Lattice dynamics of cryocrystals at high pressure
The lattice dynamics of cryocrystals is investigated from first principles in the framework of the Tolpygo model over a wide range of pressures. The phonon frequencies in rare-gas solids are calculated in terms of models that go beyond the scope of adiabatic approximation. At high pressure the pho...
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irk-123456789-1217792017-06-17T03:03:09Z Lattice dynamics of cryocrystals at high pressure Horbenko, E.E. Troitskaya, E.P. Chabanenko, Val.V. Classical Cryocrystals The lattice dynamics of cryocrystals is investigated from first principles in the framework of the Tolpygo model over a wide range of pressures. The phonon frequencies in rare-gas solids are calculated in terms of models that go beyond the scope of adiabatic approximation. At high pressure the phonon spectrum along the Δ and Σ directions is distorted and the longitudinal L- and transverse T₂-modes soften as a result of the electron–phonon interaction, with the relative contribution decreasing in the sequence Ar, Kr, Xe. The calculated phonon frequencies are in good agreement with the experimental data for argon crystals of a pressure 3.1 GPa. 2007 Article Lattice dynamics of cryocrystals at high pressure / E.E. Horbenko, E.P. Troitskaya, Val.V. Chabanenko // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 752-757. — Бібліогр.: 28 назв. — англ. 0132-6414 PACS: 63.20.Dj; 63.20.Kr http://dspace.nbuv.gov.ua/handle/123456789/121779 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Classical Cryocrystals Classical Cryocrystals |
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Classical Cryocrystals Classical Cryocrystals Horbenko, E.E. Troitskaya, E.P. Chabanenko, Val.V. Lattice dynamics of cryocrystals at high pressure Физика низких температур |
| description |
The lattice dynamics of cryocrystals is investigated from first principles in the framework of the Tolpygo
model over a wide range of pressures. The phonon frequencies in rare-gas solids are calculated in terms of
models that go beyond the scope of adiabatic approximation. At high pressure the phonon spectrum along
the Δ and Σ directions is distorted and the longitudinal L- and transverse T₂-modes soften as a result of the
electron–phonon interaction, with the relative contribution decreasing in the sequence Ar, Kr, Xe. The calculated
phonon frequencies are in good agreement with the experimental data for argon crystals of a pressure
3.1 GPa. |
| format |
Article |
| author |
Horbenko, E.E. Troitskaya, E.P. Chabanenko, Val.V. |
| author_facet |
Horbenko, E.E. Troitskaya, E.P. Chabanenko, Val.V. |
| author_sort |
Horbenko, E.E. |
| title |
Lattice dynamics of cryocrystals at high pressure |
| title_short |
Lattice dynamics of cryocrystals at high pressure |
| title_full |
Lattice dynamics of cryocrystals at high pressure |
| title_fullStr |
Lattice dynamics of cryocrystals at high pressure |
| title_full_unstemmed |
Lattice dynamics of cryocrystals at high pressure |
| title_sort |
lattice dynamics of cryocrystals at high pressure |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2007 |
| topic_facet |
Classical Cryocrystals |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121779 |
| citation_txt |
Lattice dynamics of cryocrystals at high pressure / E.E. Horbenko, E.P. Troitskaya, Val.V. Chabanenko // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 752-757. — Бібліогр.: 28 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT horbenkoee latticedynamicsofcryocrystalsathighpressure AT troitskayaep latticedynamicsofcryocrystalsathighpressure AT chabanenkovalv latticedynamicsofcryocrystalsathighpressure |
| first_indexed |
2025-07-08T20:30:39Z |
| last_indexed |
2025-07-08T20:30:39Z |
| _version_ |
1837112115629916160 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 752–757
Lattice dynamics of cryocrystals at high pressure
E.E. Horbenko1,2, E.P. Troitskaya1, and Val.V. Chabanenko1
1
Donetsk A.A. Galkin Institute of Physics and Technology of the
National Academy of Sciences of Ukraine, 72 R. Luxemburg Str., Donetsk 83114, Ukraine
2
Luhansk Taras Shevchenko National Pedagogical University, 2 Oboronna Str., Luhansk 91011, Ukraine
E-mail: e_g81@mail.ru
Received October 20, 2006
The lattice dynamics of cryocrystals is investigated from first principles in the framework of the Tolpygo
model over a wide range of pressures. The phonon frequencies in rare-gas solids are calculated in terms of
models that go beyond the scope of adiabatic approximation. At high pressure the phonon spectrum along
the � and � directions is distorted and the longitudinal L- and transverse T2-modes soften as a result of the
electron–phonon interaction, with the relative contribution decreasing in the sequence Ar, Kr, Xe. The cal-
culated phonon frequencies are in good agreement with the experimental data for argon crystals of a pressure
3.1 GPa.
PACS: 63.20.Dj Phonon states and bands, normal modes, and phonon dispersion;
63.20.Kr Phonon–electron and phonon–phonon interactions.
