Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II
The problem of the creation of high-energy phonons (h-phonons) by a pulse of low-energy phonons
 (l-phonons) moving from a heater to a detector in superfluid helium, is solved. The rate of h-phonon creation
 is obtained and it is shown that created h-phonons occupy a much smaller sol...
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| Опубліковано в: : | Физика низких температур |
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| Дата: | 2007 |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
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| Цитувати: | Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II / I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, A.F.G. Wyatt // Физика низких температур. — 2007. — Т. 33, № 5. — С. 523-537. — Бібліогр.: 22 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860241800284864512 |
|---|---|
| author | Adamenko, I.N. Kitsenko, Yu.A. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. |
| author_facet | Adamenko, I.N. Kitsenko, Yu.A. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. |
| citation_txt | Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II / I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, A.F.G. Wyatt // Физика низких температур. — 2007. — Т. 33, № 5. — С. 523-537. — Бібліогр.: 22 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | The problem of the creation of high-energy phonons (h-phonons) by a pulse of low-energy phonons
(l-phonons) moving from a heater to a detector in superfluid helium, is solved. The rate of h-phonon creation
is obtained and it is shown that created h-phonons occupy a much smaller solid angle in momentum space,
than the l-phonons. An analytical expression for the creation rate of h-phonon, along the symmetry axis of a
pulse, are derived. It allows us to get useful approximate analytical expressions for the creation rate of
h-phonons. The time dependences of the parameters which describe the l-phonon pulse are obtained. This
shows that half of the initial energy of l-phonon pulse can be transferred into h-phonons. The results of the
calculations are compared with experimental data and we show that this theory explains a number of experimental
results. The value of the momentum, which separates the l- and h-phonon subsystems, is found.
|
| first_indexed | 2025-12-07T18:30:39Z |
| format | Article |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5, p. 523–537
Creation of high-energy phonons by four-phonon
processes in anisotropic phonon systems of He II
I.N. Adamenko1, Yu.A. Kitsenko2, K.E. Nemchenko1,
V.A. Slipko1, and A.F.G. Wyatt3
1
Karazin Kharkov National University, 4, Svobody Sq., Kharkov 61077, Ukraine
2
Akhiezer Institute for Theoretical Physics National Science Center
«Kharkov Institute of Physics and Technology» National Academy of Sciences of Ukraine
1 Academicheskaya Str., Kharkov 61108, Ukraine
3
School of Physics, University of Exeter, Exeter EX4 4QL, UK
E-mail: a.f.g.wyatt@exeter.ac.uk
Received July 31, 2006
The problem of the creation of high-energy phonons (h-phonons) by a pulse of low-energy phonons
(l-phonons) moving from a heater to a detector in superfluid helium, is solved. The rate of h-phonon creation
is obtained and it is shown that created h-phonons occupy a much smaller solid angle in momentum space,
than the l-phonons. An analytical expression for the creation rate of h-phonon, along the symmetry axis of a
pulse, are derived. It allows us to get useful approximate analytical expressions for the creation rate of
h-phonons. The time dependences of the parameters which describe the l-phonon pulse are obtained. This
shows that half of the initial energy of l-phonon pulse can be transferred into h-phonons. The results of the
calculations are compared with experimental data and we show that this theory explains a number of experi-
mental results. The value of the momentum, which separates the l- and h-phonon subsystems, is found.
PACS: 67.70.+n Films (including physical adsorption);
68.08.–p Liquid–solid interfaces;
62.60.+v Acoustical properties of liquids.
Keywords: phonon–phonon interaction, liquid helium, anisotropic phonon system.
1. Introduction
The dispersion relation, which has an important role
for the processes of phonon-phonon interactions in
superfluid helium (He II), can be written as
� �� �� �cp p( ),1 (1)
where c is sound velocity, � and p correspond to phonon
energy and momentum, � �( p is a function which de-
scribes the deviation of the spectrum from linearity which
is small (| ( |� �p �� 1), but it nevertheless completely deter-
mines the possible interactions of phonons.
When p pc� , where ~pc �10 K (here and below
~ /p cp k B� ) at zero pressure, the function � 0 and the
dispersion is anomalous. Then the conservation laws of
energy and momentum allow processes which do not con-
serve the number of phonons. The three-phonon process
(3pp) is fastest of these processes and involves one
phonon decaying into two or two interacting phonons
combining into one.
When p pc the function � � 0 and the dispersion is
normal. Then the decay processes are prohibited by the
conservation laws of energy and momentum, and the fast-
est scattering is by four-phonon processes (4pp or 2 2
).
The three-phonon rate �3pp is much higher than the
four-phonon rate �4 pp
� �3 4pp pp (2)
and this inequality of rates causes the phonons in
superfluid helium to separate into two subsystems: a sub-
system 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 the equilibrium occurs relatively
quickly.
On the time scale of the concerned problem, the equilib-
rium in the subsystem of l-phonons occurs instantly and the
© I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt, 2007
phonon distribution of this anisotropic phonon system is
given by the Bose–Einstein distribution function with two
parameters, temperature and drift velocity [1,2].
The possibility of two-phonon subsystems with differ-
ent relaxation times leads to interesting phenomena in
anisotropic phonon systems of He II. Such systems are
created by a heater immersed in the liquid helium, which
is at such a low temperature that the thermal excitations
can be neglected. When a short current pulse, of duration
t p , is given to the heater, a phonon pulse in the liquid is
formed by fast three-phonon processes. This l-phonon
pulse moves in a «phonon vacuum», from the heater to the
detector, with a velocity very near to c [3].
In momentum space, the phonons in the pulse are
mainly located in a narrow cone with a solid angle
� p �� 1. The transverse dimensions of this phonon pulse,
near the heater, are about dimensions of the heater (typi-
cally 1
1 mm), and the longitudinal dimension, which is
determined by the duration of current pulse, is ct p : typi-
cally ct p � 24�m.
In experiments Ref. 4 a unique phenomenon was ob-
served: when one short current pulse was given to the
heater, two-phonon pulses, well separated in time, were
detected. The first comprised l-phonons, and the second,
h-phonons. From subsequent experiments (see Refs. 5, 6)
it was unambiguously demonstrated that the h-phonon
pulse was not injected by the heater but was created by the
l-phonon pulse during its motion from the heater to the de-
tector. The theory of this surprising phenomenon, when a
cold l-phonon pulse with temperature close to 1 K, creates
high-energy phonons with energy � � 10 K, was given in
Refs. 7–9.
In Refs. 7–9 it was shown that l-phonon pulse is in equi-
librium due to the fast 3pp. The h-phonons are created
within the l-phonon pulse by slow 4pp with rate �4 pp . The
h-phonons have a group velocity u h � 189 m/s that is
smaller than the velocity of the l-phonon pulse c � 238 m/s.
The difference in these velocities and the relatively weak in-
teraction between h- and l-phonons (see Eq. (2)) leads to the
h-phonons leaving the l-phonon pulse through its rear wall
and forming a pulse of noninteracting h-phonons which ar-
rives at the detector after the l-phonon pulse.
However the theory created in Refs. 7–9 gives no ex-
planation for some of the phenomena observed in experi-
ments. One of these is that the measured solid angle in mo-
mentum space occupied by the h-phonons is much smaller
than the solid angle occupied by l-phonons. Also it was
difficult to explain how the l-phonon signal amplitude de-
pended on heater power because the occupied solid angle
in momentum space was constant during the propagation.
The theory predicted that all pulses cool to about the same
temperature due to the creation of h-phonons; the final
temperature is where the h-phonon creation rate is negli-
gible. So if the temperature and occupied solid angle are
the same then the signal on the detector from the pulses of
different powers should be the same. But experiments
showed that the l-phonon signal amplitude increased with
increasing of heater power, so it was proposed that larger
solid angles were created at higher powers [10].
In this paper the exact local-equilibrium distribution
function, for phonons in a pulse, is used. It allows us to
obtain the angular dependence of the h-phonon creation
processes, and to explain the above mentioned experi-
mental phenomena. In Refs. 7–9 a more simple approxi-
mation of the distribution function was used, the cone ap-
proximation, in which a cone of occupied states is cut
from an isotopic distribution at tempeature T p .
Some of results presented here have been published
[11]; here we expand and extend those results, and from
this foundation obtain several new results. In particular,
the important case of creation and decay of high-energy
phonons with momentum directed along the anisotropy
axis of phonon system, is investigated for the first time.
