Apply the given small-

x

expansion with

x = T_{E} / T

U = 3 N k_{B} T_{E} [\frac{1}{2} + \frac{T}{T _{E}} - \frac{1}{2} + \frac{T _{E}}{12 T} + O (T^{- 3})] = 3 N k_{B} [T + \frac{T _{E}^{2}}{12 T} + O (T^{- 3})] .

C_{V} = 3 N k_{B} [1 - \frac{T _{E}^{2}}{12 T ^{2}} + O (T^{- 4})] .

Thus

A_{0} = 3 N k_{B}, A_{1} = 0, A_{2} = - \frac{N k _{B} T _{E}^{2}}{4} = - \frac{A _{0} T _{E}^{2}}{12} .

There is no

1/ T

correction to the heat capacity even though the internal energy has a

1/ T

correction.

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