Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen
The diffusion kinetic of classic impurity particles has been investigated in the frame of one-dimensional two-level model and applied for the explanation of solid hydrogen thermal conductivity data with extremely low con-centrations of neon impurity in samples growth at different crystallization rat...
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irk-123456789-1184752017-05-31T03:03:16Z Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen Zholonko, N.N. 9th International Conference on Cryocrystals and Quantum Crystals The diffusion kinetic of classic impurity particles has been investigated in the frame of one-dimensional two-level model and applied for the explanation of solid hydrogen thermal conductivity data with extremely low con-centrations of neon impurity in samples growth at different crystallization rates in which the plateau effect was observed. The main idea is that heavy isotopic impurities could segregate into thin long chains near dislocation cores if the growth rate is slow. Neon impurity chains can persist for a long time. Such rigid linear objects ensure inelastic scattering of phonons. The diffusion coefficient of neon atoms in (p-H₂)₁–cNec mixtures was estimated for the experimental conditions with с = 0.0001 аt. % and с = 0.0002 аt. %. 2013 Article Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen / N.N. Zholonko // Физика низких температур. — 2013. — Т. 39, № 6. — С. 722–725. — Бібліогр.: 22 назв. — англ. 0132-6414 PACS: 67.80.Gb, 65.40.–b http://dspace.nbuv.gov.ua/handle/123456789/118475 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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9th International Conference on Cryocrystals and Quantum Crystals 9th International Conference on Cryocrystals and Quantum Crystals |
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9th International Conference on Cryocrystals and Quantum Crystals 9th International Conference on Cryocrystals and Quantum Crystals Zholonko, N.N. Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen Физика низких температур |
| description |
The diffusion kinetic of classic impurity particles has been investigated in the frame of one-dimensional two-level model and applied for the explanation of solid hydrogen thermal conductivity data with extremely low con-centrations of neon impurity in samples growth at different crystallization rates in which the plateau effect was observed. The main idea is that heavy isotopic impurities could segregate into thin long chains near dislocation cores if the growth rate is slow. Neon impurity chains can persist for a long time. Such rigid linear objects ensure inelastic scattering of phonons. The diffusion coefficient of neon atoms in (p-H₂)₁–cNec mixtures was estimated for the experimental conditions with с = 0.0001 аt. % and с = 0.0002 аt. %. |
| format |
Article |
| author |
Zholonko, N.N. |
| author_facet |
Zholonko, N.N. |
| author_sort |
Zholonko, N.N. |
| title |
Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen |
| title_short |
Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen |
| title_full |
Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen |
| title_fullStr |
Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen |
| title_full_unstemmed |
Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen |
| title_sort |
diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
| topic_facet |
9th International Conference on Cryocrystals and Quantum Crystals |
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http://dspace.nbuv.gov.ua/handle/123456789/118475 |
| citation_txt |
Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen / N.N. Zholonko // Физика низких температур. — 2013. — Т. 39, № 6. — С. 722–725. — Бібліогр.: 22 назв. — англ. |
| series |
Физика низких температур |
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2025-07-08T14:03:45Z |
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2025-07-08T14:03:45Z |
| _version_ |
1837087796159840256 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6, pp. 722–725
Diffusion model of the thermal conductivity plateau of
weak solid solutions of neon in parahydrogen
N.N. Zholonko
Bogdan Khmelnitsky Cherkasy National University, 81 Shevchenko Blvd, Cherkasy 18031, Ukraine
E-mail: zholonko@yahoo.com
Received September 24, 2012
The diffusion kinetic of classic impurity particles has been investigated in the frame of one-dimensional two-
level model and applied for the explanation of solid hydrogen thermal conductivity data with extremely low con-
centrations of neon impurity in samples growth at different crystallization rates in which the plateau effect was
observed. The main idea is that heavy isotopic impurities could segregate into thin long chains near dislocation
cores if the growth rate is slow. Neon impurity chains can persist for a long time. Such rigid linear objects ensure
inelastic scattering of phonons. The diffusion coefficient of neon atoms in (p-H2)1–cNec mixtures was estimated
for the experimental conditions with с = 0.0001 аt. % and с = 0.0002 аt. %.
PACS: 67.80.Gb Thermal properties;
65.40.–b Thermal properties of crystalline solids.
Keywords: master equation, two-level model, solid hydrogen, thermal conductivity, diffusion coefficient.
Introduction
The effect of heavy impurities on the thermal conduc-
tivity of solid parahydrogen for the first time has been in-
vestigated in [1–4]. The heat transfer experimental data for
solid hydrogen have been analyzed but not to full comple-
tion up to now. It is possible to mention results for pure
hydrogen (conclusions about the possibility of Poiseuille
flow of phonons [5,6]), the huge anomaly for thermal con-
ductivity of solid parahydrogen with extremely low con-
centrations of heavy impurities [2]. So, it is necessary to
reconsider the plateau effect [7,8] as well as interesting
impurity effects with argon in solid parahydrogen, as, for
example, the chromatic polarization of light due to lattice
deformations created by this heavy impurity [3].
