Regular & Chaotic. collective modes in nuclei. Pavel Cejnar. ipnp.troja.mff.cuni.cz
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1 Pavel Cejnar Regular & Chaotic collective modes in nuclei Institute of Particle and Nuclear Physics Faculty of Mathematics and Physics Charles University, Prague, Czech Republic ipnp.troja.mff.cuni.cz ) Classical chaos & its visualization ) Classical chaos in the Geometric Collective Model & Interacting Boson Model 3) Quantum chaos: approaches by Bohigas & Peres 4) Quantum-chaos measures in GCM & application in IBM Jerusalem 0
2 a/0 Physics of the st kind encoding complex behavior simple equations rot rot E H B t D t j div D divb ρ 0
3 b/0 Physics of the st kind encoding complex behavior simple equations complex behavior simple equations decoding Physics of the nd kind Sequence of complex numbers Mandelbrot set values of c for which the sequence is bound ( < for n ) z n z z z z z 0 3 c c ( c c) c ( ) z c n n z n, c C c
4
5 /0 Classical chaos Trajectories flow in the phase space incompressible fluid volume conserved shape can get very complicated time exponenstial sensitivity on initial conditions butterfly-wing effect practical loss of exact predictability But quasi-ergodic character of trajectories in the phase space applicability statistical description
6 3 /0 Classical chaos Trajectories flow in the phase space incompressible fluid volume conserved shape can get very complicated time Visualization method for D systems Poincaré section Momentum 4D Phase Space Coordinate Fixed exponenstial sensitivity on initial conditions butterfly-wing effect order Momentum determined from E Coordinate chaos
7 4 /0 Classical chaos Billiard systems D cavities of different shapes (integrable or nonintegrable) Fraction of regular/chaotic orbits does not depend on E Eccentric annular billiard Coordinates (r, φ) Momenta (p r,l φ ) Bohigas et al. NPA 560, 97 (993) Frischat, Doron PRE 57, 4 (998) Dembowski et al. PRL 84, 867 (000) Hentschel, Richter PRE 66, (00) Poincaré sections at r R L φ φ
8 5a/0 Geometric Collective Model A. Bohr 95 Gneuss et al. 969 quadrupole tensor of collective coordinates ( shape 3 Euler angles 5D ) corresponding tensor of momenta H H Hamiltonian (0) (0) 35 [ π π ]... 5A[ α α] B α K () (0) (0) [ α α] α ] 5C( [ α ] )... 5 ' J π γ 3 4 π... A B cos3 C... x, y, z (, ) K γ I γ V T rot T vib K ( π π ) x y neglect Angular momentum J m i 0[ α π m * ] () Shape variables y sin γ Re α ± x cosγ Reα 0 PAS PAS Principal Axes System
![5b/0 Geometric Collective Model A. Bohr 95 Gneuss et al. 969 quadrupole tensor of collective coordinates ( shape 3 Euler angles 5D ) corresponding tensor of momenta H H Hamiltonian (0) (0) 35 [ π π ].](/docs-images/92/110787592/images/9-5.jpg ".. 5A[ α α] B α K () (0) (0) [ α α] α ] 5C( [ α ] )... 5 ' J π γ 3 4 π... A B cos3 C.")
