= Solution
The <energy-band widths> are the difference between their extrema:
$$
\boxed{W_s=4t_s=4|\beta_s|,\qquad W_p=4t_p=4|\beta_p|.}
$$
For the separated bands, the top of the $s$ band and bottom of the $p$ band are both at $k=\pi/a$. Hence
$$
\boxed{E_g^{\min}=\alpha_p-\alpha_s-2(t_s+t_p).}
$$
It is a \b[direct <band gap> at the zone boundary], because both limiting states have the same <wavevector>. This result uses the opposite signs of the $s$ and axial-$p$ hoppings.
Back to article page
