Study of alpha decay with WS-type mass formula and RBF Correction

Transcription

1 Proceedings of the 15th National Conference on Nuclear Structure in China Guilin Oct. 5-8th 1 Ê3IØ(Œ ø1 giø(;k?ø? 1 5-8F Study of alpha decay with WS-type mass formula and RBF Correction Nana Ma(êAA) School of Nuclear Science and echnology, Lanzhou University =²ŒÆ#Ø Æ EâÆ CollaboratorsµH.F.Zhang(Ü œ), J.M.Dong(Âï ), H.F.Zhang(Üõœ)

2 Outline 1 Introduction heoretical Framework 3 Results and Discussions 4 Summary êaa (=²ŒÆ) / 1

3 Outline Introduction 1 Introduction heoretical Framework 3 Results and Discussions 4 Summary êaa (=²ŒÆ) 3 / 1

4 Introduction Introduction he nuclear masses play key roles in study of nuclear structure and reactions, also in understanding the origin of elements in the universe. + Extracting symmetry energy. J.M. Dong, et al., PRC (13). + he evolution of shell closures. R. Kanungo, Phys. Scr (13). + he rp-process ; quantitative information about the stellar environments. H. Schatz, et al., NPA (6).; X.L. u, et al., PRL (11). Models (or approaches) for nuclear mass: + Doflo-Zuker formula (DZ) J. Duflo, et al., PRC. 5 3 (1995). + Relativistic mean-field (RMF) L. S. Geng, et al., PP. 113, 785 (5). + Hartree-Fock-Bogoliubov (HFB) S. Goriely, et al., PRC (13). + Finite-range droplet model (FRDM) Models σ(kev) Reference DZ Zuker(8). RMF 118 Geng(5). HFB-7 5 Goriely(13). FRDM 57 Mo ller(1). WS4 98 Wang(14). P. Mo ller, et al., PRL (1)). + Weizsa cker-skyrme model (WS); (WS3);......; (WS4) N. Wang, et al., (1-14). ê A A (= 4 / 1

5 Introduction Introduction Mass formula with a unified prescription for the shell and pairing corrections. + WS model with different pairing treatment: Strutinsky-like+BCS method; + Have 13 independent adjustable parameters. Haifei Zhang, et al., Nucl. Phys. A 99, 38 (14). α decay + Superheavy island; Identify new synthesized superheavy elements. Nilsson S G et al., Nucl. Phys. A 131, 1 (1969). + Models: the cluster model; the density-dependent M3Y (DDM3Y) effective interaction; the SkyrmeõHartreeõFock mean field model,... Buck (199); Basu (3); Pei (7). he generalized liquid drop model (GLDM) + he shape evolution; Proximity potential; E(r) = ELDM + EProx Precise nuclear radius formula. Royer(198); Wang(14). Study target: We will calculate the half-lives of α decay with new nuclear mass formula. êa A (= 5 / 1

6 Outline heoretical Framework 1 Introduction heoretical Framework 3 Results and Discussions 4 Summary êaa (=²ŒÆ) 6 / 1

7 heoretical Framework Nuclear mass formula In the macroscopic-microscopic method, the total energy of a nucleus consists of two parts: Y E(Z, A, βk ) = ELDM 1 + bk βk + Emic(Z, A, βk ), (1) k where: 3 ELDM =aν (1 + kν I )A + as(1 + ksi )A I Z + c1 I A + ac 1/3 (1.76Z /3), + I A A Epair =(EBCS Es.p.) (E BCS E s.p.), (3) Emic =Epair + Eshell = EBCS E BCS, (4) EBCS = M X i λbcs E BCS 1 = Z! 1 p ( i λbcs)+ M i=1 ê A A (= () 1 p i λbcs M i, G! ( i λbcs)+ M M ig ( )d. G 7 / 1

8 heoretical Framework Nuclear mass formula In the macroscopic-microscopic method, the total energy of a nucleus consists of two parts: Y E(Z, A, βk ) = ELDM 1 + bk βk + Emic(Z, A, βk ), (1) k where: 3 ELDM =aν (1 + kν I )A + as(1 + ksi )A I Z + c1 I A + ac 1/3 (1.76Z /3), + I A A () Epair =(EBCS Es.p.) (E BCS E s.p.), (3) Emic =Epair + Eshell = EBCS E BCS, (4) + 67: N, Z > 7, Deviation uncertainty on the mass 15 kev. he root-mean square (rms) deviation with respect to 67 measured nuclear masses is.493 MeV. Haifei Zhang, et al., Nucl. Phys. A 99, 38 (14). ê A A (= 7 / 1

