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Maciej Zalewski, UW

Maciej Zalewski, UW. Kazimierz Dolny, 30.09.2006. Terminating states as a unique laboratory for testing nuclear energy density functional. under supervision of W. Satuła. Outline: -fine tuning of LEDF parameters using terminating states, -time odd fields and spin-orbit strenght,

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Maciej Zalewski, UW

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  1. Maciej Zalewski, UW Kazimierz Dolny, 30.09.2006 Terminating states as a unique laboratory for testing nuclearenergy density functional under supervision of W. Satuła Outline: -fine tuning of LEDF parameters using terminating states, -time odd fields and spin-orbit strenght, -phenomenological restoring of broken rotational symmetry in Imax-1 states

  2. Y | H | Y Slater determinat Local Energy Density Functional Theorywith Skyrme force induced parameters Skyrme force parameters are fitted to set of data (nuclear matter, masses and radii of nuclei). There is no one obvious way to obtain this parameters hence there is great number of parametrizations. 20 parameters We may treat LDF as a starting point and adjust C parameters

  3. -7/2 -7/2 -5/2 -5/2 -3/2 -3/2 -1/2 +1/2 +1/2 +3/2 DE = +3/2 +5/2 +5/2 n +7/2 +7/2 f7/2 -3/2 -3/2 -1/2 -1/2 E( ) +1/2 +1/2 +3/2 +3/2 Imax -1 n+1 d3/2 f7/2 E( ) - Imax Examples of band terminating states in 46Ti24 • Terminating states: • the best example of almost unperturbed single particle motion, • uniquely defined (NZ), • configuration mixing beyond mean-field expected to be marginal, • shape-polarization effects included already at the level of the SHF, • good to test badly known time-odd fields, • seem to be ideal for fine tuning of particle-hole interaction. -1 (n=7) (n=6) d3/2 f7/2 f7/2 partially filled -1/2 14h 17h p-h 20 across the gap +3h fully filled 0h 0h proton neutron cranking The idea is to calculate the difference energy scale (bulk properties) spin-orbit dominates!!!

  4. „locked” by the local gauge invariance SLy4 SLy5 1,0 „free” i.e. not constrained by data SkO SIII MSk1 0,5 Hence, the isoscalar Landau parameters induced by the Skyrme: 0.40 SkM* SkXc 0,0 SkP g0 -0.19 -0,5 g1 are more or less „random” 0,7 0,8 0,9 1,0 m* scales with m* Skyrme force induced LEDF

  5. (N-Z)/A (N-Z)/A 2.5 2.0 1.5 (b) Landau LEDF: (a) Skyrme LEDF: SkP SkM* SkXc 1.0 SkO SLy4 SLy5 0.5 DEexp-DEth [MeV] SIII 0 45Ti 42Ca 45Ti 42Ca 47V 47V 46Ti 44Sc 45Sc 44Ca 46Ti 44Sc 45Sc 44Ca Spin fields Landau parameters: g0=0.4; g1=0.19 iduced LEDF Original Skyrme force induced LEDF - dicrease of ΔEexp-ΔEthand unification of isotopic and isotonic dependence H. Zduńczuk, W. Satuła, R. Wyss, Phys. Rev. C58 (2005) 024305

  6. * * DE [MeV] W0 W0 120 140 160 180 [MeV fm5] W0; Correlation between spin-orbit strenght and ΔE 2.0 „standard” s-o term: SkM* W0 1.5 MSk1 SkP 1.0 VERSUS SkXc „scaled” s-o term: Sly.. Sly.. 0.5 SkO SIII 0 H. Zduńczuk, W. Satuła, R. Wyss et al, Int. Jour. of Mod. Phys.A422

  7. SLy4 0.18 W=W’ W1/W0 = 1/3 0.16 s [MeV] 0.4 0.14 SkO DE 0.12 1/3 0 -1 -1.3 0.2 W1/W0 W=-W’ W1/W0 = -1 0.0 0 1/3 -0.2 W’=0 W1/W0 = 0 45Ti 42Ca 47V 46Ti 44Sc 45Sc 44Ca (N-Z)/A Modification of spin-orbit strenght Standard Skyrme s-o: Landau LEDF: SkO Non-standard Skyrme s-o: W1/W0 ~ -1.3 Reinhard/Flocard: DEexp-DEth [MeV] SkO -1 W1/W0 = -1.3 Brown (SkXc): spin-orbit reduced by 5% -further dicrease of ΔEexp-ΔEth by ~200keV to acceptable level, H. Zduńczuk, W. Satuła, R. Wyss, Phys. Rev. C58 (2005) 024305