For this case, an exact expression for the creation and de-
cay rates in an anisotropic system is obtained, together
with the relaxation rate of the isotropic phonon system.
Hence we derive an explicit analytical expression for the
rate of h-phonon creation along the anisotropy axis. We
also construct an analytical approximation for the rate in
range of angles where the collision integrals are not negli-
gible. This allows us to analyse the rate dependences on
all the parameters of the problem. It is shown that the
h-phonon creation rate strongly depends on the momen-
tum pd , which separates the l- and h-phonon subsystems.
From a comparison of the theory, developed in this paper,
with experimental data, we conclude that p pd c� .
2. The kinetic equation for the distribution function
of h-phonons
In this section we consider the four-phonon processes
(4pp) which determine the creation rate of h-phonons in
anisotropic and isotropic l-phonon systems. The conser-
vation laws of energy and momentum for 4pp can be
written as
� � � �1 2 3 4 1 2 3 4� � � � � �, .p p p p (3)
Hereinafter, the phonon with energy �1 and momentum p1
is the h-phonon and the other three phonons are l-phonons.
This is the type one 4pp scattering [12] and is the most impor-
tant process for h-phonon creation.
The kinetic equation describing the rate of change of
the distribution function, n n( )p1 1� , due to 4pp can be
written as
dn
dt
N Nb d
1
1 1� �( ) ( ),p p (4)
524 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
where N b ( )p1 and N d ( )p1 are, respectively, the rates of
increasing and decreasing numbers of h-phonons in the
state with momentum p1, due to collisions. N b d, ( )p1 can
be written as
N b d, ( )p1 �
� �W n d p d p d pb d( , ; , ( ( .,p p p p p1 2 3 4
3
2
3
3
3
4�� �� � �� �
(5)
Here W ( , ; , )p p p p1 2 3 4 defines the probability density of
the process (3), the �-functions correspond to the conser-
vation of energy and momentum; � � � � �� � � � �1 2 3 4
and p p p p p� � � � �1 2 3 4 . The factors nb d, are
n n n n nb � � �
3
0
4
0
2
0
11 1
( ) ( ) ( )
( )( ),
n n n n nd � � �1 2
0
3
0
4
0
1 1
( ) ( ) ( )
( )( ), (6)
where n n
l l
( ) ( )( )
0 0� p is the equilibrium distribution
function of l-phonons which are considered to be instan-
taneously in equilibrium due to three-phonon processes.
The equilibrium distribution function of l-phonons as
intoduced in Refs. 1, 2 can be written as
n
k Tl
l l
B
( )( ) exp0
1
1p
p u
� �
�
� �
�
� �
�
�
�
�
�
�
��
. (7)
The drift velocity u can be written as
u N� �c ( ),1 ! (8)
where N is the unit vector directed along the direction of
the total momentum of the l-phonon system; this defines
the anisotropy axis of the phonon system. This equation
defines !, the anisotropy parameter. In weakly aniso-
tropic systems, ! is close to unity. For strongly ani-
sotropic systems, which corresponds to the experiments
[4–6,13,14], ! �� 1.
In all further calculations it will be more convenient to
us to use the expression
n p
cp
k T
l l
l
B
l l l
( )( , ) exp ( )0 1" � ! " " !� � � �
�
�
��
�
�
�� �
�
�
�
�
�
�1
, (9)
where " l l lp� �1 p N / . Equation (9) was obtained by
substitution of Eqs. (1) and (8) into (7).
In an earlier description of anisotropic phonon systems we
used the approximate distribution function (see Refs. 7–9)
which can be written as
n k Tp l p l l B p
( )
/( ) ){exp( ) }0 11p � � � �#�$ $ � . (10)
This distribution function has a simple physical mean-
ing and includes all the parameters of anisotropic phonon
systems: temperature T p and the anisotropy which is de-
fined by the cone angle $ p . The values of T p and $ p can
be estimated from the experimental data. However the de-
scription of the creation of h-phonons, at any angle to the
anisotropy axis (see, for example, [1,2]), can only be
made using the exact local-equilibrium distribution func-
tion (7) for the l-phonons. As that is the purpose of this pa-
per, we shall use only the parameters! and T from now on.
The relationships between the parameters !, T and $ p , T p
come from the equating energy and momentum in the two
descriptions, and were obtained in Refs. 1, 2.
The interaction of phonons in superfluid helium is de-
scribed by the Landau Hamiltonian (see, for example,
[15]), which we write as
� � � � .H H V Vph � � �0 3 4 (11)
Here �H 0 is the Hamiltonian of noninteracting phonons.
The terms �V3 and �V4 describe the interaction of phonons
to the third and fourth orders of small deviations of a sys-
tem from an equilibrium state, respectively.
The probability density of four-phonon process fol-
lowing Refs. 9, 15, 16, can be written as
W V Hfi( , ; , ) | |
( )
p p p p1 2 3 4
2 2
6
2 1
2
�
%
%� �
. (12)
Here V is the volume of system and Hfi is the amplitude
of four-phonon process which is obtained by second or-
der perturbation theory on �V3 and first order perturbation
theory on �V4 with a help of standard procedures (see, for
example, [9,15–18]). Hence
H
V V
E E
Vfi
iQ
�
& '& '
�
� &( p p Q Q |p p
p p p
Q
3 4 3 3 2
3 4 4
, | � | | �
, | � |1,
1, p2',
(13)
where Q is an intermediate state with energy EQ, and Ei is
the energy of initial state.
The interaction of phonons with momenta p1 and p2 to
create phonons with momenta p3 and p4 , has six interme-
diate states I–VI in which phonons have momenta
I. II. III.p p p p p p p p p p1 2 2 3 1 3 2 4 1 4� � �; , , ; , , ;
IV V.. , , ; , , ;p p p p p p p p1 3 2 3 1 4 2 4� �
VI. p p p p p p1 2 3 4 1 2, , , , .� � (14)
From expressions (13) and (14) we obtain
H
p p p p
V
Mfi �
1 2 3 4
8) � , (15)
where ) �145 kg/m 3 is the density of He II,
M M M M M M M M� � � � � � � �( ) ( ) ( ) ( ) ( ) ( )1
13
2
14
3
23
3
24
3 5
4
(16)
is the matrix element for 4pp, which consists of seven terms,
six of which correspond to the six intermediate states (14),
and the seventh, to first order of perturbation theory on �V4 .
Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II7
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 525
The superscripts show the number of phonons in the inter-
mediate state. We can write the matrix elements as
M u( ) ( )1 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 , (17)
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 , (18)
M u w4
24 1� � �{( ) }, (19)
M u
13
3 1 3
1 3 1 3
1 3 1 1 3 3 1 32 1
( )
( )�
� �
� � � �
�
�
� �
�
� � �
n n n n n n
� � � �� �( )2 1 2 4 2 1 3 4 1 3u n n n n n n . (20)
Other terms of expression (16), i.e., M
23
3( )
, M
14
3( )
, M
24
3( )
can be obtained from M
13
3( )
by permuting the corres-
ponding subscripts. Here n pi i ip� / , � � � �i i� p ,
u c�
( / )) ( / ) .* * �c ) 2 84 is the Gr��uneisen constant and
w � ( / )( / ) .) )2 2 2 0 188c c* * � . The matrix elements are
given in detail in the Appendix A.
We note, that first three terms of Eq. (16) are resonant:
when � �( p � 0 their denominators can vanish, which
leads to an essential divergence of the matrix element.
These three terms give the main contribution to Eq. (16).
The rest of the terms give small contributions of the same
order of magnitude.
We define the rate of the 4pp scattering by
�1 1
3
2
3
3
3
4( )p �
� d p d p d p
� �W n n n( , ; , ) ( ( ( )( ) .
( ) ( ) ( )
p p p p p1 2 3 4 2
0
3
0
4
0
1 1� �� � �� �
(21)
Then from Eq. (5), the expression for N d can be written as
N nd � 1 1� . (22)
For N b we use the equality
n n n
k T
n n
B
3
0
4
0
2
0 1 1
2
0
1 1
( ) ( ) ( ) ( )
( ) (� � �
��
�
��
�
�
�� �exp
� p u
3
0
4
0
1
( ) ( )
)( ),� n
(23)
which follows from the Eqs. (3) and (7). Then we can
write
N
k T
nb
B
� �
��
�
��
�
�
�� �exp
�
��1 1
1 11
p u
( . (24)
For short pulses that were used in experiments
[13,14], n1 is much less than the unity. (We shall only con-
sider short pulses from now on.) Therefore the relation
N
k Tb
B
( )sh
exp� �
��
�
��
�
�
��
�
�1 1
1
p u
(25)
is always satisfied in short pulses.
