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

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

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

Band structures

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

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

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

Deformation analysis

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

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

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

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

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

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

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;  )

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

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

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

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

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 +

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

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

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.

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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