On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering
Previous experimental studies of the thermal conductivity of plastically deformed lead crystals in the superconducting state have shown strong anomalies in the thermal conductivity. Similar effects were also found for the thermal conductivity of bent ⁴He samples. Until now, a theoretical explanati...
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| Zitieren: | On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering / J.A.M. van Ostaay, S.I. Mukhin, L.P. Mezhov-Deglin // Физика низких температур. — 2012. — Т. 38, № 11. — С. 1336–1339. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1179632017-05-28T03:04:55Z On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering van Ostaay, J.A.M. Mukhin, S.I. Mezhov-Deglin, L.P. К 75-летию Л.П. Межова-Деглина Previous experimental studies of the thermal conductivity of plastically deformed lead crystals in the superconducting state have shown strong anomalies in the thermal conductivity. Similar effects were also found for the thermal conductivity of bent ⁴He samples. Until now, a theoretical explanation for these results was missing. In this paper we will introduce the process of phonon–kink scattering and show that it qualitatively explains the anomalies that experiments had found. 2012 Article On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering / J.A.M. van Ostaay, S.I. Mukhin, L.P. Mezhov-Deglin // Физика низких температур. — 2012. — Т. 38, № 11. — С. 1336–1339. — Бібліогр.: 8 назв. — англ. 0132-6414 PACS: 72.10.–d, 72.15.Eb, 66.70.–f, 61.72.Lk, 67.80.–s http://dspace.nbuv.gov.ua/handle/123456789/117963 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 75-летию Л.П. Межова-Деглина К 75-летию Л.П. Межова-Деглина |
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К 75-летию Л.П. Межова-Деглина К 75-летию Л.П. Межова-Деглина van Ostaay, J.A.M. Mukhin, S.I. Mezhov-Deglin, L.P. On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering Физика низких температур |
| description |
Previous experimental studies of the thermal conductivity of plastically deformed lead crystals in the superconducting
state have shown strong anomalies in the thermal conductivity. Similar effects were also found for
the thermal conductivity of bent ⁴He samples. Until now, a theoretical explanation for these results was missing.
In this paper we will introduce the process of phonon–kink scattering and show that it qualitatively explains the
anomalies that experiments had found. |
| format |
Article |
| author |
van Ostaay, J.A.M. Mukhin, S.I. Mezhov-Deglin, L.P. |
| author_facet |
van Ostaay, J.A.M. Mukhin, S.I. Mezhov-Deglin, L.P. |
| author_sort |
van Ostaay, J.A.M. |
| title |
On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering |
| title_short |
On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering |
| title_full |
On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering |
| title_fullStr |
On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering |
| title_full_unstemmed |
On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering |
| title_sort |
on the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon–kink scattering |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2012 |
| topic_facet |
К 75-летию Л.П. Межова-Деглина |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117963 |
| citation_txt |
On the low-temperature anomalies in the thermal
conductivity of plastically deformed crystals due
to phonon–kink scattering / J.A.M. van Ostaay, S.I. Mukhin, L.P. Mezhov-Deglin // Физика низких температур. — 2012. — Т. 38, № 11. — С. 1336–1339. — Бібліогр.: 8 назв. — англ. |
| series |
Физика низких температур |
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AT vanostaayjam onthelowtemperatureanomaliesinthethermalconductivityofplasticallydeformedcrystalsduetophononkinkscattering AT mukhinsi onthelowtemperatureanomaliesinthethermalconductivityofplasticallydeformedcrystalsduetophononkinkscattering AT mezhovdeglinlp onthelowtemperatureanomaliesinthethermalconductivityofplasticallydeformedcrystalsduetophononkinkscattering |
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2025-07-08T13:04:52Z |
| last_indexed |
2025-07-08T13:04:52Z |
| _version_ |
1837084068782538752 |
| fulltext |
© J.A.M. van Ostaay, S.I. Mukhin, and L.P. Mezhov-Deglin, 2012
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 11, pp. 1336–1339
On the low-temperature anomalies in the thermal
conductivity of plastically deformed crystals due
to phonon–kink scattering
J.A.M. van Ostaay1, S.I. Mukhin2, and L.P. Mezhov-Deglin3
1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
2Theoretical Physics and Quantum Technologies Department, NITU MISIS, 119991 Moscow, Russia
E-mail: i.m.sergei.m@gmail.com
3Institute of Solid State Physics RAS, 2 Institutskaia Str., 142432 Chernogolovka, Russia
Received August 31, 2012
Previous experimental studies of the thermal conductivity of plastically deformed lead crystals in the super-
conducting state have shown strong anomalies in the thermal conductivity. Similar effects were also found for
the thermal conductivity of bent 4He samples. Until now, a theoretical explanation for these results was missing.
