proton-neutron pairing vibrations

Transcription

1 proton-neutron pairing vibrations

2 Outline of this lecture: Basics of the vibrational modes of excitation in nuclei surface vibration, like-particle pairing vibration, and then Microscopic framework to describe the vibrations in spinisospin space Some results in N=Z nuclei: 4 Ca - 56 Ni Summary and outlook

3 Vibrational excitations in nuclei Giant resonance: high-frequency vibration of surface intuitive and classical picture of the collective modes Soft mode: low-frequency vibration associated with phase transition How do we define phases and its transition in finite nuclei?

4 Quadrupole correlation and associated collective excitation ˆQ dxr Y (r) ˆψ (x) ˆψ(x) q ˆQ β E( + ), + ˆQ +

5 Quadrupole correlation and associated collective excitation ˆQ dxr Y (r) ˆψ (x) ˆψ(x) q ˆQ β <H> q q

6 Quadrupole correlation in rare-earth nuclei Sm Sm Sm Energy (MeV) Sm Sm Sm Energy(MeV)

7 Shape evolution seen in photo absorption cross sections KY and T. Nakatsukasa, PRC83(11)134R (mb) Photoabsorption cross section 3 1 (a) 15 Nd 1 15 Nd Nd Nd Nd (b) 154 Sm 15 Sm 15 Sm 148 Sm 146 Sm ω 1 R cf. Harakeh & van der Woude, Giant Resonances E (MeV) 14 Nd E (MeV) 144 Sm

8 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

9 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 > 1 many works on the possible occurrence of the condensation, but largely unknown no evidence so far S. Frauendorf and A. O. Macchiavelli, Prog. Part. Nucl. Phys. 78 (14) 4 E. Garrido et al., PRC63(1)3734

10 Pairing phase diagram: Pairing vibration and rotation G.G.Dussel et al., NPA45(1986)164 H=N,-X,, c I$,&,-x,, 1 P;,P,,% 11 =1,f* p t,r=r,r* p (8)(f-b +n, 1 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.

11 Density functional theory 1998 Hohenberg-Kohn theorem (1964) Existence of the energy density giving the exact g.s. energy of many-body int. system Kohn :EDF Kohn-Sham theorem (1965) The exact g.s. of many-body int. system is given as a Slater determinant of the Kohn-Sham orbitals Kohn-Sham (KS) eq. particle density kinetic energy density cf. HF mean field Γ[ρ] v[ρ] gs of many-body system single-particle motions in a one-body potential

12 Skyrme energy-density functional (EDF) Energy functional: Energy density: Skyrme energy density: E = H Skyrme = dre[ρ(r)] E = T +H Skyrme +H em ( ) ( ) t t=,1 t 3 = t ( H even tt 3 ) + Htt odd 3 ρ(r) σ ˆψ(rσ) ˆψ (rσ) 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 dynamics E[ρ(r), ρ(r)] ρ(r) ˆψ(r ) ˆψ(r )

13 starting point: Skyrme + pairing EDF variation w.r.t densities KY, PTEP13,113D Self-consistent pn-qrpa for exploring vibrational modes in spin-isospin space E[ρ(r), ρ(r)] T=1(nn and pp) pairing condensates The coordinate-space Kohn-Sham-Bogoliubov-de Gennes eq. for ground states J. Dobaczewski et al., NPA4(1984)13 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

14 Recent progress EDF-based self-consistent pnqrpa for axially-deformed nuclei Prog. Theor. Exp. Phys. 13, 113D(17pages) DOI: 1.193/ptep/ptt91 Spin isospin response of deformed neutron-rich nuclei in a self-consistent Skyrme energy-density-functional approach Kenichi Yoshida 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 PHYSICAL REVIEW C 9, 438(14) 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 , U PHYSICAL REVIEW C 89, 4436(14) 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-6, Université Libre de Bruxelles, 15, Brussels, Belgium

15 Interactions employed for pn-pairing vibrations in fp-shell nuclei KSB(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

16 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, 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 8 + properties were considered to fix the coupling constants entering in the EDF [14]. For the pp channel, the densitydependent 6 contact interactions are employed: 4v 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 ρ(r) ] δ(r r ), () 3 MeV 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 =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/ M ν1f 5/ 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/ π1d 5/ ν1d 3/ 1..3 π1d 5/ ν1d 5/ ˆF t 3 K = σ,σ f=1.3 π[413]7/ ν[413]5/ 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, 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

17 1 Talk by Fujita this morning: Low-energy super GT state in 4 Sc 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) T. Adachi, Y. Fujita et al., NPA788 (7) 7c

18 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.3.3 strength. In Table I, the microscopic structure of th 6 + 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 4 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 4 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 1 backward-going amplitudes squared is.17 The low-lying 1 + state in 58 Cu is also sensitive to T = pairing interaction. As shown in Table II,

19 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 4 Ca) difference calculated to the experimental value E/ E exp. The experimental data are approaching the critical point to the T= pairing condensation

20 Enhancement of the pair transfer strengths λ ˆP T = λ ˆP T =1 λ ˆP T = unp. ˆP T = Strength enhancement f=1. f=1.3 f= K 4 Sc 54 Co 58 Cu enhancement of the cross section in 4 Ca( 3 He,p) 4 Sc σ(1 + ) exp σ(1 + ) unp = 3.9 F. Pϋhlhofer, NPA116 (1968) 516

21 Microscopic transition density of the T= pair transfer 4 Ca ( + ) 4 Sc (1 + ) r δρ add T =,Sz=(r) (fm 1 ) r =3.4 r (fm) f= f=1. f=1.3 unp.

22 pn-pairing vibrations in the open-shell nuclei w/ T=1 pairing condensation

23 pn-pairing vibrations in the open-shell nuclei 44 Ti 46 V MeV Exp. w/ T=1 pairing condensation repulsive ph interaction (GT-type) attractive pp interaction p ph int. pp int. n 44 Ti + qp excitation

24 pn-pairing vibrations in the mid-shell nuclei w/ T=1 pairing condensation and quadrupole def. 48 Cr 1 constrained HFB+pnQRPA 5 Mn T= T= T= T=

25 pn-pairing vibrations in sd-shell nuclei 4 Mg 4 Mg T= T= 6 Al Na T=1 T=1

26 Summary and outlook