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On the local field method in the electrodynamics of condensed media
Journal of Molecular Structure, 294 (1993) 83-86 Elsevier Science Publishers
83
B.V., Amsterdam
ON THE LOCAL FIELD METHOD IN THE ELECTRODYNAMICS OF CONDENSED MEDIA N.S.Tyu, A.N.Goncharuk and S.G.Ekhilevsky Department of Higher Mathematics, Polytechnical Institute, Donetsk 340000, Ukraine The present work studies the quantum-statistical theory of local field method (LFM) in the condensed media. The basic LFM equations with the account of spatial dispersion and arbitrary multipole transitions are formulated. For the perfect molecular crystals the expression for the polarizability tensor Is obtained. This formula Is used to study the effect of rotation of the polarization plane. The problems of application of the theory being developped towards the liquid and partially ordered condensed media are also discussed.
1. INTRODUCTION At present, for the investigation of the optical properties of condensed media the LFM has been widely used [l-3]. This Is mostly due to simplicity and clearness of this method and also by its applicability to disordered or partially ordered condensed media, e.g. liquid or mixed crystals. The LFM essence consists in the following. An individual structural unit (molecule or atom) constructing the particular condensed medium is considered rather than the medium itself and the difference between the macroscopic field electric satisfaying the phenomenological Maxwell equations and the electric field acting on this structural unit is taken Into account. The relationships between the macroscopic and acting fields are essential in this method and for the perfect crystals they were conventionally derived previously using Ewald technique [I]. For the disordered media these relationships has been derived relying on
some or other phenomenological considerations [3]. For the partially ordered media the deduction of abovementioned equations is being complicated by the fact, that it Is necessary to account the intermolecular correlations, which are impossible to neglect due to unadequate description of the real experimental situations [3]. The solution of this problem in the dipole approximation has been suggested recently in [4]. However, all the studies did’ not take into account the spatial dispersion and highest multipole transition moments of the separate structural units. In order to solve the problems, mentioned above, the study in question develops the general LFM theory, based on the density matrix equation and enabling by the natural way to account the statistical behaviour of the electrodynamic events in the condensed media. It should be pointed out, that the similar approach has been suggested in [5], when studying the optical effects in the perfect molecular crystals.
0022-2860/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
84
2. HAMILTONIAN OF PROBLEM AND GENERAL LFM-EQUATIONS.
distance)
Let us consider the quantum-mechanical system, comprising the neutral molecules and the transverse electromagnetic field. In the Coulomb’s calibration the system ~amlltonian can be expressed as follows H =H, where
+H, +V +W H,
noninteracting
and x~O,
If IC> 1/
a,
674”)p
w)
tensor of is electroconductivity the molecule m, N is the number of molecules in the medium,
(1)
is
the
Hamiltonian
molecules,
H,
is
of the
Hamiltonian of radiation field, V and W of the molecules are the operators interactions with the iongitud~nal and transverse parts of the electromagnetic field. Using (1) and the equation for density matrix let us calculate the quantum averages of the current operators and the total and local electromagnetic fields operators. If we ignore the correlation in orientation the positron and of the molecules and limit the approach by linear field approximation, then we receive the of equations for the given system averages. Let us exclude from these equations the average values of the current operators and extract the Fourier-component of the macroscopic field in medium Ei @II). As a result, we get the following integral equation Fourier~omponent of the acting
for field
the
function of Green the the electromagnetic field, V is the volume, occupied by one molecule, 6,, is the iS
Kronecker symbol, di
is the transverse
6
-function. The first term in (3) describe the Coulomb’s charges interaction. The second term Is due to the account of the short wavelength retarding interaction between the medium charges. In the optical range of spectrum it is less than the first one by a factor of &a)” = 10” [6], It should pointed out that function the 4 from zero only for ;c.J. KW) differs ~
tensors
(LFT)
T$)@?
o),
defining
them by means of the equality m
@) =I
T$) @? w) Ei Q?” w)
(4)
d using the tensors, defined above, the medium polarizabiiity tensor Xi, @?’ w) and where
XP = 1,
characteristic
if
dimension
/ctI/a
(a
is
the
of ~ntermolecuiar
dielectric the E, @?’ w) can
be
permittivity expressed
tensor via the
85
x,, (A&x) = v-l 2
x!J)(W’w)
stipulated by the umklapp processes [6]. If this is neglected it can be shown that the eq (6) Coincides with the expression for the crystal polarizability tensor, which follows from the theory of Frenkel excitons. with of Xi, (E?o) coincide The poles
x
K”
x T;;’ (i?“k’w)
(5)
The equations for LFT can be derived from eq (2) if substitutes eq. (4) into them and takes ‘into account the arbitrary character of the macroscopic field E?(A).
