Evolution of discrete local levels into an impurity band in solidified inert gas solution
The density of states g(w) of disordered solutions of solidified inert gases have been calculated using the Jacobian matrix method. The transformation of a discrete vibrational level into an impurity zone at a growing concentration of light impurity atoms has been investigated. It is shown that a...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | Evolution of discrete local levels into an impurity band in solidified inert gas solution / A.M. Kosevich , S.B. Feodosyev, I.A. Gospodarev, V.I. Grishaev, O.V. Kotlyar, V.O. Kruglov, E.V. Manzhelii, E.S. Syrkin // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 735-740. — Бібліогр.: 25 назв. — англ. |
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Kosevich, A.M. Feodosyev, S.B. Gospodarev, I.A. Grishaev, V.I. Kotlyar, O.V. Kruglov, V.O. Manzhelii, E.V. Syrkin, E.S. 2017-06-16T07:19:34Z 2017-06-16T07:19:34Z 2007 Evolution of discrete local levels into an impurity band in solidified inert gas solution / A.M. Kosevich , S.B. Feodosyev, I.A. Gospodarev, V.I. Grishaev, O.V. Kotlyar, V.O. Kruglov, E.V. Manzhelii, E.S. Syrkin // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 735-740. — Бібліогр.: 25 назв. — англ. 0132-6414 PACS: 63.20.–e; 63.20.Mt; 63.20.Pw; 63.50.+x https://nasplib.isofts.kiev.ua/handle/123456789/121775 The density of states g(w) of disordered solutions of solidified inert gases have been calculated using the Jacobian matrix method. The transformation of a discrete vibrational level into an impurity zone at a growing concentration of light impurity atoms has been investigated. It is shown that a 1–10% change in the impurity concentration leads to smearing the local discrete level into an impurity band. As this occurs, additional resonance levels appear which carry important information about the impurity–impurity and impurity– basic lattice force interactions in such solutions. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Classical Cryocrystals Evolution of discrete local levels into an impurity band in solidified inert gas solution Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Evolution of discrete local levels into an impurity band in solidified inert gas solution |
| spellingShingle |
Evolution of discrete local levels into an impurity band in solidified inert gas solution Kosevich, A.M. Feodosyev, S.B. Gospodarev, I.A. Grishaev, V.I. Kotlyar, O.V. Kruglov, V.O. Manzhelii, E.V. Syrkin, E.S. Classical Cryocrystals |
| title_short |
Evolution of discrete local levels into an impurity band in solidified inert gas solution |
| title_full |
Evolution of discrete local levels into an impurity band in solidified inert gas solution |
| title_fullStr |
Evolution of discrete local levels into an impurity band in solidified inert gas solution |
| title_full_unstemmed |
Evolution of discrete local levels into an impurity band in solidified inert gas solution |
| title_sort |
evolution of discrete local levels into an impurity band in solidified inert gas solution |
| author |
Kosevich, A.M. Feodosyev, S.B. Gospodarev, I.A. Grishaev, V.I. Kotlyar, O.V. Kruglov, V.O. Manzhelii, E.V. Syrkin, E.S. |
| author_facet |
Kosevich, A.M. Feodosyev, S.B. Gospodarev, I.A. Grishaev, V.I. Kotlyar, O.V. Kruglov, V.O. Manzhelii, E.V. Syrkin, E.S. |
| topic |
Classical Cryocrystals |
| topic_facet |
Classical Cryocrystals |
| publishDate |
2007 |
| language |
English |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
The density of states g(w) of disordered solutions of solidified inert gases have been calculated using the
Jacobian matrix method. The transformation of a discrete vibrational level into an impurity zone at a
growing concentration of light impurity atoms has been investigated. It is shown that a 1–10% change in the
impurity concentration leads to smearing the local discrete level into an impurity band. As this occurs,
additional resonance levels appear which carry important information about the impurity–impurity and impurity–
basic lattice force interactions in such solutions.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121775 |
