Relaxation caused by one phonon decay into three in superfluid helium
The analytical relation for the rate of one phonon decay into three is obtained. Starting from the relation obtained, the rate of spontaneous decay in the first order of perturbation theory is found. It is shown, that processes of one phonon decay into three provide a fast establishment of equilibri...
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| Zitieren: | Relaxation caused by one phonon decay into three in superfluid helium / I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, A.F.G. Wyatt // Вопросы атомной науки и техники. — 2007. — № 3. — С. 399-403. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1110462017-01-08T03:04:19Z Relaxation caused by one phonon decay into three in superfluid helium Adamenko, I.N. Kitsenko, Yu.A. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. Physics of quantum liquids The analytical relation for the rate of one phonon decay into three is obtained. Starting from the relation obtained, the rate of spontaneous decay in the first order of perturbation theory is found. It is shown, that processes of one phonon decay into three provide a fast establishment of equilibrium in anisotropic and isotropic phonon systems. It allows us to relate the momentum region in which processes of one phonon decay into three are permitted to subsystem of low-energy phonons. Отримано аналітичний вираз для частоти процесів розпаду одного фонона на три. Виходячи з отриманого виразу, знайдено частоту самовільного розпаду в першому порядку теорії збурень. Показано, що процеси розпаду одного фонона на три забезпечують швидке встановлення рівноваги в анізотропних і ізотропних фононних системах. Це дозволяє область, у якій дозволені процеси розпаду одного фонона на три, віднести до підсистеми низькоенергійних фононів. Получено аналитическое выражение для частоты процессов распада одного фонона на три. Исходя из полученного выражения, найдена частота самопроизвольного распада в первом порядке теории возмущений. Показано, что процессы распада одного фонона на три обеспечивают быстрое установление равновесия в анизотропных и изотропных фононных системах. Это позволяет область, в которой разрешены процессы распада одного фонона на три, отнести к подсистеме низкоэнергетических фононов. 2007 Article Relaxation caused by one phonon decay into three in superfluid helium / I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, A.F.G. Wyatt // Вопросы атомной науки и техники. — 2007. — № 3. — С. 399-403. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 67.40.Db, 67.40.Fd, 67.90.+z http://dspace.nbuv.gov.ua/handle/123456789/111046 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Physics of quantum liquids Physics of quantum liquids Adamenko, I.N. Kitsenko, Yu.A. Nemchenko, K.E. Slipko, V.A. Wyatt, A.F.G. Relaxation caused by one phonon decay into three in superfluid helium Вопросы атомной науки и техники |
| description |
The analytical relation for the rate of one phonon decay into three is obtained. Starting from the relation obtained, the rate of spontaneous decay in the first order of perturbation theory is found. It is shown, that processes of one phonon decay into three provide a fast establishment of equilibrium in anisotropic and isotropic phonon systems. It allows us to relate the momentum region in which processes of one phonon decay into three are permitted to subsystem of low-energy phonons. |
| format |
Article |
| 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. |
| author_sort |
Adamenko, I.N. |
| title |
Relaxation caused by one phonon decay into three in superfluid helium |
| title_short |
Relaxation caused by one phonon decay into three in superfluid helium |
| title_full |
Relaxation caused by one phonon decay into three in superfluid helium |
| title_fullStr |
Relaxation caused by one phonon decay into three in superfluid helium |
| title_full_unstemmed |
Relaxation caused by one phonon decay into three in superfluid helium |
| title_sort |
relaxation caused by one phonon decay into three in superfluid helium |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2007 |
| topic_facet |
Physics of quantum liquids |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/111046 |
