Quantum and Classical Raman theory

Quantum theory

When light interacts with matter, the photons which make up the light may be absorbed or scattered, or may not interact with the material and may pass straight through it (Smith and Dent, 2005). When monochromatic light interacts with molecules, most of the photons are scattered without any change in energy. This process is called elastic or Rayleigh scattering, and it occurs when the electrons in a molecule oscillate in resonance with the applied electric field of the incident light. However, a small number of photons (1 out of 106 to 109) are inelastically scattered and undergo a change in energy. The relationships between energy, frequency, wave number and the wavelenght are as follows:

• E = hv (h is Planck’s constant, and v is frequency in sec-1)
• E = hcν (c is the speed of light in cm/sec, and ν is wave number in cm-1)
• Ν = 1/λ (λ is wavelenght in nm) Figure 1. Diagram of Rayleigh and Raman scattering processes. The lowest energy vibrational state m is shown at the foot with states of increasing energy above it. Both the low energy (upward arrows) and the scattered energy (downward arrows) have much larger energies than the energy of a vibration (Smith and Dent, 2005).

The inelastic scattering process is called Raman scattering (Raman effect). The change in photon energy occurs because a molecule may vibrate during the time the electrons oscillate in resonance with the applied electric field. A vibrational mode that changes molecular polarizability (dipole moment induced by electric field) results in a change of incident photon energy. The difference in energy between the incident photons and inelastically scattered photons is called Raman shift. A plot of the intensity of the inelastically scattered light as a function of the energy change is called Raman spectrum.

Figure 1 shows a schematic of the Raman and Rayleigh scattering processes. There are two kinds of Raman scattering events. First event, incident photons lose energy to the molecule, causing it to go to an excited, vibrational state (m → n). As a result, scattered photons will have less energy (Stokes Raman) compared to the incident photon. In the second case, the incident photons gain energy from the molecule, because the molecules lose energy by going from a higher to lower vibrational state (n → m). In this case, the Raman scattered photon will have higher than the incident photons (anti-Stokes Raman). At room temperature, most molecules are in the vibrational ground state (m), so there is a greater probability that incident photons will lose energy to the molecules during the interaction; that is Stokes Raman. Generally, Raman spectroscopy refers to Stokes Raman unless specified otherwise. Thus, Raman scattering is observed at lower energy or longer wavelenght compared to that of the incident light (Lipták, 2003).

Classical theory

The classical theory of the Raman effect is based upon polarizability of molecules, which reflects how easy an electron cloud of a molecule can be distorted by an electric field (light).

The technique is based on molecular deformations in electric field E determined by molecular polarizability α. The laser beam can be considered as an oscillating electromagnetic wave with electrical vector E. Upon interaction with the sample it induces electric dipole moment P = αE which deforms molecules. Because of periodical deformation, molecules start vibrating with characteristic frequency (McCreery, 2000).

The scattered light can have a frequency equal to the incident light (Rayleigh), equal to the incident light minus the vibrational frequency (Stokes) and equal to the incident light plus the vibrational frequency (anti-Stokes).

[References: LIPTÁK B. G. (2003) – Process Measurement and Analysis (Fourth edition). CRC Press; McCREERY R.L. (2000) – Raman spectroscopy for chemical analysis. John Wiley and Sons, New York.; SMITH E., DENT G. (2005) – Modern Raman Spectroscopy – A Practical Approach. John Wiley and Sons, England]