On the attenuation of sound in liquid helium

Physica 32 1905-1920

Pethick, C. J. Ter Haar, D. 1966

ON THE ATTENUATION

OF SOUND IN LIQUID HELIUM

by C. J. PETHICK

and D. TER

HAAR

Department of Theoretical Physics, Oxford, England Synopsis We develop a theory for the attenuation of sound in liquid helium, valid at temperatures below about l”K, by considering three-phonon processes. We also consider some higher-order contributions to the attenuation and consider the influence of finite lifetimes on the velocity of sound. We compare our results with experimental data and also discuss theories developed by other authors. 1. Introduction. Since the pioneer measurements of Pellam and Squirel) a number of people have measured the attenuation of sound in liquid helium.

We quote in particular the experiments by Chases’), Chase and Herlins), Whitneyd), Dransfeld, Newell andwilkss) and Jeffersand Whitney6 At temperatures of the order of 3°K and above, the attenuation measurements are adequately accounted for by classical hydrodynamical theory. Near the il point the attenuation has a maximum; this has been discussed theoretically by Pippardy), and by Landau and Khalatnikovs). A second maximum is observed at lower temperatures; this occurs at about 1°K for a frequency of 10 MHz. Before experiments had been performed, Khalatnikovs) predicted this maximum; however, Khalatnikov’s theory predicts a value of the attenuation at temperatures below those where the second peak occurs which is far too low, and we shall be concerned in the present paper with that temperature range. There have been a large number of papers dealing with this low-temperature behaviours) 10-17) all of which except refs. 5 and 15 are based on a discussion of the three-phonon process and lead to an mT4-dependence of the attenuation in some temperature range, and we shall discuss the relation of those papers with our approach. A second, related topic to be discussed by us is the velocity of sound at 1 MHz, which was measured by Whitney and Chaser*). The theory of Andreev and Khalatnikovrs) which uses kinetic equations to derive an expression for the temperature-dependent part of the velocity of sound does not give satisfactory agreement with experiment. In the next section we discuss some of the earlier work on the attenuation of sound in helium, before presenting in section 3 a unified approach to the problem of acoustic waves in helium. Section 4 is devoted to a description -

1905 -

1906

C. J.

PETHICK

AND

D. TER

HAAR

of the generalised unitarity condition for bosons, and this is used in section 5 to calculate the rate of the three-phonon process. Some differences between our results and those of other authors are indicated.

A higher-order

contri-

bution to the attenuation is evaluated in section 6, and in section 7 we rederive the result of Andreev and Khalatnikovla) for the velocity of sound and show how this is modified finite lifetimes. Theory

and experiment

when the thermal excitations are compared

have

in section 8.

2. Earlier work on the attenuation of sound. All the papers referred to in the Introduction use some form of excitation model to evaluate the attenuation. We first of all note that the theoriesof Kawasakill), Dransfeldls), Woodruffls), and Kawasaki and Moril4) break down if the linewidth of thermal phonons is small compared with the energy deficit for the threephonon process *). We want to comment briefly on the papers by Dransf eld and Woodruff as their coupling constants differ from the one obtained from the quantum hydrodynamical model of liquid helium (see e.g., ref. 20, par. 17). Dransfeldlz) omits part of the kinetic energy term in the interaction. We also note that section 2 of his discussion is in error. Dransfeld states that for appreciable attenuation to occur, the mean free path of thermal excitations must be sufficiently long for a thermal phonon to be able classically to absorb one phonon of the acoustic wave between collisions. If this were true, the attenuation coefficient would vanish in the limit of zero acoustic wave amplitude. The error lies in the use of a mixture of classical and quantum mechanical arguments. Woodruff 1s) states that the superfluid and the normal velocities may be out of phase. He shows that this will lead to an alteration in the effective coupling constant. At the end of his calculation Woodruff adjusts the ratio of the velocities to obtain agreement with experiment. The ratio of velocities is not an arbitrary parameter and can be obtained from a knowledge of the coupling between thermal excitations and a density fluctuation. 3. A general discussion of sound waves in LiqzGid helium. The velocity and damping or a sound wave may be found from the position of the poles in the density-density Green function. We cannot evaluate this Green function from first principles, and we shall use the hydrodynamical model. In terms of this model the Green function we require is a single-particle function, whereas in a theory using real-particle creation and annihilation operators perturbation it would be a two-particle function. The thermodynamic

