Two-wave estimate of a Bragg-gap coupling from a pocket area ($A(v)=4{\int }_0^{q_m}y(q)dq$)

Put $C={\hslash }^2/(2m)$, $v=|V_G|$, $\mathbf{G}=(2K,0)$, and $k_x=K+q$. The upper eigenvalue of the two coupled plane waves isAt fixed Fermi energy $C{k}_F^2$, its closed Fermi contour hasThus $A(v)=4{\int }_0^{q_m}y(q)dq$ can be inverted numerically to estimate the Bragg scattering coupling. The pocket disappears at $v=C(k_F^2-K^2)$. The estimate assumes a fixed chemical potential and neglects coupling to other plane waves; a measured pocket area alone does not determine all Fourier coefficients of an arbitrary lattice potential.