Laser Raman Spectroscopy

Raman scattering may be regarded as an inelastic collision of an incident photon $\hbar\omega_i$ with a molecule in the initial state $E_i$. Following the collision a photon $\hbar\omega_s$ with different energy is detected and the molecule is found in a different energy state $E_f$.

$$\hbar\omega_i + M(E_i) \rightarrow \hbar\omega_s + M(E_f).$$ (92)

The energy difference $E_f-E_i = \hbar(\omega_i - \omega_s)$ may appear as vibrational, rotational, or electronic energy of the molecule. The intermediate state $E_i+\hbar\omega_i$ of the system "during" the scattering process is formally described as a "virtual" level, which however, is not necessary a real stationary eigenstate of the molecule. Sometimes, if the virtual state coincides with one of the molecular eigenstates, the process is called Resonance Raman Scattering.

The great number of incident photons scatter in the forward direction without change of their frequency. This radiation is called Rayleigh radiation. However, some minor part of the scattered photons change their frequency according to eq. (92). These photons consist of two groups: the Stokes radiation which frequency is smaller than the initial one, and the anti-Stokes radiation which frequency is higher than the initial one.

The Raman spectroscopy finds wide application in investigation of rotation and vibration spectra of diatomic and polyatomic molecules. It is often complementary to infrared absorption spectroscopy because different selection rules are obeyed. The necessary condition for a rotational Raman transitions is that the molecule must have anisotropic polarizability. The term polarizability means that the molecule can acquire an induced dipole moment under the influence of an external electric field $E$:

$$\mu = \alpha E.$$ (93)

The term anisotropic means that the molecular polarizability (that is, the value of the coefficient $\alpha$ in eq. (93)) must depend on the direction of the applied electric field. For instance, all atoms is isotropically polarized and therefore, they are not Raman active. Most of the molecules, including all diatomic (both homonuclear and heteronuclear) have anisotropic polarizability, and so they are rotationally Raman active. The rotational Raman selection rules are:

• Linear rotors: $\Delta J = 0, \pm 2$
• Symmetric rotors: $\Delta J = 0, \pm 1, \pm 2$; $\Delta K =0$

In addition, $\Delta J =0$ transition do not lead to any shift of the scattered photon frequency and contribute to the unshifted Rayleigh radiation.

In vibrational-rotational Raman transitions the same selection rules are valid. The Stokes lines correspond to all $\Delta v =1$ transitions and the anti-Stokes lines correspond to all $\Delta v =-1$ transitions. The $\Delta J =0$ rotational branch is called the $Q$ branch, the $\Delta J =-2$ rotational branch is called the $O$ branch, and the $\Delta J = +2$ rotational branch is called the $S$ branch.

The general disadvantage of the Raman spectroscopy is very small scattering cross section which is about $10^{-30}cm^2$. Therefore, the sensitivity of the method is not very high and the typical experimental problem is detection of a weak signal in the presence of an intense background radiation.

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