Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles
The features of phonon spectra and their effect on the vibrational heat capacity of linear chains of inert gas atoms adsorbed onto a substrate, which is the surface of nanotubes bound to a nanobundle. The influence of the substrate results both in a shift of the lower limit of the chain spectrum f...
Saved in:
| Date: | 2019 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2019
|
| Series: | Физика низких температур |
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/176075 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles / E.V. Manzhelii, S.B. Feodosyev, I.A. Gospodarev // Физика низких температур. — 2019. — Т. 45, № 3. — С. 404-412. — Бібліогр.: 21 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-176075 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1760752025-02-23T18:09:40Z Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles Фононні спектри та коливальна теплоємність квазіодновимірних структур, що утворені інертними газами на поверхні вуглецевих нанобандлів Фоннонные спектры и колебательная теплоемкость квазиодномерных структур, образуемых инертными газами на поверхности углеродных нанобандлов Manzhelii, E.V. Feodosyev, S.B. Gospodarev, I.A. Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) The features of phonon spectra and their effect on the vibrational heat capacity of linear chains of inert gas atoms adsorbed onto a substrate, which is the surface of nanotubes bound to a nanobundle. The influence of the substrate results both in a shift of the lower limit of the chain spectrum from zero, and in mechanical stress in the chain (its extension or compression) also. It is shown that in the case of a compressed chain, the non-central interaction between atoms is negative (repulsive), it results in a shift of the lower boundary of the spectrum of transverse vibrations to low frequencies and to a shortening of the part of the specific heat temperature dependence in which this dependence is close to exponential. Heterogeneity of the nanobundle structure can cause a change in the distances between atoms of the chain. It is shown both and analytically and numerically, that as a result of it, discrete levels with frequencies both above and below the quasi-continuous spectrum band can appear in the phonon spectrum of the chain. The discrete levels with frequencies below the quasi-continuous spectrum band lead to a further shortening of the temperature interval at which the temperature dependence of the specific heat is close to the exponential one. Вивчено особливості фононних спектрів, а також їх вплив на коливальну теплоємність лінійних ланцюжків атомів інертних газів, які адсорбовані на підкладку, що є поверхнею пов'язаних в нанобандл нанотрубок. Вплив підкладки призводить як до зміщення нижньої межі спектра ланцюжка від нуля, так і до виникнення механічної напруги в ланцюжку (його розтягування або стиснення). Показано, що у випадку стислого ланцюжка нецентральна взаємодія між атомами від’ємна (носить характер відштовхування), що призводить до зміщення нижньої межі спектра поперечних коливань в область низьких частот і до зменшення довжини ділянки на температурній залежності теплоємності, на якому ця залежність близька до експоненційної. Дефекти структури нанобандла можуть зумовити зміну відстаней між атомами ланцюжка. Як аналітично, так і чисельно показано, що внаслідок цього у фононному спектрі ланцюжка можуть виникнути дискретні рівні з частотами, які розташовані як вище, так і нижче смуги квазібезперервного спектра. Дискретні рівні з частотами нижче смуги квазібезперервного спектра призводять до подальшого скорочення температурного інтервалу, на якому температурна залежність теплоємності близька до експоненційної. Изучены особенности фононных спектров, а также их влияние на колебательную теплоемкость линейных цепочек атомов инертных газов, адсорбированных на подложку, представляющую собой поверхность связанных в нанобандл нанотрубок. Влияние подложки приводит как к смещению нижней границы спектра цепочки от нуля, так и к возникновению механического напряжения в цепочке (ее растяжению или сжатию). Показано, что в случае сжатой цепочки нецентральное взаимодействие между атомами отрицательно (носит характер отталкивания), что приводит к смещению нижней границы спектра поперечных колебаний в область низких частот и к уменьшению длины участка на температурной зависимости теплоемкости, на котором эта зависимость близка к экспоненциальной. Неоднородность структуры нанобандла может обусловить изменение расстояний между атомами цепочки. Как аналитически, так и численно показано, что при этом в фононном спектре цепочки могут возникнуть дискретные уровни с частотами, лежащими как выше, так и ниже полосы квазинепрерывного спектра. Дискретные уровни с частотами ниже полосы квазинепрерывного спектра приводят к дальнейшему сокращению температурного интервала, на котором температурная зависимость теплоемкости близка к экспоненциальной. This work was supported by Fundamental Investigation Foundation of NANU (Grant No. 4/18-N). 2019 Article Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles / E.V. Manzhelii, S.B. Feodosyev, I.A. Gospodarev // Физика низких температур. — 2019. — Т. 45, № 3. — С. 404-412. — Бібліогр.: 21 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/176075 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) |
| spellingShingle |
Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) Manzhelii, E.V. Feodosyev, S.B. Gospodarev, I.A. Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles Физика низких температур |
| description |
The features of phonon spectra and their effect on the vibrational heat capacity of linear chains of inert gas
atoms adsorbed onto a substrate, which is the surface of nanotubes bound to a nanobundle. The influence of the
substrate results both in a shift of the lower limit of the chain spectrum from zero, and in mechanical stress in the
chain (its extension or compression) also. It is shown that in the case of a compressed chain, the non-central interaction between atoms is negative (repulsive), it results in a shift of the lower boundary of the spectrum of
transverse vibrations to low frequencies and to a shortening of the part of the specific heat temperature dependence in which this dependence is close to exponential. Heterogeneity of the nanobundle structure can cause a
