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

SFB 634 – C4: Quantum Chaos

● 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

Lund 2005

* 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)

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

s

P(s

)

Ensemble of 18764 resonance frequencies of a 3D-microwave resonator

s

P(s

)Ensemble of 1726 highly excited

nuclear states of the same spin and parity: `Nuclear Data Ensemble´

● Highly excited nuclei (many-body quantum chaos) and chaotic microwave resonators (one-body quantum chaos) exhibit a universal (generic) behaviour

Properties of spectral fluctuations I

Properties of spectral fluctuations II

● The low-lying Scissors Mode and integrable microwave resonators exhibit the same universal (non-generic) behaviour

Ensemble of 152 1+ states in 13 heavy deformed nuclei

between 2.5 and 4 MeV

Scissors Mode in deformed nuclei

s

LP

(s)

∆3(

L)Regular (integrable) elliptic billiard

Ensemble of 300 resonance frequencies

s

L

P(s

)∆

3(L)

L

P(s

)s

∆3(

L)

Limaçon billiard of mixed dynamics

Ensemble of 800 reso-nance frequencies

● Short and long range level-level correlations lie between Poisson (integrable) and GOE (chaotic) behaviour. Do we understand this coexistence ?

Properties of spectral fluctuations III

L

P(s

)

s

∆3(

L)

Pygmy Dipole Resonance in 138Ba, 140Ce, 144Sm and 208Pb

Ensemble of 154 1- states in three semimagic (N=82) nuclei and

one magic (Z=82, N=126) nucleus between 5 and 8 MeV

np, n

1. 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´

Definition: `generic´

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

The Scissors mode DALINAC 1984

The PDR mode S-DALINAC 2002

● Level statistics

● Width distributions

● Certain POs (BBOs)

● Collective rotations and vibrations, i.e. `ordered motion´

The nuclear electric dipole response

PDR

GDR

(2+ x 3- )1-

p n

B(E1)

E (MeV)

15

3

Electric dipole response of neutrons and protons in QPM calculations for 138Ba

● Evidence for surface neutron density oscillations

● “Soft dipole mode“ at 7 MeV is dominantly isoscalar

● Influence on the spectral fluctuation properties ?

neutronsprotons

r2ρ

(r)

Photon scattering off 138Ba

138BaEmax = 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)

-3 2

● 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-

)1(

)1(log10 EB

EBz

● The strengths show Porter-Thomas (PT) statistics for the QPM, while

the experimental distribution deviates from PT.

● Experiment and QPM show spectral properties in between GOE and

Poisson statistics.

Possible interpretations of the observedfluctuation properties

● The missing levels destroy spectral correlations.

● Limited statistics (low number of levels) affect the

spectral fluctuation properties.

● Coexistence of regular nuclear and chaotic nuclear motion: intermediate

or “mixed“ statistics.

Qualitative modelling of the missing level effect

Obtain a subset of the states calculated within the QPM

by cutting away the weakest transitions below the experimentaldetection limit of about 10-3 e2 fm2

1200 levels

184 levels

184 levels

● They are close to Poisson with some remnants of level repulsion

(limited to the lowered probability in the first bin).

● All three distributions show similar behaviour for experiment,

truncated QPM and full QPM.

Transition strength distributionsRMT predicts in case of GOE correlations that

the wave function components or, equivalently, their squares follow a Gaussian or Porter-Thomas distribution, respectively.

1200 levels

184 levels

184 levels

● 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.

● Strength distribution of the full QPM agrees with PT, while experiment

and truncated QPM deviate from PT statistics, but in a similar way.

QPM matrix elements and missing strength

● Overall distribution of coupling matrix elements (for 2p2h-2p2h and 1p1h-2p2h interactions) is not a Gaussian

● Nevertheless: we have been able to understand certain statistical features of the PDR ( J. Enders, Nucl. Phys. A741 (2004) 3)

● 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)

● Few large matrix elements indicative of collective configurations lie in the tails of the distribution

How can the problem of the missing strength be overcome?

