The plateau effect in thermal conductivity of solid hydrogen with neon impurity
The thermal conductivity of solid hydrogen with 1–2 ppm Ne impurity was investigated in the temperature range 1.5–10 K on samples grown from the liquid phase at various growth rates. The result differ qualitatively from those obtained on samples grown from the gas phase: the thermal conductivity c...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
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| Цитувати: | The plateau effect in thermal conductivity of solid hydrogen with neon impurity / N.N. Zholonko // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 676-679. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-1217872017-06-17T03:03:15Z The plateau effect in thermal conductivity of solid hydrogen with neon impurity Zholonko, N.N. Quantum Crystals The thermal conductivity of solid hydrogen with 1–2 ppm Ne impurity was investigated in the temperature range 1.5–10 K on samples grown from the liquid phase at various growth rates. The result differ qualitatively from those obtained on samples grown from the gas phase: the thermal conductivity curve exhibited a dip in a broad plateau. The relaxation model is suggested to explain the effect supposedly due to linear impurity structures, arranged on dislocation lines. A comparison with the case of isolated neon atoms homogeneous distributed in the solid hydrogen is made. 2007 Article The plateau effect in thermal conductivity of solid hydrogen with neon impurity / N.N. Zholonko // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 676-679. — Бібліогр.: 13 назв. — англ. 0132-6414 PACS: 67.80.Gb; 65.40.–b http://dspace.nbuv.gov.ua/handle/123456789/121787 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Quantum Crystals Quantum Crystals |
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Quantum Crystals Quantum Crystals Zholonko, N.N. The plateau effect in thermal conductivity of solid hydrogen with neon impurity Физика низких температур |
| description |
The thermal conductivity of solid hydrogen with 1–2 ppm Ne impurity was investigated in the temperature
range 1.5–10 K on samples grown from the liquid phase at various growth rates. The result differ qualitatively
from those obtained on samples grown from the gas phase: the thermal conductivity curve exhibited
a dip in a broad plateau. The relaxation model is suggested to explain the effect supposedly due to linear impurity
structures, arranged on dislocation lines. A comparison with the case of isolated neon atoms homogeneous
distributed in the solid hydrogen is made. |
| format |
Article |
| author |
Zholonko, N.N. |
| author_facet |
Zholonko, N.N. |
| author_sort |
Zholonko, N.N. |
| title |
The plateau effect in thermal conductivity of solid hydrogen with neon impurity |
| title_short |
The plateau effect in thermal conductivity of solid hydrogen with neon impurity |
| title_full |
The plateau effect in thermal conductivity of solid hydrogen with neon impurity |
| title_fullStr |
The plateau effect in thermal conductivity of solid hydrogen with neon impurity |
| title_full_unstemmed |
The plateau effect in thermal conductivity of solid hydrogen with neon impurity |
| title_sort |
plateau effect in thermal conductivity of solid hydrogen with neon impurity |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2007 |
| topic_facet |
Quantum Crystals |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121787 |
| citation_txt |
The plateau effect in thermal conductivity of solid hydrogen with neon impurity / N.N. Zholonko // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 676-679. — Бібліогр.: 13 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT zholonkonn theplateaueffectinthermalconductivityofsolidhydrogenwithneonimpurity AT zholonkonn plateaueffectinthermalconductivityofsolidhydrogenwithneonimpurity |
| first_indexed |
2025-07-08T20:31:22Z |
| last_indexed |
2025-07-08T20:31:22Z |
| _version_ |
1837112160003555328 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 676–679
The plateau effect in thermal conductivity of solid
hydrogen with neon impurity
N.N. Zholonko
Cherkasy Bogdan Khmelnitski National University, 81 Blvd Shevchenko, Cherkasy 18031, Ukraine
E-mail: zholonko@yahoo.com
Received November 20, 2006
The thermal conductivity of solid hydrogen with 1–2 ppm Ne impurity was investigated in the tempera-
ture range 1.5–10 K on samples grown from the liquid phase at various growth rates. The result differ quali-
tatively from those obtained on samples grown from the gas phase: the thermal conductivity curve exhibited
a dip in a broad plateau. The relaxation model is suggested to explain the effect supposedly due to linear im-
purity structures, arranged on dislocation lines. A comparison with the case of isolated neon atoms homoge-
neous distributed in the solid hydrogen is made.
