Relativistic versus Non Relativistic Mean Field Models in Comparison

Transcription

1 Relativistic versus Non Relativistic Mean Field Models in Comparison 1) Sampling Importance Formal structure of nuclear energy density functionals local density approximation and gradient terms, overall performance ) More subtle observables separation energies, neutron radii, shell corrections, fission, dipole resonances 3) Beyond mean field collective correlation from soft quadrupole modes effect on ground state energies, low lying spectra Collaboration: M. Bender T. Bürvenich, D. Madland, W. Nazarewicz, M.R. Strayer, T. Cornelius, J.A. Maruhn, W. Greiner, P. Klüpfel, P. Fleischer, P. G.R. Argonne Los Almos Oak Ridge Frankfurt Erlangen

2 1) Sampling importance term by term Brief overview of energy functionals Local Density Approximation and beyond example SHF Local Density Approximation and beyond all models need for phenomenological control high quality description crucial role of gradient "corrections" Acronyms: SHF = Skyrme Hartree Fock RMF = Relativistic Mean Field " PC =... with point couplings " FR =... with finite range meson fields LDA = Local Density Approximation

3 Composition of the energy density ρ ρ τ kin ρ J ls ρ ρ ρ +α SHF RMF PC RMF FR ~ ρ ~ ρ ~ τ kin ~ ρ J ~ ls ~ ρ ~ ρ ~ ρ ρ α non relativistic expansion ρ σ ρ ~ ( δ ) ~ ρ ω ρ ρ ρ σ ρ σ ρ ω ρ ω ρ σ 3 ρ σ 4, short range exp. ρ ρ ρ ρ ~ ρ ρ ρσ ρ σ D σ ( ρ σ ) ρ σ ρ ω D ω ρ ~ ω ρρ D ~ ρ ρρ 1 D = +m different density dependence T= T=1 T= T=1 T= T=1 same in all 3: pairing energy density χχ (1 ρ/ρ ), c.m. correction

4 The Skyrme energy density functional E = E kin + τ kin J ls ~ ρ ~ τ kin J ~ ls d 3 r ε( ρ,,,,, ) +E pair +E C +Ecm kinetic energy φ α depends on s.p. wavefunctions maintains shell structure mean field part Skyrme functional density τ kin kinetic denisty J spin orbit density ls ρ = ρ p+ρn isoscalar ~ ρ = ρp ρn isovector ρ? Pairing Coulomb center of mass correction well known δ interaction density dependence yet in construction Mean field (=Kohn Sham) equations: δe δ φ α = ε α φ α

5 Check applicability of LDA for nuclei τ kin J ls ~ ρ ~ τ kin J ~ ls E = E kin + d 3 r ε( ρ,,,,, ) +E C +E pair +E cm E/A [MeV] Local Density Approximation RBHF Malfliet BHF Baldo et al ρ [fm -3 ] spin orbit BHF nuc.mat. relevant E ρ ( ρ, ~ ρ) A b 4 ρ J ls } ambigous input % error binding energy % error rms radius Pb Sn N=8 Ca Zr Ni fair, inprinciple useless, in practice total nucleon number A

6 LDA + phenomenological gradient terms τ kin J ls ~ ρ ~ τ kin J ~ ls E = E kin + d 3 r ε( ρ,,,,, ) +E C +E pair +E cm Local Density E ~ ρ ( ρ, ρ) Approximation A } spin orbit ρ E/A [MeV] RBHF Malfliet BHF Baldo et al BHF nuc.mat. relevant ρ [fm -3 ] b 4 J ls ρ ambigous input BHF gradient % error E B % error r rms Ca Ni Zr Sn N=8 E(ρ) BHF + ls + gradient total nucleon number A Pb BHF+gradient = high quality model

