Debye approximation. The first two Brillouin zones of a square lattice are replaced by a circle
with the same total area, and the entire spectrum is approximated by a linear dispersion law inside this
The maximum quantum of energy that can excite lattice vibrations expresses energy = specific heat capacity at all temperatures in terms of one empirical parameter
At very low temperatures, modes with frequencies hws(k) >> kwT make a negligible contribution
1. Even for a crystal with a polyatomic basis, the sum over s cannot be
take into account optical modes, since their frequencies are limited from below
2. Dispersion law of three acoustic branches w = ws(k) - limiting form for long wavelengths w = cs(k) k.
3. Integration over the first Brillouin zone in k-space can be replaced by integration over the entire k-space
The Debye approximation for the acoustic branch and the Einstein approximation for the optical branch.
The first Brillouin zone is replaced by a circle with the same area, the acoustic branch is approximating
is controlled by a linear branch inside the circle, and the optical branch by a branch with a constant frequency
experimental methods for determining the photon spectrum
Neutron and photon scattering studies are different ways of analyzing the phonon spectrum - they are characterized by completely different relationships between energy and momentum