Experimental evidence for closed nuclear shells

1. Experimental evidence for closed nuclear shells Deviations from Bethe-Weizsäcker mass formula: Neutron Proton 28 28 50 50 B/A (MeV per nucleon) 82 very stable: 126 82 mass number A

2. Shell structure from masses • Deviations from Weizsäcker mass formula:

3. Energy required to remove two neutrons from nuclei(2-neutron binding energies = 2-neutron “separation” energies) 25 23 21 19 17 S(2n) MeV 15 13 Sm 11 Hf 9 Ba Pb 7 Sn 5 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 Neutron Number N = 82 N = 126 N = 84

4. Shell structure from Ex(21) and B(E2;2+→0+) • high energy of first 2+ states • low reduced transition probabilities B(E2)

5. The three faces of the shell model

6. Average nuclear potential well: Woods-Saxon

7. Woods-Saxon potential • Woods-Saxon gives proper magic numbers (2, 8, 20, 28, 50, 82, 126) • Meyer und Jensen (1949):strong spin-orbit interaction Spin-orbit term has its origin in the relativistic description of the single-particle motion in the nucleus.

8. Woods-Saxon potential (jj-coupling) The nuclear potential with the spin-orbit term is spin-orbit interaction leads to a large splitting for large ℓ.

9. Woods-Saxon potential • The spin-orbit term • reduces the energy of states with spin oriented parallel to the orbital angular momentum j = ℓ+1/2 (Intruder states) • reproduces the magic numbers large energy gaps → very stable nuclei • Important consequences: • Reduced orbitals from higher lying N+1 shell have different parities than orbitals from the N shell • Strong interaction preserves their parity. The reduced orbitals • with different parity are rather pure states and do not mix • within the shell.

10. Shell model – mass dependence of single-particle energies • Mass dependence of the neutron energies: • Number of neutrons in each level:

11. ½ Nobel price in physics 1963: The nuclear shell model

12. Experimental single-particle energies 208Pb → 209Bi Elab = 5 MeV/u 1 i13/2 1609 keV 2 f7/2 896 keV 1 h9/2 0 keV γ-spectrum single-particle energies

13. Experimental single-particle energies 208Pb → 207Pb Elab = 5 MeV/u γ-spectrum single-hole energies 3 p3/2 898 keV 2 f5/2 570 keV 3 p1/2 0 keV

14. Experimental single-particle energies particle states 1 i13/2 209Pb 209Bi 1609 keV 2 f7/2 896 keV 1 h9/2 0 keV energy of shell closure: 207Tl 207Pb hole states proton

15. Level scheme of 210Pb 2846 keV 2202 keV 1558 keV 1423 keV 779 keV 0.0 keV -1304 keV (pairing energy) M. Rejmund Z.Phys. A359 (1997), 243

16. Level scheme of 206Hg 2345 keV 1348 keV 997 keV 0.0 keV B. Fornal et al., Phys.Rev.Lett. 87 (2001) 212501

17. Success of the extreme single-particle model • Ground state spin and parity: Every orbit has 2j+1 magnetic sub-states, fully occupied orbitals have spin J=0, they do not contribute to the nuclear spin. For a nucleus with one nucleon outside a completely occupied orbit the nuclear spin is given by the single nucleon. n ℓ j → J (-)ℓ = π

18. Success of the extreme single-particle model • magnetic moments: The g-factor gj is given by: with Simple relation for the g-factor of single-particle states

19. Success of the extreme single-particle model • magnetic moments: • g-faktor of nucleons: proton: gℓ = 1; gs = +5.585 neutron: gℓ = 0; gs = -3.82 proton: neutron:

20. Magnetic moments: Schmidt lines magnetic moments: proton magnetic moments: neutron