To see the important role of "1 in determining the cre-
ation rate of h-phonons, we rewrite the exponent in (25):
�
�
� � � � �
( )
( )
�
� ! " " !1 1 1
1 1 1
p u
k T
cp
k TB B
. (26)
For h-phonons � 1 is small and negative: when
10 111K � �~p K then 0 0 023 �� . . For the anisotropic
phonon systems studied experimentally, the energy densi-
ties in the liquid helium and the angular distribution of the
l-phonons are consistent with ! � 0.02 and temperature
T � 0.04 K. These values of T � 0 041. K and ! � 0 02. in the
distribution function (9) correspond to the same energy
density and momentum density as the values T p �1 K and
$ p � +12 in the cone distribution function (10). Although !
and � are small, "1 is not necessarily small; it can vary
from 0, when p1 is along the anisotropy axis, to 2, when p1
is antiparallel to the anisotropy axis. When"1 is nearly zero
then the modulus of the exponent is not large, and the expo-
nential term in (25) is not so small, and the creation rate is
not small. However when " !1 and | |� , then the expo-
nent is large and negative, and the exponential term in (25)
is very small, which makes the creation rate negligible.
In order to calculate the creation rate N b and the decay
rate N d , which appear in the kinetic equation (4), we must
find the rate �1 which is done in the next section.
3. The rate of h-phonons creation and decay in the
anisotropic l-phonon system
3.1. The general expression for the rate �1
To derive the general expression for the rate �1, we re-
write the expression (21), taking into account (12)–(15),
in spherical coordinates
�
% )
, " , " , "1
1
11 5 7 2 2 2 2 3 3 3 4 4 4 2
3
3
3
4
2
� �
p
dp d d dp d d dp d d p p p
�
3
� �M n n n� � �
2
2
0
3
0
4
0
1 1� �� � �( ( ( )( )
( ) ( ) ( )
p , (27)
where " i i ip� �1 p N / .
Without any restriction on generality, we can choose
the angle ,1 as the computing origin of angles , i . In this
case the �-functions can be written as
�� � � , , , �p� � � � �
- - - -( cos cos cosp p p p1 2 2 3 3 4 4
� �
- - -� , , , �( sin sin sinp p p2 2 3 3 4 4
� � �� ( )|| || || ||p p p p1 2 3 4 . (28)
526 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
� � � � .( ( )� � � � � �
1
1 2 3 4
c
p p p p , (29)
where p pi i i i- � �2 2" " , p pi i i|| ( )� �1 " ,
. � � � �� � � �p p p p3 3 4 4 1 1 2 2. (30)
We note that the value ., when p1 is an h-phonon, is al-
ways positive (. 0).
In order to make the integration with respect to , 3 and
, 4 we introduce new variables
X p p� �- -3 3 4 4cos cos ,, ,
Y p p� �- -3 3 4 4sin sin, , . (31)
Then the expression (27) can be written as
�
% )
" " " ,
1
1
10 5 7 2
2
3
3
3
4
3
2 3 4 2 3 4 2
2 4
�
p p p p dp dp dp d d d d dXdY
p�
3
2
4
2 2 2
3
2
4
2 2
- - - -� � � �
�
p X Y p p( )
� �M n n n� � �
2
2
0
3
0
4
0
1 1� �� � �( ( ( )( ).
( ) ( ) ( )
p (32)
Hereinafter the integration is made so that the radicands
are not negative.
As M � depends on cos, 3 and cos, 4 , to make the inte-
gration it is necessary to solve the system of the 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 ,, (33)
with respect to cos, 3 and cos, 4 .
The system of equations (33) has two solutions which
can be written as
cos
( cos ) ( ) sin( ),
, ,
3
1 2 2 3
2
4
2
2 2
32
/ - - - - -�
� � � /p p A p p p R
Ap -
,
(34)
cos
( cos )( ) sin( ),
, ,
4
1 2 2 3
2
4
2
2 2
42
/ - - - - -�
� � �p p A p p p R
Ap
�
-
,
(35)
where
A p p p p� � �- - - -1
2
2
2
1 2 22 cos ,, (36)
R p p p p p p p p� � � � � �- - - - - - - -4 23
2
4
2
1
2
2
2
3
2
4
2
1 2 2
2( cos ) .,
(37)
At first we integrate with respect to X and Y , and then
with respect to p4 and " 4 with the help of the �-functions.
As a result we have
� " " ,1 1 2 3 2 3 2
2
3
3
3
4
2
�
�Kp dp dp d d d
p p p
R
� � �� �{ } ( )( ),( ) ( )
( ) ( ) ( )
M M n n n2 2
2
0
3
0
4
0
1 1 (38)
where
K
c
�
1
210 5 7 2% )�
, (39)
M M( )
( ) ( )
(cos cos , cos cos ),/
/ /� � �� , , , ,3 3 4 4
(40)
p p p p4 1 2 3� � � �., (41)
"
" " " .
4
1 1 2 2 3 3
4
�
� � �p p p
p
. (42)
Further integration cannot be precisely made analyti-
cally because of the complicated integrand expression.
Therefore we first consider the important case of "1 0�
and secondly present some dependences of the rate �1 ob-
tained from Eq. (38).
3.2. The rate �1 for h-phonons moving
along the anisotropy axis
This case is important as it is possible to obtain an ex-
plicit analytical expression for �1. In Ref. 9 the rates of
four-phonon processes were calculated when "1 0� using
the approximate cone distribution function (10).
When "1 0� , the expression (38) becomes much sim-
pler as firstly the dependence of matrix element on, 3 and
, 4 vanishes, and secondly the dependence of the
integrand expression on , 2 disappears. So the integration
over , 2 can be easily made analytically. Hence we find
� %
" "
" " " "" "
1 1
2 3 2 3
3
0
3 3 3
0
2
1 1
�
� �
�
�
�
��Kp
dp dp d d
( ) ( )
�
�
�
�
�
�
�
�
�
�
�
1
2 1 2 2
1 2
2
1
2p p
p p
"
( )
�
� ��p p p
p p
M n n n2
3
3
2
4
2
1 2
0 2
2
0
3
0
4
01 1 1( ) ( )( ) ,
( ) ( ) ( )
�
"
(43)
where
M M M�
" " "1 0
1 10 0
�
� �� � � �( ) ( )( ) ( ) , (44)
" "
3
1 0
/
� �
�
/ � � ��
�
b p p p
a
"
"
" " " " " "1
1
0
1 2
2
3 2 2 2 2 2 2
0
8 2
2
( )( )( )
|
min max
|
.
(45)
Here
Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II7
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 527
b p p p p p p p p p p
" " .1 0
1 2
2
3 2
2
2 3 1 4 2 3 18 4 2 2 2
� � � � � � �(
� � � � �2 4 24
2
2 3 1 2 4p p p p p. . �" . � .( ( ) , (46)
a p p p
p p
p p
" "
1 0
3
2
1 2
2 1 2 2
1 2
2
4 1
2� � � � �
�
�
�
�
�
�
�
�
�
( )
( )
, (47)
"
. .
2
3 4
1 2
2 2
2
min
( )
,�
� �p p
p p
(48)
"
. .
2
3 4
1 2
2 2
2
max
( )( )
.�
� �p p
p p
(49)
We note that Eq. (43) for �1 is very important as it not
only describes the rate of h-phonon creation at $1 0� but
also gives the relaxation in isotropic phonon system, i.e.,
when ! �1, because in the isotropic case there is no de-
pendence on $1 and every direction is identical to $1 0� .
