1. 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

2. 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

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

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

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

6. 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