Consider a one$-$dimensional slab of thickness $R,$ containing a material with macroscopic absorption cross section ${}_a$ and diffusion coefficient $D,$ and internal neutron source density $s(x)=\frac{2s_{0}x}{R}.$ The left side of the slab $)=(0$ has a vacuum $($escape$)$ boundary condition where we may assume that the scalar flux is zero, $(0)=0.$ The right side of the slab $)=(R$ has a zero net current $($reflecting$)$ boundary condition, $J(R)=0.$

Verify that $(x)=\frac{2L^2{s}_{0}x}{DR}$ is a particular solution to the one$-$speed, one$-$dimensional, steadystate neutron diffusion equation for this slab.

Following the examples shown in class, solve the one$-$speed, one$-$dimensional, steady$-$state neutron diffusion equation to determine the scalar flux as a function of position, $(x),$within the slab.

Using $R=180cm,D=10cm,{}_a=0\mathrm{.}0015c{m}^{-1},$ and $s_0=1$ neutron $\frac{?}{c}\frac{m}{s},$ plot the scalar neutron flux $(x)($obtained in problem $2)$ from $x=0$ to $R$