= Concentration for appreciable donor overlap
For donor <number density> $N_D$, the mean separation is of order $\ell\sim N_D^{-1/3}$. A <donor-state overlap estimate> sets this scale comparable with the effective radius:
$$
\boxed{N_D\sim(a_B^*)^{-3}=1.65\times10^{24}\,\mathrm{m^{-3}}}.
$$
Relative to the atomic density, this is $3.77\times10^{-5}$, about $38$ parts per million. If “start to overlap” is defined geometrically by centre separation $2a_B^*$, the estimate is instead $2.06\times10^{23}\,\mathrm{m^{-3}}$, about $4.7$ parts per million. Thus the geometric estimate is of order $10^{23}$–$10^{24}\,\mathrm{m^{-3}}$. Real donor wavefunctions have tails, not hard orbit edges, so this does not define a sharp threshold or by itself locate a metal-insulator transition.
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