ABSOPTION OF SOUND IN HELIUM II

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP78-04861A000100030009-1
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 35
• Document Creation Date: November 11, 2016
• Document Release Date: November 10, 1998
• Sequence Number: 9
• Publication Date: January 1, 1950
• Content Type: REPORT

"~'s`rciAA- 25X1A2g
Sanitized - Approved For Release: CTA 78-04861A0001000
Absorption of acund in helium II
I. M.._Khalatnikov
,
T 009t
7
CPYRGHT (J. Exp. Theor.. Phys. 9 USSR, 20, 2L.3, 1950).
Summary
The present paper, is an attempt to construct the thenry
of the absorption of first and second sound in helium II..
As slow processes leading to absorption of sound in helium
II fare co sidered processes of emission and absorption of
elementary excitations (phonons and rotons The temperature
dependences of the absorption coefficients of first and second
sound are determined. The question of the second viscosity
of helium II is discussed, and expressions are obtained for
the four coefficients of se:,ond viscosity of helium II.
1. Introduction.
The fact that helium II has a non-zero viscosity must
in itself lead to absorption of sound propagated in helium
II. However the anomalously low viscosity (*) has not per-
mitted the discovery of the effects of absorption in helium
II for sound waves of relatively low frequencies.
Only recently has it been possible to discover the
absorption of ordinary sound in helium II (1). The fre-
quency of the sound used for this purpose was about 108
cyc/sec, Since in helium II the specific heats C and ,
are practically equal, the absorption of sound connected
with conductivity does not occur. Thus all the absorption
observed in,these experiments must be due to viscosity..
:Howeve,"`the calculation of the coefficient of viscosity
from the absorption data gave values roughly an order.of
magnitude higher than the values from Andronikashvili's
-measurements (2). This discrepancy suggests that besides
ordinary viscosity, helium II also has the so-called second
viscosity. Moreover the processes of establishment of
equilibrium in helium II must occur relatively slowly.
Of the order of 105 poise from 1.5 - 2?K.
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 Sanitized - Approve
CPYRGHT
1
2.
As was showh previously (3), the establishment of energy
equilibrium in the phonon and -roto.n gases takes place faster
than the scattering processes determining the first viscosity.
As regards exchange of energy between the phonon and roton
gases',: this takes place mostly by scattering of phonons and
rotons. The calculation of the-relaxation time for processes
establishing temperature` equilibrium between, the phonon and
roton gases (in the presence of energy ? ,equilibrium in each
gas separately) showed that this process is also faster than
the processes of scattering of the elementary excitations
responsible 'for ordinary viscosity., Thus the processes
indicated for the establishment. of energy equilibrium can-
not be responsible for the anomalously.; large absorption of
first sound in helium II. T.eonly relatively slow processes
of helium II are.the proce?ssses of establishment of equilibrium
with respect'to numbers bf phonons and rotons (compare (3).
6) Thus we shall start out from tho assumption that it
is just. these processes'which determine the second viscosity
o,f,helium II.
The dispersion of '*sound in helium II, which is possible
i,n pri,nciple9 (i.e. dependence of velocity on frequency), has
a'lso..not been observed up to the..present time (4). It will
be shown below that the sound waves used were not of suf-
;ficiently high frequency and so the dispersion was not achieved.
2, Absorption and emission of -phonons and rotons.
For collisions of phonons and.rotons, processes are
F i
possible as a'"result of which the total number of rotons and
phonons can change. All such processes in helium II may be
divided into three types.- (a) processes in which phonons
are emitted or absorbed; (b) processes in which rotons are
emitted or absorbed; (c) processes in. which rotons are con-
vert'ed into phonons' or. vice versa.,
We shall now consider the' rmast pr.obabl,e..,processes
each type.. Only the fastest of''the indicated processes will
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Emission and absorption of phonons
3.
In the collision of two phonons, processes are possible
leading to a change in the total number of phonons. The'
simplest of these processes -- ? the.three-phonon process is
forbidden because of the impossibility of the simultaneous
fulfilment of
energy. Thus
absorption or
two phonons.
We start
liquid (5)
the two conservation laws of momentum and
we consider a 5-phonon process -consisting of
emission of a third phonon for collision of
with the Hamiltonian function of the quantum
where Y and are respectively the velocity and den-
sity of the liquid and is tine energy per unit mass of
the liquid.
The non-vanishing terms of the matrix elements for
transitions of three phonons into two are obtained from the
cubic terms in the energy for the third approxima-
tion of the perturbation theory, from the cubic and
fourth order terms in the second approximation and from the
fifth order terms in the first approximation (} is the
deviation of the. density from its value in a stationary
liquid). We express the hamiltonian (2.1) in the form of a
~-- - H 0 + V3 + v4 --
where H 0 is the density of sei nd ?_;-,nergy containing. terms
(2.2)
quadratic in r.. and V3, V and V5 containing respectively
the terms ' of ,third, fourth and fifty. order with respect top}
With the help of (2.2) in which the terms V39.V4 and
V5
are considered as. asmall perturbation, we
matrix element of the transiti.on-
(v3)4r t'3 6~3
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CPYRGHT.
When di.$persion,is neglected9 the occurrence of ,,a denominator
of,,.the resonance type for small angles between ,, the. moments
of, the colliding phonons leads to divergence of the first
rzns: . in (2.3) (3). Taking account of the dispersion. of the
te
,phonon.part of the energy spectrum i.e. of the fact that
the energy is not a strictlylinear function of the momentum
of,th.e phonon, eliminates the divergences of the terms
indicated. It is easy to see that only those transitions
will be significant' which are described by the first sum
some of the terms of ~,hiah contain in their
denominators products of two expr_ssions ter..r?.ng to zero
for neglect of the dispersion and ` ) . T es we
shall leave in the matrix clement only the' first stun deh_:'1
% ding on the terms V~ of the
.1
'V 0,
Suppose that. in the initial
phonons with
momenta
th._r(f order in
-x-
C( A`2
sta'C' (A) we have three
9 A
, and in the final
state (F) two ph_cnbn.s with momenta and Altogether
there are possible only s veral no of intermediate i-oces-
ses (I and II.) through which the transition from state A to
state F can take place. Howe for, es has a ready been poi ntc.i
out, the si ` ificant ones will be only those intermediate
states for which at small angles 'o e7 s?efl the r omenta o:
phonons we have siwjlt`._aaeously+
i V r r .r a a~ .. ~- J o
5,
The dispersion in question here must not be confused with
the dispersion of sound' i.e. w;_th the phenomenon of the
dependence of the velocity of sounel on frequency.