| citation_txt |
Relaxation caused by one phonon decay into three in superfluid helium / I.N. Adamenko, Yu.A. Kitsenko, K.E. Nemchenko, V.A. Slipko, A.F.G. Wyatt // Вопросы атомной науки и техники. — 2007. — № 3. — С. 399-403. — Бібліогр.: 15 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT adamenkoin relaxationcausedbyonephonondecayintothreeinsuperfluidhelium AT kitsenkoyua relaxationcausedbyonephonondecayintothreeinsuperfluidhelium AT nemchenkoke relaxationcausedbyonephonondecayintothreeinsuperfluidhelium AT slipkova relaxationcausedbyonephonondecayintothreeinsuperfluidhelium AT wyattafg relaxationcausedbyonephonondecayintothreeinsuperfluidhelium |
| first_indexed |
2025-07-08T01:32:49Z |
| last_indexed |
2025-07-08T01:32:49Z |
| _version_ |
1837040528877682688 |
| fulltext |
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 399-403. 399
Section H. PHYSICS OF QUANTUM LIQUIDS
RELAXATION CAUSED BY ONE PHONON DECAY INTO THREE
IN SUPERFLUID HELIUM
I.N. Adamenko1, Yu.A. Kitsenko2, K.E. Nemchenko1, V.A. Slipko1, and A.F.G. Wyatt3
1Karazin Kharkov National University, Kharkov, Ukriane;
2National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
e-mail: ykitsenko@kipt.kharkov.ua;
3School of Physics, University of Exeter, Exeter, UK
The analytical relation for the rate of one phonon decay into three is obtained. Starting from the relation
obtained, the rate of spontaneous decay in the first order of perturbation theory is found. It is shown, that
processes of one phonon decay into three provide a fast establishment of equilibrium in anisotropic and isotropic
phonon systems. It allows us to relate the momentum region in which processes of one phonon decay into three
are permitted to subsystem of low-energy phonons.
PACS: 67.40.Db, 67.40.Fd, 67.90.+z
1. INTRODUCTION
Liquid helium is the unique medium in which a
number of interesting macroscopic quantum phenomena
are observed. One of them is the creation of high-energy
phonons with energy close to 10 K by a pulse of low-
energy phonons with typical temperature of 1 K (see
Refs. [1-3]).
The theory of this phenomenon has been developed
in Refs. [4,5]. As it followed from the theory, phonons
of superfluid helium break up into two subsystems:
1. Subsystem of low-energy phonons ( l -phonons)
with cpp < . The establishment of equilibrium in this
subsystem is due to three-phonon processes ( 21↔ )
with typical time 1
33
−= pppp ντ (see Refs. [6,7]).
2. Subsystem of high-energy phonons ( h -phonons)
with cpp > . Here decay processes are forbidden by
conservation laws of energy and momentum and the
fastest are the four-phonon processes ( 22 ↔ ) with
typical time 1
44
−= pppp ντ (see Refs. [4,5,8]).
There is strong inequality
pppp 43 ττ << (1)
between pp3τ and pp4τ . As a result the equilibrium in
l -phonon subsystem occurs very quickly in contrast to
h -phonon subsystem where it happens rather slowly.
The difference in group velocities and rather weak
connection between h - and l -phonons results to h-
phonon leaving through a rear wall of l -phonon pulse
and forming of h -phonon pulse which comes to the
detector after l -phonon pulse.
However, in Refs. [9,10] it was shown, that three-
phonon processes took place not up to cp but to
momentum cpp pp 543 = and therefore phonons with
momenta
cpp ppp <<3 (2)
should be related to h -phonon subsystem. But in
momentum region (2) processes of one phonon decay
into a greater number of phonons (decay processes) are
still allowed and if the rates of these processes appear
one order of magnitude with pp3ν then these phonons
should be related to l -phonon subsystem. It would
seem that it is not essential to what subsystem these
phonons should be related, as the momentum range (2)
is rather small. However, as it has been shown in
Ref. [11], the rate of four-phonon processes is very
sensitive to a numerical value of momentum which
delimitates l - and h -phonon subsystems. Therefore the
calculation of the rates of decay processes is of
undoubtful interest as it will allow us to answer the
question to what subsystem the mentioned momentum
range should be related. Here we consider one of these
processes such as the process of one phonon decay into
three ( 31↔ ).
