(a) The number of nuclei of a given radioactive sample at time $t=0$ and $t=T$ are $N_0$ and $\; \frac{N_0}{n}$ respectively. Obtain an expression for the half-life $(T_{1/2})$ of the nucleus in terms of $n$ and T. (b) Write the basic nuclear process underlying $\beta^{-}$ decay of a given radioactive nucleus.