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 Kamila Sieja (IPHC) / 46

Atomic nucleus: a complex object Nucleus is a quantum object composed of neutrons and protons, which themselves are composite structures, composed of quarks and gluons.

2 Atomic nucleus: a complex object Nucleus is a quantum object composed of neutrons and protons, which themselves are composite structures, composed of quarks and gluons. In low energy nuclear physics, the QCD degrees of freedom are not excited, but are manifested in a very complex internucleon force. In addition to the strong interaction in nuclear physics we have to deal with electromagnetic forces between positively charged protons and with weak interactions, which is already 3 out of the 4 fundamental forces in nature! Despite of the complexity of this many-body system itself and the interactions involved, the empirical knowledge shows that surprisingly simple structures and collective phenomena occur in nuclei. Kamila Sieja (IPHC) / 46

Nuclear many-body problem The number of nucleons in nuclei is too large for an exact solution of A-body Schrödinger equation. Still, it is much too small for statistical methods.

3 Nuclear many-body problem The number of nucleons in nuclei is too large for an exact solution of A-body Schrödinger equation. Still, it is much too small for statistical methods. Ab initio Shell Model DFT Nuclear Shell Model (SM), known as well as Configuration Interaction (CI) Density Functional Theory (DFT): E = Ψ H Ψ Ψ Ψ E EDF [ρ] ρ ij = φ a j a i φ = φ = a i. Macroscopic-microscopic models Kamila Sieja (IPHC) / 46

4 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) / 46

5 Model spaces Classical 0 hω model spaces (e.g. sd-shell, pf -shell) are successfull for the description of a low lying states of nuclei, their transition rates and weak-decays. f5/2 p1/2 p3/2 Π f7/2 f7/2 40 Ca Neutron rich nuclei require different active proton and neutron shells (e.g. sd-shell for protons, π pf -shell for neutrons). H. O. π+ ν f5/2 p1/2 p3/2 H. O. Deformed nuclei and deformed bands in spherical nuclei are due to many-particle many-hole excitations across the gaps. At least 2 hω spaces are necessary e.g. sd-pf for both neutrons and protons). 2s1/2 1d3/2 1d5/2 Π 42 Ca In simple shell-model nuclei, certain observables require going beyond 0 hω model space, e.g. parity changing Π transitions. ν p1/2 p3/2 f5/2 f7/2 d3/2 s1/2 d5/2 p1/2 p3/2 ν 2p1/2 1f5/2 1f7/2 2p3/2 Kamila Sieja (IPHC) / 46 4 He

6 Model spaces Classical 0 hω model spaces (e.g. sd-shell, pf -shell) are successfull for the description of a large number of nuclei, their transition rates and weak-decays. f5/2 p1/2 p3/2 Π f7/2 f7/2 40 Ca Neutron rich nuclei require different active proton and neutron shells (e.g. sd-shell for protons, π pf -shell for neutrons). ν f5/2 p1/2 p3/2 H. O. Deformed nuclei and deformed bands in spherical nuclei are due to many-particle many-hole excitations across the gaps. At least 2 hω spaces are necessary e.g. sd pf for both neutrons and protons). 2s1/2 1d3/2 1d5/2 Π 42 Ca In simple shell-model nuclei, certain observables alos require going beyond 0 hω model space, e.g. parity changing Π transitions. ν p1/2 p3/2 f5/2 f7/2 ν 2p1/2 1f5/2 1f7/2 2p3/2 H. O. π+ d3/2 s1/2 d5/2 For each model space an effective SM interaction has p1/2 p3/2 to be constructed Kamila Sieja (IPHC) / 46 4 He

Shell Model: giant computations Problem dimension in the m-scheme: ( ) ( ) dπ dν D p n In the pf -shell (1f 7/2, 2p 3/2, 1f 5/2, 2p 1/2 ): 48 Cr 1,963,461 56 Ni 1,087,455,228 Current diagonalization

