1. Shell Model based deformation analysis of light Cadmium isotopes T. Schmidt 1, A. Blazhev 1, K. Heyde 2, J. Jolie 1 1 Institut für Kernphysik, Universität zu Köln, Germany 2 Department of Physics and Astronomy, Ghent University, Belgium Brix Workshop, Spring 2015, Liège, Belgium

2. Outline Introduction to shell model calculations Comparison of band structures in theory and experiment ( 106 Cd, 108 Cd) Introduction to the method of Kumar and Cline (extraction of β- and γ-values) Illustration of deformation in 98 Cd Deformation analysis of 100 Cd to 108 Cd as a function of neutron number and spin

3. SM-Calculations with ANTOINE Monopole corrected, effective interaction Ref.: N. Boelaert et al. PRC 75, 014316  Protons: Z = 38 – 50  Neutrons: N = 50 – 82 Illustration: Boelaert et al. PRC75, 014316, 2007 2,23 MeV N = 50 Z = 50

4. Band structures

5. Band structures 106 Cd Defined by the B(E2)-strength in ground- and side-band 106 Cd, Theory B(E2) in W.u. 106 Cd, Experiment Data: NNDC & J.Wood (private communication) Band 1 Band 2

6. Band structures 106 Cd Data: NNDC & J.Wood (private communication) 106 Cd, Theory B(E2) in W.u. 106 Cd, Experiment Band 1 Band 2 Defined by the B(E2)-strength in ground- and side-band

7. Only B(E2)-transitions are considered Data: NNDC Band 1 Band 2 108 Cd, Theory 108 Cd, Experiment B(E2) in W.u. Band 3 Band structures 108 Cd

8. Deformation analysis

9. Motivation: extraction of - and -values Use method of K.Kumar and D.Cline Main tool for analysis: rotational invariants Invariants are model independent. Valid in laboratory and inertial frame of nucleus

10. Invariants M sr ×…×M ws Building Invariants: Couple n single-particle-transition- operators to angular momentum 0:

11. Invariants M sr ×…×M ws Building Invariants: Couple n single-particle-transition- operators to angular momentum 0:

12. Kumar-Cline-method K. Kumar: D. Cline:

13. Kumar-Cline-method K. Kumar: D. Cline:

14. Kumar-Cline-method K. Kumar: D. Cline: Kumar: uses model independent reference ellipsoid Cline: connects P (2) to deformation of ideal vibrator of Bohr & Mottelson

15. 98 Cd 100 Cd 102 Cd 104 Cd 106 Cd 108 Cd theory experient Protons give constant contribution Neutron number mainly influences collectivity Indication of collectivity; B(E2; 2 1 +  0 1 + )

16. Deformation with rising N β is rising with N, as aspected. Summed up to i=20 moving towards triaxiality state 0 1 +

17. Deformation in 98 Cd (semi-classical) Illustration: K. Heyde, Springer, 1990

18. Deformation in 98 Cd (semi-classical) Illustration: K. Heyde, Springer, 1990

19. Deformation in 98 Cd (semi-classical) Illustration: K. Heyde, Springer, 1990

20. Deformation in 98 Cd Observed problem with Kumar-method for 2 particle configuration in max. aligned spin Comparison between quadrupole moments (SM- spectroscopic, calc from spectroscopic and Kumar- method) show deviation in value and sign for 8 1 +

21. Deformation following bands Band 1 shows short increase in deformation from spin 0 + to 2 + (Cd104 increase to 4 + ). Afterwards decreases with spin. Band 2 shows similar behavior as band 1. Increase till some lower spin (2 + or 4 + ) then decreases. Cd 108: starts to be influenced by h 11/2 Cd 100 Cd 104 Cd 102 Cd 106 Cd 108

22. 108 Cd-deformation & h 11/2 occupation Correlation with deformation of band 2 an occupation of h 11/2 Further analysis necessary state Band 1Band 2

23. Summary and Conclusions SM-Calculations give good reproduction for low energy states and still acceptable up to 8 +. Deformation of 0 + 1,2 show expected increase in deformation towards the middle of the shell Also increase in γ. Band 1 & 2 show decrease in deformation at high spin after a short increase. 108 Cd starts to be occupied by h 11/2 influencing the deformation.

24. Thank you for your attention

26. backup

27. Comparison with ideal vibrator Cd108. Deviations in energy levels from ideal vibrator. Deviation in transition rates from ideal vibrator. Forbidden transitions exist. B(E2) in W.u. Daten: NNDC