Question #43831

A heavy nucleus of mass number $A$ undergoes symmetric fission into two identical nuclei of mass number $A/2$. The binding energy per nucleon, $B(A)$, in this mass region has a constant negative gradient given by $\frac{dB}{dA} = -\gamma$ (where $\gamma > 0$). The fission event occurs entirely within a semiconductor detector micro-pixel of constant heat capacity $C$, initially at temperature $T_0$. The fission fragments are completely stopped within the pixel, and all the released energy $Q$ is rapidly converted into heat, raising the pixel's temperature uniformly to $T_f$. The detector measures the energy by generating electron-hole pairs. Assume the energy required to create one electron-hole pair is exactly equal to the semiconductor's band gap $E_g$ at the final temperature $T_f$. If the band gap decreases with temperature as $E_g(T) = E_{g0} - \beta(T - T_0)$ (where $\beta > 0$ and $E_{g0}$ is the band gap at $T_0$), which of the following expressions gives the total number of electron-hole pairs $N$ generated by this single fission event?