9 5b/0 Geometric Collective Model A. Bohr 95 Gneuss et al. 969 quadrupole tensor of collective coordinates ( shape 3 Euler angles 5D ) corresponding tensor of momenta H H Hamiltonian (0) (0) 35 [ π π ]... 5A[ α α] B α K () (0) (0) [ α α] α ] 5C( [ α ] )... 5 ' J π γ 3 4 π... A B cos3 C... x, y, z (, ) K γ I γ V T rot T vib K ( π π ) x y neglect Angular momentum J 0[ * ] () m i α π m 0 effectively D system Shape variables y sin γ Re α ± x cosγ Reα 0 PAS PAS Principal Axes System
10 5c/0 Geometric Collective Model A. Bohr 95 Gneuss et al. 969 quadrupole tensor of collective coordinates ( shape 3 Euler angles 5D ) corresponding tensor of momenta H H Hamiltonian (0) (0) 35 [ π π ]... 5A[ α α] B α K () (0) (0) [ α α] α ] 5C( [ α ] )... 5 neglect higher-order terms ' J π γ 3 4 π... A B cos3 C x, y, z (, ) K γ I γ V T rot... Shape-phase diagram T vib K ( π π ) x y B A prolate spherical oblate Shape variables y sin γ Re α ± x cosγ Reα 0 PAS PAS neglect C >0 Principal Axes System
11 6 /0 Geometric Collective Model independent scales energy time coordinates but a fixed value of Planck constant Hamiltonian 5 (0) (0) 35 () (0) (0) H [ π π ]... 5A[ α α] B [ α α] α ] 5C( [ α α] )... K External parameters neglect Two essential parameters ) shape parameter τ ) classicality parameter κ B AC K B A prolate oblate spherical C >0 path crossing all parabolas integrability all equivalent classes of Hamiltonians
12 7a/0 Geometric Collective Model Classical chaos map of chaos Stránský, Cejnar : PRL 93, 050 (004), PRC 74, (006), NPNews(0) 0 >0 0 0 Regular phase space fraction f reg Ω reg / Ω tot chaos order
13 7b/0 Geometric Collective Model Classical chaos Stránský, Cejnar : PRL 93, 050 (004), PRC 74, (006), NPNews(0) map of chaos & Poincaré sections 0>0 x y0 00 x Regular phase space fraction f reg Ω reg / Ω tot chaos 50,000 passages of 5 randomly chosen order through trajectories the section y0
14 8 /0 Geometric Collective Model Classical chaos map of chaos & convex-concave transition Stránský, Cejnar : PRL 93, 050 (004), PRC 74, (006), NPNews(0) 0 >0 0 0 convex concave y change of the shape of the border of the accessible domain in the xy plane y concave convex y y Regular phase space x fraction f reg Ω reg / Ω tot x x x chaos order
 More than classical control parameter](/docs-images/92/110787592/images/15-7.jpg "multi-dimensional chaotic map.")
15 9 /0 Interacting Boson Model U(5) η0.75 η0.70 η SU(3) 0 spherical deformed χ J 0, E 0 O(6) 0 Macek, Cejnar, Jolie 007 H η nd N ( η )(Qχ Qχ ) ~ nd d d ~ ~ Qχ d s s d χ [d d ]( ) More than classical control parameter multi-dimensional chaotic map. But there exist regions of almost full compatibility with the GCM. B χ
16 0a/0 Interacting Boson Model U(5) η0.70 η0.50 η SU(3) 0 f reg spherical deformed χ O(6) 0 Macek, Cejnar, Jolie 007 Macek, Cejnar, Dobeš 009 Macek (PhD) 00 H η nd N ( η )(Qχ Qχ ) ~ nd d d ~ ~ Qχ d s s d χ [d d ]( ) f reg J0 E E η 0.5 χ η 0.7 f reg Ω reg / Ω tot χ
17 0b/0 Interacting Boson Model U(5) η0.70 η0.50 η SU(3) 0 f reg spherical J0 H η nd N ( η )(Qχ Qχ ) deformed O(6) χ 0? Macek, Cejnar, Jolie 007 Macek, Cejnar, Dobeš 009 Macek (PhD) 00 ~ nd d d ~ ~ Qχ d s s d χ [d d ]( ) f reg Change of stability of some high-energy and γ vibrations? E E ω ωγ Change of stability of low-energy and γ vibrations η 0.5 χ f reg Ω reg / Ω tot η 0.7 ω ωγ χ
18 /0 Quantum chaos No genuinely quantum definition of chaos (linearity & quasi periodicity of quantum mechanics) > chaos studied in connection with classical limit Bohigas conjecture (984): Chaos on quantum level affects statistical properties of discrete energy spectra. Chaotic systems yield strong short- and long-range spectral correlations consistent with the Gaussian random matrix model. Spectra from D billiards (Wigner) Nuclear data: ensambles of neutron & proton resonances [Bohigas, Haq, Pandey 983] low-energy levels [Von Egidy et al. 987] ω0.6 P( s) α ω Γ( s ω e ω ω ω ) Nearest-neighbor spacing ω Ei E α i ω s si E E i i Brody distribution interpolates between Poisson (ω0) order Wigner (ω) chaos
![Peres definition (984): fidelity [PRA 30, 60] Quantum chaotic systems show an exponential sensitivity to details of the Hamiltonian (control parameters) Peres visual method (984): spectral lattices](/docs-images/92/110787592/images/19-6.jpg "[PRL 53, 7] Integrals of motions for non-integrable systems: T [ P, H ] 0 P lim PH ( t) dt T T 0 Energy spectrum represented as a planar lattice i th eigenstate of H Regular system P (integrable) i")
