 Solution

For the valence band, ${\beta }_v>0$, so the electron curvature is negative. A missing electron is a positive-mass electron hole:The quadratic expansion has $E_v(\mathbf{k})\simeq E_v(0)-{\beta }_v{a}^2|\mathbf{k}{|}^2$, so each diagonal electron inverse mass is $-2{\beta }_v{a}^2/{\hslash }^2$. Reversing the sign to describe the missing electron gives the positive hole mass above. As in the conduction calculation, convert beta from eV to joules and use $a=5.42\times 10^{-10}$ m; the result applies near the band edge.