In this paper we will introduce the process of phonon–kink scattering and show that it qualitatively explains the
anomalies that experiments had found.
PACS: 72.10.–d Theory of electronic transport; scattering mechanisms;
72.15.Eb Electrical and thermal conduction in crystalline metals and alloys;
66.70.–f Nonelectronic thermal conduction and heat-pulse propagation in solids; thermal waves;
61.72.Lk Linear defects: dislocations, disclinations;
67.80.–s Quantum solids.
Keywords: phonon thermal transport, low temperatures, kinks on dislocation line, phonon–kink scattering anomaly.
1. Introduction
Previous studies of the thermal conductivity of lead
crystals in the superconducting state, which were deformed
plastically by low-temperature stretching of the initially
perfect samples, and observation of the recovery processes
on annealing of the samples at room temperatures, had
demonstrated strong anomalies in the thermal conductivity
of the deformed Pb crystals below 4 K [1]. The same ef-
fects were also seen in weakly bent Bi crystals [2]. Fur-
thermore, experiments on the thermal conductivity of hcp
4 He crystals grown from high pure 4 He in a long capil-
lary had also revealed strong anomalies in thermal conduc-
tivity of samples that were weakly deformed by bending
them at temperatures near and above 0.4 K [3,4].
Several attempts for a theoretical explanation of these
results have been made, but none have unfortunately been
completely successful [5]. In this primer paper however,
we introduce a new model for explaining the observed
anomalies in the thermal conductivity of the weakly de-
formed crystals from high pure matter. This model is based
on phonon scattering on mobile kinks on the newly in-
duced dislocation lines. Previously, a similar model, based
on scattering of electrons by mobile kinks, has been intro-
duced for the explanation of the anomaly in the electronic
contribution to the thermal conductivity of plastically de-
formed copper crystals [6]. In systems where the phonon
thermal conductivity is the main contribution to the trans-
fer of heat flux, such as quantum crystals, metal crystals in
superconducting state and nonmetals, the scattering of
thermal phonons by the mobile kinks on dislocation lines
induced under weak deformation of initially perfect sam-
ples at reduced temperatures seems to be the natural expla-
nation of the experimentally observed effects. This paper
will only introduce this process and show the main results
of detailed calculations of the thermal conductivity in dif-
ferent directions relative to the glide plane of the disloca-
tions. We have found that in the crystals where scattering
of phonons on kinks is the dominant scattering process our
theoretical results can qualitatively reproduce the experi-
mental features. The detailed calculations referred to in this
primer note and the quantitative fit of the experimental
results can be found in a paper which is soon to appear [7].
On the low-temperature anomalies in the thermal conductivity
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 11 1337
2. Kinematics
For a description of the kinematics of phonon–kink
scattering we use a similar procedure from Ref. 8. We con-
sider a crystal which contains dislocations due to an exter-
nal influence on the crystal. The dislocations lie in the xz
plane and the direction parallel to the dislocations is the z
direction.
Around a dislocation the displacement ju can be de-
composed in two components
= .s d
j j ju u u+ (1)
The “static” displacement s
ju depends on the presence
of the kinks and can be written as
( ( ))0
0 0= ( : ) ( )e ,i z z ts s
j j ju f uκ −
⊥
κ
κ ξ κ +∑ r (2)
where 0 ( )ξ κ is the Fourier transform of the dislocation's
line displacement due to the kink, ( : )jf ⊥ κr is a propor-
tionality constant and 0
s
ju is displacement around the
straight dislocation without kink. The abbreviation ⊥r
indicates ( , ).x y The “dynamical” displacement d
ju has its
origin in the phonons and can be expressed as a superposi-
tion of plane waves,
,
= ( , ) ( , )e ,d i
j j
s
u q s e s ⋅∑ k r
k
k k (3)
where s indicates the polarization of the lattice vibrations
and e is the polarization vector. Treating the kink in a
harmonic trap (potential well) with angular frequency Ω
and writing 0 ( , )sω k for the angular frequency of the pho-
nons results in the total Lagrangian
int0
= const,q zL L L L+ + + (4)
{ }* 2 *
0
,
= ( , ) ( , ) ( , ) ( , ) ( , ) ,
2q
s
VL q s q s s q s q sρ
−ω∑
k
k k k k k (5)
{ }2 2 0 2
0 0 00
= ( ) ( ( ) ) ,
2z
ML z t z t z−Ω − (6)
( )0
int 0 0
, ,
= ( )e ( ) ( ) ( , ) ( , ).ik z tz
z z j j
j s
L i V k z t k F q s e s− ∗ ∗− ρ ξ∑
k
k k k
(7)
In the equations above, ρ is the density of the crystal,
3=V L its total volume, M is the kink mass [6], 0z indi-
cates the position of the kink, 0
0z is its rest position and
( )jF k is the Fourier transform of ( : )jf ⊥ κr , being de-
fined as
2
2
1( ) e ( : ) ,i
j j zF f k d r
L
− ⋅⊥ ⊥
⊥ ⊥≡ ∫ k rk r (8)
where = ( , )x yk k⊥k .
