The Debye model treats a three-dimensional solid as an isotropic elastic continuum with three independent acoustic phonon polarizations. It assumes harmonic normal modes, a linear dispersion relation

ω = v q

, and equilibrium Bose-Einstein statistics with zero phonon chemical potential. Real anisotropy and nonlinear dispersion are replaced by one effective Debye sound speed. A cutoff restores the finite number of atomic vibrational degrees of freedom; the continuum itself would otherwise have infinitely many modes.

For sample volume

V

V / (2 π)^{3}

N (q) = 3 \frac{V}{( 2 π ) ^{3}} \frac{4 π q ^{3}}{3} = \frac{V q ^{3}}{2 π ^{2}} .

There are

3 N

modes for

N

atoms. Hence

q_{D}^{3} = 6 π^{2} \frac{N}{V}, ω_{D} = v (6 π^{2} n_{a})^{1/3}, n_{a} = N / V .

g (ω) = \frac{3 V ω ^{2}}{2 π ^{2} v ^{3}} = \frac{9 N ω ^{2}}{ω _{D}^{3}}, 0 < ω < ω_{D} .

If longitudinal and transverse sound speeds differ, use

3/ v^{3} = 1/ v_{L}^{3} + 2/ v_{T}^{3}

for the low-frequency count. The Debye frequency is an effective mode-counting cutoff, not a claim that every actual branch terminates at exactly that frequency.