Keywords: rare-gas solids, phonon spectrum, electron–phonon interaction, high pressure.
1. Introduction
The advent of new methods for studying materials has
opened up new opportunities for the experimental investi-
gation of phonon spectra under high pressures [1].
Investigations into the behavior of phonon frequencies
under pressure provide valuable information on the struc-
tural instability, the mechanisms of phase transitions, and
the interatomic interactions in the crystals. With know-
ledge of the phonon frequencies, one can easily calcu-
late the thermodynamic properties of materials at high
pressures.
The simplicity of rare-gas solids (RGSs) makes them
particularly attractive for detailed studying lattice dyna-
mics of solid Ne, Ar, Kr and Xe under pressure. It is
known that the pressures required to investigate physical
phenomena in cryocrystals are higher than those neces-
sary for other materials [1,2]. This imposes a number of
stringent requirements on the methods used for the pho-
non spectrum calculations. Therefore, it is necessary to
choose computational methods such that they 1) do not
contain fitting parameters (at least, in the short-range part
of the interatomic potential); 2) do not operate with ap-
proximations for the crystal potential, which are poorly
controllable with varying in pressure; 3) use function
basis sets that are appropriate at any compression (includ-
ing metallization pressures); 4) do not assume smallness
of the overlap integrals of localized basis orbitals; and
5) allow for the inclusion of nonadiabatic effects (elect-
ron–phonon interactions).
In the articles [3–6] first-principles calculations for
crystalline Ne, Ar, Kr, and Xe have been performed to
investigate phase relations, electronic structure, and vib-
rational properties of RGSs under pressure. These calcu-
lations are based on the density functional theory (DFT)
[7] and the local density approximation (LDA) for the
exchange–correlation potential [8] which should be rec-
ognized as a suitable method for the purpose of this study.
However, LDA is known to fail to describe weakly bond-
ed systems such as the van der Waals solids [9]. Yet, it
is expected that a compression-related increase of the
charge density in the internuclear region will be a validat-
ing argument for this approximation [10]. The authors of
Ref. 6 «report on the first ab initio investigation of the
lattice dynamics of fcc Xe. Not only is such an investiga-
tion useful for obtaining thermodynamic properties of
xenon per se, but also serves to gauge the performance
of density-functional and pseudopotential techniques for
this class of materials».
However, in our opinion, direct consistent calculation
of the dynamic-matrix elements is the most efficient
technique, because, in this case, all the approximations
© E.E. Horbenko, E.P. Troitskaya, and Val.V. Chabanenko, 2007
used in the model perform at their best. In this respect, the
development of methods appropriate for calculating the
phonon frequencies at high pressures is an important
problem.
In this paper, the lattice dynamics in RGSs has been
investigated from first principles over a wide range of
pressures with due regard for nonadiabatic effects. Ana-
lysis of these effects is of the utmost importance in de-
scribing the behavior of matter at pressures p � 0, when
the adiabaticity parameter is not small and the electron
and phonon spectra can overlap.
These calculations are based on the Tolpygo model
and its variants [11–14]. In contrast to the standard ap-
proach, which is based on Green's functions (see, for
example, [15]), the Tolpygo theory makes it possible to
investigate quantitatively the electron–phonon interac-
tion in crystals with a high binding energy over a wide
range of pressures, including those corresponding to an
insulator–metal transition.
2. Interatomic potential and the parameters
of electron–phonon interaction in compressed
cryocrystals
The dynamical theory of cryocrystal lattices with
allowances for the deformation of electron shells in atoms
(the Tolpygo model) was developed in [12,13].
The crystal energy has the form
E E El l l
l l
l l� � � � �
�
�
�
�
� �
��( ) / ( ) ( , ) ( ) ( )
P P P P
2 1 22
1
2
� � �
�
��
�
l
.