From the interaction term intL one can determine the pho-
non–kink scattering amplitude per unit time ( , ; ', ).A s s′k k
Due to phonon–kink scattering, phonons are no longer de-
scribed by the Bose–Einstein distribution 0
0( ( , ))N sω k . In
the presence of a small temperature gradient T∇ , the li-
near correction to the Bose–Einstein distribution sNδ k is
given by
0 00 0
0 02
( , ) ( , )
( ( , ))(1 ( ( , ))
B
s s
N s N s T
k T
ω ∂ω
− ω + ω ∇ ⋅ =
∂
k k
k k
k
3
'3= ( , ; ', )[ ]
(2 )
s s
s
d k s s N N ′
′
′
′ δ − δ
π
∑∫ k kk kP (9)
with
0 0
0 0
= ,
( ( , ))[1 ( ( , ))]
s
s
N
N
N s N s
δ
δ
ω + ω
k
k
k k
(10)
and
2 2
ph( , ; ', ) = | ( , ; ', ) |s s N L A s s′ ′ ×k k k kP
0 0[ ( , ) ( , ); , ]x zK s s q q′ ′× ω −ω ×k k
0 0
0 0( ( ', ))[1 ( ( , ))]N s N s′× ω + ωk k (11)
with phN the number of phonons in the crystal and
1( ; , ) = exp [ ( ) ]x z zK q q dzdz dt iq z z i t
L
′ ′ω − + ω ×∫
exp [ ( ,0)]exp [ ( , )] .x xiq z iq z t′× 〈〈 − ξ ξ 〉〉 (12)
With Eq. (9) a full kinematical treatment of the phonon–
kink scattering is possible.
3. Heat flow
With the full kinematics of the phonon–kink scattering
at our disposal we are able to study the effect of phonon–
kink scattering on the heat flow through the crystal. The
heat flux Q is given by
3
0
03
( , )
= ( , ) ,
(2 )
s
s
sd k s N T
∂ω
ω δ ≈ −χ∇
∂π
∑∫ k
k
Q k
k
(13)
where χ is the matrix of the thermal conductivity. For
simplicity, we will assume here that this matrix only has
two distinct diagonal elements and no off-diagonal ele-
ments
0 0
= 0 0 .
0 0
⊥
⊥
⎛ ⎞χ
⎜ ⎟
χ χ⎜ ⎟
⎜ ⎟χ⎝ ⎠
(14)
This implies that there two distinct heat flows. One along
the dislocation
= ( ) ,zQ T−χ ∇ (15)
and one perpendicular to,
= ( ) ,Q T⊥ ⊥ ⊥−χ ∇ (16)
with ( ) = (( ) , ( ) ,0)x yT T T⊥∇ ∇ ∇ .
J.A.M. van Ostaay, S.I. Mukhin, and L.P. Mezhov-Deglin
1338 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 11
Combined Eqs. (9) and (13) allow for a full calculation
[7] of χ and ⊥χ . This full calculation shows that there
are four different temperature regimes for the thermal con-
ductivity. These four intervals are
regime 1: ,
regime 2: ,
regime 3: ,
regime 4: ,
T T
T T T
T T T
T T
ω
ω Ω
∗
Ω
∗
(17)
Here
0 (1 / )
= ,
B
T
kω
ω = ,
B
T
kΩ
Ω
2 22= ,
B
MT
k
∗ Ω (18)
where is the typical size of the kink and 0 (1/ )ω is the
angular frequency for a phonon with a wavelength equal to
the size of the kink. The three temperatures are ordered as
follows:
.T T T∗
ω Ω (19)
In the calculations we also took into account that in real
experiments, one does not measure the thermal conductivi-
ty in one particular direction, but rather an average over
different direction as one has no perfect control of the
orientation of the kinks. As the scattering in different di-
rections is a consecutive process, the scattering rates for
the different processes add. This means that the measured
thermal conductivity χ is found from
1 1 1= (1 ) ,− − −
⊥χ βχ + −β χ (20)
where [0,1]β∈ .
