The Raman shift depends on the energy spacing of the molecules’ modes. However not all modes are “Raman active” i.e. not all appear in Raman spectra. For a mode to be Raman active it must involve a change in the polarisability, α of the molecule i.e.

\({\left( {\frac{{{\rm{d}}\alpha }}{{{\rm{dq}}}}} \right)_{\rm{e}}}\)\( \ne 0\) where q is the normal coordinate and e the equilibrium position.

This is known as spectroscopic selection. Some vibrational modes (phonons) can cause this. These are generally the most important, although electronic modes can have an effect, and rotational modes may be observed in gases at low pressure.

The spectroscopic selection rule for infrared spectroscopy is that only transitions that cause a change in dipole moment can be observed. Because this relates to different vibrational transitions than in Raman spectroscopy, the two techniques are complementary. In fact for centrosymmetric ( centre of symmetry ) molecules the Raman active modes are IR inactive, and vice versa. This is called the rule of mutual exclusion.

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The origin of Stokes and anti-Stokes scattering due to vibrational modes can be explained in terms of the oscillations involved. The polarisability (α) of the molecule depends on the bond length, with shorter bonds being harder to polarise than longer bonds. Therefore if the polarisability is changing then it will oscillate at the same frequency that the molecule is vibrating (ν_{vib}).

Polarisability of the molecule:

\[\alpha = {\alpha _0} + {\alpha _1}\sin (2\pi {\nu _{{\rm{vib}}}}t)\]

There is an external oscillating electric field from the photon, with a frequency ν_{p}:

*E = E*_{0} sin( 2 *π ν*_{p} *t*)

Therefore the induced dipole moment is:

\[{p_{{\rm{ind}}}} = \alpha E = ({\alpha _0} + {\alpha _1}\sin (2\pi {\nu _{{\rm{vib}}}}t)) \times {E_0}\sin (2\pi {\upsilon _{\rm{p}}}t)\]

Using the trigonometric identity:

\[\sin A \times \sin B = \frac{{\cos (A - B) - \cos (A + B)}}{2}\]

The induced dipole moment is:

\[{p_{{\rm{ind}}}} = {\alpha _0}{E_0}\sin (2\pi {\upsilon _{\rm{p}}}t) + \frac{{{\alpha _0}{E_0}}}{2}\cos (2\pi ({\upsilon _{\rm{p}}} - {\nu _{{\rm{vib}}}})t) - \frac{{{\alpha _0}{E_0}}}{2}\cos (2\pi ({\upsilon _{\rm{p}}} + {\nu _{{\rm{vib}}}})t)\]

A dipole moment oscillating at frequency ν results in a photon of frequency ν. Therefore in this case there are photons scattered at frequency ν_{p} (Rayleigh scattering), ν_{p} – ν_{vib} (Stokes scattering) and ν_{p} + ν_{vib} (anti-Stokes scattering).

Of course if the polarisability is not changing then the dipole moment will simply oscillate at frequency ν_{p}, and only Rayleigh scattering will occur. This is the origin of the spectroscopic selection rule for Raman scattering.

The Raman (and IR) activity of more complicated molecules can be determined using their symmetry and group theory, which goes beyond the scope of this TLP. There are links to more information in the Going further section.

The above is based on single molecules in a gas, and hence not interacting with neighbours. In materials science Raman techniques are more often used for solids, where molecules cannot be taken individually. In crystalline materials vibrations are quantised as *phonons*, modes determined by the crystal structure. The spectroscopic selection rule still applies, i.e. only phonons with a change in polarisability are Raman active. Phonons are generally of a lower frequency than the vibrations in gases, so result in lower wavenumber shifts. Structural information can therefore be determined from these shifts.

Crystal orientation can also be determined from the polarization of the scattered light.

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