Again we take the limit a0-> 1 of the first integral of eqn. (7.13) while the second and third integrals are treated with the NR approximation obtaining

7.3. Computer methods for resonance calculations in heterogeneous geometry

Various computer codes have been developed for detailed resonance-absorption calculations. Equation (7.14) is numerically solved in the General Atomic ZUT code<5) which is incorporated in the GGC-IV and GGC-V’6,7’ codes for neutron spectra calculation. The ZUT code deals only with resolved resonances while the unresolved energy range is covered by the TUZ code’51 (see § 7.12). The more sophisticated General Atomic GAROL code’8’ considers also two space regions but takes into account overlapping and mutual shadowing of resonances solving the following equations in any specified number of energy points:

VoX, o<f>o = (l-Po)Vo2j^-f ‘фо(Е’) dE’

і = 1 1 QJ і J E С

M ж t <*E/a( / T? t

+ P, V, 2 t—^ Ф.(Я’) dE’ + V0(l — Po)Qo(E) + ViP, Q,(E)

(7.17)

and another just like it but with subscripts 1 and 0 reversed. Nki is the atomic density of isotope і in region k, asi(E’) is the microscopic scattering cross-section of isotope i, Qk(E) is the source in region к, M is the total number of isotopes considered.

The collision probability method is also used in the General Atomic MICROX code<9) which solves the neutron slowing-down equations on a detailed energy grid for a two-region lattice cell.

In both GAROL and MICROX the detailed shape of the Doppler broadened cross-sections is stored on data tapes in an ultrafine grid of discrete energy mesh points covering the resolved resonance energy range in 9000-15,000 points. Because of storage limitation this treatment has to be limited to the most important nuclides.

Other codes are not limited to two regions. Even if physically only two regions are usually present in HTRs the use of more regions allows a better approximation because collision probabilities are usually calculated assuming flat sources in each region.

For example, the Ispra collision probability code PETARD’10’ can be used in typical cases with 2200 groups, 15 regions, 2 compositions and 50 nuclides. The French RACOLE code allows up to 50 concentrical regions.’10 Four regions are used in the UKAEA SDR code with a total of 120,000 energy points.“2)