2. PROCESSES OF ONE PHONON DECAY
INTO THREE
Conservation laws of energy and momentum which
should be satisfied in process of one phonon decay into
three can be written as
4321 pppp ++= , (3)
4321 εεεε ++= , (4)
where ip is the momentum of i -th phonon
participating in the process and iε is its energy which
we write as
( ) ( )( )iiii pcpp ψεε +=≡ 1 . (5)
Here 41038.2 ⋅=c cm/s is the velocity of sound and
( )pψ is a function which describes a deviation of a
spectrum from linearity which is small ( ( ) 1<<pψ ) but
nevertheless it completely determines the mechanisms
of phonon interactions. Here and below we shall use the
simple analytical approximation of function ( )pψ
obtained in [7] which is valid for cpp ≤
400
( )
−= 2
2
2
2
max 14
cc p
p
p
pp ψψ . (6)
Here maxψ is the maximum value of function ( )pψ
reached when 2cpp = . In case of saturated vapour
pressure 046.0max =ψ and 10~ == Bcc kcpp K.
From conservation laws (3) — (4) taking (5) into
account we obtain a relation on angles between phonons
with momenta 1p , 2p and 3p , 4p
( )
21
344321
12 2
222
pp
pppp ζφφ
ζ
−+−
= , (7)
where 4321 ffff −−−=φ , ( )iii ppf ψ= ,
ijij θζ cos1−= , and ijθ is the angle between phonons
with momenta ip and jp .
Having put 34ζ and 12ζ equal to zero in (7) and
taking (6) into account we derive the boundaries of
regions in which processes of one phonon decay into
three can take place
( ) ( 21213 55
10
1, ppppp −=±
)22
2212
1 121510155 cppppp +−+−± , (8)
( ) 1213 , pppp = .
From the positivity of radicand in Eq. (8) the
restrictions on momentum 2p could be obtained:
( ) ),min(,0max 2122 +− << pppp , (9)
where
( ) ( )2
1
2
max112 22
3
1 ppppp −±=± , (10)
cpp
10
9
max = . (11)
From Eqs. (8) – (11) it follows, that the momenta of
phonons participating in 31→ processes can change in
ranges (see also Ref. [12]):
min4,3,2 530 ppp c =<< , (12)
max10 pp << . (13)
At saturated vapour pressure 75.7~
min =p K and
48.9~
max =p K.
We note that when pppp 31 < the processes of one
phonon decay into three can proceed at rather big angles
between momenta of interacting phonons while when
pppp 31 > the mentioned processes are small-angle.
Interaction of phonons in superfluid helium is
described by Landau Hamiltonian which we write as
(see, for example, [13])
430
ˆˆˆˆ VVHH ph ++= . (14)
Here 0Ĥ is a Hamiltonian of noninteracting phonons
and terms 3̂V and 4V̂ describe the interaction of
phonons caused by the third and the fourth orders of
small deviations of a system from an equilibrium state
accordingly.
The probability density of process of one phonon
decay into three in the correspondence with [14] can be
written as
( )
( )6
22
4321
2
12
π
π
fiHVW =pppp . (15)
Here V is a volume of a system and fiH is an
amplitude of the process of one phonon decay into three
which can be written in the form
M
V
pppp
H fi ρ
δ
8
4321
; 4321 pppp ++= . (16)
Here 145.0=ρ g/cm3 is a density of helium and
( ) ( ) ( )
( ) ( ) ( )
4
4
14
4
13
4
12
2
14
2
13
2
12
MMMM
MMMM
+++
++= (17)
is a matrix element consisting of seven terms which can
be written as
( )
( )
( ),12
12
21221121
43443343
2121
212
12
−−
++
−
−
+++−×
+++−×
−−
=
nnnnnn
nnnnnn
u
u
M
εεε
ε
(18)
( )
( )
( ),12
12
21221121
43443343
2121
214
12
−−
++
−
−
−−+−×
−−+−×
+−
−=
nnnnnn
nnnnnn
u
u
M
εεε
ε
(19)
( ){ }wuM +−= 2
4 14 , (20)
where
i
i
i p
p
n = , ( )jiji pp −=− εε , 84.2=
∂
∂
=
ρ
ρ c
c
u ,
188.02
22
=
∂
∂
=
ρ
ρ c
c
w . The other terms of Eq. (17), can
be obtained from the mentioned by replacement of
corresponding subscripts. We note, that 4M
corresponds to the first order of perturbation theory on
4V̂ , and the others correspond to the second order of
perturbation theory on 3̂V .
The first three terms in the right-hand side of Eq.
(17) are resonant. These terms give the main
contribution to amplitude (16). In momentum range
where three-phonon processes are allowed their
denominators can vanish giving the essential divergence
in matrix element. This divergence can be eliminated by
taking the final lifetime of phonon caused by three-
phonon processes into account.
