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

[itex]P=\exp(-\sigma n x)[/itex]

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

[itex] 1- exp(- (\sigma_f + \sigma_c) nr ) [/itex]

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.

 

[paranormal]

The critical radius for fission is a fundamental concept in nuclear physics and refers to the minimum distance at which a fission reaction can be sustained. It is determined by the balance between the rate of neutron production and the rate of neutron absorption in a nuclear reactor. The equation provided in the homework statement is the basic formula for calculating the probability of a neutron interacting with a material without escaping. However, in order to determine the critical radius for fission, we need to consider the specific cross sections for fission and neutron absorption (σ_f and σ_c, respectively) and the number of target particles (n) in the material.

The fractional factor at the start of the equation, 1-exp(-(\sigma_f + \sigma_c)nr), is a correction factor that takes into account the possibility of multiple interactions between the neutron and the material. This is necessary because a neutron can undergo multiple interactions before either being absorbed or escaping.

Additionally, the total cross section in the exponent takes into account the fact that the probability of interaction is dependent on the distance traveled by the neutron (x) and the number of target particles (n) in a given volume. As the neutron travels further, the probability of interaction increases due to the higher chance of encountering target particles.

To determine the critical radius for fission, we need to find the value of r at which the probability of a neutron escaping without a reaction is equal to the probability of a neutron undergoing a fission reaction. This can be achieved by setting the equation equal to each other and solving for r. The resulting value of r represents the critical radius for fission, and any distance greater than this will result in a self-sustaining fission reaction.