change in the distances between atoms of the chain. It is shown both and analytically and numerically, that as a
result of it, discrete levels with frequencies both above and below the quasi-continuous spectrum band can appear in the phonon spectrum of the chain. The discrete levels with frequencies below the quasi-continuous spectrum band lead to a further shortening of the temperature interval at which the temperature dependence of the
specific heat is close to the exponential one. |
| format |
Article |
| author |
Manzhelii, E.V. Feodosyev, S.B. Gospodarev, I.A. |
| author_facet |
Manzhelii, E.V. Feodosyev, S.B. Gospodarev, I.A. |
| author_sort |
Manzhelii, E.V. |
| title |
Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles |
| title_short |
Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles |
| title_full |
Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles |
| title_fullStr |
Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles |
| title_full_unstemmed |
Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles |
| title_sort |
phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2019 |
| topic_facet |
Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/176075 |
| citation_txt |
Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms on the surface of carbon nanotube bundles / E.V. Manzhelii, S.B. Feodosyev, I.A. Gospodarev // Физика низких температур. — 2019. — Т. 45, № 3. — С. 404-412. — Бібліогр.: 21 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT manzheliiev phononspectraandvibrationalheatcapacityofquasionedimensionalstructuresformedbyraregasatomsonthesurfaceofcarbonnanotubebundles AT feodosyevsb phononspectraandvibrationalheatcapacityofquasionedimensionalstructuresformedbyraregasatomsonthesurfaceofcarbonnanotubebundles AT gospodarevia phononspectraandvibrationalheatcapacityofquasionedimensionalstructuresformedbyraregasatomsonthesurfaceofcarbonnanotubebundles AT manzheliiev fononníspektritakolivalʹnateploêmnístʹkvazíodnovimírnihstrukturŝoutvoreníínertnimigazaminapoverhnívuglecevihnanobandlív AT feodosyevsb fononníspektritakolivalʹnateploêmnístʹkvazíodnovimírnihstrukturŝoutvoreníínertnimigazaminapoverhnívuglecevihnanobandlív AT gospodarevia fononníspektritakolivalʹnateploêmnístʹkvazíodnovimírnihstrukturŝoutvoreníínertnimigazaminapoverhnívuglecevihnanobandlív AT manzheliiev fonnonnyespektryikolebatelʹnaâteploemkostʹkvaziodnomernyhstrukturobrazuemyhinertnymigazaminapoverhnostiuglerodnyhnanobandlov AT feodosyevsb fonnonnyespektryikolebatelʹnaâteploemkostʹkvaziodnomernyhstrukturobrazuemyhinertnymigazaminapoverhnostiuglerodnyhnanobandlov AT gospodarevia fonnonnyespektryikolebatelʹnaâteploemkostʹkvaziodnomernyhstrukturobrazuemyhinertnymigazaminapoverhnostiuglerodnyhnanobandlov |
| first_indexed |
2025-11-24T06:45:40Z |
| last_indexed |
2025-11-24T06:45:40Z |
| _version_ |
1849653188840390656 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3, pp. 404–412
Phonon spectra and vibrational heat capacity
of quasi-one-dimensional structures formed by rare gas
atoms on the surface of carbon nanotube bundles
E.V. Manzhelii, S.B. Feodosyev, and I.A. Gospodarev
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Nauky Ave., Kharkiv 61103, Ukraine
E-mail: feodosiev@ilt.kharkov.ua
Received November 11, 2018
The features of phonon spectra and their effect on the vibrational heat capacity of linear chains of inert gas
atoms adsorbed onto a substrate, which is the surface of nanotubes bound to a nanobundle. The influence of the
substrate results both in a shift of the lower limit of the chain spectrum from zero, and in mechanical stress in the
chain (its extension or compression) also. It is shown that in the case of a compressed chain, the non-central in-
teraction between atoms is negative (repulsive), it results in a shift of the lower boundary of the spectrum of
transverse vibrations to low frequencies and to a shortening of the part of the specific heat temperature depend-
ence in which this dependence is close to exponential. Heterogeneity of the nanobundle structure can cause a
change in the distances between atoms of the chain. It is shown both and analytically and numerically, that as a
result of it, discrete levels with frequencies both above and below the quasi-continuous spectrum band can ap-
pear in the phonon spectrum of the chain. The discrete levels with frequencies below the quasi-continuous spec-
trum band lead to a further shortening of the temperature interval at which the temperature dependence of the
specific heat is close to the exponential one.
Keywords: phonon spectra, phonon heat capacity, linear chain, localized states.
1. Introduction
Quasi-one-dimensional (q1D) crystalline structures at-
tract great interest from both the fundamental and applied
points of view, in particular, as promising materials for
quantum computers. The interest is caused by the unique
properties of their quasiparticle spectra, such as the root
singularities of the spectral densities at the edges of quasi-
continuous spectrum bands, the thresholdless formation of
discrete levels localized near defects, etc. Due to the Lan-
dau–Peierls instability [1], the existence of these structures
is impossible without some three-dimensional substrate,
the choice of which is associated with considerable diffi-
culties. The substrate should ensure the stability of q1D
systems of sufficient length and, at the same time, mini-
mally distort its 1D spectral peculiarities.
Recently the adsorption of rare-gas atoms onto carbon
nanotube bundles is often used to obtain stable macroscopi-
cally long q1D structures [2–8]. In the grooves between the
nanotubes, the adsorbed atoms can form linear chains of
length ~ 10 μm. The length corresponds to the number of
atoms in the chain ~ 103–104. For the chains of this length,
the boundary effects can be neglected. The one-dimensional
nature of these structures is confirmed by both neutron-
diffraction studies [2] and heat capacity data [3–7]. Neutron
diffraction studies of 4He atoms adsorbed in grooves on the
nanobundle surface have shown the periodicity of the ar-
rangement of 4He atoms in the chain [9]. Theoretical calcu-
lations have shown the presence of a periodic potential along
the grooves on the surface of nanobundles [8]. The variation
of amplitude of this potential depends on the relative orien-
tation and displacement of nanotubes forming the groove.
