Contents / Key words Self-consistent deformed pnqrpa for spin-isospin responses Self-consistency: T=1 pairing and IAS Collectivity of GT giant resonance Possible new type of collective mode: T= proton-neutron pairing vibrations
Self-consistent pnqrpa for spin-isospin responses variation w.r.t densities starting point: Skyrme EDF E[ρ(r), ρ(r)] The coordinate-space Hartree-Fock-Bogoliubov eq. for ground states J. Dobaczewski et al., NPA(19)3 s.p. hamiltonian and pair potential: h q = δe δρ q, hq = δe δ ρ q q = ν, π quasiparticle basis [ ] 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
Recent progress EDF-based self-consistent pnqrpa for axially-deformed nuclei Prog. Theor. Exp. Phys. 13, 113D(17pages) DOI:.93/ptep/ptt91 Spin isospin response of deformed neutron-rich nuclei in a self-consistent Skyrme energy-density-functional approach Kenichi Yoshida PHYSICAL REVIEW C 7, 3(13) Large-scale calculations of the double-β decay of 7 Ge, 13 Te, 13 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 7599-355, USA PHYSICAL REVIEW C 9, 3(1) Finite-amplitude method for charge-changing transitions in axially deformed nuclei M. T. Mustonen, 1,* T. Shafer, 1, Z. Zenginerler,, and J. Engel 1, epartment of Physics and Astronomy, CB 355, University of North Carolina, Chapel Hill, North Carolina 7599-355, U PHYSICAL REVIEW C 9, 3(1) Gamow-Teller strength in deformed nuclei within the self-consistent charge-exchange quasiparticle random-phase approximation with the Gogny force M. Martini, 1,,3 S. Péru, 3 and S. Goriely 1 1 Institut d Astronomie et d Astrophysique, CP-, Université Libre de Bruxelles, 5, Brussels, Belgium
[ 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, PRL(197)7 IAS appears as a NG mode in the pnrpa ρ max z max = 1.7 fm 1. fm ρ = z =. fm E qp MeV Fermi Strength S S + =.5 ~. MeV.1..3..5 E T (MeV) excitation energy w.r.t the gs of 9 Zr
Restoration of the isospin symmetry breaking (ISB) Ex. 9 Zr (N-Z=) w/o Coulomb, w/ pairing Fermi Strength SkM* + mixed-type pairing ν =. MeV π =.1 MeV S S + =..3 MeV numerical error increases??.1..3..5 E T (MeV)
Restoration of the isospin symmetry breaking (ISB) Ex. 9 Zr (N-Z=) w/o Coulomb inclusion of the S= pairing interaction in the pnqrpa v pp (r, r )=V [ 1 1 ρ(r) ρ ] δ(r r ) Fermi Strength S S + =.. MeV.1..3..5 E T (MeV)
GTGR: the need of self-consistency SLy + mixed-type pairing Zr R - (/MeV) β,ν =.39 β,π =.3 the collectivity generated by the Landau-Migdal approximation is weak v ph (r 1 r )=N 1 [f τ 1 τ + g σ 1 σ τ 1 τ ] δ(r 1 r ) R - (/MeV) QRPA LM qp 5 15 5 3 E* (MeV) 9 Zr LM parameter: M. Bender et al., PRC5()53 the self-consistent treatment of the static and dynamic calculations is needed for a quantitative description of the GTGR
proton-neutron pairing vibrations collectivity of T= and T=1 types
Pairing vibration and condensation (of neutrons) cf. Bès and Broglia neutron-pair operator; a probe to see the collectivity ˆP T =1,Tz =1,S= 1 dr ˆψ(rστ)δ σ,σ τ τ + τ ˆψ(r σ τ )= dr ˆψ ν (r ) ˆψ ν (r ) σ,σ τ,τ ˆψ(r σ τ) =( σ)( τ) ˆψ(r σ τ) pairing condensation: order parameter q ˆP T =1,Tz =1,S= = pairing gap: pairing vibration; precursory soft mode: λ w/ an enhanced transition strength λ ˆP T =1,Tz =1,S= is seen in normal nuclei (q=) dr h(r) ρ(r) E dr ρ ν (r) <H> q
Proton-neutron pairing collectivity T=1 (Tz=), S= pair ˆP T =1,Tz =,S= 1 dr ˆψ(rστ)δ σ,σ τ τ τ ˆψ(r σ τ ) T = 1 S = σ,σ τ,τ strong collectivity is expected as in nn and pp pairings T = S = 1 T=, S=1(Sz=,±1) pair ˆP T =,S=1 1 σ,σ τ,τ dr ˆψ(rστ)δ τ,τ σ σ σ ˆψ(r σ τ ) 1 > many works on the possible occurrence of the condensation, but largely unknown no experimental evidence so far S. Frauendorf and A. O. Macchiavelli, Prog. Part. Nucl. Phys. 7 (1) E. Garrido et al., PRC3(1)373
Pairing phase diagram: Pairing vibration and rotation G.G.Dussel et al., NPA5(19)1 H=N,-X,, c I$,&,-x,, 1 P;,P,,% 11 =1,f* p t,r=r,r* p ()(f-b +n, 1 OTC51 @ 35) \ / \ (T=l pairing) / XOl X;: vibrational rotational motion motion Fig. 1. The two-dimensional space of phases of the model. Various limiting schemes are indicated.
