On the suprathermal distribution in an anisotropic phonon system in He II
The equation that describes the suprathermal distribution of high-energy phonons (h phonons) created in anisotropic phonon systems in superfluid helium is obtained. The solution of this equation enables the derivation of the value of suprathermal ratio S as the ratio of the actual distribution to th...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | On the suprathermal distribution in an anisotropic phonon system in He II / I.N. Adamenko, K.E. Nemchenko, A.F. G. Wyatt // Физика низких температур. — 2003. — Т. 29, № 1. — С. 16-21. — Бібліогр.: 28 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860017318904463360 |
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| author | Adamenko, I.N. Nemchenko, K.E. G.Wyatt, A.F. |
| author_facet | Adamenko, I.N. Nemchenko, K.E. G.Wyatt, A.F. |
| citation_txt | On the suprathermal distribution in an anisotropic phonon system in He II / I.N. Adamenko, K.E. Nemchenko, A.F. G. Wyatt // Физика низких температур. — 2003. — Т. 29, № 1. — С. 16-21. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | The equation that describes the suprathermal distribution of high-energy phonons (h phonons) created in anisotropic phonon systems in superfluid helium is obtained. The solution of this equation enables the derivation of the value of suprathermal ratio S as the ratio of the actual distribution to the Bose-Einstein one, its dependences on the momentum of the h phonons, the anisotropy parameters, and the temperature of the low-energy phonons from which the h phonons are created. We analyze this equation to obtain an estimate of the value of the ratio between the h-phonon number density in anisotropic and isotropic phonon systems and draw conclusions about the dependence of S on the relevant parameters.
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Fizika Nizkikh Temperatur, 2003, v. 29, No. 1, p. 16–21
On the suprathermal distribution in an anisotropic
phonon system in He II
I. N. Adamenko1, K. E. Nemchenko1, and A. F. G. Wyatt2
1 V. N. Karazin Kharkov National University, 4 Svobody Sq., Kharkov 61077, Ukraine
2 School of Physics, University of Exeter, Exeter EX4 4QL, UK
E-mail: adamenko@pem.kharkov.ua
Received July 25, 2002, revised September 24, 2002
The equation that describes the suprathermal distribution of high-energy phonons (h phonons)
created in anisotropic phonon systems in superfluid helium is obtained. The solution of this equa-
tion enables the derivation of the value of suprathermal ratio S as the ratio of the actual distribu-
tion to the Bose—Einstein one, its dependences on the momentum of the h phonons, the anisotropy
parameters, and the temperature of the low-energy phonons from which the h phonons are created.
We analyze this equation to obtain an estimate of the value of the ratio between the h-phonon
number density in anisotropic and isotropic phonon systems and draw conclusions about the de-
pendence of S on the relevant parameters.
PACS: 67.40.Db, 67.40.Fd
1. Introduction
For a phonon system in superfluid 4He (He II) the
rates of kinetic processes are determined by the un-
usual form of the phonon energy—momentum depend-
ence. At zero pressure the phonon dispersion curve in
He II bends upwards [1–3], and this causes sponta-
neous decay of phonons with energies � �� �c 10 K
[4,5]. For these phonons, the energy and momentum
conservation laws allow processes in which the pho-
non numbers in the initial and final states do not equal
one another. The fastest process among these is the
three-phonon process (3pp), in which one phonon de-
cays to two phonons or two phonons interact to create
one phonon. The rate of these three-phonon processes
was obtained in [6,7], in the two extreme limits; and
the general case was calculated in [8].
At higher energies the phonon dispersion curve
bends downwards, and at � �� c phonon spectrum be-
comes nondecaying. Here the most rapid process is the
four-phonon process (4pp), in which there are two
phonons in the initial and final states.
The rate of three-phonon processes �3pp is obtained
using Landau’s Hamiltonian in first-order perturba-
tion theory, and the rate of four-phonon �4pp pro-
cesses is determined by second-order perturbation the-
ory [9–11]. This is quantitatively evaluated and
confirmed by experiment [12]. The difference between
the orders of perturbation theory results in the strong
inequality � �3 4pp pp�� . Thus the phonons of super-
fluid 4He form two subsystems: one of low energy (l
phonons) with � �� c, which very quickly attains equi-
librium; and the other of high-energy phonons (h
phonons), which goes to equilibrium relatively slowly.
