= Solution
The FCC conventional <unit cell> has four atoms, so the cube contains
$$
N=4\left(\frac{180}{0.409}\right)^3=3.4096\times10^8
$$
atoms, neglecting surface corrections. A <phonon> is one excitation quantum of a vibrational <normal mode>; there can be many quanta per mode, and <zero-point motion> does not add half a phonon to the thermal population.
For the <low-temperature thermal phonon number>, integrate the <Bose-Einstein distribution> against the <Debye model> <phonon density of states>:
$$
N_{\rm ph}=\int_0^{\omega_D}\frac{g(\omega)}{e^{\hbar\omega/(k_BT)}-1}\,d\omega
=9N\left(\frac{T}{\Theta_D}\right)^3
\int_0^{\Theta_D/T}\frac{x^2}{e^x-1}\,dx.
$$
At $10\,\mathrm K$, $\Theta_D/T=21.5$, so extending the upper limit to infinity has negligible effect at the precision required. The supplied integral yields
$$
\boxed{N_{\rm ph}\simeq9(3.4096\times10^8)\left(\frac{10}{215}\right)^3(2.404)
\simeq7.42\times10^5.}
$$
This is the thermally excited <phonon> number, not the number $3N$ of available modes.
The continuum approximation fails when the thermally occupied <wavelengths> become comparable with the cube size $L$. From the <Debye temperature>,
$$
v=\frac{k_B\Theta_D}{\hbar(6\pi^2n_a)^{1/3}}\simeq1.86\times10^3\,\mathrm{m\,s^{-1}}.
$$
A lowest nontrivial acoustic frequency is of order $v/L$; a standing-wave estimate $\omega_{\min}\sim\pi v/L$ corresponds to $\hbar\omega_{\min}/k_B\sim0.25\,\mathrm K$. <Periodic boundary conditions> instead give $2\pi v/L$ and $0.50\,\mathrm K$. The precise spectrum depends on mechanical boundary conditions and branch speeds, neither of which is specified.
This <finite-crystal phonon freeze-out> requires summing discrete <normal modes>. For a finite positive lowest frequency, $N_{\rm ph}\sim g_{\min}e^{-\hbar\omega_{\min}/(k_BT)}$, rather than extrapolating the bulk $T^3$ law indefinitely. Rigid translations and rotations of an isolated crystallite are not internal thermal <phonons>. Surface acoustic modes can modify the intermediate crossover, but do not justify inventing a unique boundary spectrum.
\Image[/past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2012-nanocrystal-thermal-phonon-count.png]
{title=Bulk and finite-box thermal phonon populations}
{description=The discrete curve uses an illustrative periodic cube and three equal acoustic speeds; its low-temperature suppression is not a fitted free-surface spectrum.}
Back to article page
