Effect of pairing correlations on nuclear low-energy structure: BCS and Bogoliubov descriptions

Transcription

1 Proceedings of the 15th National Conference on Nuclear Structure in China Guilin Oct. 5-8th 1 Ê3IØ(Œ? 014c10 5-8F Effect of pairing correlations on nuclear low-energy structure: BCS and Bogoliubov descriptions Jian Xiang( ê) Lanzhou University#School of Nuclear Science and Technology =²ŒÆ#Ø Æ EâÆ CollaboratorsµZ. P. Li(o ) J. M. Yao(ô²) W. H. Long(9 Ÿ) P. Ring J. Meng(Š#)

2 Outline 1 Introduction Theoretical Framework 3 Results and discussion 4 Conclusion ê (LZU) / 19

3 Outline Introduction 1 Introduction Theoretical Framework 3 Results and discussion 4 Conclusion ê (LZU) 3 / 19

4 Introduction Introduction Pairing correlation: BCS or Bogoliubov transformation. [Y. K. Gambhir(1990), H. Kucharek(1991), P. Ring(1996).] There is no essential difference between BCS and Bogoliubov methods for the descriptions of the ground-state of stable nuclei. [P. Ring and P. Schucko, The nuclear many-body problem, 1980.] Girod et al. have compared the results from HFB and HFBCS calculations. [M. Girod et al., PRC 7, 317(1983).] 1 PESs given by these two methods are very similar; the pairing gaps and energies from HFBCS are slightly smaller than those from HFB. 4 / 19

5 Introduction Introduction Pairing correlation: BCS or Bogoliubov transformation. [Y. K. Gambhir(1990), H. Kucharek(1991), P. Ring(1996).] There is no essential difference between BCS and Bogoliubov methods for the descriptions of the ground-state of stable nuclei. [P. Ring and P. Schucko, The nuclear many-body problem, 1980.] Girod et al. have compared the results from HFB and HFBCS calculations. [M. Girod et al., PRC 7, 317(1983).] 1 PESs given by these two methods are very similar; the pairing gaps and energies from HFBCS are slightly smaller than those from HFB. low-lying spectra: BCS Vs Bogoliubov transformation. 4 / 19

6 Introduction Introduction Covariant density functional theory (CDFT) is a reliable platform for studying the complicated nuclear excitation spectra and electromagnetic decay patterns [M. Bender(003), J. Meng(006), D. Vretenar(005), J. Dobaczewski(011).] 5 / 19

7 Introduction Introduction Covariant density functional theory (CDFT) is a reliable platform for studying the complicated nuclear excitation spectra and electromagnetic decay patterns [M. Bender(003), J. Meng(006), D. Vretenar(005), J. Dobaczewski(011).] Bohr collective Hamiltonian achieved great success in describing the low-lying excited states in nuclei. [T. Niks ic (009, 011), Z. P. Li(011), J.M. Yao(011), H. Mei(01), J.-P. Delaroche(010)] 5 / 19

8 Introduction Introduction Covariant density functional theory (CDFT) is a reliable platform for studying the complicated nuclear excitation spectra and electromagnetic decay patterns [M. Bender(003), J. Meng(006), D. Vretenar(005), J. Dobaczewski(011).] Bohr collective Hamiltonian achieved great success in describing the low-lying excited states in nuclei. [T. Niks ic (009, 011), Z. P. Li(011), J.M. Yao(011), H. Mei(01), J.-P. Delaroche(010)] The goal of this work To test the validity of the BCS in describing the low-energy structure of nuclei, as referred to the Bogobliubov. Taking even-even Sm isotopes as candidates, the comparisons are performed within RMFBCS and RHB based 5DCH. 5 / 19

9 Outline Theoretical Framework 1 Introduction Theoretical Framework 3 Results and discussion 4 Conclusion ê (LZU) 6 / 19

10 Theoretical Framework CDFT in zero-range approximation The relativistic point-coupling energy functional XZ ERMF = dr vkψ k (r)( iγ m)ψk (r) k Z αs βs 3 γs 4 δs dr ρ ρs ρs ρs 4ρS S 3 4 γv δv αv µ µ jµj (jµj ) jµ4j µ 4 αt V µ δt V µ jt V (jt V )µ jt V 4(jT V )µ 1 1 τ3 Fµν F µν F 0µ 0Aµ e jµaµ 4 (1) ρs, j µ, jtµv : local densities and currents; S, V,T : scalar, vector, isovector; α, β, γ, δ : coupling constant. P.W. Zhao et al., PRC 8 (010) / 19

11 Theoretical Framework Collective Hamiltonian in Five Dimension Five-dimension Bohr Collective Hamiltonian: [Bohr195] H (β, γ, Ω) = T rot(β, γ, Ω) T vib(β, γ) Vcoll(β, γ) () with collective potential Vcoll, rotational and vibrational kinetic energies 3 1 X ˆ T rot = Jk /Ik (3) k=1 r r 1 r r ~ 4 3 β B β Bβγ γ T vib = β γγ β β w w wr β 4 r r 1 r r 1 γ sin 3γBβγ β γ sin 3γBββ γ β sin 3γ w β w (4) 8 / 19

12 Theoretical Framework Collective Hamiltonian in Five Dimension Five-dimension Bohr Collective Hamiltonian: [Bohr195] H (β, γ, Ω) = T rot(β, γ, Ω) T vib(β, γ) Vcoll(β, γ) () with collective potential Vcoll, rotational and vibrational kinetic energies 3 1 X ˆ T rot = Jk /Ik (3) k=1 r r 1 r r ~ 4 3 β B β Bβγ γ T vib = β γγ β β w w wr β 4 r r 1 r r 1 γ sin 3γBβγ β γ sin 3γBββ γ β sin 3γ w β w (4) 7 collective parameters (β, γ): I1, I, I3, Bββ, Bβγ, Bγγ, and Vcoll 8 / 19