7 Shell Model: giant computations Problem dimension in the m-scheme: ( ) ( ) dπ dν D p n In the pf -shell (1f 7/2, 2p 3/2, 1f 5/2, 2p 1/2 ): 48 Cr 1,963, Ni 1,087,455,228 Current diagonalization limit in m-scheme The largest SM diagonalization up to date has been achieved by the Strasbourg group (using very modest computing resources): Phys. Rev. C82 (20) 0543, ibidem m scheme CODE ANTOINE coupled scheme CODE NATHAN Largest SM matrices we treat contain non-zero matrix elements. They can not be stored on a hard drive. It would take DVDs to store one matrix! E. Caurier et al., Rev. Mod. Phys. 77 (2005) 427; ANTOINE website Kamila Sieja (IPHC) / 46

Shell gaps in nuclei & realistic 2-body interactions V(r) N=5 N=4 N=3 3p 2f 1h 3s 2d 1g 2p 1f 3p1/2 2f5/2 1h9/2 3s1/2 2d5/2 1g9/2 1f5/2 1f7/2 1i13/2 (14) (2) 3p3/2 (4) (6) 2f7/2 (8) (10) 1h11/2 2d3/2

8 Shell gaps in nuclei & realistic 2-body interactions V(r) N=5 N=4 N=3 3p 2f 1h 3s 2d 1g 2p 1f 3p1/2 2f5/2 1h9/2 3s1/2 2d5/2 1g9/2 1f5/2 1f7/2 1i13/2 (14) (2) 3p3/2 (4) (6) 2f7/2 (8) (10) 1h11/2 2d3/2 1g7/2 2p1/2 2p3/2 (12) (2) (4) (6) (8) (12) (2) (6) (4) (8) (126) (82) (64) (50) (40) (38) (28) 28 5 short ω ρ intermediate long range σ π 1 2 r(fm) N=2 N=1 2s 1d 1p 2s 1p1/2 1d3/2 1d5/2 1p3/2 (4) (2) (6) (2) (4) (20) (14) (8) (6) 20 8 E(2 + ) (MeV) Ca, Z=20 THE EXP N=0 1s 1s (2) (2) Neutron number U(r)= 1 2 mω2 r 2 + D l 2 C l s Realistic 2N potentials produce strong h.o. closures but no spin-orbit ones... Kamila Sieja (IPHC) / 46

9 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) / 46

10 SM with empirical interactions: regions of activity Kamila Sieja (IPHC) / 46

11 Shell evolution: what have we learned? Kamila Sieja (IPHC) / 46

12 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) / 46

13 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) / 46

14 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 (22) 0513R. Kamila Sieja (IPHC) / 46

15 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) / 46

16 Shell evolution along isotopic chains and 3N forces K. Sieja et al., Phys.Rev.C85 (22) 0513(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 (20) Kamila Sieja (IPHC) / 46

17 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) / 46

18 Role of nuclei around N=90 B.S. Meyer et al., Astr. Journal 399 (1992) 656 Kamila Sieja (IPHC) / 46

19 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) / 46

20 Seniority isomers in the tin chain Exp - PhD work of G. Gey (LPSC Grenoble) G. Simpson, G. Gey et al., submitted to PRL B(E2;6 + -> 4 + ) (e 2.fm 4 ) Exp. Seniority SM A Previous SM calculations wrong by a factor 2 for the B(E2; ) in 136 Sn Important seniority mixing effects in 136 Sn Data compatible with a f 7/2 p 3/2 gap <1.5 MeV in present SM calculations ( B(E2;νJ νj 2) 1 2n ) 2 (2j+ 1) ZStudy of masses in this region in the present SM framework in progress Kamila Sieja (IPHC) / 46

21 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 )=2 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) / 46

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) / 46

23 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) / 46

24 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 (22) 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) / 46

25 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 (22) 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) / 46

26 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 (20) 4318 Evidence for the dipole nature of low energy enhancement in 56 Fe A. C. Larsen et al., Phys. Rev. Lett. 111 (23) Gamma energy (MeV) Kamila Sieja (IPHC) / 46

27 What theory says about it? E. Litvinova and N. Belov, Phys. Rev. C88 (23) 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 (23) 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) / 46

M1-strength SM calculations in 94 Mo R. Schwengner et al., Phys. Rev. Lett. C88 (23) KS, present SM <B(M1)> ( N 2 ) 0.3 0.25 0.2 0.15 0.1 0.

