I am calculating the critical mass (radius) of $U^{235}$ sphere. I want to calculate the mass for three different cases:
1. air/vacuum surrounds the sphere (diffusion coefficient is infinite)
2. ideal reflector surrounds the sphere(diffusion coefficient is zero)
3. diffusion coefficient is the same in the surrounding matter as it is in the sphere.
There is no absorption or fission outside of $U^{235}$ sphere.
The equation i am using is : $$-D\nabla^2\phi + \Sigma_a\phi = \frac{1}{k}\nu\Sigma_f\phi.$$
(D is diffusion coefficient, $\phi$ is neutron flux, $\Sigma$ is macroscopic cross section, k is multiplication factor, $\nu$ is average number of secondary neutrons per fission )
This equation can also be written as $$\nabla^2\phi + B^2\phi = 0, $$ where $B^2 = \frac{1}{L^2}(\frac{k_{\infty}}{k} -1)$, $k = \frac{\nu\Sigma_f}{\Sigma_a}$ and $L^2 = \frac{\Sigma_a}{D}$.
Solution to this equation is: $$\phi(r) = \frac{a*\sin(B*r)}{r}.$$
This solution describes neutron flux in the $U^{235}$ sphere.
My question is about boundary conditions.
Is it OK if I say that $\phi(R)=0$ in the first case and $\frac{\partial\phi}{\partial r}|_{r=R}=0$ in the second (R is radius of the sphere)? In the third case I think i should calculate $-D\nabla^2\phi=0$ for the area outside of a sphere and then use: $$\phi_s|_{r=R} = \phi_o|_{r=R}$$ and $$j_s|_{r=R} = j_o|_{r=R}$$ (s refers to sphere and o to outside, j is neutron flux current). Are these boundary conditions correct for these cases?
