Recent developments in the nuclear shell model and their interest for astrophysics

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1 Recent developments in the nuclear shell model and their interest for astrophysics Kamila Sieja Institut Pluridisciplinaire Hubert Curien, Strasbourg FUSTIPEN topical meeting "Recent Advances in the Nuclear Shell Model" Kamila Sieja (IPHC) / 30

2 Shell model approach Calculations Ab Initio Realistic NN interactions Diagonalization in N hω h.o.space define valence space H eff Ψ eff = EΨ eff INTERACTIONS CORE build and diagonalize Hamiltonian matrix CODES Weak processes: β decays ββ decays [T 0ν 1/2 ( )] 1 = G 0ν M 0ν 2 m ν 2 ZASTROPHYSICS ZPARTICLE PHYSICS Collective excitations: deformation, superdeformation superfluidity symmetries Shell evolution far from stability: Shell quenching New magic numbers ZASTROPHYSICS Kamila Sieja (IPHC) / 30

3 Shell Model & 3N forces No-core shell model calculations with 3N forces possible for light systems (A 12). In core-shell model for heavier nuclei, 3N contribution taken into account empirically. A. Cortes and A.P. Zuker, "Self-consistency and many-body monopole forces in shell model calculations" Phys. Lett. B84 (1979) 25 Ab Initio No Core Shell Model Realistic NN&NNN forces Effective interactions Diagonalization of many body Hamiltonian Core Shell Model Phenomenological effective interactions Diagonalization of the Hamiltonian for valence nucleons Many body experimental data ; ; 1 B 1/2 - ; 3/2 B C 7/2 - hω=14 5/ ; 1 ; /2 - ; 3/2 C ; 1 ; /2 - ; 3/ ; 1 3/2 - ; 3/ ; ; /2 - ; 1 5/ / /2-1/2-7/2-3/2-7/ /2-1/ / /2-7/2 - ; ; 1 5/ /2-5/ / /2-3/ /2-1/2-3/ /2-1/2 - NN+NNN Exp NN NN+NNN Exp NN NN+NNN Exp NN NN+NNN Exp NN Excitation energies in light nuclei in NCSM with chiral EFT interactions. P. Navratil et al., Phys. Rev. Lett. 99 (2007) Kamila Sieja (IPHC) / 30

4 SM with empirical interactions: regions of activity Kamila Sieja (IPHC) / 30

5 SM with empirical interactions: regions of activity PLAN: 1 Some features of shell evolution and of nuclear structure around 78 Ni 2 Non-statistical behavior of microscopic γ- strength functions and their impact on neutron capture cross sections Kamila Sieja (IPHC) / 30

6 Shell evolution: what have we learned? Kamila Sieja (IPHC) / 30

7 Stability of shell gaps in 78 Ni: the N=50 shell closure Experimental evidence at Z = 32 non conclusive Extrapolation of experimental data dangerous since shell efects are sudden Correlations in semi-magic nuclei hide the mean-field effects O.Sorlin, M.-G. Porquet, Theory essential for understanding of the shell evolution Prog. Part. Nucl. Phys. 61 (2008) Kamila Sieja (IPHC) / 30

8 Stability of shell gaps in 78 Ni: the N=50 shell closure (5,6 + ) [20(4)] (4 + ) [60(7)] (5,6 + ) [15(4)] E exc (MeV) N=49,exp (5/2 + ) 9/2 + 5/2 1/ /2-3/2-1/ Z [100(10)] 50 d5/2 g9/2 p1/2 40 g.s Ge f5/2 p3/2 f7/2 28 T. Rzaca-Urban et al., PRC 76 (2007) Kamila Sieja (IPHC) / 30

9 82 Ge and N=50 isotones protons neutrons J π f 7/2 f 5/2 p g 9/2 d 5/ E exc (MeV) EXP vs SM Z K. Sieja et al., Phys. Rev. C85 (2012) R. Kamila Sieja (IPHC) / 30

10 Levels across the N=50 gap: role of the correlations E exc (MeV) BE (MeV) /2+ (a) (b) Z Robustness of the N=50 gap in 78 Ni Z ESPE (MeV) f5/2 p3/2 p1/2 g9/2 d5/ Ni 28 N=50 84 Se Proton number No change of the g 9/2 -d 5/2 splitting between Ni and Se Possibility of evaluating (n,γ) cross sections around A= Sn 50 Kamila Sieja (IPHC) / 30

