Advantages and Limitations Of The Shell Model

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1 Advantages and Limitations Of The Shell Model F. Nowacki 1 Atelier de Structure Nucléaire Théorique April 7 th /11 th Strasbourg-Madrid-GSI Shell-Model collaboration

2 What do we mean by SHELL MODEL nowadays An approximation to the exact solution of the nuclear A-body problem Using effective interactions in restricted valence spaces (or regularized interactions in the No Core Shell Model description of very light nuclei) Where the Monopole hamiltonian is the (spherical) mean field and the Multipole hamiltonian is the correlator

3 The three pillars of the shell model The Effective Interaction Valence Spaces Algorithms and Codes E. Caurier, G. Martínez-Pinedo, F. Nowacki, A. Poves and A. P. Zuker. The Shell Model as a Unified View of Nuclear Structure, Reviews of Modern Physics, 77 (2005)

4 Shell Model Problem Define a valence space CORE

5 Shell Model Problem Define a valence space Derive an effective interaction HΨ = EΨ H eff Ψ eff = EΨ eff CORE

6 Shell Model Problem Define a valence space Derive an effective interaction HΨ = EΨ H eff Ψ eff = EΨ eff CORE Build and diagonalize the Hamiltonian matrix.

7 Shell Model Problem Define a valence space Derive an effective interaction HΨ = EΨ H eff Ψ eff = EΨ eff CORE Build and diagonalize the Hamiltonian matrix. In principle, all the spectroscopic properties are described simultaneously (Rotational band AND β decay half-life).

8 Basis Choice of the basis: m-scheme : Φ α = a i 0 = a i1...a ia 0 i=nljmτ Simple H IJ but Maximal size: D ( d πp ) (. dνn ) coupled-scheme (J or JT): [ [ (j1 ) n 1 v 1 γ 1 x 1 (j 2 ) n 2 v 2 γ 2 x 2 ] Γ 2... (j k ) n Γk k v k γ k x k ] Reduced dimensions BUT complicated and much more non zero terms

9 Diagonalization Lanczos Method: iterative process HΨ i = E i 1 i Ψ i 1 + E ii Ψ i + E i i+1 Ψ i+1 E 11 E E 12 E 22 E E 32 E 33 E E 43 E 44 E 45 0 storage of many vectors to build the eigenvectors solved by increase of disk capacity Extension RAM memory (CPU time/elapsed time) 1 regular increase of CPU power

10 Giant Matrices Giant matrices: H IJ recalculated in the iterative process. Basic idea : Factorization I iα i : proton state, α : neutron state dim(i), dim(α) dim(i) Precalculations in each separate space two body ME in m scheme (code Antoine):H IJ = V(K) the decoupled basis coupled scheme (code Nathan): H IJ = h ij.h αβ.w(k) I=i+α, J=j+β, K=r+µ Recents improvements: generalized factorization i,α ex: semi-magic nuclei N=126 i 1i 13 2, 1h 92 α 2f 72, 2f 52, 3p 32, 3p 12

11 Actual Limitations m scheme Huge dimensions of the matrices ( ) storage of Lanczos vectors on disk AMD Opteron 64bits 2.2 GHz / 8 Gb RAM 56 Ni: D=10 9 1h30/it. Coupled scheme Small dimensions of the matrices ( ) Parallelization : each processor has the initial and a final vector Ψ (k) f final vectors are added = H (k) Ψ i Very large cases: splitting of the initial and final vectors Ψ f = k Ψ (k) f Ψ (m) f Ψ i,f = m Ψ m i,f = n H (m,n) Ψ (n) i ImaBIO Cluster 24 nodes Xeon 2.8 GHz Nuc Theo Cluster 10 nodes Xeon 2.7 GHz 128 Xe GS: D= (D M=0 =10 10 ) 400 Gb precalculation storage non zero terms!!! 8h/it.(34 procs) 2 + out of reach

12 Actual Limitations m scheme Huge dimensions of the matrices ( ) storage of Lanczos vectors on disk AMD Opteron 64bits 2.2 GHz / 8 Gb RAM 56 Ni: D=10 9 1h30/it. Coupled scheme Small dimensions of the matrices ( ) Parallelization : each processor has the initial and a final vector Ψ (k) f final vectors are added = H (k) Ψ i Very large cases: splitting of the initial and final vectors Ψ f = k Ψ (k) f Ψ (m) f Ψ i,f = m Ψ m i,f = n H (m,n) Ψ (n) i ImaBIO Cluster 24 nodes Xeon 2.8 GHz Nuc Theo Cluster 10 nodes Xeon 2.7 GHz 128 Xe GS: D= (D M=0 =10 10 ) 400 Gb precalculation storage non zero terms!!! 8h/it.(34 procs) 2 + out of reach

