Unexpected behavior of neutron resonances (Pozorování neočekávaných vlastností neutronových rezonancí) Milan Krtička, František Bečvář Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic P.E. Koehler, J.A. Harvey, K.H. Guber Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA Niels Bohr - Nature (1936) 17. KČSF, Žilina, 7. 9. 2011

Outline Description of compound nucleus at high excitation energies – Random Matrix Theory Neutron resonances Experimental data on Neutron Widths from ORELA Results - Distribution of Neutron Widths in Pt isotopes 17. KČSF, Žilina, 7. 9. 2011

Neutron resonances observed in 30’s (Pre)history Neutron resonances observed in 30’s Bohr’s idea of the compound nucleus (CN): the narrowly spaced and narrow resonances are incompatible with independent–particle motion (IPM) and arise from strong interaction IPM (R  5 fm, V0  50 MeV) gives the Single-Particle states with spacing of several hundred keV and widths of the order of ten keV or larger  evident disagreement with data CN model is needed n collides with the nucleons in the target and shares its energy it takes a long time until one of nucleons acquires sufficient energy to be re–emitted from the system. Total n x-section on 232Th 17. KČSF, Žilina, 7. 9. 2011

Model of compound nucleus Bohr’s wooden toy model of the compound nucleus - Nature (1936) 17. KČSF, Žilina, 7. 9. 2011

Random Matrix Theory The CN idea implies that the states are complicated mixtures involving all the available degrees of freedom of the many-body-system 1955 – Eugene P. Wigner suggested that some features of n resonances (fluctuations in level widths and spacings) might be modeled by considering an expansion of the Hamiltonian matrix on an arbitrarily chosen finite set of basis states The strong mixing of different degrees of freedom and the randomness of the compound nucleus is expressed by choosing the matrix elements of the Hamiltonian matrix independently and randomly from an appropriate ensemble 17. KČSF, Žilina, 7. 9. 2011

Random Matrix Theory Instead of considering the actual nuclear Hamiltonian one considers an ensemble of Hamiltonians (each in matrix form) The ensemble is defined in terms of probability distribution for the matrix elements  P(H) dNH = 1 N = n(n+1)/2 … the number of independent elements Nuclei are invariant under time reversal – matrix representation of the nuclear Hamiltonian can be chosen real and symmetric The invariance under rotations is required  P(H) dNH =  P(H’) dNH’ for H’ = OHOT 17. KČSF, Žilina, 7. 9. 2011

Random Matrix Theory s 2s The requirement on the invariance against rotations and independence of matrix elements leads to the choice: s 2s Statistical Theories of Spectra:Fluctuations, edited by C.E. Porter (Academic, 1965) M.L. Mehta – Random Matrices and the Statistical Theory (Academic, 1967) Gaussian Orthogonal Ensamble (GOE) 17. KČSF, Žilina, 7. 9. 2011

Random Matrix Theory RMT - a statistical theory of spectra – only the joint probability distributions of the eigenvalues and eigenfunctions (and not the individual spectra) are predicted which can be compared with experimental data The spectral distribution functions are calculated as averages over the ensemble and are compared with the actual fluctuation properties of nuclear spectra If the observed properties agree with RMT predictions one concludes that the system is generic - no additional information cannot be deduced Disagreement of data with RMT predictions indicates that the spectrum is not generic and that the available spectral information may be used to deduce further properties of the system. Levels must have the same Jp (conserved quantum numbers) 17. KČSF, Žilina, 7. 9. 2011

Properties of Random Matrices Predictions of distribution of the nearest-neighborhood spacing of levels correlations between the nearest spacings correlations between the next-nearest spacings … the long-range correlations among energies of levels the distribution of the overlaps with a random vector 17. KČSF, Žilina, 7. 9. 2011

Properties of Spectra Spectra of 50 levels with the same mean level spacing: From right to left: 1D harmonic oscillator, a sequence of zeros of the Riemann Zeta–function, a sequence of eigenvalues of the Sinai billiard, a sequence of resonances seen in neutron scattering on 166Er, a sequence of prime numbers, a set of eigenvalues obeying Poisson statistics. Bohigas, O., M. J. Giannoni, Mathematical and Computational Methods in Nuclear Physcis, 1984 17. KČSF, Žilina, 7. 9. 2011

Nearest-neighborhood spacings (NNS) The distribution of eigenvalues is given by so-called Wigner semicircle law If the correction for average spacing is applied the distribution of NNS becomes so-called Wigner distribution every state is coupled to all other states with equal average strength which results in level repulsion between any pair of levels, and in a complete mixing of states in Hilbert space 17. KČSF, Žilina, 7. 9. 2011

