Second-Sound Waves in Cryocrystals
The self-coordinated values of parameters of cryocrystals of orthodeuterium, parahydrogen, and neon (temperature, isotope concentrations, and the sizes of samples), which define the range of second-sound waves existence, are found. The limiting isotope concentrations, below which the propagation of...
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| Цитувати: | Second-Sound Waves in Cryocrystals / V.D. Khodusov, A.A. Blinkina // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 543-546. — Бібліогр.: 12 назв. — англ. |
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| author | Khodusov, V.D. Blinkina, A.A. |
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| citation_txt | Second-Sound Waves in Cryocrystals / V.D. Khodusov, A.A. Blinkina // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 543-546. — Бібліогр.: 12 назв. — англ. |
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| description | The self-coordinated values of parameters of cryocrystals of orthodeuterium, parahydrogen, and neon (temperature, isotope concentrations, and the sizes of samples), which define the range of second-sound waves existence, are found. The limiting isotope concentrations, below which the propagation of second-sound waves is possible, are established. The sizes of samples, starting from which their increase does not essentially influence the damping of second-sound waves are found. The results are presented in threedimensional plots.
Знайдено узгодженi значення параметрiв крiокристалiв ортодейтерiю, параводню та неону (температури, концентрацiї iзотопiв, домiшок та розмiрiв зразкiв), що визначають область iснування хвиль другого звуку. Встановлено граничнi концентрацiї iзотопiв, нижче за якi можливе поширення хвиль другого звуку, а також визначено розмiри зразкiв, починаючи з яких їх збiльшення не впливає на загасання хвиль другого звуку. Результати наведено у виглядi тривимiрних графiкiв.
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SECOND-SOUND WAVES IN CRYOCRYSTALS
SECOND-SOUND WAVES IN CRYOCRYSTALS
V.D. KHODUSOV, A.A. BLINKINA
Karazin Kharkiv National University
(4, Svobody Sq., Kharkiv 61077, Ukraine; e-mail: blinkina@ gmail. com,vkhodusov@ ukr. net )
PACS 66.70.-f, 61.72.S
c©2010
The self-coordinated values of parameters of cryocrystals of orth-
odeuterium, parahydrogen, and neon (temperature, isotope con-
centrations, and the sizes of samples), which define the range of
second-sound waves existence, are found. The limiting isotope con-
centrations, below which the propagation of second-sound waves
is possible, are established. The sizes of samples, starting from
which their increase does not essentially influence the damping of
second-sound waves are found. The results are presented in three-
dimensional plots.
1. Introduction
For the first time, second-sound waves (SSW) in solid
bodies were observed experimentally in 4He cryocrystals
[1]. The question on an opportunity of the SSW propa-
gation in other cryocrystals was discussed in a number
of works [2–5]. For the experimental detection of SSW,
it is necessary to know the compatible areas of values
for the most essential parameters such as the tempera-
ture, impurity and isotope concentrations, and sizes of
samples. This compatibility of parameters can be ob-
tained (if the second sound damping factor is known)
from a condition of weakness of the damping of these
waves. It is possible to write down the damping coef-
ficient using the diffusion times for various dissipative
processes running in a phonon gas [6] which are caused
by umklapp processes, scattering on impurities and iso-
topes, and scattering on the sample boundaries. The
direct calculation of these times is rather difficult. It is
known that the SSW propagation is possible in a vicin-
ity of the maximum of the thermal conductivity coeffi-
cient of a cryocrystal. According to this, it is possible
to use phenomenological values of collision frequencies
connected with these times which are used in the Call-
away model to describe experimentally observable de-
pendences of the thermal conductivity coefficient on the
temperature in a low-temperature range.