13 The Effective Interaction(s): Key aspects The evolution of the Spherical mean field in the valence spaces. What is missing in the monople hamiltonian derived from the realistic NN interactions, be it through a G-matrix, V low k or other options? The multipole hamiltonian does not seem to demand major changes with respect to the one derived from the realistic nucleon-nucleon potentials Do we really need three body forces? Would they be reducible to simple monopole forms? Much more to come with Andres Zuker s talk

14 Ab initio calculations Valence space : all states with excitation energy in the H.O. basis until N ω N=10 for A 8 N=8 for A 16 Specific problem : number of nljm states 276 shells for protons or neutrons 4600 states in the 22 ω space (20 in fp shell) Lee-Suzuki unitary transformation to derive effective hamiltonian in the model space use of modern realistic interactions CD-Bonn, Argonne, N 3 LO, Idaho-A...

15 NSM: 4 He E [MeV] He CD-Bonn ; 0+0 exc 28; 0+0 exc 19; 0+0 exc 40; 0+0 gs 28; 0+0 gs 19; 0+0 gs N max

16 p shell: Halo nuclei 8 Li and 8 B quadrupole moments Quadrupole moment (e.fm 2 ) N=0 N=2 N=4 N=6 N=8 N=10 N=12 exp. QFMC 8 B (21) 8.2(3) 8 Li (7) 3.0(2) Two first 3 2 states in 11 Li N=2 N=4 N=6 N=8 E( 3 2 ) 1 - E( 3 2 ) n components Ψ 1 Ψ 2 Ψ 1 Ψ 2 Ψ 1 Ψ 2 Ψ 1 Ψ 2 n= n= n= n= n=

17 Spectroscopy in 10 B

18 3N: status ; ; hω= ; ; /2 - ; 3/2 5/2-5/2-7/2-1/2 - ; 3/2 5/2-5/2-2 + ; ; / /2-7/2 - ; ; 1 5/ / /2-1/2-3/ /2 - NN+NNN Exp NN NN+NNN Exp NN NN+NNN Exp NN ; 1 10 B 11 B 12 C 13 C 1 + 7/2-2 + ; /2 - ; 3/2 0 + ; ; /2-1/2-7/2-3/2-8 5/2-1/2-4 3/2-3/2-0 1/2-1/2 - NN+NNN Exp NN 3/2 - ; 3/2 5/2 - P. Navratil, V. G. Gueorgiev, J. P. Vary, W. E. Ormand, and A. Nogga Phys. Rev. Lett. 99, (2007)

19 3N: status Nucleus/property Expt. NN+NNN NN 6 Li : E(1 + 0) [MeV] Q( ) [e fm2 ] (2) µ( ) [µ N ] Ex(3 + 0) [MeV] B(E2; ) 10.69(84) B(E2; ) 4.40(2.27) B(M1; ) 15.43(32) B(M1; ) 0.149(27) B : E(3 + 0) [MeV] rp [fm] 2.30(12) Q( ) [e fm2 ] (56) µ( ) [µ N ] rms(exp Th) [MeV] B(E2; ) (6) B(E2; ) 1.71(0.26) B(GT; ) 0.083(3) B(GT; ) 0.95(13) C : E(0 + 0) [MeV] rp [fm] 2.35(2) Q( ) [e fm2 ] +6(3) rms(exp Th) [MeV] B(E2; ) 7.59(42) B(M1; ) (21) B(M1; ) 0.951(20) B(E2; ) 0.65(13) P. Navratil, V. G. Gueorgiev, J. P. Vary, W. E. Ormand, and A. Nogga Phys. Rev. Lett. 99, (2007)