| citation_txt |
Evolution of discrete local levels into an impurity band in solidified inert gas solution / A.M. Kosevich , S.B. Feodosyev, I.A. Gospodarev, V.I. Grishaev, O.V. Kotlyar, V.O. Kruglov, E.V. Manzhelii, E.S. Syrkin // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 735-740. — Бібліогр.: 25 назв. — англ. |
| work_keys_str_mv |
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2025-11-25T20:40:25Z |
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2025-11-25T20:40:25Z |
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| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 735–740
Evolution of discrete local levels into an impurity band
in solidified inert gas solution
A.M. Kosevich , S.B. Feodosyev, I.A. Gospodarev, V.I. Grishaev,
O.V. Kotlyar, V.O. Kruglov, E.V. Manzhelii, and E.S. Syrkin
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: syrkin@ilt.kharkov.ua
Received October 20, 2006
The density of states g( )� of disordered solutions of solidified inert gases have been calculated using the
Jacobian matrix method. The transformation of a discrete vibrational level into an impurity zone at a
growing concentration of light impurity atoms has been investigated. It is shown that a 1–10% change in the
impurity concentration leads to smearing the local discrete level into an impurity band. As this occurs,
additional resonance levels appear which carry important information about the impurity–impurity and im-
purity–basic lattice force interactions in such solutions.
PACS: 63.20.–e Phonons in crystal lattice;
63.20.Mt Phonon–defect interaction;
63.20.Pw Localized modes;
63.50.+x Vibrational states of disordered systems.
Keywords: phonon density of states, spectral density, Green function, Jacoby matrix, local vibration,
disordered solution.
Introduction
It is well known that impurity atoms introduced into a
crystal can cause discrete impurity levels (the so-called
local oscillations) beyond the band of the quasi-con-
tinuous phonon spectrum of an ideal lattice. This occurs
when the mass of the impurity atom is smaller than that of
the atoms in the basic lattice or when the impurity atom
basic lattice binding is stronger than the atomic bonds in
the basic lattice. The appearing oscillations are localized
at the impurity atoms and their amplitudes decrease rapid-
ly with distance from the defect. The degree of localiza-
tion is the higher, the father is the oscillation frequency
from the upper edge of the continuous spectrum band. It is
thought that the damping of the local oscillation ampli-
tude is exponential when the distance from the defect
exceeds considerably the characteristic radius of the in-
teratomic interaction in the crystal. A systematic investi-
gation of local oscillations was started by I.M. Lifshitz
[1–4]. The conditions of the formation and the charac-
teristics of such oscillations can be found in many mono-
graphs concerned with the crystal lattice dynamics (e.g.,
see [5–7]). At present there are numerous techniques of
experimental measurement of local oscillation frequen-
cies. Such frequencies were obtained for many solid
solutions [8,9]. This kind of experiments provide
abundant easily-obtainable (e.g., see [10]) information
about the parameters of defects and basic lattices.
In experiment, local frequencies can be observed in
solid solutions with a finite (and small) concentration of
impurity atoms in which the interaction of states at close-
ly-spaced defects is not always negligible. Because of this
interaction, the localized oscillation levels can transform
into impurity zones with a quasi-continuous spectrum,
i.e., they alter to delocalized states [11–15]. The degree of
the smearing of discrete localized levels into impurity
zones is dependent not only on the impurity concent-
ration, but also on the parameters of the defect, the basic
lattice and the defect–defect interaction. It is therefore
interesting to find out if the resonance character of the
impurity vibrations persists at a particular matrix. If so,
will the frequency of the corresponding resonance maxi-
mum shift away from the frequency of the local oscil-
lation induced by an isolated impurity?
