# Units of measurement in Raman spectroscopy

Raman spectroscopy is the inelastic scattering of laser photons, by excitations such as symmetric breathing modes in molecules or phonons in crystalline solids. We are measuring the loss (or gain) in photon energy upon the creation (or annihilation) of such an excitation. These energies are typically ≈ 100 meV. Why do we not use the meV as our unit of measurement instead of cm^{-1}, which is in fact a unit of inverse length? Is the “Raman shift (cm^{-1})” the phonon wavenumber? Many websites provide part of the answer to this question – here is a reasonably complete answer!

The answer lies in the experimental method of Raman spectroscopy, and the historical development of the technique. To measure the energy of the Raman scattered (and incident) photons we use a diffraction grating. So the raw data recorded is the angle through which the photon is diffracted, which is completely determined by the relation between the photon wavelength and the spacing between the grooves on the diffraction grating. This is related to the energies of the incident photon (E_{I}), scattered photon (E_{S}) and phonon (or other excitation) (E_{P}) as follows, for Stokes scattering:

E_{I} = E_{S} + E_{P}

For the incident and scattered photons the energy is related to the wavenumber via Planck’s constant and the speed of light: E = hcν. So we obtain:

hcν_{I} = hcν_{S} + E_{P}

When Raman scattering was first developed as an analytical technique it was probably desirable to cancel out the product so that the reported results remained correct independent of any change in the accepted values of these constants. However, it is ** not** correct to write E

_{P}= hcν

_{P}for the phonon as the energy and wavenumber for Raman-active phonons are generally not related by a simple linear dispersion relation. For the vibrational modes of isolated molecules and zone-centre optical phonons in crystalline solids, for instance, the energy remains finite even when ν

_{S}= 0. So to cancel out the product hc we define a quantity which is defined by E

_{P}= hcν

_{P}‘ which has dimensions of [length]

^{-1}and is directly proportional to the phonon energy. In this case we obtain:

ν_{I} = ν_{S} + ν_{P}‘

The quantity ν_{P}‘ is what is shown on the x-axis of Raman spectra, referred to as “Raman shift”, “Raman frequency” or “Wavenumber”. The fact that it is ** not** the wavenumber of the phonon can be demonstrated using the conservation of momentum

**p**. This principle is applied to Raman scattering from a phonon in the first Brillouin zone as follows:

**p**_{I} = **p**_{S} + **p**_{P}

Where this addition concerns the vectorial quantities. Clearly, according to this relation the phonon momentum (and therefore wavenumber since p = hν) depends on the geometry chosen for the Raman scattering experiment. For the most common geometry (backscattering) we obtain |ν_{I}| = |ν_{P}| – |ν_{S}|. Thus the actual phonon wavenumber ν_{P} is generally not equal to the quantity ν_{P}‘ given in Raman spectra and calculated above from the conservation of energy.

Confusing? Yes. Nowadays the value of the product hc is known to great precision so it would be more appropriate to use hcν_{I} = hcν_{S} + E_{P} to calculate the phonon energy E_{P} and display this in units of meV. Why do we not do this? Conventions and traditions are hard to change…

**The fundamental selection rule**

Having obtained |ν_{I}| = |ν_{P}| – |ν_{S}| we can also demonstrate the fundamental selection rule in Raman scattering. In Raman scattering we typically use visible laser photons (energy 1 – 2 eV) and excite vibrations with energy ≈ 10 – 100 meV. So it is reasonable to approximate ν_{I} ≈ ν_{S}. Then we obtain (for the backscattering geometry, which corresponds to the largest possible ν_{P}) ν_{P} ≈ 2ν_{I}. For a 500 nm laser photon we obtain ν_{P} ≈ 4 x 10^{6} m^{-1}. The first Brillouin zone of a crystalline solid extends – by definition – out to roughly 1 / (lattice constant) so with a lattice constant of 5 Å the first Brillouin zone extends out to about 10^{9} m^{-1}! So only a tiny proportion of the phonons, those at the very centre of the first Brillouin zone, can potentially be measured in a conventional (first order, non-resonant) Raman scattering experiment. In a nutshell, the small momentum of the incoming laser photon limits the variety of phonons that can be excited to those at the very centre of the first Brillouin zone.

This is the *fundamental selection rule* in Raman scattering. To study the remainder of the phonons it is usually necessary to resort to inelastic neutron or X-ray scattering at central facilities such as the Institut Laue Langevin or the European Synchrotron Radiation Facility.