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Physica Scripta. Vol. 42, 481-484, 1990.

Two-F lu id Hydrodynamic Description of Quasiparticle

Gas in Crystals

D. I. Pushkarov and R. D. Atanasov

Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria

Received August 11, 1989; accepted February 13, 1990

(1)

(2)

Abstract

A Bose gas of quasiparticles in crystals is described in terms of a two-fluid

hydrodynamics based of both the momentum and quasimomentum conser

vative laws. The full set of hydrodynamic equations is deduced. Under

certain conditions the quasiparticle gas possesses superfluid properties. They

appear only in the case of a non-quadratic quasiparticle dispersion law. The

possibility of a second-sound-type wave propagation is considered, and

corresponding dispersion laws are obtained for all interesting limits. The

results are expressed by means of the thermodynamic functions of the

quasiparticle gas under consideration.

1. Introduction

Since the Landau's two-fluid theory [1], a number of attempts

about the applicability of i ts ideas to the behaviour of the

quasiparticle gas in crystals have been made. In particular,

the question for a second-sound propagation in diverse media

has been considered [2-5]. Special attention has been paid to

the idea for the existence of a superfluid solid. In this sense,

the solidHe

4has

beenproposed as

amost promising

objectof study. As is known, vacancies are delocalized in it and

turn into quasiparticles-vacancions [6, 7]. In some approach,

vacancions obey Bose-Einstein statistics [6, 7] and, in prin

ciple, they could possess superfluid properties. Naturally, a

superfluid vacancy flow in some direction is equivalent to a

superfluid mass flow in the opposite direction. First con

sideration of the two-fluid properties of crystals has been

made by Andreev and Lifshitz [6], using a phenomenological

descr ip tion in close analogy to that one int roduced by

Landau for the liquid helium. The authors show that in

principle, a superfluid movement is possible, and in a par

ticular case a mass flow at fixed crystal lattice sites could exist.

Later, Saslow [8] made an a tt empt to build a microscopic

theory of the same phenomenon bu t as showed Liu [9] some

Maxwell's relations are no t satisfied in his theory, which calls

the obtained results in question. Liu's investigation is based

also on a macroscopical consideration of the problem con

cerning a superfluid sol id and the main problem which is

accentuated is the role of the transformation properties of the

superfluid velocity VS: if it obeys the Galilean transformation

then a mass superfluid flow could arise and if it doesn't

entropy superfluid flow turns ou t to be possible. But the

question concerning the physical origin of VS remains open.

In the present investigation, we consider the two-fluid

properties of an arbitrary quasiparticle Bose-gas subject to

both quasimomentum and momentum (e.g., quasiparticle

mass flow) are conserved. In principle, conservation of the

momentum means a conservation of the to tal mass flow

consisting of quasiparticle gas flow and lattice sites mass flow

[7]. But in case of fixed lattice sites, the second one vani.shes.

The analysis performed in this work shows that the account

ing of both the quasimomentum and momentum conservation

lows leads to a superfluid behaviour of the quasiparticle gas

i.e., under certain conditions, a mass flow no t attended by an

entropy flow could exist. A second-sound propagation in

such a system is considered depending on the type of the

quasiparticle dispersion law.

2. Distribution function and thermodynamic relations

We shall examine a gas of non-interacting quasiparticles in a

crystal. Every quasiparticle will be represented by its reduced

wave vector k and dispersion law W k == wk(k). If the tem

perature T(r, t) and the quasiparticle number density nCr, t)

are sufficiently small, one may regard that the frequency of

the normal processes r ; 1 is to a marked degree greater than

the frequency of the Umklapp processes r 1, and hence, the

quasimomentum can be regarded as a conservative quantity.

