MP-41 Teil 2: Physik exotischer Kerne 13.4. Einführung, Beschleuniger 20.4. Schwerionenreaktionen, Synthese superschwerer Kerne (SHE) 27.4. Kernspaltung und Produktion neutronenreicher Kerne 4.5. Fragmentation zur Erzeugung exotischer Kerne 11.5. Halo-Kerne, gebundener Betazerfall, 2-Protonenzerfall 18.5. Wechselwirkung mit Materie, Detektoren 25.5. Schalenmodell 1.6. Restwechselwirkung, Seniority 8.6. Tutorium-1 15.6. Tutorium-2 22.6. Vibrator, Rotator, Symmetrien 29.6. Schalenstruktur fernab der Stabilität 6.7. Tutorium-3 13.7. Klausur

Shell structure Experimental evidence for magic numbers close to stability Maria Goeppert-Mayer J. Hans D. Jensen

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

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

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

Level scheme of 210Pb 2846 keV 2202 keV 1558 keV 1423 keV 779 keV exp. single particle energies 1423 keV 779 keV 0.0 keV -1304 keV (pairing energy) residual interaction ! M. Rejmund Z.Phys. A359 (1997), 243

The 100Sn/132Sn region, a brief background 0.5 1.6 2.2 2.6 MeV d5/2 Single particle energies N=82 Z = 50 g7/2 d5/2 s1/2 d3/2 h11/2 Naïve single particle filling

The 100Sn/132Sn region, isomeric states d3/2 h11/2 0.5 1.6 2.2 2.6 MeV d5/2 Single particle energies N=82

Shell Model with residual interactions – mostly 2-particle systems Start with 2-particle system, that is a nucleus „doubly magic + 2“ Consider two identical valence nucleons with j1 and j2 Enormous simplifications of shell model calculations, reduction to 2-body matrix elements Energies of single magic nuclei Behaviour of g-factors g(41Ca)= g(43Ca)=g(45Ca)=g(47Ca) Parabolic systematics of intra-band B(E2) values and peaking near mid-shell Preponderance of prolate shapes at beginnings of shells and of oblate shapes near shell ends

Shell Model with residual interactions – mostly 2-particle systems Start with 2-particle system, that is a nucleus „doubly magic + 2“ Consider two identical valence nucleons with j1 and j2 Two questions: What total angular momenta j1 + j2 = J can be formed? What are the energies of states with these J values?

Coupling of two angular momenta j1+ j2 all values from: j1 – j2 to j1+ j2 (j1 = j2) Example: j1 = 3, j2 = 5: J = 2, 3, 4, 5, 6, 7, 8 BUT: For j1 = j2: J = 0, 2, 4, 6, … ( 2j – 1) (Why these?)

Several methods: easiest is the “m-scheme”. How can we know which total J values are obtained for the coupling of two identical nucleons in the same orbit with total angular momentum j? Several methods: easiest is the “m-scheme”.

Coupling of two angular momenta

Residual interaction - pairing Spectrum 210Pb: Assume pairing interaction in a single-j shell energy eigenvalue is none-zero for the ground state; all nucleons paired (ν=0) and spin J=0. The δ-interaction yields a simple geometrical expression for the coupling of two particles 2 4 6 8

Pairing: δ-interaction wave function: interaction: with and A. de-Shalit & I. Talmi: Nuclear Shell Theory, p.200

Pairing: δ-interaction wave function: interaction: with and A. de-Shalit & I. Talmi: Nuclear Shell Theory, p.200

δ-interaction (semiclassical concept) q for and θ = 00 belongs to large J, θ = 1800 belongs to small J example h11/22: J=0 θ=1800, J=2 θ~1590, J=4 θ~1370, J=6 θ~1140, J=8 θ~870, J=10 θ~490

Pairing: δ-interaction 2 4 6 8 δ-interaction yields a simple geometrical explanation for Seniority-Isomers: DE ~ -Vo·Fr· tan (q/2) for T=1, even J energy intervals between states 0+, 2+, 4+, ...(2j-1)+ decrease with increasing spin.

Generalized seniority scheme 0.5 1.6 2.2 2.6 MeV d5/2 Single particle energies N=82 Z = 50 g7/2 The 100Sn / 132Sn region d5/2 s1/2 d3/2 h11/2 Naïve single particle filling

Generalized seniority scheme 0.5 1.6 2.2 2.6 MeV d5/2 Single particle energies N=82 The 100Sn / 132Sn region

Generalized seniority scheme Seniority quantum number ν is equal to the number of unpaired particles in the jn configuration, where n is the number of valence nucleons. energy spacing between ν=2 and ground state (ν=0, J=0): independent of n energy spacing within ν=2 states: independent of n G. Racah et al., Phys. Rev. 61 (1942), 186 and Phys. Rev. 63 (1943), 367

Generalized seniority scheme Seniority quantum number ν is equal to the number of unpaired particles in the jn configuration, where n is the number of valence nucleons. E2 transition rates: for large n ≈ Nparticles*Nholes Sn isotopes

Generalized seniority scheme Seniority quantum number ν is equal to the number of unpaired particles in the jn configuration, where n is the number of valence nucleons. ≈ Nparticles*Nholes ≈ Nparticles*Nholes number of nucleons between shell closures

Signatures near closed shells Excitation energy Sn isotopes N=82 isotones N=50 isotones

Generalized seniority scheme Seniority quantum number ν is equal to the number of unpaired particles in the jn configuration, where n is the number of valence nucleons. E2 transition rates that do not change seniority (ν=2): Sn isotopes