# I0 I Probability Neutron Attenuation (revisited) X Recall t = N t

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I0 I Probability Neutron Attenuation (revisited) X Recall t = N t
mfp for scattering ls = 1/Ss mfp for absorption la = 1/Sa total mfp lt = 1/St Probability per unit path length. Probability Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Reaction Rate Rt  Ft = t  =  /t (=nvNt)
Neutron Flux and Reaction Rate Recall Ft = t I N = I t Simultaneous beams, different intensities, same energy. Ft = t (IA + IB + IC + …) = t (nA + nB + nC + …)v In a reactor, if neutrons are moving in all directions n = nA + nB + nC + … Ft = t nv neutron flux  = nv Reaction Rate Rt  Ft = t  =  /t (=nvNt) Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

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

Neutron Flux and Reaction Rate
In general, neutron flux depends on: Neutron energy, E. Neutron angular direction, W. 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. Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

What if not? Neutron Flux and Reaction Rate
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. Reference What if not? Factor 2200 m/s flux Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Neutron Moderation 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: After n s-wave collisions: where the average change in lethargy is HW 6 Collision Parameter Reference Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Neutron Moderation HW 6 (continued) Reproduce the plot. Discuss the effect of the thermal motion of the moderator atoms. Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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: 1H 2H 4He graphite 238U ? What is special about 1H? Why we considered elastic scattering? When does inelastic scattering become important? Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission Surface effect Coulomb effect ~200 MeV  Fission Fusion  Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission B.E. per nucleon for 238U (BEU) and 119Pd (BEPd) ? 2x119xBEPd – 238xBEU = ??  K.E. of the fragments   1011 J/g Burning coal  105 J/g Why not spontaneous? Two 119Pd fragments just touching  The Coulomb “barrier” is: Crude …! What if 79Zn and 159Sm? Large neutron excess, released neutrons, sharp potential edge, spherical U…! Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission 238U (t½ = 4.5x109 y) for -decay. 238U (t½  1016 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. Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission Liquid Drop Shell Activation Energy (MeV) Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission = Volume Term (the same) Surface Term Bs = - as A⅔ Coulomb Term BC = - aC Z(Z-1) / A⅓  fission Crude: QM and original shape could be different from spherical. Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission Consistent with activation energy curve for A = 300. Extrapolation to 47   s. Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission 235U + n 93Rb + 141Cs + 2n Not unique. Low-energy fission processes. Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission Z1 + Z2 = 92 Z1  37, Z2  55 A1  95, A2  140 Large neutron excess Most stable: Z=45 Z=58 Prompt neutrons within s. Number  depends on nature of fragments and on incident particle energy. The average number is characteristic of the process. Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission The average number of neutrons is different, but the distribution is Gaussian. Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Higher than Sn? Delayed neutrons ~ 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. Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission 1/v Fast neutrons should be moderated. 235U thermal cross sections fission  584 b. scattering  9 b. radiative capture  97 b. Fission Barriers Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission Fissile Q for 235U + n  236U is MeV. Table 13.1 in Krane: Activation energy EA for 236U  6.2 MeV (Liquid drop + shell)  235U can be fissioned with zero-energy neutrons. Q for 238U + n  239U is 4.??? MeV. EA for 239U  6.6 MeV  MeV neutrons are needed. Pairing term:  = ??? (Fig in Krane). What about 232Pa and 231Pa? (odd Z). Odd-N nuclei have in general much larger thermal neutron cross sections than even-N nuclei (Table 13.1 in Krane). Fissionable Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission Why not use it? f,Th x b Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission 235U + n  93Rb + 141Cs + 2n Q = ???? What if other fragments? Different number of neutrons. Take 200 MeV as a representative value. 66 MeV 98 MeV Light fragments Heavy fragments miscalibrated Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
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, 1st Semester, (Saed Dababneh).

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

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).
Nuclear Fission Segrè Lost … ! Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

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