Phonon Energy quantization of lattice vibration l=0,1,2,3 Bose distribution function for phonon number: for :zero point oscillation

Lattice heat capacity: Debye model (1) Density of states of acoustic phonos for 1 polarization Debye temperature θ N: number of unit cell N k : Allowed number of k points in a sphere with a radius k phonon dispersion relation

Thermal energy U and lattice heat capacity C V : Debye model (2) 3 polarizations for acoustic modes

・ Low temperature T<<θ ・ High temperature T>>θ Equipartition law: energy per 1 freedom is k B T/2 Debye model (3)

Heat capacity C V of the Debye approximation: Debye model (4) k B =1.38x JK -1 k B mol=7.70JK -1 3k B mol=23.1JK -1

Heat capacity of Si, Ge and solid Ar: Debye model (5) cal/mol K=4.185J/mol K 3k B mol=5.52cal K -1 Si and Ge Solid Ar