The possibilyty of second sound waves registration in isotopic enriched diamond single crystal
Using the experimental data of diamond thermal conductivity we obtain the data of normal and resistive processes of phonon interaction in Callaway approach. As the result we obtain the concentration values of ¹³C in diamond single crystal, under which the isotropic scattering is weak, and the temper...
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| Cite this: | The possibilyty of second sound waves registration in isotopic enriched diamond single crystal / V.D. Khodusov, D.M. Litvinenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 309-311. — Бібліогр.: 6 назв. — англ. |
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| citation_txt | The possibilyty of second sound waves registration in isotopic enriched diamond single crystal / V.D. Khodusov, D.M. Litvinenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 309-311. — Бібліогр.: 6 назв. — англ. |
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| description | Using the experimental data of diamond thermal conductivity we obtain the data of normal and resistive processes of phonon interaction in Callaway approach. As the result we obtain the concentration values of ¹³C in diamond single crystal, under which the isotropic scattering is weak, and the temperature range of second sound waves existence in diamond.
Використовуючи експериментальні дані теплопровідності алмазу, в моделі Калавея здобуто інформацію про нормальні та резистивні процеси фононного розсіяння. Як наслідок, здобуто величини концентрацій ізотопічних домішок ¹³C в монокристалі алмазу, за яких ізотопічне розсіяння фононів стає незначним, а також проміжок температур, в якому існують хвилі другого звуку.
Используя экспериментальные данные по теплопроводности алмаза, в модели Калавэя получена информация о нормальных и резистивных процессах фононного рассеяния. Как следствие получены концентрации примеси изотопа ¹³C в монокристалле алмаза, при которых изотопическое рассеяние фононов становится незначительным, а также промежуток температур, в котором существуют волны второго звука.
|
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THE POSSIBILITY OF SECOND SOUND WAVES REGISTRATION
IN ISOTOPIC ENRICHED DIAMOND SINGLE CRYSTAL
V. D. Khodusov1 and D. M. Litvinenko2
1Kharkiv V.N. Karazin National University, High-Technology Institute,
31 Kurchatov Ave, 61108, Kharkov, Ukraine;
e-mail: khodusov@pht.univer.kharkov.ua;
2Akhiezer Institute for Theoretical Physics,
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
e-mail: litvin3333@ukr.net
Using the experimental data of diamond thermal conductivity we obtain the data of normal and resistive proc-
esses of phonon interaction in Callaway approach. As the result we obtain the concentration values of in dia-
mond single crystal, under which the isotropic scattering is weak, and the temperature range of second sound waves
existence in diamond.
13C
PACS: 63.10.+a, 63.20.1s, 63.20.Kr
1. INTRODUCTION
Modern technologies give us opportunity to obtain
isotopic pure perfect single crystals. It’s been experi-
mentally established that the thermal conductivity coef-
ficient of such crystals has maximum in low tempera-
ture range [1]. Especially it becomes apparent in dia-
mond crystals. This fact is caused by a special role
played by normal processes (where the conservation
law of quasi-impulse is present) in interaction between
phonons. In perfect single crystals the propagation of
weakly damping second sound waves, similar to HeII
ones, is possible. The most probable at that are the nor-
mal processes of phonon interaction comparatively to
resistive ones (where the conservation law of quasi-
impulse fails). The major contribution to second sound
waves damping makes isotopic phonon scattering. For
the first time second sound waves were registered in
NaF [2] single crystal and solid [3], where the
isotopic damping is absent at the temperature range of
10…20 K.
4 He
Using the data of normal and resistive processes of
phonon interaction (umklapp processes, phonon scat-
tering on impurities, isotopes and on bounds of sam-
ple), obtained while the description of experimental
data of diamond thermal conductivity coefficient in
Callaway model, the concentration values of 13 are
obtained, under which the isotopic scattering is weak,
the temperature range of second sound waves exis-
tence in diamond is established. Starting from 13
isotope concentration equal to 0.1 % and less the sec-
ond sound waves registration is possible in the tem-
perature range near maximum of thermal conductivity
of diamond at T=104 K. This temperature is one order
higher than the temperature of second sound waves
registration in NaF and solid .
C
C
4 He
2. EQUATIONS OF GAS DYNAMICS
OF QUASI-PARTICLES WITH TAKING
INTO ACCOUNT EXTERNAL FIELDS
In the kinetic theory the state of gas of quasi-
particles is characterized by distribution function of
quasi-particles , which satisfies ki-
netic equation which has a form of Boltzmann equation:
( , ,jN N t≡ p r )
( )стN N
t
ε ∂ ∂ ∂ ∂
+ − = ∂ ∂ ∂ ∂
g
r r p
, (1)
where ( ) ( )j jε≡ = ∂ ∂g g p
( )jε ε≡
is the group velocity of
quasi-particles; is the Hamiltonian of
quasi-particle, which is equal to its local energy;
is the collision integral of quasi-particles, which takes
into account the processes of collision, merge, fission
and radiation of quasi-particles.