13 Theoretical Framework Collective parameters Collective parameter(ik, Bµν, Vcoll) Ik = X (uivj viuj ) i,j Bµν E i Ej hi Jˆk ji, i X (uivj viuj ) 1 h 1 1 M(1) M(3)M(1), M(n),µν = hi Q µ jihj Q ν ii = n µν (E E ) i j i,j Vcoll = Etot Vvib Vrot, (5) (6) (7) Vvib & Vrot: vibrational and rotational zero-point correction [D. R. Inglis (1956),S. T. Belyaev (1961); J. Dobaczewski(1981); M. Girod (1979)] Determine the collective parameters with relativistic energy density functional. Quadratic constraints on (β, γ) Etot, i, ψi, vi. PC-PK1 effective interaction in ph channel. [P.W.Zhao10] the separable pairing force in pp channel. [Y.Tian09] 9 / 19

14 Theoretical Framework Pairing correlation For open-shell nucleus, pairing correlations between nucleons are treated using the BCS and Bogoliubov with a pairing force separable in momentum space, 1 V (r1, r, r10, r0 ) = Gδ (R R0) P (r)p (r 0) (1 P σ ), (8) where R and r are the center-of-mass and the relative coordinates respectively. P (r) is taken as Gauss form P (r) = (4πa1 ) e r /4a. pairing energy is given by 3/ 0 Epair = G 0 0 Ny Nx Nz X X X (PNxNy Nz ) PNxNy Nz (9) Nx =0 Ny =0 Nz =0 with G is pairing strength, and PNxNy Nz = X N Ny Nz Vγ δ x κγ δ, (10) γδ>0 in which κγ δ is pairing tensor. [Y. Tian et al., PLB (009).] [T. Niksic et al., PRC 81, (010)] 10 / 19

15 Outline Results and discussion 1 Introduction Theoretical Framework 3 Results and discussion 4 Conclusion ê (LZU) 11 / 19

16 Results and discussion Ground-state properties of Sm isotopes (a) (MeV) RMF BCS Sm RHB v -8.1 n E/A (MeV) 1.5 (c) (b) v p 1.5 (MeV) 0.4 (d) A A Binding energies and deformations from RMFBCS are close to RHB ones. nv and pv : RHB calculations are generally larger than RMFBCS ones. J. Xiang et al., PRC (013). 1 / 19

17 Results and discussion Pairing gap for Sm isotopes 1. (a) (c) v v / Sm (d) (b) 1.1 v Sm 15 RHB BCS Neutron 0.6 (MeV) Proton A R The average pairing gaps increase almost linearly with respect to Rτ and cross the RHB results at Rτ Relative deviation between RMFBCS calculations with 6% enhanced pairing strength and the RHB results less than 5%. 13 / 19

18 Results and discussion Collective parameters of 15Sm RMFBCS (MeV) I x E tot RMFBCS(*1.06) (h /MeV) RHB 10 Sm (MeV) v n 0 (b) (e) (MeV) (d) 15 B (a) (MeV) 300 (c) B v p (MeV) 1.8 (f) The deviations eliminated by enhancing the pairing force about 6% in BCS; As pairing strength increases, spherical barrier lowered and inertia parameter reduced. 14 / 19

19 Results and discussion 15Sm Spectra of.5 15 Sm RMFBCS Energy (MeV) RMFBCS(*1.06) RHB Results of RMFBCS(*1.06) are identical with RHB calculations. Relative deviations(rhb & RMFBCS(*1.06)), intraband transitions: less than 4%; main interband transitions: within W.u.. 15 / 19

20 Results and discussion Single-particle configurations of 15Sm -4 N e u tro n (M e V ) -6 s.p. (a ) -8 R H B R M F B C S (b ) 1 5 S m E F F E E R H B R M F B C S (* ) v v Single-particle configurations: RMFBCS(*1.06) results are identical with RHB ones. 16 / 19

21 Outline Conclusion 1 Introduction Theoretical Framework 3 Results and discussion 4 Conclusion ê (LZU) 17 / 19

22 Conclusion Conclusion Compared the nuclear low-lying structure properties of Sm isotopes calculated by 5DCH based on RMFBCS and RHB approaches. The pairing correlations resulting from the RHB are generally stronger than those from the RMFBCS with the same effective pairing force. The low-energy structure given by RMFBCS(*1.06) very close to ones given by full RHB with the original pairing force. 18 / 19

23 Conclusion Conclusion Compared the nuclear low-lying structure properties of Sm isotopes calculated by 5DCH based on RMFBCS and RHB approaches. The pairing correlations resulting from the RHB are generally stronger than those from the RMFBCS with the same effective pairing force. The low-energy structure given by RMFBCS(*1.06) very close to ones given by full RHB with the original pairing force. 18 / 19

24 Spectra of 15 Sm Conclusion S m E n e rg y (M e V ) (a ) (b ) (c ) (3 0 ) 8 5 (1 4 ) 4 ( ) 1 9 (1 8 ) 4 5 (5 ) 0 9 (3 ) (3 ) 3 3 ( ) 4 0 (3 8 ) ( 7 ) (1 5 ) (1 7 ) 4 (d ) R M F B C S R M F B C S ( ) R H B E x p. ê (LZU) 19 / 19