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

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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 Ru example. Nothing is easy : vibrational energy but shapes? Typical stiff axially symmetric rotor 168 Er Transitional nuclei and important role of triaxiality Os and 194 Pt Low lying 0 + states Ge and Mo 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.

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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: B(E2; i f ) 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.

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The two basic quadrupole invariants are formed of the quadrupole operator tensor M (E2) in the following way - where [··· × ···]L stands for the vector coupling to angular momentum L. - invariants are denoted here up to coefficients as Q 2 and Q 3 cos 3δ, in order to have a correspondence with collective coordinates, is an overall quadrupole deformation parameter is a triaxiality parameter - since the components of M (E2,µ) with different µs 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

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since the components of M (E2,µ) with different µs commute with each other the 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 and for a given state S can be extracted from the experimental data.

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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

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β similar behaviour Pd, 128 Xe only 114 Cd looks like real vibrator approximation: = 3/2

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168 Er 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 Hf

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

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B o g u m i ł a B a s a j triaxial rotor, stable quadrupole deformation and triaxiality – δ 20°

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Maximal triaxiality: close to 30°

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

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

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very low second 0 +, close to first Ge: 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

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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. 0 2 : β °

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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)

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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

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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

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Rochester-Warsaw-Uppsala-Berkeley-…

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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 B(E2; if ) spectroscopic quadrupole moment

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

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± ± 0.03

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± ± 0.03

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Contribution of various matrix elements to the final result for invariant in 104 Ru the component contribution to the invariant [e 2 b 2 ] total of 4 contributions = all contributions = 0.76(8)

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SUMMARY thanks to GOSIA and model independent analysis we got sets of E2 matrix elements for many transitional nuclei thanks to the Sum Rules we experimentally deduced the 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

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V def - 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 the nuclear spectroscopy - physics of many body quantum system with finite fermions number quantum dots, molecular clusters,......,.....,.....

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