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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:Relative to the atomic density, this is $3.77\times 10^{-5}$, about $38$ parts per million. If “start to overlap” is defined geometrically by centre separation $2a_B^{\ast }$, the estimate is instead $2.06\times 10^{23}{\mathrm{m}}^{-3}$, 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.

For a hydrogenic donor state, dielectric screening replaces ${ϵ}_0$ by ${ϵ}_r{ϵ}_0$ and the electron mass by the band effective mass. The characteristic orbital radius is thereforeUsing $a_0=0.0529177\mathrm{nm}$ givesA donor-state overlap estimate compares this radius with the mean dopant spacing $d\sim N_D^{-1/3}$. Taking $d\sim a^{\ast }$ yieldsThere is no sharp geometrical onset: hydrogenic wavefunctions have exponential tails. If “overlap” is defined by neighbouring orbital diameters touching, $d\sim 2a^{\ast }$, the corresponding estimate is $4.88\times 10^{23}{\mathrm{m}}^{-3}$, smaller by a factor of eight. These are conventions for the onset of appreciable orbital overlap, not an exact critical density for a metal-insulator transition.