( , , tp r )
( )
ст
N
If we designate as a character time of phonon in-
teraction concerned with N-processes (where phonon
energy and quasi-impulse conservation laws are accom-
plished), and τ
Nτ
R as a character time of resistive proc-
esses of phonon interaction (where these laws fail), the
condition of predomination of normal processes is writ-
ten as
N Rτ τ<< . (2)
If at some moment of time the system of quasi-
particles moves out of its equilibrium state, then in time
τN quasi-local balance is established, which is character-
ized by distributive function , which turns to zero
the collision integral because of N-processes , and it is
written as :
( )
0
jN
( )
( ) ( )
( )
1
0
0
exp 1
1
j
jN
Т
ε
θ
−
− = −
+
pu
, (3)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 309-311. 309
where is drift velocity in gas of quasi-particles,
is relative temperature.
u
( -T T0 ) /TΘ = 0
In the sate of gas of quasi-particles, which is close to
local statistic balance, the solution of equation (1) in the
gas dynamical approximation we search as:
( ) ( ) ( )
0
jj jN N Nδ= + , ( ) ( )( )0
jjN Nδ << , (4)
where is the local equilibrium distributive func-
tion, which depends on gas dynamical quantities, and
δN
( )
0
jN
(j) depends on its gradients.
If we input thermodynamical potential F0 as:
( )( )0 0
,
ln 1 j
j
F T N= − +∑
k
, (5)
then knowing it as a function of T, , which satis-
fies thermodynamical identity:
, jAu
0 0dF S dT d= − −P u , (6)
we obtain impulse density , heat capacity C and en-
tropy S densities, components of density tensor of
quasi-particles
P
ijρ :
0
,T A
F∂ = − ∂
P
u
; 0
0
,A
F
S
T
∂ = − ∂ u
;
0
,A
S
C T
T
∂ = ∂ u
;
2
0
,
i
il
l i l
T A
P F
u u u
ρ
∂ ∂ = = −
∂ ∂ ⋅∂
. (7)
Applying the standard procedure [4] from kinetic
equation (1) it is possible to obtain a system of gas dy-
namical equations, that describe an isotropic gas of
quasi-particles in linear approximation on drift velocity
in reduced isotropic crystal model: u
3
ST r divηθ η ζ + ∇ = − + ∆ + + ⋅∇
P u u u
u
;
C Sdivθ + = κ∆θu , (8)
where , = ρ ⋅P ρ is the density of quasi-particles
number, r is the external friction coefficient, caused by
processes of phonon interaction without quasi-impulse
conservation (umklapp processes, phonon scattering on
impurities, isotopes and sample boundaries), ,η ζ
κ
are
the first and second viscosity coefficients, is the
thermal conductivity coefficient.
3. PROBLEM SETTING AND ITS SOLUTION
IN REDUCED ISOTROPIC CRYSTAL
AND CALLAWAY MODELS
The secondary waves (SW) propagation in Bose gas
of quasi-particles in solids was studied in work [4] bas-
ing on gas dynamics equations of quasi-particles, espe-
cially SSW in phonon gas in dielectric crystals.
The solution of equations (9) for all variables we’ll
search as with frequency Ω and wave
vector . From the compatibility condition of system
of equations we obtain the dispersion equation for sec-
ondary waves (SW) [4]:
(exp i t Ω − kr)
k
( )2 2 2 2 2
II II II2 0W k iW kΩ Ω − − Γ = . (9)
From the dispersion equation (9) follows that WII is
the phase velocity of SW and quantity ГII is the damp-
ing coefficient of SW. Expressions for WII и ГII have a
simple structure for SSW phonon gas dynamics in re-
duced isotropic crystal model [4]:
( )
1
2 2
IIW TS Cρ= ;
2
II
1 4 1
2 2 3 2
r k
C
η ζ κ
ρ ρ
Γ = + + +
, (10)
where
2 4 42
3
5 3 3
3 23 (
15(2 )
t B
t
V k T
C S
T V
πρ β
β
= = = +
+
2 ) is
obtained in reduced isotropic crystal model in low tem-
perature range. By low temperature range we mean the
range of temperature where 1>D TΘ > ;
2DΘ = t BV k aπ ,l V is the Debye temperature, tV are
the velocities of transverse and longitudinal phonons,
t lV Vβ =
1845D KΘ ≈
, a is the lattice parameter. Diamond single
crystal has the highest Debye temperature
( ) among other crystals, that’s why the
temperature in order of 100 K and lower can be consid-
ered as low for diamond.