At present there is a consistent theory of evolution of
localized oscillations into impurity zones at low impurity
© A.M. Kosevich, S.B. Feodosyev, I.A. Gospodarev, V.I. Grishaev, O.V. Kotlyar, V.O. Kruglov, E.V. Manzhelii, and E.S. Syrkin, 2007
concentrations [12–15] when the average distance bet-
ween the impurity atoms l is much larger than the atomic
spacing in the basic lattice a. The power series expansion
(the parameter of the expansion is c a l� �( ) 3) was ob-
tained for the density of states (DOS) in the impurity
zone.
Note that at low concentrations p (commonly found as
a ratio between the number of impurity atoms and the
total number of atoms in the system) the l value becomes
smaller than 2a (at p � 2% for closely-packed structures,
at p � 3% for a simple cubic structure and so on). With
this spacing between the impurity atoms, their effective
interaction involves at least the second moments of their
spectral density, which can appreciably affect (e.g., see
[10]) the frequencies of localized oscillations. Even at
such low impurity concentrations we can observe not a
discrete highly localized oscillation level, but an impurity
zone formed by delocalized oscillations.
The evolution of discrete localized levels into impurity
zones was investigated for rapidly attenuating phonons in
narrow optical bands [16,17]. However, the calculation
technique proposed in [16,17], which was based on the
Green functions and the diagram procedure, works poorly
for slowly attenuating acoustic phonons.
In this study the phonon DOS of disordered solid solu-
tions of inert gases Kr Ar1�p p have been calculated nu-
merically.
In this system the concentration p can take any value
varying from zero to unity [18]. As the concentration
changes from 1 to 10%, the smearing of the local discrete
level into an impurity band is attended by additional re-
sonance levels carrying important information about the
Kr–Ar and Ar –Ar force interactions in such crystals.
Phonon densities of states of solutions of solidified
inert gases
The computation performed in this study is based on
the method of Jacobian matrices (J matrices) [19–21]
(also see [22]). The essence of the method is the classifi-
cation of vibrations, which differs from the traditional
plane wave expansion. The corresponded basis { }
�
hn n�
�
0
can be obtained through orthonormalization of the se-
quence
{ � } , � , � , , � ,L L L L
n
n
n
� � � �
�
�
�h h h h h0 0 0 0
2
0 0�
� � , (1)
which is one of possible representations of the Huygens
principle. Here �L is the operator describing the crystal
lattice vibrations
L ik
ik
m m
( , )
( , )
( ) ( )
r r
r r
r r
� �
�
�
�
;
r and r� are the radius-vectors of the interacting atoms;
� ik ( )r r, � is the force constant matrix describing this in-
teraction; m( )r and m( )r� are the atomic masses.
�
h0 is the
vector in the space of renormalized atomic displacements
H in which the operator �L acts. The vectors of this
3N -dimensional space (N is the number of atoms in the
system) are marked with arrows to distinguish them from
ordinary «three-dimensional vectors» traditionally
shown in roman bold.
The operator �L in the basis { }
�
hn n�
�
0 is represented by a
three-diagonal (Jacobian) matrix (J matrix). Below an
and bn are used to designate the diagonal and off-diago-
nal matrix elements, respectively (n N
�[ ; ]0 3 ); the
index numbering the subspaces will be omitted. This
J matrix has a simple spectrum, which simplifies con-
siderably the computation of phonon DOS. Let � �� 2 be
the eigenvalues of the operator �L (squares of eigenfre-
quencies�). If the band of the quasi-continuous spectrum
is singly connected � � [ ; ]0 m , the following limit rela-
tions hold for the matrix elements an and bn
lim lim ( )
n
n
n
n
m
m ma b
�
�
� � �2
2
2�
� � . (2)
The arbitrary matrix elements Gmn ( )� of the resolvent
operator � ( � �)G I L� � �� 1 can be represented in terms of the
element G00( )� (Green function). For m n� we have
G Gmn m n( ) ( , �( ) )� �� �
� �
h h
� �
P Q P P Gm n m n( ) ( ) ( ) ( ) ( )� � � � �00 . (3)
Here �I is the unit operator; Pn ( )� and Q n ( )� are the
polynomians to the powers n and n �1, respectively. They
can be found in [19–22]. The polynomial Pn ( )� corres-
ponds to the determinant of the n-rank matrix of the oper-
ator � � �I L� . The polynomial Q n ( )� is the minor of the first
diagonal element of this matrix.