The angles between the phonons which take part in
3pp is small due to the smallness of the deviation of
the energy—momentum dispersion from linearity,
� � cp. Thus in isotropic phonon systems, when all di-
rections in momentum space are uniformly occupied,
equilibrium is not attained isotropically in the short
term because interactions involve only phonons within
a limited solid angle. Thus, all thermodynamic para-
meters of the l-subsystem become functions of angle
[11]. In the long term, four-phonon processes and dif-
fusion in angular space bring about complete equilib-
rium. This quasidiffusion is determined by fast
three-phonon interactions involving small angles
[13–15].
The situation is quite different in highly ani-
sotropic phonon systems. Here the momenta of all
phonons are in a narrow cone with solid angle � p ,
which has a value close to the typical angle for
three-phonon processes. Such strongly anisotropic
phonon systems have been created in liquid 4He
[16–20]. This pure and isotropic superfluid can have
such a low temperature that one can neglect the exis-
tence of thermal excitations. Low-energy phonons are
© I. N. Adamenko, K. E. Nemchenko, and A. F. G. Wyatt, 2003
injected by a heater and then propagate along the di-
rection normal to the surface of the heater. In momen-
tum space all the phonons lie in a narrow cone of solid
angle � p � 0125. sr.
In such experiments [16–20] the unusual phenome-
non of the creation of high-energy phonons was
observed. These h phonons are created by a pulse of l
phonons, which has a temperature an order of magni-
tude less than the energy of the high-energy phonons
that it creates. The theory of this unique phenomenon
was proposed in [21,22]. A further development of the
theory [23–28], shows that in strongly anisotropic
phonon systems there exists an asymmetry between
processes of creation and decay for the h phonons.
Such an asymmetry causes the distribution function of
h phonons in the anisotropic phonon system to be S
times greater than that in the Bose—Einstein distribu-
tion. Moreover, it is possible to have S �� 1. Using the
notation of [23], we will call such an unusual distri-
bution of h phonons a suprathermal distribution, and
the parameter S, the suprathermal ratio.
In this paper we derive an exact equation that al-
lows us to calculate the suprathermal ratio and to
determine its dependences on momentum, anisotropy
parameter, and temperature. We analyze the contribu-
tion of all possible processes that lead to the formation
of suprathermal distributions in anisotropic phonon
systems. From this equation we derive an estimate of
the average value of the suprathermal ratio S: the ra-
tio of the number density of h phonons in the
anisotropic and isotropic phonon systems.
2. The inequalities for determining
of the stationary distribution function
of phonons in an anisotropic phonon system
The attainment of equilibrium in the subsystem of
h phonons can be described by the kinetic equation for
the distribution functions, which can be written as
dn
dt
N Nb d
1
1 1� �( ) ( )p p , (1)
where
Nb( )p1 �
� � � �W n n n n
b
( , | , ) ( )( )p p p p1 2 3 4 3 4 1 21 1
�
� � �
� � � �( )1 2 3 4
� � �
( )p p p p1 2 3 4
3
2
3
3
3
4d p d p d p (2)
is the number of phonons with momentum p1 created
per unit time as the result of four-phonon (4pp) inter-
actions;
N W n n n nd
d
( ) ( , | , ) ( )( )p p p p p1 1 2 3 4 1 2 3 41 1� � � �
�
� � �
� � � �( )1 2 3 4
� � �
( )p p p p1 2 3 4
3
2
3
3
3
4d p d p d p (3)
is the number of phonons that decay per unit time;
W W( , | , ) ( , | , )p p p p p p p p1 2 3 4 3 4 1 2� defines the tran-
sition probability density for 4pp processes, which
have been calculated [25]; �b and � d each represent
a set of three solid angles, one for each momentum
over which the function is integrated, i.e., �bi and
� di (i = 2,3,4). These maximum angles are deter-
mined by the anisotropy of the phonon system and the
angles in the 4pp interactions. In the isotropic case
� �bi di� . In relations (1)–(3) and below we con-
sider p p k /cc B c1 � � � (i.e., phonon «1» is the h1
phonon), while the other three phonons can be l
phonons or h phonons. The stationary distribution
function is determined by the equality
N Nb d� . (4)
For isotropic phonon systems � �bi di� , and
equality (4) gives:
n n n n n n n n1 2 3 4 3 4 1 21 1 1 1( )( ) ( )( )� � � � � . (5)
The solution of this equation is the Bose—Einstein
distribution
� �� �n /Ti i
( ) exp0 1
1� �
�
� . (6)
In an anisotropic phonon system, when the initial
phonons are inside a narrow cone with solid angle
� p � 4
, the result differs from (5) and (6). In this
case the limits of integration in (2) and (3) with re-
spect to angular variables are defined by the inequali-
ties
� �i p� . (7)
Here i = 3,4 for creation processes, and i = 2 for de-
cay processes. The angular variables for the final
phonons have no such restrictions.
