Propagating collective excitations in molten salts
Longitudinal as well as transverse dynamics of a molten salt beyond the hydrodynamic region is studied within the generalized collective modes (GCM) approach. An analytical three-variable model is applied to the treatment of the coupled long- and short-time charge fluctuations. Dispersion laws o...
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Інститут фізики конденсованих систем НАН України
2003
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| Цитувати: | Propagating collective excitations in molten salts / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 395-408. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1207512017-06-13T03:03:02Z Propagating collective excitations in molten salts Bryk, T. Mryglod, I. Longitudinal as well as transverse dynamics of a molten salt beyond the hydrodynamic region is studied within the generalized collective modes (GCM) approach. An analytical three-variable model is applied to the treatment of the coupled long- and short-time charge fluctuations. Dispersion laws of propagating kinetic collective excitations such as optic phonon-like modes, heat and shear waves are obtained and analyzed for the case of molten NaCl within the eight-variable GCM scheme, combining the analytical methods and molecular dynamics simulations. У методі узагальнених колективних мод (УКМ) досліджується повздовжня та поперечна динаміка розплавів солей поза областю гідродинамічної поведінки. В рамках простої тризмінної моделі аналітично проаналізовано особливості зв’язаної динаміки короткота довго-живучих зарядових флюктуацій. В рамках восьми змінного підходу методу УКМ, комбінуючи засоби аналітичної теорії та молекулярно-динамічні розрахунки, для розплаву NaCl отримано та проаналізовано дисперсійні закони для пропагаторних кінетичних колективних збуджень типу оптичних фононних мод, теплової та зсувної хвиль. 2003 Article Propagating collective excitations in molten salts / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 395-408. — Бібліогр.: 18 назв. — англ. 1607-324X PACS: 05.20.Jj, 61.20.Lc DOI:10.5488/CMP.6.3.395 http://dspace.nbuv.gov.ua/handle/123456789/120751 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
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Longitudinal as well as transverse dynamics of a molten salt beyond the
hydrodynamic region is studied within the generalized collective modes
(GCM) approach. An analytical three-variable model is applied to the treatment
of the coupled long- and short-time charge fluctuations. Dispersion
laws of propagating kinetic collective excitations such as optic phonon-like
modes, heat and shear waves are obtained and analyzed for the case of
molten NaCl within the eight-variable GCM scheme, combining the analytical
methods and molecular dynamics simulations. |
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Article |
| author |
Bryk, T. Mryglod, I. |
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Bryk, T. Mryglod, I. Propagating collective excitations in molten salts Condensed Matter Physics |
| author_facet |
Bryk, T. Mryglod, I. |
| author_sort |
Bryk, T. |
| title |
Propagating collective excitations in molten salts |
| title_short |
Propagating collective excitations in molten salts |
| title_full |
Propagating collective excitations in molten salts |
| title_fullStr |
Propagating collective excitations in molten salts |
| title_full_unstemmed |
Propagating collective excitations in molten salts |
| title_sort |
propagating collective excitations in molten salts |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120751 |
| citation_txt |
Propagating collective excitations in molten salts / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 395-408. — Бібліогр.: 18 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT brykt propagatingcollectiveexcitationsinmoltensalts AT mryglodi propagatingcollectiveexcitationsinmoltensalts |
| first_indexed |
2025-07-08T18:31:06Z |
| last_indexed |
2025-07-08T18:31:06Z |
| _version_ |
1837104593560928256 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 3(35), pp. 395–407
Propagating collective excitations in
molten salts∗
T.Bryk, I.Mryglod
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received May 6, 2003
Longitudinal as well as transverse dynamics of a molten salt beyond the
hydrodynamic region is studied within the generalized collective modes
(GCM) approach. An analytical three-variable model is applied to the treat-
ment of the coupled long- and short-time charge fluctuations. Dispersion
laws of propagating kinetic collective excitations such as optic phonon-like
modes, heat and shear waves are obtained and analyzed for the case of
molten NaCl within the eight-variable GCM scheme, combining the analyt-
ical methods and molecular dynamics simulations.
