Statistical Mechnics of a Linear Chain

Let us return to the case of a linear chain of masses m separated by springs of force
constant B, at equilibrium distance a. The excitations of this system are phonons
which can have momenta k corresponding to energies Phonons are bosons whose number is not conserved, so they obey the Planck distribution.
Hence, the energy of a linear chain at _nite temperature is given by:



Changing variables from k to



In the limit kBT we can take the upper limit of integration to 1 and
drop the x-dependent term in the square root in the denominator of the integrand:



Hence, Cv ~ T at low temperatures In the limit kBT we can approximate :



In the intermediate temperature regime, a more careful analysis is necessary. In
particular, note that the density of states this
is an example of a van Hove singularity. If we had alternating masses on springs,
then the expression for the energy would have two integrals, one over the acoustic
modes and one over the optical modes.

Kevin M Contreras H
Electrónica del Estado Sólido
http://www.physics.ucla.edu/~nayak/solid_state.pdf