PACS: 67.80.Gb Thermal properties;
65.40.–b Thermal properties of crystalline solids.
Keywords: thermal conductivity, solid hydrogen.
Dedicated to V.A. Slyusarev's memory
Introduction
The same as helium, solid hydrogen has a very high
thermal conductivity [1,2], which near the maximum is
very sensitive to defects. For example, 1 ppm of the neon
impurity entails a more than an order of magnitude de-
crease. An impurity anomaly of the resonance type was
observed in gas-grown samples, in which the distribution
of neon is homogeneous [3,4]. It was demonstrated that
the solubility of neon in hydrogen is near 100 ppm [5]. So,
it could be presumed that experiments with neon concen-
trations about 1 ppm would not involve the decomposi-
tion of the solution. Note that in those experiments solid
parahydrogen contained an equilibrium concentration of
orthospecies (0.21%). In special experiments it was proved
[6] that by reducing considerably the ortho concentration
(down to 0.05% o-H2) did not result in noticeable changes
of the thermal conductivity. Influence of the natural iso-
tope impurity (deuterium) could also be neglected.
The resonance anomaly in the thermal conductivity
of weak Ne in p-H2 solid solutions is explained by the role
of quasi-local vibrations (QLV) due to the presence of
the heavy impurity. A specific and extremely important
role in almost ideal crystals belongs to the processes of
phonon–phonon scattering (N-processes), which redis-
tribute the phonons quasi-momentum.
This paper reports results for the system of parahydro-
gen–neon with extremely low concentrations of the heavy
neon impurity. In particular, we studied influence of the
crystal growth procedure on the thermal conductivity for the
case when samples were grown from the melt, which can re-
sults in a redistribution of impurities during solidification.
Experiment
Samples growth and thermal conductivity measure-
ments of solid hydrogen with (1–2)�10
–4
ppm neon were
performed in the cell described in [3]. The procedure was
as follows. After the thermal conductivity of a vacuum
deposited sample was measured it was melted and re-
crystallized at various growth rate. In Fig. 1, results of such
studies are shown for the concentration of neon 1 ppm for
two different growth rates in comparison with data for the
vacuum deposited sample of the same concentration. As
evident from the Fig. 1, the faster growth rate did not bring
any differences compared with the sample growth directly
from the gas phase. However, the slower rate changes the
behavior. It seems that the reason is in a different impurity
distribution in the sample. The slower growth led to the
plateau seen in the upper part of the curve.
© N.N. Zholonko, 2007
Results and discussion
The sublimation energy of neon and its melting tem-
perature are higher than those of hydrogen which limits
neon solubility in hydrogen and provokes a decomposi-
tion of the solution when the neon concentrations exceeds
a certain value. For very slow growth from the melt even
with extremely small concentrations, at least a partial seg-
regation could be expected.
On the basis of a simple gas-kinetic equation
K Cv�
1
3
2�, (1)
where � is relaxation time of the most important phonons
[7]; C is the heat capacity per unit volume; v is the average
phonon velocity, a qualitative estimate of the thermal con-
ductivity as a function of temperature could be done. If
the phonon scattering occurs primarily on certain defects
and the relaxation rate is a power dependence: �–1
~ �z
,
where � is the phonon frequency.
At sufficiently low temperatures the most important
phonons for thermal conductivity are those with the fre-
quencies of order the temperature. Then x = ���kT is ap-
proximately constant [7]. Since the low-temperature heat
capacity follows the law Ñ ~ T
3
, we get
K z� ��3 . (2)
Let us consider the case when dislocation scattering is
the main heat resistance mechanism. If the elastic fields
of dislocations scatter predominantly, � �� �1 [7]. Then
from Eq. (2) we have K ~ Ò
2
. Such dependence is really
observed, for example, after a rapid growth or a thermal
shock. For the scattering on dislocation cores we have
� ��� �1 [7]. In the latter case the thermal conductivity
is temperature independent. However, manifestations of
scattering on dislocation core was never observed in prac-
tice. Obviously it can be explained in the following way:
the elastic fields of dislocations are more long range,
while the core size is smaller (a few a lattice parameters).