3. LFM IN PERFECT MOLECULAR CRYSTALS. RELATION WITH THE THEORY OF FRENKEL EXCITONS Let us apply the approach described above towards perfect molekular crystals with several molecules in the unit cell. In this case the integration over I?’ in the eq. (2) results into the infinite sums over the vectors of the reciprocal lattice. Solving this equation and using the formula (5) we receive the following expression for the tensor of the crystal macroscopic polarizability [7]:
xi, (&) =21-l3
{g (f+
fiR)-l}$+)OO
frequencies of the mechanical excitons, which complies with the theory of the linear which the according to response, susceptibility resonances should coincide with natural frequencies of the medium. If the eq. (6) we ignored the spatial dispersion and restricted the calculations to cubic crystals with only one molecule in the unit cell, we obtained the classical Lorentz-Lorenz relationship. Generally the eq. (6) enables to study any possible effects of the spatial dispersion in the framework of LFM. in particular, from eq. be obtained the following (6) can expression for the gyrotropy tensor [7].
where
(6)
a:;’ (0)
polarizabiiity molecule
where
a
u
#I are
the
numbers
of
molecules in a cell. f is the unit matrix, fi is the Green matrix (see [7] for details). This expression takes into account the distribution of polarizability over the molekular volume, as well as contribution of the highest molecular multipoles and finiteness of the the intermolecular distances. In addition it takes into account the shortwave moiekular interaction, mentioned in [8]. It should be noted that in the dielectric crystals this interaction is
4 are Q pst
and
tensor in
the
and dipole
the derivatives
coefficients.
The
A$) (w)
first
LFT
of
the
the
a
approximation,
of the term
are
in
acting braces
field (7)
is widentiy due to the optical activity of crystal. molecules constituting a The second term describes the gyrotropy stipulated by the asymmetrical positioning of the molecules in the unit ceil. The equality (7) is the analog (7) of the Lorentz-Lorenz relationship in the theory of optical activity.
86
4. LFM IN DISORDERED CONDENSED MEDIA The results, derived in sect. 2, are also valid in case of disordered media in which the positions of the molecules and its orientation are rigidly fixed. if we perform the configurational averaging over the infinitely stall volume then the corresponding integral equation becomes the difference equation. This can be explained by the restoring of the medium translational symmetry for the configurationaf averages. From this equation the classical Lorentz-Lorenz relationships are derived, in which the effective moiekular poiarizability tensor is present. The results obtained can be easily generalized for the simple isotropic liquids. Further on the present work studies the nematic liquid crystals. in this case besides configurational averaging we perform the statistical averaging over the orientations of the separate molecules. It has been shown, that when the correlations between the local field and the molecules dipole moments have not been taken into account, the solution of the reduced integral equation is equivalent to the generalized Lorentz-Lorenz relationships for the anisotropic perfect crystals with a
simple lattice. The account of the fluctuation effects in this case, as we think, be performed the similar way, as in the work [4].
REFERENCES 1. M.Born and K.Huang. Dynamical
2.
3.
4. 5. 6. 7. 8.
Theo.ry of Crystal Lattices, Oxford University Press, Oxford, 1954. V.M.Agranovich M.D.Gaianin. and Electronic Excitation Energy Transfer in NoKh-Holland, Condensed Matter, Amsterdam, 1984. E.M.Aver’yanov M.A.Osipov, and Uspekhi Fisicheskikh Nauk, 60/5 (1990), 89. M.A.Osipov, Chem. Phys. Lett,, 133 (1985) 471. H.A.Baii and A.D.McLachian, Pros. Roy. Sot. A 272 (1964) 433. S.G.Ekhilevsky and N.S.Tyu, Phys. Lett A 158 (1991) 258. N.S.Tyu and S.G.Ekhiievsky, Sol. State Corn. (1992) in press. K.B.Toiplgo, Ukrainsky physicheskll jurnai, 3112 (1986) 178.