3. THE KINETIC EQUATION
FOR PROCESSES OF ONE PHONON
DECAY INTO THREE
The kinetic equation describing change of distribution
function ( )11 pnn ≡ of phonon with momentum 1p due
to 31↔ processes can be written as
( ) ( )11
1
2
1
!3
1 pp cd II
dt
dn
+= . (21)
Here
( ) ( ) ( )∫ Γ= ΣΣ dnWI cdcdcdcdcd ,,,,1, pp δεδ , (22)
401
where 4
3
3
3
2
3 pdpdpdd =Γ ,
( )4321 ppppWWd = ( )3214 ppppWWc = , (23)
( ) ( )( )( )43211432 1111 nnnnnnnnnd +++−+= , (24)
( )( )( ) ( )43213214 1111 nnnnnnnnnc +−+++= , (25)
4321 εεεεε −−−=Σ
d , 4321 ppppp −−−=Σ
d , (26)
3214 εεεεε −−−=Σ
c , 3214 ppppp −−−=Σ
c . (27)
We note that the first term of Eq. (21) corresponds to
phonon with momentum 1p decay into three and the
inverse process and the second term corresponds to
combining of a phonon with momentum 1p with other
two phonons and the process inverse to it.
We consider that in all momentum range (13)
phonons are in equilibrium. In this case their
equilibrium distribution function in the accordance with
Refs. [6, 15] can be written as
( ) ( )
1
0 1exp
−
−
−
=
Tk
n
B
ii
i
up
p
ε
. (28)
Here ( )χ−= 1cNu is a drift velocity, which is defined
by the unit vector N directed along the total
momentum of phonon system (an anisotropy axis of
phonon system) and parameter of anisotropy χ . In
isotropic phonon systems 1=χ . In case corresponding
to experiments [1-3] phonon pulses are strongly
anisotropic phonon systems with 1<<χ .
To obtain the relaxation rate caused by 31↔
processes we change the equilibrium number of
phonons with momentum 1p on a small value 1nδ at
equilibrium distribution of the others phonons. In this
case Eqs. (24) and (25) can be rewritten as
( )
( )( ) ( )( ) ( )( )0
4
0
3
0
20
1
1 111
1
nnn
n
n
nd +++
+
−=
δ
, (29)
( )
( ) ( ) ( )( )0
4
0
3
0
20
1
1 1
1
nnn
n
nnc +
+
−=
δ . (30)
We define the relaxation rate caused by 31↔
processes by equality
dt
nd
n
1
1
31
1 δ
δ
ν −=↔ . (31)
Substituting (29) and (30) into (21) and taking (31)
into account we have
cd ννν +=↔31 , (32)
where
( ) ( ) ( )
( )( ) ( )( ) ( )( )),111
1
1
!3
1
0
4
0
3
0
2
0
1
nnn
Wd
n
dddd
+++×
Γ
+
= ΣΣ∫ pδεδν
(33)
( ) ( ) ( )
( ) ( ) ( )( )).1
1
1
2
1
0
4
0
3
0
2
0
1
nnn
Wd
n
cccc
+×
Γ
+
= ΣΣ∫ pδεδν
(34)
We note, that in the momentum range (2) ( ) 10
1 <<n .
Thus the definition of the decay rate in Ref. [14]
actually coincides with the mentioned above.
4. THE RATE OF PROCESSES
OF ONE PHONON DECAY INTO THREE
Taking (15)-(17) into account we rewrite the relation
(33) in spherical coordinates
( ) ( ) ( )
( )( ) ( )( ) ( )( )
.
111
1
444333222
0
4
0
3
0
2
3
4
3
3
3
2
2
0
1
1
ζϕζϕζϕ×
+++×
δεδ
+
=ν ∫ ΣΣ
dddpdddpdddp
nnnppp
M
n
Kp
ddd p
(35)
Here
i
i p
Npi−=1ζ ,
2751223
1
ρπ⋅
=K .
