First and second sound hydrodynamic equations in solids

Solid State Communications,

Vol. 7, pp. 743—746, 1969.

Pergamon Press.

Printed in Great Britain

FIRST AND SECOND SOUND HYDRODYNAMIC EQUATIONS IN SOLIDS

J. Ranninger” Institut Laue

—

Langevin, 8046 Garching, Germany

(Received 20 February 1969 by J.A. Krumhansl)

The hydrodynamic equations for first and second sound derived by diagrammatic techniques are critically compared with those obtained by decoupling procedures. A simple diagrammatic description of the sum-rules is obtained.

IT HAS been pointed out that the hydrodynamic equations derived by a decoupling procedure 1 for the Green’s function equation hierarchy agreeing with the phenomenological theory2 differ from those derived by diagrammatic techniques.3’4

phonon propagators for thermal phonons. It turns out that this particular approximation has just the right analytic properties to guarantee the conservation laws of the system.6 The self-energy ~ arising from processes where a phonon spontaneously decays into two others is adequately described by the simple bubble diagram (Fig. 2).

In the following a critical comparison of the two techniques and the approximations involved in each of them is given.5 For thermal phonons the phonon propagator is adequately approximated in satisfying a Dyson equation in which the self-energy is given by the simple bubble diagram which is evaluated in the harmonic approximation (Fig. 1).

-

~

—.——

FIG. 2. Dyson equation for phonon propagator

in the hydrodynamic regime.

~ FIG. 1. Dyson equation for phonon propagator describing thermal phonons. For wave vectors and frequencies in the hydrodynamic regime the self-energy ~ (q,z) arising from processes where a phonon absorbs one and emits another one has to be approximated by the ladder diagrams. The usual argument justifying this approximation is that each of the ladder diagrams containing any arbitrary number

The Bethe—Salpeter equation derived for the ladder diagrams of the time ordered product <~3~ (t) ~ (0)> with (t)

~q

=

(k

~

k

~,

q)

(b~.,~/ 2 (t) b/,~.~/2 (t)

of rungs is of the same order of magnitude. The lines of these diagrams are represented by the *

~,

k +

c.c.)

describing the phonon 3 (k ~ self-energy ~ ~q,z) by

On leave of absence from AERE Harwell.

k

743

12 th

2

2

(1)

(2)

744

FIRST AND SECOND SOUND HYDRODYNAMIC EQUATIONS G~(k

3(q,z) 0)= /3 2S~J(2wQ)e~D~ in (6) we obtain the hydrodynamic equation

7

becomes

i (z =

—

q) Gk (q,z)

~

Vol. 7, No. 10

(9)

Ckk Gk (q,z)- iv~,q 2 S~th~(k 0

+

iz G (k0)- vk0.q(G (k0)- G~(k0)) Ck0k (G (k)— G0 (k))

~,k0 _~,q) (3)

(10)

1 derived by Götze and Michel.

where qY (k

0, k1, k2) denotes the phonon coupling constant, w~ the harmonic frequency and ~k2 the group-velocity of a phonon k the four vector k1 denoting the wave vector k~and the phonon polarization. b~ (0 and bk~(0 denote the phonon creation and annihilationoperators in the usual way. The collision operator Ck0k is defined by

collision operator in (3) we obtain G (k0) = G0(k0), which corresponds to evaluating the static part of the correlation function G (k0) in the harmonic approximation. Taking into

2 Sk0 Ck0k ~k (q,z)

function (Fig. 2)

—

Taking the limit z -~ 0 and neglecting the

account the ladder diagrams for ~ (q,z) and the bubble diagram for ~‘(q,z) we obtain the following Dyson-equation for the displacement autocorrelation

=

2(Sk k1k2~

(kQ,kl,k2)}

(Wk0— ~k1 —

+

—

~k2

0/SkSk2)x2~S

[z2~ w e~e~]D~~(q,z)= ~+ ±/3\!(2cQ)e~V(2~&)Q)eQ (2~(q,z)+ ~‘(q,z))D~~ (q,z), -

)tSk0Gk0 (q,z)— Sk G~1(q,z)

Sk2 Gk2 (q,z)1

(11)

where e~denotes the ith component of the

2(Sk

2k~[~~ (k2,kQ,k~)]

0/Sk1Sk2)x 27T~

(Wk0— W1c~+W~2)

Gk~(q,z) S~Gk1 (q,z)

~

—

+

S~G~2(q,z)~ (4)

sinh (f3~k~/2)•

with S~

phonon polarization vector. Approximating ~.‘(q,z) by >_‘(q,O) and using relations (6) and (8) we obtain 2 ~ - w~e~e~ - ~ (2w~) e~\/(2w~)e~ [z (2~ (q,0)~~‘(q,0))J D12 (q,z) = =

+

\

(2c~)e~ c~ (k 0 +~,k0-~.q)

Taking the limit z 0 in (3) and subtracting the equation thus obtained from (3) we find -‘

iz G~0(q,z)—iv~~q (Gk~(q,z)—G~0(q,0))

Ck0k (Gk (q,z)

—

Gk (q,0)).

