Kinematics

Gregora, I.

doi:10.1107/97809553602060000640

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.3, p. 314

Section 2.3.2.1. Kinematics

I. Gregora^a ^*

^a Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic
Correspondence e-mail: gregora@fzu.cz

2.3.2.1. Kinematics

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Let us consider the incident electromagnetic radiation, the scattered electromagnetic radiation and the elementary excitation to be described by plane waves. The incident radiation is characterized by frequency $[\omega{_I}]$ , wavevector $[{\bf k}{_I}]$ and polarization vector $[{\bf e}{_I}]$ . Likewise, the scattered radiation is characterized by $[{\omega}{_S}]$ , $[{\bf k}{_S}]$ and $[{\bf e}{_S}]$ : $[{\bf E}_{I,S}({\bf r},t) = E_{I,S}{\bf e}_{I,S}\exp(i{\bf k}_{I,S}{\bf r}-{\omega}t). \eqno (2.3.2.1)]$

The scattering process involves the annihilation of the incident photon, the emission or annihilation of one or more quanta of elementary excitations and the emission of a scattered photon. The scattering is characterised by a scattering frequency $[{\omega}]$ (also termed the Raman shift) corresponding to the energy transfer $[\hbar \omega]$ from the radiation field to the crystal, and by a scattering wavevector q corresponding to the respective momentum transfer $[\hbar {\bf q}]$ . Since the energy and momentum must be conserved in the scattering process, we have the conditions $[\eqalignno{{\omega}_I - {\omega}_S &= {\omega}, &\cr {\bf k}{_I} - {\bf k}{_S} & = {\bf q}.&(2.3.2.2)}]$ Strictly speaking, the momentum conservation condition is valid only for sufficiently large, perfectly periodic crystals. It is further assumed that there is no significant absorption of the incident and scattered light beams, so that the wavevectors may be considered real quantities.

Since the photon wavevectors ( $[{\bf k}{_I}]$ , $[{\bf k}{_S}]$ ) and frequencies ( $[{\omega}{_ I}]$ , $[{\omega}{_S}]$ ) are related by the dispersion relation $[{\omega}= ck/n]$ , where c is the speed of light in free space and n is the refractive index of the medium at the respective frequency, the energy and wavevector conservation conditions imply for the magnitude of the scattering wavevector q $[c^2 q^2 = n_I^2 \omega _I^2 + n_S^2 (\omega _I - \omega)^2 - 2n_I n_S \omega _I (\omega _I - \omega)\cos \varphi, \eqno (2.3.2.3)]$ where $[{\varphi}]$ is the scattering angle (the angle between $[{\bf k}{_I}]$ and $[{\bf k}{_S}]$ ). This relation defines in the ( $[\omega, q]$ ) plane the region of wavevectors and frequencies accessible to the scattering. This relation is particularly important for scattering by excitations whose frequencies depend markedly on the scattering wavevector (e.g. acoustic phonons , polaritons etc.).

References

International Tables for Crystallography (2006). Vol. D. ch. 2.3, p. 314