Key words: generalized hydrodynamics, collective excitations, ionic
liquids, molten salts, longitudinal and transverse dynamics
PACS: 05.20.Jj, 61.20.Lc
1. Introduction
So far, the picture of collective dynamics in molten salts has been well established
exclusively in the hydrodynamic regime [1–3], i.e. reflecting the processes on large
spatial and time scales. The hydrodynamic four-variable model permits to obtain
analytic expressions for hydrodynamic time correlation functions and the spectrum
of hydrodynamic collective excitations [4,5]. This model can explain a very specif-
ic feature of the relaxational behaviour of molten salts which appears due to the
fluctuations of an inner electrical field under the constraint of total electric neutral-
ity of the system and makes the main difference from the case of non-ionic binary
mixtures. In particular, in reference [4] it was found that, depending on the value
of electrical conductivity σ, two different regimes in the relaxational behaviour of a
molten salt can be observed. Namely, for the systems with large values of conduc-
tivity, the charge fluctuations are practically uncoupled from the mass fluctuations,
which makes the latter isomorphous to those of an one-component neutral fluid.
∗Dedicated to Myroslav Holovko on occasion of his 60th birthday.
c© T.Bryk, I.Mryglod 395
T.Bryk, I.Mryglod
When σ is small enough, the relaxational behaviour of a molten salt becomes iso-
morphous to that of a neutral mixture, with electrical conductivity playing a role
similar to the mutual diffusion.
The hydrodynamic set of longitudinal dynamical variables for a molten salt
consists of four microscopic operators of total mass-density nt(k, t), charge density
nq(k, t), longitudinal total mass-current JL
t (k, t) and total energy e(k, t):
A
(hyd)(k, t) =
{
nt(k, t), nq(k, t), JL
t (k, t), e(k, t)
}
. (1)
These four operators correspond to the fluctuations of conserved quantities and
describe the slowest dynamical processes in ionic melts. The mutual charge current
fluctuations are associated with the fast processes of a shorter time scale, compared
with the hydrodynamic ones, and are not taken into account within the standard
hydrodynamic treatment.
During the recent few years a systematic approach to the studies of collective
dynamics in pure and binary liquids, based on the concept of generalized collec-
tive excitations [6], was undertaken [7–11]. Within the generalized collective mode
(GCM) approach, proposed originally in reference [6] and developed then in ref-
erences [12–15], the collective excitations in the fluids are directly connected with
eigenvalues of the so-called generalized hydrodynamic matrix, which determines the
time evolution of the system. Such a definition of collective modes is in agreement
with the generally accepted principle of statistical physics, when the collective modes
are identified with the poles of the relevant Green function or generalized suscepti-
bility.
One of the most important advantages of the GCM approach is a possibility of
a detailed theoretical analysis of time correlation functions obtained in molecular
dynamics (MD) simulations. The MD-derived functions contain in their very shape
the information about all possible long- and short-time scale processes (diffusive-
like, propagating and kinetic collective modes) as well as mode-coupling effects in a
liquid. An analysis based on hydrodynamic set of equations fails to explain the shape
of MD-derived functions in a wide range of wavenumbers k and frequencies ω. In
contrast to the standard hydrodynamics, the GCM method, being one of the most
consistent ways of constructing the generalized hydrodynamics, treats on the same
footing the long- and short-time scale processes: the generalized Langevin equation
for time correlation functions is solved in the matrix form using the basis set of
dynamical variables, which apart from hydrodynamic (slow) variables contains their
time derivatives (fast variables). The time evolution of these fast or so-called “ex-
tended” variables which describe the processes of the time scale shorter than the
hydrodynamic ones, is obtained in computer experiments in order to estimate rele-
vant time correlation functions and static averages. In general, the basis set of Nv
dynamical variables generates Nv × Nv secular equation and results in Nv general-
ized collective modes (eigenvalues). Among the Nv eigenvalues, the lowest ones in
the long-wavelength region always correspond to the hydrodynamic modes, which
have a right asymptotic behaviour in the small k limit predicted by the linear hy-
drodynamics. All other eigenvalues are called kinetic modes and correspond to the
396
Propagating collective excitations in molten salts
processes of short-time scale. They cannot be obtained within the standard hydrody-
namic treatment. Kinetic collective excitations sufficiently affect all the dynamical
quantities beyond the hydrodynamic region. However, theoretical studies focused on
the kinetic collective excitations in liquids are rather rare.
