Energy Dependence of Neutron Flux B. Rouben McMaster University Course EP 4D03/6D03 Nuclear Reactor Analysis (Reactor Physics) 2015 Sept.-Dec. 2015 September 1
Contents We start the discussion of the energy dependence of the neutron flux. 2015 September 2
The Energy Variable of the Neutron Flux Except for the discussion of diffusion in 2 energy groups, we have up to now dealt mostly the space dependence of the neutron flux. However, we recall that fission neutrons are born with energies in the ~1-MeV range, and that (in thermal reactors) they are then slowed to thermal energies (a small fraction of 1 ev). The range of neutron energies in the reactor is thus very wide: 6-8 orders of magnitude. cont d 2015 September 3
The Energy Variable in the Neutron Flux (cont d) The exact dependence of the neutron flux on energy is therefore very important. This dependence is especially important to know or calculate in lattice codes, whose function is to solve the neutron-transport problem numerically within the basic lattice cell. In fact, the energy distribution of neutrons is crucial in the appropriate homogenisation and collapsing (by the lattice code) of the cell s nuclear properties for use in diffusion codes in 2 - or at most a few - energy groups. 2015 September 4
Three Energy Ranges We can subdivide the range of neutron energies into three broad regions: The fission-neutron energies (>~0.5 MeV) The thermal range (< ~0.625 ev) The slowing-down range (0.5 MeV - 0.625 ev) [Note that the region boundaries cannot be considered sharp.] Let s look at these 3 regions in turn. 2015 September 5
The Fission-Neutron-Energy Range We have seen the spectrum (energy distribution) of fission neutrons early in the course: the energies range up to several MeV, with a maximum around 0.7 MeV. The fission-neutron spectrum has the form 1.036E E 0.453e sinh 2.29E (1) where E is in MeV (Note: this is a distribution in number of neutrons, not flux) Energy Distribution of Fission Neutrons Note - Illustration copyright: Copyright 1985 by American Nuclear Society, La Grange Park, Illinois 2015 September 6
The Thermal Energy Range In the thermal energy range, the neutrons are in thermal balance with the medium at temperature T. The neutron population then has a Maxwellian (or approximately Maxwellian) distribution: In terms of number of neutrons: n E 2 And in terms of flux: E E e 2 kt kt E kt E 3/ 2 e E kt (3) (2) Room temperature is by convention taken as T = 293.6 K = 20.4 o C, which gives kt = 0.0253 ev (4) kt is the most probable neutron energy in the Maxwellian flux distribution [Eq. (3)], and the corresponding thermal neutron speed is v kt 0.0253eV 2200 m/ s (5) 2015 September 7
Slowing-Down Energy Range The slowing-down energy range is the most complicated, in view of its large size and in view of the very complex scheme of resonances presented to neutrons by heavy elements (mostly). Thus, neutron thermalization, especially when considering neutron absorption in the resonance range in fuel, can be very difficult to calculate, and this is left more and more to numerical computation in complex lattice codes. However, under certain approximations, it is possible to derive an analytic form (proportional to 1/E) for the distribution of neutrons in energy as they slow down in the moderator (with light nuclei). Even though approximate, this neutron slowing-down spectrum is instructive. 2015 September 8
The Slowing-Down Spectrum in Hydrogen It can be shown with a bit of analysis (see Duderstadt & Hamilton, Sections 8.I A &B) that the slowing-down flux (E) in hydrogen is proportional to 1/E below the fission energy E h. S E where S is the fission source and s is the scattering cross section. This provides an important, simple, basic formula for the slowing-down spectrum even if somewhat of an approximation. Another way of interpreting the relationship is that the product E(E) is nearly constant with energy below E h. E 2015 September 9 s
Non-Hydrogen Moderator The analysis of the slowing-down spectrum in a nonhydrogenous moderator is more complicated, but under similar approximations the same form of the slowing-down spectrum can be found: E S E s This has the same form for the slowing-down spectrum, with an additional factor of (the average gain in lethargy per collision) in the denominator. 2015 September 10
Relaxed Approximations By relaxing other approximations, e.g., by no longer assuming a non-negligible absorption cross section, it can be shown that the 1/E spectrum is modified (below the source energies) to: E S E t E exp E' de' E' E' The exponential factor in this equation represents the probability that the neutron survives slowing down to energy E; i.e., it is the resonance-escape probability to energy E, denoted p(e). E h E a t 2015 September 11
