Four and three-phonon scattering in isotropic superfluid helium
We analyse the important role of four-phonon processes (4pp) in isotropic phonon systems of superfluid helium. The matrix elements and the rate of four-phonon processes are calculated. Special consideration is given to the 4pp in the momentum range where three-phonon processes are allowed. In this m...
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2009
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| Cite this: | Four and three-phonon scattering in isotropic superfluid helium / I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, A.F.G. Wyatt // Физика низких температур. — 2009. — Т. 35, № 3. — С. 265-277. — Бібліогр.: 26 назв. — англ. |
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| author | Adamenko, I.N. Kitsenko, Yu.A. Nemchenko, K.E. Wyatt, A.F.G. |
| author_facet | Adamenko, I.N. Kitsenko, Yu.A. Nemchenko, K.E. Wyatt, A.F.G. |
| citation_txt | Four and three-phonon scattering in isotropic superfluid helium / I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, A.F.G. Wyatt // Физика низких температур. — 2009. — Т. 35, № 3. — С. 265-277. — Бібліогр.: 26 назв. — англ. |
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| description | We analyse the important role of four-phonon processes (4pp) in isotropic phonon systems of superfluid helium. The matrix elements and the rate of four-phonon processes are calculated. Special consideration is given to the 4pp in the momentum range where three-phonon processes are allowed. In this momentum range, we show that the 4pp scattering rate, at small angles, is equal to the scattering rate due to three-phonon processes. Then we show that the coefficient of first viscosity of superfluid helium is caused by two processes, the first is due to the transverse relaxation caused by many three-phonon processes and the second is due to four-phonon processes. The relaxation time that governs the viscosity is obtained from the sum of the rates from these two processes. The temperature dependence of the attenuation coefficient of a pulse of high-energy phonons in He II, due to scattering with thermal phonons, is also calculated. The theoretical results are compared with experimental data and found to be in good agreement.
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Fizika Nizkikh Temperatur, 2009, v. 35, No. 3, p. 265–277
Four and three-phonon scattering in isotropic superfluid
helium
I.N. Adamenko1, Yu.A. Kitsenko 2, K.E. Nemchenko1, and A.F.G. Wyatt 3
1Karazin Kharkov National University, 4 Svobody Sq., Kharkov 61077, Ukraine
2Akhiezer Institute for Theoretical Physics National Science Center «Kharkov Institute of Physics and Technology»
National Academy of Sciences of Ukraine, 1 Academicheskaya St., Kharkov 61108, Ukraine
3School of Physics, University of Exeter, Exeter EX4 4QL, UK
E-mail: a.f.g.wyatt@exeter.ac.uk
Received October 28, 2008
We analyse the important role of four-phonon processes (4pp) in isotropic phonon systems of superfluid
helium. The matrix elements and the rate of four-phonon processes are calculated. Special consideration is
given to the 4pp in the momentum range where three-phonon processes are allowed. In this momentum
range, we show that the 4pp scattering rate, at small angles, is equal to the scattering rate due to three-pho-
non processes. Then we show that the coefficient of first viscosity of superfluid helium is caused by two pro-
cesses, the first is due to the transverse relaxation caused by many three-phonon processes and the second is
due to four-phonon processes. The relaxation time that governs the viscosity is obtained from the sum of the
rates from these two processes. The temperature dependence of the attenuation coefficient of a pulse
of high-energy phonons in He II, due to scattering with thermal phonons, is also calculated. The theoretical
results are compared with experimental data and found to be in good agreement.
PACS: 47.37.+q Hydrodynamic aspects of superfluidity; quantum fluids.
Keywords: superfluid helium, phonon scattering, dispersion curve, viscosity.
1. Introduction
Many of the properties of superfluid helium are de-
scribed in terms of its excitations from the ground state,
the phonons and rotons. There are interactions between
these excitations which govern important characteristics
such as the time to reach thermodynamic equilibrium, and
the normal fluid viscosity. At low temperatures, T � 0.6 K,
there are practically no rotons so the phonons determine
the behavior of superfluid 4He. Phonons mutually scatter
by two main processes, three-phonon process (3pp), in
which one phonon decays into two phonons and vice
versa, and four-phonon processes (4pp) in which two
phonons scatter to two other phonons. Higher order pro-
cesses are weak in comparision.
There is only one phonon branch in liquid helium, un-
like solids which have transverse branches as well as a
longitudinal branch. These branches in solids have nor-
mal dispersion, so 3pp in solids involve phonons from
more than one branch. However 3pp are allowed within
the same branch in liquid helium, because the dispersion
is anomalous over a large part of the phonon momentum
range.
If we write the dispersion curve for phonons as
� �� �cp p( ( ))1 , (1)
where c is the sound velocity, � and p correspond to the
energy and momentum of a phonon, then when �( )p � 0,
the dispersion is anomalous and when �( )p � 0, the dis-
persion is normal and 3pp are not allowed. The function
| ( )|� p �� 1 and it describes the deviation of the phonon
spectrum from linearity. So we see that although the value
of �( )p is small, it completely determines the mecha-
nisms of phonon interactions.
When p pc� (at the saturated vapour pressure
~p cp /kc c B� �10 K) �( )p � 0 and the dispersion is anom-
alous. In this case, the conservation laws of energy and
momentum allow processes which do not conserve
phonon number. The fastest of these is the small-angle
© I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, and A.F.G. Wyatt, 2009
three-phonon process which has a typical rate �3pp in
which one phonon decays into two or two interacting
phonons combine into one. When the angle between the
momenta of the phonons is large, � 27°, three-phonon pro-
cesses are forbidden by the conservation laws and the
interaction between phonons is only by the slower
four-phonon processes, with the typical rate �4 pp . When
p pc� , function �( )p � 0. In this case, the dispersion is
normal and three-phonon processes are forbidden by the
conservation laws of energy and momentum and the
fastest scattering is by four-phonon processes.
There is a strong inequality between the typical values
of the rates of three-phonon and four-phonon processes
described above,
� �3 4pp pp�� (2)
which causes dynamic systems of phonons, in superfluid
helium, to separate into two subsystems: a subsystem of
high-energy phonons (h-phonons) with p pc� in which
equilibrium is attained relatively slowly and a subsystem
of low-energy phonons (l-phonons) with p pc� in which
equilibrium occurs relatively quickly (see, for example,
Refs. 1–5).
The dissipative coefficients of superfluid helium are
mainly governed by the interaction between l-phonons.
However �3pp does not enter directly into the dissipative
coefficients. Three-phonon processes cause thermal equi-
librium to be established rapidly in a small angular range
of momentum. Equilibrium over the whole angular range
can be obtained in two ways. The first is by superdif-
fusion in space with the momentum vector having angular
steps of a size which is typical of three-phonon processes.
This has a typical rate �� that is three orders of magni-
tude lower than the three-phonon process rate (see, for
example, Refs. 6–10). The second is by four-phonon pro-
cesses (see, for example, Refs. 11 and 12), which can
scatter through much larger angles than 3pp. The rate
from these important 4pp scatterings has not been ob-
tained when there is anomalous dispersion which allows
spontaneous decay.
Earlier rates were found indirectly from measurements
of the dissipative coefficients of phonon systems. These
relate to global equilibrium times. A direct measurement
of the rate of four-phonon process scattering was made by
measuring the attenuation of a pulse of h-phonon propa-
gating through superfluid helium at a finite temperature
[13]. Here the beam of h-phonons is scattered by the iso-
tropic distribution of thermal low-energy phonons. Re-
cently, direct measurements of the phonon–phonon scat-
tering rates interactions have been made by scattering two
beams of phonons [14–16].
