Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Praha

Transcription

1 and Nuclear Structure Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Praha ha,, CZ ipnp.troja.mff.cuni.cz Program: > Shape phase transitions in nuclear structure data > Models describing shape phase transitions in nuclei > Playing with the models Marie Marie & Pierre Pierre Curie Curie th th Nuclear Physics Workshop, Kazimierz 28 28

2 Part 1 of 3 Shape (phase) transitions in nuclear structure data Program: > Shape phase transitions in nuclear structure data Models describing shape phase transitions in nuclei Playing with the models Marie Marie & Pierre Pierre Curie Curie th th Nuclear Physics Workshop, Kazimierz 28 28

3 Ground-state QPT signatures in nuclei Shape-phase transition! All nuclei with + E( 2 1 ) < 6 kev Casten et al., PRL 71,227 (1993)

4 Ground-state QPT signatures in nuclei Semi-empirical criterion for a spherical-to-deformed transition: P-factor Casten et al., PRL 58,658 (1987), N McCutchan et al., PRC 69,2438 (24) pnn P N p + N n 5 ( N p, N n = numbers of valence protons, neutrons or the respective holes)

5 Ground-state QPT signatures in nuclei Energy ratio R E(4 = E ( / Zamfir et al., PRC 66,2134(22) ) ) McCutchan et al., PRC 69,2438 (24) R 4/ 2 N

6 Ground-state QPT signatures in nuclei 2n separation energies 2 S2n = M ( Z, N 2) + 2mnc M ( Z, N) Dieperink,Scholten,Iachello, PRL 44,1747(198) García-Ramos et al., NPA 688,735(21) García-Ramos et al., PRC 68,2437(23) Energy ratio R E(4 = E ( / Zamfir et al., PRC 66,2134(22) ) ) Transition strength + B( E2, ) Iachello,Zamfir,PRL92,21251(22) R 4/ 2 N N

7 Ground-state QPT signatures in nuclei 2n separation energies 2 S2n = M ( Z, N 2) + 2mnc M ( Z, N) Dieperink,Scholten,Iachello, PRL 44,1747(198) García-Ramos et al., NPA 688,735(21) García-Ramos et al., PRC 68,2437(23) magic Courtesy: F.Iachello magic N

8 Ground-state QPT signatures in nuclei Prolate oblate transition Jolie, Linnemann, PRC 68,3131 (23)

9 Part 2 of 3 Models describing shape phase transitions in nuclei Program: Shape phase transitions in nuclear structure data > Models describing shape phase transitions in nuclei Playing with the models Marie Marie & Pierre Pierre Curie Curie th th Nuclear Physics Workshop, Kazimierz 28 28

10 Geometric (Collective) Model (GCM) A.Bohr (1952) W.Greiner (1971) H oblate (2) () () [ α α ] α ] + 5C( [ α ] ) +... () () 35 = [ π π ] A[ α α ] B α 2K V = Aβ + Bβ γ + cos3 4 Cβ α quadrupole tensor of collective coordinates 2 shape parameters: β, γ 3 Euler angles π... corresponding tensor of momenta B spinodal critical γ-soft prolate spherical A

11 Geometric (Collective) Model (GCM) A.Bohr (1952) W.Greiner (1971) H 2 3 V = Aβ + Bβ γ + cos3 4 Cβ (2) () () [ α α ] α ] + 5C( [ α ] ) +... () () 35 = [ π π ] A[ α α ] B α 2K B spinodal critical β >,γ =π/3 oblate γ-soft spherical prolate β >,γ =[,2π) β = A β >,γ =

12 Geometric (Collective) Model (GCM) A.Bohr (1952) W.Greiner (1971) H (2) () () [ α α ] α ] + 5C( [ α ] ) +... () () 35 = [ π π ] A[ α α ] B α 2K β β cos3γ 2 3 V = A + B + 4 Cβ B spinodal critical 1 st order oblate 2 nd order 1 st order γ-soft prolate spherical A

13 Interacting Boson Model (IBM) F.Iachello, A.Arima (1975) + s + d m m = 2,..., + 2 Phase transitions caused by competing dynamical symmetries u + = ijbi bj vijklb i bj bkb l H = wc[ G ] i, j General H (7 parameters) U(5) i, j, k, l Phase structure i i Casimir invariants of U(6) subgroups U(5),O(6),SU(3),O(5),O(3) Simplified H (2 parameters) i O(6) SU(3) spherical SU(3) O(6) prolate oblate U(6) spectrum generating (dynamical) algebra O(3) invariant symmetry algebra 1 st order spherical deformed 2 nd order P.Cejnar,J.Jolie, Prog.Part.Nucl.Phys.(28); arxiv: [nucl-th]

14 Interacting Boson Model (IBM) F.Iachello, A.Arima (1975) + s + d m m = 2,..., + 2 Phase transitions caused by competing dynamical symmetries u + = ijbi bj vijklb i bj bkb l H = wc[ G ] i, j i, j, k, l Phase structure i i Casimir invariants of U(6) subgroups U(5),O(6),SU(3),O(5),O(3) Simplified H (2 parameters) i Ψ Ψ N ( s + ) N ( s β d ) spherical U(6) spectrum generating (dynamical) algebra O(3) invariant symmetry algebra Ψ N ( s + + β d ) prolate oblate P.Cejnar,J.Jolie, Prog.Part.Nucl.Phys.(28); arxiv: [nucl-th]