83
B.V., Amsterdam
ON THE LOCAL FIELD METHOD IN THE ELECTRODYNAMICS OF CONDENSED MEDIA N.S.Tyu, A.N.Goncharuk and S.G.Ekhilevsky Department of Higher Mathematics, Polytechnical Institute, Donetsk 340000, Ukraine The present work studies the quantum-statistical theory of local field method (LFM) in the condensed media. The basic LFM equations with the account of spatial dispersion and arbitrary multipole transitions are formulated. For the perfect molecular crystals the expression for the polarizability tensor Is obtained. This formula Is used to study the effect of rotation of the polarization plane. The problems of application of the theory being developped towards the liquid and partially ordered condensed media are also discussed.
1. INTRODUCTION At present, for the investigation of the optical properties of condensed media the LFM has been widely used [l-3]. This Is mostly due to simplicity and clearness of this method and also by its applicability to disordered or partially ordered condensed media, e.g. liquid or mixed crystals. The LFM essence consists in the following. An individual structural unit (molecule or atom) constructing the particular condensed medium is considered rather than the medium itself and the difference between the macroscopic field electric satisfaying the phenomenological Maxwell equations and the electric field acting on this structural unit is taken Into account. The relationships between the macroscopic and acting fields are essential in this method and for the perfect crystals they were conventionally derived previously using Ewald technique [I]. For the disordered media these relationships has been derived relying on
some or other phenomenological considerations [3]. For the partially ordered media the deduction of abovementioned equations is being complicated by the fact, that it Is necessary to account the intermolecular correlations, which are impossible to neglect due to unadequate description of the real experimental situations [3]. The solution of this problem in the dipole approximation has been suggested recently in [4]. However, all the studies did’ not take into account the spatial dispersion and highest multipole transition moments of the separate structural units. In order to solve the problems, mentioned above, the study in question develops the general LFM theory, based on the density matrix equation and enabling by the natural way to account the statistical behaviour of the electrodynamic events in the condensed media. It should be pointed out, that the similar approach has been suggested in [5], when studying the optical effects in the perfect molecular crystals.
0022-2860/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
84
2. HAMILTONIAN OF PROBLEM AND GENERAL LFM-EQUATIONS.
distance)
Let us consider the quantum-mechanical system, comprising the neutral molecules and the transverse electromagnetic field. In the Coulomb’s calibration the system ~amlltonian can be expressed as follows H =H, where
+H, +V +W H,
noninteracting
and x~O,
If IC> 1/
a,
674”)p
w)
tensor of is electroconductivity the molecule m, N is the number of molecules in the medium,
(1)
is
the
Hamiltonian
molecules,
H,
is
of the
Hamiltonian of radiation field, V and W of the molecules are the operators interactions with the iongitud~nal and transverse parts of the electromagnetic field. Using (1) and the equation for density matrix let us calculate the quantum averages of the current operators and the total and local electromagnetic fields operators. If we ignore the correlation in orientation the positron and of the molecules and limit the approach by linear field approximation, then we receive the of equations for the given system averages. Let us exclude from these equations the average values of the current operators and extract the Fourier-component of the macroscopic field in medium Ei @II). As a result, we get the following integral equation Fourier~omponent of the acting
for field
the
function of Green the the electromagnetic field, V is the volume, occupied by one molecule, 6,, is the iS
Kronecker symbol, di
is the transverse
6
-function. The first term in (3) describe the Coulomb’s charges interaction. The second term Is due to the account of the short wavelength retarding interaction between the medium charges. In the optical range of spectrum it is less than the first one by a factor of &a)” = 10” [6], It should pointed out that function the 4 from zero only for ;c.J. KW) differs ~
tensors
(LFT)
T$)@?
o),
defining
them by means of the equality m
@) =I
T$) @? w) Ei Q?” w)
(4)
d using the tensors, defined above, the medium polarizabiiity tensor Xi, @?’ w) and where
XP = 1,
characteristic
if
dimension
/ctI/a
(a
is
the
of ~ntermolecuiar
dielectric the E, @?’ w) can
be
permittivity expressed
tensor via the
85
x,, (A&x) = v-l 2
x!J)(W’w)
stipulated by the umklapp processes [6]. If this is neglected it can be shown that the eq (6) Coincides with the expression for the crystal polarizability tensor, which follows from the theory of Frenkel excitons. with of Xi, (E?o) coincide The poles
x
K”
x T;;’ (i?“k’w)
(5)
The equations for LFT can be derived from eq (2) if substitutes eq. (4) into them and takes ‘into account the arbitrary character of the macroscopic field E?(A).