Following Ref. 9, we can replace M �
"1 0�
with M which
can be written as
M u m m m u m� � � � � �4 1 4 12
2 3 4
2( ) ( ) ( ) . (50)
Here
m
p p
p p
2
1 2
2
1 2 2 2
1
�
�
�
( )
,
" 0
(51)
m
p p
p p
3
1 3
2
1 3 3 3
1
�
�
�
( )
,
" 0
(52)
m
p p
p p
4
1 4
2
1 4 4 4
1
�
�
�
( )
,
" 0
(53)
where 0 i are functions dependent on the dispersion:
02
1 2
1 2
1 2 1 2�
�
� � �
p p
p p
f f f( ) ,
03
1 3
1 3
3 1 3 1�
�
� ��
p p
p p
f f f( ),
(54)
0 � � �4
1 4
1 4
4 1 4 1�
�
� � ��
p p
p p
f f f f p pi i i( .) and (55)
For all further calculations it is necessary to have an
analytical approximation for the function � �( p . In this
section and further on, we shall use the approximation of
function � �( p , which was obtained in [9]
� �
1 2
(
, ,
p
c u
c
p
p
p
p
p p
p p
p
c u
c
c
c c
f
c c
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
2
1
2
2
2
2
�
3
�
4
56
7
89
� �
�
�
:
:
�
:
:
u
c
p p p p pc
c f
2
( ) , .�
(56)
Here ~pc �10 K, ~ .p f � 8 26 K, ~p� � 27 K, 1 �176. ,
2 �113. , u pc p pc
� * * ��� / | /189 m s, k u cB c3 �/
� ( / )k cB ( / )|* ��
2 2� *p p pc
– 19.8 m/(s;K).
Further integration can be made by replacing, in the
slowly varying functions of momenta and angles in the
integrand of Eq. (43), the momenta and angles by their
typical values. As a result we have
�
�
1 0
2 1 1 2
3 2
1
2
�
�
�
��
�
�
�� � � �
K T
cp
k T
m p p p p
p
B
c cexp ( , )
�
�
�
��
�
�
��
� � � �p p
p p
p p p pc
c
c c1
1
2
1 1
52
12 6
/
( ) ( ( ))< <
<
. (57)
Here
K
u
c u
k
c
c
B
0
4
4 3 7 2
2
1
2
�
�
�
�
�
�
�
�
�
( )
( )
,
% )�
(58)
m m
p p
p p
p
p p p
� �
�
�
�
�
�
��
�
�
��" . "
.
2
1 2
1 2
3 4
2
1 1 2
,
( )
,, (59)
<
� ! �
��
� �
�
c
k T
t
B
t
( )
. .2 1
2 0 02, where (60)
The result (57) allows us to analyze the dependences of
the rate �1 on p1, T , ! and the parameters of liquid helium.
So for example, the fast decrease of �1 with increasing p1
is mainly caused by the exponent. We see from (57) that
the rate is proportional to the minus one third power of the
dispersion as�
0. Moreover the form of result (57) will
be used below when obtaining an approximation for the
rate �1 in the range of not too-large "1, which is mainly re-
alized in a pulse. We note that Eq. (57) is in good agree-
ment with the results of exact numerical calculation of the
rates from Eq. (43) near T � 0 041. K and ! � 0 02. . At other
values of ! and T the numerical agreement with the rates
calculated from Eq. (43) are not so good, but the
dependences are qualitatively the same.
We note, that in obtaining the result (57), we used the
fact that the main contribution, in integration over p3, is
due to phonons with momenta close to pc , which we have
assumed separates the l- and h-phonon subsystems. This
leads to the conclusion that the rate of h-phonon creation
strongly depends on the numerical value of momentum
which separates the l- and h-phonon subsystems. The pos-
sibility that this momentum, pd , which separates the l-
and h-phonon subsystems, is not equal to pc will be
discussed in section 6.
3.3. Dependence of the rate �1 on all parameters
The dependences of �1 on p1, "1, T and !, can only be
obtained by numerical integration of Eq. (38). The results
of this are shown in Fig. 1: in Fig. 1,a is shown the angular
528 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
dependence of the rate �1 at different values of p1 and
fixed values of T � 0 041. K and ! � 0 02. , which correspond
to the conditions of the experiments [13,14]. We see that
there is a maximum when ~p1 13� K. The appearance of
this maximum at ~p1 10� K is caused by several reasons:
firstly it follows from the conservation laws of energy and
momentum that the four-phonon processes are prohibited
if the angle between the momenta of phonons p1 and p2 is
less than
" $ .12 12
1 2
1 2
1min mincos� � =
�p p
p p
. (61)
Secondly the distribution function of phonons, in a pulse,
has a sharp maximum at $2 0� . Thus, the majority of
phonons in a pulse have momentum p2 directed along the
anisotropy axis of the system. However at $1 0� phonons
with momentum p1 cannot interact with phonons having
momentum p2 directed along the anisotropy axis because
there is a minimum angle, $12min , for interaction between
them. Thus, when $1 0� , the interaction of phonons with
momentum p1 with most of the phonons of the pulse is
prohibited by the conservation of energy and momentum.
With increasing angle $1, an increasing number of phon-
ons in the pulse interact with the h-phonon. This leads to
the increasing rate in the initial part of the curve, up to
$1 40�
+, see Fig. 1,a. The following decrease in the rate
is caused by the decreasing value of the squared matrix
element with increasing angle $1. At these larger angles,
the h-phonon with momentum p1 can interact with almost
all phonons of the pulse. The squared matrix element is
approximately constant for $ $1 1 80� = +
ME and even
could be slightly increasing with the growth of $1 in this
range of angles. This is due to the cancellation of terms in
(16). Then the slow growth of the rate at these angles is
determined by the factor R in the integrand of Eq. (38)
and by the angular dependence of matrix element, see
Fig 1,a. We note that it is important to retain all terms in
M � , if only one term is considered there are serious qual-
itative errors; for example for M ( )1 , M
13
3( )
or M
14
3( )
, the
rate �1 decreases with growth $1, instead of increasing.
The situation is different when the p1 is far from pc .
From (61) we see that angle $12min increases with larger
p1 and becomes greater than the angle $1ME . Then the rate
rises monotonically with increasing $1 and there is no
range of angles where the rate decreases. This behavior is
apparent when ~p1 13� K, see Fig. 1,a.
In Fig. 1,b is shown the momentum dependence of the
rate �1 at different values of $1 (0 � , 30 � , 60 � , 90 � and
180 + corresponding to curves 1–5, respectively) with the
same values of T and ! as in Fig. 1,a. It can be seen, that
with increasing angle, the rate �1 is higher in more of the
momentum range, and at$1 180� + the dependence on mo-
mentum is actually absent. In Fig. 1,b we also see that the
curves cross each other. The physical reasons for these be-
haviors are the same as those discussed above. We note
that momentum dependence of �1, when $1 0� , was ob-
Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II7
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 529
10
8
a1
2
3
4
5
6
10
7
10
6
10
5
10
4
10
3
10
2
10
1
0 20 40 60 80 100 120 140 160 180
b
10 11 12 13 14
1'
1
2
3
4
5
10
8
$ , degree
1
cp /k , K1 B
1'
2'
3'
10
3
10
2
10
1
10
7
10
6
10
5
10
4
� 1
,
s–
1
� 1
,
s–
1
Fig. 1. a — The rate �1 is shown as a function of the angle $1
between p1 and the propagation direction calculated from Eq.
(38) for different values of ~p1 equal to 10 (1), 10.5 (2), 11
(3), 12 (4), 13 (5), and 14 (6) K. The dashed lines are
corresponding approximate rates calculated from Eq. (62).
b — The dependence of the rate �1 on momentum cp kB1 / cal-
culated from Eq. (38) for different $1 equal to 0 � (1), 30 � (2),
60 � (3), 90 � (4), and 180 + (5). The dashed lines are the corre-
sponding approximate rates calculated from Eq. (62). All
calculations had T � 0041. K and ! � 002. .
0.020 0.025 0.030 0.035 0.040
T, K
1
2
3
4
5
10
7
10
6
10
5
10
4
10
3
� 1
,
s–
1
Fig. 2. The temperature dependence of the rate
� "1 1 1 0( , )p pc� � for different values of the anisotropy parame-
ter ! equal to 0.01 (1), 0.015 (2), 0.02 (3), 0.025 (4), and 0.03
(5). The dashed lines represent the same dependences calculated
from Eq. (62).
tained in Ref. 9 with the approximate distribution func-
tion (10); the rate is similar to that presented here.
In Fig. 2 the dependence of �1 on T for different values of
! is shown for ~p1 10� K and $1 0� . From Fig. 2 it can be
seen, that the rate increases with both increasing temperature
and with decreasing values of the anisotropy parameter !.
Such behavior is completely determined by the dependence
of distribution function n
2
0( )
on the parameters ! and T .