+ It turns out that there are altogether 27 special inter-
mediate states, but owing to tee conservation laws only 15
intermediate sta ses satisfy the condition (2.5)..awhich
particular ones depend on the relations between the magnittr(91
of the momenta. schematically the transitions through the 27
,indicated intermediate states can be represented in the fol-
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5.
From what has been -----Id it is absolutely clear that the
calculation of probabilities of the transition considered
Z7r
)J? (2.6)
is an extremely complicated operation. Such a.calculation
has hardly any sense at, the present time. The point is that
the basic parameters of the theory of the differential coef-
ficients of the first three orders of r, the velocity of
sound with respect to density and especially the dispersion
parameter are known only very roughly (3). Thus we
shall limit ourselves to establishing the relations between
the probability of transition and 'the energy of. collision of
the phonons, This will completely determine the temperature
variation of the quantities which interest us.
The non-vanishing matrix elements of the Fourier
components'%of the density have, according to (3), the form
' F
ilf+ c e /r-~ ~n,+ I e) ,(2.7)
6P,
To establish the temperature variation of any average
with respect to all phonons of the probability of the process
considered in (2.6), there is no need to identify the - )lli-
ding phonons. Thus we may omit the indices in the momenta of
the phonons. In this way we have from (2.4) and (2.7)
r 3/
EA- EI
For instance, the difference of energiesffor a concrete
example when we have in the intermediate state t two phonons
with momenta and 7 may be writtei in the form
(fly: 91
where f1aJnare the unit vectors. in the directions of the
corresponding momenta, The matrix element of the transition
will contain in its denominators the products of two expres-
sions of the type (2.8).
We recall that taking account of dis.~persion, the dependence
of the energy on the momentum of th.e phonorhas the form
(see (3))
Sanitized - Approved Emf F Olea ef; IA-RDP78-04861A000100030009-1
 CPYR~MTitized - Approved For` eleas
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In the present context we are naturally not interested
in the angular distribution of phonons. Thus we average
(2.6) over the angles made by the momenta of the colliding
phanons and integrate over the phase volume of the momenta
of'the"scatt?ered. phonons. The necessary integration over
angles* proves, to b. quite simple owing to the smallness of
the terms in the denominator Hof the expression for.
wici to ccount of di.ave (Compare (3) Para.
Without:dwelling on the fairly simple'calcu.lations,
3
give the final result obtained for the probabilities averaged
over angles
2).
(2.9)
L t'..the total number of phonons per unit volume (not in
The rate. of., ,change of
general the equilibrium valu_1e) )e
Pl~
the number.'.of; phonons due to the "7-phonon process,..may be
vrr++,e-n' i.n +he form
J ~n. njn3(h4-i ~Ypi ?h~ - (n~-t + t fl{r~s+r1 < n ,y w v ~ _ :..._ (2.10 )
..If _th.etotal number of phonons is not equal to the equilibrium
value this means that the distri~;ution function y.'contains
a non-zero chemical potential ,N
For small deviations from e~iuilibrium, the, function (
can be expanded in powers of',1,$ , and-considering only the
linear terms in iJ(P we have
(2.11)
The function n with suffix 0 corresponds to the equi.li-
brium distribution for phonons (A .o). After some straight-
forward transformations with the help of (2.11), (2.10) takes
the form
41x
Y1 F Pl
~try
~
n
ry
53
l
20
iU
4D
30
t 7t q .
Denoting by r~ the coefficient in the equation connectir-,
the rate of change of and the quantity )A+y , we have
according to (2.10')
I ((((
~- kT
tondo n30 619o4 f n~~
1) W ~1 e., e P3
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4861 A000100030009-1
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7,
Without great loss of precision we may neglect the dis-
tribution functions ~(1*4,' and 'So compared with unity in the
integrands in (2.12). The integration over the phase volumes
of the momenta of the three colliding phonons ( el 9
can then be. carried out, whicta allows J 3 to be
replaced by W reducing the number of integrations by one.
We obtain
in this way
T 1H0n30 CJ (2.l, ' )
The integrand in (2.12') is proportional toAfter
integrating (2.12')' over the momenta of the,: colld'i'rig 'phonons
we obtain for the temperature law for
'
(2.13)
.T
i~q~e r
where 0., i.s a temperature4 coefficient'.,.
The expression (2.13) determines the temperature varia-
tion of just the quantity which enters significantly into
our further calculations.
r`-ion and, absorption of roto3is.
A collision between two rotons may be
associated with
emission or absorption of a third roton. Accurate expres-
sions for the probability of Y:rocesse,s of this kind cannot
be obtained since the interaction function betweeh'two rotons
is not known.,
The energy g of a roton of momentum p is known (5
to be equal to
= A+ (p-po) 2/2?
where ,po and p. are parameters.
If two rotons with momenta pl a"d. p2 coll..de a third
roton can be emitted only if the condition
(1/2k) PP1-po2 + (p2'po)2 j .~ (2.14)
is complied with; this follows from the conservation law for
the energy.
It follows from the form of the energy spectrum that
most of the rotons have momenta of the magnitude po. '
Therefore condition (2.14) is very ;ringent. 'It may be
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 CPYRG itized - Approved Fob ; t A-R13 78-04861A000100030009-1
Let P be the number of rotons per unit volume; if it
is not equal to the equilibrium density then the rate of
approach to equilibrium, Np, may in a similar way to that used
previously be written as
2915)
W ;ere ? is the chemical potential of the roton gas.