The unusual plateau effect [7,8] is closely associated
with the problem of the maximum of the thermal conductivi-
ty of pure solid hydrogen. This work proposes a simple two-
level diffusive model which describes and explains why
neon heavy impurities could segregate in chains when the
sample is slowly grown from liquid in comparison with the
average equal atomic distances when it grows from gas. It
was assumed that the impurity redistribution into chains can
explain the appearance of a symmetric plateau instead of a
resonance pit. The latter is characteristic of the average
equal distances between impurities. Since the neon impurity
is heavy, its motion is governed by classical physics.
The stochastic dynamics was traditionally treated with-
in the framework of Fokker–Planck equation [9]. This ap-
proach allows coming to several results [10,11]. However,
more concrete results are obtained within simplified mod-
els. For example, the Van-Hove autocorrelation function
( , )G tr [12] was used in Chudley–Eliott model [13]. This
function is the probability of the particle to move from
origin to point r during time t.
The most general equation within this approach is the
generalized Pauli equation (General master equation) [14].
It is possible to use simpler equation if there is the transla-
tion symmetry. Such equation is usually named Pauli equa-
tion (Master equation). For a simple case of equivalent
positions only, Master equation looks like:
( , ) [ ( ) ( , ) ( ) ( , )],
l
G l t p l l G l t p l l G l t
t ′
∂ ′ ′ ′= λ − − −
∂ ∑
where ( )p l l′− is the probability of particle displacement
from cite l′ in cite .l The function ( )p l l′− depends on
the potential barrier and is assumed to be known.
It is possible to use the continuous parameter x instead
of discrete length argument ( )a l l′− if the function ( , )G l t
changes with the length a slowly. Then the average dis-
placement and the average squared displacement as func-
tions of time become:
( ) ;x a p p t+ −= −
_
2 2 2 ,x a t Dt= λ =
where D is the diffusion coefficient. Here p+ is the prob-
ability of jump to the nearest write site, p− is the same to
© N.N. Zholonko, 2013
Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen
the left nearest position ( 1/2p p− += = if the external
force field is absent). So, in macroscopic limit the Pauli
equation describes a classical diffusion.
Let us consider possible applications of this model in
this paper. It was shown [5,6] that solid parahydrogen as
well as solid helium due to their high purity has, as dielec-
tric crystals, an unusual high thermal conductivity. It could
indicate even a so rare phenomenon as the Poisseule flow
of the phonon gas. Khodusov and Blinkina [15] formulated
the conditions for second sound to exist in quantum crys-
tals of ortho-D2, para-H2 and Ne. It was predicted exactly
the same temperature range for solid parahydrogen as was
found in [5,6]. The second sound and Poiseulle flow for
thermal conductivity have the common physical nature.
The large values of solid parahydrogen thermal conduc-
tivity near phonon maximum make it extremely high sensi-
tive to the presence of defects [2]. The solubility of neon in
solid parahydrogen is a few 0.01 at. % [16]. Therefore, it is
possible to state that experiments on samples grown from
gas (with neon contents c = 0.0001 at. % and 0.0002 at. %)
were such in which no segregation should occur. The ob-
served resonance pit-like anomaly on the thermal conduc-
tivity curve was explained by the influence of vibrations of
quasiisotopic heavy impurity. It was shown [4,17] that this
impurity subsystem modifies the vibration spectrum. It is
important to note that solid parahydrogen with argon im-
purity was also investigated in experimentally. But argon is
far from being an isotopic impurity. It was discovered this
impurity influence is substantially less [18].
In subsequent research of solid (p-H2)1–cNec with ex-
tremely small [7,8] a special attention was paid to the in-
fluence of sample preparation procedure on the thermal
conductivity. It was shown that after a slow growth instead
of a resonance pit an unusual symmetric plateau appeared.
To explain of this effect it was suggested to use other me-
chanisms of phonon scattering. It implied that slow growth
results in impurity chains due to ascending diffusion after
neon atoms segregation near dislocation cores (Gorsky
effect [19,20]). This work proposes a simple qualitative
model which explains the origin of linear impurity struc-
tures in weak solid solutions (p-H2)1–cNec.
