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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

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on some spectral properties of billiards and nuclei similarities and differences
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





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






Regular (integrable) elliptic billiard

Scissors Mode in deformed nuclei





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






Limaçon billiard of mixed dynamics

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





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



● Certain POs (BBOs)

● Collective rotations

and vibrations, i.e.

`ordered motion´

● Level statistics

● Width distributions

  • The Scissors mode


  • The PDR mode



The nuclear electric dipole response

E (MeV)






(2+ x 3-)1-




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 ?


Photon scattering off 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 zw

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


● 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



Modified strength distribution

K0 - distribution

modified strength distribution

due to experimental detection



● 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 (N1500)
  • ● 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


weakly coupled

strongly coupled


S2-statistics for different coupling strengths


weakly coupled

strongly coupled


Analysis of spectral properties

● largest coupling achieved in experiment:


















 Coulomb matrix element

● normalized spreading width:

  • every fourth state influenced

by the coupling


RMT model for the strength distributions






● 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






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


● Symmetry breaking in nuclei ( J.F. Shriner et al.,Phys. Rev. C71 (2005) 024313)

can be very effectively modelled through billiards.