(1)
Here, P
l is the dipole moment of the atom at site l induced
by moving nuclei (higher multiple moments are ne-
glected). The term � l l
P is the exchange dipole interac-
tion, and the third term is the dipole–dipole interaction.
The first three terms, which are associated with fluctu-
ating deformations of the electron shells, are responsible
for nonadiabatic effects (electron–phonon interaction).
The last two terms in Eq. (1) are the energies of short-
range repulsion of atoms and of the long-range attraction
due to many-particle effects, respectively [16,17]. They
are expressed in terms of pairwise interaction potentials
between the atoms
E T Ee sr
( )1 � � ; E V Rsr sr
ll
l l
� �
��
�1
2
( ), (2)
E V Rlr
ll
l l
( ) ( )2 1
2
� �
��
� . (3)
The term E ( )2 describes the attraction between the
atoms; it is not allowed for the one-electron (Hart-
ree–Fock) approximation and corresponds to virtual tran-
sitions of two electrons from states �1 and �2 of the
valence band to states c1 and c2 of the empty conduction
band under the action of the electron–electron interaction
operator. In the particular case of atomic shells that do not
overlap (at large interatomic spacings), the energy E ( )2
reduces to the well-known van der Waals energy. The
attraction energy between two atoms was calculated in
[17] and can be written in the form
V R
C
R
f xlr( ) [ ( )]� � �
6
1 ; f x A x( ) exp [ ( )]� � �� 1 ;
x R R� / 0,
(4)
where C is van der Waals constant, f x( ) is a function
associated with atomic electron shells overlap, R is the
nearest-neighbor distance in the compressed crystal and
R0 is the spacing between nearest neighbors in the crystal
without compression; A and� are parameters (see [17]).
The short-range repulsion V Rsr( ) in Eq. (1) is calcu-
lated without recourse to variational of fitting parameters.
This is of fundamental importance because the potential
Vsr(r) plays a decisive role in the calculation of the atomic
properties of the compressed crystals. In [18], the poten-
tial Vsr(r) was calculated from first principles at the Hart-
ree–Fock level using the basis set of exactly orthogo-
nalized atomic orbitals and the Abarenkov–Antonova
cluster expansion [19]. In this case, the potential Vsr(r)
accounts for the contributions from higher order terms in
the overlap integral of atomic orbitals S.
The short-range potential Vsr(r) in the pair approxi-
mation can be represented in the form
� �V R Hsr
ll
ll
sr( ) .�
�� � � �00 00
� � � �� �� � � � � � � � � �
�
�
�
�
� � ��2 4P l V l l l V l lll l
C
�
�
�
� � �
[ , , ]
� � � � � � � � � �� �( )( | | , | |P P P l V l l l V lll ll ll l
C�
�
� � � � �2 2 � �
��
, ) .l �
�
�
��
��
(5)
Here, V l is the potential of neutral atom l, VC � � � �1 | |r r ,
� � � � � �l l V l lC� � �
, | | ,
� � � � � � �� � �� [( ( ) ( ) ( ) ( )) | |]� � � � � �� � �
l l l l d dr r r r r r ,
and the Greek indices denote the Cartesian components.
Expression (5) transforms into the known relationship for
the pair potential originally obtained in our earlier work
[20] in the limit S << 1, i.e., with the use of the formulas
P S O Sll ll
�
�
� �� �
1
2
2( ), P S S O Sll lm lm
m
� �
�
� � ��3
8
4( ) ,
(6)
Lattice dynamics of cryocrystals at high pressure
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 753
where S ll
�
� is the overlap integral between two atomic
orbitals centered at different sites:
S dll
�
�� �� � � � �� * ( ) ( )r l r l r . (7)
Compression, like any action that does not change the
structure of the electron wave functions of an atom, af-
fects the properties of the crystal only due to a change in
the distances l–m and the overlap of the atomic orbitals.
In this sense, relations (7) for the overlap integrals S
�
lm
are the sole controlling parameters of the theory [21]. The
set of overlap integrals S
�
lm uniquely determines all the
properties and the adiabatic potential (through the matrix
P (6)). Therefore, the problem of calculating the proper-
ties of insulators is divided into two stages: (i) calculation
of the set of overlap integrals S
�
lm and (ii) calculation of
the spectra and the thermodynamic and kinetic character-
istics using at the known overlap integrals S
�
lm .
