What Determines the Critical Radius for Fission in Nuclear Physics?

[Pi-Bond]

Homework Statement

See the image below:

Homework Equations

In the previous part, it was proved that a particle moves through a distance x in a material without interacting with probability P given by

$P=\exp(-\sigma n x)$

Here σ represents the cross section of the reaction between the incoming particle and the material. n represents the target (material) particles per unit volume.

The Attempt at a Solution

The only thing I have figured out so far is that we need the probability of the neutron not escaping without a reaction, so there is a factor of

$1- exp(- (\sigma_f + \sigma_c) nr )$

in the equation.

I am not sure of where the fractional factor at the start of the equation comes from, or why the total cross section appears in the exponent.

 

[Pi-Bond]

Bump! Does no one have any ideas, I haven't been able to solve this since the last post.

 

[voko]

The sum in the exponent is simple: it is because cross-sections of different reactions are additive.

Thus (1 - exp) is the probability that the neutron will be captured or cause fission. You need to express from this the probability of fission.

 

[Pi-Bond]

Simple explanation! Looks like I wasn't thinking the right way, thanks.