1. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (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

2. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (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

3. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (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

4. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (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.

5. 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 0.0253 eV. Look for Thermal Cross Sections. Actually, look for evaluated nuclear data. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh). 5 Neutron Flux and Reaction Rate Reference 2200 m/s flux Independent of n(E).

6. 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, 2008-2009 (Saed Dababneh). Neutron Moderation Reference Average decrease in ln(E) after one collision. 1 H ?

7. Nuclear Reactors, BAU, 1st Semester, 2008-2009 (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?

8. 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, 2008-2009 (Saed Dababneh).

9. Nuclear Fission ~200 MeV  Fission Fusion  Coulomb effectSurface effect 9Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

10. 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   10 11 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, 2008-2009 (Saed Dababneh).

11. Nuclear Fission 238 U (t ½ = 4.5x10 9 y) for  -decay. 238 U (t ½  10 16 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, 2008-2009 (Saed Dababneh).

12. Nuclear Fission Liquid Drop Shell Activation Energy (MeV) 12Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

13. 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, 2008-2009 (Saed Dababneh).

14. Nuclear Fission Extrapolation to 47   10 -20 s. Consistent with activation energy curve for A = 300. 14Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

15. Nuclear Fission 235 U + n  93 Rb + 141 Cs + 2 n Not unique. Low-energy fission processes. 15Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

16. 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 10 -16 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, 2008-2009 (Saed Dababneh).

17. Nuclear Fission The average number of neutrons is different, but the distribution is Gaussian. 17Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

18. 18 Why only left side of the mass parabola?

19. 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, 2008-2009 (Saed Dababneh). Waste. Poison. In general,  decay favors high energy.

20. Nuclear Fission 20Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

21. 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, 2008-2009 (Saed Dababneh).

22. 22 Nuclear Fission Q for 235 U + n  236 U is 6.54478 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. 13.11 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, 2008-2009 (Saed Dababneh).

23. 23 Nuclear Fission  f,Th 5842.7x10 -6 7000.019 b Why not use it?