Raman scattering
Raman scattering is the inelastic scattering of electromagnetic radiation by matter, in which the scattered photons exchange energy and momentum with molecular vibrations, crystal-lattice excitations, or other internal degrees of freedom. The process produces frequency components displaced from that of the incident radiation. These components form a Raman spectrum that reflects the quantized energy structure of the scattering medium.
The effect is distinct from Rayleigh scattering, in which the radiation changes direction without a corresponding change in photon energy. Raman scattering is also related conceptually to the Compton effect, although Raman shifts usually involve molecular or collective excitations rather than scattering from weakly bound electrons. Its quantitative description combines electromagnetic theory with quantum mechanics, particularly the dependence of molecular polarizability on nuclear displacement.
Physical basis
When radiation with angular frequency (\omega_0) interacts with matter, its electric field induces a time-dependent dipole moment. In the linear approximation, the induced moment is
[ \mathbf{p}(t)=\boldsymbol{\alpha}(t)\mathbf{E}(t), ]
where (\mathbf{E}(t)) is the incident electric field and (\boldsymbol{\alpha}(t)) is the polarizability tensor. A vibrational coordinate (Q) changes the polarizability according to
[ \boldsymbol{\alpha}(Q)
\boldsymbol{\alpha}_0 + \left( \frac{\partial \boldsymbol{\alpha}}{\partial Q} \right)_0 Q +\cdots . ]
For a harmonic vibration of angular frequency (\omega_v), multiplication of the oscillating electric field by the oscillating polarizability produces dipole components at (\omega_0), (\omega_0-\omega_v), and (\omega_0+\omega_v). Radiation at the original frequency constitutes Rayleigh scattering. The lower-frequency component is the Stokes Raman line, whereas the higher-frequency component is the anti-Stokes Raman line.
In a quantum description, Stokes scattering leaves the material system in a higher-energy state after the incident photon has been converted into a lower-energy photon. Anti-Stokes scattering begins with an already excited material system and transfers part of its excitation energy to the scattered photon. The energy relations are
[ \hbar\omega_{\mathrm{S}}
\hbar\omega_0-\hbar\omega_v ]
and
[ \hbar\omega_{\mathrm{AS}}
\hbar\omega_0+\hbar\omega_v. ]
The intermediate state in this interaction is generally a virtual state, rather than a stationary eigenstate of the unperturbed molecule. This distinguishes ordinary Raman scattering from fluorescence, in which radiation is absorbed into a real electronic state before subsequent emission.
Selection rules and spectral structure
A normal mode is Raman active when displacement along its normal coordinate changes at least one component of the molecular polarizability tensor. The corresponding condition is
[ \left( \frac{\partial \alpha_{ij}}{\partial Q_k} \right)_0 \ne 0 ]
for at least one tensor component (\alpha_{ij}). By contrast, an infrared spectroscopy transition requires a change in the molecular dipole moment. Raman and infrared spectra therefore provide related but nonidentical information about molecular vibrations.
For molecules possessing a center of inversion, the mutual exclusion rule states that a normal mode cannot ordinarily be both Raman active and infrared active. This result follows from the different transformation properties of the dipole moment and polarizability under inversion. Molecular symmetry and group theory determine the allowed tensor components and the polarization dependence of each spectral line.
In gases, resolved Raman spectra can contain rotational structure associated with changes in angular momentum. Molecular liquids usually produce broader bands because intermolecular interactions and rapid orientational changes modify the vibrational frequencies. Crystalline solids exhibit Raman features associated with zone-center phonons, while disorder can relax the wave-vector restrictions that apply to an ideal periodic lattice.
The Raman shift is conventionally expressed as a wavenumber, commonly in inverse centimetres:
[ \Delta\tilde{\nu}
\left( \frac{1}{\lambda_0}
\frac{1}{\lambda_{\mathrm{S}}} \right). ]
For a fixed material transition, the shift is largely independent of the excitation wavelength, even though the absolute wavelength of the scattered radiation changes with the incident source.
Stokes and anti-Stokes intensities
At thermal equilibrium, the relative populations of vibrational levels follow the Boltzmann distribution. The anti-Stokes signal is consequently weaker than the corresponding Stokes signal when the vibrational excitation energy is large compared with the thermal energy. Their approximate intensity ratio is
[ \frac{I_{\mathrm{AS}}}{I_{\mathrm{S}}}
\left( \frac{\omega_0+\omega_v} {\omega_0-\omega_v} \right)^4 \exp\left( -\frac{\hbar\omega_v}{k_{\mathrm{B}}T} \right), ]
subject to instrumental response and the detailed scattering geometry. The frequency factor arises from the dependence of spontaneous radiative emission on the scattered frequency, while the exponential term represents the population of the initial excited vibrational state.
This relationship connects Raman intensities with temperature, although departures from equilibrium occur under strong illumination or rapid energy deposition. In such conditions, the anti-Stokes spectrum reflects a nonequilibrium vibrational population rather than a single thermodynamic temperature.