15 Fermionic Models Phenomenological Lipkin model Lipkin,Meshkov,Glick(1965) Fermion Dynamical Symmetry Model Ginocchio,Wu,Zhang,Guidry (198,86,87,88) Pairing models Chen,Rowe (199 ), Volya,Zelevinsky(23), Clark et al.(26) Microscopic * Collapse of RPA Thouless(196) Early shell model attempts Federman,Pittel,Campos(1979) Monte Carlo shell model Shimizu,Otsuka,Mizusaki,Honma PRL86,1171(21) Relativistic mean-field calculations Nikšić,Vretenar,Lalazissis,Ring PRL99,9252(27) * In the microscopic case the infinitesize limit cannot be performed all changes are smoothened by quantum fluctuations.

16 Part 3 of 3 Playing with the models Learning new physics on on quantum phase transitions Program: Shape phase transitions in nuclear structure data Models describing shape phase transitions in nuclei > Playing with the models Marie Marie & Pierre Pierre Curie Curie th th Nuclear Physics Workshop, Kazimierz 28 28

17 New types of symmetries at & around the critical point Critical-point dynamical symmetry Analytical solutions which are approximately valid at the QPT critical point First noted by Ginocchio et al.(1987) in the Fermion Dynamical Symmetry Model Cast in the geometric framework by Iachello(2,21),Bonatsos et al.(24) E(5), X(5), Z(5) Partial dynamical symmetry At the 1 st order critical point: distinct subsets of states retain competing dynamical symmetries The PDS idea originally introduced in quantum chaos Leviatan et al.( ) Dynamical Symmetry I Phase I Phase II Dynamical Symmetry II Quasidynamical Symmetry I critical point Quasidynamical Symmetry II Quasidynamical Symmetry Extensions of approximate dynamical symmetries far away from the corresponding limits QDS is an expression of the possibility that a subset of physical data may exhibit all the properties that would result if the system had a symmetry which, in fact, it does not have. Rowe et al.(1998,24,25)

18 Additional degrees of freedom, IBM extensions Proton-neutron variables (IBM-2) Caprio,Iachello (24,25) Arias,García-Ramos,Dukelsky (24) Odd fermions (IBFM) Jolie et al. (24) Iachello (25)? Alonso et al. (25,26,27) Higher order interactions Iachello (24) Jolos (24) Thiamova,Cejnar (26)? Other types of bosons Devi,Kota (199) Cejnar,Iachello (27)? Configuration mixing Frank,Van Isacker,Iachello (26) Hellemans et al. (27) External rotation Cejnar (22,23)

19 Finite-size scaling exponents Calculations beyond the mean field Dusuel, Vidal, Arias, Dukelsky, García-Ramos (25,26,27) Example: Energy gap = E1 E c e an 1 st order 2 nd order c N 1/3 Lipkin Hamiltonian Vidal et al., PRC73,5435(26)

20 Mechanism of the 1 st x 2 nd order transitions Thermodynamic analogy for quantum phase transitions Zeros of Z(T) in complex T partition function Z(T ) free energy specific heat latent heat F( T ) = T ln Z( T ) 2 C( T ) = T 2 Q ( T ) = T + ε lim ε + T ε T F( T ) C( T ') dt ' Degeneracies of H(λ) in complex λ = 1/ Ω [ D ( λ) ] [ E ( λ) E ( λ) ] U C k k k i( k ) 1 ( λ) = ln D ( λ) Ω 2 d ( ) = 2 U k λ dλ k Q ( λ ) = k λ + ε lim ε + λ ε C i ( λ) k k ( λ') dλ' 1/ Ω Ω = n 1 Yang, Lee (1952) Cejnar et al. (25,27) Example: linear arrangement of degeneracies (zeros) density near Imλ ρ (Imλ) α ρ α>1 <α<1 α= Imλ Imλ λ c Reλ Q Q k k ( λ ) c ( λ ) c = α = 1 st order α > continuous 2 d dx 4 d dx 2 4 C( λ ) = α (,1] c C( λ ) = α (1,3] c C( λ ) = α (3,7]... c

21 Excited-state quantum phase transitions Cejnar, Heinze, Macek, Jolie, Dobeš (26,27) Caprio, Cejnar, Iachello (28) Cejnar, Stránský (28) Example: 1D Cusp Hamiltonian 2 nd order 1 st order π = ± b = a = 1 π = + a b

22 Question: Do quantum shape-phase transitions really exist in nuclei? Tentative answer: They would exist if nuclei were infinite objects. Finite nuclei only show QPT precursors. Real QPTs can be studied in various nuclear models benefit for both nuclear structure theory & QPT theory (further applications in molecular & mesoscopic physics) Thanks to collaborators: P Stránský, M Macek, J Dobeš [Praha] S Heinze, J Jolie [Köln] M Caprio, F Iachello [Notre Dame, Yale] Thank you for attention. Marie Marie & Pierre Pierre Curie Curie th th Nuclear Physics Workshop, Kazimierz 28 28