The additional dynamic variables which reflect the mutual motion of species
make a noticeable difference in theoretical models for simple and binary fluids (for
molten salts these are charge density nq(k, t) and charge current Jq(k, t)). The dy-
namics of charge current (mass-concentration current in the case of non-ionic mix-
tures) fluctuations belong to kinetic-like processes. That is why the study of kinetic
collective excitations caused by kinetic processes and their interaction with sound
excitations within a generalized model would be of great interest. Recently, we stud-
ied within the GCM approach the transverse [9,10] and longitudinal [11] dynamics
of Lennard-Jones and metallic mixtures and found that the optic-like excitations,
which by dispersion law resemble the charge waves in molten salts, can exist even
in non-ionic binary liquids. It was shown that the damping of mass-concentration
propagating modes is very important in estimating their dispersion relations. In long-
wavelength limit the damping of optic-like excitations is determined by the value of
mutual diffusion and the mass-concentration static structure factor Sxx(k), which at
k = 0 is a measure of how close the binary non-ionic mixture is to the demixing. As a
consequence, it was obtained that in the dense gas mixtures He75Ar25 and He65Ne35
with high mutual diffusion and large value of Sxx(k = 0) the long-wavelength optic-
like excitations are suppressed and their branch does not exist in the spectrum of
collective excitations for small wavenumbers.
In this paper we report the GCM study of longitudinal and transverse dynamics
of a molten salt NaCl with the main attention focused on the dispersion relation
of kinetic collective excitations, and optic-like (charge) waves, in particular. The
remaining part of the paper is organized as follows. In section 2 we analytically
solve the generalized Langevin equation in Markovian approximation using a three-
variable model for charge fluctuations. In section 3 the four- and eight-variable GCM
approach is applied to the numerical analysis of the transverse and longitudinal
dynamics of molten NaCl, respectively, and section 4 contains conclusions of this
study.
2. Coupled long- and short-time charge fluctuations:
an analytical treatment
Let us start from the simplest models of collective charge dynamics in ionic liq-
uids. This can be done based on the separate treatment of longitudinal dynamics
of charge fluctuations using generalized Langevin equation in Markovian approxi-
mation. In our case, the separated treatment means that we neglect the coupling
between the charge fluctuations and slow dynamic processes of another origin, such
as thermal and mass-density fluctuations. Both kinds of fluctuations could cause
additional mechanisms of damping for the charge fluctuations. Here, in order to
investigate the physical sense of the “bare” charge collective modes, we focus exclu-
397
T.Bryk, I.Mryglod
sively on the coupling between slow and fast charge fluctuations. Such a simplified
approach permits to obtain a very precious information on the lowest kinetic modes
caused by charge fluctuations, and to analyze in detail the effects of interaction
between them.
Let us consider the generalized Langevin equations for the following basis set of
three dynamical variables
A
(3q) =
{
nq(k, t), JL
q (k, t), J̇L
q (k, t)
}
, (2)
which are responsible for the longitudinal long- and short-time scale charge fluctu-
ations. The hydrodynamic variable in this basis set is only nq(k, t), while the other
two dynamical variables, JL
q (k, t) and J̇L
q (k, t), describe faster charge fluctuations.