9 heoretical Framework Radial Basis Function Correction Radial basis function: S(x) = m X Ning Wang, Min Liu, PRC 84, ωiφ( x xi ), (5) 5133(R) (11). i=1 where xi denotes the point from the measurement, ωi is the weight of the center xi, φ is the radial basis function, x xi is the Euclidean norm. ω1 d1 φ11 φ1 φ1m d φ1 φ φm ω (6) = φm1 φm φmm ωm dm p + φ(r) = r, r = (Zi Zj ) + (Ni Nj ). RBF + BCal. = BCal. + S. RBF BCal. = BExp. Z.M. Niu, Z.L. Zhu, Y.F. Niu, B.H. Sun,.H. + Rmin r Rmax, Rmax = 1, 1, 5,... êa A (= Rmin =, Heng, and J.Y.Guo, PRC 88, 435 (13). 8 / 1

10 heoretical Framework Radial basis function.3 7 he rms deviations σ with res -pect to the known masses as a function of Rmax. When Rmax > 3,the predictive accuracy is almost negligible, especially for Rmax > 5. a l.+ R B F (M e V ).3 6 σc 355: M e V M e V.3 Rmin is fixed to be Rmin = R m a x he root-mean square (rms) deviation with respect to 67 masses had reduced from.493 to.33 MeV by up to 34%, and for 355 masses is.339 MeV. ê A A (= 9 / 1

11 Outline Results and Discussions 1 Introduction heoretical Framework 3 Results and Discussions 4 Summary êaa (=²ŒÆ) 1 / 1

12 Results and Discussions he rms deviation 1 P ro to n n u m b e r 1 B ( E x p.) - B ( + R B F ) Data from the atomic mass evaluation of 1 (AME1). G. Audi, et al., Chin. Phys. C 36, 1157 (1). Near the magic number the RBF correction is obvious N e u tro n n u m b e r S (Z,N ) P ro to n n u m b e r N e u tro n n u m b e r ê A A (= 11 / 1

13 Results and Discussions he rms deviation for Qα he rms for the 46 superheavy nuclei (Z 16). models σ(qα) (kev) WS3 48 WS4 38 WS4RBF 37 this work 3 Ning Wang,et al.,phys. Lett. B 734, 15 (14). For the 46 superheavy nuclei, the rms deviation falls from 51 to 3 kev. For the 45 ones, the rms deviation is 1 kev. ê A A (= Z-N σ(qα) (kev) Number Even-Even 93 7 Even- Odd Odd -Even 6 8 Odd - Odd / 1

14 Results and Discussions he rms deviation for Qα he rms for the 46 superheavy nuclei (Z 16). models σ(qα) (kev) WS3 48 WS4 38 WS4RBF 37 this work 3 (1) Ning Wang,et al.,phys. Lett. B 734, 15 (14). For the 46 superheavy nuclei, the rms deviation falls from 51 to 3 kev. For the 45 ones, the rms deviation is 1 kev. ê A A (= Z-N σ(qα) (kev) Number Even-Even 93 7 Even- Odd Odd -Even 6 8 Odd - Odd / 1

15 Results and Discussions α decay half-lives he combination of the GLDM and of a quasimolecular shape sequence,the barrier penetrability P : Hongfei Zhang, G. Royer Phys. Rev. C 77, (8). Z Rout p ln 1/ =, P =exp µ[e(r) E(parent)]dr (7) PυP ~ Rin Rin =R1 + R, Rout =ez1z/qα (8) Nuclei Qα(Exp.)[MeV] Qα(Cal.)[MeV] α(exp.)[s] α(cal.)[s] Os Pt Rn U Rf Xiaojun Bao, et al., Nucl. Phys. A 91 (14) 85õ95. ê A A (= 13 / 1

