on some spectral properties of billiards and nuclei – similarities and differences* sfb 634 –...
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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)
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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
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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
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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)
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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
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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´
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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).
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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´
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The nuclear electric dipole response
PDR
GDR
(2+ x 3- )1-
p n
B(E1)
E (MeV)
15
3
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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)
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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
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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
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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.
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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.
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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.
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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.
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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
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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 =̂
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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
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● Transmission measurements: relative power from antenna a b
2
abain,bout, SP/P
Resonance parameters
Very high signal to noise
ratio
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● 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:
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● 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
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● 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
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● 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
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● 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
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● 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
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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
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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
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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
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● Large number of resonances (N1500)
● Variable coupling strength resp. degree of symmetry breaking
Coupled billiards as a model for symmetry breaking
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Experimental set-up
● Coupling was achieved by a niobium pin introduced through holes into both resonators
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uncoupled
weakly coupled
strongly coupled
Changing the coupling strength
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2-statistics for different coupling strengths
uncoupled
weakly coupled
strongly coupled
22 )()( LLNL
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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
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● 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
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=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
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● 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
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● 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.