  8. Terminating states f7/2n -7/2 -7/2 Imax Imax-1 -5/2 -5/2 -3/2 -3/2 f7/2 -1/2 -1/2 +1/2 +1/2 +3/2 +3/2 17/2h 19/2h +5/2 +5/2 20 +7/2 Signature change -1h +7/2 -3/2 -3/2 d3/2 -1/2 -1/2 0h +1/2 +1/2 +3/2 +3/2 protons neutrons cranking

  9. -0.5 E(Imax)-E(Imax-1) [MeV] -1.0 EXP -1.5 SM SLy4 n SLy4 p -2.0 46V 43Sc 45Sc 45Ti -2.5 42Sc 44Sc 44Ti 46Ti 47V (spurious state) (Imax-1 state) Assuming that particles from f7/2 shell (outside the core) play role only , we may treat |Imax, Imax>state as a vacuum for creation and anihilation operators â+, â -3.0 Example for 43Sc b a Energy difference between Imax and Imax-1 f7/2n states results meanfield calculations disagree with SM and Exp. data • there should be one Imax-1 state, • mean-field solutions break rotational symmetry.

  10. e1= E(Imax)-E(ν) e1= E(Imax)-E(π) We set zero of the enargy scale at the energy of Imax state. Interaction between |π> and |ν> Energies of ‘spurious’ and Imax-1 states Method A Requiring λ1=0, we have: Method B We assume that we know a,b coefficients: Restoration of rotational symmetry – two methods

  11. phenomenology -0.5 0.70 from neutron from proton -1.0 E(Imax)-E(Imax-1) [MeV] 0.65 EXP -1.5 SM 0.60 SLy4 A -2.0 SLy4 B 0.55 46V 43Sc 45Sc 45Ti 46V 43Sc 45Sc 45Ti -2.5 42Sc 44Sc 44Ti 46Ti 47V 42Sc 44Sc 44Ti 46Ti 47V 0.50 -3.0 • good agreement with SM and Exp. • in general – these methods works really good ! • method B seems to work slightly better, p-state probability in Imax-1 Results of calculations with angular momentum projection support this conclusion H. Zduńczuk, J. Dobaczewski, W. Satuła – see poster by H. Zduńczuk Restorationsymmetry

  12. -7/2 -7/2 -7/2 -5/2 -5/2 -5/2 -3/2 -3/2 -3/2 -1/2 -1/2 -1/2 +1/2 +1/2 +1/2 +3/2 +3/2 12h 12h +3/2 12h f7/2 neutron signature change f7/2 proton signature change +5/2 +5/2 +5/2 d3/2 proton signature change +7/2 +7/2 +7/2 -3/2 -3/2 -3/2 a -1/2 -1/2 -1/2 1/2h 1/2h 1/2h protons +1/2 +1/2 b +1/2 neutrons +3/2 +3/2 +3/2 c Energy difference between Imax and Imax-1 [f7/2n+1 d3/2-1] states • Three possible mean-field Imax-1 states: • neutron signature change in f7/2shell, • proton signature change in f7/2shell, • proton signature change in d3/2shell. Now there is one ‘spurious’ state and two Imax-1 states

  13. ei= E(Imax)-E(i) i= ν, π, π, We set zero of the enargy scale at the energy of Imax state. Energies of ‘spurious’ and Imax-1 states We allow complex V and set it to obtain λ1=0(in this case – complex congugation in Hamiltonian) Method B where: Symmetry restoring – two methods again Method A We assume we know a, b, c coeficients and require λ1=0 :

  14. The lowest Imax-1 state The lowest Imax-1 state exp exp SLy4 2.0 corrected SLy4 SM SM 1.5 1.0 0.5 SkO corrected SkO 0 46Ti 44Sc 45Ti 46Ti 44Sc 47V 47V 44Ca 43Sc 45Ti 44Ca 43Sc 45Sc 45Sc 42Ca 42Ca Results of restoring rotational symmetry E(Imax) – E(Imax-1) [MeV] 455keV • Sly4 • -constant offset of ~450keV • details of isotopic and isotonic dependance reproduced remarkably well SKO -average value is good -discrepancies in isotonic and isotopic dependance

  15. Summary • terminating states are excellent for testing nuclear energy density functional, • mean field solutions of Imax states are in excellent agreement with experimental data, • Imax-1 states cannot be reproduced by mean field! They break rotational symmetry which can be easily restored,

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