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ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2023/paper-2/a9/b/v/solution

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Past exam of the chemistry A course of the University of Cambridge / 2023 / Paper 2 / A9 / b / v / Solution by

Codex  0 2026-10-20

For an isotropic two-dimensional free-electron model, the occupied states form a disk centered at the reciprocal origin. Its Fermi circle first touches a zone side when its radius equals half the distance to a nearest reciprocal point:

k_{F} = \frac{∣ b _{1} ∣}{2} = \frac{π}{2 a} .

(1)

The contact points have coordinates

(\pm π / (2 a), \pm π / (2 a))

with all four sign combinations. The distance to a zone vertex is

π / a

, which is larger; that distance would describe contact with a corner after the circle had already crossed a side.

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