedure in calculation of the transport times of normal fluid has previously proved to give better and more acceptable quantitative results [21]. On the whole one can say that the Pfitzner procedure gives more satisfactory results than s–p approximation. We obtain gxy" , gxz" , and gzz" with dependence temperatures T 4 , T 2 , and T 0 , respectively (see Eqs. (50)–(52)). It is seen that g" do not play a role in the shear viscosity components compare to g# . It is interesting to mention that interaction between superfluid and normal fluid in calculating the transition probabilities is distinct, whereas the contribution from the interaction between Bogoliubov quasiparticles at low temperatures is ignorable. Also it is mentioned that just binary processes are dominated in calculating the transition probabilities. In summary we can say that the viscosity components at low temperatures in the A1 -phase of superfluid 3 He are proportional to T
2 and the agreement between our results and the experiments is fairly good.

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