= High-temperature phonon number in the Debye model
{title2=$N_{\mathrm{ph}}\sim(9/2)N T/\Theta_D$}
For three-dimensional <Debye model> mode density $D(\omega)=9N\omega^2/\omega_D^3$, the <Bose-Einstein distribution> becomes $n_\omega\simeq k_BT/(\hbar\omega)$ when $T\gg\Theta_D$. Integrating the occupations gives $N_{\mathrm{ph}}\simeq(9/2)NT/\Theta_D$. The number of <phonon> quanta is not the number $3N$ of vibrational modes and need not be bounded by it. If a stipulated <electron-phonon scattering> model makes resistivity proportional to this <phonon> number with a temperature-independent coefficient, the high-temperature <electrical resistivity> is proportional to $T$.
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