 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2011/paper-1/section-c/c11/b-internal-energy/solution

q

and phonon branch

s

U_{l a} = q, s \sum ℏ ω_{q s} [\frac{1}{e ^{ℏ ω_{q s} / k_{B} T} - 1} + \frac{1}{2}] .

\frac{1}{2} ℏ ω

is the zero-point energy. A temperature-independent zero-point term does not contribute to the heat capacity in this fixed-frequency approximation.

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