As a starting point for the subsequent analysis one may choose the behaviour of
“bare” relaxing charge mode d0(k) and optic-like propagating excitations (or charge
waves) z0
±
(k), which can be easily found from the eigenvalues problem [8–11] for the
separate sets of dynamic variables, i.e. when the coupling between them is not taken
into account. Namely, within the GCM approach the eigenvalues d0(k) and z0
±
(k)
can be obtained from the secular equations for the corresponding generalized hydro-
dynamic matrix T(k), calculated for two separated subsets of dynamical variables,
respectively:
(i) A
(1q) = {nq(k, t)}, which includes only the hydrodynamic variable nq(k, t)
and corresponds to the slowest component of charge fluctuations; and
(ii) A
(2q) = {JL
q (k, t), J̇L
q (k, t)}, describing faster dynamical processes, caused by
charge fluctuations.
The “bare” relaxing eigenvalue (or mode) d0(k), derived for the single-variable
subset A
(1q), is given by the expression
d0(k) =
1
τqq(k)
, (3)
where τqq(k) is the generalized correlation time, associated with the solely charge-
charge time correlation function:
τqq(k) =
1
Fqq(k, t = 0)
∫
∞
0
Fqq(k, t)dt. (4)
There exists an analytical result for asymptotic behaviour (when k → 0) of the
hydrodynamic correlation time τqq(k) [4,3]. In contrast to the results known for the
correlation time, describing the concentration fluctuations in non-ionic liquids or
liquid metallic alloys [16], the correlation time τqq(k) at k → 0 tends to a nonzero
positive value, namely,
lim
k→0
τhyd
qq (k) '
ε
4π
(1
σ
+
λ2
pT
ρCp
)
≡ τqq(0) + τhq(0), (5)
defined by conductivity σ and thermoelectric effects (first and second term in (5),
respectively; see reference [3] for details).
398
Propagating collective excitations in molten salts
The “bare” optic-like eigenvalues, derived for the two-variable subset A
(2), are
purely imaginary ones
z0
±
(k) = ±i〈ω̄4
k〉
1
2 ≡ ±iω(k), (6)
where the following shortcut
〈ω̄2n
k 〉 =
〈ω2n
k 〉
〈ω2n−2
k 〉
was introduced for normalization of frequency moments of the charge-charge dy-
namical structure factor Sqq(k, ω) ([17]):
〈ω2n
k 〉 =
∫
∞
−∞
ω2nSqq(k, ω)dω.
Purely imaginary eigenvalues describe the propagating collective excitations without
damping. In real ionic systems, due to interaction with different kinds of dynamic
processes, these modes have a nonzero damping coefficient or, in other words, the
lifetime for these excitations is finite.
Our next task is to take into account the coupling between slow charge density
fluctuations and short-time charge current fluctuations, and to find out how this
coupling can affect the frequency and damping of optic-like excitations in the hy-
drodynamic limit. In the analytical approach we can restrict ourselves just to the
consideration of the charge fluctuations within the set of dynamic variables (2). In
fact, the dispersion and damping of optic-like excitations in molten salts is the re-
sult of interaction with total density and heat fluctuations as well. However, it is
extremely difficult to perform an analytical analysis by taking into account the cou-
pling between all possible “bare” excitations. All the microscopic processes between
the generalized hydrodynamic and kinetic collective excitations are taken into ac-
count in our numeric approach performed within many-variable scheme in the next
section. Simplified models of separated dynamics considered here permits to better
understand the general tendencies of spectra formation in molten salts.