The Green function of the system G G( ) ( )� �� 00 can be
written down easily as a continued fraction
G G( ) lim ( )( )� ��
�n
n ;
G
Q Q K
P P K
( )( )
( ) ( ) ( )
( ) ( ) (
n
n n n
n n n
b
b
�
� � �
� �
�
�
�
� � �
� � �
1 1
1 1 �)
.
(4)
In Eq. (4) K � ( )� is the function to which the continued
fraction corresponds to the J matrix whose elements are
equal to their asymptotic values can be reduced. For the
limiting values in Eq. (2) we have
K Z� � �
�( ) { ( ) | |}�
�
� � � � � �
4
2 2
2
m
m m , (5)
Z( ) ( ) ( ) ( )� � � � � �� � � �i m m� � � (6)
(�( )x is the Heaviside function).
The region D of existence of the imaginary part of the
function G( )� , Eq. (4), determines the band of the quasi-
continuous spectrum of the operator �L (in general non-
736 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
A.M. Kosevich et al.
singly-connected). The spectral density is estimated at
� D to be
g( ) ( ) ( )�
�
�
�
�
�� �
1
2Im ImG G . (7)
The method of J matrices does not include explicitly
the translational symmetry of the crystal lattice and
allows a straight forward computation of the spectral
densities corresponding to the displacements of the atoms
of the system along different crystallographical direc-
tions i. If the generating vector
�
h0 is the displacement of
an atom with the radius-vector r in the direction i, the
spectral density gi ( , )� r calculated by Eqs. (4)–(7) cha-
racterizes the frequency spectrum of the oscillations of
this atom in this direction. The phonon DOS of a solid
solution with the impurity concentration p is found as
� � � � �g p
N
( , ) ( � �)�
�
�
�
2 2 1Sp Im I L
and is a self-averaging value [12–15]. It can be obtained
by averaging the functions gi ( , )� r over all positions of
the atoms r and all directions i of their displacements.
For a fcc crystal with the nearest-neighbors interaction
the matrix of the operator �L can be represented as
L ik
a a
m m
r r r
r r
, ; ;
( ) ( )
� �
�
��
�
��
�
�
�
�
�
� �
�
�
2 2
0
1
0
0
0 0
� �
� �
�
�
�
�
�
�
�
�
�
. .(8)
The other matrices can be obtained through Oh -symmetry
operations, and the matrix L ik ( , )r r is ( ) / ( )8 4� !
ik m r .
The force constants �, and � characterizing the
Kr–Kr, Kr–Ar and Ar–Ar interactions in the solid
Kr Ar1�p p solution [23] were found from the elastic con-
stants [24] and experimental data on heat capacity. A ran-
dom distribution of impurities was realized using a gener-
ator of pseudo-random numbers distributed uniformly in
the interval ( ; )0 1 . The generator operates on the basis of
multiplicative congruent method [25]. We calculated the
phonon DOS for different concentrations of impurity at-
oms. At each concentration the averaging was performed
over several thousands of random configurations of im-
purity distribution. For each configuration the DOS was
found through averaging over several tens of spectral
densities corresponding to the displacements of several
tens of sequential atoms along different crystallographic
directions.
