What about energy in neutron capture?

Question

I have a problem with energy conservation in the neutron capture process. As far as I understood the matter, elements heavier than iron can not be produced in a fusion process, because after ion, heavier nuclei have more energy per nucleon than lighter nuclei. And that is why the heavier elements are produced in neutron capture processes.

Now I have read, that a neutron capture process looks like this:

$$^A_ZX+n~~~\rightarrow~~~ ^{A+1}_ZX +\gamma $$

But that means, that not only has the nucleus on the right side of the equation more energy than nucleus and neutron on the left side combined, but on the right side there is also energy in the form of gamma ray. To me that looks like energy is missing. Is that right?

If that is right, does the energy come from outside (fusion processes in the star, supernova)? And if that is the case, why is the neutron capture process possible but fusion of heavy elements is not?

Answer 1

Let us suppose that the nucleus $^{A+1}_Z\rm X$ has larger rest mass than $^A_Z\rm X + n$. Then $^{A+1}_Z\rm X$ can relax to a lower-energy state by emitting a neutron, and is beyond the neutron drip line.

You're correct that nuclides around iron and nickel have the largest binding energy per nucleon. However, a free neutron has an especially low binding energy per nucleon, being both unbound and unstable. So in general, neutron capture on stable or near-stable nuclei increases the average binding energy per nucleon on average over the entire system.

In fact, it's neutron capture (near the valley of stability, in the "s-process", and far from the stable isotopes in the "r-process") that is responsible for production of all elements heavier than iron.

Answer 2

Neutron capture drives iron-peak nuclei away from the valley of stability, so the nucleus that is produced has less binding energy per nucleon and is not as stable. In the slow neutron capture process (s-process), one or more neutron captures are usually followed by a beta decay, which moves the nucleus back towards the valley of stability, but now with an extra proton. In the rapid, high neutron flux r-process, then multiple neutron captures can take place before decay back towards the stability line.

The reason this works can be seen with an example. Consider this ($n,\gamma$) reaction. $$ ^{56}_{26}{\rm Fe} + n \rightarrow ^{57}_{26}{\rm Fe} + \gamma$$ Using this semi-empirical mass formula calculator, I find that $^{56}_{26}{\rm Fe}$ has a binding energy per nucleon of 8.762 MeV and a total binding energy of 490.68 MeV. On the other hand $^{57}_{26}{\rm Fe}$ has a binding energy per nucleon of 8.728 MeV and a total binding energy of 497.48 MeV.

Thus although the binding energy per nucleon is smaller in the product nucleus, the total binding energy is larger. If we consider only the rest mass of the neutron then the total mass-energy on the LHS of the reaction is 53030.91 MeV, whilst on the RHS it is 53024.26 MeV. Thus there is an excess of available mass-energy that is carried off by the gamma photons and/or in the respective kinetic energies of the particles.

A side issue is that the statement that a particular A/Z ratio is the most stable depends on the density and environment of the material. For example in the collapsing core of a pre-supernova star, electron degeneracy blocks beta decay and pushes the equilibrium to more neutron-rich nuclei and this aids neutron capture.

Finally, the reason why neutron captures can occur at a significant rate (in the presence of a high enough neutron flux) but that fusion reactions do not, is that (neutral) neutrons do not suffer the coulomb repulsion that prevents the initiation of alpha capture and other fusion reactions.