Therefore, one ends up with the following scaling be-
havior for 1−χ ,
4
1 5 7
1
ph 1 5
1 3 5
, regime1,
, regime 2,
(1 ) , regime 3,
(1 ) regime 4,
k
k
k
k
n T
T n T T
n
T n T T
T n T T
−
− − −
−
− −
− − −
⎧β+
⎪
⎡ ⎤⎪β + +⎣ ⎦⎪
χ ⎨ ⎡ ⎤β + −β + β⎪ ⎣ ⎦⎪
⎡ ⎤⎪β + β + −β⎣ ⎦⎩
∼
C
C D
C D
(21)
where 2
ph ph= /n N L and = /k kn N L are the phonon and
kink densities, respectively. The script letters indicate other
quantities than the ones expressed already in the equations
above.
4. Comparison with experimental data and conclusion
We compare our qualitative theoretical results with ex-
perimental data in Ref. 1. In Fig. 1 of this reference one
sees that for a sample of highly purified lead which has
been plastically stretched at low temperatures, the thermal
conductivity at low temperatures has a peculiar shape: up
to certain temperature it increases with temperature, then
starts decreasing and for even higher temperatures it starts
increasing with temperature again. Annealing can make
this effect less pronounced, but it seems not to be able to
completely remove this feature. Assuming that β is nei-
ther 0 nor 1 and taking numerical results into account [7],
one sees from Eq. (21) that for low temperatures χ scales as
4
4 ,
k
T
n T+C
(22)
for higher temperatures as
5
4 ,
k
T
n T+C
(23)
for even higher temperatures as
1
2 ,
k
T
n T
−
−+C
(24)
and at the highest temperatures as
5
4 .
k
T
n T+C
(25)
This mimics the behavior shown in the experimental data.
In the semi-highest temperature regime the thermal con-
ductivity will decrease with temperature, while in the other
regimes the thermal conductivity will increase with tem-
perature.
When comparing curves 6 and 7 in Fig. 1 [1], one sees
that curve 6 and 7 have similar behavior for higher temper-
atures. For lower temperatures though, curve 6 lies under
curve 7. As curve 6 shows the thermal conductivity for
a sample which has been deformed, while curve 7 shows
the thermal conductivity for a lead sample which has not
been deformed at all, this is in full agreement with the
theory. The power-law for the thermal conductivity for a
sample with none or very little kinks has a lower power
than that for a sample with many kinks. Therefore it makes
sense that for low temperature, the thermal conductivity for
a sample with many kinks is lower than that for a sample
with very little kinks. For this observation, we can there-
fore conclude that samples which have not been plastically
deformed at all show a much weaker version of this effect,
proving that this effect is indeed caused by phonon–kink
scattering. This also shows that only a small amount of
kinks are needed to let this effect appear.
The experimental data for the normal state does not
match with our theoretical calculations at all, since in the
normal state the phonon contribution to the heat flux trans-
port is much weaker than the electron contribution. There-
fore the effect of phonon–kink scattering is not visible in
that case.
We thus see that the results of our model qualitatively
agree with the experimental data. For a quantitative com-
parison we refer to Ref. 7. The work of S.I.M. is in part
supported by RFFI grant 12-02-01018.
On the low-temperature anomalies in the thermal conductivity
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 11 1339
1. L.P. Mezhov-Deglin, Sov. Phys. JETP 50, 733 (1979).
2. V.N. Kopylov and L.P. Mezhov-Deglin, Sov. Phys. Solid
State 15, 8 (1973).
3. L.P. Mezhov-Deglin and A.A. Levchenko, Sov. Phys. JETP
55, 166 (1982).
4. L.P. Mezhov-Deglin and A.A. Levchenko, Sov. Phys. JETP
59, 1234 (1984).
5. A.V. Markelov, Sov. Phys. JETP 61, 118 (1985).
6. S.I. Mukhin, Sov. Phys. JETP 64, 81 (1986).
7. J.A.M. van Ostaay and S.I. Mukhin, to be published else-
where soon.
8. T. Ninomiya, J. Phys. Soc. Jpn. 25, 830 (1968).
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