On Mathews and Walker's book exercise (8-2)
- We are given that the neutron density n inside $U_{235}$ obeys the differential equation $$\nabla ^2u+\lambda u=\frac{1}{k}\frac{\partial{n}}{\partial{t}} \;\;\;\;,$$ $ n=0 $ on surface
and we are ask to find the critical radius $R_0$ such that the neutron density inside a $U_{235}$ sphere of radius $R_0$ or greater is unstable and increases exponentially with time.
- Also Suppose two hemispheres, each just barely stable, are brought together to form a sphere. This sphere is unstable, with $n$~$e^{t/τ }$,where Find the “time-constant” τ of the resulting explosion.
I do the separation of variable $n=R(\vec{r})T(t)$ and I get that $$\frac{\nabla^2R}{R}+\lambda=m$$ $$\frac{T'}{T}=km$$ I choose $m>0$ so the T(t) is unstable and increase exponentially over time. So i get $$T =Ce^{km}$$ and $$R''+\frac{2}{r}R'+(\lambda-m)R=0$$ but I can't see how this can lead to find the $R_0$ which the density will increase with time , probably because I already set the time to diverge ,but i can't see how this will get to diverge otherwise.
Any hint please :) ?