Neutron Scattering
The neutron energy spectrum in a reactor lies between the extremes of fission and thermal equilibrium. It is determined largely by the competition between scattering and absorption reactions. For neutrons with energies significantly above the thermal range, a scattering collision results in degradation of the neutron energy, whereas neutrons near thermal equilibrium may either gain or lose energy in interacting with the thermal motions of the nuclei of the surrounding media. Energy degradation caused by scattering is referred to as neutron slowing down. In a medium for which the average energy loss per collision and the ratio of scattering to absorption cross section is large, the neutron spectrum will be close to thermal equilibrium and is then referred to as a soft or thermal spectrum. Conversely, in a system with small ratios of neutron degradation to absorption, neutrons are absorbed before significant slowing down takes place. The neutron spectrum then lies closer to the fission spectrum and is said to be hard or fast. To gain a more quantitative understanding of neutron energy distributions we consider first elastic and then inelastic scattering. Recall that in elastic scattering mechanical energy is conserved, that is, the sums of the kinetic energies of the neutron and the target nucleus are the same before and after the collision. In inelastic scattering the neutron leaves the target nucleus in an excited— that is, more energetic—state. Thus the sum of the neutron and nucleus kinetic energies following the collision is less than before by the amount of energy deposited to form the excited state. Both elastic and inelastic scattering are of considerable importance in nuclear reactors. We treat elastic scattering first.
Elastic Scattering
For simplicity we first consider the head-on collision between a neutron with speed v and a stationary nucleus of atomic mass A. If we take m as the neutron mass then the nuclear mass will be approximately Am. If v' and V are the neutron and nucleus speeds after the collision then conservation of momentum yields
mv= mv' +(Am)V, (2.44)
while from conservation of mechanical energy
Letting E and E' be the neutron energy before and after the collision, we may solve these equations to show that the ratio of neutron energies is
Clearly the largest neutron energy losses result from collisions with light nuclei. A neutron may lose all of its energy in a collision with a hydrogen nucleus, but at the other extreme, it can lose no more than 2% of its energy as the result of an elastic collision with uranium-238. Of course, head-on collisions cause the maximum neutron energy loss, although in reality most neutrons will make glancing collisions in which they are deflected and lose a smaller part of their energy. If elastic scattering is analyzed not in the laboratory but in the center of mass system as a collision between two spheres, all deflection angles are equally likely, and the scattering is said to be isotropic in the center of mass system (or often just isotropic). Detailed analyses found in more advanced texts result in a probability distribution for neutron energies following a collision. Suppose a neutron scatters elastically at energy E. Then the probability that its energy following collision will be between E' and E' + dE' will be
Often the need arises to combine the probability distribution for scattered neutrons with the scattering cross section. We then define
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Inelastic Scattering
The situation for inelastic scattering is quite different. Elastic scattering cross sections are significant over the entire energy range of neutrons. But whereas low atomic weight nuclei cause large energy losses for elastic scattering, heavy isotopes do not, and so the effects of their elastic scattering on reactor physics are small.Conversely, as discussed earlier, only neutrons with energies above a threshold that is a characteristic of the target isotope can scatter inelastically.Moreover, these thresholds are low enough for significant inelastic scattering to occur only for the heavier atomic weight materials, such as uranium. Inelastic scattering causes neutrons to lose substantial energy. The unique structure of energy levels that characterizes each nuclide, such as those illustrated in Fig. 2.5, determines the energies of inelastically scattered neutrons. To scatter inelastically the neutron must elevate the target nucleus to one of these states, from which it decays by emitting one or more gamma rays. The threshold for inelastic scattering is determined by the energy of the lowest excited state of the target nucleus, whereas the neutron’s energy loss is determined predominantly by the energy level of the state that it excites. For
example, if the neutron energy E is greater than the first three energy
levels E1, E2, or E3, then following the inelastic scatter the neutron
would have energy E' =E-E1, E-E2, or E-E3. This is illustrated in
Fig. 2.11. The peaks, however, are slightly smeared over energy, since
as in elastic scattering conservation of momentum requires that a
neutron deflected through a larger angle will lose more energy than
one deflected through a smaller angle. As the energy of the incident
neutron increases, the spectrum of scattered neutrons can become
quite complex if many states can be excited by the neutron’s energy.
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Elastic Scattering
For simplicity we first consider the head-on collision between a neutron with speed v and a stationary nucleus of atomic mass A. If we take m as the neutron mass then the nuclear mass will be approximately Am. If v' and V are the neutron and nucleus speeds after the collision then conservation of momentum yields
mv= mv' +(Am)V, (2.44)
while from conservation of mechanical energy
Letting E and E' be the neutron energy before and after the collision, we may solve these equations to show that the ratio of neutron energies is
Clearly the largest neutron energy losses result from collisions with light nuclei. A neutron may lose all of its energy in a collision with a hydrogen nucleus, but at the other extreme, it can lose no more than 2% of its energy as the result of an elastic collision with uranium-238. Of course, head-on collisions cause the maximum neutron energy loss, although in reality most neutrons will make glancing collisions in which they are deflected and lose a smaller part of their energy. If elastic scattering is analyzed not in the laboratory but in the center of mass system as a collision between two spheres, all deflection angles are equally likely, and the scattering is said to be isotropic in the center of mass system (or often just isotropic). Detailed analyses found in more advanced texts result in a probability distribution for neutron energies following a collision. Suppose a neutron scatters elastically at energy E. Then the probability that its energy following collision will be between E' and E' + dE' will be
Often the need arises to combine the probability distribution for scattered neutrons with the scattering cross section. We then define
Click Here to read more.....
Inelastic Scattering
The situation for inelastic scattering is quite different. Elastic scattering cross sections are significant over the entire energy range of neutrons. But whereas low atomic weight nuclei cause large energy losses for elastic scattering, heavy isotopes do not, and so the effects of their elastic scattering on reactor physics are small.Conversely, as discussed earlier, only neutrons with energies above a threshold that is a characteristic of the target isotope can scatter inelastically.Moreover, these thresholds are low enough for significant inelastic scattering to occur only for the heavier atomic weight materials, such as uranium. Inelastic scattering causes neutrons to lose substantial energy. The unique structure of energy levels that characterizes each nuclide, such as those illustrated in Fig. 2.5, determines the energies of inelastically scattered neutrons. To scatter inelastically the neutron must elevate the target nucleus to one of these states, from which it decays by emitting one or more gamma rays. The threshold for inelastic scattering is determined by the energy of the lowest excited state of the target nucleus, whereas the neutron’s energy loss is determined predominantly by the energy level of the state that it excites. For
example, if the neutron energy E is greater than the first three energy
levels E1, E2, or E3, then following the inelastic scatter the neutron
would have energy E' =E-E1, E-E2, or E-E3. This is illustrated in
Fig. 2.11. The peaks, however, are slightly smeared over energy, since
as in elastic scattering conservation of momentum requires that a
neutron deflected through a larger angle will lose more energy than
one deflected through a smaller angle. As the energy of the incident
neutron increases, the spectrum of scattered neutrons can become
quite complex if many states can be excited by the neutron’s energy.
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