 Solution

For one electron per primitive unit cell, the areal number density is $n_s=1/(ab)$. The Fermi circle has radius $k_F=\sqrt{2\pi /(ab)}$. The nearer first-zone boundary is $k_x=\pm \pi /a$, because $a>b$. The circle protrudes precisely whenAt equality the Fermi surface only touches the boundary. Crossing an extended-zone boundary alone does not make a gap: the free electron gas remains metallic when the ionic periodic potential is neglected.