Then, a localequilibrium

state ofthe

quasiparticle gas couldbe established by means of the normal processes. In this case

it is possib le to int roduce a local equil ibrium distribution

function of the quasiparticles nk == nk(k, r, t). It has to pro

vide a maximum of the entropy density S(r, t), on condition

that densities of energy E(r, t), momentum j(r, t), quasi

momentum K(r, t) and nCr, t) are conservative quantities. In

case of Bose quasiparticles, the mathematical representation

of the problem is to determine nk so that

S(r, t) = fs[nkl dk,s[nk] == (1 + nk) In (1 + nk) - nk In (nk)

is maximum (we use here k b == h == 1), subject to

E(r, t) fwknk dk

n(r, t) fnk dk

K(r, t) f knk dk

j(r, t) = m f (owk/ok)nk dk

where m is the bare mass of a real particle, dk == (11

2n)3 dk 1 dk 2 dk3and the integration over k is performed over

the first Brillouin zone. This leads to the following distri

bution function

= [exp ( _wk_-_V_·_k_-_m_r;_._(8_w_1 _ 8 k _ ) _ - _ ~ ) 1r(3)

Physica Scripta 42



482 D. I. Pushkarov and R. D. Atanasov

In th e case of a gap dispersion law W k == W o + w(k) and

T Wo the two densities are proportional to exp ( - wo/T)

bu t their ratio is a power function too. It can be shown that

(11 )

(13)

(12)

Analogously, fo r quasi-momentum one finds

K; == Pi/ V; + mno Jt';

where

Pil = - f k;k/(8nk/8wk) dk.

If th e dispersion law is of the form W k == akdthen the

effective density Pi} ex T(2-d)/d and n exo T 3/d and hence

Pi} /n ex T(2-d)/d

(4)

In fact, this is the definition of the energy as a function of the

variables S, n, K and j.

From (1) and (3) we find the following thermodynamic or

relations

TbS == bE - V - bK - W - bj - jibn

where the local t ime-dependent quant it ies ji and T corre

spond to the chemical potential and temperature of the

quasiparticle gas and Vand Ware velocities (Lagrange multi

pliers) conjugate to the quasimomentum and momentum

respectively. Note that there is only a mathematical similarity

between the distribution function (3) and t ha t one arising in

case of a Galilean transformation to a new frame of reference

moving with a velocity V or W [1, 4]. Here we have introduced a local frame connected with the crystal lattice and no

kind of transformations are supposed.

Varying (1) with the distribution function (3) we ge t

n == E - TS - V - K - W -j - jin

(7) Sil = f s [ n ~ ] v ; / dk (17)

In this way, the full hydrodynamic system consists of four

equations, which can be wri tten as

defining the thermodynamic potential n as a function of the

variables T, ji, V an d W.

3_ Two-fluid hydrodynamics

The non-dissipative two-fluid hydrodynamic equations for

the quasiparticle fluid are contained in th e following set of

conservation laws

To second order i n the velocities the quasimomentum flux

is given by

(16)

(15)

(14)

fwk(8wk/8k;)nk dk

Wov; + (TS;, + jini/)

where

Li j = fk; (8Wk/8kj)nk dk = Q0(jij

and the momentum flux has the following form

II il = -Q il = T fin (1 + nnvAk) dk.

In respect of the energy flux, we find

(6)

and respectively,

dn == - S dT - K - d V - j - dW - n dj i

n+V - J==O

E + v- Q == 0

j+v-n==o

K+ VeL 0

S+V - F 0

(9) dE == T dS + ji dn + V - dK + W - dj

where J == jIm is the number density flux. The vectors Q and

F are the energy density flux and entropy density flux respec

tively and Land n are the quasimomentum and the momen

tum flux tensors. In so fa r as we shall concern about a sound

wave propagation we can expand nk in powers of V; an d J--V

an d restrict ourselves to the l inear members . In this approxi

mation the above fluxes may be expressed by means of nO ==nCr, t, V == W == 0) and t he g loba l equ il ib rium thermo

dynamic functions n° == nCr, t, V == W == 0) and EO ==E(r, t, V == W == 0). Then we f ind to second order in the

velocities

the last of equations (18) can be replaced by entropy equation

J; == mno V; + mn;j Uj

where

nil = f n ~ v i l ( k ) dk

and

(8)

n + nV - U == 0

mnUi - (ani//ax,) == 0

p;s(ansl/ax,) - n(ano/ax;) + n2(b;1 - Pi/ )'r-Vl == 0

E + wov - U + TS(Su/S - n u / n ) ( a ~ / a x ; ) == 0

where

f3i/ == P;knktl(nO)2

and

W O == EO _ n°

is the enthalpy.

