Nuclear Reactors, BAU, 1st 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

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 2 Recall F t = n v  t 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 Not talking about a beam anymore. same energy

Nuclear Reactors, BAU, 1st 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

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

Thermal Reactorsabsorption Maxwellian 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, 1st Semester, (Saed Dababneh). 5 Neutron Flux and Reaction Rate Reference 2200 m/s flux Independent of n(E).

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, 1st Semester, (Saed Dababneh). Neutron Moderation Reference Average decrease in ln(E) after one collision. 1 H ?

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. On 12 C. Most probable and average energies?

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, 1st Semester, (Saed Dababneh).

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

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, 1st Semester, (Saed Dababneh).

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 is called activation energy. 11Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

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

Nuclear Fission Surface Term B s = - a s A ⅔ Coulomb Term B C = - a C Z(Z-1) / A ⅓ = Volume Term (the same)  fission  Crude: QM and original shape could be different from spherical. 13Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

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

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

Nuclear Fission Z 1 + Z 2 = 92 Z 1  37, Z 2  55 A 1  95, A 2  140 Large neutron excess Most stable: Z=45Z=58  Prompt neutrons Prompt neutrons within s. Number depends on nature of fragments and on incident neutron energy. The average number is characteristic of the process. 16Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

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

18 Why only left side of the mass parabola?

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. 19Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). Waste. Poison. In general,  decay favors high energy.

Nuclear Fission 20Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

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 21Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

22 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, 1st Semester, (Saed Dababneh).

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