__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 = n_{1}b_{1} + n_{2}b_{2} + n_{3}b_{3}

where n_{1}, n_{2,} n_{3} are integers
and b_{1}, b_{2}, b_{3} 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 1^{st} 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