Neutron capture calculation

Question

So, the problem state is:

Neutron beam radiates sample A with initial number of atoms $N_0$. With neutron capture nuclei (cores) of A are transitioning to nuclei B (they are just one neutron richer isotope).

A + n $\longrightarrow$ B + $\gamma$

Expected time for neutron capture on core is equal to $\tau_N$. 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 $\tau_0$ and they decay to the nuclei (cores) different then A
3. cores B are unstable with average lifetime $\tau_0$ and they decay back to the nuclei (cores) A.

There are also two hints in helping problem to solve:

Hint 1: Parameter $\tau_N$ 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.

From this textbook I do no have any solutions, so I do not know what am I supposed to get as the final solution. For any advice and help, thanks in advance!

Answer 1

You have to write a differential equation for each of the three problems and then solve it.

1. The DE for problem 1 is given to you in hint 1. You only need to solve the DE.

2. For problem 2 and 3 you have to add all contributions to the rate of change for each of the nuclei. Nothing changes for A so the DE for A is the same as in 1.

For nuclei B there are two contributions: The rate at which nuclei B decay, given by $-\frac{N_B}{\tau_0}$ (decaying means you lose nuclei, so this is a negative rate of change). The rate at which nuclei A decay into B is given by $\frac{N_A}{\tau_N}$ (this contribution will add more nuclei B to the total so this time the sign is positive). The total rate of change for B is then given by $$\frac{dN_B}{dt}=-\frac{N_B}{\tau_0}+\frac{N_A}{\tau_N}$$ You already know the solution for A from problem 1, if you put that into this equation you get a DE for B which you can solve.

3. Hint 2 is just a substitution and only useful in problem 3. Similarly to problem 2 you need to add the contribution to the rate of change for each element A and B. You get two coupled DE's, but you can decouple them by using the given substitution. Hint: You can add and substract the two DE's from each other to get two new DE's and then you can write the substitution as $N_A=\frac{\Sigma+\Delta}{2}$ and $N_B=\frac{\Sigma-\Delta}{2}$. The result are two decoupled DE's with $\Delta$ and $\Sigma$.