= Mode count in the Debye model
{title2=$\beta=\frac{12\pi^4}{5}\frac{N_mk_B}{\Theta_D^3},\quad k_D^3=6\pi^2N_m/V$}
Suppose a three-dimensional <Debye model> is normalized to $3N_m$ vibrational modes in volume $V$. Its low-temperature heat-capacity coefficient is the one displayed, and its spherical wavevector cutoff obeys $k_D^3=6\pi^2N_m/V$. The Debye temperature is $\Theta_D=\hbar v_Dk_D/k_B$. At fixed measured $\beta$ and volume, changing the counting convention by $N_m\mapsto cN_m$ raises both $\Theta_D$ and $k_D$ by $c^{1/3}$; the inferred average <sound speed> is unchanged.
For a mole of formula units containing $r$ atoms each, a conventional full atomic-mode count uses $N_m=rN_A$ and tends to lattice heat capacity $3rR$ at high temperature. A reduced convention using $N_m=N_A$ instead has three effective modes per formula unit. These are different continuum normalizations, not a claim that the real crystal has only three modes per formula unit. A real multi-atom primitive cell has three acoustic branches and additional optical branches; a Debye sphere preserving the full atomic count is an effective representation of that spectrum, rather than its literal first Brillouin zone.
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