The potential depth varies from the values slightly greater
than zero to 40 K. All this makes it possible to describe the
vibrational characteristics of the adsorbed chains within the
harmonic dynamics of the crystal lattice.
It was shown in [10] that starting with a certain fre-
quency 0ω , the vibrations of the linear chain deposited on
the crystal surface or in the bulk, actually do not extend
through the crystal matrix and are completely localized on
the chain. The frequency 0ω is determined by the contri-
bution of the interaction of an atom in the chain with the
© E.V. Manzhelii, S.B. Feodosyev, and I.A. Gospodarev, 2019
Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms
atoms of the crystal-matrix in the self-interaction matrix of
the atom in the chain, Thus, аt 0ω > ω , neither the struc-
ture nor the phonon spectrum of the crystal-matrix can
considerably change the spectral characteristics of atoms in
the chain. The effect of a crystal-matrix on the phonon
spectrum of the chain can be expressed in terms of only
one parameter, namely, the value of the initial frequency
0ω . This approximation is particularly profound for the
vibrational characteristics of the chains of inert gas atoms
adsorbed precisely on a carbon substrate because of the
large difference between the Debye temperatures of inert
gases and carbon structures.
At 0ω < ω , the vibrational spectrum of the atoms of the
adsorbed linear chains has a three-dimensional character
which determines the convergence of the mean-square dis-
placements of atoms in the chain and the stability of these
structures in a finite temperature range. The temperature
dependence of the phonon heat capacity of the adsorbed
linear chain necessarily contains a low-temperature inter-
val in which the temperature dependence of the heat capac-
ity is close to the exponential one.
At 0ω > ω , the vibrations of the atoms of the chain are
either quasi-localized or their propagation has one-
dimensional character and the spectral densities of these
atoms are well described by simple analytical expressions
obtained for one-dimensional models [11]. The bandwidth
of the quasi-continuous spectrum of the chain is deter-
mined by the interaction between the atoms of the chain
[10,11], which depends on the distance r between them.
Naturally, this distance is significantly affected by the in-
teraction of adsorbed atoms of the chain with the carbon
atoms of nanobundles. Therefore, as a rule, r does not
coincide with the distance 0r corresponding to the mini-
mum of the interatomic interaction potential in the chain.
At 0r r< , the parameter of noncentral interaction, which
determines the width of the spectrum of transverse vibra-
tions, is negative. It will result in a shift of the minimum
frequency of the quasi-continuous spectrum to the low-
frequency region [11] and, consequently, to the displace-
ment of the linear part of the heat capacity [10] to the re-
gion of lower temperatures. [11]. The formation of a com-
pressed chain 0( )r r< by atoms of inert gases seems quite
plausible because the period of the field created by nano-
tubes in the grooves is smaller than the equilibrium dis-
tance for most inert gases [9]. Note that the negative pa-
rameter of the noncentral interaction is intrinsic to many
solidified gases and metals [12,13].
The defects of nanotubes, as well as the incommensura-
bility of the periods of the adsorbed chain and of the field
along the groove between the nanotubes, can cause a local
change in the interaction of the atoms in the chain with
nanotubes. In the case of a local change in this interaction,
the distance between a pair of atoms in the chain can also
change, which, in turn, can result in the appearing of a lo-
calized state with a frequency below the lower limit of the
quasicontinuous spectrum of the chain [11]. It leads to a
shift of the linear part of the heat capacity curve to the low-
temperature region. The high-temperature part of the heat
capacity curve of atomic chains adsorbed on nanobundles
was studied in [14].
In this paper, we study the effect of the difference in the
interatomic distance r in the adsorbed chain from the dis-
tance 0r , corresponding to the minimum of the interaction
potential between the atoms in the chain, on its phonon
spectrum, formation conditions and characteristics of the
discrete vibrational levels localized at the defects. The con-
tribution of all these changes of the spectrum into the low-
temperature heat capacity is also studied.
2. Phonon spectrum and vibrational heat capacity of an
ideal adsorbed atomic chain
It was shown in [10] that a chain of atoms adsorbed in a
groove between nanotubes with a sufficiently high degree
of accuracy can be considered as a chain in an external
field that determines the initial frequency of its quasi-
continuous spectrum. Basing on this result, we will consid-
er the chain of atoms adsorbed in the grooves between
nanotubes as a chain of atoms in an external periodic field.
We take into account an interaction only between the near-
est neighbors, which is quite natural for inert gases. In this
case, the dispersion relations have the form:
( ) ( )
( ) ( )
2 2
0
2 2
0
4
sin ,
2
4
sin .
2
l
l l l
kak k
m
kak k
m
τ
τ τ τ
β
ε ≡ ω = ε +
β
ε ≡ ω = ε +
(1)
Here 2ε ≡ ω is the squared frequency, k is the wave vec-
tor, the indices l and τ correspond to the longitudinal and
transverse vibrations respectively. For a pairwise isotropic
interaction between atoms, the parameter of the central
interaction α and the parameter of the noncentral interac-
tion β are expressed in terms of the potential of this inter-
action ( )rϕ by the following range:
( ) ( ) ( ) ( )2
2
1;l
r r
r r
r rr
τ
∂ ϕ ∂ϕ
β = β =
∂∂
.
