A major problem in spectrum calculations is the treatment of the neutron slowing down in an heterogeneous reactor system in presence of leakage and anisotropic scattering.

The only rigorous treatment possible is when the flux distribution is of the form (see §4.10 and ref. 1, p. 436)

ф(г, Е,П)= Ф(в, E, £i)e, BZ

where В is the square root of the fundamental buckling. This implies that the space-dependence of the flux is of the type e’BZ for all energies. In the case the B„ approximation can be adopted using the Fourier transform to reduce the spatial dependence to a single buckling constant.

Because of the limited anisotropy of graphite scattering, the Bt approximation is sufficient for HTRs.

As pointed out in the paragraph on the B„ method, the components of the flux can be calculated exactly with this method, the only assumption being the truncation in the expansion of the scattering cross-section. No inverse Fourier transformation is needed as, in the approximation (8.1), the energy dependence of the angular flux is identical to the one of its Fourier transform.

In practice the validity of the B„ approach is limited because the single mode describes an asymptotic solution valid only if the system is homogeneous and large enough for such a distribution to be fully established. This is certainly not true for zones lying on the boundary between the reflector and an undermoderated core. In this case it is better to use the multigroup Pi approximation which allows the use of a group-dependent leakage, while this is not possible in the B„ approximation.

In the GAM-I code, for example,<2) the Pt eqns. (4.22) are integrated over a region of interest obtaining

J(E) + 1,(Е)ф(Е) = j Z,„(E’ -* Е)ф(Е’) dE’ + S(E), 2ф(Е) + 32,(E)J(E) = 3 j 2„(E’ -» E)J(E’) dE’,

where

Г V2cf> dV

EE = ————

)<t>dV

in a multi-group formulation we will then have group dependent leakage factors EE і for each group i. These factors may be obtained from few — or multi-group space-dependent calculations. It is then necessary to iterate between the multi-group spectrum calcula­tion and the space-dependent calculation in order to obtain the leakage factors EEt. An initial guess for the EE. factors is given by —B2, the buckling. If the number of groups used in the space-dependent calculation is sufficiently high, this process converges very rapidly because the multi-group constants are only weakly independent on the EE factors. In small reactors where the leakage has a strong influence on the neutron spectrum the few group cross-sections can depend greatly on the leakage and this iteration may present problems. In that case the number of groups of the space — dependent calculation has to be increased, thus reducing the leakage sensitivity of the cross-sections. Having performed the spectrum calculation one has to produce average constants for few group reactor calculations. These constants are then averaged according to eqns. (4.7) and (4.8).

where ф(Е) is obtained from (8.2). Transfer coefficients

—і———- 1———— !————- 1-

Е,-1 E, Ei-i Ei

For use in a space-dependent code it is usually necessary to form a macroscopic So- including elastic scattering, inelastic scattering and n, 2n reactors,

= 2 NkO-цсі + 2 inei + 2 2 МкО-ц(„,2n) (8.5)

к к к

where Nk is the atomic concentration of isotope к and the summation extends over all isotopes к present in the reactor.

Diffusion coefficient

In order to perform few group diffusion calculations it is necessary to calculate group diffusion coefficients using Fick’s law

HE)

grad ф(Е)

Other versions of the GAM code have been restricted to the B„ method, with all the disadvantages connected with the use of energy-independent bucklings.

The GAM code is only used to calculate the fast spectrum, while the thermal spectrum is calculated separately in the GATHER code.

The sources for the two energy ranges are fission and slowing down respectively and are treated as external sources, so that the problem of criticality is not posed and ксЯ is not calculated. The two codes have then been coupled in the various GGC versions.’2"6’

In the resonance range the 2, of eqn. (8.2) must be obtained from expressions of the type (7.1). See Chapter 7 for the description of the methods used in the various codes.