Thermal Spectra and Thermal Cross Sections
Although accurate determination of the thermal spectrum also requires advanced computational methods, average oversimplified spectra often serve as a reasonable first approximation in performing rudimentary reactor calculations. The main aspect of nuclear
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Thermal Spectra and Thermal Cross Sections
Although accurate determination of the thermal spectrum also requires advanced computational methods, average oversimplified spectra often serve as a reasonable first approximation in performing rudimentary reactor calculations. The main aspect of nuclear reactor analysis, as we have learned so far, is multigroup diffusion theory. In previous chapters, we developed the general form of multigroup diffusion equations and recommended a strategy for their solution. However, these set of equations contained various parameters known as group constants formally defined as the average over the energy-dependent intergroup flux which must be determined before these equations play formal important roles. In this chapter, we introduce the calculation of neutron energy spectrum characterizing fast neutrons, and as a result, the calculation of fast neutron spectra as well as generation of fast group constants will be of concern. At the conclusion, we will deal with the development of the theory of neutron slowing down and resonance absorption.
9.1
Coupling to Higher Energy Sources
In order to generate group constants for the thermal neutron groups, it is necessary to calculate detailed thermal neutron spectra. However, due to the fact that thermal neutron mean free paths are small, the heterogeneities that are present in typical thermal reactor cores are very significant and must be included in any reactor design calculation. Since we have not talked about heterogeneities yet, we will only treat thermal spectra in a very simplified form. Later in the section on generating cell cross sections, we will deal with the problem directly and calculate thermal group constants. The treatment we will apply here only applies to homogeneous thermal reactors like the AGN-201 reactor. Let us begin with the equation in an infinite medium. The P0 equation for an energy E in the thermal range can be written as
© Springer International Publishing Switzerland 2017 B. Zohuri, Neutronic Analysis For Nuclear Reactor Systems, DOI 10.1007/978-3-319-42964-9_9
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Σ t ðEÞϕ0 ðEÞ ¼
ð Eth
Thermal Spectra and Thermal Cross Sections
Σ s0 ðE0 ! EÞϕ0 ðE0 ÞdE0 þ SðEÞ
ð9:1Þ
0
Equation 9.1 is a significantly reduced version of transport equation that is written in the form of an integral equation in the single variable E, which will refer to as the infinite medium spectrum equation. Note that the abbreviation of th stands for thermal in all equations below. Note that both downscattering and upscattering must be included. Now if we write Σ a ðEÞϕ0 ðEÞ þ Σ s ðEÞϕ0 ðEÞ ¼
ð Eth
Σ s0 ðE0 ! EÞϕ0 ðE0 ÞdE0 þ SðEÞ
ð9:2Þ
0
and integrate the P0 equation over the thermal energy range, we will have ð Eth
Σ a ðEÞϕ0 ðEÞdEþ
0
¼
ð Eth ð Eth 0
ð Eth
Σ a ðEÞϕ0 ðEÞdE
0 0
0
0
Σ s0 ðE ! EÞϕ0 ðE ÞdEdE þ
0
ð Eth
SðEÞdE
ð9:3Þ
0
But noting that for thermal neutrons, the order of integral can be interchanged, therefore, we can write the following result: 0
Σ s ðE Þ ¼
ð Eth
Σ s0 ðE0 ! EÞdE
ð9:4Þ
0
The
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