Such asymmetry for the initial and final states re-
sults in the inequality � �b d� . In this case in
anisotropic phonon systems the equality (4) cannot be
satisfied by solution (5), and the Bose—Einstein di-
stribution is not a solution of equation (4).
In highly anisotropic phonon systems, when � p is
less than the typical solid angle for four-phonon pro-
cesses, the stationary distribution of h phonons will be
essentially different from the Bose—Einstein distribu-
tion (6) in magnitude and in momentum dependence.
The integrals (2) and (3) can be written as a sum of
five terms with definite ranges of integration. These
On the suprathermal distribution in an anisotropic phonon system in He II
Fizika Nizkikh Temperatur, 2003, v. 29, No. 1 17
terms correspond to the different processes, which are
possible for the interactions of h phonons between
themselves and with l phonons:
1) h l l l1 2 3 4� � � ; 2) h l h l1 2 3 4� � � ;
3) h l h h1 2 3 4� � � ; 4) h h h l1 2 3 4� � � ; (8)
5) h h h h1 2 3 4� � �
The arrow to the right indicates the decay of an h1
phonon and to the left creation. We define the rates
�b d
n
,
( ) of creation (b) and decay (d) processes with dis-
tribution function n for h phonons by the equalities:
N nb b
n
� �
�� 1
0( ) ( ); N nd d
n
� �
�� 1
( ); ( , , , , )� � 1 2 3 4 5 . (9)
As Nb is the sum over all Nb� , we rewrite relation (4)
as follows:
n nb
n
d
n
1
0
1
5
1
1
5
( ) ( ) ( )� �
�
�
�
�� �
� �� . (10)
We take into account that 3pp processes instanta-
neously establish equilibrium (on the time scale of
h-phonon creation and propagation) in the subsystem
of l phonons, which occupy the solid angle �3pp that
is typical for 3pp. The phonon pulses in experiments
[16–20] have � p close to �3pp [13–15]. Therefore in
this case we can consider that the l phonons in the
pulse have a Bose—Einstein distribution:
n nl l( ) ( )p � 0 , at p pl c� . (11)
For the stationary distribution of h phonons, the dis-
tribution function can be written in the form:
n S nh h h( ) ( ) ( )p p� 0 , at p ph c� . (12)
Starting from equalities (10)–(12) we have
� �
�
�
�
�
b
n
d
nS( ) ( )( )
� �
� ��
1
5
1
1
5
p . (13)
This equation is an integral equation with respect to
the unknown function S( )p1 . For decay processes
when h1 combines with an l or h phonon, the rate is
independent or a linear functional of S respectively.
For creation processes if initially there are zero, one,
or two h phonons then the rate is independent, a li-
near functional, or a quadratic functional, respec-
tively. Therefore the desired function S( )p1 is absent
in the rates �b
n
1
( ), �d
n
1
( ), �d
n
2
( ), �d
n
3
( ) if one takes into ac-
count that 1 1� �nh . The rates �b
n
2
( ), �b
n
4
( ), �d
n
4
( ), �d
n
5
( )
include linear functionals, and �b
n
3
( ), �b
n
5
( ) include
quadratic functionals. Using these facts, we present
the rates from (9) as
� �b
n
b1 1
0( ) ( )� ; � �d
n
d1 1
0( ) ( )� ;
� �d
n
d2 2
0( ) ( )� ; � �d
n
d3 3
0( ) ( )� ; (14)
� �b
n
b bS2 2 1 2
0( ) ( )( )� p ; � �b
n
b bS4 4 1 4
0( ) ( )( )� p ;
� �d
n
d dS4 4 1 4
0( ) ( )( )� p ; � �d
n
d dS5 5 1 5
0( ) ( )( )� p ;
and
� �b
n
b bS3 3
2
1 3
0( ) ( )( )� p ; � �b
n
b bS5 5
2
1 5
0( ) ( )( )� p ; (15)
where
� �
� �b d b d
n S,
( )
,
( ) ( )0 0 1� � (16)
are the rates calculated with distribution function
(6). The above equations define Sb d, �
which are
functionals of the function S h( )p .