The main contribution to the SSW damping factor
is given by the phonon scattering on impurities and iso-
topes and by the scattering on the sample boundaries. In
case of orthodeuterium (o−D2), parahydrogen (p−H2),
and neon (Ne) cryocrystals, it is possible to produce pure
enough samples, in which the role of impurities will be
insignificant; however, there remains an essential role of
isotopes. It is necessary to consider the phonon scatter-
ing on boundaries in an absolutely different way. SSW
propagate in crystals in the phonon gas, when the hy-
drodynamic regime in it is realized due to the fast nor-
mal processes of phonon interaction. In this case, before
reaching the boundary, a phonon will experience a set of
normal collisions. Therefore, the free path will increase
due to the boundary scattering, and the contribution of
these processes to the damping coefficient will decrease.
In the present work, we determined the coordinated
areas of concentrations of hydrogen and 22Ne isotope,
sizes of samples, and temperatures, where SSW exist.
The obtained results can be used for carrying out the
experiments on the SSW registration in orthodeuterium,
parahydrogen, and neon.
2. Existence of Second-Sound Waves in Solid
Bodies
In work [6], the general theory of secondary waves in
a gas of Bose quasiparticles which are, in particular,
second-sound waves in a phonon gas has been con-
structed for solid bodies. Meanwhile, the description
of SSW was established in the reduced isotropic crystal
model, by using the modulus of elasticity. The SSW dis-
persion equation has been found in the same work [6]
and has the following form:
Ω
(
Ω2 −W 2
IIk̃
2
)
− 2iW 2
IIk̃
2ΓII = 0. (1)
Here, WII is the SSW phase velocity in an isotropic
phonon gas, and ΓII is the SSW damping factor. The
expressions for WII and ΓII are as follows [6]:
WII =
(
TS2/Cρ̃
)
,
ΓII =
r
2ρ̃
+
[
1
2ρ̃
(
4
3
η̃ + ζ̃
)
+
1
2C
κ̃
]
k2, (2)
where ρ̃ – phonon density, r – coefficient of external fric-
tion caused by the phonon interaction without the quasi-
momentum conservation (umklapp processes, scattering
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 543
V.D. KHODUSOV, A.A. BLINKINA
on impurities, isotopes, and boundaries of the sample),
η̃ and ζ̃ – coefficients of first and second viscosities, re-
spectively„ and κ̃ – coefficient of hydrodynamical ther-
mal conductivity.
For a phonon gas in the low-temperature range
ΘD/T�1 in the reduced isotropic crystal model, the
following expressions for the densities of thermal ca-
pacity and entropy, phonon density, Debye tempera-
ture, and average sound velocity are obtained: C =
3S = 2π2
15
k4
BT
3
~3V 3
t
(
2 + β3
)
, ρ̃ = 2π2
45
k4
BT
4
~3V 5
t
(
2 + β5
)
, ΘD =
2π~Vs/kBa, VS = Vt
(
3
2+β3
)1/3
, where Vl, Vt – veloci-
ties of longitudinal and transversal sounds, β = Vt/Vl,
kB – Boltzmann constant, and a – lattice parameter.
The second sound velocity
WII =
V 2
t
3
(
2 + β3
2 + β5
)
. (3)
We now introduce the diffusion times by the relations
τη̃ =
η̃
ρ̃W 2
II
; τζ̃ =
ζ̃
ρ̃W 2
II
; τκ̃ =
κ̃
CW 2
II
; τR =
ρ̃
r
(4)
and the notation
τN =
4
3
τη̃ + τζ̃ + τκ̃. (5)
Then the introduction of the collision frequencies νi
connected with the diffusion times τi by the relation
νi = 1/τi allows us to write ΓII in the form
ΓII =
1
2
[
νR +
1
νN
Ω2
]
. (6)
The condition of existence of a weakly decaying SSW is
ΓII � Ω, which leads to a condition imposed on the fre-
quency Ω known as a “window” of SSW existence. The
given condition is written down in a general form. There-
fore, for a more detailed study of the range of SSW exis-
tence and for finding the coordinated boundary values of
parameters, at which the existence of a weakly decaying
SSW is possible, it is necessary to specify it. We demand
the next condition to be satisfied:
ΓII ≤ 10Ω. (7)
This leads to the following frequency “window” of SSW
existence:
0.1νN −
√
ν2
N
102
− νNνR ≤ ΩII ≤ 0.1νN+
√
ν2
N
102
− νNνR.
