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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture 97 b. Fast neutrons should be moderated. Fission Barriers Neutron Cross Section (Different Features)

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 2 Neutron Induced Reactions X ( n,b ) Y n ( E n ) b ( Q + E n ) For thermal neutrons Q >> E n b ( Q ) constant Probability to penetrate the potential barrier P o ( E thermal ) = 1 P > o ( E thermal ) = 0 Non-resonant

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 3 Neutron Induced Reactions

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 4 HW 3 Statistical Factor (revisited) Generalization

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 5 Resonance Reactions (revisited) Entrance Channel a + X Exit Channel b + Y Compound Nucleus C* Excited State ExEx J a + X Y + bQ > 0 b + Y X + aQ < 0 Inverse Reaction QM Statistical Factor ( ) Identical particles Nature of force(s). Time-reversal invariance. HW 4 More Generalization

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 6 Projectile Target Q-value Projectile Q-value Target Direct Capture (all energies) Resonant Capture (selected energies with large X-section) E = E + Q - E ex Q + E R = E r Resonance Reactions (revisited)

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 7

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8 Resonance Reactions (revisited) Damped Oscillator eigenfrequency Damping factor Oscillator strength

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 9 Resonance Reactions (revisited) Breit-Wigner formula All quantities in CM system Only for isolated resonances. Reaction Elastic scattering HW 5 HW 5 When does R take its maximum value? Usually a >> b.

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 10 Resonance Reactions J a + J X + l = J (-1) l (J a ) (J X ) = (J) (-1) l = (J) Natural parity. Exit Channel b + Y Compound Nucleus C* Excited State ExEx J Entrance Channel a + X

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 11 Resonance Reactions (revisited) Cross section ECEC a Energy What is the Resonance Strength …? What is its significance? In what units is it measured? Charged particle radiative capture ( a, ) (What about neutrons?)

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 12 Neutron Resonance Reactions (revisited)

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 13 Neutron Activation Analysis ( Z,A ) + n ( Z, A+1 ) - ( Z+1, A+1 ) ( -delayed -ray) Project 1 NAA and U

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 14 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

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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 15 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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 16 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.

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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 Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 17 Neutron Flux and Reaction Rate Reference 2200 m/s flux Independent of n(E). ENDF

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