# Deriving the critical radius of Uranium using diffusion equation

1. Feb 20, 2007

### vengeance123

1. The problem statement, all variables and given/known data

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 seperation constant) into one value so that my answer for the critical radius does not contain a sum!

2. Relevant equations

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

3. 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?

2. Feb 20, 2007

### vengeance123

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 seperation 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..

3. Feb 22, 2007