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Strength distribution in the decay out of superdeformed bands

Strength distribution in the decay out of superdeformed bands. -Introduction Experimental observations Excitation energy at decay-out Shape of decay-out spectrum Fragmentation of decay-out cascade Strength analysis: Maximum Likelihood Method Chaoticity parameter Sparse matrices

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Strength distribution in the decay out of superdeformed bands

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  1. Strength distribution in the decay out of superdeformed bands -Introduction Experimental observations Excitation energy at decay-out Shape of decay-out spectrum Fragmentation of decay-out cascade Strength analysis: Maximum Likelihood Method Chaoticity parameter Sparse matrices Conclusion & Perspectives A. Lopez-Martens CSNSM (NBI-ANL/Lund: T. Døssing, T.L. Khoo, S. Aberg, B. Herskind, T. Lauritsen) Workshop on Level Density and Gamma Strength in Continuum

  2. Epot 194Hg  Decay out Excitation energy  Collective feeding * * * * * Statistical feeding  Spin Life & Death of a superdeformed nucleus Workshop on Level Density and Gamma Strength in Continuum

  3. E E D Vt < D weak coupling Vt I-2 E* E* I-2 2 2 Statistical picture (Vigezzi, Shimizu) E. Vigezzi et al., PLB 249 (1990) Y.R Shimizu et al., PLB 274 (1992) Decay mechanism Collective picture (Bonche, Meyer) P. Bonche et al., NPA 519 (1990) J. Meyer et al., NPA 533 (1991) Decay-out occurs whenNDcan compete withSD What is the nature of the admixed ND state ? Workshop on Level Density and Gamma Strength in Continuum

  4. Excitation energy from 1-step linking transitions G. Hackman et al. PRL 79 (1997) U(11ħ) = 4.7 MeV T.L. Khoo et al. PRL 76 (1996) U(10ħ) = 4.2 MeV  Large excitation energy Workshop on Level Density and Gamma Strength in Continuum

  5. SD IR 192Hg ESD ND ER END IND ISD Excitation energy from quasicontinuum analysis centroid => <E> surface => <multiplicity> ER = <E> x <multiplicity> IR = 0.5 x <multiplicity>  U(10ħ) = 4.3 ± 0.9 MeV R. Henry et al. PRL 73 (1994) = 3.3 ± 0.5 MeV T. Lauritsen et al., PRC 62 (2000) Consistent results with 1-step linking transitions: 194HgT. Lauritsen et al., PRC 62 (2000) 191Hg S. Siem et al., PRC 70 (2004) Lower limit on excitation energy: 195Pb: U(11ħ)> 2.5 MeVM.S. Johnson et al.,PRC 71 (2005) Workshop on Level Density and Gamma Strength in Continuum

  6. Shape of the decay-out spectrum in 192Hg T. Døssing et al. PRL 75 (1995) no pairing • equidistant single particle levels: • dn=0.29 MeV, dp=0.41 MeV • statistical decay from a sharp state at U=4.3 MeV • pairing correlations included using self consistant BCS: • Gn=0.102 MeV, Gp=0.142 MeV • 0 = 0.7, 0.9, 1.1 MeV Transitions/MeV o-e nucleus with pairing e-e nucleus with pairing Transition energy (MeV) The shape of the decay-out spectrum reflects the density of ND states and the effects due to pairing Pair gaps extracted from decay-out spectra in 192,194Pb D. McNabb et al., PRC 61 (2000) Workshop on Level Density and Gamma Strength in Continuum

  7. N transitions sampled from an infinite number N transitions sampled from a finite number Nt (N>Nt) counts counts energy energy 2/1~ 1 Purely statistical fluctuations 2/1 = N/Nt+1 Enhancement of the fluctuations Fragmentation Fluctuation Analysis Method T. Døssing et al. Phys. Rep. 268 (1996) Workshop on Level Density and Gamma Strength in Continuum

  8. U Statistical feeding Decay-out U=0 A. Lopez-Martens et al. PRL 77 (1996) Fragmentation 192Hg - large fragmentation ~ 104 decay paths - 1-2 initial states  weak coupling Workshop on Level Density and Gamma Strength in Continuum

