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, 2007-2008 (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, 2007-2008 (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, 2007-2008 (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, 2007-2008 (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 0.0253 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, 2007-2008 (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, 2007-2008 (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, 2007-2008 (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, 2007-2008 (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, 2007-2008 (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, 2007-2008 (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, 2007-2008 (Saed Dababneh).

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

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). Nuclear Fission Consistent with activation energy curve for A = 300. Extrapolation to 47   10-20 s. Nuclear Reactors, BAU, 1st Semester, 2007-2008 (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, 2007-2008 (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 10-16 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, 2007-2008 (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, 2007-2008 (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, 2007-2008 (Saed Dababneh).

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

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). Nuclear Fission Fissile Q for 235U + n  236U is 6.54478 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. 13.11 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, 2007-2008 (Saed Dababneh).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). Nuclear Fission Why not use it? f,Th 584 2.7x10-6 700 0.019 b Nuclear Reactors, BAU, 1st Semester, 2007-2008 (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, 2007-2008 (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, 2007-2008 (Saed Dababneh).

Nuclear Fission Enge Distribution of fission energy Lost … ! Krane sums them up as  decays. Lost … ! Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

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