The generalized hydrodynamic matrix [15] constructed using the basis set of
dynamical variables A
(3q) for the description of solely charge fluctuations, can be
written down in the following form:
T(k) =
0 −ik 0
0 0 −1
−ik−1(〈ω̄4
k〉 − 〈ω̄2
k〉)τ
−1
qq 〈ω̄4
k〉 (〈ω̄4
k〉〈ω̄
2
k〉
−1 − 1)τ−1
qq
. (7)
Three eigenvalues of the generalized hydrodynamic matrix T(k) are rather com-
plicated for a detailed analysis. Therefore, we present them in a little bit simpler
manner. Let us denote by δ the additional damping of the relaxing mode d(k) caused
by the coupling with optic-like excitations, so that:
d(k → 0) ≡ d0 + δ =
1
τ 0
qq
+ δ. (8)
This means that instead of thermal correction τhq(k) for the correlation time (5),
which appeared due to the coupling between charge and heat fluctuations within
399
T.Bryk, I.Mryglod
the hydrodynamic model, the three-variable model (2) produces another correction
to d(k) caused by the coupling between slow and fast charge fluctuations. Hence,
immediately the eigenvalue problem for the generalized hydrodynamic matrix T(k)
can be reduced [with a known purely real eigenvalue (8)] to the quadratic equation,
and one gets for other two eigenvalues the following solutions:
z±(k)|k→0 ' Γq ±
√
√
√
√Γ2
q +
〈ω̄2
k〉 − 〈ω̄4
k〉
1 + τ 0
qqδ
. (9)
If the expression under the square root is negative, the eigenvalues z±(k) describe a
pair of complex-conjugated modes which correspond to the charge waves, propagat-
ing in opposite directions. Thus, due to the coupling with slow relaxing processes,
connected with the electric conductivity, the frequency of optic-like excitations is
somewhat modified in comparison with the “bare” solution (6), and one has:
ω(k → 0) =
√
√
√
√
〈ω̄4
k〉 − 〈ω̄2
k〉
1 + δτ 0
qq
− Γ2
q, (10)
where Γq is the damping coefficient for the optic-like excitations:
Γq =
〈ω̄4
k〉 − (2 + δτ 0
qq)〈ω̄
2
k〉
2τ 0
qq〈ω̄
2
k〉
. (11)
One can see in (11) that in a molten salt with a high conductivity, the value of
correlation time τ 0
qq at k = 0 is very small, which causes a high damping of optic-like
excitations (or charge waves). It is quite possible that optic-like excitations can be
even suppressed in some molten salts with high conductivity σ. For large values of
the damping coefficients Γq the expression under the square root in equation (10)
can become positive and this means that instead of a pair of complex-conjugated
solutions one obtains two purely real eigenvalues describing kinetic relaxing modes.
Similar results were previously found for nonionic binary mixtures (see, e.g., [11]).
Note also that the expressions (8)–(11) can be easily connected with the results
derived for mass-concentration collective excitations in nonionic binary mixtures
[11] by setting: δ → 0, 〈ω̄2
k〉 ∼ k2, and τ 0
qq ∼ k−2.
Thus, based on the simplified analytical analysis presented above, we can con-
clude that the damping of long-wavelength optic-like excitations is defined mainly
by the electric conductivity σ and the second and fourth order frequency moments.
Depending on these quantities the dispersion law ω(k) of optic-like excitations in
long-wavelength region can be essentially changed, in particular such modes can be
suppressed in some molten salts with high conductivity σ.
3. Longitudinal and transverse spectra of molten NaCl
3.1. Calculation details
Molecular dynamics simulations were performed in the standard microcanonical
ensemble on a system of 1000 particles in a cubic box in order to obtain the time
400
Propagating collective excitations in molten salts
evolution of hydrodynamic and short-time extended dynamical variables:
A(k, t) =
{
nt(k, t), nq(k, t),Jt(k, t),Jq(k, t), ε(k, t), J̇t(k, t), J̇q(k, t), ε̇(k, t)
}
, (12)
where the total number density nt(k, t), charge density nq(k, t), total mass-current
density Jt(k, t) and energy density ε(k, t) belong to the most slow hydrodynamic
variables. Time correlation functions and static averages were used to estimate the
matrix elements of the generalized hydrodynamic matrix T(k), constructed on the
variables A = {Ai(k, t)}. Eight longitudinal and four transverse eigenvalues of the
matrix T(k) were obtained for each wavenumber k sampled in MD. They form the
spectra {zj(k)} of generalized collective excitations in the GCM approach.
We have used the Tosi-Fumi potentials for simulations of NaCl at 1262 K [18].
