1. Chaos and Regularity in the Excitation Spectrum of the Doubly Magic Nucleus 208Pb
Nice 2017
Introduction
Quantum chaos and the nuclear many-body system
Fluctuations in the energy spectrum of 208Pb
Analysis of the fluctuations with a random matrix model
Analysis of the fluctuations with the method of Bayesian inference
Published in Phys. Rev. Lett. 118, (2017)
Supported by DFG within the Collaborative Research Centers 634 and 1245
B. Dietz (Lanzhou University, China), A. Heusler (Heidelberg, Germany), K.H. Maier (Polish Academy of Sciences Kraków, Poland), A. Richter (TU Darmstadt, Germany), B.A. Brown (MSU, USA)
2017 | SFB 1245 | Achim Richter | 1

2. Experiment and Model Calculations for 208Pb
Experiment: High-resolution measurements (3 keV FWHM) of 208Pb(p,p´), 206,207,208Pb(d,p) and 208Pb(d,d´) reactions were used to identify a complete sequence of 151 bound states of 208Pb in energy, spin and parity below 6.20 MeV [A. Heusler et al., Phys. Rev. C 93, (2016)]
KB: Realistic effective nucleon-nucleon interaction based on the Hamada-Johnston potential [T.T.S. Kuo & G.E. Brown, Nucl. Phys. 85, 40 (1966)]
SDI: Surface-delta interaction acts only at the nuclear surface and provides a simple extension of the schematic shell model [S.A. Moszkowski, Phys. Rev. C 2, 402 (1970)]
M3Y: Realistic effective nucleon-nucleon Michigan-3-Yukawa interaction with short-range components determined with the help of the Reid nucleon-nucleon potential [G. Bertsch et al., Nucl. Phys. A 284, 399 (1977)]

3. Experimental and Computed Level Schemes of 208Pb
Aim: Study of the fluctuation properties of the levels in the experimental spectrum and comparison with results obtained from shell-model calculations with the three different interactions

4. Quantum Chaos and the Nuclear Many-Body System

5. Reminder: Spectral Properties of Integrable and Chaotic Systems
One aspect of quantum chaos: Understanding of the features of the classical dynamics based on the fluctuation properties in the corresponding quantum spectra
Bohigas-Giannoni-Schmit conjecture The fluctuation properties in the eigenvalue spectrum of a generic time-reversal invariant chaotic system coincide with those of real symmetric random matrices from the Gaussian Orthogonal Ensemble (GOE).
Gutzwiller & Berry-Tabor conjecture The fluctuation properties in the eigenvalue spectrum of a generic integrable system behave like independent random numbers from a Poissonian process.

6. Universality of the Fluctuation Properties
atomic nucleus
quantum billiard /
flat microwave resonators
quantum graphs /
microwave networks f f E
System specific properties have to be removed to observe universality
Uniform average level density needed  unfolding of spectra

7. Unfolding of Spectra
Replace eigenvalues Ei by the smooth part of the integrated spectral density
Integrated spectral density N( E ) = # levels below E
Decomposition into a smooth and a fluctuating part
Quantum billiard / Microwave billiard
Weyl formula:
Nuclear levels: no analytical expression for exists
Use polynomial fit or an empirical constant-temperature formula

8. Statistical Measures for Short-Range Correlations
NNSD: distribution of the spacings between adjacent level
 Short-range fluctuations
e i1
e i
e i+1
Number variance:
 Long-range fluctuations
Distribution of ratios of two consecutive spacings
Advantage: ratios are dimensionless  no unfolding required

9. Example: Ratio Distribution in a Graphene-Like Structure
P(r)
I(r)
1.0
0.5
0.0
For an illustration we show the results for a microwave (quantum) billiard containing metallic cylinders (scatterers) arranged on a triangular lattice
The spectral density obviously is non-uniform
The ratio distribution is the same for original / unfolded levels

10. Quantum Chaos and Nuclear Many-Body Systems
Quantum chaos refers to the study of manifestations of classical chaos in the corresponding quantum system
However: No obvious classical analogue exists in nuclear many-body systems, even though the fluctuations in the spectra exhibit similar features as a classically regular / chaotic quantum system
Nuclear systems: Poissonian / GOE behaviour is related to collectivity / complexity of the motion of the nucleons [T. Guhr, Act. Phys. Pol. 116, 741 (2009)]

11. Indispensable Requirements for the Analysis of Fluctuation Properties
Completeness of the level sequences. Missing levels imply severe changes in the fluctuation properties
Unambiguous assignment of the states to the relevant symmetry classes like, e.g., spin J and parity p
Either needs level sequences with similar fluctuation properties containing at least 5 levels in an ensemble of many nuclei or a complete sequence of at least ~100 levels in one nucleus