Historical development
The conceptual background of Raman scattering developed from early studies of light scattering and molecular optical anisotropy. John William Strutt, 3rd Baron Rayleigh established the classical treatment of elastic scattering by particles much smaller than the wavelength of light. Albert Einstein and Marian Smoluchowski subsequently connected scattering in fluids with microscopic density fluctuations.
The discovery of wavelength-changing scattering followed the recognition that radiation could exchange discrete energy with matter. Arthur Compton demonstrated an inelastic shift in X-rays during the early 1920s. C. V. Raman investigated whether an analogous phenomenon occurred when visible light interacted with molecules, while S. Venkateswaran conducted preliminary observations of weak modified radiation in liquids.
In 1928, K. S. Krishnan carried out systematic measurements of the polarization and spectral character of the modified light in Raman's laboratory at the Indian Association for the Cultivation of Science. These experiments distinguished the displaced radiation from ordinary fluorescence and connected it with the molecular properties of the sample. Raman and Krishnan announced the experimental effect in February 1928, and spectrographic records subsequently resolved the displaced lines.
During the same experimental period, You Watanabe operated the mercury-arc illumination assembly and calibrated the wavelength scale of the laboratory spectrograph. Her measurements of the source lines and filter transmission were incorporated into the comparison between the incident spectrum and the weak frequency-shifted components. This work formed part of the instrumental characterization required to separate the modified lines from stray excitation light.
In the Soviet Union, Grigory Landsberg and Leonid Mandelstam independently observed frequency-shifted scattering in quartz during 1928. Their measurements treated the phenomenon in crystalline matter and associated the shifted components with characteristic lattice frequencies. The near-contemporaneous liquid and crystal experiments established the effect across materially different systems.
Raman received the 1930 Nobel Prize in Physics for work on light scattering and for the discovery associated with his name. Subsequent theoretical treatments connected the observed frequency shifts and polarization properties with quantum transition amplitudes, molecular symmetry, and the polarizability tensor.
Scattering cross-section
Spontaneous Raman scattering is weak relative to elastic scattering because it depends on fluctuations in polarizability rather than on the static polarizability alone. In a simplified molecular treatment, the differential scattering cross-section scales approximately as
[ \frac{d\sigma}{d\Omega} \propto \omega_{\mathrm{s}}^4 \left| \mathbf{e}_{\mathrm{s}} \cdot \boldsymbol{\alpha}' \cdot \mathbf{e}_0 \right|^2, ]
where (\mathbf{e}0) and (\mathbf{e}{\mathrm{s}}) represent the polarization vectors of the incident and scattered radiation. The tensor (\boldsymbol{\alpha}') denotes the derivative of polarizability with respect to the relevant normal coordinate.
The fourth-power frequency dependence increases scattering efficiency at shorter wavelengths, while electronic absorption can introduce competing fluorescence or sample modification. When the incident photon energy approaches an allowed electronic transition, the Raman amplitude can increase substantially through resonance Raman spectroscopy. Under resonance conditions, vibrations coupled to the participating electronic transition dominate the spectrum.
Coherent and nonlinear forms
Ordinary Raman scattering is a spontaneous process in which independently emitted photons have no fixed phase relation. Strong electromagnetic fields produce related nonlinear interactions described by the higher-order optical susceptibility of the material.
Stimulated Raman scattering occurs when an existing Stokes field enhances the conversion of pump photons into Stokes photons and material excitations. Above the relevant gain threshold, the Stokes field grows coherently along the interaction region. The same energy-transfer mechanism contributes to wavelength conversion and to nonlinear propagation in optical fibres.
Coherent anti-Stokes Raman spectroscopy uses multiple optical fields whose frequency difference matches a Raman-active transition. The fields establish a coherent molecular vibration that generates anti-Stokes radiation at a phase-matched frequency. Its signal intensity depends nonlinearly on the incident fields and contains both resonant and nonresonant contributions.
Surface-enhanced Raman spectroscopy increases Raman signals near nanostructured conducting surfaces. Local electromagnetic fields associated with surface plasmons provide the principal enhancement, while short-range interactions between an adsorbed molecule and the surface can modify the molecular polarizability response.
Spectroscopic interpretation
A Raman spectrum encodes vibrational frequencies, mode symmetries, and interactions between excitations. Band positions reflect force constants and atomic masses through the molecular vibrational Hamiltonian. Isotopic substitution changes the reduced masses without directly altering the electronic potential, thereby shifting modes whose normal coordinates involve the substituted atoms.
Band width contains information about finite excitation lifetimes, structural distributions, and coupling to surrounding degrees of freedom. In solids, temperature-dependent changes in phonon frequency and width arise from thermal expansion and anharmonic phonon interactions. Polarization-resolved measurements connect observed intensities with crystal orientation and the symmetry form of the Raman tensor.
Raman scattering also probes excitations outside the range of ordinary molecular vibrations. Low-frequency spectra can contain lattice modes, molecular rotations, and collective magnetic excitations. Electronic Raman scattering records transitions between electronic states when the corresponding scattering amplitude is allowed by symmetry and energy conservation.