Raman Spectroscopy

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Raman Spectroscopy The intensity of light
emitted from a classical induced dipole is given
by
where
is an induced dipole oscillating according to
An isotropic molecule will obey
where
is the polarizability of the molecule and
is the electric field. A general molecule, which
is anisotropic, obeys
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but it is possible to find a set of principle
axes which diagonalize
We define the spherical part,
and the anisotropy,
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With the definitions for different observed parts
of the scattered light given in the figure below,
it is found that
where
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It is customary to define the depolarization
ratio, given for a linearly polarized light
The polarizability can be expanded in terms of
the normal modes
which will lead to the following expression for
the scattering dipole
The first term is the Rayleigh scattering. The
second and the third are the anti-Stokes and
Stokes Raman scattering, respectively.
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Quantum-mechanically the Raman process goes
through a virtual intermediate level, and leads
to either de-excitation or excitation of a
vibrational level. The quantum mechanical Raman
scattering intensity will depend on an integral
of the form
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As for IR transitions, symmetry dictates (in the
case that n is totally symmetric, e.g. it is the
ground state) that
will belong to one of the irreducible
representations of n. The components of
transform exactly like gg, namely
transform like xy etc. The quantum mechanical
expression for the polarizability can be further
derived from perturbation theory and is shown to
be
where
is a certain element of the transition dipole
moment,
is the laser energy (frequency),
is the energy of vibronic level n and
is the energy of virtual vibronic level V.
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The ratio of the Stokes to anti-Stokes
intensities is found to be
The population ratio
(the Ns are populations while the gs are
degeneracies) depends on difference in thermal
populations of the final states in the Stokes and
anti-Stokes processes, and this dependence is
given by a Boltzmann factor.
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