Mode count in the Debye model ($\beta =\frac{12{\pi }^4}{5}\frac{N_m{k}_B}{{\Theta }_D^3},k_D^3=6{\pi }^2{N}_m/V$)

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 $k_D^3=6{\pi }^2{N}_m/V$. The Debye temperature is ${\Theta }_D=\hslash v_D{k}_D/k_B$. At fixed measured $\beta$ and volume, changing the counting convention by $N_m\mapsto c{N}_m$ 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 $N_m=r{N}_A$ 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.