To solve a number of problems it is convenient to use
analytical expression for �1. However it is impossible to
obtain an analytical expression for � " !1 1 1( , , , )p T from
the fivefold integral (38). In this case it is useful to have
an approximate expression for � " !1 1 1( , , , )p T .
Using the data from the numerical calculation of the
dependences of the rate on p1 and "1, which was made
with a help of Eq. (38) at different values of ! and T , then
least squares fitting and taking into account (57), it is pos-
sible to obtain a numerical approximation for the rate �1
on all parameters of the problem, which can be written as
� " ! !1 1
1 1 1
221 39 10 1 1 2 1( , , , ) .
( ) ( )
p T T
p p� ; ;
�
�
�
�
�
�
�
�
� �
e )
10 24 22
4 1 1
1 1
3 1
5 11
�
1
!
11 "T
p
p
p
T pc p
. ( )
( )
( ( ) , (62)
where T should be substituted in Kelvins and 1 i should
be written as
1 11 1
1
2 1
110 37 6 06 3 11 2 53( ) . . ( ) . .p
p
p
p
p
pc c
� � � �, , (63)
1 3 1
116 778 18 456( ) . .p
p
pc
� � , (64)
1 4 1
1
0 668
24 29 95( ) exp .
.
p
p p
p
c
c
�
��
�
��
�
�
��
�
�
�
�
�
�
�
�
, (65)
1 5 1
16 62 5 71( ) . .p
p
pc
� � . (66)
The approximation (62) can be used for 0 01 0 06. .� �!
and 0 016 0 06. .K K� �T . We note that all experiments
were carried out in this range of values of ! and T . Equa-
tion (62) is valid in the momentum range of ~p1 from ~pc up
to 11 K, where the rate N
b
( )sh
is significantly above zero.
We notice, that approximation (62) is applicable in the
range of angles where N
b
( )sh
is large. This is due to the fact
that �1 in N
b
( )sh
is multiplied by exp[ ( ]� ��1 p u) /1 k TB ,
which very quickly decreases with increasing angle, and
makes N
b
( )sh
relatively small at angles $1 10� +, see
Fig. 3,a. The comparison of the numerical calculations of
the rate with the approximation (62) one can see in Figs. 1
and 2 where the dashed curves were obtained with a help of
Eq. (62). We see that for the given range of variables, the
approximation (62) is rather good.
4. Angular distribution of l- and h-phonons
The angular distribution of l- and h-phonons can be de-
scribed by the probability density of the angular distribu-
tion of phonons. The probability density of created
h-phonons as a function of "1 is defined by expression
W T
N p T p dp
E
h
p
p
b
c( , , )
( , , , )
�
max
( )
" !
� " !
%
1
1 1 1 1
2
1
2 34
�
� sh
� h
, (67)
where
� ( , , , )
max
( )
E d N p T p dph
p
p
b
c
� � �
1
4 2 3 1
0
2
1 1 1 1
2
1
%
" � " !
�
sh
(68)
is the total energy of h-phonons, created in unit time and
unit volume, N p T
b
( )
( , , , )
sh
1 1" ! is defined by Eq. (25),
and the upper limit of integration over momentum is cho-
sen to be p pcmax .�14 . The value of the integrals are not
sensitive to the exact value of this upper limit as long as it
is well above pc : even if p pcmax .�11 then the results
530 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
a1
2
3
4
5
0 2 4 6 8 10 12 14 16 18 20
b1
2
3
10 11 12 13 14
$ , degree
1
cp /k , K1 B
b
N
,
s
(s
h
)
–
1
10
5
10
4
10
3
10
2
10
1
10
0
10
5
10
4
10
3
10
2
10
1
10
0
N
,
s
(s
h
)
–
1
Fig. 3. a — The creation rate of h-phonons, N
b
( )sh in short pulses,
is shown as a function of the angle $1 between p1 and the propa-
gation direction for different values of cp kB1 / equal to 10 (1),
10.5 (2), 11 (3), 12 (4) and 13 (5) K. b — The dependence of
N
b
( )sh on momentum cp kB1 / for different values of $1 equal to
0�(1), 6+(2), and 12+ (3). All curves have ! � 002. and T � 0041. K.
change very little. The main contribution to the integrals
is due to h-phonons with momentum close to pc because
the function N p
b
( )
( )
sh
1 rapidly decreases when p1 in-
creases, see Fig. 3,b.
Similarly we can also introduce the probability density
of l-phonons as a function of " l
W T
n p T p dp
E
l l
l
p
l l l l l
l
c
( , , )
( , , , )
( )
" !
� " !
%
�
�
0
0 2
2 34 �
, (69)
where
E d n p T p dpl l l
p
l l l l l
c
� � �
1
4 2 3
0
2
0
0 2
%
" � " !
�
( )
( , , , ) (70)
is the total energy of l-phonons in unit volume of the
pulse, and n p T
l l l
( )
( , , , )
0 " ! is defined by Eq. (9).
The results of numerical calculations of the depend-
ence ofWl andWh on $ "l h l h, ,( )� �arccos 1 , obtained from
Eqs. (67) and (69) at ! � 0 02. and T � 0 041. K, which are
values for t � 0, are given in Fig. 4. We see that Wh is a
considerably sharper function angle thanWl . For ! � 0 061.
and T � 0 058. K, which are values after 42 �s, we see that
W tl ( )� 42�s is even wider than W tl ( )� 0 .
The sharpness of functionsWl h, is defined by the angu-
lar width of the corresponding distributions which are
given by relation
" " "l h l h l h l hW d, , , ,� �
0
2
. (71)
Figure 5 shows the dependences of $l h, � arccos ( ),1�" l h
on T , at different values of !. It can be seen that the in-
equality $ $h l� is always satisfied. Thus the created
h-phonons always occupy a narrower cone than the
l-phonons. This characteristic was observed in experi-
ments [5] and was called «the concentration of h-phonons
near the anisotropy axis of the system». Experiments
showed, that h-phonons are in a cone with a cone angle
about 4+ , while l-phonons occupy a cone with a cone angle
close to 12 +. We notice, that results of calculations with
values of parameters which are typical for experiments
[13,14], T � 0 041. K and ! � 0 02. , give $l =
+12 and
$h = +5 5. near the heater, and near the bolometer we have
$ �l t( ) .� = +42 16 8s and $h is not expected to change
much. These calculated values after 42 �s are similar to
experimental values. Thus, the reasons for the observed
concentration of h-phonons near the anisotropy axis of the
system, can now be explained as follows.
The rapidly decreasing value ofWh with increasing "1,
is related to the angular dependence of N
b
( )sh
which is
shown on Fig. 3. It can be seen in Fig. 3,a, that N
b
( )sh
very
quickly decreases with increasing $1. This decrease is de-
termined by the multiplier exp )[ ( / ]� ��1 1p u k TB in ex-
pression (25), which has a sharp maximum at $1 0� , see
Eq. (26). While in Fig. 3,a the presence of the exponential
multiplier leads to the fast decrease of the function
N
b
( )
( )
sh $1 while function � $1 1( ) is increasing. In Fig. 3,b
the presence of the exponential multiplier makes the mo-
mentum dependence of N p
b
( )
( )
sh
1 more flat than the de-
Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II7
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 531
W
120
100
80
60
40
20
0
0 3 6 9 12 15 18 21 24 27 30
$, degree
W (t = 0 )h
�s
W (t = 42 s)�1
1W (t = 0 s)�
Fig. 4. The angular dependence of the probability density
W t( )� 0 for the h- and l-phonon distributions at ! � 002. and
T � 0041. K obtained from Eqs. (67) and (69), respectively.
Also W tl( � 42�s) is shown for ! � 0061. and T � 0058. K which
are the corresponding values at t � 42�s. Note the angular
distribution is narrower for h-phonons than for l-phonons.
a
3
2
1
T, K
0.015 0.020 0.025 0.030 0.035 0.040
$
,
d
eg
re
e
1
–
13
12
11
10
9
b
1
2
3
T, K
0.015 0.020 0.025 0.030 0.035 0.040
–
6.0
5.8
5.6
5.4
5.2
5.0
4.8
4.6
4.4
$
,
d
eg
re
e
h
Fig. 5. The angular widths of the l- and h-phonon distributions,
$l (a) and $h (b), are shown as functions of temperature T, for
different values of ! equal to 0.01 (1), 0.02 (2), and 0.03 (3).
pendence �1 1( )p . This can be explained by the fact that
the exponential exp )[ ( ]� ��1 1p u / k TB quickly increases
with increasing momentum p1, at fixed values of $1, due
to the increase in the absolute value of the negative func-
tion � �( p1 with increasing p1, see Eq. (26).