● Superconducting microwave resonators (Q 106) shaped as billiards allow the determination of all eigenfrequencies and resonance strengths

● 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 ̂= E =̂

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 E

● Resonance strengths are directly related to the squared wavefunction components at the positions of the antennas for microwave in- and output

● However, information on wavefunction components can also be extracted from the shape of the resonances in the measured spectra of superconducting billiards

● Transmission measurements: relative power from antenna a b

2

abain,bout, SP/P

Resonance parameters

Very high signal to noise

ratio

● Partial widths: a, b

b,a,i

μiμ

small for superconducting resonators

● Resonance strengths: a·b determined from transmission measurements

● Open scattering system: a resonator b

μμ

μbμa

abab

2ff

ΓΓS

ii

● Frequency of ‘th resonance: f

Resonance parameters

(+ dissipative terms) ● Total width:

● controls the degree of chaoticity

Billiards of mixed and chaotic dynamics

● Boundary of Limaçon billiards given by a mapping from z w

w = z + z2

● Measurements for altogether 6 antenna combinations about 5000 strengths were determined

Total widths and strengths of the Limaçon billiard

● Secular variation of the ‘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

● 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)

Resonance strengths distributions

=log10(ab)

● Strong deviations from GOE for the billiards with mixed dynamics demonstrated for the first time

● GOE prediction corresponds to the distribution of the product of two PT distributed random variables a and b: modified Bessel function K0

● RMT models must be developed to describe systems of mixed dynamics

K0 - distribution

modified strength distributiondue to experimental detectionlimit

=0.3

Modified strength distribution

● 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

● 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

Strength distribution and symmetry breaking

3+; T=0

1+; T=0

4+; T=0

2+; T=0

1+; T=0

3+; T=0

5+; T=0

0+; T=1

2+; T=1

75 levels: T=032 levels: T=1 mixing: <Hc>

● Observed statistics between 1 GOE and 2 GOE

(Mitchell et al., 1988) (Guhr and Weidenmüller, 1990)

Isospin mixing in 26Al

PT distribution

PT distribution

● Study of strength distributions of resonances in coupled microwave billiards

Transition probabilities in 26Al and 30P

● For both nuclei deviations from GOE prediction signature of isospin mixing (Mitchell et al., 1988, Grossmann et al., 2000)

● GOE prediction for the distribution of reduced transition probabilities ( partial widths) of systems without or with complete symmetry breaking is a Porter-Thomas distribution

0

0

GOE0

0GOE)(

2

1

TV

VH

● RMT model for Hamiltonian of a chaotic system with a broken symmetry

● = 0 no symmetry breaking: 2 GOE‘s

● 0 < <1 partial symmetry breaking

● = 1 complete symmetry breaking: 1 GOE

● = / D is the relevant parameter governing symmetry breaking; D is the mean level spacing

RMT model for symmetry breaking

● Large number of resonances (N1500)

● Variable coupling strength resp. degree of symmetry breaking

Coupled billiards as a model for symmetry breaking

Experimental set-up

● Coupling was achieved by a niobium pin introduced through holes into both resonators

uncoupled

weakly coupled

strongly coupled

Changing the coupling strength

2-statistics for different coupling strengths

uncoupled

weakly coupled

strongly coupled

22 )()( LLNL

Analysis of spectral properties

● normalized spreading width:

25.022

DD

every fourth state influenced by the coupling

● largest coupling achieved in experiment:

Coulomb matrix element26).0Al:(in21.0/ 26 D / D

● Resonances with small strengths cannot be detected experimental threshold of detection

RMT model for the strength distributions

experimental threshold

=0.13

=0.04

K0-Distribution=0.3

● Position of central maximum depends on coupling strength, i.e. on the symmetry breaking

=0.04 =0.09

=0.21=0.14

● Examples for one antenna combination show very good agreement with RMT fits

Experimental strength distribution for different couplings

● Symmetry breaking parameters extracted from spectral statistics (circles) agree with those from strength distribution (crosses)

001.0032.0 λ

001.0099.0 λ

001.0142.0 λ

003.0224.0 λ

Antenna combination

Comparison of results for coupling parameters

● Generic properties of the eigenfunctions of a chaotic billiard can be studied experimentally using the strength distributions for a microwave billiard.

Summary on symmetry breaking effects

● Various spectral measures can be used to extract the coupling strength and give consistent results.

● Changing the coupling strength influences the level and strength distributions of the coupled stadiums.

● Precise and significant tests of present RMT models for symmetry breaking are possible.

● Maximum normalized spreading width, i.e the deviation from generic behaviour, observed / D = 0.20 - 0.25 corresponds to the nuclear case of 26Al.

● Symmetry breaking in nuclei ( J.F. Shriner et al., Phys. Rev. C71 (2005) 024313) can be very effectively modelled through billiards.