Download A method for determining the magnitude of the Raman scattering

A method for determining the magnitude of the Raman
scattering matrix element for diamond-type crystals
E. Burstein, S. Ganesan
To cite this version:
E. Burstein, S. Ganesan. A method for determining the magnitude of the Raman scattering
matrix element for diamond-type crystals. Journal de Physique, 1965, 26 (11), pp.637-638.
<10.1051/jphys:019650026011063700>. <jpa-00206052>
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Submitted on 1 Jan 1965
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LE
JOURNAL DE
TOME
PHYSIQUE
26,
NOVEMBRE
1965,
637.
A METHOD FOR DETERMINING THE MAGNITUDE
OF THE RAMAN SCATTERING MATRIX ELEMENT FOR DIAMOND-TYPE CRYSTALS
By
Laboratory
E. BURSTEIN
for Research
on
(1)
and S. GANESAN
the Structure of Matter and
University of Pennsylvania, Philadelphia, Pa.,
(2),
Physics Department,
U. S. A.
Résumé. 2014 On discute le mécanisme qui produit la bande d’absorption infrarouge du premier
ordre induite par un champ électrique dans les cristaux du type diamant.
L’intensité de cette bande est déterminée par la relation entre la polarisation électrique et le
déplacement atomique dans la maille. La mesure de la constante d’absorption de la bande induite
devrait donc fournir des renseignements quantitatifs sur les éléments de la matrice de diffusion
Raman du premier ordre. Ce phénomène d’induction de bandes infrarouges par le champ électrique doit se produire aussi pour des modes actifs en Raman, dans d’autres structures cristallines
à centre de symétrie, et pour des modes d’impuretés actifs en Raman.
Abstract. 2014 A discussion is presented of the mechanism for the electric field induced first order
infrared absorption band in diamond-type crystals. The strength of the induced band is determined by the dependence of the electronic polarization on the relative atomic displacements in
the unit cell. A measurement of the absorption constant of the induced band should therefore
provide quantitative information about the first order Raman scattering matrix elements. The
phenomenon of field induced infrared absorption bands should also exist for Raman active modes
in other centrosymmetric crystal structures, and for Raman active impurity modes.
1. Introduction.
As part of a general theoretical and experimental investigation of "morphic"
effects in crystals induced by electric fields, it was
of interest to us to study the effect of an applied
electric field on the infrared absorption spectrum
of diamond-type crystals. In this type of crystal,
the effective charge of the atoms is zero and there
is no first order (one phonon) resonance absorption
of infrared radiation by the fundamental (q = 0)
optical modes. Such crystals do exhibit well defined, although weak, higher order absorption bands
arising from higher order terms in the electric
moment [1]. From a phenomenological point of
view, the application of an electric field removes
the center of symmetry of the diamond structure,
so that the q = 0 optical vibration modes which
are active in first order scattering also become
active in first order infrared absorption (3). As
shown in Section 2, the strength of the field induced
absorption band is determined by the linear dependence of the static electronic polarizability on the
relative displacements of the two atoms in the
(primitive) unit cell, aoco p X. Since the static and
optical frequency electronic polarizabilities are
essentially equal in diamond-type crystals
(fx(co) = x(0) mo) a measurement of the strength
-
=
(1) Research supported in part by the U. S. Office of
Naval Research.
(2) Research supported by the Advanced Research Projects Agency.
(3) We are also investigating the corresponding morphic
effect in NaCI and CsCI-type crystals in which a first order
Raman spectrum is induced by an applied electric field,
of the induced absorption band should provide
quantitative information about the magnitude of
the first order Raman matrix elements which are
determined by è)oc{ (0) IbX.
2. Theoretical.
The mechanism, for the field
induced infrared absorption band may be visualized in terms of a simple " atomic " model as
follows : The applied electric field induces a
dipole moment at each atom. The fundamental
optical vibrations of the lattice cause a change in
the electronic polarizability of the atoms so that
the induced dipole moment varies in magnitude
and orientation at the frequency of the lattice
vibrations.
The expressions for the electronic polarizability,
oc, and the induced dipole moment, ti, of the atoms,
take the form :
-
where a0 is the static (m
0) electronic polariis relative displaof
the
X
and
atoms ;
zability
cement of the two atoms in each (primitive) unit
cell, which varies at the frequency of the q sw 0
optical vibration modes (4).
