Model independent determination of quadrupole deformationparameters from Coulomb excitation measurements

XVIIth Nuclear Physics Workshop, Kazimierz Dolny 2010, September 25th

** Symmetry and symmetry breaking in nuclear physics **

Julian Srebrny( Heavy Ion Laboratory, University of Warsaw)

OUTLINE

• Introduction: K. Kumar-idea, D. Cline – the method development and realisation

• Formulae derivation, expectation value of

quadrupole deformation Q and triaxiality cos3δ

• How does it really work - 104Ru example.

Nothing is easy : vibrational energy but shapes?

• Typical stiff axially symmetric rotor 168Er

• Transitional nuclei and important role of triaxiality 186-192Os and 194Pt

• Low lying 0+ states - 72-76 Ge and 96-100Mo

• Higher order invariants - degree of stiffness or softness in Q or cos3δ

• SUMMARY: The information about charge deformation.

The quality of collective quadrupole model descriptions.

Nuclear microscope –T. Czosnyka.

A result of Coulomb excitation experiment is the set of electromagnetic matrix elements. It can be 20 ÷ 60 ME for stable beam experiments. mainly E2 collective transitional and diagonal matrix elements:

< f II E2 II i > B(E2; i → f ) < i II E2 II i > spectroscopic quadrupole moment very often signs can be determined, not only absolute values

Comparing the list of experimental E2 matrix elements with model values exhibits neither the uniqueness nor the sensitivity of the data to the collective model parameters.

Quadrupole collectivity produces strong correlations of the E2 matrix elements and the number of significant collective variables is much lower than the number of matrix elements.

The information about charge deformation parameters can be obtained using rotationally invariant products of the quadrupole operators that relate the reduced E2 matrix elements with the quadrupole deformation parameters

K. Kumar, Phys. Rev. Lett. 28 (1972) 249.D. Cline, Annu. Rev. Nucl. Part. Sci. 36 (1986) 683.

• The two basic quadrupole invariants are formed of the quadrupole operator tensorM(E2) in the following way

- where [··· × ···]L stands for the vector coupling to angular momentum L.

- invariants are denoted here up to coefficients as Q2 and Q3 cos 3δ, in order to have a correspondence with collective coordinates, < Q2 > is an overall quadrupole deformation parameter < cos 3δ > is a triaxiality parameter

- since the components of M(E2,µ) with different µ’s commute with each other the expectation values of the E2 invariants can be related to the reduced E2 matrix elements by making intermediate state expansions:

Σ I R > < R I = 1

since the components of M (E2,µ) with different µ’s commute with each otherthe expectation values of the E2 invariants can be related to the reduced matrix elements by making intermediate state expansions:

- S denotes state S and at the same time the spin of state S alone; R and T denotes intermediate states and their spins;

- having the experimental values of the reduced E2 matrix elements, the expectation values of the basic quadrupole invariants <S|Q2|S> and <S|Q3cos3δ IS> for a given state S can be extracted from the experimental data.

Nuclear Physics A 766 (2006) 25–51

J. Srebrny, T. Czosnyka, Ch. Droste, S.G. Rohozinski,L. Próchniak, K. Zajac, K. Pomorski,

D. Cline, C.Y. Wu, A. Bäcklin, L. Hasselgren , R.M. Diamond , D. Habs, H.J. Körner,

F.S. Stephens, C. Baktash, R.P. Kostecki

4 phonon multiplet

3 phonon

2 phonon

1 phonon

β ≈ 0.28 ≈ 0.26 ≈ 0.21

similar behaviour 106-110

Pd , 128

Xe

only 114

Cd looks like real vibrator

approximation: < Q3

cos3δ > = < Q2 >3/2

< cos3δ >

168Er the centre of the rare earth region

rigid axially symmetric rotor E(2+

) = 80 keV

β ≈ 0.33 , ≈ 9°

similar results for 182,184

W and 174-178

Hf

prolate – oblate transitional nuclei Z= 76( Os), 78(Pt)

• Bogumiła Basaj

triaxial rotor, stable quadrupole deformation

and triaxiality – δ ≈ 20°

Maximal triaxiality: close to 30°

by adding 2 protons ( 192

Os – 194

Pt) deformation

has jumped from prolate to oblate

prolate – oblate transitional nuclei Z= 76( Os), 78(Pt)

very low second 0+

, close to first 2+

72Ge: 0+(691 keV), 2+(834 keV)

in Ge: ground state - deformed and triaxial

excited state - spherical

in Mo: complicated picture,

see review talk of Katarzyna Wrzosek

The new generation of RIA: few order increase of intensity will allow on

comprehensive study of many new nuclei

The only results from radioactive beam experiments( SPIRAL): 74,76

Kr.