From the above we see that 4pp scattering is funda-
mentally important to understanding some important
properties of superfluid helium. In this paper we develop
the detailed theory of 4pp scattering, including the mo-
mentum range where conservation laws of energy and
momentum allow three-phonon processes. From the de-
tailed theory we find numerical values for the scattering
rates and the relaxation times in isotropic phonon sys-
tems. Hence we find the coefficient of first viscosity as a
function of temperature. There is good agreement with
the measured values.
In Sec. 2 we derive the matrix elements for 4pp, in Sec. 3
we find the relaxation in phonon system caused by
four-phonon processes, in Sec. 4 we find the 4pp rate
when 3pp are allowed, in Secs. 5 and 6 we calculate the
relaxation time relevant to viscosity. In Sec. 7 we calcu-
late the scattering of a beam of high-energy phonons by
thermal phonons. Finally in Sec. 8 we draw conclusions.
2. Matrix element for four-phonon processes with the
account of possibility of three-phonon processes
The interaction of phonons in superfluid helium can be
described by the Landau Hamiltonian which can written
as (see, for example, Ref. 17)
� � � �H H V Vph � � �0 3 4 , (3)
where �H 0 is the Hamiltonian of noninteracting phonons,
terms �V3 and �V4 describe the interaction of phonons
caused by small deviations of the system from equilib-
rium to third and fourth order, respectively.
The probability density of four-phonon process, fol-
lowing Refs. 13,17,18, can be written as
W V H fi( , | , )
( )
p p p p1 2 3 4
2 2
6
2 1
2
�
� �
, (4)
where V is a volume of a system, H fi is the amplitude of
four-phonon processes. This is obtained from �V3 with sec-
ond order perturbation theory, and from �V4 with first or-
der perturbation theory. Following standard procedures
(see, for example, Refs. 13,17–20) we have
H
V V
E E
fi
i
�
� �� �
��
p p Q Q p p
QQ
3 4 3 3 1 2, � � ,
� � �p p p p3 4 4 1 2, � ,V , (5)
where Ei is the energy of initial state and Q is the interme-
diate state with energy EQ.
There are six possible intermediate states Q :
I II.
III. IV.
. ; , , ;
, , ; , ,
p p p p p p
p p p p p p p
1 2 2 3 1 3
2 4 1 4 1 3 2
�
p
p p p p p p p p p p
3
1 4 2 4 1 2 3 4 1 2
;
, , ; , , , , .V. VI.
(6)
266 Fizika Nizkikh Temperatur, 2009, v. 35, No. 3
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, and A.F.G. Wyatt
There is an important effect when all four phonons
participating in four-phonon process are l-phonons. Then
it is possible that the matrix elements diverge. This is be-
cause the denominator of the first term in Eq. (5) can be
zero at some value of Q. This happens when the transition
from the initial state to the final state can be realized by
two sequential three-phonon processes. This divergence
can be eliminated if we take into account that the interme-
diate state has a finite lifetime. As pointed out in Ref. 21,
the energy of a system which can decay into some
quasi-stationary state, can be determined only to within
an accuracy of � � �/�, where � is the lifetime of the sys-
tem in this quasi-stationary state. In our case, the finite-
ness of the lifetime of the state Q is due to the possibility
that it can scatter by the three-phonon process. Thus EQ in
the denominator of the first term in Eq. (5), must be sub-
stituted by E iQ
Q
� ( ), where � ( )Q is defined by the life-
time of the corresponding intermediate state. The life-
times of the corresponding intermediate states are
I. II.
III.
� � � �
�
( ) ( )
( )
( ); ( );1
1
1 2 13
1
1 3
14
� � �
�
d cp p p p
� � �
� �
c c
c
�
�
1
1 4 23
1
2 3
24
1
2
( ); ( );
(
( )
( )
p p p p
p p
IV.
V. 4 5); .( )VI. � ��
(7)
Here �d ( )p is a rate of a decay of a phonon with momen-
tum p into two, and � c ( )p is the rate of the three-phonon
processes in which a phonon with momentum p combines
with the other phonon. These 3pp rates were calculated
by us in Ref. 4 for isotropic and anisotropic phonon sys-
tems. The intermediate state VI cannot be realized by
three-phonon processes and therefore the denominator
cannot be zero at any value of momentum. We therefore
consider that the lifetime of such a state is infinite and the
value of � is � � ��/� 0. If one of the four interacting pho-
nons is an h-phonon, then there are no divergences in the
first term of Eq. (5). In this case all lifetimes in Eq. (7)
should be considered infinite, and all �’s equal to 0.
From relations (5) and (6) it follows that
H
p p p p
V
Mfi � 1 2 3 4
8� � . (8)
Here � �145 kg/m 3 is a density of He II, and
M M M M M M M M� � � � � � � �( ) ( ) ( ) ( ) ( ) ( )1
13
3
14
3
23
3
24
3 5
4
(9)
is a matrix element which consists of seven terms, six of
which correspond to the six intermediate states (6), and
the seventh is determined by the contribution of �V4 to
first order. We can write these terms as
M
i
( )
( )
1 1 2
1 2 1 2
1
�
�
�
��
�
�
� � � �
�
� � � �� �( )2 1 1 2 1 1 2 2 1 2u n n n n n n
�
� � �� �( )2 1 3 4 3 3 4 4 3 4u n n n n n n , (10)
M u( ) ( )5 1 2
1 2 1 2
1 2 1 1 2 2 1 22 1�
� �
�
��
�
� �
�
� � �
n n n n n n
�
�
� �( )2 1 3 4 3 3 4 4 3 4u n n n n n n , (11)
M u w4
24 1�
�{( ) }, (12)
M
i
13
3 1 3
1 3 1 3
13
( )
( )
�
�
�
�
� � � �
�
� � � �
( )2 1 1 3 1 1 3 3 1 3u n n n n n n
�
� � �
( )2 1 2 4 2 1 3 4 1 3u n n n n n n . (13)
Here n pi i i/p� , � ( )
( )
q
q�
�� 1 , u /c c/� � � �( )� � 2.84 is a
Grüneisen constant, w /c c/� � � �( )� �2 2 2 0.188, � �i i� ( )p .
The rest of the terms in Eq. (9), i.e., M
14
3( )
, M
23
3( )
and M
24
3( )
can be obtained from M
13
3( )
by the replacement of the cor-
responding subscripts. The matrix element is given in de-
tail in Ref. 22, to which the corresponding �’s should be
added (see Eqs. (10) and (13)).
3. Relaxation in phonon system caused by
four-phonon processes
We consider four-phonon process which have two
phonons in the initial state and two phonons in the final
state. The conservation of energy and momentum is ex-
pressed as
� � � �1 2 3 4 1 2 3 4� � � � � �, p p p p . (16)
The kinetic equation describing the change of the distri-
bution function n n( )p1 1� due to 4pp scattering is
dn
dt
N Nb d
1
1 1�
( ) ( )p p , (15)
where N b ( )p1 and N d ( )p1 are the respective rates of in-
creasing and decreasing number of phonons with momen-
tum p1 in unit time due to collisions. They can be written as
N b d, ( )p1 �
� �
1
2
1 2 3 4
3
2
3
3
3
4W n d p d p d pb d( , , ) ( ) ( ) ,p p |p p p� � �� � .