20 3N: status Nucleus/property Expt. NN+NNN NN 11 B : E( 3 1 ) [MeV] rp( ) [fm] 2.24(12) Q( ) [e fm 2 ] (26) µ( ) [µ N ] rms(exp Th) [MeV] B(E2; ) 2.6(4) B(GT; ) 0.345(8) B(GT; ) 0.440(22) B(GT; ) 0.526(27) B(GT; ) 0.525(27) B(GT; ) 0.461(23) C : E( 1 1 ) [MeV] rp( ) [fm] 2.29(3) µ( ) [µ N ] rms(exp Th) [MeV] B(E2; ) 6.4(15) B(M1; ) 0.70(7) B(GT; ) 0.20(2) B(GT; ) 1.06(8) B(GT; ) 0.16(1) B(GT; ) 0.39(3) B(GT; ) 0.19(2) Total energy rms [MeV] P. Navratil, V. G. Gueorgiev, J. P. Vary, W. E. Ormand, and A. Nogga Phys. Rev. Lett. 99, (2007)

21 Valence space The choice of the valence space: In light nuclei the harmonic oscillator closures determine the natural valence spaces: 4 He 16 O 40 Ca 80 Zr p shell sd shell pf shell Cohen/ Brown/ Deformed Kurath Wildenthal

22 Valence space The choice of the valence space: In light nuclei the harmonic oscillator closures determine the natural valence spaces: 4 He 16 O 40 Ca 80 Zr p shell sd shell pf shell Cohen/ Brown/ Deformed Kurath Wildenthal In heavier nuclei: jj closures due to the spin-orbit term show up N=28, 50, 82, 126

23 Valence space The choice of the valence space: In light nuclei the harmonic oscillator closures determine the natural valence spaces: 4 He 16 O 40 Ca 80 Zr p shell sd shell pf shell Cohen/ Brown/ Deformed Kurath Wildenthal In heavier nuclei: jj closures due to the spin-orbit term show up N=28, 50, 82, 126 the transition HO jj: occurs between 40 Ca and 100 Sn where the protagonism shifts from the 1f 7/2 to the 1g 9/2

24 Valence space A valence space can be adequate to describe some properties and completely wrong for others 48 Cr (f 7 2 ) 8 (f 7 2 p 3) 8 (fp) 8 2 Q(2 + ) (e.fm 2 ) E(2 + ) (MeV ) E(4 + )/E(2 + ) BE2( ) (e 2.fm 4 ) B(GT)

25 Valence space For the quadrupole properties f 7 p 3 is a good space 2 2 (Quasi-SU3 orbitals) whereas for magnetic and Gamow-Teller processes the presence of the spin orbit partners is compulsory. In the tin isotopes the natural valence space consists in a 100 Sn core and valence orbits: (d 5 2 g 7 2 s 1 2 d 3 2 h 11) ν 2 However, numerous E1 transitions have been measured that are forbidden in this space because: (a 11/2.ã j ) λ 1, it is then necessary to incorporate the 1g 9/2 orbit.

26 Physics Goals Precision Spectroscopy towards larger masses Description of the nuclear correlations in the laboratory frame Changing Magic Numbers far from Stability: The competing roles of spherical mean field and correlations Double β decay, the key to the nature of the neutrinos, the absolute scale of their masses and their hierarchy No core shell model for light nuclei. Ab initio description of the low-lying intruder states and of the origin of the Gamow-Teller quenching Nuclear Structure and Nuclear Astrophysics

27 The most popular flaws of the standard SM description Not all the regions of the nuclear chart are amenable to a SM description yet Quadrupole effective charges are needed (But their value is universal and rather well understood) Spin operators are quenched by another universal factor which relates to the regularization of the interaction (also known as short range correlation). Indeed, BMF approaches share this shortcoming

28 Accessible Regions Ti 22 Sc 21 Ca 20 K 19 Ar 18 Cl 17 S 16 Ζ P 15 Si 14 Al 13 Mg 12 Na 11 Ne 10 F 9 O 8 N 7 C 6 B 5 16 Be Li 3 10 He 2 8 H n 0 2 Ν Dy 66 Tb 65 Gd 64 Eu Sm 62 Pm Nd 60 Pr 59 Ce 58 La 57 Ba 56 Cs 55 Xe I Te Sb Sn 50 In 49 Cd Ag Pd Rh 45 Ru Tc Mo Mt 109 Nb 41 Hs 108 Zr Ns 107 Y 39 Sg 106 Sr 38 Ha 105 Rb 37 Rf 104 Kr Lr 103 Br 35 No 102 Se Md 101 As 33 Fm 100 Ge Es Ga 31 Cf 98 Zn Bk Cu 29 Cm 96 Ni Am Co Pu 94 Fe 26 Np Mn U 92 Cr Pa V Th Ac 89 Ra 88 Fr Rn At Po Bi Pb Tl Hg Au Pt Ir Os Re W 74 Ta Hf 72 Lu 71 Yb Tm Er Ho

