Surface effect Coulomb effect ~200 MeV 1
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: 2 (46) V = 1.44MeV. fm 250MeV > 214MeV 12.2 fm Crude! What if 79 Zn and 159 Sm?? Large neutron excess, released neutrons, sharp potential edge, spherical U!! 2
238 U (t ½ = 4.5x10 9 y) for α-decay. 238 U (t 16 ½ 10 y) for spontaneous 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. 3
MeV) Energy ( Activation E Liquid Drop Shell 4
4 πr 3 3 = 4 2 3 πab 1 a Z ) 5 C > Nuclear Fission 3 2 R = ab Volume Term (the same) a b = R(1 + ε ) 2 Surface Term B s = - a s A ⅔ 2 (1 + ε + 5...) Coulomb Term B C = - a C Z(Z-1) / A ⅓ 2 (1 ε 1 2 3 2 3 ( Z 1 A a A Z 2 A 5 > ~ 47 S fission = R 1+ ε 1 + 5...) Crude: QM and original shape could be different from spherical. 5
(120) 300 2 = 48 Consistent with activation energy curve for A = 300. Extrapolation to 47 10-20 s. 6
235 U + n 93 Rb + 141 Cs + 2n Not unique. Low-energy fission processes. 7
Z 1 + Z 2 = 92 Z 1 37, Z 2 55 A 1 95, A 2 140 Large neutron excess Nuclear Fission Most stable: Z=45 Z=58 Prompt neutrons within 10-16 s. Number ν depends on nature of fragments and on incident neutron energy. The average number is characteristic of the process. 8
The average number of neutrons is different, but the distribution is Gaussian. 9
ν neutrons emitted per fission. ν depends on fissioning nuclide and on neutron energy inducing i fission. India? 10
Why only left side of the mass parabola? 11
Higher than S n? Delayed neutrons In general, β decay favors high energy. ~ 1 delayed neutron per 100 fissions, i but essential for control of the reactor. Waste. Poison. 12
13
1/v Fast neutrons should be moderated. d 235 U thermal cross sections σ fission 584 b. σ scattering 9 b. σ radiative capture 97 b. Fission Barriers 14
Q for 235 U + n 236 U is 6.54478 MeV. Table 13.11 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: δ =??? (Fig. 13.1111 in Krane). What about 232 Pa and 231 Pa? (odd Z). Odd-N nuclei have in general much larger thermal fission cross sections than even-n nuclei (Table 13.1 in Krane). 15
Why not use it? σ f,th 584 2.7x10-6 700 0.019 b 16
Thermal neutron fission of 235 U forms compound nucleus that splits up in more than 40 different ways, yielding over 80 primary fission fragments (products). 235 92 U + 1 0n 90 37Rb + 144 55Cs + 2 1 0n 235 92 U + 1 0n 87 35Br + 146 57La + 3 1 0n 235 92 U + 1 0n 72 30Zn + 160 62Sm + 4 1 0n The fission yield is defined as the proportion (percentage) of the total nuclear fissions that form products of a given mass number. Nuclear Reactors, Theory, BAU, BAU, 1 st Semester, Second Semester, 2007-20082011-2012 (Saed Dababneh). 17
Remember neutron excess. (A,Z) (A,Z+1) or(a1 (A-1,Z). Only left side of the mass parabola. 18
235 U + n 93 Rb + 141 Cs + 2n Q =???? What if other fragments? Different number of neutrons. Take 200 MeV as a representative value. 165 MeV average kinetic energy carried by fission fragments per fission. 66 MeV 98 MeV Heavy fragments Light fragments miscalibrated 19
Mean neutron energy 2 MeV. 2.5 neutrons per fission (average) 5 MeV average kinetic energy carried by prompt neutrons per fission. Nuclear 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. E1 m2 95 E 66 E m 98 140 2 1 20
HW 7 1.036E χ( E ) = 0.453 e sinh 2. 29 E The experimental spectrum of prompt neutrons is fitted by the above equation. Calculate the mean and the most probable neutron energies. 21
The fission gamma radiation Prompt with average energy of 0.9 MeV. β delayed gammas. HW 8 Investigate how prompt gammas interact with water, uranium and lead. Nuclear Reactors, Theory, BAU, BAU, 1 st Semester, Second Semester, 2007-20082011-2012 (Saed Dababneh). 22