28 M1-strength SM calculations in 94 Mo R. Schwengner et al., Phys. Rev. Lett. C88 (23) KS, present SM <B(M1)> ( N 2 ) Mo - J=0i to 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 ) 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 ) ZLow energy M1 enhancement independent on details of calculations: model space, interaction & number of states Kamila Sieja (IPHC) / 46

29 M1-strength SM calculations in sd-pf shell nuclei <B(M1)> ( N 2 ) Open questions: 43 Sc - and + 40 states, J=1/2-9/ E γ (MeV) <B(M1)> ( N 2 ) Mechanism of enhancement? The same in all nuclei? EM character, multipolarity. Which nuclei? 56 Fe + 70 states, J= E γ (MeV) ZFurther theoretical effort necessary to describe consistently E1/M1/E2 transitions in different regions Kamila Sieja (IPHC) / 46

30 Summary Thanks to the progress in computing more and more regions of nuclei are accessible to SM diagonalizations, which provide currently the most accurate description of nuclear properties. Understanding of the mechanisms driving shell evolution (3N forces, tensor forces) allows for a better modelling all over the nuclear chart. Consistent description of gamma-excitation M1/E1/E2 strength functions is possible in the SM, for the moment in sd nuclei, soon also in the sd-pf region. Thanks to: E. Caurier, H. Naidja, F. Nowacki (IPHC Strasbourg) G. Martinez-Pinedo, J. Mendoza-Temis (TU Darmstadt) Kamila Sieja (IPHC) / 46

31 Modern theory of nuclear forces QCD is a gauge field theory which describes the strong interactions of color quarks and gluons (one universal coupling constant). There are six quarks q =(u,d,s,c,b,t) and 8 gluons. The QCD Lagrangian can be written L QCD = L 0 QCD qm q In the massless case, the Lagrangian is invariant under separate unitary global transformations of left and right-hand quark fields, so-called chiral rotations To describe the strong interactions, it is necessary that the chiral symmetry is spontaneously broken: indeed, the chiral symmetry is not a symmetry of the ground state (no parity doublets). Chiral symmetry is also explicitly broken (quark mass 0). Kamila Sieja (IPHC) / 46

32 Modern nuclear forces: N3LO We use pions and nucleons as degrees of freedom. The effective Lagrangian is classified using a systematic expansion based on a power counting in terms of (Q/Λ χ ) ν, where ν is called chiral order and Λ χ is the hard scale ( 700MeV) ν=0 is called leading order, ν > 1 are called next-to-ν 1 leading orders. Note hierarchy of nuclear forces. Coupling constants (LEC) adjusted to phase shifts and deuteron properties. E. Epelbaum et al., Rev. Mod. Phys. 81 (2009) Kamila Sieja (IPHC) / 46

33 Modern nuclear forces: N3LO The important role of 3N forces has been proven in light systems, e.g. in the no-core shell model calculations ; 1 hω= ; 1 ; ; ; ; NN+NNN Exp NN 1/2 - ; 3/2 5/2-16 1/2 - ; 3/2 12 5/2-5/2-5/2-7/2-8 3/2-5/2-7/2-5/2-4 3/2-1/2-3/2-1/2-3/2-0 NN+NNN Exp NN 0 + ; 1 10 B 11 B 12 C 13 C 2 + ; ; NN+NNN Exp NN 7/2-2 + ; /2 - ; 3/2 0 + ; ; 1 3/2 - ; 3/2 3/2-1/2-7/2-3/2-1/2-5/2-3/2-5/2-3/2-1/2-1/2 - NN+NNN Exp NN Excitation energies in light nuclei in NCSM with chiral EFT interactions. P. Navratil et al., Phys. Rev. Lett. 99 (2007) In heavier mass nuclei (shell model with the core) only 2-body interactions are nowadays calculated. The missing 3N part is taken into account by phenomenological corrections of TBME. Experimental information is crucial to constrain the interactions! Kamila Sieja (IPHC) / 46

34 3N force from N2LO V (2π) 3N = i j k g 2 A 8f 4 π σ i q i σ j q j ( q i 2 + mπ)( q 2 j 2 + mπ) F 2 ijk αβ τα i τ β j F αβ ijk = δ αβ( 4c 1 m 2 π+ 2c 3 q i q j )+c 4 ε αβγ τ γ k σ k ( q i q j ) V 1π 3N = i j k g A c D σ i q j σ i q 8fπΛ 4 χ q j 2 + m 2 i τ i τ j π V3N ct = g A c E τ i τ i j k 8fπΛ 4 j χ g A = 1.29,f π = 92.4MeV,Λ χ = 700MeV,m π = Mev/c 2, q i = p i p i Kamila Sieja (IPHC) / 46