(8)
Equation (8) yields the desired relation between the pa-
rameters
νN ≥ 100νR. (9)
3. Definition of the Coordinated Values of
Parameters, at Which SSW Exist in
Cryocrystals
The expression for the scattering frequency due to resis-
tive processes is as follows:
νR = νiso + νBeff + νU. (10)
Here, νiso – frequency of phonon scattering on isotopes,
νBeff – frequency of scattering on boundaries of a sample,
νU – scattering frequency due to umklapp processes.
The collision frequency on isotopes is taken into ac-
count for all crystals with the use of the formula [7,8]
νiso (x) =
a3
4πV 3
S
(
ΔM
M
)2
x4T 4Cd, (11)
where Cd = Nd/N – concentration of isotopes, ΔM – a
difference between nuclear weights of isotopes and that
of the basic substance, M – weight of basic atoms, and
x – dimensionless variable.
We considered the phonon scattering on boundaries
which depends on a relation between the free path length
of phonons due to normal processes lN = τNVS and that
in the sample bulk D. SSW exist in dielectrics if the
hydrodynamical mode in the description of phonons is
realized. Thus, the inequality lN � D is valid. Phonons
that are in the sample bulk will experience a set of nor-
mal collisions before reaching the boundary. As a re-
sult, the way which is passed between two collisions with
boundaries essentially increases. Using the known for-
mulas for the Brown movement, it is easy to show that
the trajectory between two collisions with a boundary
will be of the order of D2/lN [9]. This implies that the
effective frequency of phonon collisions with the bound-
ary is
νBeff =
V 2
S
D2νN
. (12)
For the numerical estimations and the calculations of
conditions for the existence of weakly decaying SSW
in cryocrystalls of orthodeuterium and parahydrogen
and in crystals of neon, we used data given in works
[3,4,10,11] and presented in a tabular form in work [12].
We give here only Table for the averaged frequencies of
phonon collisions.
544 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
SECOND-SOUND WAVES IN CRYOCRYSTALS
Expressions for the average collision frequencies in the Callaway model for various crystals
Crystal o−D2 p−H2 Ne
νN , s−1 1.4× 106T 5 1.9× 106T 5 1.3× 107T 4
νU, s−1 1.3× 109T 3e
[
−37
T
]
1.4× 109T 3e
[
−40
T
]
4× 106T 5e
[
−13
T
]
νiso, s−1 1.3× 109T 4Cd 2.3× 108T 4Cd 9.5× 107T 4Cd
νBeff , s−1 9× 103T−5/D2 4.3× 103T−5/D2 0.2× 103T−4/D2.
Fig. 1. Range of the self-coordinated parameters for crystals of
orthodeuterium
Using these average values of frequencies of phonon
collisions and inequality (9), we build the plots of self-
coordinated values of the parameters (T,Cd, D) deter-
mining the range of SSW existence in the considered
cryocrystals. These plots are presented in Figs. 1–3.
4. Conclusions
The values of parameters lying on surfaces are limitind
values of the self-coordinated parameters, at which the
SSW propagation is possible. The range below these sur-
faces corresponds to the SSW existence in cryocrystals.
Fig. 2. Range of the self-coordinated parameters for crystals of
parahydrogen
The analysis of the resulted plots enables us to estab-
lish the limiting values of isotope concentrations, below
which the SSW propagation is possible. For orthodeu-
terium, parahydrogen, and neon, Cdlim are, respectively,
3.1 × 10−5, 2.7 × 10−4, and 1.3 × 10−3. In addition, it
is seen from plots that, starting from the sizes of the
order of 0.5 cm, an increase in the sample size weakly
influences the range, where SSW exist.
Choosing the certain parameters in the range of SSW
existence from inequality (8), it is possible to find a fre-
quency spectrum of weakly decaying SSW. For example,
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 545
V.D. KHODUSOV, A.A. BLINKINA
Fig. 3. Range of the self-coordinated parameters for crystals of
neon
let us consider parahydrogen: if we take a crystal with
D = 0.7 cm, the isotope concentration Cd = 3× 10−5 at
T = 2 K, the frequency range is 0 ≤ ΩII ≤ 4.9×106 s−1.