*) For a discussion refer

of the concepts

to the paper by Leggett

important

and Ter

Haarl’).

in the discussion

of the three-phonon

processes

we

ON THE

theory

described

ATTENUATION

by

may be generalized

Abrikosov,

OF SOUND

Gor’kov

to the phonon-phonon

IN LIQUID

1907

HELIUM

and Dzyaloshinskiizr)

interaction

problem,

*)

and we use

this method to calculate the position of poles in the single-particle Green function. We refer to AGD for the notation to be used and for a definition of the various Green functions. The single-particle Green function D(k, con) may be expanded in terms of its unperturbed

value &(k,

con) by using the relation

D-l(k, con) = Dol(k,

cum)- X(k, con);

(3-l)

(#I = 1/knT; Kn : Boltzmann’s constant ; T: absolute temperature). wn is an even multiple of ni/b, and Z(k, con) is the sum of all linked graphs which contain a single incoming and a single outgoing line of momentum k and frequency con, and which cannot be separated into two parts by cutting a single Do(k, con) line. The hydrodynamical approach is not expected to be valid for momenta of the same order of magnitude as, or greater than, the inverse inter-particle spacing in the liquid. However, one soon runs into difficulties in the low momentum region: low-order calculations of shifts of phonon frequencies due to the strictly linear unperturbed excitation lead to divergences, spectrum which gives rise to a large number of multi-phonon states almost degenerate with a given single-phonon state. To surmount these difficulties one should resum the perturbation expansion and attempt some other expansion procedure. The resummation gives results similar to those of Hugenholtz and Pinesss) for the imperfect Bose gas at zero temperature In the resummation not only D(k, con)but also the Green function B(k, con) corresponding to the annihilation or creation of two phonons will occur, as well as the irreducible self-energy parts corresponding to D and D. The function Z occurring in (3.1) can be expressed in terms of these irreducible self-energy parts and Da. As the self-energy parts are functions of D and a one might try an iterative procedure, using De as the zeroth-order propagator This is, however, impracticable as it leads to the divergences mentioned a moment ago. On the other hand, using a different zeroth-order propagator may lead to a convergent expansion. One possibility is to use as such a zeroth-order propagator [won- E(k)]-1 wh ere E(k) is some phenomenological energy spectrum, for instance, one obtained from neutron scattering experiments. Other iteration procedures, starting from Green functions having a cut along the real axis are also possible; for example, we can choose a Lorentzian spectral weight function, corresponding to an exponentially decaying excitation. These approaches will be illustrated by what follows.

*) We refer to this monograph as AGD.

1908

C. J. PETHICK

AND

D. TER HAAR

These resummation and iteration methods lead to results related to these of Landau and Khalatnikovss) who used a phenomenological excitation spectrum and an interaction given by the quantum to estimate excitation lifetimes.

hydrodynamical

model

4. The generalized unitarity condition for bosom. We wish to discuss the rules for evaluating discontinuities across cuts in contributions from thermodynamic perturbation theory graphs. These rules were derived by Lange r 24) for a special class of graphs in the theory of many-fermion systems and the results are easily extended to the many-boson problem. Consider a graph for some vertex function. Let the energies of external lines be expressed in terms of the variables vi. The vi are multiples of 2ni//3 in thermodynamic perturbation theory, but functions may be analytically continued in the various complex planes. Let us calculate the discontinuity across the cut given by the condition Im x{=, vi = 0. We first of all discuss which energy denominators will appear in the expression for the graph. Energy denominators are associated with intermediate states which have the property that, if the graph is divided by cutting all lines in this state, it will fall into just two pieces. Moreover, for the energy denominator to give a discontinuity when Im xi=, vi = 0, the total energy going between the two parts of the graph obtained by division as indicated above must be x<=i vi. Each energy denominator is associated with a reduced graph which is obtained by omitting all lines except those giving rise to the energy denominator under consideration, and fusing the remaining vertices. Thus the thermodynamic perturbation theory graph shown in figure 1 gives rise to the reduced graphs shown in figure 2 when we calculate the discontinuity

Fig.

1. A typical

vertex

diagram.

Fig. 2. Some reduced with the diagram

graphs associated shown in fig. 1.