If as asserted, the probability for a five-roton-
._..
process is very small, then the quantity must also be
small. It is accordingly assumed that
2.16)
at .all temperatures below the X-point.
Transformation of phonons into rotons.
Transformations of rotons' into phonons are forbidden on
account of the.conservationof'momentum, in so far as the
momentum of Phonons (2) satisfy the condition
and simultaneously the momentum of the rotons is almost
equal to po. This selection rule applies to the collision
between two rotons if the angles between the momenta of the
coliiaing rotons are small and hence their transformation
into a scattered roton and phonon pis impossible. If,%however,
the above angle is not too- small the selection rule does not
apply;, collisions between two rotons are permitted in which
they are transformed into a'phonon and a roton and even into
two phonons (if the angles between the momenta of the col-
liding rotons are near to 7c). For the reasons given a coupl
of paragraphs back, it is impossible to calculate the pro-
babilities of these processes, It is possible to carry out
estimates, just as before; the essential points of them are
presented in the following.
Consider, for the sake of expediency, :a process which is
the inverse to the above cod lision between two rotons; let a
phonon of momentum 21 and alroton of momentum R2 collide, and
let this process result in two rotons of momenta ]-and p
respectively. The energy condition demands that the energy
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of the phonon is not smaller titan . The phonon must
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accordingly have high energy. It is to be expeeted.that,.
phonons of energies as high as those considered differ from
rotons.,only by the way in which the energy depends upon the
momentum. Their interaction with rotons; is, in particular,
of the same kind as the interaction between rotons. Hence
calculations can be carried out in full analogy to those
appertaining to the scattering of.a roton by a roton (3).
Similarly as in this case the phonon-rot-on interaction
energy v(the phonon having large energy, in any case greater
than 4 ) can be represented es being proportional to a delta-
,function of the distance between. pho. ,oar and. roton
V = Vo (rl - r2) (2,17)
(hers rl,r2 are the osition vectors of the phonon and..roton
respectively). It was stated preciously (3) that an expres-
6-ion of this kind for the eiergg e f interaction is chosen
because it makes it possible to determine the temperature
effect on the probability of a. corresronding collision pro-
cess . Just as before the rate of aprroach from non- equilibriiiie.
to equilibrium in a phonon-roton gas car. be written In the form
- ;/U(2,18)
provided that the approach to equllibriu.m depends on process
of transformation of rotons into phonon and vice-Versa.
The coefficient 1 is dooorei_n d by means of a. complete
calculation of the collision intc,=alp Omitting inter-
mediate steps in the calculation, we'L7ive the final result
as obtained with the interaction fur,;tion (2,17)
I V
(2,19)
The i,iagnltude of the amplitude Vo i s tot known for the
IL~Poces6 under discussion, For the purpose of making an
estimate one can use the value for tiro derived for the roton-
roton interaction from the viscosity of, he1i urr~ 1
(3) :. it may be noted. that this est r.+-c provides an upper
limit for ~a > for it is impli.r,d? that the. transformation
of a phonon into a roton occurs in every process in which the
phonon has sufficient energy and the angle between the moments
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pj!~fi
1050 e-2.4/T . (2
20)
3
9
.
t:
phonon process is obviously :less probable than
process. For the latter we made previously
of sufficient a(.cu-.. ac-T, thereby providin,-,
an ,upper limit for r . Con : ri sc o" these upper lip J. t.s
( aV r( ) shows that they are of c.imilar magnitude for
wide 'range of temperature. ker.co the two coefficient:
themselves are possibly of magnitudes similar to
each other.
The numerical coefficient entering into.'.the expression
for !r which is for the time considered as unknown,
is denoted by by so that (2918) rends
p/
b 72 (2 2I)
The quantity b and the numeri,,i1 coefficient a of equation
(2,13) are subsequently obtained fro: empirical data, the
absorption coefficient ~f-~--or first so:,nd. in helium II. It
will then be seen that l~~ and have in fact the ame
order of magnitude.`)
3. Equations governing the ro agatizTn of sound in hliu.m
clearly this is a necessary but not a sufficient condition,
fob t'allows'also simple scattooring of the phonon by the
rot on.
Inserting into the right hand side of (2, 19) the values
of ail parameters
erg/cm3) We find
A five --
e four-phonon
(3) estimates
set of hydrodynamical equations of. notion (5)
The propagation of sound in hr~lium II.-is determined by
equations which may be written in the form of a linearized
+) In so far as it is due to pho-eons, viscosity is, as shown
in (3), for temperatures between ;-) and 0.7 deg.K dercndent
on the fundamental process of sc Ltter-ng of phonons by rotons.
The, probability of this process is known., Calculation of, the
collision integral for that process and the corresponding
coefficient r and comparison with the upper limits of
shows that r In this way it is justified to neglert th"
phonon-roton xansformation in calculating the viscosity
according to (3).
Sonitized. -Approved For
and the assumed value for V0 (= 5 . 10-
i:"CfA-RDP78-04861 A000100030009-1
 CPYRSaHT.. ._ f- i~ ' =11..
1-'AbP78-04861 A000100030009-1
-~~v
C)
Here density of helium,' ,or) and density and
vn and vs = velocity of the normal and superfluid
fraction respectively, S'= entropy per unit volume,
thermodynamical potential per unit mass,
~,~!r,~VS = mass flow density.
In equilibrium, all thermodynamic functions appearing in
11.
equations depend on two variables only, which
may be chosen to be r and S. If a sound wave of low fre-
quency is propagated through helium II, equilibrium is re-
established at such a high rate(as shown by a calculation)
that the rate of approach to equilibrium follows instan-
taneously after the change of state due to wave propagation.
Hence the state. is at every instant fully determined by the
thermodynamical functions appropriate to equilibrium and
dependent on and S.