7 Fit parameters to ground state properties τ kin J ls ~ ρ ~ τ kin J ~ ls E = E d 3 r ε( ρ, kin +,,,, ) +E C +E pair +E cm two body density dep. spin orbit gradient kinetic ρ ~ ~ b ρ ρ + b +α ~ ~ b ρ ρ ρ α 3 + b 3 ~ ~ b ρ J ~ 4 ρ J + ls b 4 ( ) ls b ( ~ ( ρ) ~ + b ρ ) ρ τ ~ ρ ~ 1 kin + b ~ τ kin b 1 isoscalar T= = ρ +ρ p n isovector T=1 ~ ρ = ρ ρ p n % error E B % error r rms Ca Ni Zr Sn N=8 only ρ+ls +grad.terms all terms total nucleon number A Pb gradient terms are most crucial

8 Composition of the energy density ρ SHF RMF PC RMF FR ~ ρ ρ σ ρ +α ~ ρ ρ α ρ ω ρ J ~ ls ρ J ~ ls ρ ρ ~ ρ ~ ρ ρ τ kin ~ ρ ~ τ kin ρ σ 3 ρ σ 4, ρ σ ρ σ ρ ρ ω ω ρ ~ ( δ ) ~ ρ ρ ~ ρ ρ ρσ ρ ρ ρ ρ ρ σ D σ ( ρ σ ) ρ σ m < σ m = LDA = gradient = full model ρ ω D ω ρ ~ ω ρρ D ~ ρ ρρ 1 D = +m T= T=1 T= T=1 T= T=1

9 LDA gradients full model ρ ρ m * ρ ρ ρ σ ρ ρ σ ρ ρ T=1 all m * ρ ρ σ ω ω T=1 ω m σ ρ ρ ρ σ ω m σ all %error r rms % error r rms % error E % error R diffr %error R diffr 1 %error E % error σ %error σ SHF RMF PC FR SHF RMF PC FR

10 Conclusion on formal aspects: (holds for all three approaches SHF, RMF PC, and RMF FR) LDA = pure ρ dependence gradient terms all further terms &: gradient terms ρ gross features o.k., compatible with BHF most crucial for quantitative success some fine tuning not yet under control in ab initio theories strong phenomenologial control required in model development properly adjusted models deliver high precision in describing nuclear bulk properties, and more... try to access more observables

11 ) More subtle observables Separation energies ( see also "beyond m.f.) Neutron radii and isovector information Shell corrections in superheavy elements Fission barriers in sperheavy elements Dipole excitations

12 Observables from mean field calculations Energy: E = E kin + E pot ( ρ,...) + E Coul + E pair + E cm S n = E(Z,N ) E(Z,N), s.p. energies: δ n = E(Z,N+) E(Z,N)+E(Z,N ) ε α l*s splitting ε n l = ε n j+1/ l εn j 1/ l Density: ρ(r) deformation β lm = 4π <r l Y lm > 3AR l Formfactor: F(q) = d 3 r e iqr ρ(r) (nucl) F charge (q) = [ G proton (q) (nucl) F p (q) + G neutron (q) F n (q) ] G cm (q) r rms, R diffr, σ surf, R n R 5 skin p, halo 3 r R Giant resonances: ρ ω ω ρ (r, ) = dt e i t (r,t) B(Elm) strength

13 Two neutron separation energy in Sn S n = E(Z,N ) E(Z,N) = measure of HOMO & stability S n [MeV] 15 1 shell gap NL-Z NL3 SLy6 SkI1 SkI4 exp. } RMF } SHF 5 agreement for known nuclei different halting points neutron number N different driplines neutron rich region dominated by shell effects

14 SHF & g.s. data allow broad variation Symmetry energy and neutron radii E neutron skin r n -r-p [fm] a sym ( p p ρ ρ ρ +ρ n n SHF RMF exp. ) ties proton and neutrons together fixed in RMF adopted uncertainty neutron skin uniquely related to symmetry energy.1 8 more Pb reliable.5 neutron radii desirable symmetry energy a sym [MeV]

15 Typical fission paths in (super )heavy nuclei deformation energy heavy nucleus V( β) = < Φ β H Φ β > with Φ β > from CHF: H superheavy nucleus H λq symmetric triaxial asymmetric double (triple) humped barrier only single barrier simply to characterize by one barrier height

