1.3.2 Microscopic cross-section

For neutrons of a given energy one or more of the foregoing nuclear reactions can occur. It is necessary to have a method of calculating how many neutrons are undergoing which reaction.

Clearly, the rate R at which neutrons undergo any reaction is dependent on the number of target nuclei and the number of bombarding neutrons. We can say that R is proportional to N0 where N is the number of nuclei/m3 for the target material and Ф is the intensity of the neutron radiation measured in neu — trons/sm:. о is called neutron flux and is the product of the number of neutrons/m3 and the average neu­tron speed і/, which may be in any direction, i. e., ф = nv neutrons/sm-. Let the constant of proportionality be a, so that

R = a N ф

This is usually written in the order R — Na ф

———— і————— —————————————————

and the value of о for each type of nuclear reaction is dependent on the target material and the neutron speed, a may be regarded as a measure of the like­lihood, or the probability in the non-mathematical sense, of a given reaction occurring. Alternatively, because о has the unit of m2, it is also thought of as the ‘effective’ area presented to the incident neutron by the target nucleus. Hence a is called the micro­scopic cross-section for the neutron reaction but it must be emphasised that it is not the physical area of the nucleus (a can have values orders of magnitude greater or less than the physical area). Generally, but not always, microscopic cross-section values are in the range 10"26 to 10"30 m2 and so it is convenient to have a new unit for cross-sections. This is called the barn and is defined by 1 barn = 10“28 m2.

Since neutrons can have more than one type of re­action the total microscopic cross-section at, the like­lihood of the neutrons interacting with the nuclei, can be expressed as the sum of a number of partial cross-sections.

Thus:

at — <7S + aa

= Oe + Cq + a^ + CTf

where the subscripts refer to total, scattering, absorp­tion, elastic, inelastic, capture and fission respectively. To indicate the usefulness of these ideas Fig 1.6 shows examples of how the microscopic cross-section depends on the material, the neutron energy and of course the particular nuclear reaction. In referring to the figure it is convenient to regard the neutron energy as being in three parts:

• Low energy, slow neutrons — energies up to a

few eV.

• Intermediate energy — few eV to 100 keV,

say.

• High energy, fast neutrons — energies greater

than 100 keV.

From Fig 1.6, then, the following may be inferred:

• Boron 10 capture cross-section, Fig 1.6 (a).

For slow neutrons the capture cross-section for B-10 is very large (thousands of barns) and decreases progressively over the slow to intermediate energy range. In fact ac, using logarithmic scale, decreases linearly with increasing neutron speed. This linearity is not unusual and such materials are known as ‘/v absorbers’. For the lighter dements the /v dependence of ac may persist to several hundred eV but only to a few eV for the heavier elements. The extremely large values of ac for B-10 explains why boron, which contains 19.6% B-10, is incor-

porated for neutron absorption purposes in control rods, in the coolant of water reactors and in fuel storage ponds. [2] energy range but there is a series of peaks in the intermediate energy range where ac has extremely large values, in the thousands of barns. These peaks are called resonance capture peaks and are associated with discrete energy values of the ura­nium nucleus. It will be seen later that the resonance capture of neutrons has important implications in reactor design.

• Uranium 235 and 238 fission cross-sections, Fig

1.5 (c).

Resonance cross-section peaks are not unusual and are seen in the fission cross-section values for U-235. Much more important here however is to note the large of values for slow neutrons, several hundreds of barns, in contrast to fractions of a barn for very fast neutrons. Figure 1.6 (c) also gives the fission cross-section fff for U-238. This shows that fission of U-238 is possible but only by high energy neu­trons and even then is not very likely. There is a threshold value of 1.1 MeV below which fission of U-238 will not occur. An explanation for this is given in Section 3 of this chapter.

Graphs similar to those given in Fig 1.6 are more or less readily available in the literature for a wide range of materials and applicable to neutron and other nuclear reactions.