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Nuclear Binding Energy

Nuclear Binding Energy. B tot (A,Z) = [ Zm H + Nm n - m(A,Z) ] c 2 B m B ave (A,Z) = B tot (A,Z) / A HW 8 Krane 3.9 Atomic masses from: http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele=&all=all&ascii=ascii&isotype=all Separation Energy

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Nuclear Binding Energy

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  1. Nuclear Binding Energy Btot(A,Z) = [ ZmH + Nmn - m(A,Z) ] c2 Bm Bave(A,Z) = Btot(A,Z) / A HW 8Krane 3.9 Atomic masses from: 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 9Show that HW 10 Similarly, find Sp and S. HW 11 Krane 3.13 HW 12 Krane 3.14 Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  2. Nuclear Binding Energy In general X  Y + 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, 1st Semester, 2006-2007 (Saed Dababneh).

  3. Nuclear Binding Energy Surface effect Coulomb effect ~200 MeV  Fission Fusion  Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  4. Nuclear Binding Energy • HW 13 • A typical research reactor has power on the order of 10 MW. • 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, 1st Semester, 2006-2007 (Saed Dababneh).

  5. Nuclear Binding Energy Is the nucleon bounded equally to every other nucleon? C ≡ this presumed binding energy. Btot = ½ CA(A-1) 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, 1st Semester, 2006-2007 (Saed Dababneh).

  6. Nuclear Binding Energy Lead isotopes Z = 82 For constant Z Sn (even N) > Sn (odd N) For constant N Sp (even Z) > Sp (odd Z) Remember HW 12 (Krane 3.14). 208Pb (doubly magic)  can then easily remove the “extra” neutron in 209Pb. 208Pb Neutron Separation Energy Sn (MeV) Neutron Number N Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (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, 1st Semester, 2006-2007 (Saed Dababneh).

  8. Neutron Excess Remember HWc 1. Asymmetry Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  9. Abundance Systematics HWc 1\ • 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, 1st Semester, 2006-2007 (Saed Dababneh).

  10. Abundance Systematics Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  11. Abundance Systematics NEUTRON CAPTURE CROSS SECTION Formation process  Abundance NEUTRON NUMBER ABUNDANCE r s r s MASS NUMBER Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  12. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  13. The Semi-empirical Mass Formula • von Weizsäcker in 1935. • Liquid drop. • Main assumptions: • Incompressible matter of the nucleus  R  A⅓. • Nuclear force saturates. • Binding energy is the sum of terms: • Volume term. 4. Asymmetry term. • Surface term. 5. Pairing term. • Coulomb term. 6. Closed shell term. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  14. 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. constant The other terms are “corrections” to this term. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  15. 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: Nucleons. Remember t/R A-1/3 Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  16. 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  Remember HW 7 … ?! Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  17. The Semi-empirical Mass Formula Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  18. The Semi-empirical Mass Formula HW 14 Show that from our information so far we can write: For A = 125, what value of Z makes M(A,Z) a minimum? Is this reasonable…??? So …..!!!! Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  19. The Semi-empirical Mass Formula Asymmetry Term Ba = - aa (A-2Z)2 / A • Light nuclei: N = Z = A/2 (preferable). • Deviation from this “symmetry”  less BE and stability. • Neutron excess (N-Z) is necessary for heavier nuclei. • Fraction affected = |N-Z| / A • Total decrease in BE  fraction x excess. • Ba / A = - aa (N-Z)2 / A2. • Back to this when we talk about • the shell model. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  20. The Semi-empirical Mass Formula Pairing Term Bp =  Extra Binding between pairs of identical nucleons in the same state (Pauli !)  Stability (e.g. -particle, N=2, Z=2). even-even more stable than even-odd or odd-even and these are more tightly bound than odd-odd nuclei. Remember HWc 1\ ….?! Bp expected to decrease with A; effect of unpaired nucleon decrease with total number of nucleons. But empirical evidence show that:   A-¾ . • Effect on: • Fission. • Magnetic moment. • Effect of high angular momentum. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  21. The Semi-empirical Mass Formula Closed Shell Term Bshell =  • Extra binding energy for magic numbers of N and Z. • Shell model. • 1 – 2 MeV more binding. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

  22. The Semi-empirical Mass Formula • Fitting to experimental data. • More than one set of constants av, as ….. • In what constants does r0 appear? • Accuracy to ~ 1% of experimental values (BE). • Atomic masses 1 part in 104. • Uncertainties at magic numbers. • Additional term for deformation. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

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