AE-145 UDC 681.177 621.039.538 LLJ A User's Manual for the NRN Shield Design Method L. iniarn I * •• e AKTIEBOLAGET ATOMENERG STOCKHOLM, SWEDEN 1964 AE-145 A USER'S MANUAL FOR THE NRN SHIELD DESIGN METHOD Editor: Leif Hjärne Authors: E Aalto, R Fräki, L Hjärne, M Leimdörfer, K Lindblom S Linde, K Målen, K Nyman Summary: This report describes a code system for bulk shield design written for a Ferranti Mercury computer and is intended as a ma- nual for those using the programme. The idea of an "almost direct" flux, as in the removal theory serves as a basis for further deve- lopment of the theory. An important aspiration has been to mini- mize the manual work of administering the codes. The codes con- cerned are: NECO, computing necessary group constants from pri- mary data, REFUSE and REBOX (infinite plane or cylindrical, and box geometry, respectively), computing removal flux, NEDI a one- dimensional (plane, spherical, cylindrical) diffusion multigroup code, and SALOME a Monte Carlo code computing the gamma flux. Output tapes are constructed for direct use as input tapes, when required, for a following code. Printed and distributed in June 1964 Contents DESCRIPTION OF PRINCIPLES 1. Introduction 1 2. Cross-sections 1 3. The deep penetration of neutrons 5 4. Neutron diffusion and slowing down 9 5. Gamma penetration and energy deposition 11 DATA PREPARATION 6. General instructions 13 7. Data for NE CO 14 8. a. Data for REFUSE 26 8.b. Data for REBOX 31 9. Data for NEDI 33 10. Data for SALOME 44 OPERATION INSTRUCTIONS 11. Operation of NEC O 47 12. a. Operation of REFUSE . 49 12.b. Operation of REBOX 49 13. Operation of NEDI 50 14. Operation of SALOME 50 15. Error prints 52 LAY-OUTS OF DATA 16. Lay-out of data for NECO 56 1 7. a. Lay-out of data for REFUSE 60 17.b. Lay-out of data for REBOX 60 18. Lay-out of data for NEDI 64 19. Lay-out of data for SALOME 69 APPENDICES Appendix A 72 Appendix B. Tables 103 References 107 DESCRIPTION OF PRINCIPLES ]_, Introduction The diffusion theory, based on the so called diffusion approxima- tion, is accurate only far away from sources and material surfaces. It has,, however, certain advantages over the strict transport theory, i.e. its relative simplicity and its applicability to geometrically more complicated problems. For deep penetration problems, however, diffusion theory requires some complementary information about the neutrons coming directly from the sources at a comparably high energy (about 1 Mev). This is because of the decreasing cross-sections and the increasing scattering anisotropy with higher energies. The methods used hitherto for neutron penetration computa- tions in large geometries are impaired by some serious defects, essentially inherent in the semi-empirical character of the removal theory. The method which will be described here is a further develop- ment of the course suggested by the removal theory, but more strictly based on theoretical elements. In this report we shall describe a code system for shielding purposes.An important aspiration has been to minimize the manual work of administering the codes. The codes concerned are: NECO, computing necessary group constants from primary data, REFUSE and RE BOX (infinite plane or cylindrical, and box geometry, respec- tively), computing removal flux, NEDI a one-dimensional (plane, spherical, cylindrical)diffusion multigroup code, and SALOME, a Monte Carlo code computing the gamma flux. Output tapes are con- structed for direct use as input tapes, when required, for a following code. 2. Cross-sections In solving neutron attenuation problems one needs a rather detailed knowledge of elementary data, such as cross-sections as functions of energy, from which the constants occurring in the transport formalism can be computed. This work can be accurately executed by NECO, a code which reads raw data from a tape library, containing experimental and theoretical information which can easily be kept up to date. The output tapes from NECO can be used directly as data tapes for the following codes. - 2 - In the data library for NECO the cross-sections are organized from the pqint of view of neutron transport. Therefore all reactions of the type (n, y) (n,p) (n,OJ) must be looked upon as absorptions. The angular distribution of elastic scattering is very important owing to its effect on deep penetration and the energy degradation. In the data library these distributions are given in the form of Legendre poly- nomial expansions: 0(E;li) = (2 I + l) ^ (E) P fti) (2. with the first coefficient normalized f = 1 o Taking data from various compilations one must keep in mind that the Legendre coefficients, £„ , can be defined in a number of ways, al- though lately this form (eq. 2. l) has predominated. Directly from the experimental data often some kind of fitting by least squares must be accomplished. For this purpose the code COFFEE (ref. 1) for the IBM 7090 has been developed in cooperation with FOA (Res. Inst. Natl. Def.)