Consider a one$-$dimensional slab, containing a fissile material with macroscopic absorption cross section ${}_a=0\mathrm{.}1\frac{1}{(cm)},$ macroscopic fission production cross section $v{}_f=0\mathrm{.}13\frac{1}{(cm)},$ and diffusion coefficient $D=10cm.$ Both the left $)=(0cm$ and right $)=(100cm$ sides of the slab have escape boundary conditions.

a$.)$ Determine the infinite $($no leakage$)$ multiplication factor $(k_{})$ for the system.

b$.)$ Determine the fundamental mode $)=(1$ multiplication factor $(k_{\text{e}\text{f}\text{f}})$ for the slab.

c$.)$ Determine the fundamental mode $)=(1$ flux solution, $(x),$ within the slab. Please choose the normalization constant for your solution so that the maximum flux in the slab is equal to $1\mathrm{.}0.$

d$.)$ Calculate $J(K),$ the net neutron current at $)=(100cm.$ This value gives the rate at which neutrons are leaking out of the right$-$hand side of the reactor.

e$.)$ Calculate a new thickness $R^{\text{'}}$ for the slab that will cause the system to be exactly critical.

f$.)$ Assuming that the width of the reactor is adjusted to $R^{\text{'}}($the critical thickness calculated in part $($e$)),$ calculate $J(R^{\text{'}}),$ which is the leakage at the right side of the reactor. Has the leakage increased or decreased from the original solution in part $($d$)?$ Does this change make physical sense? Why or why not?