# Deriving the critical radius of Uranium using diffusion equation

• vengeance123
In summary, the student attempted to solve the equation for the neutron density as a function of position and time, but ran into a problem because the coefficient of sin(kr)/r + cos(kr)/r in the equation includes m, which is an integer. The student then attempted to solve for the critical radius using boundary conditions, but ran into another problem because the expression for the critical radius contains m.

## Homework Statement

I have solved the equation for the neutron density as a function of position and time. I need the boundary conditions to change my infinite number of solutions (the varying separation constant) into one value so that my answer for the critical radius does not contain a sum!

## Homework Equations

del squared (n) - A(dn/dt) = -Bn (which i have solved)

## The Attempt at a Solution

i assumed spherically symmetric solutions so using spherical polar coordinates n varies only (spatially) from the distance to the centre of the sphere. The r dependence is of form cos(kr)/r + sin(kr)/r. So the coefficients of cos term must all be 0 (as the density at the centre of any given sphere cannot be infinite). I thought that at the surface, the density is 0 as neutrons do not diffuse back into the sphere once they are out. Is this right?

maybe i should add a little more to this. The time dependence part of the neutron density is found to be exp(B-K)t where K=k*k (k is the separation constant). now at the critical radius i know that the neutron density cannot be increasing or decreasing with time (it must be stable). So i reasoned that B-K = 0. If also we reason that the density of neutrons on the surface of a sphere R must be zero, then we have that sin(kR)=0 or kR=m*pi where m is an integer. So k=m*pi/R and subbing this to B-K=0, i have an expression for the critical radius in terms of B (which are the product of some intrinsic properties of uranium like mean free path etc). The only problem I have now is that the expression for R, the critical radius, contains m! so I have infinitely many expressions! How do I fix the value of m using the boundary conditions?! this is driving me insane..