  9. Similarity with (n,) spectrum Résumé of Observations • Large excitation energy • Spectrum well reproduced by statistical decay calculations including the effects of pairing on the density of ND states • -Large fragmentation • The SD state couples to a hot (complex?) state  The primary strengths should follow a Porter Thomas distribution Workshop on Level Density and Gamma Strength in Continuum

  10. Porter-Thomas Distribution Direct consequence of Random Matrix Theory (GOE): In a truly compound state, any amplitude can be viewed as being selected from a Gaussian distribution centred around 0 E. Wigner, Proc. Cambridge Philos. Soc. 47 (1951) • since the reduced transition strengths for the decay to a particular final state are proportional to the square of one amplitude, they follow a 2 distribution with =1 degree of freedom • C. Porter et R. Thomas, Phys. Rev. 104 (1956) Workshop on Level Density and Gamma Strength in Continuum

  11. i/i Strength fluctuations 22 1- resonances H. Jackson et al. PRL 17 (1966) Large strength fluctuations Workshop on Level Density and Gamma Strength in Continuum

  12. wlow/ w/ What is the nature of the distribution of primary strengths in the decay-out of 194Hg ? Analysis in terms of 2 distributions:  - the SD state couples to a chaotic state   incoherent amplitudes to one final state (strong coupling limit ?) - Selection rules ? Experimentally, the weakest transitions are not identified  Strength analysis in terms of truncated 2 distributions: A. Lopez-Martens et al. NPA 647 (1999) Workshop on Level Density and Gamma Strength in Continuum

  13. Condition of Maximum Likelihood: <log w>exp Minimun variance bound estimators (sufficient statistics) =9wlow =1 <w>exp Maximum Likelihood Method Probability that a sample of N0 strengths consists precisely of the measured values (Likelihood): Workshop on Level Density and Gamma Strength in Continuum

  14. Experimental Strengths • U(decay out) = 4.2 MeV • discrete transitions with E > 2.6 MeV are assumed to be 1st step transitions • primary character verified for the most intense transitions Assuming E1 character for the primary lines: w=I x (E/E0)3 wlow 19 strengths >wlow Workshop on Level Density and Gamma Strength in Continuum

  15. +20 =1,wlow/= 4.2 -1 theoretical distribution (Nt≈ 400 ) +1.4 -2.9 Simulation data Most Likely Distribution A. Lopez-Martens et al. NPA 647 (1999) Large error bar: only the tail of the distribution is accessible Other observations in support of most likely distribution: Fluctuation analysis method: Nt= 530 Quasicontinuum analysis: total strength = 0.34  Nt = 550 Probability to have 2 1-step links with strengths > 8 = 10-4 -10-5 ! Workshop on Level Density and Gamma Strength in Continuum

  16. What do ~600 primary lines look like ? 192Hg strengthssampled from PT distribution E sampled from inverse level density distribution + exp resolution, feeding & compton background, counting statistics 194Hg  Some simulated spectra are very similar to experimental 192Hg and 194Hg Workshop on Level Density and Gamma Strength in Continuum

  17. H = H = Vt Vt Vt Vt What does ≠1 mean ? => Chaoticity parameter « Bridge » between ordered/chaotic pictures S. Åberg, Phys. Rev. Lett. 82 (1999) |n> |> |sd> D |sd> Vt |d> E 2 * SD state couples only to specific ND states (doorway |d>) * ND spectrum described by a large GOE matrix (NxN) off diagonal elements x (scaled GOE)  = 1 = 0 Workshop on Level Density and Gamma Strength in Continuum

  18. N=400, NSD=N/2, Ndoorway=N/4 |> |<|S>|2/Vt2 |d> |n> |<|S>|2/Vt2 |sd> N states contribute with the same admixture |S> Admixtures into the superdeformed state • Hamiltonian H : NxN GOE matrix • Off diagonal elements x  • Diagonalise H • Add SD state + weak coupling Vt to • doorway, rediagonalise H  ‘chaos-assisted’ tunneling Workshop on Level Density and Gamma Strength in Continuum

  19. Conjecture admixture  strength (each basis state |> is connected to 1 final state |‘> at lower energy) basis states: mixed states: 1 2 2 2 1 1 Workshop on Level Density and Gamma Strength in Continuum

