After studding this lecture, the student should be able to: define main principles of the neutron behavior in thermal range; explain the ideas to compute the thermal neutron flux — Maxwell’s spectrum; define an average cross section, Westcott factors and its dependency on the ambient temperature. The scattering process of neutrons on nuclei in thermal energy range is called Thermalization. Features of the thermal range: the first, the neutron kinetic energy is comparable to the thermal oscillation movement of nuclei — 3/2kT where the K is the Boltzmann constant, T — is the ambient temperature. The second, the neutron’s de Broile wave length of neutron is comparable to the molecule size or the crystal structures. The third for most isotopes the cross section in the thermal range behaves as 1/v Law (it means that absorption cross sections increases proportionally as the velocity of the neutron decreases). So the direct description of the neutrons behavior is too complicated. We have to make an assumption: concerning the neutron to nuclei interactions as the interaction of two single atom gases: the heavy (nuclei of medium) and the light one (neutrons). If we have an infinite homogeneous medium we can form the balance equation for neutron's spectrum in the thermal energy range. The source for the range is slowed down by neutrons with higher energies. In the thermal energy range: the probability density to get certain energy E prime after scattering at the energy E is more complicated. Look at the drawing. If the kinetic energy is much more than the thermal oscillation movement of nuclei, then the probability density is the elastic scattering law. But if the neutron energy is comparable to the thermal oscillation movement of nuclei the following function arisen — it means that the energy after collision can be both above and below of the current energy. In comparison with the slowing down range it is difficult to find directly the neutron spectrum function. For the neutron flux searching of the equilibrium condition method is used. In the framework of the two single atom gases model let’s imagine the following experiment. In a closed volume consisting of non-absorbing medium a number of neutrons is placed there. So there are no absorption processes and no thermal neutron source is present in the medium. In that case thermal equilibrium takes place. That means the neutron energy distribution coincides with the energy distribution of the molecules in a single atom gas which is well known as Maxwell’s spectrum as it is presented here. Where n is the neutron density, m = mass of the neutron, k — the Boltzmann constant. In reality the appearance of the neutron absorption (more for the lower energy than for higher one) and of the neutron thermal source leads to the real neutron spectrum which is very similar to Maxwell’s distribution but the maximum of the distribution is shifted to the right. The temperature linked to the new maximum energy is the so called temperature of the neutron gas. The temperature of the neutron gas depends on the ambient temperature and the ratio of absorption cross section and slowing down density. It can be a bit above the ambient temperature in a moderator and it differs in hundred times for fuel. In the case of a finite medium with dependency of the neutron flux on both the space and energy variables. First of all we will integrate over energy for the whole thermal range and take into consideration new quantities: the average cross section and the thermal flux. Like this. So finally we got the one–speed diffusion equation for the thermal energy, which can be solved as we did in the previous chapters. The ratio of diffusion coefficient and absorption cross section is the diffusion length squared for thermal neutrons. But there is one feature characterizing the thermal energy range I mean the dependency of averaged cross section in the balance equation on the ambient temperature due to the shifting of neutron spectrum by the ambient temperature. Because the cross section in the thermal energy depends as 1/v, there is easy-to-use the representation of the dependency by reference energy — the thermal point, which equals 25 mille eV and is denoted as a quantity with the lower index zero. After changing the ambient temperature the neutron flux shifted too. In turn, it is changing the macroscopic cross sections: skipping the mathematical derivation we can write the final formula, which is presented here. For 1/v absorbers, the absorption rate is independent of the energy distribution. Equivalently, the absorption rate is the same as that for a monoenergetic beam of neutrons of arbitrary energy E0 and intensity v0 × n. The Westcott factor. In the last slide we assumed that the spectrum is maxwellian and the absorbers have 1/v dependency of cross section. But for certain heavy-nuclei there are resonances settled down in the thermal area that break dependence 1/v, it is necessary to apply the following formula. Where g — the Westcott factors which takes into account a deviation of dependence of the effective microscopic section in thermal energy range from dependence 1/v; and takes into account deviation of the real neutron spectrum from Maxwell‘s. For example look the dependency of the Westcott factor on the ambient temperature for 239 Pu. One can see the growth of the Westcott factor of fission and absorption with the increasing of temperature. That means increasing the temperature shifts the spectrum to the right and one goes over the big resonance located at 0.2 eV.
