Skyrme-EDF for charge-changing excitation modes

Transcription

1 ICNT Skyrme-EDF for charge-changing excitation modes Niigata Univ. Kenichi Yoshida

2 Outline Self-consistent Skyrme-pnQRPA approach GT strengths in neutron-rich deformed Zr isotopes deformation T= pairing pn-pair transfer strengths in N=Z nuclei in fp-shell collectivity of the pn-pairing vibration

3 Skyrme energy-density functional (EDF) approach Energy functional: E = drh(r) Energy density: Skyrme energy density: ( ) H = H kin +H Skyrme +H em H Skyrme = t ( ( Htt even 3 ( ) t=,1 t 3 = t ) + Htt odd 3 H even tt 3 = C ρ t ρ tt 3 + C ρ t ρ tt3 ρ tt3 + C τ t ρ tt3 τ tt3 + C J t ρ tt3 J tt3 + C J t J tt3 H odd tt 3 = Ct s s tt 3 +Ct s s tt3 s tt3 +Ct T s tt3 T tt3 +Ct s ( s tt3 ) +C j t j tt 3 +C j t s tt3 j tt3 T-odd densities vanish in g.s of e-e nuclei T-odd Skyrme energy density is not well constrained, but plays a role in studying compact star EoS of polarized neutron matter: ( ) H 3π = 4/3 /3 [ ρ m + ( C τ + C1 τ + C T + C1 T T-odd components ] ) ρ ρ /3 + ( C ρ + Cρ 1 + Cs + C1) s ρ isovector components

4 T-odd energy density seen in nuclear responses ( ) Collective motion is generated by the residual interaction ] v res (r 1, r ) δ E δρ(r 1 )δρ(r ) : self-consistency spin density spin excitation modes Nuclear response to the IV external field: drr L Y L ŝ 1t IV spin density: ŝ 1t = ψ (rστ) σ σ σ τ τ t τ ψ(rσ τ ) Gamow-Teller, Spin dipole,...

5 Self-consistent Skyrme-EDF approach for spin-isospin excitations in superfluid systems North Carolina + Oak Ridge group PHYSICAL REVIEW C, VOLUME 6, 143 b decay rates of r-process waiting-point nuclei in a self-consistent approach J. Engel, 1 M. Bender, 1, J. Dobaczewski,,3,4 W. Nazarewicz,,3,5 and R. Surman 1 Skyrme-pnQRPA (1999) coordinate-space 1D-HFB + QRPA in canonical basis Application to fixing the coupling constants Gamow-Teller strength and the spin-isospin coupling constants of the Skyrme energy functional 1 PHYSICAL REVIEW C, VOLUME 65, 543 M. Bender, 1,,3,4 J. Dobaczewski, 5,6 J. Engel, 3 and W. Nazarewicz 1,,5 Milano group PHYSICAL REVIEW C 87, 643(13) Large-scale calculations of the double-β decay of 76 Ge, 13 Te, 136 Xe, and 15 Nd in the deformed self-consistent Skyrme quasiparticle random-phase approximation M. T. Mustonen 1,,* and J. Engel 1, 1 Department of Physics and Astronomy, CB 355, University of North Carolina, Chapel Hill, North Carolina , USA The first deformed Skyrme-pnQRPA coordinate-space D-HFB + QRPA in canonical basis PHYSICAL REVIEW C 76,4437(7) Spin-isospin nuclear response using the existing microscopic Skyrme functionals 1D-HFBCS + QRPA S. Fracasso and G. Colò Dipartmento di Fisica, Università degli Studi and INFN, Sezione di Milano, I-133 Milano, Italy Sichuan + Wako group Physics Letters B 719 (13) 116 Role of T = pairingingamow TellerstatesinN = Z nuclei coordinate-space 1D-HFB+ QRPA C.L. Bai a, H. Sagawa b,c,,m.sasano c, T. Uesaka c,k.hagino d,h.q.zhang e,x.z.zhang e,f.r.xu f