(16)
The probability density W ( , ),p p |p p1 2 3 4 is defined by
Eq. (4) and determines the probability of the process.
The factor 1/2 is due to the identity of processes
Four and three-phonon scattering in isotropic superfluid helium
Fizika Nizkikh Temperatur, 2009, v. 35, No. 3 267
p p p p1 2 3 4� � � and p p p p1 2 4 3� � � , the �-func-
tions correspond to the conservation laws of energy �� �
� �
� � � �1 2 3 4 and momentum p p p p p� � �
1 2 3 4 ,
and
n n n n n n n n n nb d� � � � � �3 4 2 1 1 2 3 41 1 1 1( )( ), ( )( ).
(17)
To determine the typical rate of four-phonon processes
in phonon systems in superfluid helium, we take the dis-
tribution functions in Eq. (15) as
n n n n n n n n n1 1
0
1 2 2
0
3 3
0
4 4
0� � � � �( ) ( ) ( ) ( )
, , ,� . (18)
In Eq. (18) superscript «0» corresponds to the equilibrium
distribution function, and �n is the deviation of the distri-
bution function from equilibrium.
The equilibrium distribution function of phonons, ac-
cording to Refs. 4, 23, can be written as
n
k TB
( )( ) exp0
1
1p
pu
�
�
�
��
�
�
��
�
!
"
#
$
�
, (19)
where
u N�
c( )1 % (20)
is a drift velocity, N is a unit vector directed along the to-
tal momentum of phonon system. This defines an aniso-
tropy axis of phonon system and % is the anisotropy pa-
rameter.
For isotropic systems the parameter % �1 and u � 0 in
Eq. (19). For weakly anisotropic systems % is close to 1.
However for strongly anisotropic phonon systems % �� 1.
Such systems can be created experimentally [14–16].
The typical rate of relaxation caused by four-phonon
processes �4 pp , according to kinetic theory in the relax-
ation time approximation, can be defined as
�
�
�
4 1
1
11
pp
n
d n
dt
( )p �
. (21)
From Eqs. (15)–( 21) we have
�4 1
1
0
3
2
3
3
3
4
1
2
1
1
pp
n
d p d p d p( )
( )
p �
�
��
� � �W n n n( , | , ) ( ) ( ) ( )( )
( ) ( ) ( )
p p p p p1 2 3 4 2
0
3
0
4
0
1 1� � �� � .
(22)
Equation (22) can be integrated with a help of the
�-functions. For this we rewrite Eq. (22) taking into ac-
count (4), (8)–(13) in a spherical coordinate system
�
�
4
1
11 5 7 2
1
02
1
1
pp
p
n
�
�
�
�
( )
� �� dp d d dp d d dp d d p p p2 2 2 3 3 3 4 4 4 2
3
3
3
4
3& ' & ' & '
� � �M n n n� � �
2
2
0
3
0
4
0
1 1� � �( ) ( ) ( )( )
( ) ( ) ( )
p , (23)
where ' i i i/p�
1 ( )p N and & i is the azimuth angle of the
phonon with momentum p i . Without loss of generality,
we assume that p p3 4� for the integration.
Making the integration in Eq. (23), with a help of the
�-functions (see Appendix A) we have
�
�
4 10 5 7 2
1
1
0
1
2 1
pp
c
p
n
�
�
�
�
( )
� �� dp dp d d d
p p p
R
2 3 2 3 2
2
3
3
3
4
2
' ' &
� � � ��
{ } ( )( )( ) ( )
( ) ( ) ( )
M M n n n2 2
2
0
3
0
4
0
1 1 , (24)
where
p p p p
p p p
p
4 1 2 3 4
1 1 2 2 3 3
4
� �
( �
�
(
, ,'
' ' '
(25)
( � �
�p p p p pi i3 3 4 4 1 1 2 2� � � � � �, ( ), (26)
M M( )
( ) ( )
(cos cos ,cos cos ))
) )� � �� & & & &3 3 4 4
(27)
and cos
,
( )&
3 4
)
are determined by Eqs. (88), (89) of Appen-
dix A.
Further integration of Eq. (24) cannot be precisely
made analytically, because of the complexity of the in-
tegrand. The results of a numerical calculation of the
l-phonon relaxation rate from Eq. (24), for the isotropic
case at temperature T �0.6 K, are shown in Fig. 1, curve 1.
The limits of integration in Eq. (24), with the condition
p p3 4� , are the momenta given by
p
p p
p p p pc3
1 2
3 1 2
2
low up�
�
� �, min( , ), (28)
p p pc2 0low 2up� �, . (29)
For comparison, the rate of three-phonon processes in the
isotropic case with T � 0.6 K calculated by us in Ref. 4 is
shown by curve 5.
From Fig. 1, it can be seen that in practically all the
momentum range, where three-phonon processes are al-
lowed, the rate of four-phonon processes is almost the
same as the rate of three-phonon processes. To further un-
derstand this result, the contribution of the different angu-
lar groups of phonons with momentum p2 is calculated.
268 Fizika Nizkikh Temperatur, 2009, v. 35, No. 3
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, and A.F.G. Wyatt
Curves 2–4 show the contributions to the rate of phonons
with momentum p2 having angles with a phonon p1:
0–30° (curve 2), 30–60° (curve 3) and 60–180° (curve 4).
From Fig. 1 it can be seen, that the main contribution to
the rate of four-phonon processes involves phonons with
momentum p2, at an angle up to 30° to phonon p1. We
conclude that the rate of these four-phonon processes is
practically equal to the rate of three-phonon processes.
From this analysis, it follows that the main contribution to
�4 pp , where three-phonon processes are allowed, is
caused by small angle scattering, and then �4 pp appears
practically equal to �3pp .
In order to understand why there is a coincidence be-
tween �3pp and �4 pp for angles where three-phonon pro-
cesses are permitted, it is necessary to make analytical
calculations which we do in the next section.
4. The rate of four-phonon processes in the range of
angles where three-phonon processes are permitted
To calculate the rate of four-phonon processes, at
small angles where three-phonon processes are allowed,
we start from Eq. (22) for the rate and Eq. (4) for the prob-
ability density of the four-phonon processes. At small an-
gles, the main contribution to the matrix element (9), is
given by the five resonant terms corresponding to the in-
termediate states I–V (see (6)). Our calculations show
that after squaring the absolute value of the matrix ele-
ment, cross-terms give a small contribution to the inte-
gral. So the main contribution is caused by the squares of
the five resonant terms.
We now calculate the contribution to the rate of each of
the five terms mentioned above. The probability density
WI of the four-phonon process which is caused by the in-
termediate state I, is conveniently written in the form
W VI( , | , )
( )
p p p p1 2 3 4
2
6
2 1
2
� �
� �
�
� �� �
�
�
��
p p q q| p p
q
p p
q
3 4 3 3 1 2
1 2
2
3
, | � | � | ,
( )
(
V V
i d� � �
�
� 4
3
q) ,d q
(30)
where
�d d( ) ( )q q� �� . (31)
The probability density of three-phonon processes is
given by (see, for example, Ref. 4)
W V Vi j i j( , | ) , | � |p p q p p q� � �
2
3
2
�
. (32)
Taking (32) into account, we can rewrite Eq. (30) as
W I( , | , )
( )
p p p p1 2 3 4 62
1
2
� �
�
�
�
�
�
�
�
W W
d
( , | ) ( | , )
( ) ( )
( )
p p q q p p
q
p p q
q
3 4 1 2
1 2
2 2 3 4
� � �
�
�
d q3 . (33)
As
1
1 2
2 2( ) ( )� � ��
�
�
q q�d
�
�
�
* � � ��
�
�d
d
d
( )
( )
( ) ( )}q
q
qq1 2
2 2
, (34)
then for small values of �, due to the equality
lim
( )
( )
+
+
, +
� ,
- �
�
0 2 2
(35)
the second factor in the right-hand side of Eq. (34) can be
approximately replaced by a �-function.