The knowledge of this spectrum is very essential for at-
tempts to register SSW in experiments on the observa-
tion of the thermal pulse evolution.
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2. N.N. Zholonko, B.Ya. Gorodilov, and A.I. Krivchikov,
JETP Lett. 55, 167 (1992).
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Phys. 6, 2279 (1973).
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Temp. Phys. 20, 939 (1994); Low Temp. Phys. 21, 1
(1995).
7. R. Berman, Thermal Conductivity in Solids (Oxford Uni-
versity Press, Oxford, 1976).
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45, 573 (2002).
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10. Properties of the Condensed Phases of Hydrogen and
Oxygen (Reference Book) (Naukova Dumka, Kiev, 1984).
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451 (2009).
Received 22.12.09
ХВИЛI ДРУГОГО ЗВУКУ У КРIОКРИСТАЛАХ
В.Д. Ходусов, А.А. Блiнкiна
Р е з ю м е
Знайдено узгодженi значення параметрiв крiокристалiв орто-
дейтерiю, параводню та неону (температури, концентрацiї iзо-
топiв, домiшок та розмiрiв зразкiв), що визначають область
iснування хвиль другого звуку. Встановлено граничнi концен-
трацiї iзотопiв, нижче за якi можливе поширення хвиль друго-
го звуку, а також визначено розмiри зразкiв, починаючи з яких
їх збiльшення не впливає на загасання хвиль другого звуку. Ре-
зультати наведено у виглядi тривимiрних графiкiв.
546 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
|
| id | nasplib_isofts_kiev_ua-123456789-56196 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2071-0194 |
| language | English |
| last_indexed | 2025-12-07T18:13:43Z |
| publishDate | 2010 |
| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Khodusov, V.D. Blinkina, A.A. 2014-02-13T20:18:00Z 2014-02-13T20:18:00Z 2010 Second-Sound Waves in Cryocrystals / V.D. Khodusov, A.A. Blinkina // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 543-546. — Бібліогр.: 12 назв. — англ. 2071-0194 PACS 66.70.-f, 61.72.S https://nasplib.isofts.kiev.ua/handle/123456789/56196 The self-coordinated values of parameters of cryocrystals of orthodeuterium, parahydrogen, and neon (temperature, isotope concentrations, and the sizes of samples), which define the range of second-sound waves existence, are found. The limiting isotope concentrations, below which the propagation of second-sound waves is possible, are established. The sizes of samples, starting from which their increase does not essentially influence the damping of second-sound waves are found. The results are presented in threedimensional plots. Знайдено узгодженi значення параметрiв крiокристалiв ортодейтерiю, параводню та неону (температури, концентрацiї iзотопiв, домiшок та розмiрiв зразкiв), що визначають область iснування хвиль другого звуку. Встановлено граничнi концентрацiї iзотопiв, нижче за якi можливе поширення хвиль другого звуку, а також визначено розмiри зразкiв, починаючи з яких їх збiльшення не впливає на загасання хвиль другого звуку. Результати наведено у виглядi тривимiрних графiкiв. en Відділення фізики і астрономії НАН України Український фізичний журнал Тверде тіло Second-Sound Waves in Cryocrystals Хвилі другого звуку у кріокристалах Article published earlier |
| spellingShingle | Second-Sound Waves in Cryocrystals Khodusov, V.D. Blinkina, A.A. Тверде тіло |
| title | Second-Sound Waves in Cryocrystals |
| title_alt | Хвилі другого звуку у кріокристалах |
| title_full | Second-Sound Waves in Cryocrystals |
| title_fullStr | Second-Sound Waves in Cryocrystals |
| title_full_unstemmed | Second-Sound Waves in Cryocrystals |
| title_short | Second-Sound Waves in Cryocrystals |
| title_sort | second-sound waves in cryocrystals |
| topic | Тверде тіло |
| topic_facet | Тверде тіло |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/56196 |
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