ON THE ATTENUATION

OF SOUND

IN LIQUID

1909

HELIUM

across the cut given by the condition

Im(Yr + ~2) = 0. In calculating

the

contribution

1 we have not included

for

from the graph in figure

the external lines. The contribution tained by: (A) Multiplying

to the discontinuity the following

given by a reduced

factors

graph is ob-

factors :

i) A (pa, E,) for the cr-th line in the reduced graph, pa being its momentum and 5, its energy; A@, t,) is the spectral values of the energy and momentum.

function

for the appropriate

ii) Functions for the two vertices calculated using the usual rules of thermodynamic perturbation theory. These are to be evaluated with the severed internal lines having energies 5,. iii) 1 + %(Eoc)or PZ(~~)for the cc-the line, if it travels from left to right or from right to left respectively (n(5) = [eflc - 11-l). The energy x;ii=r vi is assumed to travel from left to right in the graph. iv) 2741 - e-0”) 6(,Z f ta - v) where v = Re &, vg. The plus and minus signs in the delta function are for lines travelling from left to right or from right to left respectively. (B) Integrating over the 5a and summing over internal momenta. Integrals along the real 5, axes must be interpreted as principal value integrals. We must be careful about overlapping cuts. Langer24)25) shows that these may be taken care of by evaluating the left-hand vertex function at C{=, vi = v + is and the right-hand vertex function at &, vi = v - is. Using this generalized unitarity condition enables us to calculate imaginary parts of contributions from graphs easily, without it being necessary to introduce thermal factors for many internal lines. If ordinary perturbation theory is used, it is often found that contributions cause thermal factors for internal lines to vanish.

from different

graphs

S. The attenuation of sound due to the three-phonon process. Figure 3 shows the self-energy graph whose imaginary part gives the rate of the threephonon process. We shall assume that the self-energy part corresponding

Fig. 3. The simplest

graph contributing

part gives the lifetime

to the phonon self-energy

of the phonon

part. Its imaginary

due to the three-phonon

process.

1910

C. J. PETHICK

AND

D. TER

HAAR

to B is small compared with D;’ (-k, -mn) so that .Z is simply equal to the self-energy part corresponding to D. The only reduced graph contributing to the imaginary part of 2 is the same as the graph itself. Using the form of the interaction Hamiltonian given in the appendix and the rules given in the previous section, we find Im Z(p, E) = j-c

(24 +

1)2

(1 -

e-P&)

(27&)3

2P +

p,

cP

s

52) *n(E1)[1

d3& dh d&z fi;A +

n(fs)l w2

-

(~1,61)A (pi t1 -

4.

+

(5.1)

The imaginary part of Z(p, E) is just half the discontinuity across the cut. The linewidth of long wavelength phonons is strongly dependent on the spectral weight function of thermal phonons, which will not be determined by the three-phonon process. Consequently it is incorrect to solve equation (5.1) and the corresponding equation for Re Z(p, E) in a self-consistent fashion. We therefore adopt a plausible form for the spectral weight function of thermal phonons and calculate the first-order contribution to Im X(p, E) for acoustic phonons. In our calculation we assume E(p, E) for thermal phonons to be independent of E; thus the spectral function is of the Lorentzian form :

r(P’)/n A(p’,

4

=

[m -

4p’)]2

+

F2(p’)

-

The energy of the phonon is E(#‘) and its lifetime AlaT( that the energy of a phonon may be approximated by &(P’) = cP’(l -

We assume

yp’s)

(5.3)

which is the form used by Landau and Khalatnikovss); c is the velocity of sound and y is a positive constant. Equation (5.3) should be a reasonable approximation at low temperatures where effects due to the roton part of the dispersion curve may be neglected. Performing the [I-, &- and angular integrations

we find

Im Z (p, E) = 2-c

(24+

1)2

1-

e-P&

@lP$Wl)[l

+ n (cPd1 * (5.4)

We have used the fact that p
ON THE ATTENUATION

OF SOUND

IN LIQUID

HELIUM

1911

lifetime T of a phonon is given by Thus from a =

1 7

= +

Im Z (p, c$)

(5.5)

(5.5) and (5.4) we find the attenuation 7d3

(@ ’

1)2

P

60

_!% mT4 ti3c3

arctan!%

_

coefficient arctan

_??!!%

2r

r

~(=I/~cT):