We note that the symmetry condition for the tensor of elas-
tic moduli should be applied to the whole system (includ-
ing not only the chain, but also the substrate), since it is the
interaction of the chain with the substrate that ensures the
stability of the chain. Therefore, the transverse vibrations
of the atoms in the chain (1) are not flexural vibrations, so
their dispersion relation is not 4( ) ~k kτε in the long-
wavelength region.
Unlike the positive parameter of the central interaction
lβ , the parameter of the noncentral interaction τβ changes
its sign according to the relative position of the atom with
respect to the minimum of the interatomic pairwise poten-
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3 405
E.V. Manzhelii, S.B. Feodosyev, and I.A. Gospodarev
tial of the interaction ( )rϕ . For the case of a compressed
chain 0( ),r r< the parameter of the noncentral interaction
is negative. Dispersion relations for 0r r> and 0r r< are
presented in Fig. 1. In what follows, the width of the con-
tinuous spectrum of longitudinal oscillations will be denot-
ed by l∆ , and the width of the continuous spectrum of
transverse oscillations, by ( 4 / , 4 / ).l l m mτ τ τ∆ ∆ = β ∆ = β
In [11] the problem of the vibrational properties of both
the ideal chain of atoms in an external field and a chain
with a defect is solved by using the Jacobian matrix meth-
od [15–17]. The space of displacements of atoms of a
chain is represented as a direct sum of orthogonal subspa-
ces ( ) ( ) ,H H H− += ⊕ where the subspace ( )H − is the
subspace of in-phase displacements of atoms, and the sub-
space ( )H + is the subspace of anti-phase displacements.
Each of them is a linear span of the sequence of vectors
{ }( )
0 0
n
n
h
∞−
=
L and { }( )
0 0
,n
n
h
∞+
=
L respectively. Here, ( )
0h −
and ( )
0h + are the so-called generating vectors, correspond-
ing to in-phase and anti-phase displacements of two neigh-
boring atoms, respectively, and L is the dynamic operator
which has the form:
( ) 0 , ,( 2 ) ( )
, l l RR l R R a R R a
ik ix ikR R
m
+ −′ ′ ′ε + β δ − β δ + δ
= δ δ +′L
0 , ,( 2 ) ( )
( )RR R R a R R a
iy iz ikm
τ τ τ + −′ ′ ′ε + β δ − β δ + δ
+ δ + δ δ
(2)
(R and R′ are the coordinates of the chain atoms).
Complete information about the vibrational spectrum of
the chain is contained in its Green function ˆ ( )G ε =
1ˆ ˆ( ) ,I L −= ε − where Î is the unit operator. In the formalism
of the Jacobian matrices, all matrix elements of the operators
( )ˆ ( )G ± ε are expressed in terms of elements
( ) ( ) ( )
00 0 0
ˆ( ) , ( )( )i iG h G h
± ± ±
ε = ε [18], for which we obtain [11]:
____________________________________________________
in the subspace ( ) :H −
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
max
000
0
min
000
0
2 1 sgn , 0,
2 1 sgn , 0,
i
i i ii
i i
i
i i ii
i i
G Z
G Z
−
−
ε − ε
ε = + ε ε − ε ∆ >
∆ ε − ε
ε − ε
ε = − + ε ε − ε ∆ <
∆ ε − ε
(3)
Fig. 1. (Color online) The dispersion relations (1) are shown by the curves 1 and 2 in both fragments, for the cases r1 > r0 (a) and r2 < r0 (b),
respectively.
406 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3
Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms
in the subspace ( )H + :
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0
max00
max
0
min00
min
2 1 sgn , 0,
2 1 sgn , 0.
i
i i ii
i i
i
i i ii
i i
G Z
G Z
+
+
ε − ε ε = − − ε ε − ε ∆ >
∆ ε − ε
ε − ε ε = + ε ε − ε ∆ <
∆ ε − ε
(4)
_______________________________________________
Here ( ) min min max( ) ( ) ( )i i i iZ iε ≡ Θ ε − ε + Θ ε − ε Θ ε − ε −
max( ),i− Θ ε − ε and ( )xΘ is the Heaviside Θ-function.
The values miniε and maxiε are squares of the minimum
and maximum vibration frequencies respectively. Figures 2
and 3 show the Green functions in the subspaces and for
the longitudinal and transverse vibrations of the chain at
0r r> and at 0r r< . We emphasize that for transverse vi-
brations 0τ∆ < and min 0 | |τ τ τε = ε − ∆ for 0r r< .
The density of the vibrational states of the chain has the
form:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }1 Im 2 .
6 l lg G G G G− + − +
τ τ
ε = ε + ε + ε + ε π
Since this function contains the spectral densities of
transverse and longitudinal vibrations, it has square root
singularities not only at the edges of the spectrum, but also
within it. Figure 4 shows the dependencies ( )g ε for 0r r>
and 0r r< . For 0r r< , the band of the quasi-continuous
spectrum is much wider because of high values of the in-
teraction parameter lβ , peculiar to these interatomic dis-
tances, which determines the longitudinal vibrations, and
of negativity of the parameter τβ , which determines the
transverse vibrations.
