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Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh). 1 Neutron Attenuation (revisited) Recall t = N t Probability per unit path length. X I0I0 I Probability mfp for scattering s = 1/ s mfp for absorption a = 1/ a total mfp t = 1/ t

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Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh). 2 Recall F t = t I N = I t same energy Simultaneous beams, different intensities, same energy. F t = t (I A + I B + I C + …) = t (n A + n B + n C + …)v reactorall directions In a reactor, if neutrons are moving in all directions n = n A + n B + n C + … F t = t nv neutron flux = nv Reaction Rate R t F t = t = / t (=nvN t ) Neutron Flux and Reaction Rate

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Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh). 3 Different energies Density of neutrons with energy between E and E+dE n(E)dE Reaction rate for those monoenergetic neutrons dR t = t (E) n(E)dE v(E) Neutron Flux and Reaction Rate

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Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh). 4 Neutron Flux and Reaction Rate In general, neutron flux depends on: Neutron energy, E. Neutron angular direction, Neutron spatial position, r. Time, t. Various kinds of neutron fluxes (depending on the degree of detail needed). Time-dependent and time-independent angular neutron flux.

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Thermal Reactorsabsorption In Thermal Reactors, the absorption rate in a medium of thermal (Maxwellian) neutrons Usually 1/v cross section, thus then The reference energy is chosen at eV. Look for Thermal Cross Sections. Actually, look for evaluated nuclear data. Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh). 5 Neutron Flux and Reaction Rate Reference 2200 m/s flux

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elastic Show that, after elastic scattering the ratio between the final neutron energy E \ and its initial energy E is given by: For a head-on collision: s -wave After n s -wave collisions: lethargy where the average change in lethargy is HW 6 6Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh). Neutron Moderation Reference

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Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 7 Neutron Moderation HW 6 (continued) Reproduce the plot. Discuss the effect of the thermal motion of the moderator atoms.

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Neutron Moderation HW 6 (continued) Neutron scattering by light nuclei then the average energy loss and the average fractional energy loss How many collisions are needed to thermalize a 2 MeV neutron if the moderator was: 1 H 2 H 4 Hegraphite 238 U? What is special about 1 H? Why we considered elastic scattering? When does inelastic scattering become important? 8Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission ~200 MeV Fission Fusion Coulomb effectSurface effect 9Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission B.E. per nucleon for 238 U (BE U ) and 119 Pd (BE Pd ) ? 2x119xBE Pd – 238xBE U = ?? K.E. of the fragments J/g Burning coal 10 5 J/g Why not spontaneous? Two 119 Pd fragments just touching The Coulomb barrier is: Crude …! What if 79 Zn and 159 Sm ? Large neutron excess, released neutrons, sharp potential edge, spherical U …! 10Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission 238 U (t ½ = 4.5x10 9 y) for -decay. 238 U (t ½ y) for fission. Heavier nuclei?? Energy absorption from a neutron (for example) could form an intermediate state probably above barrier induced fission. Height of barrier above g.s. is called activation energy. 11Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission Liquid Drop Shell Activation Energy (MeV) 12Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission Surface Term B s = - a s A Coulomb Term B C = - a C Z(Z-1) / A = Volume Term (the same) fission Crude: QM and original shape could be different from spherical. 13Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission Extrapolation to s. Consistent with activation energy curve for A = Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission 235 U + n 93 Rb Cs + 2 n Not unique. Low-energy fission processes. 15Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission Z 1 + Z 2 = 92 Z 1 37, Z 2 55 A 1 95, A Large neutron excess Most stable: Z=45Z=58 Prompt neutrons Prompt neutrons within s. Number depends on nature of fragments and on incident particle energy. The average number is characteristic of the process. 16Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission The average number of neutrons is different, but the distribution is Gaussian. 17Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Delayed neutrons Higher than S n ? ~ 1 delayed neutron per 100 fissions, but essential for control of the reactor. Follow -decay and find the most long-lived isotope (waste) in this case. 18Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission 19Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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Nuclear Fission 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture 97 b. Fast neutrons should be moderated. Fission Barriers 20Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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21 Nuclear Fission Q for 235 U + n 236 U is MeV. Table 13.1 in Krane: Activation energy E A for 236 U 6.2 MeV (Liquid drop + shell) 235 U can be fissioned with zero-energy neutrons. Q for 238 U + n 239 U is 4.??? MeV. E A for 239 U 6.6 MeV MeV neutrons are needed. Pairing term: = ??? (Fig in Krane). What about 232 Pa and 231 Pa ? (odd Z). Odd-N nuclei have in general much larger thermal neutron cross sections than even-N nuclei (Table 13.1 in Krane). Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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22 Nuclear Fission f,Th x b Why not use it?

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23 Nuclear Fission 235 U + n 93 Rb Cs + 2 n Q = ???? What if other fragments? Different number of neutrons. Take 200 MeV as a representative value. 66 MeV98 MeV miscalibrated Heavy fragments Light fragments Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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24 Nuclear Fission Mean neutron energy 2 MeV. 2.4 neutrons per fission (average) 5 MeV average kinetic energy carried by prompt neutrons per fission. Show that the average momentum carried by a neutron is only 1.5 % that carried by a fragment. Thus neglecting neutron momenta, show that the ratio between kinetic energies of the two fragments is the inverse of the ratio of their masses. Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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25 Nuclear Fission Distribution of fission energy Krane sums them up as decays. Lost … ! Enge Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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26 Nuclear Fission Segrè Lost … ! Nuclear Reactors, BAU, 1 st Semester, (Saed Dababneh).

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