3. Phonons and electron–phonon interaction in RGSs
under high pressure
The equations of motion for RGSs were obtained in the
harmonic approximation in [12]. Belogolovskii et al. [13]
analyzed these equations and derived analytical expres-
sions for the squares of the phonon frequencies as applied
to the symmetric directions of the wave vector K. These
expressions for the [00�] direction are as follows:
� L z
zz
zH G k
h g
A
k2
2
1
22 1
2 2
1� � � �
�
�
� �
�
( )( cos )
( )
( cos )
�
� � �( ) sinF E k Bz zz2 2 � ; (8)
�T z
xx
zG H k
h g
A
k2
2
1
22 1
2
1� � � �
�
�
� �
�
( )( cos )
( )
( cos )
�
� �2 2F k Bz xxsin � , (9)
where k K� a (a is half of the cube edge) and � i ik� � .
Similar relationships are available for the [���] and [��0]
directions.
The tensors � ij and � ij are lattice sums which depend
on the wave vector K to be found in [22] and [13], respec-
tively. In Eqs. (8) and (9), A is the atomic polarizability
divided by à
3
and related to the permittivity by the Clau-
sius–Mossotti equation; G, H, E, and F are the force pa-
rameters of the short-range interaction between the near-
est and next-nearest neighbors, respectively (H and F are
the transverse elastic coefficients); B is the Van der Waals
constant divided by ( )1 6 2 5� e a ; and g and h are the pa-
rameters of the electron–ion interaction.
754 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
E.E. Horbenko, E.P. Troitskaya, and Val.V. Chabanenko
0
40
80
120
160
Ne
T
L
T
L
[110][001][111]
T1 T1
T1
L
T2 T2
T2
�
�
�
�
K
K
K
K
X
X
X
X
�
�
�
�
L
L
L
L
0
40
80
120
160
Ar
T
L
L
T
T1
L T2
[110][001][111]
0
20
40
60
80
T
L
Kr
L
[110][001][111]
0
10
20
30
40
Xe
L
T
T
T
L
L
L
[111] [001] [110]
a c
db
!
!
!
!
"
"
"
"
"
"
"
"
�
,
m
eV
#
�
,
m
eV
#
�
,
m
eV
#
�
,
m
eV
#
Fig. 1. Phonon dispersion curves for Ne (a), Ar (b), Kr (c), and Xe (d) crystals along high-symmetry directions of the wave vector
k for compression ratios �V V� �0 0.7. Solid curves indicate the transverse T T T( , )1 2 and longitudinal L-branches along the ! �, and
� directions as calculated within the M3 model. Dashed curves represent the transverse T T T( , )1 2 and longitudinal L- branches calcu-
lated within the M3a model.
The relationships for the parameters G H E, , , and F, de-
scribing the repulsion, their calculated values, and the
van der Waals parameters of RGSs for compressions
�V V� 0 in the range from 0 to 0.8 (where V0 and V are the
unit cell volumes at p � 0 and p � 0, respectively) are
given in [18,21,23]. The parameters of the electron–pho-
non interaction g, h were determined in [24,25].
Let us now analyze the phonon frequencies of the
neon, argon, krypton, and xenon crystals. As in [25], the
simplest model M2 corresponds to the nearest-neighbor
approximation (E F� � 0) without account for the non-
adiabatic contributions (g h� � 0) in which the potential
Vsr (1) is calculated in the S 2 approximation. The M3
model additionally includes the next-to-nearest neigh-
bors. The M3a model takes into account the nearest and
next-nearest neighbors and the nonadiabatic contribu-
tions (Vsr ~ S 2). Figure 1 schematically depicts the pho-
non dispersion curves for the neon, argon, krypton, and
xenon crystals at a compression of 0.7.
We showed in [24,25] that, as the compression ratio
increases to below 0.6, the frequencies �#$( )k increase
by approximately one order of magnitude but the phonon
dispersion curves remain smooth; by contrast, the
electronic spectra for the neon crystal at a compressions
above 0.6 exhibit distortions and «humps». In the phonon
spectrum, as it can be seen from Fig. 1, the distortion is
clearly pronounced in the longitudinal L- and transverse
T2-modes along the � and � directions for a compression
of 0.7 when the nonadiabatic effects are taken into ac-
count (the calculation within the M3a model). This impli-
es a softening of these modes. The relative contribution �
of these effects decreases in the sequence Ar, Kr, Xe.