E(E) From the general form of the slowing-down spectrum, the following simplified statements on the product E(E) can be deduced: 1) If absorption is neglected, and with the assumption that the scattering cross section does not depend on energy, E(E) is constant (flat) with E. 2) Including absorption, and if the absorption cross section is smooth with E, then E(E) will decrease smoothly for decreasing E. These statements are shown in graphical form on the following slide. 2015 September 12
E(E) with Smooth Absorption 2015 September 13
Resonances (cont.) At the resonance energy, the neutron flux goes through a significant dip on account of the very high absorption cross section. As a result of the dip in flux, the total absorption in the resonance is reduced relative to the background flux value, i.e., the product a is smaller than if the flux were unaffected. This is called resonance self shielding. On account of resonances then, the slowing-down flux will be deformed relative to the smooth curve in the previous slide, as shown in the next figure. 2015 September 14
E(E) with Resonances 2015 September 15
Temperature Dependence of Resonances Resonances are Doppler broadened as temperature increases. This comes about as the result of the greater random motion of nuclei at higher temperatures. Because of this greater random motion, there is a greater possibility for the neutron speed (relative to the nucleus) to correspond exactly to the energy of the resonance. As a result, the resonance broadens (in terms of the neutron energy E), while its peak is reduced: in other words, absorptions in the resonance are smeared (redistributed) over a wider range of neutron energies. cont d 2015 September 16
Temperature Dependence of Resonances (cont.) The actual area under the resonance is not changed by the Doppler broadening. However, since the resonance peak is reduced by the Doppler broadening, the flux dip within the resonance is reduced, and as a result, the self-shielding within the resonance is reduced, i.e., the number of interactions () within the resonance is increased. If the resonance is a capture resonance, the number of captures is increased at higher temperatures. If it s a fission resonance, fissions are increased. In most reactors, capture resonances are more important than fission resonances, and consequently system reactivity is reduced at higher temperatures. 2015 September 17
Lumping of Fuel It was discovered early in the history of reactor technology that criticality was easier to attain if the fuel were lumped (e.g., into channels) rather than homogeneously mixed with the moderator. Can you think of reasons why reactivity is increased when lumping the fuel? Which factors are affected by the lumping, and in what direction? 2015 September 18
Summary on Effects of Resonances Resonance absorption is an important player in the reactivity balance within the neutron cycle. While there are fission resonances, resonance capture dominates in most thermal reactors. In the standard CANDU reactor, resonance capture amounts to about 90 milli-k of negative reactivity. Doppler effects broaden resonances. In most thermal reactors, Doppler broadening results in a negative reactivity component as the fuel temperature increases. 2015 September 19
Flux Spectrum Over Full Energy Range Once the (approximate) slowing-down spectrum is available, we are able to piece together the neutron flux over the energy range from fission energies to the thermal range, using: the fission spectrum at energies above about 50-100 kev the slowing-down spectrum to about 1 ev the Maxwellian spectrum at thermal energies, below about 1 ev [note that in the thermal energy range neutrons can gain as well as lose energy in collisions; to be consistent with the approximation of no upscattering in the derivation of the slowing-down spectrum, the boundary between thermal and epithermal energies should be selected sufficiently high to ensure negligible upscattering from the thermal region to the epithermal: in many applications and computer codes, this boundary is taken as 0.625 ev]. 2015 September 20
Flux Spectrum Over Full Energy Range The piecing together of the neutron spectrum results in the sketch in the next slide. Note that the thermal spectrum is not a perfect Maxwellian (which applies to a gas, without absorption); the Maxwellian is deformed somewhat by neutron absorption. 2015 September 21
Flux Spectrum Over Full Energy Range 2015 September 22
Numerical Spectrum Calculations The analytical derivations above are useful to acquire a general understanding of the dependence of the neutron flux on energy. However, since they do depend on some approximations, the analytical forms may not be sufficiently accurate for precise reactor design and analysis. Therefore, most modern reactor analysis relies on numerical solutions of the neutron-transport equation in the basic lattice, either multigroup calculations using a very large number of energy groups (several dozen to several hundred), or Monte Carlo calculations in continuous energy. 2015 September 23
END 2015 September 24