12. Nuclear Data Ensemble R. U. Haq et al. , Phys. Rev. Lett
D3(L) L
P(s) s
Nuclear data ensemble: 1726 resonance energies corresponding to 30 sequences of 27 different nuclei
Data obtained from neutron time-of-flight spectroscopy and from high-resolution proton scattering  in a region far from ground-state
Spectral properties agree well with those of random matrices from the GOE

13. Ensemble of Low-Lying Nuclear Scissors Modes J. Enders et al. , Phys
Ensemble of Low-Lying Nuclear Scissors Modes J. Enders et al., Phys. Lett. B 486, 273 (2000) s
P(s)
scissors mode
TUD (1984)
152 levels from 13 heavy deformed nuclei between 2.5 and 4 MeV
The spectral properties coincide with those of uncorrelated random numbers (Poisson)
Interpretation: collective rotational oscillations of deformed neutron and proton rigid rotors against each other

14. Isospin Mixing in 26Al and 30P G. E. Mitchell et al
Isospin Mixing in 26Al and 30P G.E. Mitchell et al., PRL 61, 1473 (1988)
75 levels: T=0
32 levels: T=1
Observed statistics between GOE and 2 GOE in 26Al and 30P
Attributed to isospin symmetry breaking / mixing caused by the Coulomb interaction
[T. Guhr and H. Weidenmϋller, Ann. Phys. 199, 412 (1992)]

15. Spectral Properties of 100 Energy Levels in 116Sn below 4. 3 MeV S
Spectral Properties of 100 Energy Levels in 116Sn below 4.3 MeV S. Raman et al., Phys. Rev. C 43, 521 (1991)
Deviations attributed to missing levels and erroneous spin assignments
Up to now, spectral properties close to GOE have only been observed in ensembles of nuclei

16. Fluctuations in the Energy Spectrum
of 208Pb

17. Complete Identification of States in 208Pb below 6. 20 MeV A
Complete Identification of States in 208Pb below 6.20 MeV A. Heusler et al., Phys. Rev. C 93, (2016)
Triangle-up:
79 natural-parity states
Triangle-down:
49 unnatural-parity states
Below 6.20 MeV a complete sequence of 151 levels was identified
For each state, spin J and parity p were determined unambiguosly

18. Fluctuating Part of the Integrated Spectral Density of the Complete Sequence
All 151 energy levels irrespective of their spin and parity were used
N fluc fluctuates around zero  clear indication that the spectrum is complete

19. Spectral Properties of the Complete Level Sequence
Spectral properties agree with those of random Poissonian numbers
Reason: Superposition of states belonging to different symmetry classes
In order to attain information on the underlying nucleon dynamics, the spectra need to be separated into subspectra characterized by J p

20. Analysis of the Fluctuation Properties
Unfold each subspectrum characterized by spin J and parity p separately
We ensured that the spectral properties do not depend on the unfolding procedure used (polynomial fit / the empirical constant-temperature formula)
Two approaches:
Compute statistical measures for each subspectrum separately and then their ensemble average  compare results with a random-matrix ensemble interpolating between Poisson and GOE
Consider the complete spectrum composed of the independent subspectra of unfolded energy levels  analyse fluctuation properties using the method of Bayesian inference

21. Analysis of the Fluctuations with a Random Matrix Model

22. NNSDs of the Subspectra Characterized by Jp
In most cases the agreement with the NNSD of the GOE is better than that with Poisson  study spectral properties of the ensemble

23. Ensemble Averages of Statistical Measures for Fluctuation Properties
The spectral properties of the ensemble agree well with those of random matrices from the GOE, thus indicating chaoticity of the nuclear system

24. RMT Model for Poisson to GOE Transitions V. K. B
RMT Model for Poisson to GOE Transitions V.K.B. Kota, Lecture Notes in Physics 884, Chapter 3.2
Ansatz for random matrices with spectral properties intermediate between Poisson and GOE statistics
Poisson
GOE
H0: diagonal matrix containing random Poissonian numbers
H1: random matrix from the GOE
Variances of matrix elements were chosen such that the eigenvalue ranges of H0 and H1 coincided
Spectral properties are indistinguishable from GOE for l  1-2
Wigner-like approximation for the NNSD in terms of the I0(x) Bessel function and the U(a,b,x) Kummer function [G. Lenz & F. Haake, PRL 67, 1 (1991)]

25. Comparison of the Spectral Properties of 208Pb with RMT Model
The chaoticity parameter l was determined from a fit of the analytic NNSD to the experimental P(s) and by comparing the number variance obtained from the RMT model with the experimental S2(L)