6 ( ) ( Intermezzo -constrained by ab-initio calculations- Skyrme-EDF for matter H = H kin + (C ρ t ρ t + Ct τ ρ t τ t + Ct s s t + Ct T s t T t ) t=,1 w/ density dependence Configuration Interaction QMC cal. by Trento group A. Roggero et al., arxiv: C ρ t = A ρ t + B ρ t ρ α C ρ s = A ρ s + B ρ s ρ α TOV-min Polarized neutron matter w/ impurity (ε p -ε p )/ E F -. SkP SkM* SAMi impurity energy spin-up N neutrons, and a spinup (-down) proton -.4 BSK1 SLy4 SkO " p" " p# = 4m(Cs C1) s m(c T C1 T ) E F 3 ~ k F 5 ~ k 3 F purely T-odd k F [fm -1 ]

7 Self-consistent pnqrpa for spin-isospin excitations in deformed nuclei variation w.r.t densities starting point: Skyrme EDF E[ρ(r), ρ(r)] KY, PTEP13,113D The coordinate-space Hartree-Fock-Bogoliubov eq. for ground states J. Dobaczewski et al., NPA4(1984)3 s.p. hamiltonian and pair potential: quasiparticle basis h q = δe δρ q, hq = δe δ ρ q q = ν, π [ ] The proton-neutron quasiparticle RPA eq. for excited states Collective excitation = coherent superposition of qp excitations: residual interactions derived self-consistently : v res (r 1, r )= δ E δρ 1t3 (r 1 )δρ 1t3 (r ) τ 1 τ + Ô λ = αβ δ E δs 1t3 (r 1 )δs 1t3 (r ) σ 1 σ τ 1 τ X λ αβâ α,νâ β,π Y λ αβâ β,π âᾱ,ν

8 Canonical basis and quasiparticle basis J. Engel et al., PRC6 (1999) 143 A pn,p 8n8 5E p,p8 d n,n8 1E n,n8 d p,p8 1Ṽ pn,p8n8 ~u pv n u p 8 v n8 1v p u n v p 8 u n8!1v pn,p8n8 ~u pu n u p 8 u n8 1v p v n v p 8 v n8! ~16! B pn,p 8n8 5Ṽ pn,p8n8 ~v pu n u p 8 v n8 1u pv n v p 8 u n8! V pn,p 8n8 ~u pu n v p 8 v n8 1v pv n u p 8 u n8!. ~17! calculation cost ~ BCS-QRPA introduction of cutoff occupation probability ( )( ) ( )( ) calculation cost ~ 4*BCS-QRPA introduction of cutoff simply by the qp energy pp(hh) and ph excitations are treated on the same footing A αβα β = (E α + E β )δ αα δ ββ + d1dd1 d {ϕ1,α ν (r 1σ 1 )ϕ1,β π (r σ ) v pp (1; 1 )ϕ1,α ν (r 1σ 1 )ϕπ 1,β (r σ ) B αβα β = + ϕ ν,α (r 1σ 1 )ϕ π,β (r σ ) v pp (1; 1 )ϕ ν,α (r 1 σ 1 )ϕπ,β (r σ ) + ϕ1,α ν (r 1σ 1 )ϕ,β π (r σ ) v ph (1; 1 )ϕ,β π (r 1 σ 1 )ϕν 1,α (r σ ) + ϕ1,β π (r 1σ 1 )ϕ,α ν (r σ ) v ph (1; 1 )ϕ,α ν (r 1 σ 1 )ϕπ 1,β (r σ )}, (7) d1dd1 d { ϕ ν 1,α (r 1 σ 1 )ϕ π 1,β (r σ ) v pp (1; 1 )ϕ ν,ᾱ (r 1 σ 1 )ϕπ, β (r σ ) ϕ ν,α (r 1σ 1 )ϕ π,β (r σ ) v pp (1; 1 )ϕ ν 1,ᾱ (r 1 σ 1 )ϕπ 1, β (r σ ) ϕ ν 1,α (r 1 σ 1 )ϕ π 1, β (r σ ) v ph (1; 1 )ϕ π,β (r 1 σ 1 )ϕν,ᾱ (r σ ) KY, PTEP (13 )113D ϕ1,β π (r 1 σ 1 )ϕ 1,ᾱ (r ν σ ) v ph (1; 1 )ϕ,α ν (r 1 σ 1 )ϕπ, β (r σ )}. (8)

