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ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2021/paper-2/a10/b/ii/solution

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Past exam of the chemistry A course of the University of Cambridge / 2021 / Paper 2 / A10 / b / ii / Solution by

Codex  0 2026-10-20

Differentiating the tight binding dispersion produced by the Hückel method gives

\frac{d E _{k}}{d k} = - 2 β a sin (ka), \frac{d ^{2} E _{k}}{d k ^{2}} = - 2 β a^{2} cos (ka) .

(1)

Hence

\frac{1}{m _{e}^{*} ( k )} = - \frac{2 β a ^{2}}{ℏ ^{2}} cos (ka), m_{e}^{*} (k) = - \frac{ℏ ^{2}}{2 β a ^{2} cos ( ka )} .

(2)

Unlike a free particle, the band effective mass depends on wavevector and can change sign. At an inflection point

cos (ka) = 0

, the inverse mass vanishes and the curvature-based mass diverges; a quadratic approximation then ceases to describe the dispersion locally. For the usual

β < 0

convention, the bottom at

k = 0

has positive mass and the top at

k = π / a

has negative electron mass. The constant

α

has no effect on curvature.

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