If to introduce diffusive lengths of free path and re-
lated with them diffusive times by following expres-
sions:
2
IIW
η
ητ
ρ
= ; 2
IIWζ
ζτ
ρ
= ; 2
IICW
κ
κτ = ; R r
ρτ = , (11)
ГII can be written in the following way:
2
II
1 1 4
2 3R
η ζτ τ τ
τ
Γ = + + Ω + Ω
2
κ . (12)
The condition of weak damping SSW existence
( ) leads to next condition imposed on fre-
quency Ω , known as SSW existence “window” [4]:
ΙΙΓ << Ω
min{ , , } Rη κζν ν ν ν>> Ω >> , (13)
where the collision frequencies ν are related to diffu-
sive times by the expression
i
iτ 1i iτν = .
The calculation of kinetic coefficients of phonon gas
dynamics in the model of reduced isotropic crystal rela-
tively to elasticity coefficient in [4] shows that basic
role plays phonon viscosity. The number, similar to
Prantl number in gas theory Pr Cη
ρκ
=
2
, for phonon gas
for most of crystals is equal to 10 with a great degree
of accuracy.
Further on we’ll take into account that in reduced
isotropic crystal model energy and quasi-impulse con-
servation laws allow t t and
umklapp processes.
l g+ ↔ + l t l g+ ↔ +
Using the Callaway [5] model for description of ex-
perimental data of thermal conductivity of diamond in
low temperature range [6], we can calculate frequencies
of phonon collisions, caused by normal and umklapp
processes and also by scattering on isotopes and sample
boundaries:
310
4
6 5! (5)
2
B
N
k A
T
a
ν ζ
π ρ
= ;
24 3
4
4 3 8! (8)
4
B
iso i
s
k a Mc T
Mv
ν
π
∆ =
ζ ; (14)
2
3
2 6! (6)
(2 )
C
B TU
s
k B
T e
v
ν ζ
π
−
= ; 4! (4)s
b
v
D
ν ζ= ,
where i
i
N
c
N
= is the isotope concentration, is
the difference in isotope and the basic atom masses,
is the mass of basic atom,
M∆
M
( )nζ is the Riemann zeta-
function, ρ is the diamond crystal density,
, cm/K, C K ,
cm are phenomenological constants obtained
from experiment [6].
11 32 10A K− −= ⋅
33
7,
0,D =
1210−= ⋅1,5B 670=
4. RESULTS ANALYSIS AND CONCLUSION
For making numerical estimations and obtaining the
condition of weakly damping second sound waves exis-
tence in diamond single crystal we use the following
data:
3,512ρ = g/cm2, , V cm/s, 0,68β = 61.81 10l = ⋅
61.235 10tV = ⋅ cm/s, W cm/s, 6
II 0.76 10= ⋅
where , and W were calculated in reduced iso-
tropic crystal model [4]. Using the Callaway model we
examine inequality (13) to obtain the “window” of SSW
existence or at least its low limit and the isotope
concentrations (the main isotope is , ) of
such “window”. Substituting expressions (14) into
inequality (13) we obtain SSW existence “window” in
diamond single crystal. It’s convenient to introduce
function
lV tV II
13C M∆
)
1=
(f( = ν + ν
f(T) >>
T)
(T)
N Uν iso + ν
K
b , which value range
will provide the SSW existence “window”.
The following figure gives temperature dependence of
function for different concentrations of 13 iso-
tope:
1
f C
In the figure we can see that diamond has the SSW
existence “window” in the region of T , start-
ing from and less. Such form of curves is
provided by competition of normal and resistive proc
80 K≈
410ic −≈
esses. In high temperature range (higher than
) umklapp processes predominate, in tem-
perature range lower than T normal processes
and processes of phonon scattering on impurities and
boundaries predominate.
80T K≈
80≈
Solid line – , dash-dot line – c ,
dash line – , dot line –
45 10ic −= ⋅
510−
410i
−=
ic = 610ic −=
Thus the carried out analysis gives us reason to af-
firm that SSW in diamond crystal can be registered at
temperatures in the range of , if isotope
concentration is lower than 10 .
100T K≈
4−
13C
REFERENCES
1. A.P. Jernov, A.V. Inushkin. Kinetic coefficients in
isotopically disordered crystals //Phisics – Uspekhi.
2002, v. 172, N5, p. 573-599.
2. H.E. Jackson, C.T. Walker. Thermal conductivity,
second sound and phonon-phonon interaction in
NaF // Phys. Rev. B. 1971, v. 3, N 4, p. 1428-1439.
3. С.С. Ackerman, B. Bertman, H.A. Fairbank,
P.A. Guyer. Second sound in solid helium //Phys.
Rev. Lett. 1996, v. 16, p. 789-801.