Using (14)–(16) the relation (13) can be written
as
� � � �SS S SS Sd d d b d d b b5 5
0
5
2
5
0
4 4
0
3
2
3
0� � � �( ) ( ) ( ) ( )� � � �
� � � �� � � � �S S S Sd b b d b b� � � �3
0
4 4
0
2
0
2
2
2
0( ) ( ) ( ) ( )
� �� �b dS1
0
1
0( ) ( ). (17)
For isotropic phonon systems, when� �bi di� , re-
lations (2), (3) give � � �
� � �b d
( ) ( ) ( )0 0� � isotr . In this case
Eq. (17) has the solution S h( )p � 1, and, according to
(12), we get the Bose—Einstein distribution (6) for
all phonons in an isotropic system.
3. Asymmetry of processes of h-phonon creation
and decay, resulting in a suprathermal
distribution on an anisotropic phonon system
In anisotropic phonon systems, when � �bi di�
the creation rate �
�b
( )0 can be significantly different
from the decay rate �
�d
( )0 . In Refs. [25,26] the rates of
all five processes of creation and decay are calculated
for phonons with momentum p1 directed along the
axis of symmetry of the pulse, chosen as the Z axis, so
�1 0� . These rates we denote �b d, , where the super-
script (0) is understood.
The main role in (17) is played by a type-1 process
that describes the exchange of phonons between the l
and h systems. For a pulse typically used in the experi-
ments [16–20] the values are: anisotropy parameter
� p � 0125. sr and temperature T = 1 K. Then the min-
imum value of the ratio � � �b d1 1 30� at p pc1 � ; it
grows quickly with increasing p1 and becomes equal
to infinity at p p1 0� (see Fig. 1). The limiting mo-
mentum p0 is determined by the solid angle � p and
18 Fizika Nizkikh Temperatur, 2003, v. 29, No. 1
I. N. Adamenko, K. E. Nemchenko, and A. F. G. Wyatt
the conservation laws of energy and momentum,
which govern the interaction of l phonons with such
h1 phonons. The corresponding analytical expressions
and detailed discussion of the rates and their de-
pendences on momentum shown in Fig. 1, are given in
Refs. 25, 26.
An infinite lifetime coupled with a finite creation
rate of h phonons with p p1 0� means that in aniso-
tropic phonon systems, type-1 processes cannot effect
a dynamic equilibrium between the h and l subsys-
tems. However such an equilibrium can be provided
by type-4 processes, because � �d b4 4� in the momen-
tum region where �d1 0� (see Fig. 1). Using the nu-
merical values for rates calculated for the anisotropic
phonon system (see Fig. 1), equation (17) can be sat-
isfied for S �� 1. As a result, in anisotropic phonon
systems, the stationary distribution function of such h
phonons is many times greater than the Bose—Ein-
stein one (6) and has a different energy dependence
which is determined by the momentum-dependent
rates shown in Fig. 1.
In the left-hand side of (17) the rates that have the
same power of S and describe mutually compensating
processes are separated in curved braces. These com-
pensating processes in (17) are of two fundamentally
different types. The second and the third braces de-
scribe processes which exchange phonons between the
l and h systems. At the same time, type-4 decay pro-
cesses are partly compensated by type-3 creation pro-
cesses (but not by type-4 creation processes), and
type-3 decay processes partly compensate type-4 cre-
ation process (but not type-3 creation process). The
first and fourth curved braces describe processes that
conserve the number of h phonons. Here decay is com-
pensated by creation in processes of the same type.
The presence in (17) of processes that conserve and
do not conserve the number of h phonons makes it use-
ful to consider the expression obtained from (17) by
averaging the anisotropic phonon system over p1. We
define the average as
A
An d p
n d p
p
p
�
�
�
1
0 3
1
1
0 3
1
( )
( )
�
�
. (18)
Then one should take into account the following
equalities, which are obtained by reindexing the vari-
ables of integration:
N d p N d pd b
p
d
2 1
3
1 2 1
3
1
2
3
( ) ( )
( )
p p
� �
� �� , (19)
N d p N d pd b
p
d
5 1
3
1 5 1
3
1
5
3
( ) ( )
( )
p p
� �
� �� , (20)
N d p N d pd b
p
d
3 1
3
1 4 1
3
1
3
3
( ) ( )
( )
p p
� �
� �� , (21)
N d p N d pd b
p
d
4 1
3
1 3 1
3
1
4
3
( ) ( )
( )
p p
� �
� �� , (22)
where� d�
( )3 is the solid angle of the created p3 phonon
in decay processes of the type �.