9 QRPA for heavy axially-deformed nuclei with HPC HFB cal. (64 CPUs) Box size: Cut-off: QRPA cal. (51 CPUs) Cut-off: HFB Hamiltonian: block diagonal in Ω π distributed to cores Matrix elements of A and B: D-numerical integration independent of each qp configuration For each K π # of qp excitation: ~5, matrix elements: 59 core hours diagonalization: 33 core hours with the help of ScaLapack

10 GT strengths and beta-decay properties of neutron-rich Zr isotopes

11 β-decay half-lives in neutron-rich Zr isotopes S. Nishimura et al., PRL6(11)55 3 (a) Kr (Z=36) (b) Sr (Z=38) (c) Zr (Z=4) (d) Mo (Z=4) Half-life (ms) FRDM+QRPA [16] FRDM+QRPA [1] KTUY+GT Previous This work 3 (e) Rb (Z=37) (f) Y (Z=39) (g) Nb (Z=41) (h) Tc (Z=43) Half-life (ms) Number of Neutrons

12 β-decay half-lives in neutron-rich Zr isotopes S. Nishimura et al., PRL6(11)55 3 (a) Kr (Z=36) (b) Sr (Z=38) (c) Zr (Z=4) (d) Mo (Z=4) Half-life (ms) 3 (e) Rb (Z=37) (f) Y (Z=39) (g) Nb (Z=41) FRDM+QRPA [16] FRDM+QRPA [1] KTUY+GT Previous This work (h) Tc (Z=43) short half-lives were measured Half-life (ms) Number of Neutrons

13 Deformed Zr isotopes on the r-process path proton number spherical deformed spherical magic number deformed magic number neutron number Zr isotopes r-process path spherical magic number 6 neutron drip line 1/E( + 1 ) (MeV 1 ) 8 4 systematics of the + energy 64 Kr (Z = 36) Sr (Z = 38) Zr (Z = 4) Mo (Z = 4) Ru (Z = 44) Pd (Z = 46) Cd (Z = 48) Sn (Z = 5) Neutron Number 8 T. Sumikama et al., PRL6(11)51

14 GT giant resonance ˆF t 3 K = σ,σ τ,τ dr ˆψ (rστ) σ σ K σ τ τ t3 τ ˆψ(rσ τ ) sudden onset of deformation at N=6 R - (/MeV) SkM* + mixed-type pairing β,ν =.38 β,π =.41 Zr SLy4: A. Blazkiewicz et al., PRC71(5)5431 fragmentation of strength distribution due to deformation separable pnrpa: P. Urkedal et al., PRC64(1)5434 R - (/MeV) Total K= K= MeV smearing width E T (MeV) 98 Zr SkM* and SLy4 give almost the same (ΔE<1 MeV) excitation energies of GTGR g =.94 (SkM*),.9 (SLy4) E* GR (MeV) SkM* SLy excitation energy w.r.t. the g.s of daughter A

15 GTGR: the need of self-consistency 8 SLy4 + mixed-type pairing 4 Zr a repulsive character of the residual interaction raises the peak energy R - (/MeV) 6 4 β,ν =.39 β,π =.43 low-lying strengths are absorbed to the highenergy peak strong collectivity of the GTGR the collectivity generated by the Landau-Migdal approximation is weak R - (/MeV) QRPA LM qp 98 Zr v ph (r 1 r )=N 1 [f τ 1 τ + g σ 1 σ τ 1 τ ] δ(r 1 r ) LM parameter: M. Bender et al., PRC65() E* (MeV) self-consistency is required for a quantitative description of the GTGR

