= Two-wave estimate of a Bragg-gap coupling from a pocket area
{title2=$A(v)=4\int_0^{q_m}y(q)\,dq$}
Put $C=\hbar^2/(2m)$, $v=|V_G|$, $\mathbf G=(2K,0)$, and $k_x=K+q$. The upper <eigenvalue> of the two coupled <plane waves> is
$$
E_+(q,k_y)=C(K^2+q^2+k_y^2)+\sqrt{4C^2K^2q^2+v^2}.
$$
At fixed <Fermi energy> $Ck_F^2$, its closed <Fermi contour> has
$$
y(q)^2=k_F^2-K^2-q^2-\sqrt{4K^2q^2+(v/C)^2},\qquad
q_m^2=k_F^2+K^2-\sqrt{4k_F^2K^2+(v/C)^2}.
$$
Thus $A(v)=4\int_0^{q_m}y(q)\,dq$ 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.
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