- Inelastic scattering is treated in different ways in different energy regions. In the energy region where the energy levels are known (generally below 4-5 Mev) we give cross-sections for excita- tion of each level. At higher energies the level density grows very fast, and we use an evaporation model to describe the secondary energy distribution. Let E be the primary energy E1 the secondary energy. The distribution is then described by: E' E' e + -2" e (2.2) T n2n nn1 T~ The "temperatures" T are defined according to - 3 - B is the (n, 2n) threshold. The constants a. and a? can be deter- mined from experiments, but very roughly they are one tenth of the atomic weight of the residual nucleus (ref. 2), Inelastic scattering is assumed to be isotrbpic. The group constants are defined as usual as weighted averages over the groups. The weight function must be the expected form of the neutron flux spectrum within the group. This spectrum can be de- scribed in an approximate way as varying as E' & for the diffusion group constants. If the a> are not given, they are all set equal to - 1, m.aking collision density constant. For the removal group con- stants the weight function is the source spectrum, a Watt type fission spectrum: E sin h yE • (2.3) being a normalization constant. The two constants P and y a.xe for u ] .036 and 2.29, respectively, for Pu 7 1.0 and 2.0 and for Pu 1 . 0 and 2. 2 (see table II in app. B). Thermal cross-sections are treated in a special way. For the library we need elastic and capture cross-sections at 2200 m/sec (0.0253 ev). The elastic cross-section is assumed to be constant the capture cross-section to vary as (Energy) ' . Now the thermal neutron spectrum is assumed to be a Maxwellian spectrum with temperature T , which usually can be set to some fifty degrees above the temperature of the medium: E E " kT~ ,- ^ _ e (2.4) (kT)2 Observe that T is given in centrigrades for NECO, not as the absolute temperature as in (2.4). In the data for the programme a maximum step length in lethargy (Au) for the group averaging integrations must be speci- fied for each group g . The programme selects the "basic step length" (Au) slightly below the given value so that it goes an inte- gral number of times into the group width. Close to the group boun- dary certain functions can vary quickly with the energy: - 4 2 (E) = / 2 (E-»E') dE' (2.5) v f v x ; s,g ' J s ' E g S^ (E) .= 2(E) - J 2s(E-*E') dE' (2.6) E • g This applies also, for example, to (E-»E«) dE' (2.7) The variation is particularly rapid in the event of strong an- isotropy in heavy nuclei. To obtain reasonable accuracy with a simple integration formula, the basic step lowest in energy in each group is divided into seven "whole integration steps" (ÅU)1 .These are, in order from the group boundary, u Au Au Vi k >g 54 < >g lz ^>g T6 < 'g i Mg In the remainder of the energy group a !'whole integration step" is equal to the "basic step length". In the integration the flux per lethargy unit is regarded as constant over a "whole integration step" and cross- sections and Legendre coefficients over a "basic step length". The functions are integrated as step functions or with Simpson's formula except as regards the distribution of the secondary neutrons from the inelastic scattering process amongst discrete levels, where an analytical approach is used. The remarks above concerning step length etc. apply likewise to the integration over primary groups. The distribution functions are integrated analytically over the secondary groups. - 5 3. The deep penetration of neutrons The treatment of neutron transport is divided into two stages. The first stage is the direct flux attenuation which is of a simple ex- ponential type, with its sources situated within the core. Our "direct flux" is not exactly uncollided, as a small energy loss and small angle scattering do not necessarily remove a neutron from the direct flux, for which reason we shall retain the expression "removal flux". The theory underlying this method is presented in ref. 3. The relation to the old removal theory can be illustrated by fig. 1 which demonstrates the organization of the rectangular removal matrix of cross-sections. R R D ~Fixed energy boundary» —• -s boundary of tfree choice.