The time step in MD simulations was 2·10−15 s. The long-range Coulomb interaction
was taken into account via Ewald summation. Eleven wavenumbers with the small-
est value of kmin = 0.197 Å−1 were sampled in computer simulations. Additional
averaging over all possible directions of vectors k were taken. The time evolution of
basis dynamical variables was recorded over the production run of 3 ·105 time steps.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3 3.5
S
ta
tic
s
tru
ct
ur
e
fa
ct
or
s
k / A�°-1
Snn
Sqq
Figure 1. Density-density Snn(k) and charge-charge Sqq(k) static structure fac-
tors directly evaluated in molecular dynamics.
In figure 1 the static structure factors “total density-total density”
Snn(k) = 〈nt(k, 0)n∗
t (k, 0)〉
and “charge density-charge density”
Sqq(k) = 4 〈nq(k, 0)n∗
q(k, 0)〉
are shown at the wavenumbers sampled in MD simulations. The factor 4 in the Sqq(k)
structure factor was used to normalize its short-wavelength asymptotics to unity. It
401
T.Bryk, I.Mryglod
is important to note the structureless feature of the Snn(k) structure factor, while
charge fluctuations have a well-pronounced first peak of the relevant structure factor
located at k = 1.7 Å−1. This makes the main difference in the behaviour of static
structure factors in molten NaCl in comparison, for instance, with the behaviour
of Snn(k) and “mass-concentration – mass-concentration” structure factor Sxx(k)
in a Lennard-Jones equimolar KrAr mixture [9–11]. In the later case, the function
Sxx(k) exhibited almost a structureless shape and Stt(k) had a sharp first maximum.
The effect of such a different behaviour of static structure factors on the spectra of
longitudinal excitations will be discussed in the next paragraphs.
3.2. Transverse dynamics
Transverse dynamics in liquids is much simpler for the analysis than the longitu-
dinal one because, in particular, the transverse components of dynamical variables
do not interact with the heat fluctuations. In the case of transverse dynamics the
basis set within the model (12) consists of four dynamical variables:
A
(4T ) =
{
JT
t (k, t), JT
q (k, t), J̇T
t (k, t), J̇T
q (k, t)
}
. (13)
Hydrodynamic treatment of the transverse dynamics is based only on a single vari-
able JT
t (k, t) which is the transverse component of total mass-current. The imaginary
parts of four eigenvalues, which was obtained for the generalized hydrodynamic ma-
trix T(k), constructed on the basis set (13), is shown in figure 2.
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5 3 3.5
Im
z
j /
p
s-1
k / A�°-1
t
q
Na
Cl
Figure 2. Dispersion of shear waves (crosses) and transverse optic-like excitations
(symbols “+”) obtained numerically in the four-variable (A(4T )) GCM approach
for NaCl. Lines correspond to the eigenvalues obtained by separated treatment of
the “t-q” and “partial” dynamics with the help of the following sets of transverse
variables: A
(2,t) (solid line), A
(2,q) (dashed line), A
(2,Na) (dotted line), A
(2,Cl)
(dash-dotted line).
402
Propagating collective excitations in molten salts
For all the wavenumbers sampled in this study, we obtained two complex-conju-
gated pairs of eigenvalues which correspond to the propagating collective excitations.
Since the hydrodynamic picture of transverse dynamics is reflected by a single re-
laxing mode, one can note that the smallest wavenumber reached in this study is
located, in fact, outside the hydrodynamic region which is very small. However, the
kmin is close enough to the hydrodynamic region because the dispersion of the low-
frequency branch in figure 2 implies the boundary of a propagation gap for shear
waves for k < kmin. To find the origin of each branch we have used the same treat-
ment of transverse dynamics as we did before for the case of metallic alloys and
Lennard-Jones mixtures (see, for details, [9]). Namely, the basis set A
(4T ) was divid-
ed into two subsets in two different ways: (i) we separated the dynamical variables
reflecting mass-current and charge-current fluctuations
A
(2,t) =
{
JT
t (k, t), J̇T
t (k, t)
}
,
A
(2,q) =
{
JT
q (k, t), J̇T
q (k, t)
}
,
(14)
and, (ii) we split up the set A
(4T ) on the partial dynamical variables as follows:
A
(2,Na) =
{
JT
Na(k, t), J̇T
Na(k, t)
}
,
A
(2,Cl) =
{
JT
Cl(k, t), J̇T
Cl(k, t)
}
.