If, however, the frequency of the sound waves is so
high that the approach to equilibrium cannot follow the
change of state, then processes arise which have the effect
of an approach to equilibrium and result in abs.oVption of
sound. In this range of fr,.oquencies', the number of phonons
and rotons at any instant differs from-the equilibrium value
and the thermodynamic functions must rtaw be considered as
depending.. not only on and... S but also' upon the number of
phonons and rotons present (or rather upon the corresponding
chemiea.l:..~potentials). The set.,,6f equat.ons (3,l)-(3,4) is
accordingly to be considered .s incomplete and should be
supplemented by equations expressing the rate of change of
the numbers of phonons and rotons.
Let Np and N be the number of phonons and rotons
respectively per unit voli.me of helium II,,,~ and /kA their
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p o In equilibrium t - ~i - 0, and the
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number.of-phonons and rotons denoted by N and N are
functions.., of and only.
If equilibrium is perturbed, Np and N, change in time
approaching their equilibrium values NPo an.d N YO . Equations
determining the rate of approach ,to `equilibrium can be obtained
by expressing.-the time derivatives NP and:N as functions of
the chemical potentials.Expanding this function in powers
jiA.
and concluding with the. term linear in:?,,.and
N +'div Nvn pP?P Y
p p P
+ div In = Y4P ?p Y J' (3,6)
where y pp, O . , YPp, Y are the
kinetic coefficients +)
which are symmetric with respect to
In these
equations the terms o,f,the
the fact that
of the normal
the suffixes, and p.
form div Nv__. account for
phonons and roton.s participate in the movement
fraction !of ht'!l.ium II with the. velocity -vn.
?Neglecting,second order effects, we substitute in the
corresponding terms the equilibrium values of the N, so that
(3,5) and (3,6) become
Np + Npo div -vn
`div v
+ N
T
-n
The set of equations (3,l)-(3,4), (3,7),(3,8) determines the
propagation of sound in helium II taking due account of the
change in time of the numbers of rotons and phonons.
Elimination of two variable's,
C
YPP~t P .+ Ypo 40
YOP aP W?,
--
vn and vs gives
+) Of. L. Landau and E. Lifshitz,-StatisticaltPhysics, ? 41-42.
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In equations (3,.9)-(3,12) all.thermodynamic;functions`can be
expressed as depending upon S, ?p, u,A as independent
variables. Entropy and density can be represented as sums of
the equilibrium expression and a small increment due to the
'sound waves; the increments are denoted by 10 A solu-
tion of this set of simultaneous equations is required which
ha..s the form of a plane wave ..such, that S' up and ?
contain a factor exp i41(t - x/u) , where u is. the velocity
of sound. The compatibility conditions for this set of
equations assumes the form of a determinant of 4th order being
equal to 0; solution of the latter equation provides 4 values
for the velocity of sound.'_)
In order to avoid a determinant of the 4th order we
proceed in a slightly different way, by expressing all
functions in (3,9) and (3,10) in terms of S and 14 as
,independent variables. Hereby the dependance of the function,
upon the variables u,p and ? has to be taken into account at
all differentiations. The partial derivative of some thermo-
dynamic function f with respect to is written
r )~
Similarly
3,13)
+ .(3,14)
The derivatives at constant
chemical potentials (which
are equal to 0) are seen--to be}equal to the equilibrium value
of the derivative under consideration. 'In,(3,13) and (3,14)
+)Attention is drawn to the fact that equations (3,9)-(3,12)
lead to a determinant of the. 4th order, but that neverthelesp
the equation determining the velocity u is a quadratic aqua-
,tion for u2. Equations (3,11) and (3,12), which express the
variations of the number of quantized excitations . do . not
contain any differentiations of the variables with respect
to the coordinates. If a solution is'sought which is propor--
.:tional. to exp iW(t - x/u) ,,, then the. velocity u does not
enter into theseequations.The square of the. velocity enters
only the first two lines of the determinant; ;it follows that
the characteristic equation is quadratic in u2.
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BE * Ri
 the derivatives of the chemical potentials u and;?
w,ith respoct to P and S are used. They are obtained. by.
meansi of equat..ions (3,11) and (3,12) which establish rela=
tions between ? ? and ? cn the one hand and the ,small.varia-
tions of e and.S'in the plane wave on the other hand.
.:V`e,.express the small variations of and.S. as before
including a factor exp iW(t - x/u) The compatibility
condition for (3,9) and (3,10) is now obtained in the form
of a determinant of second order being.,equal to 0.
ua
(3915)
The quadratic equation which is obtained. from (3,15) contair.-~
terms of different magnitudes. All teDms?arising from the
tixrn out t o b e small,.. .
differential coefficient (s)
This comes from.the.,,;?fact that the specific heats cp and cw of
helium II are almost equal Go each other at all. temperatures
and that .the derivative 0S), ( is, agoord.ing to a familiar
thermodynamic theorem, proportional to (cp - cv)2. On account
of this, the roots of (3,15) can be written in a remarkably
concise form
The chemical potentials ?, and t can be expressed in
terms o`f the small increments of entropy and density (S' and
by means of equations (3,11) and (3,12), from which`tho
following two equations are derived
(L
Ire) A r
~Pr
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 (`~1 4W>-~
L_` Y
(3,19)
r)(W)4 ~OT))
cPYRGHTSanitized - Approve
It is readily found that
4Af re
;D 0
For further calculation the thermodynamic identity
T dS ~- d. - Np d i - N~ d L = J g (3,20)
is required.
The flunction/\pia- s a semi ar pars a Ut1G Gi1G, y
and, if the chemical potentials are constant, b,-oomes -equal to
the energy per unit of volume of the liquid. A connection
between the pressure p and the function is derived by
remembering that the derivative of the total energy with
respect to the volume at constant total entropy ane mass_is
'equal to minus the pressure.
It then follows that p - --~-- S 1
(3,21)
Here follow a number of thex?modyanical relations which
follow from (3,20), (3,21) and are required for calculating
As the mass and the total entropy Tr are constant:
r 3_
t10"OI RDP78-04861A0001000300091
(3,23)
As a matter of print p - ;! Y
diFc Rel ` T: CtA OIF 7 4S6? 0 '10603 09 1(?