Consequently, the main dislocation contribution in the
low-temperature thermal conductivity will be due to the
elastic fields of dislocations. However, the situation
might change, if the cores became more intensive scat-
ters, for example, chains of with impurities present. For a
fixed linear cylinder of a radius r, Relay’s expression be-
comes ( )� �� r /v4 3 3, that is the relaxation rate will be
�
�� �1
4 3
2
SN
r
v
, (3)
where N is the number of rigid cylindrical objects per unit
area, S is an empiric constant (we put in this work S = 1,
see Table 1). More detailed calculations of the thermal
conductivity requires taking into account the contribution
of all phonon modes and of the N-processes, which are
nonresistive, but actively redistribute the energy of ele-
mentary excitations.
It could be done within the framework of the relax-
ation model with N-processes [7,11]:
K K K�
1 2, (4)
K
k
v
kT x
dxC
x
x
T
1 2
3 4
2
0
2 1
� �
�
�
�
�
���
�
��
�
e
e( )
,
K
k
v
kT
2 2
3
2
� �
�
�
�
� �
� �
�
�
�
�
�
�
�
�� ��
�
�
�
� �
��
C
N
x
x
T
C
N R
x
x
x
dx
x4
2
0
2
4
1 1
e
e
e
e( ) ( ) 2
0
1
dx
T��
�
�
�
�
�
�
��
�
,
where � is Planck’s constant, � is the Debye temperature
(for solid hydrogen, � = 118.5 K [5]), � � �C R N
� � ��
1 1 1 is
the combined relaxation rate of resistive (� R
�1) and normal
(� N
�1) processes. In our work, in addition to (3), the fol-
The plateau effect in thermal conductivity of solid hydrogen with neon impurity
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 677
+ +
+
+
+
+
+++
+++
+
10
2
10
0
10
1
1 2 4 6 8
T, K
K
,
W
/(
m
)
·K
10
Fig. 1. Experimental temperature dependences of thermal con-
ductivity (p-H Ne2 1) �c c for various Ne concentrations and dif-
ferent manner of growing (received common with B.Ya. Goro-
dilov and A.I. Krivchikov) (c = 1 ððm for all samples besides
triangles): sublimation from gas phase (�); from liquid after
remelting under the rate of growth 8 mm/h (�); from liquid
after remelting under the rate of growth 16 mm/h (+); (c = 2 ppm)
from liquid under the rate of sample growth 8 mm/h (�).
lowing relaxation rates of resistive and normal processes
were chosen [3,12]:
� � �U U
E/T
N N BA T x A T x
v
L
� � � �� � �1 3 2 1 6 2 1e , , ; (5)
A N
V
N A� 9916 5 3
2
2 3 5
. /
/
�
�
� �
,
where � is the molar mass, � is the Grüneisen constant,
V is the molar volume, NA is the Avogadro number. In
Eq. (5) L is the dimension of the crystals, E � � �/ is the
constant of U-process activation. The relaxation rates of
U-processes in parahydrogen with the heat flow oriented
transversally and parallel to the crystallographic axis of hcp
lattice, the following expressions have been obtained [12]:
� � �UT T /T� �� �1 15 22 4 10 39 4( ) . exp( . ),
� � �U T /T||( ) . exp( . )� �� �1 15 214 10 318
The values of the constants are placed in Table 1.
Let us assume that during slow crystallization a distri-
bution of impurity atoms takes place in chains. For the new
mechanism of phonon scattering by linear structures to be
efficient the average distance between nearest impurities in
a chain must be at least comparable with the wave lengths
of the most important phonons at the proper temperature.
For estimations we consider the left part of the plateau in
Fig. 1. We assume that the start of the plateau from the
low-temperature side occurs when the role of impurity
scattering becomes predominant compared to boundary
scattering. Then we could assume that � �bound imp
� ��1 1 , ar-
riving at the estimated density N ~ 10
13
m
–2
for linear
chains. A similar estimate can be obtained for the right
edge of the plateau, but this time, we must compare � imp
�1
with the inelastic phonon–phonon processes: � �U
� ��1 1
imp .
It is possible to take for �U
�1 the same parameters as for
pure parahydrogen or the optimum for H2–Ne mixtures [5].
