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Nuclear Binding Energy Btot(A,Z) = [ ZmH + Nmn - m(A,Z) ] c2 Bm Bave(A,Z) = Btot(A,Z) / A HW 9 Krane 3.9 Atomic masses from: HW 10 Krane 3.12 http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele=&all=all&ascii=ascii&isotype=all Separation Energy Neutron separation energy: (BE of last neutron) Sn = [ m(A-1,Z) + mn – m(A,Z) ] c2 = Btot(A,Z) - Btot(A-1,Z) HW 11 Show that HW 12 Similarly, find Sp and S. Magic HW 13 Krane 3.13 HW 14 Krane 3.14 numbers Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 1 Nuclear Binding Energy Magic numbers Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 2 Nuclear Binding Energy In general XY+a Sa(X) = (ma + mY –mX) c2 = BX –BY –Ba The energy needed to remove a nucleon from a nucleus ~ 8 MeV average binding energy per nucleon (Exceptions???). Mass spectroscopy B. Nuclear reactions S. Nuclear reactions Q-value Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 3 Nuclear Binding Energy Surface effect Coulomb effect ~200 MeV HWc 4 Think of a computer program to reproduce this graph. Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 4 Nuclear Binding Energy HW 15 A typical research reactor has power on the order of 10 MW. a) Estimate the number of 235U fission events that occur in the reactor per second. b) Estimate the fuel-burning rate in g/s. Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 5 Nuclear Binding Energy Is the nucleon bounded equally to every other nucleon? C ≡ this presumed binding energy. Btot = C(A-1) A ½ Bave = ½ C(A-1) Linear ??!!! Directly proportional ??!!! Clearly wrong … ! wrong assumption finite range of strong force, and force saturation. Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 6 For constant Z Sn (even N) > Sn (odd N) For constant N Sp (even Z) > Sp (odd Z) Remember HW 14 (Krane 3.14). 208Pb (doubly magic) can then easily remove the “extra” neutron in 209Pb. Neutron Separation Energy Sn (MeV) Nuclear Binding Energy Lead isotopes Z = 82 Neutron Number N Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 7 Nuclear Binding Energy Extra Binding between pairs of “identical” nucleons in the same state (Pauli … !) Stability (e.g. -particle, N=2, Z=2). Sn (A, Z, even N) – Sn (A-1, Z, N-1) This is the neutron pairing energy. even-even more stable than even-odd or odd-even and these are more tightly bound than odd-odd nuclei. Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 8 Abundance Systematics Odd N HWc 1\ Even N Total Odd Z Even Z Total Compare: • even Z to odd Z. • even N to odd N. • even A to odd A. • even-even to even-odd to odd-even to odd-odd. Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 9 Neutron Excess Z Vs N (For Stable Isotopes) 90 Remember HWc 1. 80 70 Z=N 60 Z 50 40 30 20 Odd A 10 Even A 0 0 20 40 60 N 80 Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 100 120 140 10 Neutron Excess Remember HWc 1. Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 11 Abundance Systematics Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 12 ABUNDANCE Formation process Abundance NEUTRON CAPTURE CROSS SECTION Abundance Systematics NEUTRON NUMBER r s r s MASS NUMBER Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 13 Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 14 The Semi-empirical Mass Formula • von Weizsäcker in 1935. • Liquid drop. Shell structure. • Main assumptions: 1. Incompressible matter of the nucleus R A⅓. 2. Nuclear force saturates. • Binding energy is the sum of terms: 1. Volume term. 2. Surface term. 3. Coulomb term. ….. Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 4. Asymmetry term. 5. Pairing term. 6. Closed shell term. 15 The Semi-empirical Mass Formula Volume Term Bv = + av A Bv volume R3 A Bv / A is a constant i.e. number of neighbors of each nucleon is independent of the overall size of the nucleus. BV constant A The other terms are “corrections” to this term. Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 16 The Semi-empirical Mass Formula Surface Term Bs = - as A⅔ • Binding energy of inner nucleons is higher than that at the surface. • Light nuclei contain larger number (per total) at the surface. • At the surface there are: 4r A ro2 2 0 2 3 4A 2 3 Nucleons. Bs 1 1 A A3 Remember t/R A-1/3 Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 17 The Semi-empirical Mass Formula Coulomb Term BC = - aC Z(Z-1) / A⅓ • Charge density Z / R3. • W 2 R5. Why ??? • W Z2 / R. • Actually: W Z(Z-1) / R. • BC / A = - aC Z(Z-1) / A4/3 4r 2 dr 4 3 r 3 Remember HW 8 … ?! Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 18 The Semi-empirical Mass Formula Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 19 The Semi-empirical Mass Formula Quiz 1 From our information so far we can write: M ( A, Z ) AM n Z ( M n M H ) aV A a S A 2 3 aC Z ( Z 1) A 1 3 ... For A = 125, what value of Z makes M(A,Z) a minimum? Is this reasonable…??? So …..!!!! Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh). 20