From the conservation laws of energy and momen-
tum it follows that at the typical momenta of the
phonons taking part in the decay processes, the cre-
ated p3 phonon remains inside the initial phonon
pulse. This fact allows us to consider approximately
that � �d p�
( )3 � . In this case the averaging of (17) ta-
king into account (19)–(22) gives:
SS S Sd d b b b d4 4
0
4 4
0
1
0
1
02 2� � � �( ) ( ) ( ) ( )� � � . (23)
On the suprathermal distribution in an anisotropic phonon system in He II
Fizika Nizkikh Temperatur, 2003, v. 29, No. 1 19
Fig. 1. The momentum dependences of the creation rates
�b and decay �d rates, at T = 1 K and �p � 0125. sr, for
all the processes which exchange phonons between the l-
and h-phonon systems.
This result can be rewritten in the form
S S Sd b b d
2
4
0
4
0
1
0
1
02 2~ ~( ) ( ) ( ) ( )� � � �� � � , (24)
where � �b b1
0
1
0( ) ( )� , and the other average values of
the rates ~
,
( )�b d1 4
0 are determined by the obvious equali-
ties of the respective terms in Eqs. (23) and (24). The
solution of Eq. (24) is
S b
d b d b d
�
� � �
4
2 8 2
1
0
1
0
4
0 2
4
0
1
0
1
�
� � � � �
( )
( ) ( ) ( ) ( ) (( ~ ~ ) ~ ~ 0
4
0) ( )~� �b
.
(25)
Using this solution we can estimate the value of S
if we substitute the rates in (25) (shown in Fig. 1)
with their average values calculated over the range
10 201K K� �cp /kB , at �1 0� . These rates we de-
note �b d, . In this case
2 81 4 1 4� � � �d b b d� �� (26)
and from (25) we have
S b
d
� �2 301
0
4
0
�
�
( )
( )
. (27)
This relation has a simple physical meaning. The
value of S is defined as the square root of the ratio
of the rate of growth of the number of h phonons by
type-1 processes to the rate of decrease of the number
of h phonons by type-4 processes, which are partly
compensated by type-3 processes.
According to (17) and results obtained for �b d2 2,
and �b d5 5, [25,26], it follows that S depends strongly
on the angle �1 between the direction of the phonon
p1 1 1 1( , , )p � � and the Z axis of symmetry of the
anisotropic phonon system. Therefore, according to
[25,26], over a relatively wide region of momentum p1
the creation rate of h phonons with �1 0� , for the se-
cond and the fifth processes, is greater than decay
rate. Since the total number of h phonons is separately
conserved in processes 2 and 5, the h phonons will
concentrate in momentum space near the Z axis. With
increasing �1, the number of h phonons decreases, so
that there will be relatively few h1 phonons at large
� �1 � p in the pulse. This result agrees with the re-
sults of experiments [17,19], where the h-phonon cone
is found to be narrower than the l-phonon cone.
The suprathermal ratio S is also a function of the
temperature of the l phonons, because according to
[25,26] the rates of creation and decay of all five pro-
cesses become smaller, at different rates, with decreas-
ing temperature. Thus, at � p � 0125. sr and p pc1 � ,
according to [26], with decreasing temperature from
1 K to 0.7 K the rates �b1 and �d1 becomes smaller by
� 5 and � 6 times, respectively, the rates �b2 and �d2
by � 9 and � 6 times, the rates �b3 and �d3 by � 70 and
� 65 times, the rates �b4 and �d4 by � 80 and
� 95 times, and the rates �b5 and �d5 by � 85 and
� 100 times, respectively. In general, the suprather-
mal ratio increases as the temperature decreases.
Although in this paper we are concerned with the
value of S averaged over momentum, we do expect
that S is strongly peaked just above pc, where
� �d d4 3� is small and �b3 has a maximum [26]. How-
ever, the situation is complicated, as S is also expected
to vary with angle within the beam.
4. Conclusion
In this paper we have shown that the asymmetry
between the processes of decay and creation of
high-energy phonons in long enough phonon pulses
created in experiments [16–20] in superfluid helium
results in a suprathermal distribution. Then the quasi-
equilibrium distribution function of the h phonons dif-
fers from the Bose—Einstein distribution by a factor
S(p).
