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  1. On some spectral properties of billiards and nuclei – similarities and differences* Lund 2005 ●Generic and non-generic features of billiards and nuclei ● The Scissors Mode and regularity ● The Pygmy Dipole Resonance (PDR) and “mixed“ statistics ●Resonance strengths in microwave billiards of mixed dynamics ●Isospin symmetry breaking in nuclei and its modelling with coupled billiards * Supported by the SFB 634 of the Deutsche Forschungsgemeinschaft C. Dembowski, B. Dietz, J. Enders, T. Friedrich, H.-D. Gräf, A. Heine, M. Miski-Oglu, P. von Neumann-Cosel, V.Yu. Ponomarev, A. R., N. Ryezayeva, F. Schäfer, A. Shevchenko and J. Wambach (Darmstadt) T. Guhr (Lund), H.L. Harney (Heidelberg) SFB 634 – C4: Quantum Chaos

  2. Stadium billiard  n +232Th Transmission spectrum of a 3D-stadium billiard T = 4.2 K Spectrum of neutron resonances in 232Th + n ●Great similarities between the two spectra: universal behaviour

  3. Properties of spectral fluctuations I P(s) P(s) s s Ensemble of 1726 highly excited nuclear states of the same spin and parity: `Nuclear Data Ensemble´ Ensemble of 18764 resonance frequencies of a 3D-microwave resonator ●Highly excited nuclei (many-body quantum chaos) and chaotic microwave resonators (one-body quantum chaos) exhibit a universal (generic) behaviour

  4. P(s) P(s) s s Regular (integrable) elliptic billiard Scissors Mode in deformed nuclei ∆3(L) ∆3(L) L L Ensemble of 300 resonance frequencies Ensemble of 152 1+ states in 13 heavy deformed nuclei between 2.5 and 4 MeV Properties of spectral fluctuations II ●The low-lying Scissors Mode and integrable microwave resonators exhibit the same universal (non-generic) behaviour

  5. Properties of spectral fluctuations III p, n n P(s) P(s) s s Limaçon billiard of mixed dynamics Pygmy Dipole Resonance in 138Ba, 140Ce, 144Sm and 208Pb ∆3(L) ∆3(L) L L Ensemble of 800 reso- nance frequencies Ensemble of 154 1- states in three semimagic (N=82) nuclei and one magic (Z=82, N=126) nucleus between 5 and 8 MeV ● Short and long range level-level correlations lie between Poisson (integrable) and GOE (chaotic) behaviour. Do we understand this coexistence ?

  6. Definition: `generic´ • 2nd Concise Edition of Webster's New World • Dictionary of the American language (1975): • `referring to a whole kind, class, or group´ • `something inclusive or general´ • 2. Oriol Bohigas: • `opposite of specific´ • `non-particular´ • `common to all members of a large class´ • more specific (Bohigas‘ conjecture): • `A classical chaotic system after being quantized results in • a quantum system which can be described by Random Matrix Theory. • All systems for which this is true are called generic, the behaviour • of the rest is called non-generic´. • 3. Thomas Seligman: • `structurally stable against small perturbations´

  7. Definition: `generic´ 4. Hanns Ludwig Harney: `there is a minimal number of symmetries in the system´ • Example: • (i) An ensemble of levels with given isospin is generic. • (ii) An ensemble of levels without taking notice of • the isospin quantum number is non-generic. • (iii) An ensemble of levels with broken isospin is • non-generic too, and the deviation from the • generic behaviour yields the isospin breaking • matrix element (→ 26Al, 30P and its modelling • with coupled billiards).

  8. Generic and non-generic features of billiards and nuclei Generic Non-generic ● Certain POs (BBOs) ● Collective rotations and vibrations, i.e. `ordered motion´ ● Level statistics ● Width distributions • The Scissors mode DALINAC 1984 • The PDR mode S-DALINAC 2002

  9. The nuclear electric dipole response E (MeV) p n GDR 15 PDR (2+ x 3-)1- 3 B(E1)

  10. Electric dipole response of neutrons and protons in QPM calculations for 138Ba neutrons protons r2ρ(r) ● Evidence for surface neutron density oscillations ● “Soft dipole mode“ at 7 MeV is dominantly isoscalar ● Influence on the spectral fluctuation properties ?