  20. Method N=400 Nt=200 N0=10 exponential different tails NS Simulation: - select Nt strengths - identify the N0 strongest => equivalent of wlow - normalisation to wlow power law Workshop on Level Density and Gamma Strength in Continuum

  21. simulation Cum. Dist. wlow strength data banana - shaped curve Results Ns=500 N=1000 N0=19 > 0.08, Nt > 250 Workshop on Level Density and Gamma Strength in Continuum

  22. Ns=500 N=1000 N0=19 Success rate of the experiment fraction of simulations, which have a worse square difference to the average cumulative distribution than the experiment (= succes rate of the experiment) Data 194Hg Average over all simulations Workshop on Level Density and Gamma Strength in Continuum

  23. Strength analysis with sparse matrices Motivation: 2body <  deff = 3 Sparcity = fraction of non zero independent off-diagonal matrix elements Effective dimensionality deff = N x sparcity (=average number of off-diagonal elements in a row) Workshop on Level Density and Gamma Strength in Continuum

  24. banana - shaped curve One would need to investigate spectral correlations over more than 10 levels around the SD state to obtain similar information on the chaoticiy of ND states Results Ns=500 N=1000 N0=19 Workshop on Level Density and Gamma Strength in Continuum

  25. Result for Jackson’s neutron resonances 196Pt, N=1000 • Nt = 70 ± 6 • no banana shape Workshop on Level Density and Gamma Strength in Continuum

  26. Conclusions and perspectives • the decay-out from SD bands occurs at large excitation energy and is highly fragmented • 3 different methods have been investigated to determine the strength distribution of the primary lines in 194Hg • the three methods give consistent results  relationship between parameters • improved experimental data with Agata  increased possibilities for studying nuclear “chaoticity” Workshop on Level Density and Gamma Strength in Continuum

  27. HD decay-out ? SD decay-out (A~150) SD decay-out (A~190) Decay-back (fission isomers) Investigations of ‘chaoticity’ Energy Neutron & proton resonances Levels up to ~Sn Near yrast levels Masses Spin Workshop on Level Density and Gamma Strength in Continuum

  28. Pandemonium and spurious lines 192Hg 194Hg Sim1 Sim1’ Pandemonium: Overlapping of transitions and creation of ‘funny-shaped’ peaks Counting statistics: Destruction or alteration of ‘peaks’ Workshop on Level Density and Gamma Strength in Continuum

  29. T. Lauritsen et al. PRL 88 (2002) U(28ħ) = 3.7 MeV Workshop on Level Density and Gamma Strength in Continuum

  30. Excitation energy A. Wilson et al, PRL 90 (2003) U(10ħ) = 2.0 MeV A. Lopez-Martens et al. PLB 380 (1996) K. Hauschild et al. PRC 55 (1997) U(6ħ) = 2.7 MeV 192Pb U(6ħ in 196Pb) = 3.4 MeV A. Wilson et al., PRL 95 (2005) Workshop on Level Density and Gamma Strength in Continuum

  31. Uncertainties n=100 transitions =1 /wlow = 1, 10, 100 Dispersion ellipses for the estimators The number of degrees of freedom most clearly reveals itself for small strengths Scatter of determined parameters (,) becomes more Gaussian-like as n and wlow  Workshop on Level Density and Gamma Strength in Continuum

  32. Long-range correlations - Spectral rigidity Number variance = fluctuations in the total number of levels found in an energy interval [L0-L/2,L0+L/2] -uncorrelated levels  number variance is linear in L -random matrices  number variance grows logarithmically with L In sparse GOE matrices, the number variance is described by GOE up to a certain scale Lc Workshop on Level Density and Gamma Strength in Continuum

  33. 2 distributions - for a given , Ntrans are sampled from the corresponding 2 distribution - the smallest of the strongest 19 lines defines wlow Workshop on Level Density and Gamma Strength in Continuum

  34. Sparse matrices in the case of neutron resonances Only one doorway is kept for any low-lying state. The doorway is taken as state number 100 The interval around Sn is taken to cover states 200-299 Workshop on Level Density and Gamma Strength in Continuum

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