1

(5.6)

where r is r(SkBT/c) and o is the angular frequency of the sound wave. Due to the incorrect evaluation of an integral, previous perturbation theory calculationsl6) 17) failed to give the arctan(&o/r) term in (5.6). Equation (5.6) is similar to a result obtained by Kwok, Martin and Millerss). We postpone a discussion of (5.6) until the final section, where we compare theoretical predictions with experimental data. 6. A higher-order contribution to the attenuation. If r vanishes, the threephonon process will not occur. Alternatively one may argue that as r -+ 0, the Lorentzian approximation for the spectral function becomes inadequate. Khalatnikovls) maintains that the linewidth of an acoustic phonon is then determined by the self-energy graph shown in figure 4. This graph does not occur in the functional for the self-energy part in terms of renormalised Green functions, but will occur in the second iteration of this equation for 2: If we use zeroth-order propagators of the form [We - E($)]-1, the graph in figure 3 will not contribute to Im Z (6, cfi). Consequently the graph shown

Fig. 4

Fig. 5

Fig. 4. A more complicated imaginary

part

gives

diagram

the lifetime

contributing of the

process, which was calculated Fig. 5. A further

phonon

to the self-energy due to the

by Khalatnikov

contribution

to the self-energy

in figure 4 gives one of the lowest-order contributions. to CLfrom the graph in figure 4 is proportional to

of a phonon.

indirect

The

four-phonon

(Ref. 29). part.

The contribution

(6-l) A second graph giving

a contribution

of the same order in the coupling

1912

C. J. PETHICK

AND

D. TER

HAAR

Fig. 6

Fig. 7

Fig. 6. A reduced graph obtained from the diagram in fig. 5. The only contribution to the imaginary part of 2Y($, cp) calculated from the diagram in fig. 5 comes from this reduced graph. Fig. 7. A diagram giving the same contribution to 2 as that shown in fig. 5.

constant, u + 1, is shown in figure 5. The only to Im Z(P, cP) f rom this graphs is shown in tribution is given by the graph shown in figure our calculation. Application of the generalised that

In-lJJPJCP)=

(24,+ 1)4

-22

Tj-

(1 -

e-pep)

P/d%

(2nti)6

p2

reduced graph contributing figure 6. An identical con7, and this is included in unitarity condition shows

d3p’ P:P’” Ipi -

~‘1.

~fi(CPl)[l + 4cP’)l[l + fib IPl - P’IJI d[E(P’ + P) + 4Pl -P) - 4Pl) - CPI C4Pl+ P) - 4Pl) - cPl[dP + PI - 4P’) - CPI (6.2) We now investigate the frequency first order in P we find E(P1

+

p)

-

4Pl)

-

and temperature

cp

=

-cP(l

dependence

of (6.2). To

- p1 + 3YPf).

(6.3)

The polar angles (Or, $I), (13’,4’) and (0, @) are used to describe the directions of p1 and p’ relative to p, and pl relative to p’, and we write ,ur = cos 81. The argument of the Dirac delta function is given by

4P’ + PI -

4Pl -

-

P’) -

&(Pl) -

w

cp = c __ 2

PlP’ Pl

-P’

-

3YlPl - P’l PlP’

I

(6.4)

to first order in &Jq,8’2, 02, P and yPf. Substituting (6.4) and (6.3) into (6.2), we find

Im -%k cp) = -

(N+ II4 p2

GPl)[l

c2n;4 j-dh ~6

dP'

dm W

d@P;P'3(Pl

-

+ W’)l[1 + fl(c IPl - P’IH d(@ - @, ’ (1 - Pl + 3YPT) (1 - Lc’ + 3YP’2)

where 0: = 6y(Pr - P’)2. Since 81 and 0’ are small when the energy denominators

P')' (6.5)

in (6.5) are small,

ON THE

ATTENUATION

OF SOUND

IN LIQUID

1913

HELIUM

we must use the approximations 0s = e; + 8'2 - 2e1e' cog to be consistent,

Q,

p1

=

since we have already

i -

+e;,

PI =

i -

gel2

made similar approximations

(6.6)

in

evaluating the argument of the delta function. As a result of using the delta function in (6.5) in performing the @ integration, the angular integration in (6.5) is now e1 de1 8’ de’ s (:e:

+ 3rfi;)($e’s

+ 3yp’s)[{(el

+ et)2 -

egj{e; -

(ei -

e’)s)lh ’

