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On the Calculation of the Fast Fission Factor AE-27 On the Calculation of the w < Fast Fission Factor B. Almgren AKTIEBOLAGET ATOMENERGI STOCKHOLM • SWEDEN • I960 AE-27 ON THE CALCULATION OF THE FAST FISSION FACTOR E. Almgren Summary: Definitions of the fast fission factor e ars discussed. Different methods of calculation of e are compared. Group constants for one -, two- and three- group calculations have been evaluated using the best obtainable basic data. The effects of back-scattering, coupling and (n, 2n)-reactions are discussed. Completion of manuscript in June 1960 Printed and distributed in November 1960 LIST OF CONTENTS Page 1. Definitions 3 2. Formulas for the fast fission factor e and the fast fission ratio R 3 3. Calculation of group constants 7 4. Collision probabilities 10 5. Numerical results 12 6. Discussion 12 7. Acknowledgements 13 Preferences 14 On the calculation of the fast fission factor. 1. Definitions The fast fission factor e may be defined in different ways. Carlvik and Pershagen (3) have defined e as "the number of neutrons which are either slowed down below 0, 1 MeV in the fuel or leave the fuel, per primary neu- tron from thermal fission". This definition has been used in Sweden, since it was proposed (1956). Another commonly used definition is due to Spinrad (2) "the number of neutrons making first collision with moderator per neutron arising from thermal fission". The choice of definition must be consistent with the definition of the resonance escape probability. The above-mentioned definitions include in e some of the capture below the fast fission threshold. However, they do not include the effects of (n, 2n)-reactions or capture of high energy neutrons in the moderator. A definition which includes these effects is the following "the number of neutrons that are slowed down past a given energy per primary fission neutron". In this case € could be called the fast multiplication factor. Different energies can be chosen as boundary energy. However, the capture below the fission threshold could, be included in the resonance escape proba- bility. If so the definition of e is identical with that of Glas stone-Edlund (1) p. 276. 2. Formulas for the fast fission factor e and the fast fission ratio R The following general expressions for R and s which include moderator effects above the boundary energy can be used for different definitions of e and different number of groups. e = no + > N. a.n + ) M. b.A 0 ' iL x i0 L'-~ !•i0 i= 1 i= 1 (2) N. = n. P. + P. > a:. N. + (1 - Q.) ) b.. M. (3) M. = n. (1 - P.) + (1 - P.) > a.. N. + Q. > b.. M. (4) where n. is tke probability for a fission neutron to be born in group j ' "> N. and M. are the numbers of collisions per unit time in group i in the fuel element and the moderator respectively, if the number of ne-atrons per ".nit time emitted from thermal is taken to be one a., and b.. are the numbers of neutrons that are transferred to group j altar a collision in group i in the fuel element and the moderator respec- tively • • P. is the probability for a fission neutron that has not collided and belonging to group j to make a. collision in the fuel P. is the probability for a neutron having collided in the fuel and be- longing to grou.p j to make its next collision in the fuel Q. is the probability for a neutron having collided in the moderator and belonging to group j to make its next collision in the moderator v is the number of fission neutrons pex* thermal fission cr is the fission cross-section in group j «r. is the total cross-section in group j The following assximptions are made, when the above formulas are derived. 1) The form of the fission spectrum is the same for fast and thermal fission 2) The fuel element and the moderator can be treated as homogeneous medias. Additional assumptions are needed for the calculation of the group constants. Usually moderator effects such as back-scattering and (n, 2h)~reac- tions are neglected. This can be done here by putting Q. = 1 andS b.. = 1, J J Then the formulas are identical with Carlvik-Pershagen's (3), if the neutron groups are: Group 0 Neutrons with energies below 0, 1 MeV 11 1 " " " between 0, 1 - 15 49 MeV " 2 " " " above 1, 49 MeV They are identical with Glasstone-Edlund's formula (1), if the neutron groups are: Group 0 Neutrons .with energies below the fast fission threshold " 2 Neutrons with energies