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

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

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

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

Heat capacity CV of the Debye approximation: Debye model (4) kB=1.38x10-23JK-1 kBmol=7.70JK-1 3kBmol=23.1JK-1

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