Theory
Let us investigate of the one-dimensional problem of
particle diffusion in the field of alternating equally
spaced potential (Fig. 1). We will make sure in that this
model has an exact solution. Let for definiteness regular-
ly alternating potential pits of the first type (less deep,
Fig. 1) will be odd and the other one is even. The func-
tions 1(2 1, )G s t+ and 2 (2 , )G s t are the probabilities to
find of particle at moment t at the proper site (odd or even,
here 0, 1, 2, 3, ...).s = ± ± ± Let us make the set of differen-
tial Pauli equations with the particle at moment t = 0 in
even site (see the initial conditions):
____________________________________________________
1
1 1 1 1
(2 1, )
[(2 1) (2 1)] (2 1, ) [(2 1) (2 1)] (2 1, )
s
G s t p s s G s t p s s G s t
t ′
∂ + ′ ′ ′= λ + − + + −λ + − + + +
∂ ∑
2 2 1 1[(2 1) 2 )] (2 , ) [2 (2 1)] (2 1, );
s
p s s G s t p s s G s t
′
′ ′ ′+ λ + − −λ − + +∑
2
1 1 2 2
(2 , )
[2 (2 1)] (2 1, ) [(2 1) 2 ] (2 , )
s
G s t p s s G s t p s s G s t
t ′
∂ ′ ′ ′= λ − + + −λ + − +
∂ ∑
2 2 2 2[2 2 ] (2 , ) [2 2 ] (2 , );
s
p s s G s t p s s G s t
′
′ ′ ′+ λ − −λ −∑
the initial conditions are: 2 ,0(2 ,0) ;sG s = δ 1(2 1,0) 0.G s + =
_______________________________________________
We suppose that the particle can accomplish jumps only
to a nearby site by exchanging places with the molecule of
the matrix. Then the set of equations simplifies and we go
to its Laplace image to get an explicit solution:
1 1
0
exp ( ) ,G G ut dt
∞
= −∫ 2 2
0
exp ( ) ,G G ut dt
∞
= −∫
where u is a complex argument.
Then we Fourier transform the set of Laplace images:
(2 1)
1 1( , ) (2 1, )e ,ik s
s
U k u G s u += +∑
(2 )
2 2( , ) (2 , )e .ik s
s
U k u G s u= ∑
And obtain the set of algebraic equations for Fourier
and Laplace images:
Fig. 1. Model of two-level alternating potential pits. The particle
is at the origin (even site).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6 723
N.N. Zholonko
1 2 2 1 1( , ) ( e e ) ( , ) ( , ),ik ikuU k u p p U k u U k u−
+ −= λ + −λ
2 1 1 2 21 ( , ) ( e e ) ( , ) ( , ).ik ikuU k u p p U k u U k u−
+ −− + = λ + −λ
The zeros of denominators of functions 1 2,U U , which are
the singular points, determine the roots of the equation:
2 2
1,2 1 2 1 2 1 2
1 ( ( ) ( ) 4 (1 )).
2
u r= − λ + λ ± λ + λ − λ λ −
Here e e .ik ikr p p −
+ −= + The simple poles give the steady
solutions because its real parts are negative. We obtain the
originals 1 2( , ), ( , )U k t U k t after applying of the Riman–
Mellin’s formula. So, the solution to the initial set of Pauli
equations are obtained as Fourier series:
1 1
1(2 1, ) ( , )exp ( (2 1)),
k
G s t U k t ik s
N
+ = − +∑
2 2
1(2 , ) ( , )exp ( (2 )).
k
G s t U k t ik s
N
= −∑
Finding of the functions 1 2(2 1, ), (2 , )G s t G s t+ explicit-
ly is connected with large difficulties. However it is enough
to know the analytical dependences of the Fourier images
1( , ),U k t 2 ( , ).U k t Then we can deduce physically interest-
ing results without inverting the Fourier transform. Howev-
er, the model must assume that these probabilities are conti-
nuous functions of the site coordinates. As an example let us
investigate the model of equivalent sites. We can calculate
path ( )x t al= for time t where ( , )
l
l lG l t= ∑ is the aver-
age number of site. Then, using decomposition in Fourier
series, in case of large N we will have:
1( ) ( , ) e ( , ) ( )ikl
k l k
d dl t U k t i i U k t k
N dk dk
= = δ∑ ∑ ∑ .