11 Shell evolution along isotopic chains and 3N forces K. Sieja et al., Phys.Rev.C85 (2012) (R) ESPE (MeV) ds 0d5/2 1s1/ Neutron number f7/2 1p3/2 fp Neutron number g9/2 gd 1d5/ Neutron number ν ν p1/2 ν ν ν ν d3/2 s1/2 N=14 f5/2 p3/2 N=28 s1/2 d5/2 N=50 d5/2 f7/2 g9/2 16 O O Ca 68 Ca Ni 78 Ni ZShell model predictions for the N = 50 gap consistent with 3N effects observed in light nuclei! A. P. Zuker, Phys. Rev. Lett. 90 (2003) ZUnderstanding three-body mechanisms essential for modeling of shell evolution in nuclear physics. Ideas how to improve DZ mass formula based on what we know about the role of 3N forces in the creation of shell gaps J. Mendoza-Temis et al., Nuc. Phys. A843 (2010) Kamila Sieja (IPHC) / 30

12 3N forces: a new shell closure at N=90? Π ν 0f5/2 1p1/2 1p3/2 0f7/2 0f5/2 1p1/2 1p3/2 0f7/2 40 Ca Π ν 0h11/2 1d3/2 3s1/2 1d5/2 0g7/2 0h9/2 1f5/2 2p1/2 2p3/2 1f7/2 132 Sn N=90 gap =2.5MeV Can one make analogy between 48 Ca and 140 Sn? Kamila Sieja (IPHC) / 30

13 Structure at N=90 and r-process Masses around N = 90 determine the mass flow from the second to third r-process peaks (NS mergers) Calculations from PhD thesis of J. Mendoza-Temis (TU Darmstadt) Final abundances solar r-process abundances FRDM WS3 DZ A Kamila Sieja (IPHC) / 30

14 Structure at N=90 and r-process Masses around N = 90 determine the mass flow from the second to third r-process peaks (NS mergers) Calculations from PhD thesis of J. Mendoza-Temis (TU Darmstadt) Final abundances See next talk for the recent SM 10 studies around N -5 = 90 solar r-process abundances FRDM WS3 DZ A Kamila Sieja (IPHC) / 30

15 Nuclei above 78 Ni g9/2 p1/2 p3/2 f5/2 Π Ni ν h11/2 d3/2 s1/2 g7/2 d5/2 ZKnowing s.p. structure of 79 Ni will be useful to validate the model assumptions Interaction: based on realistic G-matrix with CD-Bonn potential, empirically corrected. Proven sucessful and predictive in a large number of applications: Structure, mixed symmetry states in Zr isotopes, shell evolution between 91 Zr and 101 Sn K. Sieja et al., Phys. Rev. C79 (2009) Isomers and medium-spin structures of 95 Y, Rb, Sr PRC85 (2012) , PRC79 (2009) , PRC82 (2010) , PRC79 (2009) Collectivity and j-1 anomaly of 87 Se PRC88 (2013) β-decays of Ga nuclei and structure of N = 52,54 isotones PRC88 (2013) , PRC88 (2013) , PRC88 (2013) Magnetic moments of 86 Kr, 88 Sr, PRC 80 (2014) Collectivity of N = 52,54 nuclei PRC88 (2013) Kamila Sieja (IPHC) / 30

16 Simple perspective: predictions of the pseudo-su(3) scheme Pseudo SU3 Q + k g9/2 p1/2 p3/2 f5/2 Π 5/2 ν 7/2 h11/2 d3/2 g7/2 s1/2 d5/2 9/2 Pseudo SU3 Predictions of pseudo-su(3) model for N=52,54: Nucleus Q 0 B(E2; ) 82 Zn Ge Se Kr Zn Ge Se Kr /2 3/2 maximal prolate deformation in 88 Se possible triaxiality in 86 Ge Kamila Sieja (IPHC) / 30

