The cross sections stored in the evaluated nuclear data file are continuous — energy data, which are converted to multi-group cross sections through energy discretization in formats suitable for most of nuclear design calculation codes, as shown in Fig. 2.5. The number of energy groups (N) is dependent on nuclear design codes and is generally within 50-200.

Multi-group cross sections are defined that integral reaction rates for reac­tions (x = s, a, f,—) of a target nuclide i) should be conserved within the range

9.223500+4 2.330250+2 0

1.935800+8 1.935800+8 0

140 2 0

1.000000-5 0.000000-0 2.530000-2 3.000000+4 2.007500+0 3.250000+4 3.750000+4 1.886600+0 4.000000+4 5.000000+4 1.808200+0 5.500000+4 6.500000+4 1.757500+0 6.999990+4 7.999990+4 1.670400+0 8.499990+4 9.500000+4 1.572400+0 1.000000+5 1.250000+5 1.479300+0 1.375000+5 1.625000+5 1.445300+0 1.750000+5 2.000000+5 1.354000+0 2.250000+5 2.750000+5 1.250200+0 3.000000+5 3.500000+5 1.231800+0 3.750000+5 4.250000+5 1.203400+0 4.500000+5 5.000000+5 1.147100+0 5.250000+5 5.750000+5 1.135500+0 6.000000+5 6.500000+5 1.133440+0 6.750000+5

Energy

Discretization

Fig. 2.5 Multi-group cross section by energy discretization

of energy group g (AEg), as shown in Eq. (2.13). The neutron flux and cross section in group g are defined as Eqs. (2.14) and (2.15), respectively.

/йЕд аІ (Е)ф(Е)сІЕ аІдфк

Фя=/ле, Ф^е^Е

, ІЛЕоІІЕ)ф{Е)йЕ

L*(M)dE

The neutron spectrum of Fig. 2.1 can be used as ф(Е) in Eq. (2.15) for thermal reactors. However, while those multi-group cross sections are intro­duced to calculate neutron flux in reactors, it is apparent that the neutron flux is necessary to define the multi-group cross sections. This is recursive and con­tradictory. For example, the depression of neutron flux due to resonance, shown in the 1/E region of Fig. 2.1, depends on fuel composition (concentrations of resonance nuclides) and temperature (the Doppler effect). It is hard to accu­rately estimate the neutron flux before design calculation. In actual practice, consideration is made for a representative neutron spectrum (фw (E): called the “weighting spectrum”) which is not affected by the resonance causing the depression and is also independent of position. Applying it to Eq. (2.15) gives Eq. (2.16).

(2.16)

Since these microscopic cross sections are not concerned with detailed design specifications of fuel and operating conditions of the reactor, it is possible to prepare them beforehand using the evaluated nuclear data file. The depression or distortion of neutron flux does not occur for resonance nuclides at sufficiently dilute concentrations and this is called the infinite dilution cross section (oiooxg). On the other hand, the cross section prepared in Eq. (2.15) compensates for the neutron spectrum depression due to resonance in the 1/E region by some method, and is called the “effective (microscopic) cross section” (&le/f xg). There are several approximation methods for the effec­tive resonance cross section: NR approximation, WR approximation, NRIM approximation, and IR approximation. For more details, references [1, 6] on reactor physics should be consulted. The relationship between the effective cross section and infinite dilution cross section is simply discussed here.

Since neutron mean free path is long in the resonance energy region, a homogenized mixture of fuel and moderator nuclides, ignoring detailed struc­ture of the materials, can be considered. If a target nuclide (i) in the mixture has a resonance, it leads to a depression in the neutron flux at the corresponding resonance energy as shown in Fig. 2.6. The reaction rate in the energy group including the resonance becomes smaller than that in the case of no flux depression (infinite dilution). Hence, the effective cross section is generally smaller than its infinite dilution cross section. The following definition is the ratio of the effective cross section to the infinite dilution cross section and called the self-shielding factor.

Fig. 2.6 Neutron flux depression in resonance

a о: Intermediate

Small

Depression of ф(Е) with the

Background Cross Section a,

In Fig. 2.6, the neutron flux depression becomes large as the macroscopic cross section (No*) of the target nuclide becomes large. On the other hand, if the macroscopic total cross section {^.NJaj) of other nuclides (assuming that they have no resonance at the same energy) becomes large, the resonance cross section of the target nuclide is buried in the total cross section of other nuclides and then the flux depression finally disappears when the total cross section is large enough. To treat this effect quantitatively, a virtual cross section (the background cross section) of the target nuclide, converted from the macro­scopic total cross section, is defined as Eq. (2.18).

Ы=^NjaCN‘(2 18)

j*i

The self-shielding factor is dependent on the background cross section and the temperature of resonance material because the resonance width varies with the Doppler effect. In fact, the background cross section and material temper­ature are unknown before design specifications and operating conditions of a reactor are determined. Hence, self-shielding factors are calculated in advance and tabulated at combinations of representative background cross sections and temperatures. Figure 2.7 shows a part of the self-shielding factor table (f table) for the capture cross section of 238U. The self-shielding factor in design calculation is interpolated from the f table at a practical background cross section (o0) and temperature (T). The effective microscopic cross section for neutron transport calculation can be obtained from the self-shielding factor as

where T0 is normally 300 K as a reference temperature at which the multi-group infinite dilution cross sections are prepared from the evaluated nuclear data file.

Recent remarkable advances in computers have made it possible that continuous-energy data or their equivalent ultra-fine-groups (groups of several tens of thousands to several millions) are employed in solving the neutron slowing-down equation and the effective cross sections of resonance nuclides are directly calculated based on Eq. (2.15). Some open codes, SRAC [7] (using the PEACO option) and SLAROM-UF [8] for fast reactors, adopt this method. This method can result in a high accuracy for treating even interference effects of multiple resonances which cannot be considered by the interpolation method of a self-shielding factor table, though a long calculation time is required for a complicated system.