1. 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
Klausur

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

3. 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

4. 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

5. 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

6. 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

8. 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

9. 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

11. 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

12. 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?

13. Coupling of two angular momenta
j1+ j 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?)

14. 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”.

15. Coupling of two angular momenta

16. 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

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

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

19. 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

20. Pairing: δ-interaction2 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.

21. 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

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

23. 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

24. 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

25. 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

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

27. 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