# Neutron capture calculation

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!

## 1 Answer

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 $$\Sigma$$.

• 2. You meant $\frac{dN_A}{dt}=-\frac{N_A}{\tau_N}$ right? Commented Jun 30, 2019 at 17:23
• 3. You can gain nuclei of type B by neutron capture and you can lose them by decay to A. So you have to add two terms to get the rate of change. You can gain A by decay of B and you can lose them by neutron capture. So you have two contributions too. If you write down both DE's you can solve them with eigenvalues if you want, or alternatively if you use the substitution with $\Sigma$ and $\Delta$ you don't need eigenvalues, because the DE for $\Sigma$ decouples (becomes independent of $\Delta$) and you can solve that DE directly. After solving both DE's you can just substitute back. Commented Jun 30, 2019 at 17:29
• You can write the DE's for 3 here and I will check them if you want. Commented Jun 30, 2019 at 17:30
• No, in your first equation $N_B/\tau_0$ represents the rate at which nuclei B decays, and it needs a negative sign, because you lose nuclei B by that process. The equation you have written says that you gain more of B if a nucleus B decays. Your second equation has the same problem, and it is missing a second term. Commented Jul 3, 2019 at 13:48
• The first equation describes the total rate of change for B. It does not describe only a single process. The neutron capture is important for the rate of change for B, because you can gain B by it, and the decay process back is also important, because you lose B by it. Both processes play a role in each of the equations! The first equation is right. The second equation has wrong signs. Commented Jul 3, 2019 at 14:43