An important characteristic of h-phonon creation is the
relation E El h/ � . It is the characteristic time it takes for the
energy of the created h-phonons to become equal to the
initial energy of l-phonon pulse, if the parameters ! and T
of l-phonon pulse are assumed constant. This characteris-
tic time shows the intensity of the h-phonon creation
process.
Comparing this time with the time of the pulse propa-
gation
> �p � = ; ��10
238
4 2 10 425mm
m s
s s
/
. , (72)
we can determine whether l-phonons have time to trans-
fer a significant part of their energy to h-phonons during
their propagation from the heater to the detector.
Figure 6 shows the dependence of E El h/ � on temperature
T for different values of !. One can see that when ! � 0 01. ,
the h-phonon creation is rapid at temperatures T 0 02. K,
when! � 0 02. atT 0 032. K, and when! � 0 03. there is prac-
tically no h-phonon creation at any temperature.
5. Evolution of a short l-phonon pulse caused
by h-phonon creation
A short phonon pulse is such that all the created
h-phonons are lost from the l-phonon pulse well before
equilibrium between l- and h-phonons is established. The
kinetic equation (4) can be written as
dn
dt
N
b
1 � ( )sh
. (73)
We multiply the right and left part of Eq. (73) by �1 and
integrate over all d p3
1
32/ ( )%� . As a result we have
d
dt
Eh
h
�
� � , (74)
where �Eh is defined by expression (68), and
�
%
" �h
p
p
d dp p n
c
� � �
1
4 2 3 1
0
2
1 1
2
1 1
�
max
. (75)
From conservation of energy it follows that
d
dt
dE
dt
h l�
� � , (76)
where El is defined by expression (70). From (74) and
(76) we have
� �
dE
dt
El
h
� . (77)
Now we multiply the right and left part of Eq. (73) by
p z1 and in a similar way obtain
� �
dP
dt
Pl
h
� , (78)
where
P d n p T p dpl l l
p
l l l l l
c
� �� �
1
4
1
2 3
0
2
0
0
3
%
" " ! "
�
( )
( , , , ) ( ) ,
(79)
� ( , , , ) ( )
( )
max
P d N p T p dh b
p
p
c
� �� �
1
4
1
2 3 1
0
2
1 1 1
3
1
%
" " ! "
�
sh
p1.
(80)
Thus, we have obtained two equations (77) and (78)
that must be solved with respect to functions ! �( t and T t( )
with the initial conditions ! � !( t � �0 0 and T t T( )� �0 0.
The solution of combined equations (77) and (78) is dis-
cussed in detail in the Appendix B.
Figure 7,a represents the temperature T t( ) dependence
on time calculated with T0 0 041� . and ! 0 0 02� . , 0.025,
0.03, 0.035 (curves 1–4, respectively). In Fig. 7,b is
shown the anisotropy parameter ! �( t as a function of time,
for ! 0 0 02� . and T0 0 025� . , 0.030, 0.036, 0.041 (curves
1–4, respectively). It can be seen from Fig. 7 that
h-phonon creation leads not only to increasing ! but also
to increasing T ; nevertheless the total energy density of
l-phonon pulse decreases with time (see Fig. 8).
In Fig. 8 the time dependence of the l-phonons energy
density E t E t T tl l( ) ( ( , ( ))� ! � and the l-phonon pulse an-
gular width $ $ ! �l lt t T t( ) ( ( , ( ))� are shown. The broaden-
ing of a pulse with time, seen in Fig. 8, is due to the con-
532 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
T, K
3
2
1
10
10
10
10
10
10
0.015 0.020 0.025 0.030 0.035 0.040
>p
3
1
–1
–3
–5
–7
E
/E
,
s
l
h.
Fig. 6. The characteristic time for l-phonon energy to be con-
verted to h-phonon energy is given by the ratio E El h/ � is
shown as a function of temperature T for different values of !
equal to 0.01 (1), 0.02 (2), and 0.03 (3). The dashed line
shows the time for a pulse to travel 10 mm, i.e. 42 �s, from
Eq. (72). We see more energy is converted when ! is small,
i.e., most anisotropic.
servation of momentum when h-phonons are created in
the l-phonon pulse. The created h-phonons, as can be seen
in Fig. 4, have a sharper angular dependence and that is
why they carry away momentum mainly parallel to the
anisotropy axis. To compensate for this, the l-phonon an-
gular distribution must broaden.
For other values of ! 0 and T0, all the dependencies are
qualitatively the same as shown in the figures.
We introduce
? �(
( , ) ( ( , ( ))
( ,
t
E T E t T t
E T
l l
l
�
�! ! �
!
0 0
0 0 )
, (81)
which shows the fraction of the l-phonon energy which
transforms into h-phonons.
In Fig. 9 the function ? �( t is shown at ! 0 0 02� . for five
values of the initial temperature T0 0 041� . , 0.036, 0.031,
0.026, and 0.021 K (curves 1–5, respectively). It can be
seen from Fig. 9 that the process of h-phonon creation is
more rapid at higher values of the initial temperature. The
dependence ? �( t shown in Fig. 9 is close to that obtained
in [7,8], which was derived with the approximate cone
distribution function (10), and with corresponding values
of T p and " p .
In Fig. 10,a the dependences E tl ( )� 0 (curve 1) and
E tl ( )� 42 �s (curve 2) on the initial temperature T0, with
! 0 kept constant, are shown. We see, that with increasing
of T0, the energy density E tl ( )� 42 �s also increases. In
Fig. 10,b similar dependences of the angular widths
$l t( )� 0 (curve 1) and $ �l t( )� 42 s (curve 2) are shown.
All the curves shown on Fig. 10 were calculated with the
value of ! 0 0 02� . which is a typical measured value.
From Fig. 10 we see that with increasing initial tempera-
ture, we expect wider and higher energy pulses on the detec-
tor. Such behavior was observed in experiments when the
initial power was increased, see Figs. 3 and 7 in Ref. 10.
Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II7
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 533
a
1
2
3
4
0 5 10 15 20 25 30 35 40
b
4
3
2
1
0 5 10 15 20 25 30 35 40
T
,
K
t, s�
t, s�
0.060
0.056
0.052
0.048
0.044
0.06
0.05
0.04
0.03
0.02
Fig. 7. a — The temperature T t( ) is shown as a function of time,
calculated with T0 0041� . K and ! 0 002� . (1), 0.025 (2), 0.03
(3), 0.035 (4). b — The anisotropy parameter ! �(t is shown as a
function of time, calculated with ! 0 002� . and T0 0025� . (1),
0.030 (2), 0.036 (3), 0.041 (4) K.
8
7
6
5
4
t, s�
0 5 10 15 20 25 30 35 40
$
,
d
eg
re
e
l
17
16
15
14
13
12
E
,
J/
m
3
l
Fig. 8. The l-phonon energy density E tl( ) and the angular
width of the l-phonons in momentum space $l t( ) are shown as
functions of propagation time, calculated with initial values
of ! 0 002� . , T0 0041� . K.
1
2
3
4
5
0.6
0.5
0.4
0.3
0.2
0.1
t, s�
0 5 10 15 20 25 30 35 40
Fig. 9. The energy density lost by the l-phonons, due to
h-phonon creation, relative to the initial l-phonon energy den-
sity, ? �(t , is shown as a function of time calculated with
! 0 002� . for different values of T0 equal to 0.041 (1), 0.036
(2), 0.031 (3), 0.026 (4), and 0.021 (5) K.
6. The momentum which separates the
l- and h-phonon subsystems
The calculations made above are based on the fact that
phonons in superfluid helium separate into two subsystems
according to their relaxation rate: l-phonon subsystem in
which equilibrium occurs quickly and h-phonon subsystem
in which equilibrium occurs slowly. The basis for such di-
vision is the strong inequality (2). In this paper, as well as
in previous ones (see Refs. 7, 8), it was supposed, that the
momentum pd separating l- and h-phonon subsystems is
equal to pc , after which the spontaneous decay of phonons
is forbidden. However, according to Refs. 1, 19 the conser-
vation laws of energy and momentum allow three-phonon
processes ( )1 2
only up to momentum p pc1 2 4 5
� /
(at the saturated vapour pressure ~p1 2
� 8.94 K with our
parametrization of the dispersion curve). In the momentum
range from p1 2
to pc processes of one phonon decaying
into more than two phonons are allowed. So, for example,
one phonon decay into three is allowed by conservation
laws up to momentum p pc1 3 9 10
� / ( ~p1 3
� 9.49 K),
into four — up to 9.7 K, into five — up to 9.81 K and so on
up to ~pc � 10.0 K.
