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VIBRATIONAL STATES

Assuming that within a crystal structure each atom or group of atoms is positioned at a lattice point. The atoms of these solids do not remain stationary but move in a region of space central on their lattice point. This results in the propagation of atomic vibrations through the crystal as weakly interacting waves with wave vectors k and frequency w .

There are 3N types of vibrational waves that can propagate in a crystal. Out of the 3N types of waves 3 are acoustical modes the rest are optical modes. Each of these wave functions denoted by w j(k) where j is the wave mode equal to 1,2,3…3N, is a periodic function of its argument; that is,

w j(k + 2p b) = w j(k)                 b = n1b1 + n2b2 + n3b3

where n1, n2, n3 are integers and b1, b2, b3 are the basis vectors of reciprocal space.

The periodic arrangement of atoms in real space generates a periodic reciprocal lattice in ‘k’ space; which allows us to limit analysis to a unit cell in reciprocal space; that is to the 1st Brillouin zone with boundary conditions k = ± p/a. That is no waves with wavelength smaller than the spacing between neighbouring atoms can propagate in the crystal. Brillouin zones are illustrated in the section ‘Electronic States’.

According to de Broglie to each wave we can assign a quasiparticle, that is a phonon that has both energy and momentum. E = h/2p · w j(k) where j enumerates the number of wave modes j = 1,2,3…3N where N = number of atoms per unit cell and p = h/2p · k.

Prepared by Laura Malcolm -- last update : March 27, 1997