16 Results and Discussions decay half-lives he combination of the GLDM and of a quasimolecular shape sequence,the barrier penetrability P : Hongfei Zhang, G. Royer Phys. Rev. C 77, (8). Z Rout p ln 1/ =, P =exp µ[e(r) E(parent)]dr (7) PυP ~ Rin Rin =R1 + R, Rout =ez1z/qα (8) Comparison of the experimental and calculated α decay half-lives for even-even nuclei: distinct shell effect at N = P t E x p. 1 / P b 1 / 5 1 / h E x p. 1 / U C m R a 1 W R n H g 1 5 L o g α P o C f O s F m R f N o S g H s ê A A (= N / 1

17 Results and Discussions decay half-lives he combination of the GLDM and of a quasimolecular shape sequence,the barrier penetrability P : Hongfei Zhang, G. Royer Phys. Rev. C 77, (8). Z Rout p ln 1/ =, P =exp µ[e(r) E(parent)]dr (7) PυP ~ Rin Rin =R1 + R, Rout =ez1z/qα (8) Comparison of the experimental and calculated α decay half-lives for even-even nuclei: distinct shell effect at N = P t E x p. 1 / P b 1 / 1 / E x p. h 1 / U C m R a 1 W R n H g 1 5 P o C f O s F m 5 L o g α R f N o S g H s ê A A (= N 13 / 1

18 Results and Discussions decay half-lives he combination of the GLDM and of a quasimolecular shape sequence,the barrier penetrability P : Hongfei Zhang, G. Royer Phys. Rev. C 77, (8). Z Rout p ln 1/ =, P =exp µ[e(r) E(parent)]dr (7) PυP ~ Rin Rin =R1 + R, Rout =ez1z/qα (8) Comparison of the experimental and calculated α decay half-lives for even-even nuclei: distinct shell effect at N = P t E x p. 1 / 1 / R n H g E x p. h 1 / U 1 5 P b 1 / W C m R a 1 P o C f O s F m 5 L o g α R f N o S g H s ê A A (= N / 1

19 Results and Discussions α decay half-lives (Z=even) L o g 1 / E x p. 1 / 1 / 1 / 1 / E x p. O r ig. H f a W R e O s P b P o R n R a h L o g 1 / U P u (a ) C m C f F m N o R f S g H s (b ) D s N (c ) he calculated half-lives within the GLDM are in agreement with the experimental data. An abrupt drop reflects the information of shell structure. êaa (=²ŒÆ) 14 / 1 N (d )

20 Results and Discussions α decay half-lives (Z=even) L o g 1 / E x p. 1 / 1 / 1 / 1 / E x p. O r ig. H f a W R e O s P b 1 6 P o R n R a h L o g 1 / U P u (a ) C m C f F m N o 1 6 R f S g H s (b ) D s N (c ) he calculated half-lives within the GLDM are in agreement with the experimental data. An abrupt drop reflects the information of shell structure. êaa (=²ŒÆ) 15 / 1 N (d )

21 Results and Discussions α decay half-lives (Z=odd) L o g 1 / L o g 1 / E x p. 1 / 1 / 1 / 1 / E x p. O r ig N L u a P a R e N p A m Ir B k (c ) A u E s l B i he calculated half-lives within the GLDM are in agreement with the experimental data. An abrupt drop reflects the information of shell structure. êaa (=²ŒÆ) 16 / 1 N A t M d L r D b B h F r A c (b ) (d ) M t

22 Results and Discussions α decay half-lives (Z=odd) L o g 1 / L o g 1 / E x p. 1 / 1 / 1 / 1 / E x p. O r ig N L u a P a R e N p A m Ir B k (c ) A u E s l B i he calculated half-lives within the GLDM are in agreement with the experimental data. An abrupt drop reflects the information of shell structure. êaa (=²ŒÆ) 17 / N A t M d L r D b B h F r A c (b ) (d ) M t

23 Outline Summary 1 Introduction heoretical Framework 3 Results and Discussions 4 Summary êaa (=²ŒÆ) 18 / 1

24 Summary he current results and future improvement Utilizing a Weizsa cker-skyrme-type nuclear mass formula, Qα of heavy nuclei are systematically calculated [the root-mean square (rms) deviation is.51 MeV]. Incorporating the Radial Basis Function method, rms value of Qα is reduced from.51 MeV to.3 MeV. In the framework of GLDM within the above Qα values as input, the calculated α decay half-lives are in appropraite agreement with the data. êa A (= 19 / 1

