= Solution
For $0<\gamma<t$, the positive branch ranges from $2\gamma$ to $2t$, and the negative branch from $-2t$ to $-2\gamma$. Therefore
$$
\boxed{W_1=W_2=2(t-\gamma)=2\big(|\beta|-|\beta_{sp}|\big).}
$$
The smallest separation is between the upper-band minimum and lower-band maximum at the same <wavevector> $k=\pi/(2a)$:
$$
\boxed{E_g^{\min}=4\gamma=4|\beta_{sp}|.}
$$
It is a \b[direct <band gap> at the interior <avoided crossing>]. The zero-mixing limit has zero gap; the width is not $4|\beta|$ because these are the mixed energy-ordered branches, each turning around at the middle of the plotted interval.
Back to article page