4. A.I. Akhiezer, V.F. Aleksin, V.D. Khodusov. Gas
dynamics of quasiparticles //Low. Temp. Phys. 1994,
v. 20, p. 939-970.
5. R. Berman. Thermal conduction in solids. M.:
“Mir”, 1979, 286 р. (in Russian).
6. Lanhua Wei, P.K. Kuo, R.L. Thomas, T.R. Antony,
W.F. Banholzer. Thermal conductivity of isotopi-
cally modified single crystal diamond
//Phys.Rev.Lett. 1993, v. 70, p. 3764-3767.
ВОЗМОЖНОСТЬ НАБЛЮДЕНИЯ ВОЛН ВТОРОГО ЗВУКА В ИЗОТОПИЧЕСКИ ОБОГАЩЁННОМ
МОНОКРИСТАЛЛЕ АЛМАЗА
В. Д. Ходусов, Д.М. Литвиненко
Используя экспериментальные данные по теплопроводности алмаза, в модели Калавэя получена информация о нор-
мальных и резистивных процессах фононного рассеяния. Как следствие получены концентрации примеси изотопа 13 в
монокристалле алмаза, при которых изотопическое рассеяние фононов становится незначительным, а также промежу-
ток температур, в котором существуют волны второго звука.
C
МОЖЛИВІСТЬ СПОСТЕРЕЖЕННЯ ХВИЛЬ ДРУГОГО ЗВУКУ В ІЗОТОПІЧНО ЗБАГАЧЕНОМУ
МОНОКРИСТАЛІ АЛМАЗУ
В. Д. Ходусов, Д.М. Литвиненко
Використовуючи експериментальні дані теплопровідності алмазу, в моделі Калавея здобуто інформацію про норма-
льні та резистивні процеси фононного розсіяння. Як наслідок, здобуто величини концентрацій ізотопічних домішок
в монокристалі алмазу, за яких ізотопічне розсіяння фононів стає незначним, а також проміжок температур, в якому
існують хвилі другого звуку.
13C
311
|
| id | nasplib_isofts_kiev_ua-123456789-111021 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-28T09:20:02Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Khodusov, V.D. Litvinenko, D.M. 2017-01-07T17:38:10Z 2017-01-07T17:38:10Z 2007 The possibilyty of second sound waves registration in isotopic enriched diamond single crystal / V.D. Khodusov, D.M. Litvinenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 309-311. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 63.10.+a, 63.20.1s, 63.20.Kr https://nasplib.isofts.kiev.ua/handle/123456789/111021 Using the experimental data of diamond thermal conductivity we obtain the data of normal and resistive processes of phonon interaction in Callaway approach. As the result we obtain the concentration values of ¹³C in diamond single crystal, under which the isotropic scattering is weak, and the temperature range of second sound waves existence in diamond. Використовуючи експериментальні дані теплопровідності алмазу, в моделі Калавея здобуто інформацію про нормальні та резистивні процеси фононного розсіяння. Як наслідок, здобуто величини концентрацій ізотопічних домішок ¹³C в монокристалі алмазу, за яких ізотопічне розсіяння фононів стає незначним, а також проміжок температур, в якому існують хвилі другого звуку. Используя экспериментальные данные по теплопроводности алмаза, в модели Калавэя получена информация о нормальных и резистивных процессах фононного рассеяния. Как следствие получены концентрации примеси изотопа ¹³C в монокристалле алмаза, при которых изотопическое рассеяние фононов становится незначительным, а также промежуток температур, в котором существуют волны второго звука. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory The possibilyty of second sound waves registration in isotopic enriched diamond single crystal C можливість спостереження хвиль другого звуку в ізотопічно збагаченому монокристалі алмазу Возможность наблюдения волн второго звука в изотопически обогащённом монокристалле алмаза Article published earlier |
| spellingShingle | The possibilyty of second sound waves registration in isotopic enriched diamond single crystal Khodusov, V.D. Litvinenko, D.M. Kinetic theory |
| title | The possibilyty of second sound waves registration in isotopic enriched diamond single crystal |
| title_alt | C можливість спостереження хвиль другого звуку в ізотопічно збагаченому монокристалі алмазу Возможность наблюдения волн второго звука в изотопически обогащённом монокристалле алмаза |
| title_full | The possibilyty of second sound waves registration in isotopic enriched diamond single crystal |
| title_fullStr | The possibilyty of second sound waves registration in isotopic enriched diamond single crystal |
| title_full_unstemmed | The possibilyty of second sound waves registration in isotopic enriched diamond single crystal |
| title_short | The possibilyty of second sound waves registration in isotopic enriched diamond single crystal |
| title_sort | possibilyty of second sound waves registration in isotopic enriched diamond single crystal |
| topic | Kinetic theory |
| topic_facet | Kinetic theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/111021 |
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