Taking into account the thermodynamic identity

(18)

(19)

In general, the velocities V and Ware not collinear, conse

quently, t he mass flux j is no t parallel to anyone of them.

Obviously, the mass flux and the entropy flux depend on

different velocities both in value and in direction. In this way,

a mass flux without an entropy transport may occur. That

means the existence of a superfluid density pS•

In case of quadratic dispersion law, however , the super -

vi/(k)==

m(a

2

w(k)/ak;ak,).Therefore, the local drift velocity U is given by

U; == V; + ( n i / / n O ) ~ . (10)

S + SV - U + SCSi/IS -n i / / n ) ( a ~ / a X i ) ==

0 (20)

Physica Scripta 42



Two-Fluid Hydrodynamic Description of Quasiparticle Gas in Crystals 483

and the third of eqs. (18) together with the additional entropy

flux disappears. Then K becomes proportional to j i.e.,

(29)

(28)

o

o

o

o

(8n/8T)/l

(8S/8T)/l'

a t + f3 jl + nSV • VS + nnV · vn

yt + ajl + S V · vn

S VT + n V J.1 + pS VS+ pn Vn

nS VT + nnn VJ.1 + pnnsVs + pnnnvn

where

4. Second-sound propagation

With the aid of the derived equations, let us consider the

problem of the spectrum of the sound waves in the quasiparti

cle gas under consideration. Linearizing (25) we obtain the

following system

(22)

(21)

In order to see in an explicit form the superfluid behaviour

of the quasiparticle gas under consideration we shall rewrite

the full hydrodynamic system (18) in terms of the superfluid

hydrodynamics. Fo r simplicity of the notation we shall con

sider cubic crystals. Then we find to second order in the

velocities

S,.dS == nu/n

f luid density vanishes, in sense that

Pu == mnobu

where b is the unit tensor and the following notation has been

introduced:

and ~ == (8/8k)2. The above quantities are connected by the

relations:

P/l == (8p/8J.1)T == V,

PT == (8p/8T)/l == (p + q - J.1v)/T.

They have no t an exact physical sense, except for the case of

quadratic dispersion law W k == k2/2m* when p == - (m/m*)Qo,

v == (m/m*)n and q == (m/m*)Eo.

Substituting F == SVn

and V == V Sinto the above expres

sions for the fluxes, we obtain the full set of hydrodynamic

equations 9f the quasiparticle gas consisting of equation (19)

and

(31)

(30)

o

o

o

a t + f3jl + nV · vnyt + ajl + SV · vnSVT + nVJ.1 + pV

n

The introduced effective densities:

4.1. Quasiparticles with quadratic dispersion law

As was noted in the previous section the quadratic dispersion

law is critical for the superfluid behavior of quasiparticle gas

under consideration. It is easy to see that pS == nS == 0 when

W k ex k2• Then the general system of hydrodynamic equations

(25) is reduced to the following one:

pn == mnS/PT,

p f k2nO ( l + nO) dk,

pS P _ pn

obey the following equation

K == pV + mnW == pSVs + pnvn.

Now we shall examine a few particular cases of the second

sound propagation, according to the type of the quasiparticle

dispersion law and the value of the chemical potential.

(24)

(23)

tm f (owk jok)2 n2 dk,

tm f (Akwk)n2 dk,tm fwk(Akwk)n2 dk,

v(r, t)

q(r, t)

p(r, t)

J(r, t) == nV + vW, Q(r, t) == WO V + W(p + q),

n == pb, L == -Qob, F == SV + pTW

S + s v · v n ~ oK + SVT + nVJ.1 == 0

J+ PTVT + PpVJ.1 == 0

oNote tha t a t m == 0 the same system is valid also for massless

quasiparticles (e.g., phonons) with arbitrary dispersion law.

4.1.1. The number of quasiparticles is not conserved. If the

(25) quasipart ic le number is no t constant then J.1 == 0 and the

number conservation law (the first one of (31)) is to be

excluded from the system (31). Hence, the standard pro

cedure yields the following second-sound dispersion law

where the following notation has been introduced (32)

(34)

(33)

where Cy is the specific heat of the quasiparticle gas.