'
1 1
('11
1 r- 13
 CPYRGitized - Approved For
The derivatives of the
ands; on which the
according to (3913)9
(3918)9 (3,19):
- t i8-04861 A000100030008-1
i _ '-~ -;,
chemical potentials with respect to
Fop rA
veloc. t~,ies of sound u and u2!
'14) 9 are obtai (-..d byrnea1ie off.,
Expressions (3,22)-(3
thermodynamical quantities
the velocities of sound ul
I
4
r I . Q -I- ~ 1
Sanitized - Approved F qge
25) for the derivatives of the
are r,,r-,di U,.y used for obtain -ag
jr) It
Js S/1?
P78-04861 A000100.030009-1
 5anitizea - Hpprov
CPYRGHT
If the chemical potentials are set equal to 0, the above
expressions for the velocity of lst and 2nd sound in helium
II apply.to thermal equilibrium; this corresponds to extr.mely
low frequencies. These velocities a're denoted by a suffix 0;.
their values follow from (3926), (3,27):
U 0
OPP
(3y 9) are equal to
Here = S/ Expressions (3,2), 2
expressions as obtained previously (5). According to (3,26),
(3, 27) the velocities ul and u2 are complex. The wave vecto~?s
as defined.according to k = 1/u are also complex. The
existence of an imaginary part cf 'the `wave vectors is known to
be an expression of the fact that there is absorption of sound,
The real part of the wave vector determines the variation of
phase with the distance, whereas the imaginary part determines
the absorption coefficient.
If the freque_ncies.ar.c`.Extremelyhigh, the expressions
(3,26), (3,27) approach the limits.:
These velocities of sound apply to such high frequencies thai
there is not time enough for processes in which the numbers of
phonons and rotons change; these numbers are then constant.
The kinetic coefficients y which appear in the formulae
of this section are readily expressed in terms of those
quantities which are obtained in the preceding section, and
tiAnITI7Pflt`xnnrnyP -
- r L ,M / l i dJ4,p /
lee- 1_ 1) 21) '
 CPYRG nitized - Approved1Pbr'el~a:a J. kDP78=04861A000104930009-1
I~,~ ~
By comparison of equations'(3,5) and (3,6).with equations,
15) and (2,18), and considering the meaning of/,,, we,find
{
In the subsequent. calculations it is convenient to employ
.instead of the coefficients y and lthe relaxation times
de'fined according to
By means of the definitions (3,3.3), the expressions on the
right-hand side of (3,26) and (3,27) are converted to
`y
+ ffi'~'
Sanitized - Approved r 8e :RI Pj DP78-04861A000100030009-1
 Sanitized - ADDroveU'"C~
li64-RDP78-04861 A000100030009-1
CPY,GHT
24
By these equations the velocities of 1st and 2nd sound
are determined as a function of the frequericy. In this way
the coefficient of absorpton..of 1st and 2nd sound in helium
II can .be. cal~cu3a,ted.
It has, however, to be pointed out that equations
(3,34) and (3,35) ar¬ sufficient for. determining the
dispersion of sound in helium II. Th~;p,roblem of dispersion
is complicated by the.. rap.;d increase of the mean free path
of the quantized excitations at decreasing temperature.
Comparing the time constant by which the first
viscosity is specified (3)., with the time constants Qo
and )it is found that these time constants are at all
temperatures of equal order of magnitude. The range of ids-
persion for sound coincides with'the range of dispersion of
the first viscosity. If this first fact is taken in account,
it is necessary to apply some speci,alized_arguments which we
are,., going to postpone ` for.: a f.ture paper. Equations (3,34)
ahd (3,35) are accordingly applied to the calculation of the,
absorption coefficient only; they are considered: to be valid
.for 1st and 2nd sound at low frequencies.+)
The range of frequencies where the. formulae for u
100
and u2oo are valid is presumably out of reach. Even at fre-
quencies lower than these any propagation of sound waves is
proolula.d.by the phenomenon of the mean free path of phonons
and rotons.approaching the same magnitude as the.wave length
of the sound wave. This is another point to be studied more
closely in some other paper.
It may beTrioted that in first approximation with respect.
,,to & there'is no dispersion,
Sanitized - Approved Fo E i-678 t04861AO001 00030009-1
 Sanitized - Approved For re
CPYRGHT
Absorpti,on of first sound i, helium II.
Consider the range of frequencies complying with the
conditions
It will:be shown subsequently that the derivatives
and are of similar magnitude.
What has-been said in section 2 with respect to the relative
magnitudes of the coefficient accordingly remains
valid for the reciprocal. time constant which is determined
according to (3,33). For this reason we are entitled to
neglect in (3,34) and (3,35) the terms: that are due to the
five-roto?n process. Expression'(3,35) then assumes at low
frequencies the simplified form
U Oft
I I: ) q(tw,
~_, , (~, M-S-4 ~Pv
The imaginary part of the wave-cector can be obtained
by means of the last equationn'the coefficient of absorption
for first sound) denoted by q,-, is given by
rV
,,,/ I
Tft) + - t *)jr-,S
In. order to compare (4,3) with the experimental data
given in (1), it is necessary to evaluate the differential
coefficients entering into that expression. It is convenient
to'introduce and T as independent variables. From (3,22)
we derive
+) The symbol u10 is in the following replaced by the
conventional C,
\ 1,1 1 ? I . ) 1 ~w (41,, )
R 78-04861 A000100030009-1
Sanitized - Approved FCIP78-04861 A000100030009-1
 Sanitized - Approved For Release : CIA-RDP78-04861A000100030009-1
CRYRGHT 21,
Calculations are fairly simple (cf. appendix).
Here are the results for +,p( and
It should be noted that the number of rotons between
1.5 and 2 deg is markedly higher than the number of phonons
but that the derivative
smaller than jU ; :this is due to the factor
in the denominator of (97).
VVe require expressions for the entropy S and for the
number of phonons and rotons per volume unit in equilibrium
IN
(41
. Omitting intermediate,steps in the calculation, we present
the results:
-- - T
LE'
!,- - j -:~ -4 4 i-.