(15)
These cases of two-variable subsets (14)-(15) were considered separately and the
obtained numerical results are shown by lines in figure 2 (explanation is given in
the capture). It is seen in figure 2 that in some restricted regions of wavenumbers
a very good correspondence between two-variable and four-variable treatment (13)
is observed. Namely, one can immediately conclude that there exist two different
domains in the k-space, where such a correlation is very good. In long-wavelength
region the transverse dynamics of a molten salt can be nicely reproduced in terms
of weakly interacted shear waves (low-frequency branch, solid line) and optic-like
excitations (high-frequency branch with almost flat dispersion, dashed line). In this
region one can claim about collective dynamic behaviour. For larger k, k > 1.2 Å−1,
the transverse dynamics can be represented as a superposition of the partial trans-
verse waves defined by the motion of light (Na) and heavy (Cl) components [dotted
and dashed-dotted lines in figure 2, respectively]. This is the region of the so-called
partial dynamics. It is interesting that such a picture for the transverse dynamics
in molten salts corresponds quite well to acoustic and optic phonons in ionic crystal
where on the first Brillouin zone boundary the relevant branches are mainly defined
by heavy and light components, respectively.
3.3. Longitudinal dynamics
The spectrum of longitudinal collective excitations was obtained by solving the
eigenvalues problem for the 8×8 generalized hydrodynamic matrix T(k), constructed
on the following basis set of dynamic variables:
A
(8L) =
{
nt(k, t), nq(k, t), JL
t (k, t), JL
q (k, t), ε(k, t), J̇L
t (k, t), J̇L
q (k, t), ε̇(k, t)
}
. (16)
403
T.Bryk, I.Mryglod
The results of our calculations, obtained for the dispersion of propagating longitudi-
nal modes, are shown in figure 3. In contrast to the case of transverse dynamics we
have obtained three branches of propagating collective excitations for k > 0.3 Å−1.
The lowest-frequency branch (shown by boxes) corresponds to heat waves, because
it can be quite well reproduced by considering a separated treatment of the heat fluc-
tuations on the basis subset
{
h(k, t), ḣ(k, t)
}
, where h(k, t) is a dynamical variable
of heat density. The branch of heat waves has a propagation gap in long-wavelength
range in complete agreement with the standard hydrodynamics, because the heat
waves belong to a class of kinetic collective excitations and cannot be obtained
within the linear hydrodynamic approach. Another branch of collective excitations
shown by crosses abides by almost linear dispersion law in long-wavelength region
and describes the sound modes. One can see in figure 3 a small deviation from the
linear dispersion law csk (dotted line) with cs = 3400 m/s, which can be attributed
to the so-called positive dispersion due to the mode-coupling effects. This branch
reaches at k ≈ 1.2 Å−1 a region with a rather flat dispersion.
0
10
20
30
40
50
0 0.5 1 1.5 2 2.5 3 3.5
Im
z
j(k
) /
p
s-1
k / A�°-1
Figure 3. Dispersion of longitudinal collective excitations for molten NaCl, ob-
tained for the basis set A
(8L)(k, t) [see (16)]: generalized sound modes (crosses),
longitudinal optic-like excitations (symbols “+”), and heat waves (boxes), respec-
tively. Dotted line shows the linear hydrodynamic dispersion law with the speed
of sound cs = 3400 m/s.