Substituting expression (34) into (33), taking into ac-
count (35), we obtain
W I( , | , )
( )
p p p p1 2 3 4 62
1
2
� �
�
�
� �
�
�
W W
d
d
( , ) ( , )
( )
( ) ( )
p p |q q|p p
q
p p qq
3 4 1 2
1 2 3 4
3
�
� � � � � q.
(36)
Then substituting (36) into (22) and assuming the an-
gles are small, we have
Four and three-phonon scattering in isotropic superfluid helium
Fizika Nizkikh Temperatur, 2009, v. 35, No. 3 269
0 2 4 6 8 10
1
2
54
3
cp /k , K1 B
10
4
10
6
10
8
10
10
�,
s–
1
Fig. 1. The rate of four-phonon processes in the isotropic
l-phonon system with T � 0.6 K (curve 1) as a function of mo-
mentum p1; curves 2–4 show contributions to the relaxation rate
by phonons p2 having angles with phonon p1: 0–30° (curve 2),
30–60° (curve 3) and 60–180° (curve 4). Curve 5 shows the rate
of three-phonon processes in the isotropic case for T � 0.6 K. It
can be seen that curves 1 and 5 are almost coincident for p pc1� .
�
4 1 6
1
0
3 3
2
3
3
3
4
4
1
2
1
1
pp
n
d qd p d p d pI ( )
( ) ( )
p �
�
��
�
�
� � � �
W W
n n n
d
( , | ) ( | , )
( )
( )( )
( ) ( ) ( )p p q q p p
q
3 4 1 2
2
0
3
0
4
0
1 1
�
� �
�
�
�
� � � � � � � � � �( ) ( ) ( ) ( ).p p q p p q q q1 2 3 4 1 2 3 4
(37)
Taking the �-functions into account, together with the
properties of the equilibrium distribution functions, the
combination of distribution functions in Eq. (37) can be
rewritten in the form
1
1
1 1
1
0 2
0
3
0
4
0
�
� � �
n
n n n
( )
( ) ( ) ( )
( )( )
�
� �( )( )
( ) ( ) ( ) ( )
n n n n
2
0 0
3
0
4
0
1q . (38)
The rate of decay of a phonon with momentum p i , due
to the three-phonon process p p pi j k� � is, according to
Ref. 4, equal to
�
d i j k
i j k
j k
d p d p
W
n n( )
( | , )
( )
( )
( ) ( )
p
p p p
� � � �
1
2 2
13 3
3
0 0
�
�
�
� � � � �( ) ( )p p pi j k i j k , (39)
and the rate of combining with the phonon with momen-
tum p j is
�
c j k i
j k i
k id p d p
W
n n( )
( , | )
( )
( )
( ) ( )
p
p p p
�
�� 3 3
3
0 0
2 �
� �
�
� � � � �( ) ( )p p pj k i j k i . (40)
Tak ing in to accoun t the equa l i ty W i j k( | , )p p p �
�W j k i( , | )p p p we have
�
4 1 3
3 3
2 2
0 0 1 2
2 2
pp d qd p n n
WI ( )
( )
( )
( | , )( ) ( )
p
q p p
q�
�
�
� �d ( )q
�
� �
�
� � � � � �d ( ) ( ) ( )q p p q q1 2 1 2 . (41)
On substituting (31) into (41) and taking (40) into ac-
count, we finally have
� �4 1 1
1
2
pp c
I ( ) ( )p p� . (42)
Now we calculate the contribution to rate due to the in-
termediate state II. The probability density of a four-pho-
non process caused by the intermediate state II is conve-
niently written in the form
W VII( , | , )
( )
p p p p1 2 3 4
2
6
2 1
2
� �
� �
�
� �� �
�
p p q p p q p p p p
q
3 4 3 2 3 2 3 3 1 2
1 3
, | � | , , , , | � | ,V V
i� � � �c ( )q� �
2
�
�( ) ,p q p1 3
3d q (43)
where
�c c( ) ( )q q� �� . (44)
Taking (32) into account we can rewrite (43) as
W II( , | , )
( )
p p p p1 2 3 4 62
1
2
� �
�
�
�
�
W W
d
c
( | , ) ( , )
( ) ( )
( )
p q p q p |p
q
p q p
q
4 2 3 1
1 3
2 2 1 3
� � �
�
�
3q� . (45)
Replacing the resonant factor in Eq. (45) by a �-func-
tion, in a similar way to the previous case, we obtain
W II( , | , )
( )
p p p p1 2 3 4 62
1
2
� �
�
�
�
W W
d q
c
( | , ) ( , | )
( )
( ) ( )
p q p q p p
q
p q pq
4 2 3 1
1 3 1 3
3
�
� � � � �� .
(46)
Substituting (46) into (22), and assuming the angles are
small, we have
�
4 1 6
1
0
3 3
2
3
3
3
4 2
0
4
1
2
1
1
pp
n
d qd p d p d p nII ( )
( ) ( )
( )
p �
�
��
�
�
� � �
( )( )
( | , ) ( , | )
( )
(
( ) ( )
1 1
3
0
4
0 4 2 3 1
1n n
W W
c
p q p q p p
q
p q
�
�
�p3 )
�
� � � � � � � � �( ) ( ) ( )p q p q q4 2 1 3 4 2 . (47)
Taking the �-functions and the properties of the equilib-
rium distribution functions into account, the combination
of distribution functions in the integral can be rewritten in
the form
1
1
1 1
1
0 2
0
3
0
4
0
�
� � �
n
n n n
( )
( ) ( ) ( )
( )( )
�
� �( )( )
( ) ( ) ( ) ( )n n n n
2
0
4
0
3
0 01 q . (48)
Taking (48) into account we can rewrite (47) as
�
4 1 3
3 3
3 3
0 0
4 2
1pp d qd p n nII ( )
( )
( )
( ) ( )
p q� � � ��
�
�
�
W
c
c
( , | )
( )
( ) ( ) ( )
q p p
q
q p q p q
3 1
1 3 1 3�
� � � � � � .
(49)
270 Fizika Nizkikh Temperatur, 2009, v. 35, No. 3
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, and A.F.G. Wyatt
Having substituted (44) into (49) we finally have
� �4 1 1
1
2
pp d
II ( ) ( )p p� . (50)
We draw the reader’s attention to an apparent contra-
diction. Firstly when n
2
0( )
tends to zero in Eq. (47), then it
seems that the rate �4 1pp
II ( )p , also tends to zero too. Sec-
ondly, the rate of decay �d ( )p1 does not tend to zero when
n
2
0( )
goes to zero. So we arrive at the contradiction that the
left-hand side of Eq. (50) is equal to zero while the
right-hand side is nonzero. However, if n
2
0( )
goes to zero,
then �c ( )q goes to zero too. So after cancelling n
2
0( )
with
the same term in the expression for �c (see Eqs. (44) and
(40)) we find that the rate �4 1pp
II ( )p is nonzero. Conse-
quently we are convinced in correctness of Eq. (50), which
is valid at all values of n
2
0( )
which are not exactly zero.
