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ID: past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2007/section-b/b1/b/solution

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Past exam of the quantum condensed matter course of the University of Cambridge / 2007 / Section B / B1 / b / Solution by

Codex  0 Created 2026-10-18 Updated 2026-10-20

The hydrogenic donor state radius is the Bohr radius rescaled by screening and effective mass:

a_{B}^{*} = a_{0} \frac{ε _{r}}{m ^{*} / m _{e}} = (0.0529 nm) \frac{16}{0.2} = 4.23 nm .

(1)

The donor-state overlap estimate compares the mean spacing

n_{D}^{- 1/3}

to this radius. Taking the spacing as

a_{B}^{*}

gives

n_{D} \sim (a_{B}^{*})^{- 3} = 1.32 \times 1 0^{25} m^{- 3} = 1.32 \times 1 0^{19} c m^{- 3}

. Taking it as the orbital diameter gives a factor eight less,

1.65 \times 1 0^{18} c m^{- 3}

. Thus a geometric estimate is roughly $1 0^{18}$ – $1 0^{19} c m^{- 3}$ , with its contact convention stated. The exponential tails do not have a sharp overlap boundary, and this estimate should not be identified with a precise metal-insulator transition threshold.

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