Interactions employed for pn-pairing vibrations in fp-shell nuclei KSB(HFB) eq: pnqrpa eq: [ v pp (r, r )=V 1 ρ(r) ] δ(r r ) ρ [ v pp (r, r )=fv 1 ρ(r) ] δ(r r ) ρ Ti Δn = 1. MeV Δp = 1.7 MeV [ ] cf. C. Bai et al., PLB719(13)11
Ca Sc 5 3 FIG. 1: (Color online) pn pair-addition strengths of Ca Sc and 5 Ni 5 Cu in the J π = [(a), (b)] and J π = + [(c), (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, the unperturbed single-particle transition strengths are also shown 1 by a dotted line. + f= f=1. f=1.3 unp. for the particle-hole (ph) channel because the spin-isospin + properties were considered to fix the coupling constants entering in the EDF [1]. For the pp channel, the densitydependent contact interactions are employed: v pp T = (rστ, r σ τ ) 1+P σ = f V v T =1 1 P τ. pp (rστ, r σ τ ) [ 1 P σ = V 1+P τ 1 ρ(r) ] δ(r r ), () 3 MeV.1 Exp. [ 1 ρ(r) ] δ(r r ), (1) ρ ρ 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 = state, the J z =componentisonlysho Sc J π = J π = + configuration E α + E β M S=1,S z= αβ Mαβ S= Qπ1f β 7/ ν1f 7/ 7.5 1.7.5 π1f 7/ M ν1f 5/ 15.. π1f 5/ ν1f n H + λ 7/ 1.7 ν λ.51 π E πp E 3/ =min[e νp 3/ ν 1. E π ].17. π1d 3/ ν1d 3/..5. πs 1/ νs 1/..5 π1d 3/ ν1d 5/.1.3 π1d 5/ ν1d 3/..3 π1d 5/ ν1d 5/ 1.1.1.31 ˆF t 3 K = σ,σ f=1.3 π[13]7/ ν[13]5/ αβ M αβ.3 5.7 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, Z τ and the L = T = 1 pn-pair-addition operator as ˆP T =1,T z =,S= = 1 dr ˆψ (rστ) τ τ τ ˆψ (r σ τ ) σ dr ˆψ (rστ) σ σ K σ τ τ t3 τ ˆψ Transition matrix element ττ λ ˆP T,S = αβ M T,S αβ in terms of the nucleon field operator, where ˆψ (r σ τ ( σ)( τ) ˆψ (r σ τ). Note that the absolute val
1 1 1 3 1 f= f=1. f=1.3 5 15 pn-pairing more effective Y. Fujita et al, PRL11(1)115 C. L. Bai et al, PRC9(1)5335 Low-energy super GT state in Sc B(GT) B(GT) B(GT) B(GT) 3.. 1............. (a) Ca( 3 He,t) Sc (b) Ti( 3 He,t) V (c) 5 Cr( 3 He,t) 5 Mn (d) 5 Fe( 3 He,t) 5 Co. 1 E x (MeV) T. Adachi, Y. Fujita et al., NPA7 (7) 7c
5 Ni 5 Cu 15 5 +. MeV f=1.3 TABLE II: Same as Table I but for 5 Cu. 5 Cu J π = J π = + configuration E α + E β M S=1,S z= αβ Mαβ S= πp 3/ νp 3/.5 1. 1.9 πp 1/ νp 3/..39 πp 3/ νp 1/.5.37 πp 1/ νp 1/ 7.9. π1f 5/ ν1f 5/ 9.7.15.55 π1f 7/ ν1f 7/ 5.1.17.5 αβ M αβ.5. ij Y ij.3.3 strength. In Table I, the microscopic structure of th + state obtained by setting f to 1.3 is summarized. state is constructed by many pp excitations inv ing an f Exp. 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 fro large amount of the ground-state correlation: A su the backward-going amplitudes squared is.17 The low-lying state in 5 Cu is also sensitive to T = pairing interaction. As shown in Table II,
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 ω + in 3 K, Sc, 5 Co and 5 Cu calculated with f =, 1., 1.3, and 1.5. (b) Ratio fc=1.53 of the( energy Ca) difference calculated to the experimental value E/ E exp. The experimental data are approaching the critical point to the T= pairing condensation
Interaction dependence: no qualitative difference pair-transfer strength (MeV -1 ) 3 5 15 5 1 1 1 (a) Sc (c) Sc + 1 3 5. 3 5 15 5 RPA frequency (MeV) (b) 3 K 1 (d) 3 K + 1 3 pair-transfer strength (MeV -1 ) 3 1 (a) Sc (c) Sc +. 15 5 1 (d) RPA frequency (MeV) (b) 3 K 3 K + 1 3 5
B(M1)exp=.3±.3 μn : PRC75(7)31 g l (i) l(i) i gs IV τ z (i) s(i) g IS s i s(i) i B(M1 )(µ N) (*)Lisetskiy et al.,prc(1999)3 (**)Tanimura et al., PTEP(1)53D
pn-pairing vibrations in 1 O: another example of LS-closed system 5 pair-transfer strength (MeV -1 ) 15 5 + 1. MeV Exp. (a) 1 F (c) 1 F + 1. 3 1 3 1 RPA frequency (MeV) (b) 1 N (d).31 MeV + Exp. 1 N + 1 3 5
pn-pairing vibrations in the open-shell nuclei Ti V 3 5 15 5 + +.99 MeV Exp. w/ T=1 pairing condensation repulsive ph interaction (GT-type) attractive pp interaction p ph int. pp int. n Ti + qp excitation
pn-pairing vibrations in the mid-shell nuclei w/ T=1 pairing condensation and quadrupole def. Cr constrained HFB+pnQRPA 5 Mn 3 T= T=1 15 5 7 5 3 T= T=1 1 1
Summary