16 IV giant dipole resonance: Anti-analog GDR ˆF t 3 1K = σ τ,τ drry 1K ˆψ (rστ) τ τ t3 τ ˆψ(rστ ) 14 SLy4 + mixed-type pairing R - (/MeV) ΔTz = -1 Zr deformation splitting (Ksplitting) is clearly seen as in GDR R - (/MeV) Total K= K=1 3 4 E T (MeV) 98 Zr

17 AGDR and neutron skin thickness linear relationship between the neutron-skin thickness and the AGDR energy E(AGDR)-E(IAS) (MeV) a (MeV) * 8 Pb Exp. result R pn =.161(4) R pn (fm) A. Krasznahorkay, N. Paar, D. Vretenar, M.N.Harakeh, Phys. Scr. T154(13)418 J.Yasuda, T.Wakasa et al., PTEP13, 63D

18 T= (S=1) pairing affects the GT response if we have (a) T=1 pairing condensate(s) due to the coupling between the p-h and p-p excitations we may see the effect in the low-lying states that are generated by qp excitations around the Fermi levels does not affect the gs properties in N>Z nuclei a form of the interaction or an np-pairing EDF is seldom known Take the simplest one; vpp T = (r, r )= 1+P σ 1 P τ V δ(r r ) the pairing strength determined to reproduce the β-decay half-life of Zr (7.1 s)

19 Low-lying GT states SLy4+Mixed-type pairing.1 MeV smearing width.5. 4 Zr w/ T= pairing w/o T= pairing 1 Zr Strength (/MeV) Zr 8 Zr.5 Zr 6 Zr MeV smearing width ET (MeV)

20 Low-lying GT states Strength (/MeV) Zr w/ T= pairing w/o T= pairing Zr Zr Zr 8 Zr 6 Zr ET (MeV) selection rule for GT- π[nn 3 Λ]Ω = Λ +1/ t σ +1 ν[nn 3 Λ]Ω = Λ 1/ = 6 Zr constructed dominantly by π[413]7/ ν[413]5/ particle-like particle-like T= pairing effective neutrons added 8, 1 Zr π[413]7/ ν[413]5/ particle-like hole-like T= pairing ineffective

21 Beta-decay half-lives SkM* (w/o T= pairing) SLy4 (w/o T= pairing) Exp. Fermi s golden rule N. B. Gove, M. J. Martin, At. Data Nucl. Data Tables (1971)5 Fermi and Gamow-Teller strengths included T 1/ (s) 1 SkM* produces longer half-lives primarily due to a large Q-value approximate Q-value Q β = M n H + B(A, Z + 1) B(A, Z).1 M n H + λ ν λ π E E =min[e ν + E π ] cf. J. Engel et al., PRC6(1999) N

22 Beta-decay half-lives with T= pairing SkM* (w/ T= pairing) SkM* (w/o T= pairing) SLy4 (w/ T= pairing) SLy4 (w/o T= pairing) Exp. Strength of T= pairing determined at N=6 SLy4 T 1/ (s) 1 reproduces well the observed isotopic dependence with T= pairing Effect of the T= pairing is small beyond N=68 SkM* gives a strong deformed gap at N= N

23 pn-pair transfer strengths in N=Z nuclei in fp-shell

24 Proton-neutron pair excitations IS spin triplet ˆP T =,S=1,S z = = 1 dr ˆψ (rστ) σ σ σ ˆψ (r σ τ) σ,σ τ ˆP T =,S=1,S z =±1 = 1 σ,σ τ dr ˆψ (rστ) σ σ ±1 σ ˆψ (r σ τ) IV spin singlet ˆP T =1,T z =,S= = 1 σ τ,τ dr ˆψ (rστ) τ τ τ ˆψ (r σ τ )