(5)

Defining the three-point function G (k0) for the

=

/3 S~(2w~)e~ G~~(q,z)D~(q,z) (6)

iz G (k~) ivk

0. q (G (k0) with

—

—

~ (k0))

~ (k))

1~(q,z) /3 S~J(2wq)e~G~(q,0)D and D1~(q,z) denoting the displacement autocorrelation4unction.’ Replacing G (k 0) by G0(k0)

of (12) with 2 [z

~

~

qmq~~)D&7(q,z)

(13)

8 the elastic sum-rule ~li~n~ ~ qThqr~D~’~q,z)=

—

Ck0k (G (k)

Here again, only after replacing ~ (k0) by G0 (?<~)in (12)do we obtain the corresponding hydrodynamic equation for the displacement

identifying the expression on the left hand side

we~an re-write (5)in the form

=

(12)

autocorrelation derived by Gdtze and Michel.’ Defining the elastic coefficients ~ by

phonon vertex by G (k0)

(G (k~)- G (k0))

(7)

8

3~J

(14)

follows the collision immediately term in since (12) vanishes the contribution identically. from The elastic coefficients thus defined (14, 15) are evaluated in the ladder diagram approximation. The approximation used by Götze and Michel1 leads to the same sum-rule (15), yet in their case

Vol.7, No. 10

FIRST AND SECOND SOUND HYDRODYNAMIC EQUATIONS

the elastic coefficients are evaluated in the

745

The improper contribution to the energy

3 is given by density autocorrelation function ~2w’ tDtJI t t””2 ~

quasiharmonic approximation.

Finally some comments on the energy density autocorrelation function should be made,

/

k /

~

V\

~q,

—

<~@‘) Eq(t)>

restricting ourselves to the harmonic approximation for the energy density operator.

r~. w~je~ (17)

.

and thus we obtain the total contribution to the energy density autocorrelation function

Given the above approximation for the phonon

G~(q,z)=G~(q,z) 0+

self-energy the consistent approximation for the two particle and Green’s function is automatically determined is derived in the usual way by

12 t2(q,z) + /3[~~~Cj<~(t)~~(z)>~(2w0)e~D

functional derivatives of the phonon Green’s function. The dominant contributions from the

\/(2wQ)e’Q <~(t) c~(t)>.

~‘

\

(18)

diagrams obtained in this way for the two-particle

Taking the limit z~0,q~0in (18) we obtain the

Green’s function is again given by the ladder diagrams.

sum-rule previously derived by Götze and Michel .

(

.

2

lim lim G (q,z)= C~T+T q Q90

The proper part of the energy density autok

correlation function

G~(q,z)~

I

~ G~(q,z)~= Sd(

etztc

tj,rni~

q q

q

ac-’~

T2~(2wQ)e~ = q’

(15) with the energy density operator EQ

(0

~

[41

(~k*(q~~+ ~k-(q’~ ~ \‘(wk(Q:~

9c~ —

2-90

19 ~-‘

(20)

aT where ~

denotes the stress tensor1 and C, the

specific heat at constant volume.

)1 1 )J2

.4ckriowledgements

—

The author is indebted to

(b1(~~ 2~ (t) bk_(Q

~2) (t)

—

c.c.)

(16)

satisfies a Bethe—Salpeter equation for G~0(q,z) which is obtained from (3) by replacing the inhomogeneous term in the equation by iv~0.q 2 S~ and where G~0(q,z) in the collision operator is weighted by the factor c~k./ci~k.Wk~ is given by the expression in the square bracket

of (16).

Drs. K.H. Michel and F. Schwabl for illuminating discussions.

746

FIRST AND SECOND SOUND HYDRODYNAMIC EQUATIONS

Vol. 7, No. 10

REFERENCES 1.

GOTZE W. and MICHEL K.H., Phys. Rev. 157, 738 (1967).

2. 3. 4. 5.

GOTZE W. and MICHEL K.H., Phys. Rev. 156, 963 (1967). SHAM L.J., Phys. Rev. 156, 494 (1967); Ibid. 163, 401 (1967). KLEIN R. and WEHNER R.K., Phys. condensed. Matter., 8, 141 (1968). The following discussion is given for a phonon system interacting via 3-phonon processes only. Where certain quantities are not specifically defined the reader is referred to the notations of references 2 and 3.

6. 7.

RANNINGER J., PToc. phys. Soc. (to be published). The equation for Gk0(q,z) = . q (Z~0(q,z)+ Z~0(q,z))derived in reference 3 differs from (3) by the occurrence of ~k0 . q in the inhomogeneous term. Nevertheless these equations are equivalent to each other which is easily seen by iterating the respective solutions of the equations.

8.

GOTZE W., Phys. Rev. 156, 951 (1967).

Die auf diagrammatischem Weg hergeleiteten hydrodynamischen Gleichungen für ersten und zweiten Schall werden mit den durch Entkopplungsverfahren ermittelten Resultaten verglichen. Die Summenregein werden auf diagrammatischem Weg hergeleitet.