from the entropy

To find the entropy consider a solid made of atoms, each of which has 3 degrees of freedom. So there are quantum harmonic oscillators (hereafter SHOs for "Simple Harmonic Oscillators").

Possible energies of an SHO are given by

where the n of SHO is usually interpreted as the excitation state of the oscillating mass but here n is usually interpreted as the number of phonons (bosons) occupying that vibrational mode (frequency).

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