Raman scattering

Raman scattering generally gives information about vibrations within a molecule. In the case of gases, information about rotational energy can also be gleaned.[12] For solids, phonon modes may also be observed.[13] The basics of infrared absorption regarding molecular vibrations apply to Raman scattering although the selection rules are different.

For any given molecule, there are a total of 3N degrees of freedom, where N is the number of atoms. This number arises from the ability of each atom in a molecule to move in three dimensions.[14] When dealing with molecules, it is more common to consider the movement of the molecule as a whole. Consequently, the 3N degrees of freedom are partitioned into molecular translational, rotational, and vibrational motion. Three of the degrees of freedom correspond to translational motion of the molecule as a whole (along each of the three spatial dimensions). Similarly, three degrees of freedom correspond to rotations of the molecule about the ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$-axes. Linear molecules only have two rotations because rotations along the bond axis do not change the positions of the atoms in the molecule. The remaining degrees of freedom correspond to molecular vibrational modes. These modes include stretching and bending motions of the chemical bonds of the molecule. For a linear molecule, the number of vibrational modes is 3N-5, whereas for a non-linear molecule the number of vibrational modes is 3N-6.[14]

Molecular vibrational energy is known to be quantized and can be modeled using the quantum harmonic oscillator (QHO) approximation or a Dunham expansion when anharmonicity is important. The vibrational energy levels according to the QHO are

${\displaystyle E_{n}=h\left(n+{1 \over 2}\right)\nu =h\left(n+{1 \over 2}\right){1 \over {2\pi }}{\sqrt {k \over m}}\!}$,

where n is a quantum number. Since the selection rules for Raman and infrared absorption generally dictate that only fundamental vibrations are observed, infrared excitation or Stokes Raman excitation results in an energy change of ${\displaystyle E=h\nu ={h \over {2\pi }}{\sqrt {k \over m}}}$

The energy range for vibrations is in the range of approximately 5 to 3500 cm⁻¹. The fraction of molecules occupying a given vibrational mode at a given temperature follows a Boltzmann distribution. A molecule can be excited to a higher vibrational mode through the direct absorption of a photon of the appropriate energy, which falls in the terahertz or infrared range. This forms the basis of infrared spectroscopy. Alternatively, the same vibrational excitation can be produced by an inelastic scattering process. This is called Stokes Raman scattering, by analogy with the Stokes shift in fluorescence discovered by George Stokes in 1852, with light emission at longer wavelength (now known to correspond to lower energy) than the absorbed incident light. Conceptually similar effects can be caused by neutrons or electrons rather than light.[15] An increase in photon energy which leaves the molecule in a lower vibrational energy state is called anti-Stokes scattering.

Raman scattering is conceptualized as involving a virtual electronic energy level which corresponds to the energy of the exciting laser photons. Absorption of a photon excites the molecule to the imaginary state and re-emission leads to Raman or Rayleigh scattering. In all three cases the final state has the same electronic energy as the initial state but is higher in vibrational energy in the case of Stokes Raman scattering, lower in the case of anti-Stokes Raman scattering or the same in the case of Rayleigh scattering. Normally this is thought of in terms of wavenumbers, where ${\displaystyle {\tilde {\nu }}_{0}}$ is the wavenumber of the laser and ${\displaystyle {\tilde {\nu }}_{M}}$ is the wavenumber of the vibrational transition. Thus Stokes scattering gives a wavenumber of ${\displaystyle {\tilde {\nu }}_{0}-{\tilde {\nu }}_{M}}$ while ${\textstyle {\tilde {\nu }}_{0}+{\tilde {\nu }}_{M}}$ is given for anti-Stokes. When the exciting laser energy corresponds to an actual electronic excitation of the molecule then the resonance Raman effect occurs.

A classical physics based model is able to account for Raman scattering and predicts an increase in the intensity which scales with the fourth-power of the light frequency. Light scattering by a molecule is associated with oscillations of an induced electric dipole. The oscillating electric field component of electromagnetic radiation may bring about an induced dipole in a molecule which follows the alternating electric field which is modulated by the molecular vibrations. Oscillations at the external field frequency are therefore observed along with beat frequencies resulting from the external field and normal mode vibrations.[10][2]

The different possibilities of light scattering: Rayleigh scattering (no exchange of energy: incident and scattered photons have the same energy), Stokes Raman scattering (atom or molecule absorbs energy: scattered photon has less energy than the incident photon) and anti-Stokes Raman scattering (atom or molecule loses energy: scattered photon has more energy than the incident photon)

The spectrum of the scattered photons is termed the Raman spectrum. It shows the intensity of the scattered light as a function of its frequency difference Δν to the incident photons, more commonly called a Raman shift. The locations of corresponding Stokes and anti-Stokes peaks form a symmetric pattern around the RayleighΔν=0 line. The frequency shifts are symmetric because they correspond to the energy difference between the same upper and lower resonant states. The intensities of the pairs of features will typically differ, though. They depend on the populations of the initial states of the material, which in turn depend on the temperature. In thermodynamic equilibrium, the lower state will be more populated than the upper state. Therefore, the rate of transitions from the more populated lower state to the upper state (Stokes transitions) will be higher than in the opposite direction (anti-Stokes transitions). Correspondingly, Stokes scattering peaks are stronger than anti-Stokes scattering peaks. Their ratio depends on the temperature, and can therefore be exploited to measure it:

${\displaystyle {\frac {I_{\text{Stokes}}}{I_{\text{anti-Stokes}}}}={\frac {({\tilde {\nu }}_{0}-{\tilde {\nu }}_{M})^{4}}{({\tilde {\nu }}_{0}+{\tilde {\nu }}_{M})^{4}}}\exp \left({\frac {hc\,{\tilde {\nu }}_{M}}{k_{B}T}}\right)}$

In contrast to IR spectroscopy, where there is a requirement for a change in dipole moment for vibrational excitation to take place, Raman scattering requires a change in polarizability. A Raman transition from one state to another is allowed only if the molecular polarizability of those states is different. For a vibration, this means that the derivative of the polarizability with respect to the normal coordinate associated to the vibration is non-zero: ${\displaystyle {\frac {\partial \alpha }{\partial Q}}\neq 0}$. In general, a normal mode is Raman active if it transforms with the same symmetry of the quadratic forms ${\displaystyle (x^{2},y^{2},z^{2},xy,xz,yz)}$, which can be verified from the character table of the molecule's point group. As with IR spectroscopy, only fundamental excitations (${\displaystyle \Delta \nu =\pm 1}$) are allowed according to the QHO. There are however many cases where overtones are observed. The rule of mutual exclusion, which states that vibrational modes cannot be both IR and Raman active, applies to certain molecules.

The specific selection rules state that the allowed rotational transitions are ${\displaystyle \Delta J=\pm 2}$, where ${\displaystyle J}$ is the rotational state. This generally is only relevant to molecules in the gas phase where the Raman linewidths are small enough for rotational transitions to be resolved.

A selection rule relevant only to ordered solid materials states that only phonons with zero phase angle can be observed by IR and Raman, except when phonon confinement is manifest.[13]

Monitoring the polarization of the scattered photons is useful for understanding the connections between molecular symmetry and Raman activity which may assist in assigning peaks in Raman spectra.[16] Light polarized in a single direction only gives access to some Raman–active modes, but rotating the polarization gives access to other modes. Each mode is separated according to its symmetry.[17]

The symmetry of a vibrational mode is deduced from the depolarization ratio ρ, which is the ratio of the Raman scattering with polarization orthogonal to the incident laser and the Raman scattering with the same polarization as the incident laser: ${\displaystyle \rho ={\frac {I_{r}}{I_{u}}}}$ Here ${\displaystyle I_{r}}$ is the intensity of Raman scattering when the analyzer is rotated 90 degrees with respect to the incident light's polarization axis, and ${\displaystyle I_{u}}$ the intensity of Raman scattering when the analyzer is aligned with the polarization of the incident laser.[18] When polarized light interacts with a molecule, it distorts the molecule which induces an equal and opposite effect in the plane-wave, causing it to be rotated by the difference between the orientation of the molecule and the angle of polarization of the light wave. If ${\displaystyle \rho \geq {\frac {3}{4}}}$, then the vibrations at that frequency are depolarized; meaning they are not totally symmetric.[19][18]

Suppose that the distance between two points A and B of an exciting beam is x. Generally, as the exciting frequency is not equal to the scattered Raman frequency, the corresponding relative wavelengths λ and λ' are not equal. Thus, a phase-shift Θ = 2πx(1/λ − 1/λ') appears. For Θ = π, the scattered amplitudes are opposite, so that the Raman scattered beam remains weak.

• A crossing of the beams may limit the path x.

Several tricks may be used to get a larger amplitude:

• In an optically anisotropic crystal, a light ray may have two modes of propagation with different polarizations and different indices of refraction. If energy may be transferred between these modes by a quadrupolar (Raman) resonance, phases remain coherent along the whole path, transfer of energy may be large. It is an Optical parametric generation.
• Light may be pulsed, so that beats do not appear. In Impulsive Stimulated Raman Scattering (ISRS),[20][21][22][23] the length of the pulses must be shorter than all relevant time constants.[24] Interference of Raman and incident lights is too short to allow beats, so that it produces a frequency shift roughly, in best conditions, inversely proportional to the cube of the pulse length.

In labs, femtosecond laser pulses must be used because the ISRS becomes very weak if the pulses are too long. Thus ISRS cannot be observed using nanosecond pulses making ordinary time-incoherent light.

For high-intensity continuous wave (CW) lasers, stimulated Raman scattering can be used to produce a broad bandwidth supercontinuum. This process can also be seen as a special case of four-wave mixing, in which the frequencies of the two incident photons are equal and the emitted spectra are found in two bands separated from the incident light by the phonon energies. The initial Raman spectrum is built up with spontaneous emission and is amplified later on. At high pumping levels in long fibers, higher-order Raman spectra can be generated by using the Raman spectrum as a new starting point, thereby building a chain of new spectra with decreasing amplitude. The disadvantage of intrinsic noise due to the initial spontaneous process can be overcome by seeding a spectrum at the beginning, or even using a feedback loop as in a resonator to stabilize the process. Since this technology easily fits into the fast evolving fiber laser field and there is demand for transversal coherent high-intensity light sources (i.e., broadband telecommunication, imaging applications), Raman amplification and spectrum generation might be widely used in the near-future.