In our model, the compression 0.7 corresponds to a
pressure of 136.6 GPa for Ne, 287.8 GPa for Ar, 212.7
GPa for Kr, 128.6 GPA for Xe. For these crystals this is
the region of insulator-metal phase transition, when the
fordidden band gap vanishes (EG % 0) [2,18].
4. Discussion
Unfortunately, there are few experimental works
where phonon spectra at high pressures were studied
[1,26]. In this respect, our calculations will be compared
with the sole experiment available for argon. It was
shown [26] that the phonon branches for a single crystal
compressed in a diamond-anvil cell, can be in principle,
accurately measured using inelastic x-ray scattering. The
measurements were performed at pressures 3.1 and
20 GPa and could be extended to Mbar pressures. How-
ever, the best results were obtained at 3.1 GPa.
The experimental [26] and theoretical phonon fre-
quencies calculated in the M3a model are presented in
Fig. 2. The compression ratio is chosen to be equal 0.246,
which corresponds to the experimental lattice parameter
(i.e., the cube edge aexp � 4.845 �, p � 3.1 GPa [26],
a theor � 4.842 �, p � 2.6 GPa [18]). It can be seen from
Fig. 2 that the calculations agree well with experiment.
The calculations performed with an allowance for non-
adiabatic contributions lead to the smallest relative error
�. The X point is especially characteristic: at this point, the
inclusion of the electron–phonon interaction results in an
almost ideal agreement between the longitudional fre-
quencies #L
theor and #
L
exp
(the relative error � decreases
from 2.9 to 0.9%). The transverse branch is described
somewhat worse.
Dewhurst et al. [16] used DFT to obtain phonon spect-
ra of fcc xenon under pressure. They found that all pho-
non modes monotonically increase with pressure up to
100 GPa beyond which the transverse acoustic modes
at the X- and L-points start to soften (�#T X( ) �
= 21.3 meV; 20.9 meV; 19.0 meV and �#T L( ) .�131meV;
13.1 meV; 13.0 meV at p � 100 GPa; 110 GPa; 120 GPa,
respectively).
The values of the frequencies are approximately the
same as ours (within Ì3à, �#T X( ) � 23.2 meV and
�#T L( ) .�141 meV at p � 128 GPa, within the Ì3 model
�#T X L( , ) is somewhat lower, see Fig 1,d). However, we
have shown that at the Õ- and L- points the longitudinal
modes «soften» and the electron–phonon interaction con-
tributes positively to the transverse modes. For longitudi-
nal and transverse modes one can easily determine the
sign of the contribution from the electron–phonon inter-
action ( )*�$
2 into �$
2 at the Õ -point. From Eqs. (6) and
(7) for the direction of k| | [ ]00� we have:
Lattice dynamics of cryocrystals at high pressure
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 755
0
2
4
6
8
10
12
14
16
18
20
Ar
T branch
L branch
X�"
�
,
m
eV
#
Fig. 2. Experimental and theoretical phonon frequencies of the
Ar crystal along the [00�] direction at the compression ratio
�V V� �0 0.246. Solid curves with closed squares indicate the
results of calculations within the M3a model. The dotted line
with open circles corresponds to the experimental data obtain-
ed for the first Brillouin zone [26]. The dotted line with clos-
ed circles represents the experimental data obtained for the
second Brillouin zone [26].
( )
( )
( cos )*� L
zz
z
h g
A
k2
2
1
22 2
1� �
�
�
�
� �
, (10)
( )
( )
( cos )*�T
xx
z
h g
A
k2
2
1
22
1� �
�
�
�
� �
. (11)
For all RGSs, with p &20 GPa, the polarizability
A > 1.0 and the sign of the contribution is determined by
the sign of �
�. At the Õ-point in crystals with fcc struc-
ture [22], � zz ' 0, � xx & 0, and � ��
� & �A 1, therefore
( )*� L
2 0' , ( )*�T
2 & 0 a t t h e Õ - p o i n t . S i m i l a r l y,
( )*� L
2 0' , ( )*�T
2 & 0 at the L-point.