The third branch of propagating excitations, shown in figure 3 by symbols “plus”,
tends to a value of ∼ 53 ps−1 at k → 0, and in the region of k ≈ 1.7 Å−1 it has
a wide minimum. In a similar manner as was described above, using the separated
treatment of the total density and charge density fluctuations, it has been shown
that the longitudinal propagating branches, shown in figure 3 by symbols “cross”
404
Propagating collective excitations in molten salts
and “plus”, correspond to the acoustic and optic-like excitations and could be well
reproduced on these separated subsets up to k ≈ 2 Å−1. The wide minimum in
the dispersion of optic-like excitations is closely related to a position of the first
maximum in the static structure factor Sqq(k). It is worth noting that in nonionic
binary mixtures, for instance in the case of KrAr mixture [11], the wide minimum
in the dispersion law is attributed to the acoustic branch and is connected mainly
with the k-dependence of static structure factor Snn(k) (e.g., for KrAr this function
has a sharp first maximum).
It is also interesting to compare the dispersion curves for transverse and longi-
tudinal optic-like excitations in molten NaCl, shown in figures 2 and 3 by symbols
“plus”. In contrast with nonionic binary mixtures, the main conclusion of such a
comparison is the existence of a gap between L and T branches in long-wavelength
limit, which is approximately equal to 25 ps−1 for the case of NaCl studied. Such
a gap in molten salts is a consequence of long-range Coulomb interaction between
ions and has a complete analogy in ionic NaCl crystal.
4. Conclusions
Using the generalized collective modes approach, we have investigated the specific
features of collective dynamics in a molten salt NaCl beyond the hydrodynamic
region. We focused on the study of the propagating kinetic collective excitations
which cannot be found within the standard hydrodynamic theory but play a crucial
role in the dynamics of the system. In this way, three types of propagating collective
modes were studied more in detail. Namely, optic-like excitations, heat and shear
waves. The main results of this study can be summarized as follows:
(i) An analytical condition for the existence of long-wavelength optic-like exci-
tations is derived within the three-variable analytical model describing the charge
fluctuations. According to this condition, a high conductivity in molten salts can
suppress the charge waves;
(ii) The spectrum of longitudinal collective excitations in NaCl was calculated
numerically based on the eight-variable scheme of the generalized collective modes
approach. It is shown, that this spectrum contains three branches of generalized
propagating modes beyond the hydrodynamic region. Namely, we found one acoustic
and two branches of kinetic propagating excitations describing the low-frequency
heat waves and optic-like excitations;
(iii) Using the four-variable set for the case of transverse dynamics in NaCl,
we studied the spectrum of the generalized transverse excitations and found two
branches of propagating collective modes: shear waves and optic-like excitations.
Comparing the results obtained with those found for the separated subsets of dy-
namic variables, there is established the existence of two domains in k-space, in
which the branches are of different character: short-wavelength domain of “partial”
dynamics, and long-wavelength domain, where the collective type of dynamics is
dominant;
(iv) In complete agreement with the results known for ionic crystals, we have
405
T.Bryk, I.Mryglod
found a gap between longitudinal and transverse optic-like excitations in molten
NaCl in long-wavelength limit, caused by the long-range Coulomb interaction be-
tween ions. The obtained results are discussed in comparison with those known for
nonionic binary mixtures.
Acknowledgements
I.M. thanks the Fonds zur Förderung der wissenschaftlichen Forschung (Austria)
for financial support under Project No. P15247.
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406
Propagating collective excitations in molten salts
Узагальнені колективні збудження у розплавах
солей
Т.Брик, І.Мриглод
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 6 травня 2003 р.
У методі узагальнених колективних мод (УКМ) досліджується повз-
довжня та поперечна динаміка розплавів солей поза областю гід-
родинамічної поведінки. В рамках простої тризмінної моделі ана-
літично проаналізовано особливості зв’язаної динаміки коротко-
та довго-живучих зарядових флюктуацій. В рамках восьми змінно-
го підходу методу УКМ, комбінуючи засоби аналітичної теорії та
молекулярно-динамічні розрахунки, для розплаву NaCl отримано та
проаналізовано дисперсійні закони для пропагаторних кінетичних
колективних збуджень типу оптичних фононних мод, теплової та
зсувної хвиль.
Ключові слова: узагальнена гідродинаміка, колективні збудження,
іонні рідини, розплави солей, поперечна і повздовжня динаміка
PACS: 05.20.Jj, 61.20.Lc
407
408
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