What are the rate equations for neutron capture on unstable nuclei B?

[solidbastard]

Summary: Problem: nuclear physics, neutron capture

In the problem one should calculate time dependence of number of nuclei.

Problem statement:

Neutron beam radiates sample A with initial number of atoms N₀. With neutron capture nuclei (cores) of A are transitioning to nuclei B (they are just one neutron richer isotope).

A + n → B + ϒ

Expected time for neutron capture on core is equal to τ_N. With an assumption that neutrons do not affect the sample B, calculate time dependence number of nuclei B if:

1. cores B are stable
2. cores B are unstable with average lifetime of τ₀ and they decay to the nuclei (cores) different then A
3. cores B are unstable with average lifetime τ₀ and they decay back to the nuclei (cores) A.

There are also two hints in helping problem to solve:
Hint 1:
Parameter τ_N considers that contribution to the destroying of nuclei A with neutron captures is described as:
$(\dfrac{dN_A}{dt})_{capture} = \dfrac{-N_A}{\tau_N}$

Hint 2:
Sometimes it is useful to assume solution in advance, but sometimes it is easier to switch to the new set of variables like:
$\Sigma = N_A+N_B$ and $\Delta = N_A-N_B$So, this is the problem. It is hard for me to actually attack it anyhow, because problem is generalized and what bothers me the most are conditions for 1, 2 an 3. On the other side, kind of confused with hint 2.
How should I treat here stable and unstable nuclei B. To just assume N/Z ratio, like even - even nuclei or similar. But the also to assume the same for nuclei A.
For any advice and help, thanks in advance!

[Moderator's note: Moved from a technical forum and thus no template.]

 

[vanhees71]

Then you just write down rate equations. For case (3)
$$\dot{N}_A=-\lambda_N N_A+\lambda_0 N_B,\\
\dot{N}_B=\lambda_N N_A -\lambda_0 N_B.$$