, -4, -43 )L ?4) 3 .1 L-11f
- Approv rtRs-he I P78k0 1?A000100030009-1
 Sanitized - Approved F' r lea je~ :,Q4A : P78-04861A000100030009-1
..~ 1 rtJ i-.. 22.
CPYRGHT
The data of Peshkov & Ziii.o'vievh (") concerning the
velocity of second. sound under pressure can be used for
:.calculating those derivatives of parameters which determine
the value of The calculations have been given in a
previous paper (3). Assuming that isvirtually independent
of'10 , the result is:
a
On account oFthis and of equation (4,9), it is found that
The differential coefficient of the velocity of first
sound c with respect to the density can be derived from
Keesom's density - pressure curve for helium II, which yields
~
?Jc)(cJ4)
V
rNHS
and consequently
Using these values we obtain finally:
~,, _ N e~~~~3a was
It is necessary to remember
A
05)
that equation (4,3) provides
only that contribution to the absorption
due to relaxation in connection with the
coefficient which is
creation of phonons
and rotons. ' The complete ab sorption.coefficient consists of
two terms, of which one is given by (4,3).and the other is due
to the effect of first visc~osity.'.and is equal to = 2t~/3~ c~
F~;ellam & Squire (1) made measurements of the absorption
coefficient of first sound at a frE~quency of 1.5. 107 sec
and at temperatures between 1.57 and 2.0 deg.K. Their data
are, in spite of their small number, sufficient for calculating
the coefficients a and b with sufficient accuracy. Introducing
Sanitized - Approved. For
q P78-04861 A000100030009-1
'L U.
 CPYRG itized - Approved F ij RDP78-04861A000100030009-1
: j 23.
these parameters into the formulae (2,13), M19
,are -obtained.
expression for F~~ is of course smaller than
the upper
limit given by (2,20). By means of (4918) and. the derivatives
of the chemical.potontials (496)5 (497) the. relaxation times
can be derived as function of the temperature.
In figure ,1 the reciprocal
relaxation times and
are plotted against the
temperature. This graph
shows that in the experiments
of.Pellam and Squire (1) con-
dition (491) was complied with
at all temperatures.
frequency applied by
authors the.greatest
in the neighbourhood
At the
these
absorption
a, ~? a J ! 4. 1 C 1' s .1? o
should have been observed
of 1?K. It is now easy to understand
why in the experiments of Peshkov (4)9 which were made at
frequencies between 10 and 104 sec-19 no dispersion of
second sound. could be observed. In those experiments the
temperature was appreciably higher (1.63 deg) so that the
relaxation times and are as low. as 10-10'sec..-`Hence
and &7~ were so small that there was not only` no di s-
persion9 but hardly any absorption of sound in.helium.;II.
We add a table of the difference ulM u10 as bcalcu:.lated
T (deg K)
l Co
Sanitized - Approved F r
u -10)
0.6 0.7 0.8 0.9 1.0 1.1 1.2
0.175. 1.45 5.80 13.5 23.0 37.5 51.0
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
" u1 73.5 90.5 ?115 140 165 195 240 275
The difference u1 - u10 is seen. to be small) .
00
It vanishes when absolute zero is approached.
jj 'Ael RDP78-04861A000100030009-1
 Sanitized - ADDroved ;f or rRelease CIA..RDP78-04861 A00010003000.9-1
CPYRGHT
L i 2~4.
Absorption of second sound in helium II.
Using the results obtained in section 4, predictions
concerning the effect of temperature on the absorption coef-
ficiont of second sound in helium II can be made. If the
.frequencies are small, equation (3,35) may be written, on
account of condition (4,i)
Hence the absorption coefficient for second sound is readily
derived as
OR
According to (3,22) a 9d (3,23):
lr.
simple calculations:
Evaluation of these expressions gives by means of relatively
and tll ,
T 72. T
4- rm
which are required for evaluating
(5,2), are according to (5)
The right hand side
L M + al, 2
(5,2) expresses the rather
involved relation between the temperature and the absorp-
tion coefficient. It must be noted that the absorption
Satriliwmc; A 'oYodFor 1ea!etaaCb4 RDPi8EO48S1JA?00tfl003OOO941
7,
 CPYRGI-ffanitized - Approved kI eiiisii~ : DP78-04861A0001000300.1
to which it is due, to the processes of creation of phonons
and rotors. In the following section it will be shown that
absorption of second sound owing to ordinary viscosity is
similar magnitude as that calculated in this section.
We add a table giving the velocity u2 a
t various
temperatures
T (deg.)
0.6 0.7 0.8 0.9 1.0 1.1 1
.2
u20 (m/sec)
101 59.5 31.8 24.9 20.6 19.1 1
8.7
u2co (m/sec)
103 65.0 37.8 29.6 25.5 23.7 2
2.7
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
.0
U
0 19.5 20.1 20'.4 20.5 20..3 19.6
19-
18.3
20
.
u
23.8 20.8
9 23.1 23.9 24.1 24.5 24.3
22
2oo
,
.
6. Second viscosity
A classical liquid has two coefficients of viscosity
such that the dissipation function appropriate to viscous'
flow has the form
i ~ ( I!') K + ~V C -bV
k
where dV is a volume clement in the liquid. 0winq to the
effect of dissipation there arc 2 supplementary terms in
the equations of ihotion, which have to be written:
~Y -t uv)y +v . +V4wvv
Usually the first coefficient of viscosity and the
second coefficient are of equal magnitude.
Matters are different with respect to helium II. Itm
hydrodynamics has to consider two velocities9 that appro-
priate to normal movement vn and the velocity of superfluid
movement v This opens an unusual., perspective. Whereas
the first viscosity affects only the normal movement this is
+) In the neighbourhood of absolute zero the velo.ci v u c
assumes the same value as u 9 which is equal to c) . This
is not a coincidence. At t i~se temperatures the n{ ,-ber of
rotons is negligible as,copared with the number of phonons-
the value of all thermodynamic quantities is determined by
phonons only. Absorption of n sound is ccording to (592)
determined. by the difference This, however,
vents es. In fa according o (5,5
8 6 A000100030009-1
Saniti Kp~ro O Rel a "CI KD#
 CPYRG?I itized - Approved Fo
RL, D78-04861 A000100030009-1
of -
25.
not at all obvious with regard to the second coefficient.