Then we get the same order of N for impurity strings. Taking
into account that the radius r of area of dislocation cores is
equal to a few lattice constants, we can state that the present
estimate of N is consistent with literary data [13].
To evaluate the average distance a� between impurities
in a string we assume that all impurities are in linear struc-
tures. Then we get a simple formula � �a N N cA� � �( ) =
= 10
–9
m, where � is the molar density of hydrogen, c is
the neon concentration. It is less than the wave lengths of
the most important phonons in the temperature range of
experiment. Since at lowermost temperatures the length
of such waves is large, the boundary scattering is the most
effective resistive mechanism. Above or near 4–5 K the
wave length of such phonons decreases. Therefore impu-
rities here are main phonon scatterers in the resonance
mechanism of QLV with only one frequency (Table 1).
However, as impurities aggregate in chains, such reso-
nance «hole» evidently cannot exist. This is what we have
in Figs. 1, 2,a for the concentration of 1 ppm: the transi-
tion on both sides of the plateau occurs with a dip without
any previous growth. If the concentration of neon is twice
as large (Fig. 2,b), there is a tendency of the thermal con-
ductivity to increase at the right side. It might be related
to some insignificant part of impurities which did not en-
ter in chains in the process of slow growth and remained
isolated and randomly distributed in the matrix of solid
parahydrogen.
Calculated thermal conductivities are shown in Fig. 2.
Here the relaxation model (5) was employed with the im-
purity mechanism (3) in comparison with the impurity
mechanism of QLV with only one resonance frequency.
One can see that the concentrations of 1 and 2 ppm result
actually in a plateau.
678 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
N.N. Zholonko
Table 1. Parameters of phonon–phonon scattering and of other processes
Scattering mechanism Parameter Scattering rate Notes
Boundaries L = 3 mm �
imp
� �1 v/L [1,2,5]
N-processes AN = (1–4)�10
5
�
N
1� � ANT x6 2 [5,12]
U-processes AU = 2�10
–15
, E = 38.4 K �
U
AUT x E T� � �1 3 2e / [5,12]
Impurity strings const � Nr4� 10 24� �
�� � �1
3
2
const
v
[7]
Resonant impurity scattering
� = 9, c1= 10
–6
, c2 = 2�10
–6
,
�
�
�
0
3
� D = 2.98�10
12
�
� �
� � ��
� �
�
1
6
1 0
2 2 2 2 2
f / D
/ /
( )
[ ( ) ] ( )
, f c /
D
�
3
2
2 4 3�� � � [3,13]
We feel that we can formulate the following physical
interpretation of the plateau. Unlike in samples grown
from gas with homogeneously distributed isolated neon
impurities, characterized by a unique frequency, strings
form a spectrum in slowly grown samples prepared.
Therefore, it is these frequencies that are solely important
for heat transport, the thermal conductivity becoming a
linear function temperature. In the low temperature, the
thermal conductivity (proportional to T
3
) due to boundary
scattering can be observed. On the other hand, the
high-temperature edge of the plateau will be «cut» by
U-processes.
Conclusions
Our results give evidence that the effect discovered
weak (p-H Ne2 1) �c c solutions in the thermal conductivity
(an unusual symmetric plateau) can be described and
explained as being due to linear aggregates of impurities.
This effect can be observed in slow grown samples. Such a
transition from the spatially homogeneous distribution of
isolated atomic impurities to their segregation in chains pro-
vides a new collective mechanism of phonon scattering.
Acknowledgments
Author expresses gratitude to B.Ya. Gorodilov and
A.I. Krivchikov for the everyday help in experimental
works and fruitful discussions.
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The plateau effect in thermal conductivity of solid hydrogen with neon impurity
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 679
1 2
T, K
K
,
W
/(
m
)
·K
3 4 5
10
30
50
70
90
1 2
T, K
K
,
W
/(
m
)
·K
3 4 5
10
30
50
20
40
a
b
Fig. 2. Calculated temperature dependencies of the thermal
conductivity (p-H Ne2 1) �c c. On Fig. 2,a: c = 1 ððm, up curve is
for resonant case with only one impurity frequency; bottom
curve — for linear impurity model. On Fig. 2,b: up curve —
linear impurity structures with c = 1 ppm; bottom curve — lin-
ear impurity structures with c = 2 ppm.
|