= Solution
Use <thermal de Broglie wavelength> $\lambda_T=h/\sqrt{2\pi mk_BT}$ and spacing $a=n^{-1/3}$. <Degeneracy onset from thermal wavelength> occurs at
$$
\boxed{T_\lambda=\frac{h^2n^{2/3}}{2\pi mk_B}.}
$$
The numerical factor depends on the convention for a typical <momentum>, but the scale $\hbar^2n^{2/3}/(mk_B)$ does not. For <fermions> the <Fermi temperature> $T_F=\hbar^2(3\pi^2n)^{2/3}/(2mk_B)$ gives a comparable scale for spin degeneracy two.
Representative estimates, with the assumed <densities> stated explicitly, are:
|| System
|| Assumed <number density>
|| Assumed particle <mass>
|| $T_\lambda$
| <Conduction electrons>
| $10^{29}\,\mathrm{m^{-3}}$
| $m_e$
| $1.2\times10^5\,\mathrm K$
| Dilute heavy atomic cloud
| $10^{19}\,\mathrm{m^{-3}}$
| $87u$
| $1.6\times10^{-7}\,\mathrm K$
| <Neutrons> at nuclear-scale <density>
| $10^{44}\,\mathrm{m^{-3}}$
| $m_n$
| $6.5\times10^{11}\,\mathrm K$
Here $u$ is the atomic <mass> constant. Thus the characteristic orders are \b[$10^5\,\mathrm K$ for <electrons>, sub-microkelvin for a dilute heavy atomic gas, and $10^{12}\,\mathrm K$ for <neutrons>]. There is no density-independent quantum-degeneracy <temperature> for atomic gases: for example, taking $n=2.5\times10^{25}\,\mathrm{m^{-3}}$ and <mass> $4u$ instead gives about $0.065\,\mathrm K$. These are illustrative input estimates, not values supplied by the paper. Denser neutron-star cores require relativistic and interaction corrections; the estimate identifies the degeneracy scale rather than a precise stellar <temperature>.
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