25 Summary he current results and future improvement Utilizing a Weizsa cker-skyrme-type nuclear mass formula, Qα of heavy nuclei are systematically calculated [the root-mean square (rms) deviation is.51 MeV]. Incorporating the Radial Basis Function method, rms value of Qα is reduced from.51 MeV to.3 MeV. In the framework of GLDM within the above Qα values as input, the calculated α decay half-lives are in appropraite agreement with the data. Further improvement In the BCS method, both for neutrons and protons the force strength Z +7.5) ( 1 N A G are derived from 4 =. A1/3 êa A (= 19 / 1

26 Summary he current results and future improvement Utilizing a Weizsa cker-skyrme-type nuclear mass formula, Qα of heavy nuclei are systematically calculated [the root-mean square (rms) deviation is.51 MeV]. Incorporating the Radial Basis Function method, rms value of Qα is reduced from.51 MeV to.3 MeV. In the framework of GLDM within the above Qα values as input, the calculated α decay half-lives are in appropraite agreement with the data. Further improvement In the BCS method, both for neutrons and protons the force strength Z +7.5) ( 1 N A G are derived from 4 =. A1/3 hank you for your attentions! êa A (= 19 / 1

27 he total energy Summary E(Z, A, β k ) = E LDM b k = k ( ) k g 1 A 1/3 + the macroscopic energy (liquid drop energy ): E LDM =a ν (1 + k ν I )A + a s (1 + k s I )A 3 ( ) 1 + b k βk + Emic (Z, A, β k ), (9) ( ) k g A 1/3, K =, 4, 6. (1) + c 1 I + I A I A + a c Z A 1/3(1.76Z /3 ), (11) As for the microscopic part,the Dirac generalized single-particle level density(spld): g(ε) = he corresponding smoothed SPLD: ḡ(ε) = 1 γ M d i δ(ε ε i ), (1) i=1 M ( ) ε εi d i K γ i=1 (13) êaa (=²ŒÆ) / 1

28 he total energy Considering only values at discrete points: Summary M G = N = i=1 [ M 1 i=1 1 (14) (ɛi λ BCS ) +, ] ɛ i λ BCS (ɛi λ BCS ) + (15) and the corresponding BCS energy ( ) M ɛ i λ BCS E BCS = 1 ɛ i (ɛi λ BCS ) + i=1 G (16) êaa (=²ŒÆ) 1 / 1

29 he total energy Summary Considering only values at discrete points: the continuous version of the gap equations take form: M G = N = i=1 [ M 1 i=1 1 (14) (ɛi λ BCS ) +, ] ɛ i λ BCS (ɛi λ BCS ) + (15) and the corresponding BCS energy ( ) M ɛ i λ BCS E BCS = 1 ɛ i (ɛi λ BCS ) + i=1 G (16) G = 1 N = 1 1 (ɛi λ BCS ) + [ 1 ḡ(ɛ)dɛ, (17) ɛ i λ BCS (ɛi λ BCS ) + ] ḡ(ɛ)dɛ (18) and the corresponding BCS energy Ē BCS = 1 ( ) ɛ i λ BCS 1 ɛ i ḡ(ɛ)dɛ (ɛi λ BCS ) + G (19) êaa (=²ŒÆ) 1 / 1

30 he total energy Summary Considering only values at discrete points: the continuous version of the gap equations take form: M G = N = i=1 [ M 1 i=1 1 (14) (ɛi λ BCS ) +, ] ɛ i λ BCS (ɛi λ BCS ) + (15) and the corresponding BCS energy ( ) M ɛ i λ BCS E BCS = 1 ɛ i (ɛi λ BCS ) + i=1 G (16) G = 1 N = 1 1 (ɛi λ BCS ) + [ 1 ḡ(ɛ)dɛ, (17) ɛ i λ BCS (ɛi λ BCS ) + ] ḡ(ɛ)dɛ (18) and the corresponding BCS energy Ē BCS = 1 ( ) ɛ i λ BCS 1 ɛ i ḡ(ɛ)dɛ (ɛi λ BCS ) + G (19) thus, E pair = (E BCS ĒBCS) E shell () E mic = E pair + E shell = E BCS ĒBCS (1) êaa (=²ŒÆ) 1 / 1