RMT and “chaos” 1984 - Bohigas conjecture (O. Bohigas, M.-J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52 (1984) 1) According to the so-called Bohigas-conjecture quantum systems whose classical counterparts are chaotic have a nearest-neighbor spacing distribution given by RMT whereas systems whose classical counterparts are integrable obey a Poisson distribution There are exceptions (harmonic oscillator) Sinai billiard 17. KČSF, Žilina, 7. 9. 2011

RMT and “chaos” In CM, the case of complete chaos and completely regular motions are limiting cases In QM, regular motion corresponds to the existence of a complete set of quantum numbers that label every state There is no level repulsion, and no correlation between levels This case generically yields a spectrum where the spacings have an exponential (Poisson) distribution NDE -Nuclear Data Ensemble 17. KČSF, Žilina, 7. 9. 2011

Long-range correlations Often D3(L) statistics is used For GOE 17. KČSF, Žilina, 7. 9. 2011

Porter-Thomas distribution Distribution of overlaps of wave functions y with a randomly selected vector f In the limit n   the distribution of amplitudes y|f becomes a gaussian  distribution of overlaps y|f2 is a c2 distribution with one (n = 1) degree of freedom (PT distribution) PT distribution – obtained earlier using some simple physical arguments It is believed to apply for distribution of “overlaps” in nuclear reactions in region of “high” level density 17. KČSF, Žilina, 7. 9. 2011

Neutron resonances 17. KČSF, Žilina, 7. 9. 2011

Neutron resonances Resonance shape of x-section described by Breit-Wigner formula Reduced neutron width Interference with potential scattering for s-wave resonances Only elastic neutron scattering and radiative neutron capture allowed at low neutron energies 17. KČSF, Žilina, 7. 9. 2011

Neutron resonances At neutron energies of interest only neutrons with a low orbital momentum l = 0 (s-wave) and l = 1 (p-wave) can enter the nucleus Reduced neutron widths are expected to follow the PT distribution - the same average value must be assumed No evidence for changes of average values found so far in the energy region used in analyses. Gn ~ En1/2 Gn(0) for s-wave Gn ~ En3/2 Gn(1) for p-wave 17. KČSF, Žilina, 7. 9. 2011

Properties of neutron resonances Consensus View from Last ~50 Years: Reduced Neutron Widths Follow the PT distribution Spacing of Resonances Follows the Wigner distribution The most comprehensive test is assumed to come from so-called nuclear data ensemble (NDE) a set of resonance energies consisting of 30 sequences in 27 different nuclides Fluctuation properties of resonance energies in the NDE were found to be in remarkably close agreement with GOE predictions. Hence, the NDE often is cited as providing striking confirmation of RMT predictions for the GOE 17. KČSF, Žilina, 7. 9. 2011

Problems with analysis of data Analysis of data extremely complicated - main problems Completeness. No missing resonances. e.g., due to finite detection threshold 17. KČSF, Žilina, 7. 9. 2011

Effect of Missing Small Widths Missing small widths changes shape of the distribution Gn(0)/< Gn(0) > = T * E/Emax T… threshold If it’s assumed that all widths were observed, one obtains larger n from maximum-likelihood (ML) analysis ntrue = 1.0 x nML = 1.9 Experiment threshold must be accounted for in comparison to theory 17. KČSF, Žilina, 7. 9. 2011

Problems with analysis of data Analysis of data extremely complicated - main problems Completeness. No missing resonances. e.g., due to finite detection threshold Purity. i.e., no p-wave resonances in s-wave set 17. KČSF, Žilina, 7. 9. 2011

Effect of p-wave Contamination p-wave widths smaller than s-wave ones G(0)n,s ~ constant G(1)n,p ~ En Simulation for 232Th (assuming GOE) Assuming all widths above NDE threshold are s wave results in smaller n from ML analysis. ntrue = 1.0 x nML = 0.6 Comparison to theory must assess purity of the data 17. KČSF, Žilina, 7. 9. 2011

Problems with analysis of data Analysis of data extremely complicated - main problems Completeness. No missing resonances. e.g., due to finite detection threshold Purity. i.e., no p-wave resonances in s-wave set Limited number of resonances. Data spread out over broad PT distribution Always fighting limited statistical precision P.Koehler – EPJ Conferences (2010) Many problems with the NDE – properties (Jp) of some of the resonances were ascribed to follow the predictions of RMT 17. KČSF, Žilina, 7. 9. 2011

ORELA The Oak Ridge Electron Linear Accelerator The facility consists of a 180-MeV electron accelerator neutron producing targets buried and evacuated flight tubes up to 200 m long leading to underground detector locations detectors, data acquisition and analysis systems Neutrons are produced by bremsstrahlung from a Ta radiator Moderated or unmoderated neutrons are available (further tailoring of the spectral shape is done with movable filters). Pulse widths from 4 - 30 ns are available at a repetition rates 12 - 1000 Hz. 17. KČSF, Žilina, 7. 9. 2011

