# Burnout time

The half-life of a radioactive nuclide, t, is defined as the time it takes for half of the atoms of a radioactive source to undergo transformation. If one takes the standard decay equation:

where is the number of atoms at the time t, and is the decay constant, the solution is . It follows that and hence . In a reactor, the rate of disappearance of a nuclide is given by

where is the number of atoms at the time t, the absorption cross section and the neutron flux of the reactor. The absorption cross section is the sum of the capture and fission cross-sections and in addition the cross sections for n,2n, n,3n, n,p and n,alpha reactions i.e.

.

Similarly to the definition of the half-life, a "burnout time" can defined as the time it takes for half the atoms of a fuel to transmute i.e. . The above relations can be combined to give a relation for the overall rate of change due to decay and reaction of a nuclide in a neutron flux i.e.:

In this equation, when (long half-life, short burnout time), one can neglect the decay process as the absorption of a neutron occurs much faster than the decay, and when (long burnout time and short half-life), one can neglect the nuclear reaction as the nuclides has decayed long before a neutron is absorbed.

In the case of a ^{238}U nucleus in a reactor, the ^{238}U will transform mainly by neutron absorption to give ^{239}U, since the burnout time for ^{238}U is much shorter than its half-life of 4.47x10^{9} years. This ^{239}U then decays to ^{239}Np since the half-life of ^{239}U (23.4 minutes) is shorter than its burnout time. The ^{239}Np than decays to ^{239}Pu as its half-life of 2.35 days is much shorter than its burnout time (for a standard neutron flux of 5x10^{13} neutrons cm^{-2} s^{-1}) of 1.18 years. The ^{239}Pu then absorbs neutrons and goes on to ^{240}Pu and ^{241}Pu which all have a half-life of several years, so that the burnout time is much shorter and the neutron reaction takes place.

One has to be careful with the neglection of a particular process. If the half-life and the burnout time for the standard neutron flux of 5x10^{13} neutrons cm^{-2} s^{-1} aren't very different, only a little change in the neutron flux will change the behaviour of the nuclide. Changing the neutron flux can change the reaction path. This can be illustrated with the nuclide ^{233}Pa. This nuclide has a half-life of 26.9 days and a burnout time of 1.09 years. In this case, the ^{233}Pa decays to ^{233}U. But if the neutron flux is increased to 8x10^{14} neutrons cm^{-2} s^{-1}, the burnout time decreases to 25 days, so that the probability for the neutron reaction is higher than for the decay. This means that ^{233}Pa absorbs a neutron and gives ^{234}Pa more often than decaying to ^{233}U.

Reference: J. Magill and J. Galy, **Radioactivity Radionuclides Radiation** Springer Verlag, 2005