How Is the Critical Radius Determined in Nuclear Chain Reactions?

[vengeance123]

Homework Statement

I am trying to solve the diffusion equation for a sphere of fissile material. I then have to derive an expression for the radius above which a chain reaction will occur (the critical radius). . My trouble then, is finding a boundary condition other than at the surface of the sphere the neutron density is 0. what could be happening in the centre? i know that the B coefficients must be 0 because at r=0 the cos term is infinite. I also know that using my boundary condition k*r=m*pi. the critical radius is when n doesn't vary with time so i set D-k^2=0 to obtain an expression for the critical radius. My problem is that it contains this m value (which is ANY integer). How does one fix m so the critical radius is single-valued? Any help or hints would be greatly appreciated..

Homework Equations

del squared (n) - 1/C*(dn/dt) = -n/D where n is the neutron density n(r,t).
if sin(k*r)=0 then k*r=m*pi , m an integer

The Attempt at a Solution

I have already solved the governing equation and have the neutron density n(r,t) in its most general form which is the sum over all k of some time dependence (exp(D-k^2)t) times some spatial dependence (Asin(k*r)/r + Bcos(k*r)/r)

I also know that using my boundary condition kr=m*pi. the critical radius is when n doesn't vary with time so i set D-k^2=0 to obtain an expression for the critical radius. My problem is that it contains this m value (which is ANY integer).

 

[HallsofIvy]

Typically, problems involving spheres or circles, in addition to the condition on the boundary, have the condition that value at the center, r= 0, must be finite. That would mean that your B in B cos(kr)/r must be 0.

 

[vengeance123]

Thanks but i have already figured that out. When i set the density at the surface to 0, i get kr=m*pi, i know this is heading towards the correct answer because the answer is of the from r=pi*sqrt(D).