4.1.2. The number of quasiparticles is conserved. In this

case J.1 =1= 0 and the specific heat is expressed by Cy ==

T(y - a2/ f3). Then the condition for a non-zero solution of

the full system (31) yields

w2(q) =:= w6(q) {[I - an/(f3S)]2 + Cyn2/(Tf3S2)}

(26)

(27)

In these variables, the number density flux is expressed as

The first and second of equations (25) as well (26) and (27)are analogous to those arising in the classic theory of super- The above equation may be rewritten as

fluidity [1]. In this sense we may regard the quantities nS and 2

V S as the number density and the velocity of the superfluid oi(q) == [(T/Cy)(8s/8vo)} - (8J.1/8vo)T] 'Lflow whereas nn and Vn are the corresponding ones of the p

normal flow. where s == Sin an d Vo == l/ n are the entropy and the volume

Physica Scripta 42



(35)

484 D. I. Pushkarov and R. D. Atanasov

per one quasiparticle. Making use of the following Maxwell

relations:

(oPjoT)vo;

Vo [(OP/Ovo)s + f (OP/OT)vo]

where P is the presence of the quasiparticle gas and Cy==

Cy jnis the specific heat per one quasiparticle, we find at last

2 (OP) q2W (q) == - n-

on S p

In the event W k == k 2 (2m) from (30) one finds p == nm

and (35) coincides formally with the sound dispersion low in

the ordinary gases.

4.2. Quasiparticles with non-quadratic dispersion law

This type of dispersion law keeps the superfluid densities nS

and pS non-vanishing, and hence, we need the full system (28).

Westart

again with a second-soundpropagation

in casethe quasiparticle number is not constant. Then substituting

J1 == 0 into (28) and excluding the first of them, one finds the

following second-sound dispersion law

where

bn== K

n2j(KP - Kn).

The system (38) determines two eigen frequencies W I,2(Q) for

each value of the wave vector q. The precise expressions of

WI ,2(q) are especially simple in the case when the density

ratios KP, K

n, and hence bP and bn are small in comparison

with the unity. Then we find the frequencies W I ,2 in l inearapproximation in the small bP and bn as follows

W 2I(q) == bn[2( == n) _ ( TS)2 ( o P j a T ) ~ q2] (39)

W o q, P P n Cy (oPjon)s pn

and

w ~ ( q )

(40)

where wo, wand WI are given by (32), (35) and (39) respectively.

Obviously, w2(q) is greater than WI (q), and hence, the corres

ponding velocity of the sound wave C2 is greater than C I for

any q -# O.

(36) References

where

bP == KPKnj(KP - Kn), KP == pSjpn, Kn == nSjnn (37)

The general spectrum of the sound propagating modes

when J1 -# 0 is obtained by solving equations (28) simul

taneously. Assuming that all quantities depend on the coor

dinates and time by means of the factor exp (iq · r - iwt)

and eliminating VSand vn with the aid of the last two vector

equations of (28), we find

[w2(a - nnyjS) + ( n n b n S j p n ) ~ ] T + w2(fJ - nnajS)J1 0

[yw2 - (1 + bP)[Sqfjpn]T + [aw 2 - (Snj pn)q2]J1 == 0

(38)

Physica Scripta 42

1.

2.

3.

4.

5.

6.

7.

8.

9.

Landau, L. D., Zh. Eks p. Teo r. Fiz . 11, 592 (1941).

Gurzhi, R. N., Zh. Eksp. Teor. Fiz. 46,719 (1964). [Sov. Phys. - JETP

24,1146 (1967)].

Gurevich, V. L. and Efros, A. L., Zh. Eksp. Teor. Fiz. 51,1693 (1966).

Enz, C. P., Rev. Mod. Phys. 46, 705 (1974).

Pushkarov, D. I., Preprint-P 17-85 24 of the J INR, Dubna (1985).

Andreev, A. F. , and Lifshits, I. M., Sov. Phys. - JETP 29, 1107

(1969).

Pushkarov, D. I., Defectons in Crystals (in Russian) PI7-87-177

Dubna, 1987.

Saslow, W. M., Phys. Rev. 15,173 (1977).

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