The calculation of �4 1pp
III ( )p can be made in a way sim-
ilar to the calculation of �4 1pp
II ( )p , if we replace p p3 4�
in the integrand. Thus we obtain
� �4 1 1
1
2
pp d
III ( ) ( )p p� . (51)
Analogous calculations for the fourth and fifth inter-
mediate states give
� � �4 1 4 1 1
1
4
pp pp c
IV V( ) ( ) ( )p p p� � . (52)
Finally we have
� � � �4 1 1 1 3 1pp c d pp( ) ( ) ( ) ( )p p p p� � � . (53)
Thus, the rate of four-phonon processes is equal to the
rate of three-phonon processes, in the momentum range
where three-phonon processes are allowed.
Thus, for small angle scattering, the main contribution
to the 4pp rate is caused by processes which can be repre-
sented as two consecutive three-phonon processes. At
larger angles between the momenta of phonons p1 and p2,
this mechanism becomes forbidden by the conservation
laws of energy and momentum and the scattering is
caused by «exclusive» four-phonon processes. These
processes exclude 4pp which can be represented by two
consequative 3pp. As a result, the scattering rate obtained
by us is the sum of the three-phonon process rate at small
angles (� 30°), and the rate of «exclusive» four-phonon
processes at larger angles.
5. The calculation of the viscous relaxation time
In this section we calculate the relaxation time that de-
termines the coefficient of first viscosity of superfluid he-
lium. We start from the stationary kinetic equation for
phonons. Following Ref. 12, we consider a macroscopic
nonuniform flow of fluid with velocity u(r) directed along
z axis. The velocity gradient is taken to be sufficiently
small so that in every moving volume element, there is a
local equilibrium. We consider that the velocity gradient
is directed along the x axis which is perpendicular to the
z axis. The distribution function ni of phonons with mo-
mentum p i can be represented as a sum of a local-equilib-
rium distribution function ni
( )0
, and a small deviation �ni :
n n ni i i� �( )
.
0 � (54)
Taking all the above into account, the kinetic equation
can be written as
n n
cp
k T
u
x
I n
B
1
0
1
0
11
( ) ( )
( ) cos sin cos ( )�
�
�
�. . & , (55)
where . is the polar angle between the phonon momentum
and the z axis, and & is the azimuth angle of the phonon
momentum relative to the x axis.
The collision integral, in the relaxation time � approxi-
mation, can be written as
I n
n
( )1
1�
�
�
. (56)
From the kinetic equation (55), the deviation of the distri-
bution function from its equilibrium value, which is caused
by the macroscopic velocity gradient, can be written as
�
.
�n n n
P
k T
a bcpi i i
i
B
i i� � �( ) ( )
( )
(cos )
( )
0 0 21 , (57)
where P x2( ) is the second order Legendre polynomial, a
and b are parameters which define the deviation of the
distribution function of the phonons moving in the given
direction, from its equilibrium value. We note, that the pa-
rameters a and b are not independent. As equilibrium in
the given direction is attained quickly, by fast three-pho-
non processes, there is a relation between the parameters
a and b which can be written as follows (see, for example,
Refs. 8, 10)
b a�
3
2
. (58)
Having integrated the left hand and right hand sides of
Eq. (56) over all phonon energies, and taking (57) and
(58) into account, we obtain the relaxation time which ap-
pears in the expression for the first viscosity,
�
.
�
�1
4
4
5
1 1
3
1
2 1
15
2
c
k T
I p p dp
P
B( )
' ( )
(cos )
, (59)
where, from (15)–(17), (54), (57), and (58), the collision
integral can be written as
Four and three-phonon scattering in isotropic superfluid helium
Fizika Nizkikh Temperatur, 2009, v. 35, No. 3 271
/ � � �I p d p d p d p Wn n n n( ) ( )( )
( ) ( ) ( ) ( )
1
3
2
3
3
3
4 1
0
2
0
3
0
4
01
2
1 1� �
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
� . � .1 1 2 1 2 2 2 2
3
2
3
2
cp P cp P(cos ) (cos )
!
�
�
�
�
�
�
�
�
�
�
�
�
"
#� . � .3 3 2 3 4 4 2 4
3
2
3
2
cp P cp P(cos ) (cos )
$
�
�� � �( ) ( )� �p . (60)
The probability densityW , in Eq. (60), is defined by Eq. (4).
Equation (59) differs from that obtained in Ref. 12 for
a nondecaying phonon spectrum, due to the terms in the
curly brackets of the integrand in Eq. (60). They are the
small terms containing �( )pi � 0, which occur in � i . As
will be shown below, these terms determine the relaxation
rate caused by phonons interacting at small angles (see re-
sult (75)). At small angles, the large terms in Eq. (60) are
equal to zero and only the small terms containing�( )p are
left (see result (75)) .
In Eq. (59) for the relaxation time, which determines
the coefficient of first viscosity, the integration can be
made with the help of the �-functions, as in the third sec-
tion. Further integration can only be precisely made nu-
merically. The results of this integration are shown in
Fig. 2 by curve 1. However the contribution of large and
small angles to Eq. (60) could be approximately calcu-
lated analytically; this will be done in the next section
where we also discuss the results shown in Fig. 2.
6. Calculation of the contribution of small and large
angle scattering to the relaxation time which deter-
mines the coefficient of first viscosity
To calculate the contribution of scattering at small an-
gles between the interacting phonons in Eq. (60), as well
as in the fourth section, we consider the contribution to
Eq. (60) of the five main terms, which correspond to the
five intermediate states I–V (see (6)).
We begin with the first intermediate state. The proba-
bility density of four-phonon process caused by the inter-
mediate state I, at small angles between the momenta of
the interacting phonons, is given by Eq. (36). After sub-
stituting Eq. (36) into Eq. (60) we obtain
/ � ��I d qd p d p d pI( )
( )
p1 6
3 3
2
3
3
3
4
4
1
2
�
�
� �
�
�
�
�
�
�
W W
cp P
d
( , | ) ( , )
( )
(cos )
p p q q|p p
q
3 4 1 2
1 1 2 1
3
2�
� . �
�
!
�
�
�
�
�
�
�
�
�
�
�
�
�
� . � .2 2 2 2 3 3 2 3
3
2
3
2
cp P cp P(cos ) (cos )
�
�
�
�
�
�
"
#
$
� �� .4 4 2 4 1
0
2
0
3
03
2
1 1cp P n n n n(cos ) ( )(
( ) ( ) ( )
4
0( )
)�
� �
�
�
�
� � � � � � � � � �( ) ( ) ( ) ( ).p p q p p q1 2 3 4 1 2 3 4q q
(61)
The curly brackets in Eq. (61) can be rewritten in the form
{ (cos ) (cos ) }| |
P A P Aq q q
q
2 1 12
1
2 34
. .� , (62)
where
A cp cp Pq12
1
1 1 2 2 2 21
3
2
3
2
| (cos )�
�
�
�
�
�
� �
�
�
�
�
�
�
� � .
