Section 4.4.4.2. Scattering and absorption cross sections

When a thermal neutron collides with a nucleus, it may be either scattered or absorbed. By absorption, we mean reactions such as , (n, p), or (n, α), in which there is no neutron in the final state. The effect of absorption can be included by allowing the bound scattering length to be complex, The total bound scattering cross section is then given by in which denotes a statistical average over the neutron and nuclear spins and the absorption cross section is given by where k = 2π/λ is the wavevector of the incident neutron and λ is the wavelength.

If the neutron and/or the nucleus is unpolarized, then the total bound scattering cross section is of the form in which and are called the bound coherent and incoherent scattering cross sections and are given by Also, so that the absorption cross section is given by The absorption cross section is therefore uniquely determined by the imaginary part of the bound coherent scattering length. It is only when the neutron and the nucleus are both polarized that the imaginary part of the bound incoherent scattering length contributes to the value of .

For most nuclides, the scattering lengths and, hence, the scattering cross sections are constant in the thermal-neutron region, and the absorption cross sections are inversely proportional to k. Since k is proportional to the neutron velocity v, the absorption is said to obey a 1/v law. By convention, absorption cross sections are tabulated for a velocity v = 2200 m s⁻¹, which corresponds to a wavevector k = 3.494 Å⁻¹, a wavelength λ = 1.798 Å, or an energy E = 25.30 meV.

The only major deviations from the 1/v law are for a few heavy nuclides (specifically, ¹¹³Cd, ¹⁴⁹Sm, ¹⁵¹Eu, ¹⁵⁵Gd, ¹⁵⁷Gd, ¹⁷⁶Lu, and ¹⁸⁰Ta), which have an (n, γ) resonance at thermal-neutron energies. For these nuclides (which are indicated by the symbol * in Table 4.4.4.1), the scattering lengths and cross sections are strongly energy dependent. The scattering lengths of the resonant rare-earth nuclides have been tabulated as a function of energy by Lynn & Seeger (1990).