25 Interactions employed for pn-pairing vibration in fp-shell nuclei HFB eq: pnqrpa eq: 44 Ti Δn = 1.8 MeV Δp = 1.87 MeV vpp T = (rστ, r σ τ 1+P σ )= f V vpp T =1 (rστ, r σ τ 1 P σ )= V 1+P τ 1 P τ [ 1 ρ(r) [ 1 ρ(r) ρ ] ρ ] δ(r r ) δ(r r ) cf. C. Bai et al., PLB719(13)116

26 4 Ca 4 Sc FIG. 1: (Color online) pn pair-addition strengths of 4 Ca 4 Sc and 56 Ni 58 Cu in the J π =1 + [(a), (b)] and J π = + [(c), 1(d)] states smeared with a width of.1 MeV. For the (J, T )=(1, ) channel, shown are the strengths obtained with factors f =, 1., 1.3, and 1.5. For the (J, T )=(, 1) channel, theunperturbed single-particle transition strengths 1 are also shown by a dotted line. 8 6 for the particle-hole (ph) channel because the spin-isospin properties were considered to fix the coupling constants entering 4 in the EDF [14]. For the pp channel, the densitydependent contact interactions are employed: v T = f= f=1. f=1.3 unp MeV Exp. TABLE I: Microscopic structure of the collective J π = and + states in 4 Sc calculated with f =1.3. Listed are configuration, its excitation energy, and the matrix elem The excitation energies are given in MeV. The pp and excitations possessing the amplitude X Y greater t.1 are only shown. Sums of the backward-going amplitu squared and the matrix elements are shown in the last li For the J π =1 + state, the J z =componentisonlysho 4 Sc J π =1 + J π = + configuration E α + E β M S=1,S z= αβ Mαβ S= Qπ1f β 7/ ν1f 7/ π1f 7/ ν1f 5/ M π1f 5/ ν1f n H + λ 7/ 14.7 ν λ.51 π E πp E 3/ =min[e νp 3/ ν E π ].17. π1d 3/ ν1d 3/ πs 1/ νs 1/ π1d 3/ ν1d 5/.1.3 π1d 5/ ν1d 3/..3 π1d 5/ ν1d 5/ ˆF t 3 K = σ,σ f=1.3 π[413]7/ ν[413]5/ pp (rστ, r σ τ ) [ 1+P σ 1 P τ = f V 4 1 ρ(r) ] and the L = T = 1pn-pair-addition operator as 6 δ(r r ), 8 (1) ρ ˆP T =1,T z =,S= v = pp T =1 (rστ, r σ τ ) 1 [ ] dr ˆψ (rστ) τ τ τ ˆψ (r σ τ ) αβ M αβ ij Y ij.17.9 T = pn-pair-addition operators are defined as ˆP T =,S=1,S z = 1 dr ˆψ (rστ) σ σ Sz σ ˆψ (r σ τ) σσ = M n H + B(A, Z + 1) B(A, τ dr ˆψ (rστ) σ σ K σ τ τ t3 τ ˆψ Transition matrix element λ ˆP T,S = αβ M T,S αβ

27 56 Ni 58 Cu MeV f=1.3 TABLE II: Same as Table I but for 58 Cu. 58 Cu J π =1 + J π = + configuration E α + E β M S=1,S z= αβ Mαβ S= πp 3/ νp 3/ πp 1/ νp 3/ πp 3/ νp 1/ πp 1/ νp 1/ π1f 5/ ν1f 5/ π1f 7/ ν1f 7/ αβ M αβ ij Y ij Exp. strength. In Table I, the microscopic structure of th 4 state obtained by setting f to 1.3 is summarized. 1 + state is constructed by many pp excitations in ing an f 5/ and a p 3/ orbitals located above the F levels as well as the πf 7/ νf 7/ excitation. It is ticularly worth noting that the hh excitations from sd-shell have an appreciable contribution to generate T = pn-pair-addition vibrational mode, indicati Ca core-breaking. Furthermore, all the pp and hh tations listed in the table construct the vibrational m in phase. The strong collectivity can be also seen fr large amount of the ground-state correlation: A su