�
�
�
�
�
�� .q qcq P
3
2
2 1(cos ), (63)
A cq cp P
q
q
q q|
(cos )
34 3 3 2 3
3
2
3
2
�
�
�
�
�
�
�
�
�
�
�
�
�
� � .
�
�
�
�
�
�� .4 4 2 4
3
2
cp P q(cos ), (64)
where . ij is the angle between phonons with momenta p i
and p j . The subscripts of A indicates the corresponding
process (for example, 12| q corresponds to p p q1 2� - ,
the superscript shows the momentum direction to which
all angles are referenced. We note, that for the A quanti-
ties the following relations are valid
A A A Aq q q
q
q
q
12
1
12
1
34 34| | | |
;�
�
. (65)
272 Fizika Nizkikh Temperatur, 2009, v. 35, No. 3
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, and A.F.G. Wyatt
3
2
1
0.2 0.4 0.6 0.8 1.0
10
–4
10
–2
10
2
10
4
l,
cm
10
0
T, K
Fig. 2. The phonon mean free path l c� � as a function of tem-
perature T. Curve 1 is calculated from the relaxation time
defined by Eq. (59); curve 2 represents Benin’s result (see
Ref. 7); curve 3 is a numerical calculation of the relaxation
time from Eq. (59) without the contribution of small angles be-
tween p1 and p2 (i.e., the lower limit of integration over angles
is equal to 30°); experimental points from Greywall [24] are
marked by triangles, the squares show the experimental data
from Zadorozhko et al. [25].
When deriving Eq. (62) we used the result of the theo-
rem about the addition of Legendre functions. This states
that if two directions in space are given by their polar an-
gles 0, /0 and their azimuth angles&, /& then the Legendre
function of the n-th order, which depends on the cosine of
the angle //0 between these directions, satisfies the fol-
lowing integral relation:
P d P Pn n n
0
2
2
&
� //0 � 0 /0(cos ) (cos ) (cos ). (66)
Taking the above into account, the collision integral
(61) can be written as
/ � ��I d qd p d p d pI( )
( )
p1 6
3 3
2
3
3
3
4
4
1
2
�
�
� � �
W W
n n n n
d
( , ) ( , )
( )
( )(
( ) ( ) ( ) (p p q q p p
q
3 4 1 2
1
0
2
0
3
0
4
0
1 1
�
)
)�
� � �{ (cos ) (cos ) }| |
P A P Aq q q
q
2 1 12
1
2 34
. .
� �
�
�
�
� � � � � � � � � �( ) ( ) ( ) ( ).p p q p p q1 2 3 4 1 2 3 4q q
(67)
The integral from the second term in curly brackets is
equal to zero because
P dq q q2
0
0(cos ) sin. . .
�� . (68)
Integration over p3 and p4 in Eq. (67) can be precisely
made analytically. As a result, we have
/ �
I p PI( ) ( ) (cos ),p1 1 2 1
1
2
1 . (69)
where
1
( )
( | , )
( )
( )(( ) ( ) (
p d qd p
W
n n nq1
3 3
2
1 2
3
0
1
0
2
2
1 1� � ��
q p p
�
0)
)�
� �
�
Aq q| ( ) ( )12
1
1 2 1 2� � � � �p p q . (70)
Similarly for the remaining terms we have
/ � / �
I I p PII III( ) ( ) ( ) (cos )p p1 1 1 2 1
1
4
2 . , (71)
/ � / �
I I p PIV V( ) ( ) ( ) (cos )p p1 1 1 2 1
1
4
1 . , (72)
where
2
( )
( , | )
( )
( )(
( ) ( )
p d qd p
W
n n nq1
3 3
3
3 1
3 1
0
3
0
2
1 1�
� ��
q p p
�
( ) )0 �
�
A q q13
1
1 3 1 3| ( ) ( )� � � � �p p q . (73)
Substituting (69), (71)–(72) into (59), we obtain the
relaxation time � sm due to the interaction of phonons at
small angles
�
2 1sm
B
c
k T
p p p dp
� ��
!
"
#
$�1
4
4
5 1 1 1
3
1
15
2
1
2( )
( ) ( ) . (74)
Integrating Eq. (74) (see Appendix B) we finally derive
�
�
sm
B
u c
k T
�
�
�1
2
8 5 4
5
5
45 1
2
( )
( )�
�� �dp dp
p p p p p p p
k TB
1 3
1
2
2
2
3
2
1 3
2
1 3
1
+ 3 +
�
( , ) ( ( , ))
sinh sinh sinh
� �2 3
k T k TB B
, (75)
where
+ � � �( , ) ( ) ( ) ( ) ( )p p p p p p p p p p1 3 1 1 3 3 1 3 1 3�
(76)
and 3( )x is the Heaviside function, which is equal to unity
when x 4 0 and to zero when x � 0.
Equation (75) is identical with the result in Ref. 7 for
the relaxation time of the second spherical harmonic,
which defines the coefficient of the first viscosity. The
time � sm from Eq. (75) is the same as �� /6 in Refs. 8, 9.
The presence of the numerical factor of 1 6/ is connected
with the definition of the time of traverse relaxation
which was given in Refs. 8, 9. However the coefficients
of first viscosity are the same in our and their theories.
Our relaxation time, given by Eq. (59) at small angles, au-
tomatically includes the process of transverse relaxation
of Refs. 8 and 9.
At larger angles, the denominators of the matrix ele-
ments cease to be resonant. Then the matrix element actu-
ally ceases to depend on the exact form of the function
�( )p . In this case, the integration of Eq. (60), with the ap-
proximate simplified matrix element, was made in Ref. 12
where the following relation for the 4pp rate was obtained
1 9 13 1
2 2
4
4
13 7 7 2 10
9
�
�
pp
L B
u
c
k T
( )
!( )
( )
( )�
5 �
�
. (77)
Numerical calculations with the exact matrix element
from Eq. (59), in the region of angles where .12 30� 6,
gives practically the same numerical value for the relation
time as that obtained from Eq. (77) when T � 0.7 K. When
T � 0.7 K the exact calculation of the relaxation time be-
gins to deviate from the approximate relation for �
4 pp
L( )
(see (77)) to larger values of time, so that when T �1 K,
the exact value of the relaxation time appears twice as
large as the approximate time �
4 pp
L( )
.
As a result, the rate �
1 which defines the coefficient
of the first viscosity is equal to the sum of the small-angle
rate � sm
1 and of rate � 4
1
pp
of the «exclusive» four-phonon
Four and three-phonon scattering in isotropic superfluid helium
Fizika Nizkikh Temperatur, 2009, v. 35, No. 3 273
processes which occur at larger angles between the scat-
tering phonons
� � �
� �1 1
4
1
sm pp . (78)
Here � sm
1 is given by Eq. (75) and � 4
1
pp
is obtained from
Eq. (59) where we omit the integration on small angles and
is rather well described by the analytical relation (77).
We see from Fig. 2, that at low temperatures (T < 0.5 K),
the main contribution to the viscosity is due to three-phonon
processes. At higher temperatures (T > 0.9 K), the situation
changes and the contribution of the exclusive four-pho-
non processes predominates over the contribution of
three-phonon processes. However in this temperature re-
gion, there is a considerable contribution to the viscosity
coefficient from phonon-roton interactions and it is neces-
sary to take these processes into account. The deviation of
the experimental points [24, 25] from the theoretical curve
at temperatures higher 0.7 K is due exactly to ignoring
these phonon-roton interactions. In the intermediate tem-
perature range (from 0.5 to 0.9 K) the contributions of
three-phonon and four-phonon relaxation have the same
order of magnitude and both processes should be taken into
account for calculating the first viscosity coefficient.