29 The most popular flaws of the standard SM description Not all the regions of the nuclear chart are amenable to a SM description yet Quadrupole effective charges are needed (But their value is universal and rather well understood) Spin operators are quenched by another universal factor which relates to the regularization of the interaction (also known as short range correlation). Indeed, BMF approaches share this shortcoming

30 Quadrupole excitations on CS nuclei (igds) N=6 (gds) N=4 (hfp) N=5 (fp) N=3 (sd) N=2 (gds) N=4 40 Ca 100 Sn

31 SD band in 36 Ar C. E. Svensson, et al., Phys. Rev. Lett. 85, 2693 (2000) C. E. Svensson, et al., Phys. Rev. C63, (2001) good description in terms of ph excitations

32 SD band in 36 Ar C. E. Svensson, et al., Phys. Rev. Lett. 85, 2693 (2000) C. E. Svensson, et al., Phys. Rev. C63, (2001) good description in terms of ph excitations Decay of the SD bands shows the mixing between npnh configurations

33 SD band in 36 Ar C. E. Svensson, et al., Phys. Rev. Lett. 85, 2693 (2000) C. E. Svensson, et al., Phys. Rev. C63, (2001) good description in terms of ph excitations Decay of the SD bands shows the mixing between npnh configurations Mixing should not destroy the agreement of 4p4h calculations

34 SD band in 36 Ar C. E. Svensson, et al., Phys. Rev. Lett. 85, 2693 (2000) C. E. Svensson, et al., Phys. Rev. C63, (2001) good description in terms of ph excitations Decay of the SD bands shows the mixing between npnh configurations Mixing should not destroy the agreement of 4p4h calculations Complex mecanism and theoretical challenge

35 Decay out of the SD band in 36 Ar J exp. 4p-4h SDPF E γ (kev) improvment upon 4p-4h calculations J 10 backbending reproduced and correct moment of inertia E. Caurier, F. Nowacki, and A. Poves Phys. Rev. Lett. 95, (2005) exp. sdpf E γ (kev)

36 Decay out of the SD band in 36 Ar B(E2) in e2 fm exp. 4p-4h sdpf J good overall agreement between experiment and calculations

37 Decay out of the SD band in 36 Ar B(E2) in e2 fm exp. 4p-4h sdpf J good overall agreement between experiment and calculations reconstruction of 4p-4h results due to mixing with 6p-6h states (as deformed as 4p-4h ones)

38 Decay out of the SD band in 36 Ar B(E2) in e2 fm exp. 4p-4h sdpf J good overall agreement between experiment and calculations reconstruction of 4p-4h results due to mixing with 6p-6h states (as deformed as 4p-4h ones) E. Caurier, F. Nowacki, and A. Poves Phys. Rev. Lett. 95, (2005)

39 Decay out of the SD band in 36 Ar Out-band transitions in the SD band of 36 Ar (B(E2) s in e 2 fm 4 and energies in kev) E γ B(E2) experiment theory experiment theory 2 + SD (23) SD (4) SD (30) SD (8) SD (74) SD (30) 1.5 E. Caurier, F. Nowacki, and A. Poves Phys. Rev. Lett. 95, (2005)

40 The Superdeformed band of 40 Ca J exp sdpf-full E γ (in kev) exp: E. Ideguchi et al., Phys. Rev. Lett. 87, (2001)

41 Transition Quadrupole Moments Transition quadrupole moment (e fm 2 ) J

42 B(E2) s in Tin isotopes exp t=0 t=2 t=4 g9g7d5 t=4 gds B(E2) (e 2 fm 4 ) Neutron number A. Banu et al., Phys. Rev. C72, (R) (2005)

43 The most popular flaws of the standard SM description Not all the regions of the nuclear chart are amenable to a SM description yet Quadrupole effective charges are needed (But their value is universal and rather well understood) Spin operators are quenched by another universal factor which relates to the regularization of the interaction (also known as short range correlation). Indeed, BMF approaches share this shortcoming

44 Renormalization of the spin operator in the pf -shell Nucleus Uncorrelated Correlated Expt. Unquenched Q = V ± Fe ± Mn ± Fe ± Ni ± Co ± Ni ± 0.1