  11. Photon scattering off 138Ba 138Ba Emax = 9.2 MeV E1 excitations A. Zilges et al., Phys. Lett. B 542, 43 (2002) ● Large number of resolved J = 1-states

  12. E1 strength distribution in N = 82 nuclei: experiment  QPM calculation (1p1h-2p2h) ● Experimental # of levels (~ 50 per nucleus) < # of levels in the QPM (~ 300 per nucleus) ● B(E1)exp< B(E1)QPM ● Missing levels and strengths

  13. Ensemble of E1 transitions: 138Ba, 140Ce, 142Nd, 144Sm If the PDR is a truly collective mode one may see this in the spectral properties: 184 levels of J = 1- ● Experiment and QPM show spectral properties in between GOE and Poisson statistics. ● The strengths show Porter-Thomas (PT) statistics for the QPM, while the experimental distribution deviates from PT.

  14. Possible interpretations of the observed fluctuation properties ● Coexistence of regular nuclear and chaotic nuclear motion: intermediate or “mixed“ statistics. ● The missing levels destroy spectral correlations. ● Limited statistics (low number of levels) affect the spectral fluctuation properties.

  15. Qualitative modelling of the missing level effect 1200 levels 184 levels 184 levels Obtain a subset of the states calculated within the QPM by cutting away the weakest transitions below the experimental detection limit of about 10-3 e2 fm2 ● All three distributions show similar behaviour for experiment, truncated QPM and full QPM. ● They are close to Poisson with some remnants of level repulsion (limited to the lowered probability in the first bin).

  16. Transition strength distributions 1200 levels 184 levels 184 levels RMT predicts in case of GOE correlations that the wave function components or, equivalently, their squares follow a Gaussian or Porter-Thomas distribution, respectively. ● Strength distribution of the full QPM agrees with PT, while experiment and truncated QPM deviate from PT statistics, but in a similar way. ● If the large fraction of missing levels (~30% in the QPM and ~90% in the experiment) is taken into account the deviation from PT statistics can be explained qualitatively by including into the PT distribution an appropriate threshold function for detection.

  17. QPM matrix elements and missing strength ● Overall distribution of coupling matrix elements (for 2p2h-2p2h and 1p1h-2p2h interactions) is not a Gaussian ● Few large matrix elements indicative of collective configurations lie in the tails of the distribution ● Many extremely small non-collective matrix elements (almost pure 2p2h phonon states which do not interact with each other and which cannot be excited easily electromagnetically) ● Nevertheless: we have been able to understand certain statistical features of the PDR ( J. Enders, Nucl. Phys. A741 (2004) 3)

  18. How can the problem of the missing strength be overcome? ● Remember: highly excited nuclei (many-body quantum chaos) and chaotic microwave resonators (one-body chaos) exhibit a universal (generic) behaviour ● For flat microwave resonators the scalar Helmholtz equation is mathematically fully equivalent to the Schrödinger equation: e.m. eigenfrequencies q.m. eigenvalues and ● Superconducting microwave resonators (Q  106) shaped as billiards allow the determination of all eigenfrequencies and resonance strengths

  19. Resonance strengths in microwave billiards of mixed dynamics ● Direct measurement of the wavefunctions in terms of the intensity distributions of the - field is presently only possible in normal conducting billiards ● However, information on wavefunction components can also be extracted from the shape of the resonances in the measured spectra of superconducting billiards ● Resonance strengths are directly related to the squared wavefunction components at the positions of the antennas for microwave in- and output

  20. Resonance parameters Very high signal to noise ratio ● Transmission measurements: relative power from antenna a  b

  21. Resonance parameters  small for superconducting resonators ● Open scattering system: a  resonator  b ● Frequency of m‘th resonance: fm ● Partial widths: Gma, Gmb ●Total width: (+ dissipative terms) ● Resonance strengths: Gma·Gmbdetermined from transmission measurements

  22. ● Boundary of Limaçon billiards given byamapping from zw w = z + l z2 Billiards of mixed and chaotic dynamics ● l controls the degree of chaoticity

  23. Total widths and strengths of the Limaçon billiard ● Secular variation of the Gm‘s and strengths due to rf losses in the cavity walls and to the frequency dependence of the coupling of the antennas to the cavity ● Large fluctuations of widths and strengths ● Measurements for altogether6 antenna combinations  about 5000 strengths were determined

