If you are wondering where I am going with this it is because it is relevant to nuclear weapons proliferation, thorium fuel cycles and the problems with high burnup fuel.

In a previous post I explained a bit about neutron capture, cross sections etc. Now I am going to consider a simplified model for neutron capture/ fission of, for example Uranium 238. Where A undergoes neutron capture to form B which can either undergo neutron capture itself to form C or undergo fission.

where

is the capture cross-section of isotope A etc

is the fission cross-section of isotope B etc

I have only shown fission for every other isotope since only isotopes with odd atomic mass are fissile. For example U-238 is not fissile, Pu-239 is, Pu-240 is not, Pu-241 is.

Don’t worry, I am not going to go through a lot of maths since I do not want to loose too many people. However if you are interested (skip the bit down to OK if you want) then the rate of formation of isotope B can be written as:

where φ is the neutron flux.

This can be written as

we can then define

this is very similar to the formula I derived earlier for radioactive decay and can be treated similarly.

OK mathematics over and most people can come back now.

Let us consider the case where there was no fission, i.e. all the cross-sections _xN_c are zero. Then the total number of atoms with mass greater or equal to that of A will be constant.

If fission does occur then the number of atoms with mass greater or equal to that of A will be less than the original number of atoms of A – some of them would have fissioned into two atoms with mass less than A. That is a good thing since these transuranics tend to have long half-lives (e.g Pu-239 24,100 years, Pu-240 6500 years) and are an important factor in how long term radioactivity of spent fuel.  Another way of reducing the amount of these transuranics is to fission U-233 produced from Thorium, since we are already starting at a lower atomic mass. This is sometimes why the thorium cycle is called ‘clean’. It must be noted that there are also several long-lived fission products such as ⁹⁹Tc and ¹²⁹I and fission of U233 also produces the same dangerous short-lived fission products such as ¹³¹I and ¹³⁷Cs

Therefore, using higher burnup fuels (i.e. keeping it exposed to a neutron flux for longer) decreases the number of uranium 238 and transuranics in the spent fuel.

Let us look how the various isotopes vary with time. I have plotted a hypothetical graph of the number of atoms of different species with time. I have left out A which I assumed to have a very long lifetime and be present in large quantities – basically it is taking the place of U-238. However I have not made any attempt to use real constants since it is the relative shape of the curves that are important for discussion.

I have assumed that only isotope A is present at the beginning and that there is no other way of the isotopes decaying other than by neutron capture or induced fission.

B increases steadily with time and then its production tails off as its rate of loss equals its rate of production.

With C there is a short period when its rate of production is quite low (since there is not enough of B yet to create it). Then it builds us to reach equilibrium.

With D the period of low production remains low for longer and then it again builds up to reach equilibrium.

I have not plotted E since already with D there are 48 terms in the formula on the spreadsheet and there are even more with E. However, what will happen is that the period of low production remains even longer – it has a similar shape to the curve to D but is stretched further to the right.

I will come back to this point in a later post but note that the ratios of the various isotopes vary greatly with time in the early period. Early on there is relatively a lot more of B than C compared with later when they both reach equilibrium.

Coming back to reality rather than this hypothetical system what does that mean?

In higher burnup (the same as a longer exposure to a neutron flux) we get a large increase in some isotopes of high atomic mass. Although, as mentioned before the odd atomic mass isotopes – Pu-239, Pu-241 and Am-241 undergo fission the even mass isotopes – Pu-240, Pu-242, cm-242, cm-246 and cm-248 undergo spontaneous fission and therefore produce neutrons.

We would therefore expect higher burnup spent fuel to produce a lot more neutrons, which is indeed the case:

I shall discuss the issues around the production of neutrons in high burnup fuel in a later post.