In distinction from classical hydrodynamics, the hydrodynamics
of helium II involves two equations of motion, one referring
to, the ?ei tare liquid
one for the superfluid fraction
(where is the thermodynamic potential for the unit of mass
of the. liquid).
Consider now-the equations of motion if second viscosity
is explicitly accounted for. In introducing the terms
depending on the second viscosity into (6,3), (6,4) we shall
for sake of simplicity omit'at first the terms depending on
the first viscosity.
It. is very obvicus that the terms `-div V. require the
introduction of four coefficients of second viscosity.
Following the classical example (6,2), the right hand side
of (6,3) and (694)
div vs . Hence
The unusual features in the hydrodynamics of helium
require accordingly the introduction of 4 "second" coeffic'ents
of visc.osity.. These are, however, not independent. TherQ-
is a relation between them due to the fact that the part of
the dissipation function depending u~?on the second viscosity
should be quadratic function of the two variables div vn and
div vs. It is therefore to be written in the form
rg :2 S I vV)
I'V V" +A
Sanitized Approved 'Fc r Re 6 kLgl 4Li bP78-04861A000100030009-1
 CPYR 1ftized - Approved Ford a ease : CFA-F DP78-04861A000100030W,9;1
where An, Ans, A. are three coefficients which can be expros--
sed in terns of the four viscosity coefficients,
The density of the negative derivative of f with respect
should give
and v
to v
s
n
of motion for the
tivoly.
(6.6) is the
tho.right hand side of the equations
and the suporfluid fraction respcc-
equation of motion for the
-s) V V .n +
fraction. The equation of motion for the normal fraction is
obtained by subtracting (6,6)-from (6,5).
The derivatives
suporfluid
f>9 -F
V, ( )
and ~) Vs are readily cal-
culated. For this purpose the variation of f in terms of
variations of
V (~1Jn ~ i Y `Jr 1 Y 1VS +~ V VS 41 V !1 i ~
vn and vs is. derived:.
+ Y ~+~ bGt d c Vn
This expression is re-arranged:
-1 , A !/1 . IC, L1 A- %I
4
14
Ev?vtYy~
1VV ~.VnV VS) VS V
-~-- IVY ~lnV VS) g -7 VS VS t d r
All integrals which, according to the theorems of Gauss and
Ostrogradsky, can be transformed into surface integrals, arc;
considered to vanish. In this way we find
+ Terms of the form vrl~~
in deriving (6,8).
sVvVS 4V
and vs are neglected
Sanitized - Approved c r I see, hA-RDP78-04861A000100030009-1
"i
 CPYRWi.tized Approv
normal
rospoctivoly
,ficients
The first of tht three enu no fields recurire- rely.
tion between the co6ffici-_,nts r )f s::cor_c=t viscosit~T.
We a- r(3 going to ex-la,'.}i relations betwecii the
thermody_-_,. ' c du-int Lice aid the second viscosities. S _-^c
the second v: scosity of he- LUrn_ _T in c-ao to processes
involving changes in the total am or of phonons and rotcns4
d that thy:. tc :r-s containing div vn and
it can be concl ,-a
div vs in (6,5) ..d (6,6) ar d jcn(1.art on the relation of
the pressure p a,u.
~Lt and idly! .
Therei'_ _ o the ri h
equal to
The right hand sides of the! quations of
and the s_.je rfluid fraction are
U V~ -r is` e~'v Y; l
)1 s
Comparison of. (6,6) and (6;8) with (6,64
By moans of the. expressions
considering the ~nes y'ity (L ,
a
whereN=N
. By mtano
r
and (6,8' )
motion for the
gives finally the relations between
Tr
of (3,1)
which art-
and the coefficients 'of second viscosity
I
-RDP78-04861A00010003000;1
and the chemical potc:.,
,,5) and (6,6) ar(:~
*,, 4 L~~ ~7~4 )
rc>pectiv(;1:
(3:,18) 9 (3,19) for )-(r and )4
i i.c found that
and '(3,2) .fin, o.mmall t:ncrement s 8 4 and
to . `he ,,-cscnco of a sound wave are expres
a
and div v . Introducing these
sod in terms of div TT
n
sions int6"~ (6,11) gives
Sanitized - Approvea or Kfl e..,
 s4,1vS --
1DP78-04861A000100030009-1
(P - C~) S ~e
the right hand sides of (695.') and (6,6)
with the above expressions, and?th.e insertion of expressions
CPPYRGHianitized - Approved For
(6,12) for P. and jCP
yield two equations
Avv,:_
Sanitized, Approved Fox . e ease ,; PJA RDP78-04861A000100030009-1
 CPYRGHT
fit.
Since these equations are supposed To hole bur any
arbitrary value$ of vn and v
it follovws that
PS Ci) (ov _
It would not be too difficult to evaluate these expres-
sions. We are, however, not going to proceed in this way,
since experiments on sound absorption are not sufficient
for determining all coefficients of viscosity. Only two
combinations of these coefficients can be derived from the
absorption coefficients of first and second sound. It is easy
to verify by means of (6,13) and (6,14) that the coefficients
t4YS .9 9 satisfy in fact the first of the relations (6910).
J c
We have still to express the absorption coefficients of
first and second sound in terms of the coefficients of first
and second viscosity, For that purpose'the complete hydro--
dynamical equations, including viscosity terms, are written
out. According to (3,1), ('3,2), (6,5) and (6,6)~
's
SaAitze V pjTro~M ReleLe C-R&P 8-048-b1 A00010003009_1
at
 CPYR(8j iitized r Approved For
AA- DP`78-04861 A0001000300A9-1
By eliminating from these equations vny Z. and expressing
all thermodynamic functions in terms of r and S, two simul-
taneous equations are obtained. The compatibility condition.,
similarly to (3,15), has the form of a determinant set equal
to 0. Since the determinant would have a complicated form it
will: not be formulated explicitly. The roots of the deter-
minantal equation are obtained by neglecting terms of the
magnitude (,o and noting the relatively small magnitude of
S,) h~V .. _ it,,
r~jj V3.