The analytical properties of our calculated Jacobian
matrices at p� 0.1% suggest unambiguously that the
band of the quasi-continuous phonon spectrum of dis-
ordered solid solutions is singly connected. The gap se-
parating the continuous spectrum band from the local
frequency in the case of an isolated impurity is filled with
phonons even at limiting low concentrations of impurity
atoms. The eigenfrequencies are in the interval [ , ( )]0 �m p ,
where the frequency �m p( ) is determined by the asymp-
totic behavior of the matrix elements [19–21]. It exceeds
the local vibration frequency corresponding to the iso-
lated impurity with the same mass defect and it is howe-
ver smaller than the so-called natural spectrum edge (e.g.,
see [15]), i.e., smaller than the highest vibration frequen-
cy of an ideal crystal lattice consisting of atoms which we
consider as light impurity. The later fact is the result of
the finiteness of the rank of the J matrices (in our cal-
culation it is 60), which prohibits the occurrence of an
«arbitrarily large» region occupied only by impurity in
the investigated configurations (covering slightly fewer
than 10 6 atoms). At p � 50% the behavior of the spectral
densities near �m p( ) can be thought of as exponential
attenuation, which is also suggested by the general theory
of phonon spectra of disordered solid solutions [12–15].
The single-connectedness of the quasi-continuous spect-
ral region in the systems analyzed permits us to calculate
the Green functions and the spectral densities using their
analytical approximation by a continued fraction [21,22].
Such approximation enables us to calculate with accuracy
the above functions at any frequency, which is parti-
cularly important in this case when the phonon DOS
spectral densities contain sharp resonance peaks.
Discussion. Additional resonance levels at finite
impurity concentrations
Figures 1–4 show the evolution of the phonon densi-
ties � �g p( , )� in Kr Ar1�p p solutions at growing concentra-
tion p of argon atoms. The fragments b, Figs. 1, 2, are the
regions of these densities corresponding to the values
� �" m (�m is band edge of the quasi-continuous spect-
rum of Kr ideal lattice) at which these densities are signi-
ficantly nonzero. Thus, the figure illustrates transforma-
tion of the local frequency into an impurity band.
The oscillations of the impurity atoms are strongly
localized at p � 0.5% (Fig. 1,a–c) Their frequencies are
within a very narrow ( # $ �2 10 6�m) band near the fre-
quency of the local oscillation (�0) caused by one isolated
impurity atom. This is described with high accuracy
( %# 25 ) within a «two-moment approximation» proposed
in [10]. The local frequency calculated on the basis of
such approximation is shown in Figs. 1–3 (heavy dashed
line).
It is seen in both fragments of the Fig. 2 that the local
level is smeared at p � 1–5%. The shapes of the impurity
bands at these concentrations are in good agreement with
the general results [12–15]. Besides, as was mentioned in
the Introduction, at p�2% the average distance between
the impurity atoms does not exceed the doubled atomic
spacing in the lattice. In this case the influence of most
impurities upon one another starts to manifest itself in the
DOS at the second moment. The number of impurity pairs
(the impurity atoms interacting directly with each other
Evolution of discrete local levels into an impurity band in solidified inert gas solution
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 737
738 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
A.M. Kosevich et al.
Fig. 2. Phonon densities of solid Kr Ar1�p p solutions for p � 001. , 0.025, and 0.05: a — the whole frequency interval; b — beyond
the quasicontinuous spectrum band, pure Kr. Solid lines (in a and b fragments) correspond to the functions � �g p( )� ; thin dashed
vertical straight lines are the local frequencies calculated within the «two-moment approximation». Dashed curve in fragment a is
the phonon density of pure Kr.
0.2 0.6 1.0 1.4
0.5
1.0
1.5
2.0
1095
g
,
T
H
z–
1
g
,
T
H
z–
1
g
,
T
H
z–
1
Kr0.995 0.005Ar
a
1.635 1.640 1.645 1.650
200
400
600
800
1000
% , THz % , THz % , THz
�0
b c
1.640795 1.640800
200
400
600
800
1000
Fig. 1. Phonon density of solid Kr Ar0 995 0 005. . solution: a — the whole frequency interval; b and c — fragments of Fig. 1,a near the
local frequency. Solid lines in a, b and c correspond to function � � �g p( ) .� 0 005; dashed line in fragment a is the phonon density of
pure Kr; thin dashed vertical straight lines (fragments a and b) are the local frequencies calculated using the «two-moment approxi-
mation» (% �& �� 2 ).