35 Soft-core potential V lowk I Separation of scales: Integrate out the high momenta and require that the effective interaction V lowk reproduces the low-momentum scattering amplitude (calculated from V NN ) 0 0 V lowk To obtain V lowk potential we solve a RG flow equation in the momentum space: d dλ V lowk(k,k,λ)= 2 V lowk (k,k,λ)t(λ,k, h2 Λ 2 2m ) π 1 ( k Λ )2 starting from a given NN force V NN, with the initial condition: V lowk (k,k ;Λ) Λ V NN (k,k ) Λ Λ V NN Kamila Sieja (IPHC) / 46

36 Calculation of the effective Hamiltonian We know, that the realistic V NN potential is not a good candidate for V in the perturbation theory. Thus we replace it with the G-matrix or V lowk V eff = G+ G Q V E H eff 0 First-order Second-order diagrams G-matrix 2p ladder core-polarization 4p-2h Kamila Sieja (IPHC) / 46

37 Beta-decay calculations ZTo calculate beta decay between two states one needs: accurate value of the decay energy (T 1/2 E 5 ) matrix elements of Gamow-Teller ( J π = 0 +,1 + ) first forbidden ( J π = 0,1,2 ) transition operators ZNuclear model has to provide good description of masses, spectra and wavefunctions In r-process nuclei, GT is not enough, as protons and neutrons occupy different parity levels: The beta half-life to a given state is gives as: ft = 6146, where the space phase factor f is given as H. O. H. O. W0 f = C(W)F(Z,W)(W 2 1) 1/2 W(W 0 W) 2 dw 1 For allowed transitions: C(W) = B(F) +B(GT) For forbidden C(W) is a function of the energy, and we have to calculate the integral C(W)=k(1+aW + b W + cw 2 ) Kamila Sieja (IPHC) / 46

38 Coeficients a,b,c k = ka = [ ζ ] (0) [ 9 w 2 + ζ (x+ u)2 4 9 µ 1γ 1 u(x+ u) W 0 2 (2x+ u)2 1 ] (1) [ ] 1 (2) 18 λ 2(2x u) z2 (W0 2 λ 2), [ 4 3 uy 1 (1) [ ] 1 (2) 9 W 0(4x 2 + 5u )] 2 6 z2 W 0, kb = 2 3 µ 1γ 1 { [ζ 0 w] (0) +[ζ 1 (x+ u)] (1)}, kc = 1 18 [ 8u 2 +(2x+ u) 2 + λ 2 (2x u) 2] (1) 1 [ (2). + z 2 (1+λ 2 )] 12 Kamila Sieja (IPHC) / 46

39 Matrix elements w = R A F 0 1 = λ 3 J f T f ir[c 1 σ] 0 τ J i T i C, x = 1 3 R V F = J f T f ir C 1 τ J i T i C, u = 2 3 R A F111 0 z = = λ 2 J f T f ir[c 1 σ] 1 τ J i T i C, 2 3 R A F = 2λ J f T f ir[c 1 σ] 2 τ J i T i C, w = 2 3 R A F 0 1 (1,1,1,1) = λ 3 J f T f ir 2 3 I(1,1,1,1,r)[C 1 σ] 0 τ J i T i C, x = R V F (1,1,1,1) = J f T f ir 2 3 I(1,1,1,1,r)C 1 τ J i T i C, u = R A F (1,1,1,1) = λ 2 J f T f ir 2 3 I(1,1,1,1,r)[C 1 σ] 1 τ J i T i C, Kamila Sieja (IPHC) / 46

