Thermodynamics, Statistical Physics, and Quantum Mechanics

230 SOLUTIONS

The specific heat at low temperatures) is

At low temperatures, we can extend the upperlimit ofintegration

to infinity:

Therefore, at

where

and is the Riemann function. Notethat thespecific heat in 2D is

(see alsoProblem4.75). Notealso that you cansolve asomewhat differ-
ent problem: When atoms are confined to the surface but still have three
degrees offreedom, theresultswill, ofcourse, bedifferent.

4.101 Einstein Specific Heat (Maryland, Boston)

a) For a harmonicoscillator with frequency the energy

where

So,