Nonadiabatic effects are enhanced more clearly when
electrons interact stronger with phonon branches, i.e. the
L-modes along all high-symmetry directions and
T2-mode along the � direction where � �# #T L2
& . This is
also confirmed by the fact that in the sequence Ne to Xe
series the relative contribution of the electron–phonon
interaction � is the largest for Ar (see Fig.1,a–d). For
example, at the X-point (L-mode), for compression
�V V� �0 0 7. , � � 38.8%, 57%, 24.9% and 5.5% for Ne,
Ar, Kr and Xe, respectively. In our opinion, theory [16]
lacks explanation of the mechanism and the extent of the
phonon «softening», and this can be done by using the
theory proposed here.
5. Conclusions
In this study, as in [18,24,25,27], the adiabatic poten-
tial E was constructed using a general approach. This ap-
proach as applied for rare-gases crystals in the sequence
Ne–Xe makes it possible to determine the most important
interactions in these crystals, i.e., the structure of the in-
teratomic potentials. For this purpose, we analyzed six
models of interatomic interaction in rare-gas crystals
[24,25]. It was demonstrated that the M2 model for argon,
krypton, and xenon and the model M4 for neon are quite
consistent; they are based on clear physical principles and
well formulated approximations, and adequately describe
the phonon frequencies at low pressures and tempera-
tures. At high pressures, the models allowing for the elec-
tron-phonon interaction (i.e., the models M3a and M5 for
neon and the model M3a for other rare-gas crystals) yield
the best results.
In order to construct the potential for neon, it is neces-
sary to take into account the pair terms of higher orders in
S, whereas the potentials for other crystals can be con-
structed with the inclusion of the terms ~ S 2. This is
explained by the fact that the short range potential Vsr is a
small difference between large quantities [20]:
V V Vsr sr sr� �� � .
Moreover, for argon, krypton, and xenon crystals, the
short-range potential Vsr amounts to 40–50% of the
potential Vsr
� . However, the ratio V Vsr sr� � for neon is
20–25%. Therefore, the terms of higher orders in S for
argon, krypton, and xenon are small corrections, whereas
their contribution to the potential for neon is comparable
to the terms ~ S 2.
Thus, the theory developed makes it possible to calcu-
late the short-range repulsive potential for all rare gas
crystals without recourse to variational or fitting proce-
dures.
In our opinion, the ab initio calculation of the short-
range repulsive potential is the main requirement for a
theory that claims to adequately describe the properties of
materials under pressure.
The calculations of the phonon frequencies for all
RGSs made it possible to determine the contributions
from the various interactions in these crystals. Neon is a
typical representative of low-Z materials. Apart from the
quantum effects observed at T p� � 0 [28], neon is char-
acterized by effects that manifest themselves at high pres-
sures, such as nonadiabatic effects and contributions from
the terms of higher orders in S to the potential Vsr . For
other rare-gas cryocrystals, it suffices to use the appro-
ximation V Ssr ~ 2. The contribution from the elect-
ron–phonon interaction to the phonon frequencies is also
large for Ar and is considerably smaller for the krypton
and xenon crystals.
Note that in the classical case of metals the shift of the
phonon frequency �#$q due to the electron–phonon in-
teraction (see, for example, [15, p. 40]) is determined by
the expression
� ��
�
�#
( ( # ( ($ $
$
q
k
kq
k k q
k k q q
k k q
k k q
�
�
� �
�
�
�
)
)
)� �
�
�
�
M
n n n n2 )
)
).
The second term describes the adiabatic contribution
of electrons, to the phonon frequency, i.e., the contribu-
tion that is associated with the account of the electron
energy in the equation for the diagonal ionic wave
function and which is of the same order as the phonon
frequency #$q.
The phonon frequency is a very sensitive characteris-
tic (in contrast to the thermodynamic properties, which
are integral functions of frequency). By using this spe-
cific feature, it has become possible to determine the
contribution from terms of higher orders in S to the poten-
tial Vsr even at low compressions more accurately than in
the band-structure calculations of the neon crystal [2,21].
In conclusion, we note that in the adiabatic theory of
perturbations in the adiabaticity parameter �2 [15] there
are two types of terms of different physical origins, na-
mely, the electron–phonon and anharmonic (pho-
non–phonon) corrections to the energy of the electronic
subsystem. The quantitative investigations into the non-
adiabatic effects at high pressures allowed us to draw the
756 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7
E.E. Horbenko, E.P. Troitskaya, and Val.V. Chabanenko
conclusion that the structural instability and the genera-
tion of a soft mode in crystals with a high binding energy
are caused by the electron–phonon interaction.
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