(6*7)

where we have used (6.6). The domain of integration in (6.7) is governed by the conditions \er + e’l 2 8e 2 jer - 8’1, 8r > 0, 0’ > 0. Since ei = 6y(p1 - p’)2, (6.7) may b e expressed in terms of the dimensionless variables 8i/0a and W/80, and the resulting integral is a homogeneous function of $1 and 9 of degree zero. By using this fact and expressing the integrals in (6.5) as functions of the dimensionless variables @pi and /3c$‘, we find

Im z(p, cp) = -

(6.8)

where B is a positive numerical constant obtained space integrations. The contribution to the attenuation is therefore tc=

Im 2 (ib,4

=

--

tic

from the momentum

(6.9)

It may appear surprising that the contribution (6.9) to CLhas a negative sign. However, this is because we have calculated a term which cannot be written as the squared modulus of a matrix element, times a density of states. The sum of all contributions to the linewidth of an acoustic phonon must be positive, but we have evaluated an interference term which is not necessarily positive. We conclude that interference terms may reduce the attenuation appreciably, and that it may be misleading to neglect these contributions, as is usually done. The ratio of the expressions (6.9) and (6.1) is proportional to yc[knT/c13/fi~>

(6.10)

which gives an estimate of the deviation of the thermal phonon energy from the linear spectrum in terms of the acoustic phonon energy. These results will be discussed in the final section. 7. The velocity of sowzd. In this section we rederive the results of Andreev and Khalatnikovla), and shown how their calculations are modified if thermal phonons have finite lifetimes.

1914

C. 1. PETHICK AND D. TER HAAR

The results we require may be derived by using thermodynamic tion theory

to calculate

approximation,

the real part

perturba-

of the graph in figure 3. To a first

the shift in the energy of a sound wave is Z(p, cp). Applying

the usual rules to the graph in figure 3, we find

-b

Z_(P, 4=

+ 112 2Q

CP ___

(2di) 3 s

dPl dpl P;’ ;

C Wpl, Wn

in)

WI

+

mn + 8). We approximate for the moment D(pr, COG)by [olz the sum in (7.1) to a contour integral, we have

and see that the shift in the sound velocity, &

=

Q1)]-1.

P,

(7-f)

Converting

6c, is given by

CP)

Z(P,

P where we have evaluated the fir The expression (7.3) is identical Khalatnikovrs). We now perform a calculation The propagator is related to the

integral in the same way as in Section 5. with the one obtained by Andreev and using spectral functions of the form (5.2). spectral function by the equation

The summation in (7.1) is converted the sum of propagators becomes

to an integration

&!I dE2(52 (52-h-

5

fi(CPl)[l

+

+oh-

~@1)12+

where D = I

+

+ @2)1 &(Pl +P)la

r2(P1))@2-

(E2-

51-

Q){g

+

r2@1))@;

4cPdl P2&Q2 + ;r2($I) 2 + E-

+ r2(Pl)]

d51 d&z

4CPl)l -$i-ca -ce

= -GPd[l

El) fi(h)[l

in the usual way, and

&(Pr + p). In deriving efl(52-E1)-

1 = p(t2 -

+

~2(?w)

(7.4)

(7.4) we have put tl)

ON THE

since B(& neglected Putting

51) <

ATTENUATION

OF SOUND

1 when the rest of the integrand

a term not containing

1915

HELIUM

is large, and we have

the energy denominator

Es -

51 -

Q.

(7.4) into (7.1) gives

Re QJ CP)=

/w

(u + lJ2 8n2p

yj-

s

dP1dQ~:MU-l

The limits integration,

IN LIQUID

+

J-2 Ml)1

Q2 + 4r2(p1)

*

of the Q integration are 3@c# and 2c$. Performing and approximating the pi integration as we did earlier

(7.5) the 52 on, we

find

8. Discussion. From equation (5.6) we see clearly that the three-phonon process only leads to an oT4 dependence of the attenuation if &w > r > > 3yj%%iw. It is thus important to obtain an estimate of r for thermal phonons. There are three scattering processes which may contribute to r. Kramerss7) suggested that the short mean free paths observed in second sound experiments might be due to scattering by aHe impurities; so far no quantitative theoretical results have been obtained from a discussion of this process. The main scattering process is the phonon-phonon scattering, but Dransfeldss) has pointed out that even if that process is negligible, r would still be finite due to scattering at the walls of the container; this would occur at low temperatures, where we thus would expect r to be approximately constant. Later in this section we shall estimate the effect of scattering by the walls. As far as phonon-phonon scattering is concerned, Landau and Khalatnikovss) estimated the linewidth of thermal phonons due to the indirect four-phonon process, illustrated by fig. 4, and found 2r __ ti

= 2 x 10s T7 s-1.

