TEMPERATURE DEPENDENCE OF Lattice heterogeneity is treated with special reference to the NEUTRON CROSS-SECTIONS AND simplifications needed to replace a unit cell by a homogeneous mixture with RESONANCE INTEGRALS, AND SAFETY PROBLEMS an equivalent effective background cross section involving the Dancoff factor. The limitations of using the Bell factor to preserve the one W. ROTHENSTEIN term equivalence relations between homogeneous mixtures and isolated Tedinion, fuel lumps are discussed, as well as the empirical dependence of Haifa, Israel the Bell factor on the Dancoff factor. ABSTRACT Ill) Multigroup spectrum calculations at epithermal energies are discussed mainly with reference to the MUFT and GAM procedures for homogeneous The material covered in the lectures consists of the followin" topics: mixtures. I) A description is given of the bssic nuclear data needed for resonance absorption calculations with particular referenc. to commonly used The direct application of the Nordheim Integral Transport calculation rer.onance formalisms and the resonance parameter files in the basic of effective resonance integrals in the MUFT code is described, data libraries. Doppler Broadening techniques are discussed together together with its artificial separation of the shielding calculation with the construction of detailed temperature dependent resonance near the resonance peaks and the unshielded wing corrections, and cross section tabulations for use in accurate resonance absorption the use of smoothly varying corrections to the simple Breit Wignîr calculations. resonance shapes. A method is described for the modification of the approximate resonance integrals to correlate the resulting epithermal 113 The resonance shielding problem is treated fcr homogeneous mixtures to thermal absorption rates for reactor lattice unit cells with of a resonance absorber in infinite moderating media with hydrogen experiment. moderation only, moderation by a heavier nuclide, and moderation by mixtures of nuclides. The use of resonance shielding factors in the GAM and WIMS multigroup The narrow resonance approximation (NR), the infinite mass epithermal spectrum calculations is discussed. In the former the approximation (IM), and the X parameter of the intermediate resonance shielding factors refer to the group cross sections and group dependent approximation (IR) are defined. lattice disadvantage factors must be applied. The WIMS code uses The construction of group effective resonance integral or shielded cross effective resonance integrals which must be translated into group cross section tabulations is described for individual resonance nuclides sections with due regard to average flux depressions in each resonance mixed homogeneously with moderating materials characterized by a single group. The approximations encountered in defining the effective fuel background cross section. disadvantage factors and flux depressions are emphasised. 207 The method used in the LATREP code for handling fuel clusters is Lecture 1 treated briefly. Introduction IV) Detailed resonance reaction rate calculations in lattice unit cells Direct utilisation-of.basic nuclear data libraries for reactor analysis using Doppler broadened resonance cross section tabulations are is restricted to simple systems or special problems. Among these, Monte Carlo discussed with full allowance for all resonance interference phenomena. calculations for Benchmark Analysis, and the validation of nuclear data and The One dimensional OZMA code whicu treats this problem by integr reactor design cedes , may be mentioned. Such calculations cannot be applied transport or 3 transport theory on an ultrafine energy mesh is to the stjdy of the behaviour of large reactor cores containing fuel assemblies det-cribed with special reference to the rapid calculation of the of different types, undergoing burn-up at different rates, and containing sis'", elastic slowing down sources, as well as P. scattering in the S a variety of structural and control materials. Here core analysis must be option. broken up into several subprograms which lead in the final instance to the calculation of the detailed flux and power distributions in the entire re:ctor The methods of studying the effects of such detailed transport calcu- throughout the complete fuel cycle. lations in the resolved resonance region on standard multigroup reactor analysis are described. Ihe final determination of the flux and power distribution is in itself a problem of considerable complexity requiring two- or three-dimensional cal- Results of lattice analysis studies with accurately calculated reson- culations, usually of the diffusion type. Little emphasis can be placed on the ance reaction rates are given for a number of problems. neutron energy variable in such analyses; the nuclear characteristics which vary from location to location throughout the core must be represented by para- . Temperature variation of effective resonance integrals for isolated meters covering very wide energy bands. Multi-dimensional few group diffusion fuel rods. and transport codes usually handle about two to four energy groups. Doppler reactivity coefficients of typical BWR sad PWR unit cells and their variation with fuel depletion. One is therefore faced with the problem of collapsing the vast amount of . Analysis of some thermal reactor lattice benchmarks information contained in basic nuclear data libraries into very few groups . Analysis of some D.,0 moderated fuel clusters. without losing its essential content. Computer codes which perform this task Comparison of multigroup cross sections, cell fluxes and disad- must be applied on a routine basis and very rapidly, since they have to supply vantage factors calculated for a UO,/K2O lattice benchmark in the the local parameters representing the different parts of the reactor Lore. resolved resonsice region by detailed transport calculations and They must be updated periodically as the fuel gets depleted. by standard very rapid multigrcup methods for identical group reaction rates. Local few group data in the reactor core are averages over regions covered A final discussion follows to indicate how accurate resonance reaction by the mesh cells used in the determination of the flux and power maps for the rate calculations may be used to validate or correct simple resonance entire reactor. They must, in general, eliminate the internal heterogeneity of shielding algorithms. the fuel assemblies, in addition to being few group averages of the basic nuclear data. Thus, the preparation of the broad group data is in itself a procédure requiring space and energy dependent solutions of the neutron transport equation. Fuel assembly of a PWR Even at this level, i.e., for a local solution of the transport equation o o with 35x15 fuel rod unit in a relatively small region, usually with simple gcomptry, direct utilisation o o cells except the loca- of the basic neutron cross section libraries is impracticable. The purpose of o o o o tions indicated by the calculation is to obtain a space energy dependent solution for the neutron circles which contain flux which can be used for Jiomogenisatioi. of a fuel assembly, or part of it, o poison rods or and for collapse of the energy deper-ic 'ce into very few groups. The numerous instrumentation. few group parameters which must be supplied for the different fuel assemblies o o o o in the entire reactor core make it essential to use a multigroup, rather than o o point cross section, procedure to solve the transport equation in each assembly o o and subregion within it. The solutions of the transport equation are usually obtained in about one hundred to two hundred fine energy groups. The hetero- geneity of the fuel assemblies may be simple or more complicated. The basic geometry is the unit cell consisting of fuel, clad and associated coolant or moderator in an infinite regular array of similar cells, the fuel being in the form of i-ods or plates. The arrays may also be more complicated, for example, in fuel assemblies containing some burnable poison or control rods in addition to the fuel rods themselves. In heavy water moderated reactors, the basic cell is a cluster of fuel rods in the coolant, together with the external moderator Moderator outer associated with it. These types of systems, shown in Fig. 1, have a double boundary for one heterogeneity, in the first case for the poison rods interspersed between the, fuel cluster. fuel, in the second for the rods within the cluster, and for the array of complete clusters separated by the moderator. Heterogeneous analysis of fuel assemblies by multigroup methods for the Fuel rod cluster preparation of homogenised few group parameters refers in the first instance in D.O nraderator. to regular arrays of fuel rods or plates in an infinite lattice since unit cells of these geometries can be handled quite simply. In the present course such arrays will be the principal ones discussed, although some attention will also be paid to systems with double heterogeneity. In the unit cell codes used, in practice several tens of fine energy groups cover the thermal energy region from about 10 to about 1 eV, and a similar number of fine groups span the epithermal energy region, in which there is neutron down-scattering only, from abnut 1 eV to about 10 Mi-V. Fig. 1 Assemblies Kith more than one type of heterogeneity 209 The preparation of few group parameters representing the nuclear charac- spectrum at high energies, a 1/1". flux in the slowing down region,