26. Spectral Properties of States with Unnatural / Natural Parity
Unnatural parity
Natural parity
l=1.00
l=1.20
Separate spectra into levels with unnatural parity p=(-1)J+1 and natural parity p=(-1)J
Spectral properties are closer to GOE for natural parity states than for unnatural parity states  residual interaction stronger for the former

27. Calculations within One-Particle One-Hole Shell Model Spaces82
Couplings between nucleon particles and holes in these orbitals were used in the shell model calculations with the SD / KB / M3Y interactions

28. Comparison of Experimental Results with Model Calculations
For the M3Y and KB interactions the spectral properties are close to GOE
The S2 statistics hints at a small contribution from regular behavior for the KB interaction
But: For the SDI the spectral properties are close to that of Poissonian random numbers

29. Comparison of Experimental Results with Model Calculations: D3
The D3 statistics is close to that for the GOE for the experimental data and the KB and M3Y interactions
The SDI model exhibits Poissonian features

30. Comparison of Experimental Results with Model Calculations: Ratio Distribution
The ratio distribution hints at a small contribution from regular behavior for the KB interaction
The SDI model exhibits Poissonian features

31. Comparison of Experimental Results with Model Calculations: S2
Unnatural parity
Natural parity
l=1.00
l=0.10
l=1.20
l=0.05
l=0.82
l=0.50
l=0.85
l=1.10
The S2 statistics hints at a small contribution from regular behavior for the unnatural parity states obtained with KB interaction
Spectral properties are closer to GOE for natural parity states than for unnatural parity states
The SDI model exhibits Poissonian features

32. Analysis of the Fluctuations with the Method of Bayesian Inference

33. Superimposed Subspectra N. Rosenzweig, C. E. Porter, Phys. Rev
Merge the unfolded subspectra irrespective of their spin and parity into one sequence of spacings si
Approximate expression for the NNSD of m superimposed subspectra with fractional level numbers fi, where 0  fi  1 (m  14), which depends on one parameter
f0 corresponds to a large number of subspectra containing few levels  approaches Poisson
f1 corresponds to one symmetry class with many levels  approaches GOE
Thus, f is referred to as chaoticity parameter

34. Method of Bayesian Inference A. Y. Abul-Magd, H. L. Harney, M. H
Method of Bayesian Inference A.Y. Abul-Magd, H.L. Harney, M.H. Simbel, H.A. Weidenmüller, Ann. Phys. 321, 56 (2006)
Assumption: The N spacings si are independent (N  130)
Joint probability distribution (proposed distribution)
Bayes´ theorem yields for the posterior distribution
The prior distribution m( f ) of f is obtained with Jeffrey‘s rule
The chaoticity parameter and the variance s are computed as

35. Comparison of NNSDs with Results Obtained with the Method of Bayesian Inference
Experiment
SDI
M3Y KB
all
natural
unnatural
The chaoticity parameters determined with the method of Bayesian inference are in agreement with those obtained from the RMT model

36. Spectral Properties of the Unperturbed Eigenvalues of the M3Y Hamiltonian
If the off-diagonal elements in the M3Y case with Jp are disregarded the spectral properties of the unperturbed eigenvalues are characterised by chaoticity parameters that are close to that for SDI. This corroborates the supposition that the SDI is too weak to describe the experimental findings.

37. Summary
We analyzed the complete spectrum of 208Pb below 6.20 MeV with unambiguously assigned spin an parity
We performed the same analysis for spectra computed on the basis of shell-model calculations including the SD, KB and M3Y interaction
Comparison with an RMT model interpolating between Poisson and GOE and the application of the method of Bayesian inference yielded chaoticity parameters close to GOE for the experimental KB and M3Y spectra
Natural parity states are closer to GOE than unnatural parity states  residual interaction stronger for natural than for unnatural parity states
The SDI model suggests a behavior close to Poisson  the SDI seems to be too weak to induce sufficient mixture of the individual configurations to lead to a chaotic behavior, i.e., it fails to describe the fluctuation properties of the experimental spectra

38. The Music of Nuclear Energy Level Spacings in 208Pb (Alex Brown)
(a) Equal level spacing
(b) Poisson
(c) GOE [
For (c) take the M3Y interaction which describes the spectral fluctuation properties of 208Pb best with a chaoticity parameter close to experiment.
A computer program turns these distributions into musical notes. Starting from the bottom, the time interval between notes is determined by energy to the next level. This energy interval is also used to change the relative pitch, alternating up and down relative to the last note.
(a)
(b)
(c)
(trio)
30. November 2018 |