(8.1)

Khalatnikovsa) has also estimated the mean free path for thermal conductivity, but his results do not agree even qualitatively with the measurements of W hitwor thae). Consequently it is difficult to know whether or not (8.1) gives an accurate estimate of r. The expression (8.1) was derived using the value 2.8 x 1037 c.g.s.u. for y. This was obtained by fitting the spectrum required to account for the specific heat data to a simple power law spectrum, and is in reasonable agreement with the value obtained from neutron scattering dataas). However, both these estimates depend on the shape of the spectrum near the maximum, whereas our results require the value of y near the origin; inspection of the dispersion

1916 curve suggests quite different.

C. J. PETHICK

that the effective In his calculation

AND

D. TER

HAAR

values of y in these two regions may be of CL,Khalatnikov15) uses a value for

y which differs by a factor 200 from the one used in the calculation of 6c by Andreev and Khalatnikovla). The authors do not comment on this fact, and we conclude that their results do not give a consistent account of the experimental data. From the above discussion we expect r to tend to a constant at low temperatures, and to increase rapidly with increasing temperature, provided we neglect scattering of phonons by impurities. This means, however, that CLis still a complicated function of o and T, even though we have made approximations when evaluating integrals, and no simple power law should be expected to represent the data over an appreciable temperature range. Let us now estimate when the mean free path due to phonon-phonon scattering is equal to the linear dimensions of the container. If a typical linear dimension of the container is 1 cm we see that this is equal to the mean free path obtained from (8.1) when T mO.3’K. Above 0.3”K the mean free path will be limited by phonon-phonon scattering. Two important parameters in (5.6) are tio/2r and 3yj%~/2r. These are equal to unity, at T = 0.6”K and 0.2”K, respectively, for a frequency of 1 MHz, and at T = 0.8”K and 0.4”K for a frequency of 12 MHz, where I’ has been assumed to have the form (8.1). At 1 MHz we see that 3y$%%0 is always less than r, since r is limited at 0.2”K by scattering from the container walls, not by phonon scattering as we assumed in deriving these estimates. At 12 MHz there is only a small region between about 0.3”K and 0.4”K where 3yjSiu, is comparable with r. At higher frequencies the region will be larger - at 100 MHz it probably extends up to 0.6”K. When 3y$2h cu 2 r, the Lorentzian form of the spectral functions will not be sufficiently accurate, since the rate of the three-phonon process will depend on the details of the functions. Khalatnikov’s workis) may be regarded as an attempt to improve the propagators - he calculates the three-phonon process graph with a self-energy insertion in one of the internal lines (figure 4). The graph we calculated in section 6 also gives a contribution of the same order in the coupling constant u + 1. As we noted in (6. IO) the ratio of these two contributions is proportional to yc(k~T/c)~/Sico, which has a value of 0.02 for cu = 109 s-l and T = 05°K. If kBT/c is replaced by 3kBT/c, the mean thermal phonon momentum, the ratio is increased by a factor 27. Thus at high frequencies the contribution we have calculated may be important. When fico/I’ is approximately equal to unity, cx will decrease due to the decrease in the first arctan function in (5.6). If the result is formally extrapolated into the region where &o/r < 1, CLwill show a maximum. This agrees qualitatively with the experimental results. This seems an improvement on earlier perturbation theory results which failed to show that CLshould fall below the wT~ law when r is comparable with ho.