We have obtained an equation (17) whose solution
determines the value of suprathermal ratio S and its
dependence on momentum p1, anisotropy parameter
� p , and temperature T. Expressions that describe mu-
tually compensating processes are separated in (17) by
curved braces. These compensated processes have two
different principal types: the first type describes pro-
cesses that exchange phonons between the l and h sys-
tems, and the second type conserves the number of h
phonons. That is why we consider the expressions
(23), obtained from (17) by averaging with respect to
all p1 of the anisotropic phonon system.
Starting from relation (23) and the available re-
sults for the rates of creation and decay of phonons
with momentum p1 1 1 10( , , )p � �� directed along the
symmetry axis Z (see Fig.1), an estimate is made of
the average value S of the suprathermal ratio. The
full evaluation of the suprathermal ratio S and its de-
pendence on the parameters p1, � p , and T will only
be possible after calculation of all the rates in (17) at
arbitrary angles �1. We plan to carry out this calcula-
tion. At present we have only the values of all the
rates �b d, for the case �1 0� . This estimation of and
the analysis of Eqs. (17) and (23) indicates that the
distribution function of h phonons can exceed the
Bose–Einstein energy distribution by two orders of
magnitude in anisotropic phonon systems. We find
that S depends strongly on the parameters p1, � p ,
and T. Besides the creation S of a more complete the-
ory of the suprathermal distribution, we plan to carry
20 Fizika Nizkikh Temperatur, 2003, v. 29, No. 1
I. N. Adamenko, K. E. Nemchenko, and A. F. G. Wyatt
out experiments to observe this very unusual pheno-
menon occuring in phonon pulses in He II.
Acknowledgements
We express our gratitude to EPSRC of the UK
(grant GR/N18796 and GR/N20225), and to GFFI
of Ukraine (grant N02.07/000372) for support for
this work.
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On the suprathermal distribution in an anisotropic phonon system in He II
Fizika Nizkikh Temperatur, 2003, v. 29, No. 1 21
|
| id | nasplib_isofts_kiev_ua-123456789-128779 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T16:45:53Z |
| publishDate | 2003 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Adamenko, I.N. Nemchenko, K.E. G.Wyatt, A.F. 2018-01-13T19:49:20Z 2018-01-13T19:49:20Z 2003 On the suprathermal distribution in an anisotropic phonon system in He II / I.N. Adamenko, K.E. Nemchenko, A.F. G. Wyatt // Физика низких температур. — 2003. — Т. 29, № 1. — С. 16-21. — Бібліогр.: 28 назв. — англ. 0132-6414 PACS: 67.40.Db, 67.40.Fd https://nasplib.isofts.kiev.ua/handle/123456789/128779 The equation that describes the suprathermal distribution of high-energy phonons (h phonons) created in anisotropic phonon systems in superfluid helium is obtained. The solution of this equation enables the derivation of the value of suprathermal ratio S as the ratio of the actual distribution to the Bose-Einstein one, its dependences on the momentum of the h phonons, the anisotropy parameters, and the temperature of the low-energy phonons from which the h phonons are created. We analyze this equation to obtain an estimate of the value of the ratio between the h-phonon number density in anisotropic and isotropic phonon systems and draw conclusions about the dependence of S on the relevant parameters. We express our gratitude to EPSRC of the UK (grant GR/N18796 and GR/N20225), and to GFFI of Ukraine (grant N02.07/000372) for support for this work. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Квантовые жидкости и квантовые кpисталлы On the suprathermal distribution in an anisotropic phonon system in He II Article published earlier |
| spellingShingle | On the suprathermal distribution in an anisotropic phonon system in He II Adamenko, I.N. Nemchenko, K.E. G.Wyatt, A.F. Квантовые жидкости и квантовые кpисталлы |
| title | On the suprathermal distribution in an anisotropic phonon system in He II |
| title_full | On the suprathermal distribution in an anisotropic phonon system in He II |
| title_fullStr | On the suprathermal distribution in an anisotropic phonon system in He II |
| title_full_unstemmed | On the suprathermal distribution in an anisotropic phonon system in He II |
| title_short | On the suprathermal distribution in an anisotropic phonon system in He II |
| title_sort | on the suprathermal distribution in an anisotropic phonon system in he ii |
| topic | Квантовые жидкости и квантовые кpисталлы |
| topic_facet | Квантовые жидкости и квантовые кpисталлы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/128779 |
| work_keys_str_mv | AT adamenkoin onthesuprathermaldistributioninananisotropicphononsysteminheii AT nemchenkoke onthesuprathermaldistributioninananisotropicphononsysteminheii AT gwyattaf onthesuprathermaldistributioninananisotropicphononsysteminheii |