# Single phonon absorption

Single phonon Reststrahl absorption can occur in any material possessing an ionic character with an alternating pattern of positive and negative ions. This fundamental one-phonon absorption process is associated with the electrostatic motions of opposite charges which produce an oscillating electric field with which the incident radiation can couple.

The wave vectors associated with this absorption only follow the longitudinal and transverse optical branches of the phonon dispersion curves as there exists two or more atoms per unit cell. In diatomic ionic crystals, when the interaction between the photon and phonon conserve the wave vector momentum, such that *k* = 2π/λ ≅ 0, the theory predicts the strongest absorption will be present, such that the crystal becomes totally reflecting, between the transverse and longitudinal optical vibration frequencies at a resonant frequency that corresponds to the following equation;

where *m* and *M* are the masses of the two ions. If one ion is much heavier than the other, the smaller of the two masses will determine the value of the bond strength (*F*). Therefore to achieve transparency to the longest wavelength, requires both ions to be as heavy as possible.

The behaviour of this type of absorption is most suitably described as a damped Lorentz classical oscillator. This is based on the assumption that the material contains charged particles which are bound to equilibrium positions by Hooke's law forces (i.e. for a certain range of atomic stresses (vibrations), the strain produced is proportional to the stress applied). If the magnitude of the force is assumed to be inversely proportional to the square of the distance between the atoms (Coulombic), the resonant frequencies for materials with different atomic masses can be predicted from empirical estimations of *F*.

In general, ionic crystals exhibit good transmission with constant refractive index and low absorption coefficient up to the lattice absorption band (typically beyond 6µm) at which point the single phonon produces a heavily absorbing mode of vibration and subsequent strong reflection coefficient. The refractive index undergoes a rapid change forcing the Fresnel reflection coefficient to become quiet high. The extinction coefficient also rises rapidly. At wavelengths longer than the resonant Reststrahl frequency, the absorption coefficient decreases, and the refractive index falls to a level slightly higher than on the short wavelength side of the absorption band. The difference in refractive index is characteristic of this absorption mechanism in ionic crystals. The long wavelength limit of transparency is therefore set by the Reststrahl frequency with the absorption falling rapidly at higher frequencies. For most ionic materials more than one absorption peak is present. As the temperature of the material is reduced, the Reststrahl frequency moves slightly towards shorter wavelengths and the peak reflection increases. The refractive index however is unaffected, other than by the characteristic change defined by the temperature-dependent dispersion coefficients.

In homopolar crystals (Ge, Si) where there is an absence of polar electric field interactions, the atomic motions are determined only by the local elastic restoring forces, and as such there is no single phonon interactive coupling and the longitudinal vibration then equals the transverse vibration mode. Hence only weak multi-phonon absorption harmonics are present.

### Infrared material absorption theory

- Introduction
- Electronic absorption
- Lattice absorption
- Single phonon absorption
- Multi-phonon absorption
- Dielectric dispersion
- Thermal vibrations

#### Contact us

- Email:

irfilters@reading.ac.uk - Telephone:

+44 (0) 118 378 8224