ON THE

ATTENUATION

OF SOUND

Let us now turn to the experimental of Chase

and Herlins),

Dransfeld,

IN LIQUID

HELIUM

1917

results. We consider the measurements Newell

and Wilkss),

and Jeffers

and Whit n e ye). Considerable difficulties are encountered when measuring c( at low temperatures: firstly, CLis very small, and secondly, absolute measurements of CYare not possible. To evaluate cc one needs to know the signal amplitude in the absence of any attenuation. Chase and Herlina) and Dransfeld, Newell and Wilkss) do this by measuring the signal amplitude at some temperature where the absolute value of the attenuation is known. However, small errors in the signal amplitude in the absence of attenuation lead to large errors in c( at the lowest temperatures, and Jeff ers and Whitney6) adopt another method for finding the signal amplitude in the absence of attenuation. They assume that tcis proportional to some power of T and obtain the signal amplitude in the absence of attenuation by extrapolation of the finite temperature results. According to these authors the results of Chase and Herlins) may be fitted to a simple power law expression if the signal amplitude in the absence of attenuation is adjusted slightly. Jeffers and Whitneye) state that their results for frequencies of 1, 2, 4, 6, 10 and 12 MHz and for temperatures between 0.2”K and 0.6”K may be fitted by an dT3 dependence, but we are not convinced that their data really lead to this dependences; moreover, the warm-up times in their experiments were very short. Their results show that the w dependence is certainly not given by cc)Q; between 2 MHz and 4 MHz the exponent is about 1.25, while it increases to about 1.85 between 10 MHz and 12 MHz. Our result (5.6) predicts that the frequency dependence of CLshould decrease with increasing frequency; this disagreeswith the results of Jef f ers andWhit ney. However, our result shows that the frequency dependence should increase as the temperature rises, which agrees qualitatively with the measurements of Dransfeld, Newell and Wilkss) at frequencies of 6 MHz and 14.4 MHz. At 0.4”K the exponent derived from their measurements is about 1.3 and it increases to about 1.5 at 0.6”K. Recently we have received a report by J. Vignos, Y. Eckstein, J. B. Ketterson and B. M. Abraham of the Argonne National Laboratory on preliminary measurements of a at 30, 90 and 150 MHz at temperatures down to 0.125”K. They find a fairly broad temperature range below 0.5”K over which the attenuation varies as T4. We have also received a communication from M. A. Woolf of Bell Telephone Laboratories who, using stimulated Brillouin scattering, has measured the attenuation at frequencies of 535 and 710 MHz at temperatures of 1.2”K and above. At 1.25”K, where it seems that Ao/r > 1, the attenuation appears to scale as o. However, scattering of acoustic phonons by rotons may be important in this region. The measurements of Harding and Wilksal) on dilute solutions of aHe in 4He are qualitatively accounted for by our expression (5.6). For small

1918

C. J. PETHICK

AND

D. TER

HAAR

concentrations of sHe, the main effect is a reduction in the mean free path of thermal excitations; this will cause a decrease in the attenuation, as is observed

experimentally.

We conclude that our expression (5.6) is able to account for some but not all features of the experimental data. None of the higher-order processes can be compared with experiments since the linewidth of thermal phonons is too large; in the future, when detailed results of experiments at higher frequencies become available these processes may be important. We conclude by discussing the velocity of sound calculations. Earlier we noted that at 1 MHz 3y$Viu, is always less than r, and therefore Andreev and Khalatnikov’s resultra) is not valid. Our expression (7.6) may be approximated by the equation.

(8.2) This is always smaller than Andreev and Khalatnikov’s expression. According to (8.2), 6c increases with T at low temperatures and then falls to almost zero when tiu~/r = 1. Thus we might expect a maximum near 0.6”K, as is indeed found experimentally - the maximum actually occurs at 0.65”K. At higher temperatures the value of SC decreases and passes through zero at about 0.75”K. Other processes will be needed to explain the data in this region. We wish to express our gratitude to the Science Acknowledgements. Research Council for the award of a maintenance grant to one of us (C. J.P.), to Dr. W. E. Parry, Prof. R. E. Peierls and other members of the Physics Department at Oxford for useful discussions, and to Dr B. H. Abraham and Dr M. A. Woolf for communicating to us preliminary results of their measurements.

APPENDIX

The interaction Hamiltonian of liquid helium. According to the hydrodynamical model, the correlations in the liquid depend only on the local density. The Hamiltonian density H’ may be expressed in terms of the local density p’ and local velocity potential 4 as follows : H’ = &p’(V+)2 + p’J$ P

-

PO) dp’/p’2.

64.1)

P(p’) is the pressure in the liquid, and P(p) = PO, where p is the equilibrium density (see ref. 20, $ 17). The second term in (A. 1) may be expanded in terms of p’ - p, which is the conjugate variable to 4. The expression (A.l)

ON THE

ATTENUATION

OF SOUND

IN LIQUID

1919

HELIUM

is then expressed in terms of the spatial Fourier transforms of these variables : p’(r) -

_

P =

1

x pp p

dv

e-i(P’w

(A4 4(r) =

---&

$ c$~eicp*+)/*.

pp and h are conjugate variables, and to quantise the system we treat them as operators obeying the commutation relations [p,,

#Jp,l= -4% d,,,.

