Raman Spectroscopy: Tutorial
When electromagnetic radiation is scattered by a molecule or by a crystal, one photon of the incident radiation is annihilated and, at the same time, one photon of the scattered radiation is created.
The scattering mechanisms can be classified on the basis of the difference between the energies of the incident and scattered photons.
If the energy of the incident photon is equal to that of the scattered one, the process is called Rayleigh scattering.
If the energy of the incident photon is different to that of the scattered one, the process is called Raman scattering.
The Raman scattering (or Raman effect) was discovered in 1928 by V. C. Raman. Thanks to the development of the laser light, it is now a very important technique in the study of the matter structure. "If the substance being studied is illuminated by monocromatic light of frequency w_{0}, the spectra of the scattered light consists of a strong line (the exciting line) of the same frequency as the incident illumination together with weaker lines on either side shifted from the strong line by frequencies ranging from a few to about 3500 cm^{1}. The lines of frequency less than the exciting lines are called Stokes lines, the others antiStokes lines."(1)
 Raman spectroscopy on molecular compounds
 Raman spectroscopy on crystalline solids
Raman spectroscopy on molecular compounds
"The spectral lines differing in frequency from the exciting line, the Raman lines, have their origin in an interchange of energy between the incident photons and the molecules of the substance scattering the light. The lines wich appear very near the exciting line are correlated with changes in the rotational energy states of the molecules without changes in the vibrational energy states and form the pure rotational Raman spectrum. The lines farther from the exciting line are really bands of unresolved lines and are associated with simultaneous changes in the vibrational an rotational energy states.
The frequency shifts, that is, the difference between the frequencies of the Raman lines and the exciting line, are independent of the frequency of the exciting line."(1)
 Classic Theory  The electromagnetic field induces in the molecule an electric dipole moment, wich oscillates at the same frequency of the incident field and, then, emits radiation in all directions. The electric dipole moment induced by an electromagnetic field E=E_{0}cos(w_{0}t) is:
[1] m = a : E + (1/2) b : EE + (1/3!) c : EEE + ...
where : is the tensorial product. "a" is a second order tensor (polarizability); "b" and "c" are tensor of, respectively, third and fourth order (hyperpolarizability). The first term in [1] produces the linear Raman effect; the subsequent terms give higer order Raman effects (iperRaman effects).
A dipole oscillating at the frequency w_{0} emits, in all directions, radiation of the same frequency with intensity:
[2] I = [w_{0}^{4}/3c^{3}] * m^{2} = [w_{0}^{4}/3c^{3}] * a^{2}E_{0}^{2}cos^{2}(w_{0}t),
if the polarizability in constant. The molecular polarizability depends on the distribution of the electrons in the molecule, that changes with normal coordinates during vibrations and rotations. If a molecular vibration, or rotation is activated at a frequency w_{k}, the polarizability oscillates with the same frequency:
[3] a = a_{0} + a_{k}cos(w_{k}t + f_{k})
and the dipole moment results
[4] m = a_{0}E_{0}cos(w_{0}t) + (1/2)a_{k}E_{0}cos[(w_{0}+w_{k})t + f_{k}] + (1/2)a_{k}E_{0}cos[(w_{0}w_{k})t  f_{k}].
The intensity irradiated is then:
[5] I = [E_{0}^{2}/(3c^{2})] {w_{0}^{4}a_{0}^{4}cos^{2}(w_{0}t) + (w_{0}+w_{k})^{4} (a_{k}^{2}/4)cos^{2}[(w_{0}+w_{k})t+f_{k}] + (w_{0}w_{k})^{4} (a_{k}^{2}/4)cos^{2}[(w_{0}w_{k})tf_{k}]}=
= I_{R} + I_{as} + I_{s}.
The first term in [5] is the intensity scattered at the frequency w_{0} (Rayleigh scattering), the second and the third terms are respectively the intensities scattered at the frequency w_{0}+w_{k} (antiStokes Raman scattering) and at the frequency w_{0}+w_{k} (Stokes Raman scattering), with a ratio
[6] I_{s} / I_{as} = (w_{0}w_{k})^{4} / (w_{0}+w_{k})^{4} < 1.
The Stokes lines are, instead, more intense than the antiStokes lines. This experimental fact is correctly described only in the quantum theory of Raman scattering.
In molecular compounds is often possible to associate the different Raman peaks with the characteristic vibrational modes of interatomic bonds or functional groups (d = bending,
n = stretching).
Raman spectroscopy on crystalline solids
In Crystalline Solids, the Raman effect deals with phonons, instead of molecular vibration.
A phonon is Ramanactive only if the first derivative of the polarizability with respect the vibrational normal coordinate has a nonzero value, and this in turns depends by the crystal symmetry. A phonon can be either IR and Raman active only in crystals without center of inversion. For every crystal simmetry class, is possible to calculate which phonons are Raman active, and in which measurement geometry, i.e. for which direction of polarization of the incident and scattered light, relative to the crystallographic axis, also using the Raman tensors tabulated in many texts. Performing measurements in controlled polarization configurations, is possible to obtain informations about the simmetry of the crystalline lattice.
The Raman signal is very weak: only 1 photon in 10exp7 give rise to the Raman effect. The Raman spectra is usually plotted in intensity vs. the difference in wabenumber between the incident beam and the scattered light, and so the peak are in corrispondance to the phonon frequency.
Due to the small wavevector of the optical photons, the phonons involved in the Raman scattering of crystalline solids have (for the wavevector conservation law) a very small momentum compared with the Brillouin zone, so only the zonecenter phonons participate to the Raman scattering.
In disordered solids this rule is no longer valid, and all the phonons contribute to the Raman spectrum, leading to large bands that reflect the vibrational density of states.
In nanocrystalline materials, the situation is intermediate between ideal crystals and amorphous, and the Raman spectrum displays the crystalline Raman features broadened and shifted by the phonon confinement. By using an adequate model, is possible to estimate the size of the nanocrystals.
From the bandshifts and the presence of "forbidden" peaks, using the Raman spectroscopy is also possible to obtain informations on the disorder and the strains present in the crystalline lattice.
(1) Molecular Vibrations, E. BRIGHT WILSON, J.C. DECIUS, P.C. CROSS, McGrawHill Book Company, 1955.
