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

Shell structure from masses Deviations from Weizsäcker mass formula:

Energy required to remove two neutrons from nuclei (2-neutron binding energies = 2-neutron “separation” energies) N = 82 N = 84 N = 126

Shell structure from masses Neutron-separation energy:

Shell structure from masses Neutron-separation energy: N=81 N=83 N=82 N=84

Shell structure from E x (2 1 ) and B(E2;2 + →0 + )  high energy of first 2 + states  low reduced transition probabilities B(E2)

Shell structure from E x (2 1 ) High E x (2 1 ) indicates stable shell structure:

Shell structure from β -decay Plot of the β-transition energy for nuclei in the region 28≤Z≤64 which have the same neutron excess and which undergo the dacy process with Z and N even.

The three faces of the shell model

Average nuclear potential well: Woods-Saxon

The radial potential for nucleon-nucleon interactions m(π) ≈ 140 MeV/c 2 m(σ) ≈ MeV/c 2 m(ω) ≈ 784 MeV/c 2 Yukawa Potential: repulsive core ω-exchange Pauli principle: long range part 1π-exchange medium range part σ-exchange

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.

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.

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.

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

½ Nobel price in physics 1963: The nuclear shell model

Single-particle energies Single-particle states observed in odd-A nuclei (in particular, one nucleon + doubly magic nuclei like 4 He, 16 O, 40 Ca) characterizes single-particle energies of the shell-model picture.

Experimental single-particle energies 208 Pb → 209 Bi E lab = 5 MeV/u 1 h 9/2 2 f 7/2 1 i 13/ keV 896 keV 0 keV γ-spectrum single-particle energies

Experimental single-particle energies 208 Pb → 207 Pb E lab = 5 MeV/u γ-spectrum single-hole energies 3 p 1/2 2 f 5/2 3 p 3/2 898 keV 570 keV 0 keV

Experimental single-particle energies 209 Pb 209 Bi 207 Pb 207 Tl energy of shell closure: 1 h 9/2 2 f 7/2 1 i 13/ keV 896 keV 0 keV particle states hole states proton

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

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

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

Success of the extreme single-particle model  magnetic moments: The g-factor g j is given by: with Simple relation for the g-factor of single-particle states

Success of the extreme single-particle model  magnetic moments:  g-faktor of nucleons: proton: g ℓ = 1; g s = neutron: g ℓ = 0; g s = proton: neutron:

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