At present there are no publications of calculations of
the rates of the mentioned above processes, although 1 to
3 is submitted [20]. Hence there is the question of which
subsystem phonons with momenta
p p pc1 2
� � (82)
should be assigned. If the rates of one phonon decay into
three, four, etc. appear much higher than the rate �1 then
these phonons should be part of the l-phonon subsystem,
and so p pd c� . Otherwise these phonons should be part
of the h-phonon subsystem and in this case p pd c� .
The answer to this question is important as the numeri-
cal value of the rate �1, for phonons moving at small an-
gles to the anisotropy axis, is very sensitive to the value of
pd when p pd c� . Such sensitivity is because the main
contribution in Eq. (43) for �1 is given by phonons with
momentum p3 close to the upper limit of integration pd .
This reason for this is now explained. The distribution
function n
2
0( )
exponentially decreases with increasing " 2.
As a result, the main contribution to the integration over
" 2 is given by phonons with " "2 12� min (see (61)). Hence
the integration over " 2 leads to the following factor in the
integrand of (43)
exp expmin�
�
�
��
�
�
�� = �
�
�
��
�
�
��
cp
k T
c
k TB B
2
12"
.
, (83)
which for values of T expected in experiments, quickly
increases with decreasing ..
It is not difficult to check that ., determined by Eq.
(30), monotonically decreases with increasing p3 and is
minimal (and the exponent function is accordingly maxi-
mal) when p pd3 � . So if T � 0 041. K, p pd c� then for
~p1 � 10.001 K and the typical value of ~p2 � 1.58 K when
~p3 � 10 K, the exponent of the exponential function
� � � ; �c k TB. / .5 35 10 3, and when ~p3 � 8.95 K, we have
� � �c k TB. / .4 67; i.e., the value of the exponential func-
tion increases by two orders of magnitude. We see that re-
534 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
1
2
3
cp /k , K1 B
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
8.5 9.0 9.5 10.0 10.5 11.0 11.5
� 1
,
s–
1
Fig. 11. The momentum dependence of the rate �1 at $1 0� for
different values of ~pd equal to 8.94 (1), 9.49 (2), and 10 (3)
K. These dependences were obtained from Eq. (43) with
T � 0041. K and ! � 002. .
a
0T , K
0.020 0.025 0.030 0.035 0.040
1
2
8
6
4
2
0
b
l$
,
d
eg
re
e
2
1
18
16
14
12
10
0.020 0.025 0.030 0.035 0.040
0T , K
l
E
,
J/
m
3
Fig. 10. a — The initial l-phonon energy density, E tl ( )� 0
(curve 1), and the energy density after 42 �s, E tl ( )� 42�s
(curve 2), are shown as functions of the initial temperature T0.
b — The initial angular width of the l-phonons in momentum
space, $l t( )� 0 (curve 1), and after 42 �s, $ �l t s( )� 42 (curve
2), are shown as functions of the initial temperature T0. All
calculations were made with ! 0 002� . .
ducing pd , and correspondly reducing the top limit of in-
tegration over p3, leads to an exponential decrease of the
rate �1( )p1 , at small $1.
In Fig. 11 the momentum dependence of the rate
�1 1( )p is shown at three different values of ~pd equal to
8.94, 9.49, and 10 K (curves 1–3, respectively) at $1 0� .
These dependences were obtained from expression (43) at
values of T � 0 041. K and ! � 0 02. . From Fig. 11 one can
see, that with decreasing ~pd from 10 to 9.5 K, the rate
�1 1 10( ~ )p � K decreases an order of magnitude.
The physical causes for such a strong decrease are the
strong anisotropy of l-phonon subsystem and the conser-
vation laws (3) which make process 2 2
forbidden for
angles $ $12 12� min . Such a high sensitivity to the choice
of pd is absent in isotropic phonon systems. So when
! �1, the calculation of the rate with a help of formula
(43), the rate �1 in isotropic phonon system, only changes
1.5 times, when pd goes from pc to p1 2
.
So in order to answer the question of the value of pd in
superfluid helium, it is necessary to compare results of the
theory presented in this paper with experimental data [21],
where the shape and the arrival time of h-phonon signal cre-
ated by l-phonon pulse, at the bolometer, were investigated.
It follows from the Eq. (25), that the exponential factor in
N
b
( )sh
, cuts off at large angles $1 (see Fig. 3). In connection
with this the rate �1, which is in N
b
( )sh
(see Eq. (25)) is of
great interest especially at small values of $1. We note that
in the case of small$1 at different values of pd , the rate �1 is
described by curves similar to those in Fig. 11 for $1 0� .
From Fig. 11 we see that changing pd , shifts the curves
�1 1( )p to lower momenta by amounts equal to the differ-
ences in the values of pd . Therefore if p pd c� , the maxi-
mum h-phonon signal should be formed by phonons with
momentum ~p1 = 10.0 K and phonons with ~p1 � 8.95 K
should be absent. When p pd �
1 2 the maximum h-phonon
signal should be due to phonons with ~p1 =8.95 K and phon-
ons with momentum ~p1 �10 K should be practically absent.
The group velocity vgr of phonons depends on momen-
tum. From the data in Ref. 22, at the saturated vapour
pressure, v pgr K( ~ . )� �8 95 207 m/s and v pgr K( ~ )� �10
= 189 m/s. Therefore the position of the maximum of the
h-phonon signal, which in experiments [6,21] is measured
with good accuracy, allows us to understand which phonons
form this maximum and accordingly the correct value of pd .
Experiments [6,21] show that the maximum of the h-phonon
signal moves with a speed of 186.4 m/s, which corresponds to
a group velocity of phonons with ~p1 � 10.15 K.
Calculations similar to those presented in the previous
section, with p pd c� , give the shape of h-phonon signal
close to that observed in Refs. 6, 21, and are distinctly dif-
ferent from the shape with smaller values of pd . From this
section we see that in superfluid helium, the momentum
pd separating the l- and h-phonon subsystems is equal to
~ .pc �10 0 K, as it was supposed in this paper and earlier
ones, Refs. 7, 8.
7. Conclusion
The general expression (38) for the rate �1 which de-
scribes the decay (22) and creation (24) of high-energy
phonons in a pulse of low-energy phonons moving from a
heater to the detector, was found. Also the exact analytical
expression for the rate �1 when $1 0� (see (43)) was
found. This expression also describes the relaxation in
isotropic phonon systems, i.e., when ! �1. From Eq. (43)
we derive a simple and explicit analytical expression for
the rate �1 for the case when h-phonons move along the
anisotropy axis ("1 0� ) (see (57)). This allowed us to get
an analytical approximation for �1 in the most interesting
range of angles "1 (see (62)).
Starting from Eq. (38), the rates were a numerically
calculated. This allowed us to obtain the momentum and
angular dependences of the rate �1 (see Fig. 1). Also the
dependences of the rate �1 on the parameters of l-phonon
pulse were obtained (see Fig. 2). It was shown, that angu-
lar dependence of the rate �1 has a maximum at ~p1 13� K
(see Fig. 1,a). The physical reasons of this maximum were
given and the detailed analysis of such behavior of the rate
�1 was made.
The calculation of the rate �1 allowed us to get the an-
gular distribution of created h-phonons and show that
they occupy a solid angle, in momentum space, which is
much smaller than the solid angle for l-phonons (see
Figs. 4 and 5). This result explains the observation that the
created h-phonon are concentrated near the anisotropy
axis of the system [5].
Starting from the kinetic equation (73), the problem of
the evolution of a short l-phonon pulse, caused by the cre-
ation of h-phonons, was solved. For this, the principal
consideration was the angular dependence of the collision
integral N
b
( )sh
in (73), which quickly decreases with in-
creasing angle (see Fig. 3). The physical reasons of such
behavior were analyzed, and it was shown that it is con-
nected with the exponential factor in N
b
( )sh
(see Eq. (25)).