7. The attenuation of a beam of h-phonons by ther-
mal phonons
In the experiment [13] a pulse of h-phonons was pro-
pagated through superfluid helium over different path
lengths, to a detector. The amplitude of the pulse was
measured as a function of the temperature T of the liquid
helium, in the range 0.07 K 7 7T 0.21 K. The experimen-
tal data from Ref. 13 are shown in Fig. 3, the different sets
of points are for different path lengths of the h-phonon
signal.
The theory given above allows us to obtain an analyti-
cal description of the experiment [13]. As the phonons in
the h-phonon pulse have energies close to 10 K (see, for
example, Ref. 26), then our problem is reduced to finding
the relaxation time of an h-phonon, with energy 10 K, in
an isotropic phonon system. The liquid He II at a given
temperature T is an isotropic phonon system described by
the Bose distribution function for all momenta of phonons
up to pmax. The rate of relaxation �h is defined by
Eq. (24) with % �1 and integration limits given by the
relations
p
p p
p p p p3
1 2
1 2
2
low 3up�
�
� �, min( , )max , (79)
p p p2 0low 2up� �, ,max (80)
where pmax � 20 K is the maximal momentum of phonons
in liquid helium.
If we know the rate �h for this process, and taking into
account that h-phonons in a pulse have momentum close
to 10 K, we can calculate the attenuation coefficient A of
h-phonons in an isotropic phonon system using the
relation
A
l
v
h
c
�
1 exp( )He� . (81)
Here �h is the relaxation rate (24) of phonons with energy
of 10 K, in an isotropic phonon system at temperature T
and % �1, lHe is the path length of the h-phonon in the liq-
uid helium, and vc is the group velocity of the h-phonons.
The calculated attenuation coefficients for an h-pho-
non signal are shown in Fig. 3. The solid curves are for
the different h-phonon path lengths, three of which were
used in the experiment. We see from Fig. 3, that the calcu-
lated results from the analytic expressions (24) and (81)
are in good agreement with the experimental data points
of Ref. 13. In Ref. 13 the attenuation was calculated using
a computer simulation with a simplified model matrix ele-
ment. These calculations are not in quite such good
agreement with the experimental data.
8. Conclusion
In this paper, we have investigated four-phonon scat-
tering processes, in superfluid helium, when the phonon
systems is isotropic. The matrix element (9) for four-pho-
non processes is derived for the whole range of momen-
tum including the range where three-phonon processes
are allowed. The problem is unusual in this range as the
274 Fizika Nizkikh Temperatur, 2009, v. 35, No. 3
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, and A.F.G. Wyatt
3
2
1
0
0.1 0.2
0.5
1.0
T, K
4A
m
p
li
tu
d
e,
ar
b
.
u
n
it
s
Fig. 3. The amplitude of the h-phonon signal as a function of
temperature T. The solid curves show the results of the theoreti-
cal calculation: curve 1 corresponds to the propagation path
length lHe � 4 mm, curve 2 to 5 mm, curve 3 to 10.2 mm and
curve 4 to 15.7 mm. Experimental data from Ref. 13 at different
values of the h-phonon path length in liquid helium are shown
by the sets of points. Circles correspond to lHe � 5 mm, triangles
to 10.2 mm and squares to the path length equal to 15.7 mm.
matrix element has resonances when the angle between
the interacting phonons is small. We explain how to solve
this problem.
The 4pp scattering rate is found from the kinetic equa-
tion for phonons in liquid helium (24). The contribution
of different angular groups of phonons to the 4pp rate is
evaluated numerically (see Fig. 1). This analysis shows
that the contribution of small angle scattering to this 4pp
rate, is almost exactly the same as the 3pp rate. Then it is
shown analytically, that at small angles, the four-phonon
process can be represented as two consecutive three-pho-
non processes, and moreover, that the rate of such pro-
cesses is given by the rate of three-phonon processes. The
contribution of larger angles to the scattering rate, using
Eq. (24), gives the rate �4 pp of «exclusive» four-phonon
processes. These processes exclude those 4pp scatterings
that can be represented by 3pp scattering.
From the matrix element expressed in Eq. (9), we de-
rive the relaxation time (59) which appears in the expres-
sion for the coefficient of first viscosity of superfluid he-
lium. The evaluation of the rate shows that is the same as
the sum of the rate for traverse relaxation caused by
three-phonon processes (75), and the rate for exclusive
four-phonon processes (77) (see Fig. 2). It is shown that
at low temperatures (T < 0.5 K), the main contribution to
the viscosity is given by three-phonon processes, and
at higher temperature (T > 0.9 K), the contribution of
four-phonon processes is much larger than the contribution
from three-phonon processes. In the intermediate range of
temperatures 0.5 K < T < 0.7 K, both rates have the same
order of magnitude and both processes give a significant
contribution to the viscosity coefficient. We find generally
very good agreement with the measured values of the vis-
cosity in the temperature range where phonons are the
dominant excitation, see Fig. 2. At higher temperatures,
rotons begin to become increasingly important.
It is noted that the present analytical formulation auto-
matically includes all the processes in transverse relax-
ation, which previously was treated as a separate mecha-
nism in Refs. 8 and 9.
Finally we calculate the attenuation of an h-phonon
pulse propagating through liquid helium at a finite tem-
perature. The attenuation is due to the h-phonons being
scattered by the isotropic thermal phonons. The calcu-
lated the results are in good agreement with experimental
data (see Fig. 3).
We conclude that the analysis in this paper gives a very
good theoretical description of the dissipative relaxation
of liquid helium which is only slightly perturbed from iso-
tropic thermal equilibrium. We have shown that the the-
ory gives a very good quantitative explanation of the mea-
sured values of viscosity and the attenuation of beams of
high-energy phonons.
We are grateful to the EPSRC for supporting this work
through the grant EP/F 019157/1.
Appendix A: Calculation of the rate of four-phonon
processes �4pp
In the integrand of Eq. (23), without any loss of gener-
ality, we can choose the angle &1 as the computing origin
of angles& i . In this case �-functions in Eq. (23) can be re-
written as
� � & & &( ) ( cos cos cos )p� � �
�� � � �p p p p1 2 2 3 3 4 4
�
�� � �� & & &( sin sin sin )p p p2 2 3 3 4 4
� �
�( )|| || || ||p p p p1 2 3 4 , (82)
� � �( ) ( )� � �
(
1
1 2 3 4
c
p p p p . (83)
Here p pi i i i� �
2 2' ' , p pi i i|| ( )�
1 ' and ( is defined
by Eq. (26).
In order to integrate Eq. (23) over & 3 and & 4 we intro-
duce new variables
X p p� �� �3 3 4 4cos cos& & , (84)
Y p p� �� �3 3 4 4sin sin& & . (85)
Taking Eqs. (84) and (85) into account, Eq. (23) can be
rewritten as
�
�
4
1
10 5 7 2
1
02
1
1
pp
p
n
�
�
�
� ( )
( )
�
�
� � �
dp dp dp d d d d dXdY
p p X Y p p
2 3 4 2 3 4 2
3
2
4
2 2 2
3
2
44
' ' ' &
( �
�� 2 2)
� � �M n n n p p p� � �
2
2
0
3
0
4
0
2
3
3
3
4
31 1� � �( ) ( ) ( )( )
( ) ( ) ( )
p .
