4.6.1: Essential Cross Sections

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The key for understanding the physical foundations of the operation of a nuclear reactor is the energy dependence of relevant cross sections of U-235 and \(U-238\). The plots are shown in Fig. \(\PageIndex{1}\).

In the process of U-235 fission, high energy neutrons are created - with average energy of about \(1.6 \mathrm{MeV}\). For such energies, the U-238 cross section for fission is somewhat lower (about 50\%) than that of U-235 (see the red and the green curves on the right-side part of Fig. \(\PageIndex{1}\)). Yet, there are 140 atoms of U-238 per one atom of U-235, so the neutron has a very small chance, one-out-of-seventy to hit another U-235 nucleus and thus to support the chain reaction - and if it hits a U-238 nucleus, it may trigger a fission, but - as was stated before - the U-238 fission does not release any neutrons, so that it does not support the chain reaction.

The situation drastically changes in the region of very low energies, of \(0.1 \mathrm{eV}\) and less. Here, the fission cross section for U-238 is of the order of micro-barns only. The dominant is the cross section for "radiative capture of neutron", i.e., the U-238 nucleus emits a gamma photon and changes into U-239 . But now the cross section for U-235 fission is about 3 timeshigher, so that the \(0.7 \%\) of U-235 nuclei have a five-to-one chance over the \(99.3 \%\) of U-238 to capture the neutron. So, the chain reaction may be saved - if the fission neutron could be removed from the uranium fuel, forced to give away its high energy - or lowering it about ten million times (from \(1 \mathrm{MeV}\) down to \(0.1 \mathrm{eV}\) ) - and only then returned to the fuel.

Figure \(\PageIndex{1}\): Upper curves: The cross sections for fission vs. neutron energy for U-235 and two other fissile isotopes (U-233 and plutonium Pu-239) as a function of incident neutron energy – note the plot is in a “log-log” scale. Lower curve: the same for the non-fissile U-238. Note that the vertical scale in the lower plot is more compressed than in the upper plot.Note that in certain energy regions the cross sections “oscillate” in a very irregular way. But below the neutron energy of about 1 eV ( = 10⁻⁶ MeV the cross section of the three fissile nuclei are more than a million times higher than that for U-238.

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\({ }^{1} \mathrm{U}-239\) after half-life of only \(2.35\) minutes changes via a \(\beta\) decay to Neptune, and after another \(\beta\) decay with half-life of \(2.4\) days changes to Plutonium-239, with half-life over 20,000 years - not a nice element, as it is the essential component in the A-bombs and \(\mathrm{H}\) bombs.