We investigated how the h-phonon creation rate de-
pended on the parameters of the l-phonon pulse. It was
shown that h-phonon creation not only increased the ani-
sotropy parameter ! of l-phonon pulse but also increased
the temperature T (see Fig. 7). However the energy den-
sity of l-phonon pulse decreased as it must (see Fig. 8).
For the conditions used in the experiments [13,14], the
theory showed that up to 50% of initial energy of
l-phonon pulse could be transferred into h-phonon
creation (see Fig. 9).
Also the dependence of the energy El and the angular
width $l of the l-phonon pulse at detector, on the initial
temperature T0, were investigated (see Fig. 10). It was
shown, that with increasing initial temperature there will
Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II7
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 535
be wider and higher energy density pulses at the detector.
Such phenomenon were observed in experiments [10]
when different heater powers were used.
It was shown that the rate �1, for small angles $1,
strongly depends on momentum pd which separates the l-
and h-phonon subsystems (see Fig. 11). From a compari-
son of the present theory with experimental data we see
that p pd c� .
Acknowledgments
We express our gratitude to EPSRC of the UK (grant
EP/C 523199/1) for support of this work.
Appendix A.
Expressions for the matrix elements
Making the dot products in (17), (18), (20) and taking
into account the conservation laws of energy and momen-
tum (3) we have
M u
p p
p p
( )
( )
1 1 2
1 2 1 2
12
12
1 2 12
1 2
2
2
2
1
2
�
� �
� �
�
�
�
�
�
�
�
� � �
"
"
"
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
�
�
�
2
2
1
2
34
34
3 4 34
3 4
2
u
p p
p p
"
"
"
( )
�
�
�
�
�
�
�
(A.1)
M u
p p
p p
( )
( )
5 1 2
1 2 1 2
12
12
1 2 12
1 2
2
2
2
1
2
� �
� �
� �
�
�
�
�
�
�
� � �
"
"
"
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
�
�
2
2
1
2
34
34
3 4 34
3 4
2
u
p p
p p
"
"
"
( )
�
�
�
�
�
�
�
�
, (A.2)
M u
p p
p p
13
3 1 3
1 3 1 3
13
13
1 3 13
1 3
2
2
1
2
( )
( )
�
� �
� �
�
�
�
�
�
�
� � �
"
"
"
2
24
24
2 4 24
2 4
2
2
2
1
2
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
� u
p p
p p
"
"
"
( )
�
�
�
�
�
�
�
�
�
, (A.3)
M u
p p
p p
24
3 1 3
1 3 1 3
13
13
1 3 13
1 3
2
2
1
2
( )
(
� �
� �
� �
�
�
�
�
�
�
� � �
"
"
"
) ( )2
24
24
2 4 24
2 4
2
2
2
1
2
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
u
p p
p p
"
"
"
�
�
�
�
�
�
�
�
�
�
, (A.4)
M u
p p
p p
14
3 1 4
1 4 1 4
14
14
1 4 14
1 4
2
2
1
2
( )
( )
�
� �
� �
�
�
�
�
�
�
� � �
"
"
"
2
23
23
2 3 23
2 3
2
2
2
1
2
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
� u
p p
p p
"
"
"
( )
�
�
�
�
�
�
�
�
�
, (A.5)
M u
p p
p p
23
3 1 4
1 4 1 4
14
14
1 4 14
1 4
2
2
1
2
( )
(
� �
� �
� �
�
�
�
�
�
�
� � �
"
"
"
) ( )2
23
23
2 3 23
2 3
2
2
2
1
2
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
u
p p
p p
"
"
"
�
�
�
�
�
�
�
�
�
�
, (A.6)
where
"
"
34
3 4
2
1 2
2
1 2 12
3 4
2
2
�
� � � �( ) ( )
,
p p p p p p
p p
(A.7)
"
"
23
1 4
2
2 3
2
1 4 14
2 3
2
2
�
� � � �( ) ( )
,
p p p p p p
p p
(A.8)
536 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, and A.F.G. Wyatt
"
"
24
1 3
2
2 4
2
1 3 13
2 4
2
2
�
� � � �( ) ( )
,
p p p p p p
p p
(A.9)
" " " " " " " " " , ,1 1 1 1 1
2 2
12 2i i i i i i� � � � � � �cos ( ),
(A.10)
Appendix B.
Solution of combined equations (77) and (78)
To solve combined equations (77) and (78) it is conve-
nient to write them as
� �
� � � �
�
�
:
�
:
dE
dt
E
d
dt
E cP E cP
l
h
l l h h
� ,
( ) ( � � ).
(B.1)
The system of equations (B.1) can be solved for � (! �t and
� ( )T t . This system of two equations with initial conditions
! � !( t � �0 0 and T t T( )� �0 0 can be solved numerically
and the functions ! �( t and T t( ) found. However it is always
convenient to carry out calculations with simple analytical
expressions, which approximate the dependences of the
functions in (94) on ! and T . Such expressions can be intro-
duced as products of power functions and exponents. The
form of these functions was obtained from (68), (70), (79),
(80). The exponents of the power function and the coeffi-
cients in exponential functions, were calculated by a least
squares fitting.
For the intervals corresponding to the conditions in ex-
periments, 0 01 0 06. .� �! and 0 016 0 06. .K K� �T , ap-
proximations of the functions contained in (B.1) can be
written in SI system of units as
E T Tl
T T( , ) .
. .
. .! !
!
�
� � �3713 36
0 503 0 0135
1
107 3 25e , (B.2)
E T cP Tl l
T T( , ) ( , ) .
. .
.! ! !
!
� �
� � �445 856
0 435 0 00112
1
0 785e T 3 24. ,
(B.3)
� ( , ) .
. .
. .E T Th
T T! !
!
� ;
� � �8 51 1013
1135 0 00648
1
0 831 4 75e , (B.4)
� ( , ) � ( , ) .
. .
.E T cP Th h
T T! ! !
!
� � ;
� � �2 59 1012
1118 0 0158
1
1e 15 5 96T . .
(B.5)
Having substituted (B.2)–(B.5) into (B.1) and made
the differentiation, we obtain the system of two equations
with respect to �! and �T . The numerical solution of this sys-
tem gives ! ! �( , ,t T0 0 and T t T( , , )! 0 0 .
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Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II7
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 537
|
| id | nasplib_isofts_kiev_ua-123456789-127809 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T18:30:39Z |
| publishDate | 2007 |
| 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. Slipko, V.A. Wyatt, A.F.G. 2017-12-28T12:54:17Z 2017-12-28T12:54:17Z 2007 Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II / I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, A.F.G. Wyatt // Физика низких температур. — 2007. — Т. 33, № 5. — С. 523-537. — Бібліогр.: 22 назв. — англ. 0132-6414 PACS: 67.70.+n, 68.08.–p, 62.60.+v https://nasplib.isofts.kiev.ua/handle/123456789/127809 The problem of the creation of high-energy phonons (h-phonons) by a pulse of low-energy phonons
 (l-phonons) moving from a heater to a detector in superfluid helium, is solved. The rate of h-phonon creation
 is obtained and it is shown that created h-phonons occupy a much smaller solid angle in momentum space,
 than the l-phonons. An analytical expression for the creation rate of h-phonon, along the symmetry axis of a
 pulse, are derived. It allows us to get useful approximate analytical expressions for the creation rate of
 h-phonons. The time dependences of the parameters which describe the l-phonon pulse are obtained. This
 shows that half of the initial energy of l-phonon pulse can be transferred into h-phonons. The results of the
 calculations are compared with experimental data and we show that this theory explains a number of experimental
 results. The value of the momentum, which separates the l- and h-phonon subsystems, is found. We express our gratitude to EPSRC of the UK (grant
 EP/C 523199/1) for support of this work. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Квантовые жидкости и квантовые кpисталлы Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II Article published earlier |
| spellingShingle | Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II Adamenko, I.N. Kitsenko, Yu.A. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. Квантовые жидкости и квантовые кpисталлы |
| title | Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II |
| title_full | Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II |
| title_fullStr | Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II |
| title_full_unstemmed | Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II |
| title_short | Creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of He II |
| title_sort | creation of high-energy phonons by four-phonon processes in anisotropic phonon systems of he ii |
| topic | Квантовые жидкости и квантовые кpисталлы |
| topic_facet | Квантовые жидкости и квантовые кpисталлы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/127809 |
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