17 Predictions of the pseudo-su(3) model vs SM diagonalization B(E2; 2 + -> 0 + ) (e 2.fm 4 ) N=52 SM p-su(3) Z B(E2; 2 + -> 0 + ) (e 2.fm 4 ) N=54 SM p-su(3) Zpseudo-SU(3) is a good approximation for the proton mid-shell maximal prolate deformation in 88 Se possible triaxiality in 86 Ge Z Kamila Sieja (IPHC) / 30

18 Triaxiality above 78 Ni? (a) SM Ge Shell Model using 78 Ni core GCM-Gogny Agreement of excitation energies for the 1st excited band within kev! -matrices dimension feasible on a laptop -typical time of calculations: 5min to 4h on one processor -symmetry conserved (particle number and angular momentum) -cluster of 140 CPUs -typical time of calculations: 1 month K. Sieja, T.R. Rodriguez, K. Kolos and D. Verney, Phys. Rev. C88 (2013) Kamila Sieja (IPHC) / 30

19 Triaxiality above 78 Ni? (a) SM Ge Shell Model using 78 Ni core GCM-Gogny Agreement of excitation energies for the 1st excited band within kev! -matrices dimension feasible on a laptop -typical time of calculations: 5min to 4h on one processor -symmetry conserved (particle number and angular momentum) -cluster of 140 CPUs -typical time of calculations: 1 month K. Sieja, T.R. Rodriguez, K. Kolos and D. Verney, Phys. Rev. C88 (2013) Kamila Sieja (IPHC) / 30

20 Neutron capture cross sections σ µν (n,γ) (E i,n)= π h2 1 2M i,n E i,n (2J µ (2J+ 1) T n µ Tγ ν, i + 1)(2J n + 1) T J,π tot where: E i,n,m i,n - center-of-mass energy, reduced mass of the system J n = 1/2-neutron spin transmission coefficients: T n µ = T n (E,J,π;E µ i,j µ i,π µ i ) Tγ ν = T γ (E,J,π;Em,J ν m,π ν m) ν For a given multipolarity T XL (E,J,π,E ν,j ν,π ν )=2πE 2L+1 γ f XL (E,E γ ) Key ingredients in Hauser-Feschbach calculations: description of gamma emission spectra of a compound nucleus Brink-Axel hypothesis Kamila Sieja (IPHC) / 30

21 Lanczos strength function method S = Ô ψ i = ψ i Ô2 ψ i The operator Ô does not commute with H and Ô ψ i is not necessarily the eigenstate of the Hamiltonian. But it can be developed in the basis of energy eigenstates: Ô ψ i = S(E f ) E f, f where S(E f ) = E f Ô ψ i is called strength function. If we carry Lanczos procedure using O =Ô ψ i as initial vector then H is diagonalized to obtain eigenvalues E f and after N iterations we have the also the strength function: S(E f )= E f O = E f Ô ψ i. How good is the strength function S after N iterations compared to the exact one S? Kamila Sieja (IPHC) / 30

22 Lanczos strength function method S = Ô ψ i = ψ i Ô2 ψ i The operator Ô does not commute with H and Ô ψ i is not necessarily the eigenstate of the Hamiltonian. But it can be developed in the basis of energy eigenstates: Ô ψ i = S(E f ) E f, f where S(E f ) = E f Ô ψ i is called strength function. If we carry Lanczos procedure using O =Ô ψ i as initial vector then H is diagonalized to obtain eigenvalues E f and after N iterations we have the also the strength function: S(E f )= E f O = E f Ô ψ i. How good is the strength function S after N iterations compared to the exact one S? Kamila Sieja (IPHC) / 30

23 Impact of the realistic M1 fragmentation on the neutron capture cross sections M1 microscopic strength functions in iron chain ( 53 Fe- 70 Fe), impact on (n,γ) cross sections, tests of Brink-Axel hypothesis H.-P. Loens K. Langanke, G. Martinez-Pinedo and K. Sieja, EPJ A48 (2012) 48 f M1 [1/MeV -3 ] 5.0e e-08 Fe 3.0e e e e E γ [MeV] σ [b] Fe(n,γ) 60 Fe E n [MeV] State-by-state cross section 2 times larger than using Brink hypothesis Using SF of 2 + state instead of 0 + leads to larger cross sections Kamila Sieja (IPHC) / 30