The first term in the expression for the induced
==
(4 ) For the present purposes, we neglect the effect of the
applied field on the amplitude and frequency of vibration
modes.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019650026011063700
638
dipole moment, (1.0
electric moment.
=
ao
Eo represents
the static
The second term
I
---
represents the time varying electric moment which
provides the coupling to the infrared radiation.
The second term can also be expressed as
where
represents the field induced effective charge
of the
atoms. We see that ei is proportional to boco/ZX,
the first order dependence of the static electronic
polarizability
relative
displacement amplimagnitude of the infrared
absorption constant depends on the square of the
electric moment, the strength of the induced
absorption band will be proportional to
It WI
will vary
as E2
2
el
i.e., it
o.
vary as
zx 0 I.e.,
tude.
=
on
Since the
? Eo
The matrix element for first order Raman scattering depends on the magnitude of boc(co) IbX
where cx(co) is the electronic polarizability of the
atom at the frequency of the exciting radiation.
However, in the case of diamond-type crystals, the
static and optical frequency values of the electronic
polarizability are essentially identical, except when
the excitation frequency is close to a resonance
frequency. Thus boco/bX -- 3«( m) j3 X, and a
measurement of the absorption constant of the
induced first order infrared absorption band should
provide quantitative information about the coefficients which determine the intensity of first order
scattering (5).
The strength of the field induced first order
infrared absorption band in diamond-type crystals
is determined by the factor ei. 03BC1
el. a1 eo Eo
where ei is the (unit) polarization vector of the
=
03BC1= «1 eo Eo, has a component along the direction of polarization of the induced radiation. In
addition, one has the requirement of energy and
momentum conservation. Thus, the selection
rules are determined by the first order term in the
electronic polarizability and are therefore similar
to those governing the first order Raman effect.
We may accordingly expect the (q
k sw 0) longitudinal as well as the (q == k -- 0) transverse optical modes to contribute to the induced absorption,
and that the absorption bands will exhibit welldefined polarization effects which depend on the
direction of the incident radiation and the orientation of the crystal and that of the applied field.
It is of interest to note that electric field induced
infrared absorption bands in homonuclear molecules (6) were predicted by Condon in 1932 [2] and
experimentally observed for H2 molecules at fields
of 105 volts/cm by Crawford and Dagg in 1953 [3].
The field induced absorption data obtained by
Crawford and MacDonald in 1958 [4] actually provide the most reliable values for the polarizability
matrix elements of H 2 (7).
It should be pointed out that field induced
infrared absorption bands associated with Raman
active modes of vibration; should also occur for
other crystal structures having a center of symmetry, and for impurity modes. In the case of
polar crystals such as CaF2 which exhibit a first
order Raman active mode, the strength of the
induced infrared absorption band will be determined by contributions from the atomic as well as
from the electronic polarizability. Since the
strength of the second order infrared absorption
bands in polar crystals is appreciably greater than
in diamond-type crystals, it will be somewhat
more difflcult to observe the induced absorption
bands in polar crystals unless the induced bands
fall in wavelength regions where there are no
major contributions from second order processes.
=
electronic field of the infrared radiation and eo is
the unit polarization vector of the applied electric
field. An absorption band can be observed only
when the varying induced dipole moment,
We wish to thank ProAcknowledgements.
fessor J. Birman and Professor A. Maradudin for
valuable discussions of electric field induced pheno-
(5) Although higher order terms in the electronic polarizability and anharmonic coupling effects will generally be
small, it may be possible, under optimum conditions, to
observe a field induced second order infrared absorption
spectrum whose strength will be related to the matrix
(6) We are grateful to H. L. Welsh for informing us
about the theoretical and experimental work on the electric
field effects in homonuclear molecules.
(7) In the case of the homonuclear molecules, the rotational, as well as vibrational, transitions of the molecules
lead to electric field induced absorption bands.
elements for second order Raman
scattering.
-
mena.
REFERENCES
[1] LAX (M.) and BURSTEIN (E.), Phys. Rev., 1955, 97, 39.
[2] CONDON (E. U.), Phys. Rev., 1939, 41, 759.
[3] CRAWFORD (M. F.) and DAGG (I. R.), Phys. Rev., 1953,
91, 1569.
[4] CRAWFORD (M. F.) and MAcDONALD (R. E.), Can. J.
Phys., 1958, 36, 1022.