(86)
Here and below the integration is made so that radicands
are nonnegative.
As M � depends on cos& 3 and cos& 4 , then to make the
integration over & 3 and & 4 , it is necessary to solve the set
of equations
p p p p
p p
1 2 2 3 3 4 4
2 2 3 3
0� � � �
� �
�
�
cos cos cos ,
sin sin
& & &
& &
�
�
! �p4 4 0sin ,&
(87)
with regard to cos& 3 and cos& 4 .
The set of Eqs. (87) has two solutions which can be
written as
cos
( )&
3
) �
�
� �
)� � � � �
�
( cos )( ) sinp p A p p p R
Ap
1 2 2 3
2
4
2
2 2
32
& &
,
(88)
Four and three-phonon scattering in isotropic superfluid helium
Fizika Nizkikh Temperatur, 2009, v. 35, No. 3 275
cos
( )&
4
) �
�
�
�� � � � �
�
( cos )( ) sinp p A p p p R
Ap
1 2 2 3
2
4
2
2 2
42
& &�
.
(89)
Here
R p p A p p�
� � � �4 3
2
4
2
3
2
4
2 2( ) ,
A p p p p� � �� � � �1
2
2
2
1 2 22 cos .& (90)
Making the integration in (86) with a help �-functions
(82), (83) over & 3, & 4 and p4 , ' 4 we obtain Eq. (24).
Appendix B: Derivation of � sm
1
For the further integration of Eq. (74) it is convenient
to symmetrize Eq. (70) for1( )p1 :
1
1 1
( )
( )
~
( )
p
p p
1
1 1
2
�
�
. (91)
Here
~
( )1 p1 is derived from 1( )p1 by interchanging
q p� 2 in the integrand of Eq. (70). Nevertheless
~
( )1 p1 is
of course equal to1( )p1 .
As the integrand of Eq. (74) does not depend on&1 and
.1, we can rewrite Eq. (74) taking (91) into account as
�
2 1 1sm
B
c
k T
p p p p d p
� � ��1
5
4
5 1 1 1 1
3
1
15
16 ( )
{ ( ) ( )
~
( )} .
(92)
The integral in Eq. (92) can be represented as a sum of
three integrals
I p p d p p p d p p p d p� 2 1 1� � �� �� ( ) ( )
~
( )1 1
3
1 1 1
3
1 1 1
3
1.
(93)
We replace the variables 2( )p1 , 1( )p1 ,
~
( )1 p1 by their ex-
pressions. After that we replace q with p2 in the first inte-
gral, in the second and third integrals we replace q with
p3. As a result, the integration in all three integrals will be
over variables p1, p2 and p3. Further, in the second inte-
gral we rename p1 and p3 and in the third we rename p1
and p2. Thus in the second integral all angles will be
counted off p3 and in the third counted off p2. As a result
of all the above mentioned transformations, we obtain
I
W
�
� ��
( | , )
( )
p p p1 2 3
32 �
�
� � � � �
� � �
( ) ( )
sinh sinh sinh
p p p1 2 3 1 2 3
1 2 38
k T k T kB B BT
d p d p d p8 3
3
3
2
3
1,
(94)
8 �
�
�
�
�
�
� �
�
�
�
�
�
� �
�
!
p cp cp P1 1 1 2 2 2 12
3
2
3
2
� � .(cos )
�
�
�
�
�
�
�
"
#
$
�
�
�
�
�
�
�
�
!
� . �3 3 2 13 3 1 1 2
3
2
3
2
cp P p cp P(cos ) (cos ).13
�
�
�
�
�
�
�
�
�
�
�
�
"
#
$
�
� � . �3 3 2 2 2 23 2 1
3
2
3
2
3
cp cp P p(cos )
2
1cp
�
�
�
�
�
� �
�
!
�
�
�
�
�
�
�
�
�
�
�
�
�P cp cp P2 12 2 2 3 3 2 23
3
2
3
2
(cos ) (cos. � � . )
"
#
$
.
(95)
We make the integration in Eq. (94) on p2 and ' 3 with
a help of �-functions. As the integrand does not depend
on angles, then the integration over the remaining angles
is trivial. Finally for the rate � sm
1 we obtain Eq. (75).
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Four and three-phonon scattering in isotropic superfluid helium
Fizika Nizkikh Temperatur, 2009, v. 35, No. 3 277
|
| id | nasplib_isofts_kiev_ua-123456789-116992 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| language | English |
| last_indexed | 2025-12-01T16:36:24Z |
| publishDate | 2009 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Adamenko, I.N. Kitsenko, Yu.A. Nemchenko, K.E. Wyatt, A.F.G. 2017-05-18T17:37:47Z 2017-05-18T17:37:47Z 2009 Four and three-phonon scattering in isotropic superfluid helium / I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, A.F.G. Wyatt // Физика низких температур. — 2009. — Т. 35, № 3. — С. 265-277. — Бібліогр.: 26 назв. — англ. PACS: 47.37.+q https://nasplib.isofts.kiev.ua/handle/123456789/116992 We analyse the important role of four-phonon processes (4pp) in isotropic phonon systems of superfluid helium. The matrix elements and the rate of four-phonon processes are calculated. Special consideration is given to the 4pp in the momentum range where three-phonon processes are allowed. In this momentum range, we show that the 4pp scattering rate, at small angles, is equal to the scattering rate due to three-phonon processes. Then we show that the coefficient of first viscosity of superfluid helium is caused by two processes, the first is due to the transverse relaxation caused by many three-phonon processes and the second is due to four-phonon processes. The relaxation time that governs the viscosity is obtained from the sum of the rates from these two processes. The temperature dependence of the attenuation coefficient of a pulse of high-energy phonons in He II, due to scattering with thermal phonons, is also calculated. The theoretical results are compared with experimental data and found to be in good agreement. We are grateful to the EPSRC for supporting this work through the grant EP/F 019157/1. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Квантовые жидкости и квантовые кpисталлы Four and three-phonon scattering in isotropic superfluid helium Article published earlier |
| spellingShingle | Four and three-phonon scattering in isotropic superfluid helium Adamenko, I.N. Kitsenko, Yu.A. Nemchenko, K.E. Wyatt, A.F.G. Квантовые жидкости и квантовые кpисталлы |
| title | Four and three-phonon scattering in isotropic superfluid helium |
| title_full | Four and three-phonon scattering in isotropic superfluid helium |
| title_fullStr | Four and three-phonon scattering in isotropic superfluid helium |
| title_full_unstemmed | Four and three-phonon scattering in isotropic superfluid helium |
| title_short | Four and three-phonon scattering in isotropic superfluid helium |
| title_sort | four and three-phonon scattering in isotropic superfluid helium |
| topic | Квантовые жидкости и квантовые кpисталлы |
| topic_facet | Квантовые жидкости и квантовые кpисталлы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/116992 |
| work_keys_str_mv | AT adamenkoin fourandthreephononscatteringinisotropicsuperfluidhelium AT kitsenkoyua fourandthreephononscatteringinisotropicsuperfluidhelium AT nemchenkoke fourandthreephononscatteringinisotropicsuperfluidhelium AT wyattafg fourandthreephononscatteringinisotropicsuperfluidhelium |