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

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:The contact points have coordinates $(\pm \pi /(2a),\pm \pi /(2a))$ with all four sign combinations. The distance to a zone vertex is $\pi /a$, which is larger; that distance would describe contact with a corner after the circle had already crossed a side.