The spectral densities of oscillations shown in Fig. 4 cor-
respond to heat capacity temperature dependences with dif-
ferent lengths of the parts that are close to exponential. Figu-
re 5 shows the temperature dependences of the contributions
to the low temperature heat capacity of the chains ( )VC T =
( ) 2 ( )Vl VC T C Tτ= + of the longitudinal ( )VlC T and trans-
verse vibrations ( )VC Tτ (curves 1 and 2, respectively), and
also CV(T) — curves 3. Here T is the temperature. The
Fig. 2. (Color online) Real (1) and imaginary (2) parts of Green's function (3) for the cases r > r0 (a) and r < r0 (b). The upper parts of
the both fragments show the functions ( ) ( )lG − ε ; the lower parts show the functions ( ) ( )G −
τ ε .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3 407
E.V. Manzhelii, S.B. Feodosyev, and I.A. Gospodarev
position of the linear-like part of ( )VC T is determined by
the position of the inflection point. When 0r r< , the initial
frequency of the continuous spectrum shifts to low frequen-
cies. It results in the fact that the linear-like part of the heat
capacity curve starts at temperatures lower than for 0r r> .
When 0r r< , the heat capacity curve ( )VlC T is flattened
because of the large width of the quasi-continuous spectrum.
At relatively high temperatures in both cases, the heat capac-
ities ( )VC T have close values.
3. The effect of a local change in the distance between a
pair of atoms on the vibrational spectrum
and the phonon heat capacity
Consider an adsorbed chain with a local defect, namely,
an isolated pair of atoms the distance between which dif-
fers from the distance between the other neighboring at-
oms. In the technique of the Jacobian matrices this defect
is a degenerate local perturbation [17,19]. The frequencies
of local vibrations caused by this defect can be obtained
from the Lifshitz equation [20], which, applied to the prob-
lem in question, can be written as:
( ) ( ) ( )Re 1/i iG ± ±ε = λ . (5)
Here ( )
i
±λ are the operators describing the contribution of
the defect to the operators ( )
iL − and ( ).iL + The operators
( )
iL − and ( )
iL + are induced by operator (2) in the subspaces
( )H − and ( )H + respectively for displacement along the
crystallographic direction i .
In the subspace of the in-phase vibrations ( )H − , the
change in the interaction between the atoms of the chain
defect is not shown. Therefore, the operator ( )
i
−λ is deter-
mined only by the change in the interaction of the chain
with the substrate, that is, the change in the initial frequen-
cy of the quasi-continuous spectrum 0 def 0 (1 )i i iε = ε + δ .
In this case, the operator ( )
0i ii
−λ = ε δ .
In the subspace ( ),H + both the change in the value of
the interaction between the atoms of the defect with the
substrate, and the change in the interaction between the
atoms of the defect, is shown. This change can be written
as def (1 )i i i∆ = + η ∆ , and the operator ( )
i
+λ can be writ-
ten as ( )
0( /2)i i i ii
+λ = ε δ + ∆ η .
When studying the low-temperature heat capacity, the
localized states lying below the initial frequency of the
quasi-continuous spectrum are of considerable interest. For
convenience, we will call them gap localized states. De-
note the square of the frequency of the gap state by gε .
The conditions for the existence of solutions of the Lifshitz
equation (5) can be easily obtained from the Green func-
Fig. 3. (Color online) Real (1) and imaginary (2) parts of Green's functions ( ) ( )lG + ε (4). Notation is completely analogous to the nota-
tion used in the previous figure.
408 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3
Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms
tions (3), (4) shown in Figs. 2, 3. The detailed description
of these conditions is given in [11]. In this paper, we dwell
only on the conditions of their existence. Thus, because the
value of ( )
minRe ( )iG − ε tends to −∞ at 0i∆ > , the gap
levels appear at any negative values of ( )
i
−λ (i.e., values of
).iδ When 0i∆ < (transverse vibrations at 0r r< ), the
gap vibrations localized on the in-phase vibrations of at-
oms of the chain appear at 0/(2 ).i i iδ < −∆ ε For anti-phase
vibrations, the gap levels appear at any negative values of
( )
i
+λ at 0τ∆ < . In other cases, in the subspace ( )H + gap
vibrations occur at 0(1 ) /(2 )i i i iδ < − + η ∆ ε .
The energy values of the gap vibrational levels are the
poles of the Green function of the perturbed system
( ) ( ) ( ) ( ) ( ) ( )1
00 0 0( , ) ( , [ ] ).i i iiG h I L h± ± ± ± ± ±−ε λ = ε − − λ The resi-
dues at these poles are the so-called intensities 0dµ of
these levels. If the local spectral density in a defect-free
chain ( ) ( )1
00 00( ) Im ( )i iG± ±−ρ ε = π ε is normalized to unit, then
the relation ( )
000 ( ) 1 di
D
d±ρ ε ε = − µ∫ is valid for the local
spectral density of the defect ( )
00 ( )i
±ρ ε . The positive values
of intensities 0idµ > correspond to the existence of dis-
crete vibrational levels. Hence, the energy values of the
gap levels are:
( )2( ) 2
( ) ( )
( )
ii
ig i
i
b
a
±
± ±
±
λ + ε = + λ
, ( ) 0i
±λ < .
The condition for the existence of the gap levels ( 0)idµ >
has the form:
| |i ibλ < − .
It was also shown in [11] that in each of the subspaces,
one phonon splits off from the quasi-continuous spectrum
band to the gap level.
From the Lennard–Jones potential and the condition for
the existence of gap levels, one can obtain a diagram of the
existence of gap levels localized in anti-phase vibrations of
the chain (Fig. 6). Vibrations localized on defects cause a
change in thermodynamic quantities and, in particular, in
Fig. 4. Density of phonon states for r > r0 (a) and r < r0 (b).
