Symmetry and Crystal Vibrations

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Abstract
As explained in the previous chapter, the only crystal vibrations which, to a first approximation, can be active in the infrared and Raman are those of long wavelength, comparable to that of the exciting radiation. Their frequency is essentially identical with those of infinite wavelength (k=0) in which all unit cells execute identical vibrations, so that the vibrations of a single unit cell define those of the crystal as a whole. The number of these potentially active vibrations—the limiting optical vibrations—is given by 3n–3, where n is the number of atoms per unit cell. The assignment of a spectrum consists in correlating each infrared absorption band or Raman scattering frequency with one of these optical vibrations (or to some combination of them), and this is a problem whose difficulty obviously increases rapidly with increasing complexity of the unit cell. This task is often considerably simplified if the unit cell has some elements of symmetry, as its vibrations can then be classified in distinct symmetry species; its symmetry species determines whether or not a vibration can be active in the infrared and Raman, and also the directional properties of its infrared absorption and Raman scattering. Moreover, the symmetry species of a vibration limits the direction of motion of the atoms, and so helps us to guess the nature of the vibration. Indeed if there is only one vibration in a given symmetry species, its form can be exactly defined without calculation Given the symmetry of the unit cell as a whole

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