= Low-temperature thermal phonon number
{title2=$N_{\mathrm{ph}}\sim18\zeta(3)N(T/\Theta_D)^3$}
Integrating the <Bose-Einstein distribution> against the three-dimensional <Debye model> <phonon density of states> gives $N_{\mathrm{ph}}=9N(T/\Theta_D)^3\int_0^{\Theta_D/T}x^2/(e^x-1)\,dx$. For $T\ll\Theta_D$, expand $(e^x-1)^{-1}=\sum_{j\geq1}e^{-jx}$ and integrate termwise: the limiting integral is $2\sum_{j\geq1}j^{-3}=2\zeta(3)$. This counts thermal quanta, not available modes or half quanta of <zero-point energy>.
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