Having made the integration in (35) with the help of
δ -functions we get
( )( )
{ } ( )( ) ( )( ) ( )( ),111
1
2
0
4
0
3
0
2
2
4
3
3
3
222
232320
1
1
nnn
R
ppp
MM
ddddpdp
nc
Kp
d
++++×
+⋅
=
−+
∫ ϕζζν
(36)
where
( ) ( )( )±±
± === 4433 coscos,coscos ϕϕϕϕMM , (37)
( ) ( )
⊥
⊥⊥± −+
=
3
2
2
4
2
31
3 2
cos
Ap
RppA αα
ϕ
∓
, (38)
( ) ( )
⊥
⊥⊥± ±+−
=
4
2
2
4
2
31
4 2
cos
Ap
RppA αα
ϕ , (39)
2211 cosϕα ⊥⊥ −= pp 222 sinϕα ⊥= p , (40)
( )22
4
2
3
2
4
2
34 ⊥⊥⊥⊥ −−−= ppAppR , (41)
221
2
2
2
1 cos2 ϕ⊥⊥⊥⊥ −+= ppppA , (42)
φ−−+= 3214 pppp 22 iiii pp ζζ −=⊥ , (43)
4
332211
4 p
ppp φζζζ
ζ
−−+
= . (44)
The rate of processes of one phonon decay into three
defined by relation (36) consists of two compounds. The
first of them is connected with a spontaneous decay of a
phonon with momentum 1p , and the other is caused by
stimulation of a phonon with momentum 1p decay due
to the presence of phonon system, i.e. nonzero functions
( )0
in . As in prevailing momentum range distribution
functions ( )0
in are much less than the unity we will be
interested in the rate of spontaneous decay.
We shall find the decay rate I
dν caused by the first
order of perturbation theory on 4V̂ . In this case
4MMM == −+ and we can get an exact analytical
expression for the rate I
dν :
402
( )
( )
( )
≥
<≤
<<
≤
⋅=
.,0
,,
,,
,,
max1
max1313
31min12
min111
pp
ppppg
ppppg
pppg
N
pp
ppI
dν (45)
Here
( ){ }
cp
wu
N
c
4273
max
22
3265720500
1
ρπ
ψ+−
= , (46)
( ) ( )2
1
211
111 136117
64064
3492075 ppppg c −= , (47)
( ) ( )
( ) ( )
( )
( )
,
45
2735
arctan
2
1601600
9
2
1
2
3
2
min
2
1
1
22
1
122
22
1
2
max
3
1
2
1
2
3
121
12
−−
−
+
⋅−+
−=
pppp
ppp
pgppp
pppgpg
pp
c
pp
π (48)
( ) ( ) ( )122
22
1
2
max
3
113 pgppppg −⋅= π , (49)
where
( )
,4232632326231375360
0336189868208826244785
50121429933658359634287
2254448500
12102
1
84
1
66
1
48
1
210
1
12
1121
cc
cc
cc
ppp
pppp
pppp
ppg
+−
+−
+−
=
(50)
( )
.1530987480
15795093500
642
1
24
1
6
1122
cc
c
ppp
ppppg
−+
−= (51)
Big numerical coefficients in expressions (46) - (51) are
caused by repeated integration of high powers of
momentum.
We note, that the obtained relation for the rate is
valid for all pressures up to 19 bar at which the
dispersion becomes normal and decay processes are
forbidden by conservation laws.
In Fig. 1 the rate I
dν calculated with a help of Eq.
(45) is shown. We see, that the rate sharply vanishes
near max1 pp = . Such behavior of the rate could be
explained by factor ( )22
1
2
max pp − in (49).
Fig. 1. The rate of spontaneous phonon decay into
three in the first order of perturbation theory on 4V̂
calculated from Eq. (45)
The integration in case of the second order of
perturbation theory on 3V̂ cannot be exactly made
analytically due to the complexity of integrand in (36).
However in this case the integration can be made
numerically. The result of this integration in momentum
range (2) is shown in Fig. 2 (curve 2). As it follows
from the comparison of curve 2 in Fig. 2 with (45), the
main contribution in the momentum range (2) is due to
the second order of perturbation theory on 3V̂ .
Fig. 2. Momentum dependences of the rates of three-
phonon processes 21→ (curve 1), processes of one
phonon decay into three 31→ (curve 2), h -phonon
creation for three values of delimitating momentum dp
equal to cp , ppp3 and maxp (curves 3, 3′ and 3″).