From (6,16), (6,17) the absorption coefficients for 1st and
2nd sound are readily obtained;-._.
A) I
rc3
By means of (6,13) and (6,14) it is easy to show that the
terms in the absorption coefficient which depend upon the
second viscosity are the same as expressions (4,3) and
respectively.
We find in fact from (6,13) and (6,14)0
(5,1)
, '.
In figure 2 the combination of coefficients of second
viscosity determining the absorption of 1st and 2nd sound
i.e plotted against temperature, For the purpose of compari-
son, the viscosity coefficient i7 is also shown in this figure.
M" AUUMUMSUMP-1
Sanitized - Approvea or a .ease .; _ , , y?. ,
 CPYR Vized - Approved Fgr$
I LRQP78-04861 A00010003000~9-1 `
32.
It is clearly seen that the absorption of first sound ..s
mainly due to second viscosity. The sum SI is. at,
all temperatures larger than the coefficient
It was pointed out before that the coefficients of firet
and second viscosity are usually of comparable magnitude.
The marked prevlance of the effect of second viscosity with
respect to the absorption of first sound is explained by an
anomalously large value of the differential coefficient
C (ep on which the magnitude
depends essentially.
The absorption coefficient for second sound shows a
different relation, Here the' coefficient is of sim .lar
magnitude as at all temperatures. The absorption of scco,:
. sound is s,cccrdingly deter._mtnecd to equal parts by the first
and second viscosity.
If absolute, zero is approached the coefficients n?
increase rapidly. By means of (6920) (6,21)y (4,17)
and (4,16) it is possible to obtain their limiting values at
extremely low temperatures.
r,
It is soon that in the vicinity of absolute zero the
effect of the second viscosity O:n the absorption of first seen,,
1
times larger than the corresponding effect fo
is J J4 ==-
second sound.
Comparison of (6,18) and (6919) shows that owing to th
factorin (6919), the absor;.)tion coefficient of second
sound, increases at a higher rate than the absorption
coefficient of first sound, when the temperature is decreased.
Near abs-lute zero the absorption coefficient cal, is markedly
higher than C(
Sanitized - Approved For 4 5
78-04861 A000100030009-1
 mental data are available. Here the theoretical graph as
shown is a complicated. curve; the points shown are experi-
mental values (1).
the slight
In calculating the values of cF)and of
variation of the density. with. temperature is neglected; the
density of helium II is assumed to be constant and equal to
0,145 gram/cm3.
In conclusion we should lake to make some remarks con-
cerning the absorption of sound near the X-point. The above
relations are obtained on the assumption that phonons and
rotors, are forming an ideal gas. It has been pointed out
earlier (3) that this ass,?.mptio:i is no longer valid near the
X-point.
The effect of the femporature on the absorption coef-
ficient cannot be. determined in a simple way by means of the
above arguments. Nevertheless it should be admissible to
draw at least a qualitative conclusion from (6,19)? the
absorption coefficient of second sound. is found to increase
at a high rate if.the 1k-point is approached. The velocity
of second sound approaches' zero if the 7~-point is approached;
the same holds for the ratio
The factor 44 appearing in (6,19) as ...factor
of the bracket, will, in approaching the A-point be?come
infinite of the order on. account of equation (3829).
As the bracket itself in (6,19) does not vanish, the absorp-
tion;co.efficient of second sound will increase somewhat like:
I if the k-point is approached. Experimental data (7)
concerning the absorption of second sound are inconclusive.
CPYRGIHTitized - Approved For no-me MA 8-04861A000100030(9-1
In figure 3 the ratio o;r and the ratio o(,/ 13 are
plotted against the temperature. In applying this diagram
it must be remembered that the coefficients of absorption are
proportional to r< only for those frequencies for which con-
dition (491) is valid, On the same diagram absorption coof-
ficients for first sound are given corresponding to a frequency
of 1.5.107 so C-1 ift a range of temperatures for which experi--
,r - Y C5IfB8P78-04861A0001 I-
 anitized - Approved For R
CPYRGH~
04861 A000100030009-1
The only inference they admit is the statement that absorp-
tion of second sound increapes appreciably if the X-point is
approached.
I should like to express my sincere thanks to
'Academician L.D. Landau for continual advice and to
Professor K.M. Li.fshitz for.helpful discussion..
.Lppendix
:The derivatives ands are
conveniently calculated in terms of the independent variables
p and T. Transft?t;.~-mation is carried out by means of the
Jacobians:
From (3,, 20
.: . , ~' b) ~) =
), A i
and transform (1A,i into the form
Similarly we find
Since the distribution
UVV
function for rotons has to be in
accord^uee with Boltzmann statistics, it is readily found
tha., )D+ T is equal to N p /kT.
The total number of phonons per unit volume of the liquid
in a state of non-equilibrium specified by the chemical
potential ,4A~, is equal to
Differentiation of (A,4) with respect to at const ant T
and'subsequent partial integration gives
-a~) T kT 36
obtain (where r is held constant)
1 - WT'j
(A s
 T fvn
anitized - Approved For
L 16
, iA-ROE78-04861 A00010DWO009-1
The remaining differential coefficients appropriate to
equilibrium are obtained by means of (4,8)-4;4,10)?
4- -,-1 ~ +
By means of (A,4) - (A,6) the final expressions as given in
section 4, i.e. (4,6), (4,7), are derived from (A,2), (A,3).
Academy of Science of the USSR;
Institute for Problems in Physics
Received 10 November, 191.9.
REFERENCES
1. J. Pellam & C. Squire, Phys. Rev. 22, 1245, 1947?
2. E. Andronikash.vili, J. Exp. Thcor. Phys. USSR 18,
429, 1948
3. L. Landau & I. Khalatnikov, J. Exp. Theor. Phys. USSR
12, 637 and 7099 1949
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