45 Quenching of GT strength in the pf -shell

46 48 Ca(p,n) 48 Sc Strength Function

47 48 Ca(p,n) 48 Sc Strength Function

48 Quenching of GT operator in the pf -shell î = α 0 ω + n 0 β n n ω, ˆf = α 0 ω + n 0 β n n ω then ˆf T î 2 = αα T 0 + β n β n T n, n 0 2 n 0 contributions negligible α α projection of the physical wavefunction in the 0 ω space is Q α 2 transition quenched by Q 2

49 Quenching of M1 operator in the pf -shell Shell Model Total Orbital Spin K. Langanke, G. Martinez-Pinedo, B(M1) (µ N 2 ) Expt. 52 Cr P. Von Neumann-Cosel, and A. Richter Phys. Rev. Lett. 93, (2004) KB3G interaction Excitation Energy (MeV)

50 Quenching of M1 operator in the pf -shell KB3 interaction Neumann-Cosel et al. Phys. Lett. B433 1 (1998)

51 Correlations in nuclei V. R. Pandharipande, I. Sick and P. K. A. dewitt Huberts, Rev. mod. Phys. 69 (1997) 981

52 50 Z, N 82 region

53 β decay systematics Nucleus 128 Sn 130 Sn 132 Sb 132 Te 133 Te Transition , 4, , 3 2, T 1/2 exp m 3.72m 2.79m 3.2d 12.5m T 1/2 calc. (0.74) 32.21m 2.47m 1.56m 1.73d 6.42m Renorm Te 135 Xe 136 Cs , 3 2, , 5, m 9.14h 13.16d 29.19m 7.07h 8.1d Our valence space leads to a renormalization of the στ operator of a factor 0.57

54 Integrated strength Integrated B(GT) values for Te 52 (left) and Te 52 (right) calculated within the (g 72 d 52 d 32 s 12 h 112 ) space (experimental data are from (p,n) reactions) exp. théo. (espace 50-82) 20 exp. théo. (espace 50-82) B(GT) cumul B(GT) cumul Excitation energy (MeV) Excitation energy (MeV)

55 A= Cs half-life Gamow-Teller strength function from the 5 + ground state of 136 Cs : T 1/2 th. T 1/2 exp. 13 d d

56 Shell Model at the limits of the stability At the very neutron rich or very proton rich edges, the T=0 and T=1 channels of the effective nuclear interaction weight very differently than they do at the stability line. Therefore the effective single particle structure may suffer important changes, leading in some cases to the vanishing of established shell closures or to the appearance of new ones.

57 Neutron rich nuclei: N=20 10 ESPE (MeV) d5/2 s1/2 d3/2 f7/2 p3/2 p1/2 f5/ Proton number 20

58 Dipole response in Neon chain Neons bound from stability to n-rich Shell evolution Shape evolution SU3 Deformation (N=Z) Sphericity (N=14) Intruder Deformation (N=20) Island of inversion Full sd diagonalization + full 1 ω excitations Exact removal of Center of Mass spurious components

59 Neons Dipole Strength 20 Ne

60 Neons Dipole Strength 22 Ne

61 Neons Dipole Strength 24 Ne

62 Neons Dipole Strength 26 Ne

63 Neons Dipole Strength 28 Ne

64 Neons Dipole Strength 30 Ne

65 Neutron rich nuclei: N= f5 Energies in MeV p1 f7 p3 d3 s1 d (1) 028 (2) Si34 (3) S36 (4) Ca40

66 26 Ne Pygmy Resonance recently observed at RIKEN: E exp= 9 MeV E SM = 8.5 MeV

67 26 Ne Pygmy Resonance B(E1) exp= 0.60 ± 0.06 e 2.fm 2, 5.9 ± 1.0 % S TRK B(E1) SM = 0.42 e 2.fm 2, 4.0 % S TRK

68 26 Ne Pygmy Resonance Main contributions : ν(2s 1 1/2 2p 3/2 ) ν(2s 1 1/2 2p 1/2 )

69 Summary Thanks to: E. Caurier, K. Langanke, G. Martinez-Pinedo, J. Menedez, A. Poves, K. Sieja, O. Sorlin, A. Zuker

The Periodic Table. Periodic Properties. Can you explain this graph? Valence Electrons. Valence Electrons. Paramagnetism

The Periodic Table. Periodic Properties. Can you explain this graph? Valence Electrons. Valence Electrons. Paramagnetism Periodic Properties Atomic & Ionic Radius Energy Electron Affinity We want to understand the variations in these properties in terms of electron configurations. The Periodic Table Elements in a column