ORELA 17. KČSF, Žilina, 7. 9. 2011

Transmission Setup at ORELA ORELA Measurements Transmission Setup at ORELA Electron Beam Filters T.O.F. Blank Neutrons Sample T.O.F. 6 Li-glass Neutron Production Collimator Detector Target 9 m 80 m 17. KČSF, Žilina, 7. 9. 2011

Capture Setup at ORELA ORELA Measurements Neutrons Electron Beam Neutron Production Target Collimator Neutrons Filters Deuterated Benzene Detectors 40 m Sample Capture Setup at ORELA 17. KČSF, Žilina, 7. 9. 2011

ORELA Measurements Resonance parameters have to be deduced from measured data 17. KČSF, Žilina, 7. 9. 2011

ORELA Measurements Response of the measuring system must be simulated Only elastic n scattering and (n,g) reaction are allowed at low neutron energies Simultaneous fitting of transmission and capture data allow to get all the parameters, i.e. also neutron widths Shapes of strong resonances in transmission experiment allow to distinguish s- from p-wave neutron resonances Identification of weak resonances is problematic 17. KČSF, Žilina, 7. 9. 2011

Testing the PTD Using 192,194Pt+n ORELA Data 192,194,196Pt+n ORELA data are excellent - better in many ways than other experiments: More resonances Better sensitivity (~10x) Better separation of s and p waves (<Gn(0)>≈ 10 <Gn(1)>) Better Jp assignments Improved Maximum-Likelihood analysis: Used energy-dependent threshold Maximizes statistical significance while eliminating p-wave contamination Analysis threshold T0 much higher than experimental one 17. KČSF, Žilina, 7. 9. 2011

Testing the PTD Using 192,194Pt+n ORELA Data (Phys. Rev. Lett. 105, 072502 (2010)) Maximum-Likelihood (ML) analysis: 192Pt: n = 0.57±0.16 194Pt: n = 0.47±0.19 196Pt: n = 0.60±0.28 The confidence levels (CL) from ML analysis are sometimes problematic 17. KČSF, Žilina, 7. 9. 2011

Testing the PTD Using 192,194Pt+n ORELA Data The confidence levels (CL) from ML analysis are sometimes problematic 17. KČSF, Žilina, 7. 9. 2011

Testing the PTD Using 192,194Pt+n ORELA Data (Phys. Rev. Lett. 105, 072502 (2010)) Maximum-Likelihood (ML) analysis: 192Pt: n = 0.57±0.16 194Pt: n = 0.47±0.19 196Pt: n = 0.60±0.28 Additional simulations assuming n = 1 performed and the probability that the low value of n obtained from ML analysis checked as a function of <Gn0> 17. KČSF, Žilina, 7. 9. 2011

Testing the PTD Using 192,194Pt+n ORELA Data (Phys. Rev. Lett. 105, 072502 (2010)) CL is very high but depends on <Gn0> due to the experimental threshold applied - <Gn0> was “fixed” in simulations Two additional statistics applied to limit the range of <Gn0> n = 1 assumed in all the statistics 17. KČSF, Žilina, 7. 9. 2011

Testing the PTD Using 192,194Pt+n ORELA Data (Phys. Rev. Lett. 105, 072502 (2010)) Auxiliary ML analysis to verify that p-wave contamination is negligibly small The probability 0.069 for 192Pt and 0.0047% for 194Pt obtained At most one p-wave resonance can occur above the threshold p-wave s-wave 17. KČSF, Žilina, 7. 9. 2011

Testing the PTD Using 192,194Pt+n ORELA Data Combining all the restrictions for two nuclei PTD rejected at 99.997% CL 17. KČSF, Žilina, 7. 9. 2011

Conclusions Our data reject the validity of the Porter-Thomas Distribution with a statistical significance of at least 99.997% This inescapable conclusion has been made thanks to rich experimental data obtained using state-of-the-art neutron spectroscopy, and the implementation of a novel approach for testing the PT distribution Many possible consequences Basic assumptions used in the theory of nuclear reactions seem to be violated but the degree of violation is unknown The violation can hold not only for neutron widths but also transition probabilities in different channels Might influence many results 17. KČSF, Žilina, 7. 9. 2011

Testing the PTD Using 192,194Pt+n ORELA Data 17. KČSF, Žilina, 7. 9. 2011

Testing the PTD Using 192,194Pt+n ORELA Data (Phys. Rev. Lett. 105, 072502 (2010)) Limits on <Gn0> obtained 17. KČSF, Žilina, 7. 9. 2011