Calculations were made with 01 =θ , 02.0=χ and
041.0=T
5. MOMENTUM THAT DELIMITATES
l- AND h-PHONON SUBSYSTEMS
As it has been already told in introduction the
important question is where we should delimitate l -
and h-phonon subsystems. To answer this question we
start from Fig. 2. In Fig. 2 momentum dependences of
the rates of three-phonon processes (curve 1), processes
of one phonon decay into three (curve 2), four-phonon
processes of high-energy phonons creation with
momenta dpp > for three different values of
momentum dp which delimitates l - and h -phonon
subsystems (curves 3, 3′, 3″) are represented.
From Fig. 2 it can be seen, that the rates of three-
phonon processes ( 21→ ) and processes of one phonon
decay into three ( 31→ ) are comparable and appear
much greater than the rate of four-phonon processes of
high-energy phonons creation. As a result the
momentum range (2) should be related to l -phonon
subsystem in which equilibrium occurs quickly in
contrast to h -phonon subsystem with cpp > where the
decay processes are forbidden and equilibrium occurs
slowly.
Thus there is a question where we should relate
phonons with momenta from maxp up to cp . The
calculation of the rates of decay processes in this range
is rather difficult problem as the order of integrals in
this case increases strongly. It is possible only to state,
that due to decay processes the time of establishment of
equilibrium in this momentum range will be less than in
the range of cpp > where the decay processes are
forbidden. Besides this we must take into account that,
as it can be seen from Fig. 2, the rate of four-phonon
processes decreases quickly enough with increasing of
momentum. As a result the momentum dp , that
403
delimitates phonons of superfluid helium into two
subsystems with different relaxation times can be
considered to be equal to cp , as it was supposed in
Refs. [4,5,8].
6. CONCLUSION
In paper the processes of one phonon decay into
three in anisotropic and isotropic phonon systems of
superfluid helium are investigated. The restrictions on
momenta of phonons which can participate in the
mentioned processes are obtained.
The general relation (35) for the rate dν of
processes of one phonon decay into three is derived.
With a help of numerical integration of relation (35) the
numerical value of the rate dν caused by the second
order of perturbation theory on 3V̂ is found. Starting
from the general relation (35) the analytical relation (45)
for the rate of spontaneous decay in the first order of
perturbation theory on 4V̂ is derived.
The question about the value of momentum dp ,
which delimitates l - and h -phonon subsystems is
considered and it is shown, that dp is equal to cp .
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РЕЛАКСАЦИЯ, ОБУСЛОВЛЕННАЯ ПРОЦЕССАМИ РАСПАДА ОДНОГО ФОНОНА
НА ТРИ В СВЕРХТЕКУЧЕМ ГЕЛИИ
И.Н. Адаменко, Ю.А. Киценко, К.Э. Немченко, В.А. Слипко, A.F.G. Wyatt
Получено аналитическое выражение для частоты процессов распада одного фонона на три. Исходя из
полученного выражения, найдена частота самопроизвольного распада в первом порядке теории
возмущений. Показано, что процессы распада одного фонона на три обеспечивают быстрое установление
равновесия в анизотропных и изотропных фононных системах. Это позволяет область, в которой
разрешены процессы распада одного фонона на три, отнести к подсистеме низкоэнергетических фононов.
РЕЛАКСАЦІЯ, ЩО ОБУМОВЛЕНА ПРОЦЕСАМИ РОЗПАДУ ОДНОГО ФОНОНА
НА ТРИ У НАДПЛИННОМУ ГЕЛІЇ
І.Н. Адаменко, Ю.О. Кіценко, К.Е. Немченко, В.А. Сліпко, A.F.G. Wyatt
Отримано аналітичний вираз для частоти процесів розпаду одного фонона на три. Виходячи з
отриманого виразу, знайдено частоту самовільного розпаду в першому порядку теорії збурень. Показано,
що процеси розпаду одного фонона на три забезпечують швидке встановлення рівноваги в анізотропних і
ізотропних фононних системах. Це дозволяє область, у якій дозволені процеси розпаду одного фонона на
три, віднести до підсистеми низькоенергійних фононів.
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