24 Impact of the realistic M1 fragmentation on the neutron capture cross sections f M1 [10-8 MeV -3 ] f M1 [10-8 MeV -3 ] f M1 [10-8 MeV -3 ] J=0 + H.-P. Loens K. Langanke, G. Martinez-Pinedo and K. Sieja, EPJ A48 (2012) J=2 + E γ [MeV] 68 Fe J=4 + E γ [MeV] 68 Fe E γ [MeV] 68 Fe σ [b] Fe(n,γ) 68 Fe J=0 + J=2 + J= E n [MeV] State-by-state cross section 2 times larger than using Brink hypothesis Using SF of 2 + state instead of 0 + leads to larger cross sections Kamila Sieja (IPHC) / 30

25 Low energy enhancement of the γ-strength function Data from Oslo group Gamma strength (MeV 3 ) Microscopic strength functions are different from global parametrizations Low energy enhancement of γ-strength observed in different regions of nuclei It can influence the (n,γ) rates of the r-process by a factor of 10! A.C. Larsen and S. Goriely, Phys. Rev. C82 (2010) Evidence for the dipole nature of low energy enhancement in 56 Fe A. C. Larsen et al., Phys. Rev. Lett. 111 (2013) Gamma energy (MeV) Kamila Sieja (IPHC) / 30

26 What theory says about it? E. Litvinova and N. Belov, Phys. Rev. C88 (2013) R Thermal continuum QRPA calculations Enhancement due to transitions between thermally unblocked s.p. states and the continuum Note the difference between T = 0 (ground state) and T > 0 (excited state) E1 strength distribution R. Schwengner et al., Phys. Rev. Lett. C88 (2013) Shell model transitions between a large amount of states (more than transitions for each parity) Enhancement due to the transitions between states in the region near the quasicontinuum A general mechanism that can be found in other mass regions Kamila Sieja (IPHC) / 30

27 M1-strength SM calculations in 94 Mo R. Schwengner et al., Phys. Rev. Lett. C88 (2013) KS, present SM <B(M1)> ( N 2 ) Mo - J=0i -6i, i=1, E γ (MeV) Model space: π(0f 5/2,1p 3/2,1p 1/2,0g 9/2 ) ν(0g 9/2,1d 5/2,0g 7/2 ) truncations of allowed p-h excitations SMI: Model space: π(0f 5/2,1p 3/2,1p 1/2,0g 9/2 ) ν(1d 5/2,0g 7/2,1d 3/2,2s 1/2,0h 11/2 ) full space diagonalization ZLow energy M1 enhancement looks independent on model space & effective interaction Kamila Sieja (IPHC) / 30

28 M1-strength SM calculations in 94 Mo R. Schwengner et al., Phys. Rev. Lett. C88 (2013) KS, present SM <B(M1)> ( N 2 ) Mo - J=0i -6i, i=1, E γ (MeV) Model space: π(0f 5/2,1p 3/2,1p 1/2,0g 9/2 ) ν(0g 9/2,1d 5/2,0g 7/2 ) truncations of allowed p-h excitations SMII: Model space: π(1p 1/2,0g 9/2 ) ν(0g 9/2,1d 5/2,0g 7/2,1d 3/2,2s 1/2 ) 1p1h from νg 9/2 to the rest of shells ZLow energy M1 enhancement looks independent on model space & effective interaction Kamila Sieja (IPHC) / 30

29 γ-strength in SM calculations f 1 (10-9 MeV -3 ) Sc 60 Fe 94 Mo f 1 (10-9 MeV -3 ) Mo SM I SM II E γ (MeV) E γ (MeV) ZToo low SM level density in 43 Sc Open questions: EM character, multipolarity. role of E1 transitions Which nuclei? Kamila Sieja (IPHC) / 30

30 Summary Understanding of the mechanisms driving shell evolution (3N forces, tensor forces) allows for a better modelling all over the nuclear chart. Pseudo-SU(3) model predictions are in good agreement with SM diagonalization for N = 52, 54 nuclei. Consistent description of gamma-excitation M1/E1/E2 strength functions is necessary to explain low energy enhancement of radiation strength functions. The E1 calculations in sd pf gds model space are under way. Thanks to: E. Caurier, H. Naidja, F. Nowacki (IPHC Strasbourg) G. Martinez-Pinedo, J. Mendoza-Temis (TU Darmstadt) Kamila Sieja (IPHC) / 30