Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 34E / c / iii

Near $k=0$, $E=E_0-6\mu +\mu {\sum }_i(k\cdot a_i{)}^2+O(k^4)$. The quadratic form has eigenvalues $a^2/4,a^2/4,a^2$ along its principal axes, so comparison with ${\hslash }^2{k}_j^2/(2m_j^{\ast })$ gives $m^{\ast }=2{\hslash }^2/(\mu a^2)$ twice and $m^{\ast }={\hslash }^2/(2\mu a^2)$ once (with signs interpreted at the selected band extremum).