19 /0 Quantum chaos No genuinely quantum definition of chaos (linearity & quasi periodicity of quantum mechanics) > chaos studied in connection with classical limit Bohigas conjecture (984): Chaos on quantum level affects statistical properties of discrete energy spectra. Chaotic systems yield strong short- and long-range spectral correlations consistent with the Gaussian random matrix model. Peres definition (984): fidelity [PRA 30, 60] Quantum chaotic systems show an exponential sensitivity to details of the Hamiltonian (control parameters) Peres visual method (984): spectral lattices [PRL 53, 7] Integrals of motions for non-integrable systems: T [ P, H ] 0 P lim PH ( t) dt T T 0 Energy spectrum represented as a planar lattice i th eigenstate of H Regular system P (integrable) i ordered lattice Chaotic system disordered lattice Asher Peres ( ) Mixed system (regular & chaotic) mixed lattice E i expectation value Pi ψ i P ψ i of an arbitrary observable P i E i
20 J 0 ( ) vib π π π π γ K K T y x Two quantization options cos3 ) ( ) 3 ( ) ( γ C B A y x C x y x B y x A V D system (c) 5D system restricted to D (true geometric model of nuclei) H vib γ K y x K T γ γ γ γ sin 3 sin vib K T The options differ also in the metric (measure) for calculating matrix elements. Possibility to test the Bohigas conjecture and Peres ideas in different quantization schemes. with additional constraints (to avoid quasi-degeneracies due to the symmetry of V ) ), ( ), ( ), ( ), ( 3 γ γ γ γ π ±Ψ Ψ Ψ Ψ k (a) even (b) odd K κ classicality parameter Geometric Collective Model Quantum Hamiltonian 3 /0 Stránský, Hruška,Cejnar 009 PRE 79, 0460, 0660 JPconf 00, NPNews 0
21 4a/0 Stránský, Hruška,Cejnar 009 PRE 79, 0460, 0660 Geometric Collective Model JPconf 00, NPNews 0 A B.09 C / K.5 0 3
22 4b/0 Stránský, Hruška,Cejnar 009 PRE 79, 0460, 0660 Geometric Collective Model JPconf 00, NPNews 0 A B.09 C / K.5 0 3
23 5 /0 Geometric Collective Model Stránský, Hruška,Cejnar 009 PRE 79, 0460, 0660 Bohigas Conjecture Nearest Neighbor Spacing Distribution compared with a classical chaos measure f reg classical regular fraction ω adjunct of Brody parameter A C A C JPconf 00, NPNews 0 κ / K κ κ 5 0 6
24 6 /0 Stránský, Hruška,Cejnar 009 Geometric Collective Model Peres Lattices ψ P ψ i i P i PRE 79, 0460, 0660 JPconf 00, NPNews 0 ) <L > ) <H > ) Quasi-D angular momentum κ ) Hamiltonian perturbation κ V B 3 cos3γ B.09 κ 4 / K κ
25 7 /0 Stránský, Hruška,Cejnar 009 Geometric Collective Model Peres Lattices ψ P ψ i i P i ) <L > PRE 79, 0460, 0660 JPconf 00, NPNews 0 ) <H > ) Quasi-D angular momentum ) Hamiltonian perturbation V B 3 cos3γ 4 κ E i
26 8a/0 Stránský, Hruška,Cejnar 009 PRE 79, 0460, 0660 Geometric Collective Model JPconf 00, NPNews 0 Peres Lattices ) <L > ) <H > B0.6 κ 5 0 6
27 8b/0 Stránský, Hruška,Cejnar 009 PRE 79, 0460, 0660 Geometric Collective Model JPconf 00, NPNews 0 Peres Lattices ) <L > ) <H > B0.6
28 9 /0 Interacting Boson Model Consequences of regularity for the adiabatic separation of intrinsic and collective motions: rotational bands exist even at very high excitation energies if the corresponding region of the J 0 spectrum is regular. Selection of hypothetical I N 30 bands of rotational states based on the maximal correlation of the intrinsic λ, SU(3) structures J ( µ ) ψ Macek, Dobeš, Cejnar 00 PRL 05, intrinsic wave functions for various band members in the SU(3) basis Product of 0 - and 0-4 correlation coefficients for intrinsic wave functions in the given band Energy ratio for 4 and states in a given band Classical regular fraction in the respective energy region
29 0 /0 Conclusions Nuclear collective motions exhibit an intricate interplay of regular & chaotic features Models of collective motions may serve as a theoretical laboratory for general studies of chaos Order/Chaos have relevant nuclear-structure consequences (e.g. for the adiabatic separation of collective and intrinsic motions) Thanks to GCM J 0, E 0 Pavel Stránský (now ECT* Trento) Michal Macek (now Uni Jerusalem) Thank you for attention!
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