3. LFM IN PERFECT MOLECULAR CRYSTALS. RELATION WITH THE THEORY OF FRENKEL EXCITONS Let us apply the approach described above towards perfect molekular crystals with several molecules in the unit cell. In this case the integration over I?’ in the eq. (2) results into the infinite sums over the vectors of the reciprocal lattice. Solving this equation and using the formula (5) we receive the following expression for the tensor of the crystal macroscopic polarizability [7]:
xi, (&) =21-l3
{g (f+
fiR)-l}$+)OO
frequencies of the mechanical excitons, which complies with the theory of the linear which the according to response, susceptibility resonances should coincide with natural frequencies of the medium. If the eq. (6) we ignored the spatial dispersion and restricted the calculations to cubic crystals with only one molecule in the unit cell, we obtained the classical Lorentz-Lorenz relationship. Generally the eq. (6) enables to study any possible effects of the spatial dispersion in the framework of LFM. in particular, from eq. be obtained the following (6) can expression for the gyrotropy tensor [7].
where
(6)
a:;’ (0)
polarizabiiity molecule
where
a
u
#I are
the
numbers
of
molecules in a cell. f is the unit matrix, fi is the Green matrix (see [7] for details). This expression takes into account the distribution of polarizability over the molekular volume, as well as contribution of the highest molecular multipoles and finiteness of the the intermolecular distances. In addition it takes into account the shortwave moiekular interaction, mentioned in [8]. It should be noted that in the dielectric crystals this interaction is
4 are Q pst
and
tensor in
the
and dipole
the derivatives
coefficients.
The
A$) (w)
first
LFT
of
the
the
a
approximation,
of the term
are
in
acting braces
field (7)
is widentiy due to the optical activity of crystal. molecules constituting a The second term describes the gyrotropy stipulated by the asymmetrical positioning of the molecules in the unit ceil. The equality (7) is the analog (7) of the Lorentz-Lorenz relationship in the theory of optical activity.
86
4. LFM IN DISORDERED CONDENSED MEDIA The results, derived in sect. 2, are also valid in case of disordered media in which the positions of the molecules and its orientation are rigidly fixed. if we perform the configurational averaging over the infinitely stall volume then the corresponding integral equation becomes the difference equation. This can be explained by the restoring of the medium translational symmetry for the configurationaf averages. From this equation the classical Lorentz-Lorenz relationships are derived, in which the effective moiekular poiarizability tensor is present. The results obtained can be easily generalized for the simple isotropic liquids. Further on the present work studies the nematic liquid crystals. in this case besides configurational averaging we perform the statistical averaging over the orientations of the separate molecules. It has been shown, that when the correlations between the local field and the molecules dipole moments have not been taken into account, the solution of the reduced integral equation is equivalent to the generalized Lorentz-Lorenz relationships for the anisotropic perfect crystals with a
simple lattice. The account of the fluctuation effects in this case, as we think, be performed the similar way, as in the work [4].
REFERENCES 1. M.Born and K.Huang. Dynamical
2.
3.
4. 5. 6. 7. 8.
Theo.ry of Crystal Lattices, Oxford University Press, Oxford, 1954. V.M.Agranovich M.D.Gaianin. and Electronic Excitation Energy Transfer in NoKh-Holland, Condensed Matter, Amsterdam, 1984. E.M.Aver’yanov M.A.Osipov, and Uspekhi Fisicheskikh Nauk, 60/5 (1990), 89. M.A.Osipov, Chem. Phys. Lett,, 133 (1985) 471. H.A.Baii and A.D.McLachian, Pros. Roy. Sot. A 272 (1964) 433. S.G.Ekhilevsky and N.S.Tyu, Phys. Lett A 158 (1991) 258. N.S.Tyu and S.G.Ekhiievsky, Sol. State Corn. (1992) in press. K.B.Toiplgo, Ukrainsky physicheskll jurnai, 3112 (1986) 178.