16 Fission through oblate state double constraint: r Y, r Y axial triaxial quadrupole PES: V (β,γ) fission path

17 Systematics of prolate fission barriers in SHE Z 1 Proton Number Z symmetric barrier [MeV] with ski4 9MeV 5 MeV 1 MeV SkI Neutron Number N N 1 Z Proton Number Z symmetric barrier [MeV] with nlz Neutron Number N 164 9MeV 5 MeV 1 MeV NL Z 18 N Z 1 Proton Number Z symmetric barrier [MeV] with sly6 9MeV 5 MeV 1 MeV SLy Neutron Number N N Z 1 Proton Number Z MeV 5 MeV 1 MeV symmetric barrier [MeV] with nl3 NL Neutron Number N N

18 Performance of SHF for Dipole Giant Resonances systematically too low frequencies for light nuclei GDR well adjustable by SHF for heavy nuclei

19 Shell structure and low lying dipoles dipole strenght RPA spectrum in isovector dipole channel for a variety of SHF forces with different m m*/m=1. m*/m=.81 m*/m=.68 8 Pb frequency ω [MeV] * /m level density little but sufficient strength in low ωmodes crucial spectral information but masked by residual interaction

20 dipole strength. super-deformed GDR at different deformations test case = 15 Dy three minima: prolate, oblate, super deformed x mode y mode z mode ω [MeV] prolate oblate PES [MeV] x, y z = deformation splitting pattern x, y, z separately = 1ph fragmentation oblate iso. prolate g.s β 15 Dy, SLy6 super-def. iso. fragmentation can mask splitting for small deformation

21 GDR: triaxial splitting in 188 Os dipole strength Os β=.34 γ=17.5 Sly6 x y z total ω [MeV]

22 Conclusions on several observables: Neutron radii symmetry energy more reaction theory Separation energies shell structure, but... Superheavy elements broad islands of shell stabilization & fission barriers differences SHF RMF Dipole resonances problems with light nuclei mix of 1ph structures and deformation low lying dipole shell structure

23 3) Beyond mean field Collective correlations from superposition of deformed states Effect on ground state: example two proton shell gap in Sn Systematics of low lying + : example E( + ) and B(E) in Sn

24 Beyond m.f.: collective correlations soft mode: collectivc deformation path coherent superposition: variation: δ ψ = <Ψ H^ Ψ> <Ψ Ψ> Ψ> = dβ Φ > ψ(β) β Gaussian overlaps ( ^ H λ ^ Q) Φβ > = E Φ β > approx. collective wavefunction collective Schrödinger eq. (CHF) V = <Φ β H ^ Φ β > + + correlation energy Quantum corrections: ^ ^ ^ <P col H P col > V = V Σ col <P ^ col P ^ col > ^ P col =, i, i β γ ^ ^ ^ J x, J y, J z } quadrupole triaxiality ~ang.mom.projection

25 Two proton shell gap as measure δ p = E(Z+,N) E(Z,N)+E(Z,N) twice HOMO LUMO gap modified by ground state deformation (of Z+ ) and by ground state correlations δ p MeV] Z=5 levels near Fermi surface + corrections spherical MF deformed MF mf + GSC Exp Neutron Number N SkI3

26 Information from low lying + states Tool: collective path { Φ >} from CHF, GCM superposition Ψ> = dq Φ q > q f(q) doubly magic 13 Sn shell gap res.int. far off N=8 pairing gap transition strengths basically correct but could be better (better collective path?, ATDHF instead of CHF) E( + ) (MeV) B(E) (e b ) SkI3 SLy6 SkM* exp. surface pairing Sn Neutrons

27 Effect of pairing on low + E( + ) for N<8 sensitive on pairing strenght volume (DI) and surface (DDDI) pairing comparable N<8 different N>8 E( + ) (MeV) B(E) (e b ) neutr. gap [MeV] SkI3 (DI*.75) SkI3 (DI) SkI3 (DI*1.5) SkI3 (DDDI) exp. Sn Neutrons

28 Conclusions on correlations Ground state correlations are important when looking at it with amplifying glasses (e.g. separation energies, shell gaps) Low lying E( + ) and B(E) sensitive to force parameters pairing B(E) yet to be improved near magic off magic