where N is the number of terms in the sum, Θ Θ k , i ( V ) = ℏ ℏ ω ω k , i ( V ) / k B is introduced as the characteristic temperature for a phonon with wave vector k in the i -th band at volume V and n k , i ( T , V ) is shorthand for the number of (k, i)-phonons at temperature T and volume V. As is conventional, ℏ ℏ is the reduced Planck constant and k B is the Boltzmann constant. The first term in U vib is the zero-point energy of the phonon system and contributes to the thermal expansion as a zero-point thermal pressure.