  24. Resonance strengths distributions =log10(GmaGmb) ● GOE prediction corresponds to the distribution of the product of two PT distributed random variables Gma and Gmb: modified Bessel function K0 ● Strong deviations from GOEfor thebilliards with mixed dynamics demonstrated for the first time ● Agreement with RMT prediction over more than 6 orders of magnitude for the fully chaotic billiard ( in nuclei a comparison over only about 2 orders of magnitude is possible)

  25. Modified strength distribution K0 - distribution modified strength distribution due to experimental detection limit l=0.3 ● Very good agreement between the theoretical and the experimental strength distribution ● Strength distributions provide a statistical measure for the properties of the eigenfunctions of chaotic systems ● RMT models must be developed to describe systems of mixed dynamics

  26. Strength distribution and symmetry breaking ● Isospin symmetry breaking in nuclei ● RMT model for symmetry breaking ● Coupled microwave billiards as an analog system for symmetry breaking ● Experimental results ● Strength distribution for systems with a broken symmetry

  27. Isospin mixing in 26Al 3+; T=0 1+; T=0 2+; T=1 4+; T=0 2+; T=0 1+; T=0 3+; T=0 0+; T=1 5+; T=0 75 levels: T=0 32 levels: T=1 mixing: <Hc> ● Observed statistics between 1 GOE and 2 GOE (Mitchell et al., 1988) (Guhr and Weidenmüller, 1990)

  28. Transition probabilities in 26Al and 30P PT distribution PT distribution ● GOE prediction for the distribution of reduced transition probabilities ( partial widths) of systems without or with complete symmetry breaking is a Porter-Thomas distribution ● For both nuclei deviations from GOE prediction  signature of isospin mixing (Mitchell et al., 1988, Grossmann et al., 2000) ● Study of strength distributions of resonances in coupled microwave billiards

  29. RMT model for symmetry breaking ● RMT model for Hamiltonian of a chaotic system with a broken symmetry ● l=a / D is the relevant parameter governing symmetry breaking; D is the mean level spacing • ● a = 0 no symmetry breaking: 2 GOE‘s • ● 0 < a <1  partial symmetry breaking • ● a =1 complete symmetry breaking: 1 GOE

  30. Coupled billiards as a model for symmetry breaking • ● Large number of resonances (N1500) • ● Variable coupling strength resp. degree of symmetry breaking

  31. Experimental set-up ● Coupling was achieved by a niobium pin introduced through holes into both resonators

  32. Changing the coupling strength uncoupled weakly coupled strongly coupled

  33. S2-statistics for different coupling strengths uncoupled weakly coupled strongly coupled

  34. Analysis of spectral properties ● largest coupling achieved in experiment: a a / D = 0 . 21 (in Al: / D = 0 . 26) 26  Coulomb matrix element ● normalized spreading width: • every fourth state influenced by the coupling

  35. RMT model for the strength distributions K0-Distribution l=0.3 l=0.13 =0.04 experimentalthreshold ● Position of central maximum depends on coupling strength, i.e. on the symmetry breaking ● Resonances with small strengths cannot be detected  experimental threshold of detection

  36. l=0.04 l=0.09 l=0.14 l=0.21 Experimental strength distribution for different couplings ● Examples for one antenna combination show very good agreement with RMT fits

  37. Comparison of results for coupling parameters Antenna combination ● Symmetry breaking parameters extracted from spectral statistics (circles) agree with those from strength distribution (crosses)

  38. Summary on symmetry breaking effects ● Generic properties of the eigenfunctions of a chaotic billiard can be studied experimentally using the strength distributions for a microwave billiard. ● Changing the coupling strength influences the level and strength distributions of the coupled stadiums. ● Various spectral measures can be used to extract the coupling strength and give consistent results. ● Maximum normalized spreading width, i.ethe deviation from generic behaviour, observed G / D = 0.20 - 0.25 corresponds to the nuclear case of 26Al. ● Precise and significant tests of present RMT models for symmetry breaking are possible. ● Symmetry breaking in nuclei ( J.F. Shriner et al.,Phys. Rev. C71 (2005) 024313) can be very effectively modelled through billiards.