Fig. 5. (Color online) Phonon heat capacity of the chain for r > r0
(a) and for r < r0 (b). In both fragments curves 1 are the contribu-
tions to the heat capacity of longitudinal vibrations; curves (2) are
the contributions to the heat capacity of transverse vibrations; and
curves (3) are the total heat capacities. The dashed line is the
value proportional to ∂2CV(T)/∂T2 The dash-dotted line is the
tangent to CV(T) at the inflection point.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3 409
E.V. Manzhelii, S.B. Feodosyev, and I.A. Gospodarev
heat capacity. The change in the phonon spectrum due to a
defect can lead to simultaneous appearance of discrete le-
vels and deformation of the quasi-continuous spectrum due
to the localized states that split off from the boundary of
the spectrum. The deformation can be described by the
shift function [20].
The change in heat capacity due to one gap level is de-
scribed by the expression:
____________________________________________________
( ) ( ) ( ) ( ) ( ) ( )
max
min
2
2, , , , , sinh
2 2
i
i
i
V i g i min B
B Bi
C T F T F T F T d F T k
k T k T
ε
−
ε
∂ξ ε ε ε ∆ = ε − ε − ε ε ε = ∂ ε
∑ ∫
. (6)
_______________________________________________
The main change in the low-temperature heat capacity
(Fig. 7) is determined by the direct contribution of the gap
energy levels. We consider the chains of atoms with a low
concentration of defects. In this case, we can consider the
problem in a linear approximation with respect to the con-
centration p. The defect in question has the greatest overall
effect on the change in the ratio of the lengths of parts of
the heat capacity temperature dependence curve in the case
of a compressed chain ( 0, 0).lτβ < β >
The shape of the curves is also influenced by the term
(6), which contains the shift function ( )ξ ε . The expression
for the shift function of the linear chains in terms of the
Jacobian matrix method was obtained in [21]. In this arti-
cle, we give the expressions for the shift functions generat-
ed by the defect in question. In the subspace ( )H − , the
shift function has the form:
( ) ( )
( )
( )
min
max
21 arccot ii i
i
ii
b −
−
−
ε − ε− λ
ξ ε =
π ε − ε λ
, 0iβ > , (7)
( ) ( )
( )
( )
max
min
21 arccot ii i
i
ii
b −
−
−
ε − ε+ λ
ξ ε =
π ε − ε λ
, 0iβ < . (8)
In the subspace ( )H + , the following expressions are ob-
tained for the shift function:
( ) ( )
( )
( )
max
min
21 arccot ii i
i
ii
b +
+
+
ε − ε+ λ
ξ ε =
π ε − ε λ
, 0iβ > ,
(9)
( ) ( )
( )
( )
min
0
21 arccot ii i
i
ii
b +
+
+
ε − ε− λ
ξ ε =
π ε − ε λ
, 0iβ < .
(10)
The full shift function can be written as follows:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2l l
− + − +
τ τ
ξ ε = ξ ε + ξ ε + ξ ε + ξ ε . (11)
The density of states of a chain with defects in the linear
approximation with respect to the concentration p is:
( ) ( ) ( )i
i d i
d
g g p
d
ξ ε
ε = ε −
ε
, (12)
where ( )g ε is the density of states in the absence of defects.
The case of a compressed chain is of greatest interest in the
Fig. 6. (Color online) The areas of parameters δ and γ under which
there exist the gap discrete levels in the subspaces generated by
antiphase longitudinal and transverse displacements of the atoms of
the defect.
Fig. 7. (Color online) Phonon heat capacity of a compressed chain
(r < r0). Curve 1 is the contribution to the heat capacity of the
longitudinal vibrations; curve 2 — the contribution of the trans-
verse vibrations; curve 3 is the total heat capacity. Dashed line
(green) the value proportional to ∂2CV(T)/∂T2 for the chain with-
out defects. Dash-dotted line (red) is the tangent to VC at the
inflection point. Short dash-dotted line (brown) is the value pro-
portional to ∂2CV(T)/∂T2.
410 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3
Phonon spectra and vibrational heat capacity of quasi-one-dimensional structures formed by rare gas atoms
problem of the shift of the linear-like part of the heat capacity
temperature dependence. In the case of a small perturbation
( )( )ii b±λ in a compressed chain, the change of its vibra-
tional density of states is:
( ) ( ) ( )
( ) ( )
( )
0 max
l l
d l
l l
g g g p g
− + λ λ
∆ ε = ε − ε = − ε +
ε − ε ε − ε
( ) ( )
( )
min max
2 p g
+ −
τ τ
τ
τ τ
λ λ
+ − ε ε − ε ε − ε
, (13)
where
( ) ( ) ( ) ( ) ( ){ }1 Im , , .
2i i ig G G i l− +ε = ε + ε = τ
π
(14)
It is seen that due to the presence of defects, the vibra-
tional density of states changes mainly at the points of sin-
gularity. It directly changes the type of the singularities.
In the case of not a small perturbation created by the de-
fect, the change in the vibrational densities of a stressed
chain in the vicinity of the singularity points for each of the
vibration branches is proportional to the vibrational density
of the unperturbed chain for each of the vibration branches.
At 0i∆ > :
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
max
min
max
min
2
lim ,
22
2
lim ,
22
i
i
i iii i
i
i i i
i ii i i
i
ii i
gb
g p
b
gb
g p
b
− +
− +ε→ε
− +
− +ε→ε
ελ + λ
∆ ε = −
− λ λ
ε− λ λ
∆ ε = −
λ + λ
(15)
at 0i∆ < :
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( )
max
min
max
min
2
lim ,
22
2
lim .
