Shell Model based deformation analysis of light Cadmium isotopes

T. Schmidt1, A. Blazhev1, K. Heyde2, J. Jolie1

1Institut für Kernphysik , Universität zu Köln, Germany2Department 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 (106Cd, 108Cd)• Introduction to the method of Kumar and Cline

(extraction of β- and γ-values)• Illustration of deformation in 98Cd• Deformation analysis of 100Cd to 108Cd as a function of

neutron number and spin

SM-Calculations with ANTOINE

• Monopole corrected, effective interaction

• Ref.: N. Boelaert et al. PRC 75, 014316

Þ Protons: Z = 38 – 50Þ Neutrons: N = 50 – 82

Illustration: Boelaert et al. PRC75, 014316, 2007

2,23 MeV

N = 50

Z = 50

Band structures

Band structures 106CdDefined by the B(E2)-strength in ground- and side-band

106Cd, Theory

B(E2) in W.u.

106Cd, Experiment

Data: NNDC & J.Wood (private communication)

Band 1

Band 2

Band structures 106Cd

Data: NNDC & J.Wood (private communication)

106Cd, Theory

B(E2) in W.u. B(E2) in W.u.

106Cd, 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 1Band 2

108Cd, Theory 108Cd, Experiment

B(E2) in W.u. B(E2) in W.u.

Band 3

Band structures 108Cd

Deformation analysis

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

Msr×…×Mws

Building Invariants: Couple n single-particle-transition-operators to angular momentum 0:

Invariants

Msr×…×Mws

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

98Cd 100Cd 102Cd 104Cd 106Cd 108Cd

theory

theory

theory

experient • Protons give constant contribution

• Neutron number mainly influences collectivity

Indication of collectivity; B(E2; 21+ 01

+)

Deformation with rising N

• β is rising with N, as aspected.

• Summed up to i=20• moving towards triaxiality

state 01+

Deformation in 98Cd (semi-classical)

Illustration: K. Heyde, Springer, 1990

Deformation in 98Cd (semi-classical)

Illustration: K. Heyde, Springer, 1990

Deformation in 98Cd (semi-classical)

Illustration: K. Heyde, Springer, 1990

Deformation in 98Cd• 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 81

+

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 h11/2

Cd 100

Cd 104

Cd 102

Cd 106

Cd 108

108Cd-deformation & h11/2 occupation

• Correlation with deformation of band 2 an occupation of h11/2

• Further analysis necessary

state

state

Band 1 Band 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.

• 108Cd starts to be occupied by h11/2 influencing the deformation.

Thank you for your attention

backup

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