40 Results: half-lives of r-process nuclei T 1/2 (s) 10 2 FRDM+QRPA 10 1 DF3+QRPA HFB+QRPA SM Exp T 1/2 (ms) FRDM+QRPA DF3+QRPA HFB+QRPA SM Exp N=50 N=82 T 1/2 (s) Charge number Z FRDM+QRPA DF3+QRPA (I) DF3+QRPA (II) SM Suzuki et al N= Charge number Z Accurate description of Q β, neutron emission probabilities and half-lives in the known region Universal quenching factors on GT and FF operators in all mass regions Q. Zhi, J. Cuenca, E. Caurier, K. Langanke, G Martinez-Pinedo, K. Sieja, Phys. Rev. C87 (23) Charge Number Z Kamila Sieja (IPHC) / 46

41 Lanczos strength function method We can be interested in calculation of transition matrix elements ψ f Ô ψ i to many final states. For example in β-decay half-life calculation: λ = ln2 = T 1/2 λ f f To compute B(GT) one needs: λ f = ln2 K f(z,w 0)[B f (F)+B f (GT)] to determine the initial state ψ i find all possible final states ψ f compute all matrix elements ψ i M GT ψ f The same holds for any other transition operator between states in the same nucleus: M1, E1, E2... or between different nuclei: GT, ββ, a, a,... Kamila Sieja (IPHC) / 46

42 Lanczos Strength Function Let O be an operator acting on some initial state Φ ini, we obtain the state O Φ ini whose norm is the sum rule of the operator O in the initial state: S = O Φ ini = Φ ini O 2 Φ ini Depending on the nature of the operator O, the state O Φ ini belongs to the same nucleus (if O is a e.m transition operator) or to another (Gamow-Teller, nucleon transfer, a j /ã j, β β,...) If the operator O does not commute with H, O Φ ini is not necessarily an eigenvector of the system BUT it can be developped in energy eigenstates: O Φ ini = f S(E f ) E f and Φ ini O 2 Φ ini = S 2 (E f ) f with {S(E f )}={ E f O Φ ini } being the strength function (or structure function) Kamila Sieja (IPHC) / 46

43 Lanczos Structure Function If we carry on the Lanczos procedure using O = O Φ ini as initial pivot. then H is again diagonalized to obtain the eigenvalues E f U is the unitary matrix that diagonalizes H and gives the expression of the eigenvectors in terms of the Lanczos vectors: U = O 2 3 : : N E 1 E 2 E 3... E N S(E f )=U(1,f)= E f O = E f O Φ ini How good is the Strength function S obtained at iteration N compared to the exact one S? Kamila Sieja (IPHC) / 46

44 Lanczos Structure Function Any distribution can be characterized by the moments of the distribution. Ē m n = O H O = E i E i O Φ ini 2 i = O (H Ē)n O = (E i Ē)n E i O Φ ini 2 i Gaussian distribution characterized by two moments (Ē,σ2 = m 2 ) g(e)= 1 σ 2π exp( (E Ē)2 ) 2σ 2 g(e) 2σ E Kamila Sieja (IPHC) / 46

45 Lanczos Structure Function Lanczos method provides a natural way of determining the basis α. Initial vector 1 = O O O E 12 2 =(H E 11 ) 1 E 23 3 =(H E 22 ) 2 E E NN+1 N+ 1 =(H E NN ) N E N 1N N 1 where E NN = N H N, E NN+1 = E N+1N Each Lanczos iteration gives information about two new moments of the distribution. E 11 = 1 H 1 =Ē E 2 12 = O (H E 11) 2 O =m 2 E 22 = m 3 m 2 + Ē E23 2 = m 4 m2 3 m 2 m2 2 m 2 Diagonalizing Lanczos matrix after N iterations gives an approximation to the distribution with the same lowest 2N moments. Kamila Sieja (IPHC) / 46

46 Lanczos Structure Function Initial vector 1 = Ω Ω Ω. E 11 = 1 H 1 E 12 2 =(H E 11 ) 1 E 23 3 =(H E 22 ) 2 E E NN+1 N+ 1 =(H E NN ) N E N 1N N 1 where E NN = N H N, E NN+1 = E N+1N Each Lanczos iteration gives information about two new moments of the distribution. E 11 = 1 H 1 =m 1 = Ē E 2 12 = 1 (H E 11) 2 1 =m 2 E 22 = m 3 m 2 + Ē E23 2 = m 4 m2 3 m 2 m2 2 m 2 E 11 E E 12 E 22 E E 32 E 33 E E 43 E 44 E 45 0 Kamila Sieja (IPHC) / 46

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

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