Using transition ... ...
k
dk→∑ ∫ with integration of the
first area of reverse space [19] after integrating the expres-
sion with a derivative of δ-function we get:
( ) (0, ),kl t iU t′= − where
0
(0, ) ( , ) .k
k
dU t U k t
dk =
′ =
We take into account that in our two-level model the
average sum over all positions splits into two independent
parts:
1 2( ) ( ) ( ),l t l t l t= +
where
1 1( ) (2 1) (2 1, ),
s
l t s G s t= + +∑ 2 2( ) (2 ) (2 , ).
s
l t s G s t=∑
Then the average travel path of the particle becomes:
1 2 1 2( ) ( ) ( (0, ) (0, )).x t a l l ia U t U t′ ′= + = − +
We calculate the average quadratic displacement 2x of
particle in absence of external field. It can be done as for x :
2 2 2 2 2
1 2 1 2( ) ( ) [ (0, ) (0, )].x t a l l a U t U t′′ ′′= + = − +
Results and discussion
Taking into consideration the expressions for 1( , )U k t
and 2 ( , ),U k t after their double differentiation we have
2 ( )x t for two cases of initial conditions (Fig. 2). This result
is in good agreement with the one-level model because it has
the classic diffusion character for the model of identical
sites. However the deviations at small and large times differ
substantially for the model of alternating sites. Thus for
1 2
1t >> τ =
λ + λ
the motion slows down if the particle at origin is in a shal-
low potential pit (Fig. 2(b)).
Formation of impurity chains near the dislocation cores
in solid hydrogen under slow sample growth can be de-
scribed by this phenomena, at least at a qualitative level.
We will suppose that а is of order of the average distance
between impurity chains. Their density was estimated to be
[7,8]: 13 210 m .N −= Then we have 73 10 m.a N −≈ = ⋅
The average atomic impurity displacement in classic diffu-
sion requires /2x a= to reach a dislocation core during
sample growth. The proper average quadratic displacement
then is 2 2 /4.x a= We suppose that the average time to
Fig. 2. The impurity particle starts from a deep potential pit (it
is not actual for this paper) (a); the particle starts from a shal-
low pit (b).
724 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6
Diffusion model of the thermal conductivity plateau of weak solid solutions of neon in parahydrogen
reach of dislocation core is /(2 ).t a v= Here v = 8 mm/h ≈
62 10−≈ ⋅ m/s is the sample growth rate for crystal in which
the plateau appears. Making us of the expression
2 2 ,x Dt= we get for the diffusion coefficient of neon in
parahydrogen near the triple point (13.96 K [4]):
2 13/ 1.6 10D x v a −= = ⋅ m2/s.
Let us compare this result with the self-diffusion coeffi-
cient for parahydrogen at the same conditions [4] evaluated
from NMR experiments. We get
13
0exp (– / ) 1.8 10D D E kT −= = ⋅ m2/s,
where 7
0 3 10D −= ⋅ m2/s, / 200E k = K.
It is also possible to use theoretical calculations
by Ebner and Sang [21]. Then we obtain the diffusion co-
efficient versus temperature near melting point:
8 197/6 10 e TD − −= ⋅ ⋅ and finally 144.5 10D −≈ ⋅ m2/s.
Therefore the heavy isotopic neon impurity in the ab-
sence of other admixtures has the diffusivity near the melt-
ing temperature close to that of the self-diffusion of solid
hydrogen. It was shown in [7,8] that a twice as least growth
rate results in a disappearance of the plateau feature for the
same Ne fraction. Presumably, impurities have no enough
time to form linear structures under these conditions because
larger diffusion coefficients are needed for this purpose.
Therefore it is possible to assume that for simple estimate
the neon diffusion coefficients in solid parahydrogen near
the melting point does not exceed 132 10−⋅ m2/s.
After the particle reaches the dislocation core, its mobil-
ity can increase by several orders [22]. It is the reason why
impurities segregation in linear chains which results in the
plateau in the thermal conductivity as a result of the redi-
stribution in the impurity subsystem.
Conclusions
Our theoretical model allows us to study at qualitative
level the diffusive kinetics of heavy substitution impurities
accomplishing stochastic jumps. The classical one-dimen-
sional model of nearest jumps of a solitary heavy particle
with the use of the Pauli equation (Master equation) gives
an exact solution to the problem of the average displace-
ments in the field of alternating potential pits of two types
with equal distances.
The results help to understand the origin of the symme-
tric plateau in the temperature dependence of the thermal
conductivity for bulk samples slowly grown. The main
idea is that hard quasiisotopic impurity could segregate
into thin long chains near dislocation cores. Neon impurity
chains can be confined for a long time. Such a transition
from the spatially homogeneous distribution of isolated
atomic impurities to their segregation in chains provides a
new collective mechanism of phonon scattering [8]. The
result of such interactions leads to a temperature-inde-
pendent thermal conductivity. The estimates of the neon
diffusion coefficient for (p-H2)1–cNec solutions with
c = 0.0001 at. %; 0.0002 at. % is close to the coefficients
of self-diffusion of hydrogen. Thus our calculations indi-
rectly confirm the model of phonons scattering on the
heavy impurity chains suggested previously for the expla-
nation of the plateau.
Acknowledgments
The author expresses his gratitude to O.V. Usatenko
and V.V. Ulyanov for useful discussions in the process of
theoretical model construction and to A.M. Kosevich for
moral support.
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