22
i
i
i ii ii
i
i i i
i ii i i
i
i i i
gb
g p
b
gb
g p
b
−+
+ −ε→ε
+ −
+ − −ε→ε
ε+ λλ
∆ ε = −
− λ λ
ε− λ λ ∆ ε = −
λ + λ
(16)
Figure 7 shows the shift of the linear part of the tempera-
ture dependence of the heat capacity of a compressed chain
with defects ( ) ( ) ( )V V VC T C T p C T= + ∆ toward low tem-
perature due to the direct contribution of defects at
0.05p = , 0.61τδ = − , 0.65τη = − .
4. Conclusions
The stability of structure of the adsorbed linear chains
is conditioned mainly by their interaction with the sub-
strate. Thus the interatomic distance in the chains differs
from the equilibrium distance of the potential of the in-
teratomic interaction between atoms of the gas forming
the chain, and the chain itself is stressed. If the distance
between atoms in the chain is smaller than the equilibri-
um one (the chain is “compressed”), then the quasi-
continuous spectrum band of its transverse vibrations
shifts to low frequencies. In this case, the linear part of
the temperature dependence of the heat capacity shifts to
lower temperatures.
Both the stressed state of the adsorbed chain and de-
fects of nanotubes lead to the appearance of isolated de-
fects in the chain, which are local changes in the intera-
tomic distance. For transverse vibrations of the
“compressed” chains it leads to a non-threshold for-
mation of discrete vibrational levels with frequencies
lying below the quasi-continuous spectrum band, which
results in further extension of the linear part on the tem-
perature dependence of the heat capacity to an even lower
temperatures.
This work was supported by Fundamental Investigation
Foundation of NANU (Grant No. 4/18-N).
________
1. L.D. Landau, JETP 7, 627 (1937).
2. J.V. Pearce, M.A. Adams, O.E. Vilches, M.R. Jonson, and
H.R. Glyde, Phys. Rev. Lett. 95, 185302 (2005).
3. M.I. Bagatskii, M.S. Barabashko, A.V. Dolbin, and V.V.
Sumarokov, Fiz. Nizk. Temp. 38, 667 (2012) [Low Temp.
Phys. 38, 523 (2012)].
4. M.I. Bagatskii, M.S. Barabashko, and V.V. Sumarokov, Fiz.
Nizk. Temp. 39, 568 (2013) [Low Temp. Phys. 39, 441
(2013)].
5. M.I. Bagatskii, V.G. Manzhelii, V.V. Sumarokov, and M.S.
Barabashko, Fiz. Nizk. Temp. 39, 801 (2013) [Low Temp.
Phys. 39, 618 (2013)].
6. M.I. Bagatskii, M.S. Barabashko, and V.V. Sumarokov,
JETP Lett. 99, 461 (2014).
7. M.I. Bagatskii, M.S. Barabashko, V.V. Sumarokov,
A. Jeżowski, and P. Stachowiak, J. Low Temp. Phys. 187,
113 (2017).
8. A. Šiber, Phys. Rev. B 66, 235414 (2002).
9. M. Aichinger, S. Kilić, E. Krotscheck, and L. Vranješ, Phys.
Rev. B 70, 155412 (2004).
10. E.V. Manzhelii, S.B. Feodosyev, I.A. Gospodarev, E.S.
Syrkin, and K.A. Minakova, Fiz. Nizk. Temp. 41, 718 (2015)
[Low Temp. Phys. 41, 557 (2015)].
11. E.V. Manzhelii, J. Low Temp. Phys. 187, 105 (2017).
12. E.V. Manzhelii, Fiz. Nizk. Temp. 29, 443 (2003) [Low Temp.
Phys. 29, 333 (2003)].
13. S.B. Feodosyev, I.A. Gospodarev, V.O. Kruglov, and E.V.
Manzhelii, J. Low Temp. Phys. 139, 651 (2005).
14. K.A. Chishko and E.S. Sokolova, Fiz. Nizk. Temp. 42, 116
(2016) [Low. Temp. Phys. 42, 85 (2016)].
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3 411
https://doi.org/10.1103/PhysRevLett.95.185302
https://doi.org/10.1063/1.4723677
https://doi.org/10.1063/1.4723677
https://doi.org/10.1063/1.4807048
https://doi.org/10.1063/1.4807048
https://doi.org/10.1063/1.4816120
https://doi.org/10.1063/1.4816120
https://doi.org/10.1134/S0021364014080049
https://doi.org/10.1007/s10909-016-1737-z
https://doi.org/10.1103/PhysRevB.66.235414
https://doi.org/10.1103/PhysRevB.70.155412
https://doi.org/10.1103/PhysRevB.70.155412
https://doi.org/10.1063/1.4927047
https://doi.org/10.1007/s10909-016-1699-1
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=2ahUKEwinvr7a--DfAhVR5eAKHVy-D0EQFjABegQICRAB&url=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1134%2F1.1913998&usg=AOvVaw3b2CWWlO2TcRrZAGTW378t
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=2ahUKEwinvr7a--DfAhVR5eAKHVy-D0EQFjABegQICRAB&url=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1134%2F1.1913998&usg=AOvVaw3b2CWWlO2TcRrZAGTW378t
https://doi.org/10.1007/s10909-005-5453-3
https://doi.org/10.1063/1.4941962
E.V. Manzhelii, S.B. Feodosyev, and I.A. Gospodarev
15. V. I. Peresada, V. N. Afanas’ev, and V. S. Borovikov, Fiz.
Nizk. Temp. 1, 461 (1975) [Sov. J. Low Temp. Phys. 1, 227
(1975)].
