equation (see §4.11). Integral transport theory is also used in the General Atomic MICROX code<10> which solves the neutron slowing down and thermalization equations in 200 energy groups for a two-region lattice cell. The fluxes in the two regions are coupled by collision probabilities computed with flat flux approximation. Grain structure of the fuel is taken into account with the method developed by Walti (see §7.13).

The leakage in this code is treated with a diffusion formulation derived from an approximation of the В, equations. The term

which appears in the Вi equations [see eqn. (4.50)] is substituted by

where фітсі(Е) is a fixed reference current spectrum (since in the Pi approximation the ф, component can be interpreted as a neutron current). With this assumption it is possible to eliminate ф, and have an equation for ф0 where the leakage term can be expressed in the form О(Е)В2ф0(Е) (see ref. 10, pp. 35-39). In this way the Bt equations are transformed into the multi-group diffusion equation and the transport cross-section is calculated weighting the scattering matrices on the reference current spectrum. It must be noted that this approximation is rather similar to the one made for obtaining the multi-group diffusion equation from the Pi equations. The use of energy-dependent bucklings becomes then possible. As expected this approximation is rather crude for hydrogen but sufficient for heavy nuclides. As in the case of the MUPO code this allows a much faster calculation. The term D(E)B2(E) refers to the cell averaged leakage and is added to the removal cross-sections of each region treated in this code.

Another peculiarity of the MICROX code is the replacement of the elastic and inelastic slowing-down kernels of each individual nuclide by synthetic scattering kernels of mathematically more tractable form. This permits a very rapid evaluation of the slowing-down integrals. The scattering kernel 2s(u->u’) is expressed as

2s(m -» и’) = ^ Nkak, s(u)g(u -> u’)

where и = lethargy, Nk = atomic density of nuclide k, ak, s = scattering cross-section of isotope k, and the distributions gk are defined by

gk(u -» u’) = #(u’ — u)R'[Zkpke-pt, u’-u>

where #(x) is the Heaviside function (= 1 for x > 0; =0 for x < 0). The parameters Zk and pk that characterize the function gk are complex numbers chosen in such a way as to conserve the number of neutrons in a scattering collision, to give the correct mean lethargy gain per collision, and to have the minimum mean square deviation between the synthetic and the actual kernel (for details see ref. 10).

Another method of performing spectrum calculations in heterogeneous cells is given by the WIMS code. The last versions of WIMS have actually evolved to a modular system including almost all types of neutronics design calculations, but we will only deal here with the spectrum calculations."11

The programme includes a sixty-nine group library. Between 4 eV and 9.118 keV are thirteen resonance groups whose cross-sections are calculated by means of the equivalence theorem (see § 7.11) using a library of resonance integrals. This library of resonance integrals is obtained by solving the slowing-down equations for a homoge­neous mixture of moderator and resonance absorber with the SDR code in some 12,000 energy mesh points. This sixty-nine group library is then condensed to fewer groups using the SPECTROX technique developed by Leslie for the calculations of heavy water systems."21 In this technique each of the principal regions of the lattice are coupled by means of collision probabilities.

The collision probability equations can be modified to account for flux shape in large moderator regions supposing that the flux rise in the moderator at any energy is proportional to the current leaving the moderator at that energy, the constant of proportionality being taken from one-group diffusion theory. After having condensed the data to a lower number of groups the WIMS code can perform a cell transport calculation with either collision probability or S„ methods.

A unique feature of WIMS is the possibility of expressing the effect of the environment on the neutron spectrum of the region considered through collision probabilities (MULTICELL approximation)."31 The multi-cell model describes the core of a reactor in terms of a number of different cell types, each being weighted in proportion to its frequency of occurrence in the system. The cells are effectively combined for a single collision probability transport solution by specifying the matrix of probabilities that a neutron leaving each cell type will enter a cell of the same or any other type."41

The simplest assumption is that these probabilities are directly proportional to the contiguous surfaces between the cells. The multi-cell WIMS method assumes that the emission density is uniform in space and angle across the region associated with each cell type. The spatial distribution is, however, often significantly non-uniform when regions are more than one or two mean free paths in thickness. For these situations a coarse mesh correction has been developed"51 in terms of a multiplicative factor on the surface-to-surface probabilities. The simplest correction factor is a constant factor /3 (=0.7 say) to reduce the overall leakage."61 Improved values of /3 can be obtained by an iterative process with a space-dependent few-group diffusion calculation for the whole reactor. This can form a possible alternative to the buckling iteration. The advantage of the multi-cell approach is that very large meshes can be used in regions of constant spectrum, allowing a finer mesh in the regions where spectrum variations are important.

A multi-cell layout for an HTR core is represented in Fig. 8.1.031

Heterogeneous cell calculations are performed in France with the APOLLO code"71 which calculates the space and energy-dependent flux for any one-dimensional medium with collision probability methods. Various options are possible. Collision probabilities can be calculated with the flat flux assumption in each region, or in large media the flux can be considered to vary linearly within each region.

Anisotropic collision probabilities can be used to treat linear anisotropic scattering. The multi-cell formulation of WIMS has been included in APOLLO to provide a simplified treatment of two-dimensional geometries. Two libraries are available with 186 or 99 groups.