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On some spectral properties of billiards and nuclei – similarities and differences*

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## On some spectral properties of billiards and nuclei – similarities and differences*

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**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**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**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**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**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 ?**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´**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).**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**The nuclear electric dipole response**E (MeV) p n GDR 15 PDR (2+ x 3-)1- 3 B(E1)**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 ?**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**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**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.**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.**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).**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.**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)**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**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**Resonance parameters**Very high signal to noise ratio ● Transmission measurements: relative power from antenna a b**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**● Boundary of Limaçon billiards given byamapping from**zw w = z + l z2 Billiards of mixed and chaotic dynamics ● l controls the degree of chaoticity**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**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)**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**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**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)**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**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**Coupled billiards as a model for symmetry breaking**• ● Large number of resonances (N1500) • ● Variable coupling strength resp. degree of symmetry breaking**Experimental set-up**● Coupling was achieved by a niobium pin introduced through holes into both resonators**Changing the coupling strength**uncoupled weakly coupled strongly coupled**S2-statistics for different coupling strengths**uncoupled weakly coupled strongly coupled**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**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**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**Comparison of results for coupling parameters**Antenna combination ● Symmetry breaking parameters extracted from spectral statistics (circles) agree with those from strength distribution (crosses)**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.