16. M.A. Mamalui, E.S. Syrkin, and S.B. Feodosyev, Fiz. Nizk.
Temp. 25, 976 (1999) [Low Temp. Phys. 25, 732 (1999)].
17. M.A. Mamalui, E.S. Syrkin, and S.B. Feodosyev, Fiz. Niz.
Temp. 25, 72 (1999) [Low Temp. Phys. 25, 55 (1999)].
18. A.V. Kotlyar and S.B. Feodosyev, Fiz. Nizk. Temp. 32, 343
(2006) [Low Temp. Phys. 32, 256 (2006)].
19. I.A. Gospodarev, A.V. Grishaev, E.S. Syrkin, and S.B.
Feodosyev, Phys. Solid State 42, 2217 (2000) [Fiz. Tverd.
Tela 42, 2153 (2000)].
20. I.M. Lifshitz, Usp. Mat. Nauk 7, 171 (1952) [in Russian].
21. E.S. Syrkin and S.B. Feodosyev, Fiz. Nizk. Temp. 20, 586
(1994) [Low Temp. Phys. 20, 463 (1994)].
___________________________
Фононні спектри та коливальна теплоємність
квазіодновимірних структур, що утворені інертними
газами на поверхні вуглецевих нанобандлів
О.В. Манжелій, С.Б. Феодосьєв, І.А. Господарьов
Вивчено особливості фононних спектрів, а також їх вплив
на коливальну теплоємність лінійних ланцюжків атомів інерт-
них газів, які адсорбовані на підкладку, що є поверхнею пов'я-
заних в нанобандл нанотрубок. Вплив підкладки призводить як
до зміщення нижньої межі спектра ланцюжка від нуля, так і до
виникнення механічної напруги в ланцюжку (його розтягуван-
ня або стиснення). Показано, що у випадку стислого ланцюжка
нецентральна взаємодія між атомами від’ємна (носить характер
відштовхування), що призводить до зміщення нижньої межі
спектра поперечних коливань в область низьких частот і до
зменшення довжини ділянки на температурній залежності теп-
лоємності, на якому ця залежність близька до експоненційної.
Дефекти структури нанобандла можуть зумовити зміну відста-
ней між атомами ланцюжка. Як аналітично, так і чисельно по-
казано, що внаслідок цього у фононному спектрі ланцюжка
можуть виникнути дискретні рівні з частотами, які розташовані
як вище, так і нижче смуги квазібезперервного спектра. Дис-
кретні рівні з частотами нижче смуги квазібезперервного спект-
ра призводять до подальшого скорочення температурного інте-
рвалу, на якому температурна залежність теплоємності близька
до експоненційної.
Ключові слова: фононні спектри, фононна теплоємність,
лінійний ланцюжок, локалізовані стани.
Фоннонные спектры и колебательная теплоемкость
квазиодномерных структур, образуемых инертными
газами на поверхности углеродных нанобандлов
Е.В. Манжелий, С.Б. Феодосьев, И.А. Господарев
Изучены особенности фононных спектров, а также их влия-
ние на колебательную теплоемкость линейных цепочек атомов
инертных газов, адсорбированных на подложку, представляю-
щую собой поверхность связанных в нанобандл нанотрубок.
Влияние подложки приводит как к смещению нижней границы
спектра цепочки от нуля, так и к возникновению механическо-
го напряжения в цепочке (ее растяжению или сжатию). Показа-
но, что в случае сжатой цепочки нецентральное взаимодейст-
вие между атомами отрицательно (носит характер
отталкивания), что приводит к смещению нижней границы
спектра поперечных колебаний в область низких частот и к
уменьшению длины участка на температурной зависимости
теплоемкости, на котором эта зависимость близка к экспонен-
циальной. Неоднородность структуры нанобандла может обу-
словить изменение расстояний между атомами цепочки. Как
аналитически, так и численно показано, что при этом в фонон-
ном спектре цепочки могут возникнуть дискретные уровни с
частотами, лежащими как выше, так и ниже полосы квазине-
прерывного спектра. Дискретные уровни с частотами ниже
полосы квазинепрерывного спектра приводят к дальнейшему
сокращению температурного интервала, на котором темпера-
турная зависимость теплоемкости близка к экспоненциальной.
Ключевые слова: фононные спектры, фононная теплоем-
кость, линейная цепочка, локализованные состояния.
412 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&uact=8&ved=2ahUKEwj7wd6O_ODfAhURDxQKHQfMAhUQFjACegQIABAB&url=https%3A%2F%2Faip.scitation.org%2Fdoi%2F10.1063%2F1.1421462&usg=AOvVaw2XUaArWBmCLoQucEPKLnEy
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=2ahUKEwiX38-7_ODfAhUh2-AKHevoD2wQFjAAegQICRAB&url=https%3A%2F%2Ffnte.ilt.kharkov.ua%2Fabstract.php%3Fuid%3Df41-0718e&usg=AOvVaw0t5DpMaM3aBj_JYifCn_sb
https://doi.org/10.1063/1.593707
https://doi.org/10.1063/1.2178484
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&cad=rja&uact=8&ved=2ahUKEwi8tY_-_eDfAhUIORQKHbz1DqIQFjADegQIBxAB&url=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1134%2F1.1332143&usg=AOvVaw3A7d--AY-1SP7wgS3b2c3s
1. Introduction
2. Phonon spectrum and vibrational heat capacity of an ideal adsorbed atomic chain
3. The effect of a local change in the distance between a pair of atoms on the vibrational spectrum and the phonon heat capacity
4. Conclusions
|