28 Collective pn-pairing vibration mode precursory to FIG. : (Color online) Same as Fig. 1 but for the pn pairremoval strengths. the T= pairing condensation ΔE=ω1+ - ω+ FIG. 3: (Color online) (a) Energy difference E = ω 1 + ω + in 38 K, 4 Sc, 54 Co and 58 Cu calculated with f =, 1., 1.3, and 1.5. (b) Ratio fc=1.53 of the energy difference calculated to the experimental value E/ E exp. The experimental data are approaching the critical point to the T= pairing condensation

29 Enhancement of the transfer strengths λ ˆP T = λ ˆP T =1 λ ˆP T = unp. ˆP T = Strength enhancement 1 f=1. f=1.3 f= K 4 Sc 54 Co 58 Cu d-tranfer: 4 Ca( 3 He,p) 4 Sc σ(1 + ) exp σ(1 + ) unp = 3.9 F. Pϋhlhofer, NPA116 (1968) 516

30 1 RCNP f= f=1. f= pn-pairing more effective B(GT) B(GT) B(GT) B(GT) (a) 4 Ca( 3 He,t) 4 Sc (b) 46 Ti( 3 He,t) 46 V (c) 5 Cr( 3 He,t) 5 Mn (d) 54 Fe( 3 He,t) 54 Co E x (MeV) Y. Fujita et al.,prl11 (14) 115 T. Adachi, Y. Fujita et al., NPA788 (7) 7c

31 Summary Fully-selfconsistent deformed pnqrpa is developed in a Skyrme EDF framework Microscopic and quantitative description of spin-isospin excitations in nuclei with arbitrary mass number whichever they are spherical or deformed, located around the stability line or close to the drip line provide a microscopic input to the astrophysical simulation Deformation effects on spin-isospin responses Tiny deformation splitting in Gamow-Teller excitation Clear deformation splitting in Anti-analog GDR as seen in IVGDR Effects of T= pairing Fragmentation of GTGR Low-lying GT states are sensitive to the location of the Fermi levels, and the beta-decay half-lives are shortened Proton-neutron pair-transfer strengths Strong collectivity due to the T= pairing suggests emergence of a soft mode toward the T= pairing condensation

32 [ Restoration of the isospin symmetry breaking (ISB) Even w/o the Coulomb int., the ISB occurs in N>Z nuclei in a MFA Ex. 9 Zr (N-Z=) w/o Coulomb [H MF,T ] SkM* w/o pairing C. A. Engelbrecht and R. H. Lemmer, PRL4(197)67 IAS appears as a NG mode in the pnrpa ρ max z max = 14.7 fm 14.4 fm ρ = z =.6 fm E qp 6 MeV Fermi Strength S S + =.5 ~.6 MeV E T (MeV) excitation energy w.r.t the gs of 9 Zr

33 Restoration of the isospin symmetry breaking (ISB) Ex. 9 Zr (N-Z=) w/o Coulomb protons are paired Fermi Strength SkM* + mixed-type pairing ν =. MeV π =.41 MeV S S + =.6.3 MeV numerical error increases?? E T (MeV)

34 Restoration of the isospin symmetry breaking (ISB) Ex. 9 Zr (N-Z=) w/o Coulomb inclusion of the T=1 (S=) pairing interaction in the pnqrpa vpp T =1 (r, r )= 1 P σ 1+P τ V [ 1 1 ρ (r) ρ ] δ(r r ) Fermi Strength S S + =.6.8 MeV E T (MeV)