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

1. XVIIth Nuclear Physics Workshop, Kazimierz Dolny 2010, September 25th ** Symmetry and symmetry breaking in nuclear physics ** Julian Srebrny( Heavy Ion Laboratory, University of Warsaw) Model independent determination of quadrupole deformationparameters from Coulomb excitation measurements

2. 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 - 104Ruexample. Nothing is easy : vibrational energy but shapes? • Typical stiff axially symmetric rotor168Er • 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.

3. A result of Coulomb excitation experiment is the set of electromagnetic matrixelements.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 )< iII E2 II i >spectroscopic quadrupole momentvery often signscan be determined, not only absolute valuesComparing the list of experimental E2 matrix elements with model values exhibitsneither theuniqueness nor the sensitivity of the data to the collective model parameters.Quadrupole collectivity produces strong correlations of the E2 matrix elementsand the numberof significant collective variables is much lower than the number of matrix elements.The informationabout charge deformationparameters can be obtained using rotationally invariant products of the quadrupole operators that relate the reduced E2 matrix elements with the quadrupole deformationparameters K. Kumar, Phys. Rev. Lett. 28 (1972) 249.D. Cline, Annu. Rev. Nucl. Part. Sci. 36 (1986) 683.

4. 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 theE2invariants can be related to the • reducedE2 matrix elements by making intermediate state expansions: • ΣI R > < RI = 1

5. since the components ofM(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: - Sdenotes state S and at the same time the spin of state Salone; R and Tdenotes 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|Q3cos3δIS> for a given stateS can be extracted from the experimental data.

6. 4 phonon multiplet 3 phonon 2 phonon 1 phonon 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

8. β ≈ 0.28 ≈ 0.26 ≈ 0.21 similar behaviour 106-110Pd , 128Xe only 114Cd looks like real vibrator approximation: < Q3 cos3δ > = < Q2 >3/2 < cos3δ >

9. 168Er the centre of the rare earth region rigid axially symmetric rotorE(2+) = 80 keV β ≈ 0.33 ,d≈ 9° similar results for 182,184W and 174-178Hf

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

11. Bogumiła Basaj triaxial rotor,stable quadrupole deformation and triaxiality – δ ≈ 20°

12. Maximal triaxiality:dclose to 30°

13. by adding 2 protons ( 192Os – 194Pt) deformation has jumped from prolate to oblate

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

15. 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

16. 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): 74,76Kr. E. CLEMENT et al. 02 :β≈ 0.6 d ≈ 40°

17. Higher order invariants allow to measure a softness of Q 2andcos3δ the need of longer excitation pass: 3 intermediate states for σ( Q2) and 5 intermediate states forσ(cos3δ)

18. SUMMARY 1. Model independent analysis of Coulomb Excitation experiment (GOSIA) combined with non energy weighted Sum Rules - powerful tool for quadrupole deformation parameters determination Summation over double, triple or higherproducts of E2 matrix elements allowed to measure in model independent way expectation values of quadrupole deformation parameters. 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. Tools are ready for RIA of the new generation Nuclear microscope- Tomasz Czosnyka

19. main authors D. Cline, T. Czosnyka, C.Y.Wu B. Kotlinski, R. W. Ibbotson,J.SNSRL 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. RohozinskiUW, L. Prochniak UMCS

21. ≈0.16

28. Rochester-Warsaw-Uppsala-Berkeley-…

30. 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 <f II E2 II i >B(E2; i→f ) <i II E2 II i >spectroscopic quadrupole moment

32. 98Mo Magda Zielińska PhD Thesis, Warsaw University 2005 Nucl. Phys. A712 (2002) 3

33. 0.28 0.01 0.29 ± 0.02 ------------------------------------ 0.10 0.09 0.06 0.25 ± 0.03

34. -0.03 0.02 -0.01 ± 0.01 ---------------------------------------------------------------- 0.11 -0.04 0.02 0.09 ± 0.03

35. 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 E2II 2g+><2g+II E2II 22+>0.113 <22+II E2II 31+>< 31+II E2II 22+> 0.298 <22+II E2II 42+>< 42+II E2II 22+> 0.251 <22+II E2II 22+>< 22+II E2II 22+> 0.077 total of 4 contributions = 0.739 all contributions = 0.76(8)

38. 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 nucleiin 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

39. the nuclear spectroscopy - physics of many body quantum system with finite fermions number quantum dots, molecular clusters, ......, ....., ..... 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) 25–51 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