above the fast fission threshold They are identical with Spinrad's formula (2), if n« = 0 arid a._ = 0 and the neutron groups are: Group 1 Fission neutrons with energies above the fast fission threshold " 2, Fission neutrons with energies below the fast fission threshold " 3 Neutrons which have been scattered .out of group 1 The general multigroup formulas are difficult to handle without fast computers. Simplified formulas for one group are therefore given below. The group is called group 2 to get the nomenclature to agree with that of the group constants later in this report. Glass tone -Edlund's definition of e is -ased. 6. P2(l - 6 = nO + n2 a20 (1 - a22 P2) (1 - b,, Q2) - a,, b22 (P2 - 1) (Q2 - 1) (5) 1 P t ~ 2> - a22 P2 n2 b20 (1 - a?? P ) (1 - h?? Q ) - a b (P - 1) {Q - l) Q2) - b22 (1 - P2) (QQ2 - - b 22 Q 22 <P2 If we put P? = P_ and use Q ~ 1 and that the neutron multiplication in the moderator has a small importance (b77 + b_n), we get nZ P2 (a22 + a20 b22 + b?,0 1 " a22 P2 a22 P2 1 ~ b22 b (1 - P ) (1 - Q ) <P2 22 " a22>. P b (1 - P) (1 - Q-) (P + 1 - a P) R = n The first term in equation 7 comes from fast fission in the fuel ele- ment and it is the usual term found for instance by Glasctone-Sdlund (1). The second term comes from (n, 2n)-reactions and capture in the modera- tor. The third term represents the effect of the neutrons that are scattered back into the fuel elements after collisions in the moderator. 3. Calculation of group constants Formulas for a., and b.. The transfer coefficients a., and b.. can be expressed in the following way cr.. + n. v. o- + 2 m.. cr. (n, 2n) a.. - where the cross-sections of the fuel element are used cr.. + 2 m.. cr. (n, 2n) b..=-J yJ (10) where the cross-sections of the moderator are used. Nomenclatur e ij is the average transfer cross-section for a ne^^tron colliding in group i to be transferred to group j cr. is the average capture cross-section for group i <r.f " " " fission " " " " i ar.f _ » " " " (n, 2n) " " " " i i(n, 2n) v ' <s. it ii total cross-section for group i S. " " " 'macroscopic cross-section for group i v. " " average number of fission neutrons produced by a fission in group i m.. is the probability for a neutron from a (n, 2n)-reaktion in group i to be emitted in group j 8. Neutron spectrum Unfortunately the energy spectrum of fast neutrons in a reactor is badly known. For high energies the fission spectrum is a rather good approxi- mation, but for energies below 1 MeV it is certainly a bad one. It is, however, assumed that the fission spectrum can be used when the average cross-sections are calculated. The following fission spectrum given by (4) has been used: N(E) = 0, 45270 exp ( - Q ^5) sinh J 2, 29E (11) Number of fission neLxtrons per fast fission According to (17) vZ38 (£) = 2, 65 + 0, 141 (E-l, 5) (12) The average over the fissions spectrum for energies above 1, 5 MeV is v = 2, 86. This may be compared with the value recommended in (19) 2, 80 + 0, 05 if vf'35 = 2, 43. Calculation of the transfer cross-sections The cross-sections have been taken from references 5, 6, 7, 8, 9 and 18. The cross-sections of the different references agree quite well. We have looked for newer cross-section data in Nuclear Science Abstracts 1958 and 1959, but none of importance for this work has been found. When calculating the transfer probabilities of elastically scattered neutrons, anisotropic scattering has been considered for the light elements D, C and O. The trans- fer function then is • • 00 ) (2L + 1) fL (E) P 1 -± (1 -£-) (13) where fT (IC) are the coefficients of the expansion /._,% CO f E F cos e 14 cr(E, cos 6) = ^ \ (2 L + 1) L( ) Lj ( ) ( ) 9. and P, are Legendre polynomials. For inelastic scattering the transfer function is V f.n(E5E-) = | E' e (15) has been used for U, Zr, Al and at high energies for C. The constants a have been taken from ref. (10) and (11). The transfer cross-sections are then of the type fN(E){<ro1 (S)[ff (E.E')dE'T + o-. (S) f Cf R E') dE'l} dE " <r.. = : (i6) j N(E) dS The cross-sections have been calculated for the following energy groups: - Group 1 Neutrons with energies between 0, 03 and 1, 5 MeV 11 2 " " - . .» M 1} 5 ii ,6j 5 Mey- "3' " " " above 6, 5 MeV • " - 2 " "• ' '' " • " -.1, 5 MeV :" •. The small contribution to the fast-fission cross:-section for energies below 1, 5 MeV has been included in the cross-sections cr andcr^.
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