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ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-2/a9/c/i/solution

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Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 2 / A9 / c / i / Solution by

Codex  0 2026-10-20

b_{1} = \frac{2 π}{a} \hat{x}, b_{2} = \frac{2 π}{a} \hat{y}, G_{hk} = h b_{1} + k b_{2} .

(1)

Its points are spaced by

2 π / a

along both reciprocal axes. Perpendicular bisectors between the origin and its four nearest reciprocal neighbours enclose the first Brillouin zone,

- \frac{π}{a} \leq k_{x} \leq \frac{π}{a}, - \frac{π}{a} \leq k_{y} \leq \frac{π}{a} .

(2)

The original drawing labels both the reciprocal-lattice spacing and the nearer zone boundaries. Its shaded disk is the occupied region for one electron per real-space cell.

Figure 1.
Original square reciprocal lattice, first Brillouin zone and half-filled spin-degenerate Fermi disk
. **Details**.

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