E. CLEMENT et al. 02 : β ≈ 0.6 ≈ 40°

Higher order invariants allow to measure a softness of Q 2

and cos3δ

the need of longer excitation pass:

3 intermediate states for σ( Q2) and 5 intermediate states for σ(cos3δ)

SUMMARY

1. Model independent analysis of Coulomb Excitation experiment

(GOSIA) combined with non energy weighted Sum Rules

- powerful tool for quadrupole deformation parameters determination

2. Summation over double, triple or higher products of E2 matrix elements

allowed to measure in model independent way expectation values of

quadrupole deformation parameters.

3. In the future by more complicated excitation paths degree of softness

or stiffness in particular state

4. Nowadays possible mainly for stable nuclei. We got information

for more than 20 cases, including transitional nuclei.

5. Tools are ready for RIA of the new generation

6. Nuclear microscope- Tomasz Czosnyka

main authors

D. Cline, T. Czosnyka, C.Y.Wu B. Kotlinski, R. W. Ibbotson, J.S NSRL Rochester

L. Hasselgren, A. Backlin, C. Fahlander, L.-E. Svensson, A. Kavka TAL Uppsala

P. J. Napiorkowski, M. Zielinska, K. Wrzosek- Lipska, K. Hadynska-Klek, J.S. HIL Warsaw

D. Diamond, F. Stephens LBL Berkeley

C. Baktash, BNL Brookhaven

E. Clement GANIL

S. G. Rohozinski UW, L. Prochniak UMCS

≈ 0.16

Rochester-Warsaw-Uppsala-Berkeley-…

Nuclear Physics A 766 (2006) 25–51J. Srebrny, T. Czosnyka, Ch. Droste, S.G. Rohozinski,L. Próchniak, K. Zajac, K. Pomorski, D. Cline, C.Y. Wu,A. Bäcklin, L. Hasselgren , R.M. Diamond , D. Habs,H.J. Körner, F.S. Stephens, C. Baktash, R.P. Kostecki

<f II E2 II i > B(E2; i→f )

<i II E2 II i > spectroscopic quadrupole moment

98

Mo

Magda Zielińska PhD Thesis, Warsaw University 2005

Nucl. Phys. A712 (2002) 3

0.28

0.01

0.29 ± 0.02------------------------------------

0.10

0.09

0.06

0.25 ± 0.03

-0.03

0.02

-0.01 ± 0.01----------------------------------------------------------------

0.11

-0.04

0.02

0.09 ± 0.03

Contribution of various matrix elements to the final result

for < 22+|Q2| 22

+ > invariant in 104Ru

the component contribution to the invariant [e2b2]

<22+ II E2 II 2g+> <2g+ II E2 II 22

+> 0.113

<22+ II E2 II 31

+ > < 31+II E2 II 22

+> 0.298

<22+ II E2 II 42

+ > < 42+ II E2 II 22

+> 0.251

<22+ II E2 II 22

+ > < 22+ II E2 II 22

+> 0.077

total of 4 contributions = 0.739

all contributions = 0.76(8)

SUMMARY

● thanks to GOSIA and model independent analysis we got sets of 20-50 E2 matrix elements for many transitional nuclei

● thanks to the Sum Rules we experimentally deducedthe shapes of many nuclei in their ground and excited states in a model independent way:

nuclear microscope (de Broglie wavelength 0.5 fm much smaller than radius of nucleus)

● stringent test of sophisticated microscopic collective Q + P models, otherwise impossible

Vdef - the quadrupole deformation potential, the dynamical variables: β, γ - two Bohr shape deformation parameters, Ω - three Euler angles,Q + P microscopic calculations of potential and all the inertial functions, starting from the Nilsson model

Nuclear Physics A 766 (2006) 25–51J. Srebrny, T. Czosnyka, Ch. Droste, S.G. Rohozinski,L. Próchniak, K. Zajac, K. Pomorski, D. Cline, C.Y. Wu,A. Bäcklin, L. Hasselgren , R.M. Diamond , D. Habs,H.J. Körner, F.S. Stephens, C. Baktash, R.P. Kostecki

the nuclear spectroscopy

- physics of many body quantum system with finite fermions number

quantum dots, molecular clusters, ......, ....., .....