The phonon dispersion relation gives all possible phonon modes within the Brillouin zone (zone within the primitive cell in reciprocal space), and the phonon density of states D p (the number density of possible phonon modes). The phonon group velocity u p,g is the slope of the dispersion curve, dω p / d κ p. Since phonon is a boson particle, its occupancy follows the Bose–Einstein distribution { f p = [exp(ħω p / k B T)-1], k B : Boltzmann constant }. Using the phonon density of states and this occupancy distribution, the phonon energy is E p (T) = ∫ D p (ω p) f p (ω p,T) ħω p dω p, and the phonon density is n p (T) = ∫ D p (ω p) f p (ω p,T) dω p. Phonon heat capacity c v,p (in solid c v,p = c p,p, c v,p : constant-volume heat capacity, c p,p : constant-pressure heat capacity) is the temperature derivatives of phonon energy for the Debye model (linear dispersion model), is c v , p = d E p d T | v = 9 k B m ( T T D ) 3 n ∫ ∫ 0 T D / T x 4 e x ( e x − − 1 ) 2 d x ( x = ℏ ℏ ω ω k B T ) , where T D is the Debye temperature, m is atomic mass, and n is the atomic number density (number density of phonon modes for the crystal 3 n). This gives the Debye T law at low temperature and Dulong-Petit law at high temperatures.