(A-3)

Finally pp and $p are expressed in terms of the boson creation and annihilation operators ui and up by the relations

zg I/-

2-

pp =

C

(u,’ +

iA&

$6p=

U-&J

(A.4 (up - a’,).

The Hamiltonian density (A.l) may be integrated to give the Hamiltonian of the system. Expressing this Hamiltonian in terms of the creation and annihilation operators defined by (A.4), we find the coefficient of LZ~+~LZ~U~, in the Hamiltonian as 1

@P’

cv*

IP +

BP

*

P’ I >

[cos (I&p’)

+ cos(p’,p

+ 2~ -

where u = (p/c)(i?c/ap), c the velocity system. When p < $1 and cos(p, p’) mated by 1

cv which is the expression

+ P’) + cos(p +

(

11,

(AS)

of sound and V the volume of the w 1, expression (AS) may be approxi-

c3PP’IP + P’I * (u + 1)) 2P >

we used in our calculations.

Received 29-3-66

REFERENCES 1) Pellam, J. R. andSquire, C. F., Phys. Rev. 72 (1947) 1245. 2) Chase, C. E., Proc. my. Sot. A220 (1953) 116. 3) 4) 5) 6)

P’, P) +

Chase, C. E. and Herlin, M. A., Phys. Rev. 97 (1955) 1447. Whitney, W. M., Phys. Rev. 105 (1957) 38. Dransfeld, K., Newell, J. A. and Wilks, J., Proc. my. Sm. A243 (1958) 500. Jeffers, W. A. and Whitney, W. M., Phys. Rev. 139 (1965) A1082.

1920 7) Pippard, 8) Landau, 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31)

ON THE ATTENUATION

OF SOUND IN LIQUID

HELIUM

A. B., Phil. Mag. 42 (1951) 1209. L. D. and Khalatnikov, I. M., Dokl. Akad. Nauk SSSR 96 (1954) 469. (Collected Papers of L. D. Landau, Pergamon, Oxford (1965), p. 626). Khalatnikov, I. M., JETP 20 (1950) 243. Kawasaki, K., Prog. theor. Phys. 26 (1961) 793. Kawasaki, K., Prog. theor. Phys. 26 (1961) 795. Dransfeld, K., Phys. Rev. 127 (1962) 17. Woodruff, T. O., Phys. Rev. 127 (1962) 682. Kawasaki, K. and Mori, H., Prog. theor. Phys. 29 (1962) 784. Khalatnikov, I. M., JETP 44 (1963) 769. (Soviet Physics JETP 17 (1963) 519). Simons, S., Proc. Phys. Sot. (London) 62 (1963) 401. Leggett, A. J. and Ter Haar, D., Phys. Rev. 139 (1965) A779. Whitney, W. M. and Chase, C. E., Phys. Rev. Letters 9 (1962) 243. Andreev, A. and Khalatnikov, I., JETP 44 (1963) 2058. (Soviet Physics JETP 17 (1963) 1384). London, I;., Superfluids Vol. II, Wiley, New York (1954). Abrikosov, A. A., Gor’kov, L. P. and Dzyaloshinskii, I. E., Quantum Field Theoretical Methods in Statistical Physics, 2nd Edition, Pergamon, Oxford (1965). Hugenholtz, N. M. and Pines, D., Phys. Rev. 116 (1959) 489. Landau, L. D. and Khalatnikov, I. M., JETP 19 (1949) 637 and 709. (Collected Papers of L. D. Landau, Pergamon, Oxford (1965), pp. 494 and 511). Langer, J. S., Phys. Rev. 127 (1962) 5. Langer, J. S., Phys. Rev. 124 (1961) 997. Kwok, P. C., Martin, P. C. and Miller, P. B., Solid State Communications 3 (1965) 181. Kramers, H. C., Thesis, Leiden University (1955). Proc. Kon. Ned. Akad. Wetensch. B59 (1956) 48. Dransfeld, K., 2. Phys. 179 (1964) 525. Khalatnikov, I. M., Usp. Fiz. Nauk 59 (1956) 673. (Fortschr. Physik 5 (1957) 211). Whitworth, R. W., Proc. roy. Sot. A246 (1958) 390. Harding, G. 0. and Wilks, J., Proc. VIIth Int. Conf. Low Temp. Phys. (1960), Univ. of Toronto Press (1961), p. 617.