— the nearest neighbors in our case) becomes sufficient
to show up in the phonon spectrum. For these pairs the
impurity interaction is observable even at the first mo-
ment of the spectral density. It is shown in [10] that a
change in the second moment of the spectral density leads
to a displacement of the local level by ' (1–3)%. Such
displacements are shown in Figs. 2, 3 (dashed lines) near
�0 (not specified). A change in the first moment shifts the
local level by # ' (10–20)%. These levels (
( )�l
�
, �(
( )�
and �n
( )� ) are shown in the same figures by thin dashed
lines.
The levels �l
( )�
occur on co- and anti-phase displace-
ments, respectively, of two adjacent impurity atoms along
the straight line connecting them. The levels �(
( )� and
�n
( )� correspond to the displacements of two adjacent
impurity atoms that are perpendicular to the above
straight line.
When the adjacent atoms build up triangles, additional
resonance peaks appear. The local frequencies calculated
in the two-moment approximation are shown in Figs. 2, 3
(thin dashed lines). The frequency corresponds to small
rotational displacements of an equilateral triangle about
the three-fold axis; the frequency correlates with the dis-
placement of the triangle as a whole and its uniform
compression.
In the two-moment approximation the relation bet-
ween these frequencies and the force constants character-
izing the Kr–Kr, Kr–Ar and Ar–Ar interactions [10],
which enable us to calculate the force constants from
the measured frequencies of the corresponding resonance
peaks.
On a further growth of the concentration ( p �10–15,
and 25%, Fig. 3), the impurity pairs start to interact (at the
second-moment level) both with single impurity atoms
and with one another. With the mass and force constants
ratios describing the atomic interaction in the Kr Ar1�p p
solutions, the interaction at the level of the second mo-
ments causes the formation of a single band of the quasi-
continuous spectrum at these concentrations. However, at
the expression for � �� m the DOS has a nonanalytic
form. The corresponding oscillations are quasi-localized.
Their delocalization occurs as the impurity concent-
ration continues to increase. The phonon DOS of the
Evolution of discrete local levels into an impurity band in solidified inert gas solution
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 739
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
0.8
0.8
1.0
1.0
1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.6
1.6
1.6
0.5
1.0
1.5
2.0 Kr0.9 0.1Ar
Kr0.85 0.15Ar
Kr0.75 0.25Ar
0.5
1.0
1.5
2.0
0.5
1.0
1.5
2.0
%) THz
%) THz
%) THz
g
,
T
H
z–
1
g
,
T
H
z–
1
g
,
T
H
z–
1
0
0
0
Fig. 3. Phonon densities of solid Kr Ar1�p p solutions at p � 0.1,
0.15, and 0.25.
Kr0.5 0.5Ar
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0.5
1.0
1.5
2.0
%) THz
g
,
T
H
z–
1
0
Fig. 4. Phonon density of solid Kr Ar0 5 0 5. . solution (solid curve).
Dashed curves are the phonon densities of pure Kr and Ar.
Kr Ar0 5 0 5. . solution has no resonance peaks at � �" m, and
the singularity present in this frequency interval agrees
with the van Hove singularity for pure Ar. It suggests that
such a solution contains rather large clusters of each
component, which is typical for this concentration.
Conclusions
The densities of states obtained in this study for disor-
dered solid solutions (in particular, for solidified inert
gases) with a fcc lattice and an interaction of the nearest
neighbors provide at least a qualitative picture of trans-
formation of discrete oscillation levels localized at impu-
rity atoms into an impurity band formed by delocalized
states. The main feature of the transformation is the ap-
pearance of additional impurity — induced resonance
peaks at increasing impurity concentrations. The peaks
are due to the oscillations of impurity pairs and impurity
clusters. The adequate description of such oscillations
with the two-moment approximation [10] enables one to
restore in a rather simple way the parameters of the defec-
tive lattice from the measured frequencies of resonance
peaks in solid solutions.
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