1. Bulk Properties of Nuclear Mass Formulae 2. R-Process Nucleosynthesis (3

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New Era of Nuclear Physics in the Cosmos -the r-process nucleosynthesis, RIKEN, Sep. 25, 25, 2008 核物理から見た宇宙 -rプロセス研究の新時代に向かって理研2008年9月25-26日原子核質量公式とr-過程元素合成 Nuclear mass formulae and the r-process nucleosynthesis

Hiroyuki KOURA Japan Atomic Energy Agency (JAEA)

1. Bulk properties of nuclear mass formulae 2. r-process nucleosynthesis (3. Effect of β-delayed ﬁssion to r-process abundance)

1 Z=114

100 Known nuclides

Z=82 80

60 Z=50 -1984(2204 nucl.) 40 -1992(2523 nucl.)

Z=28 -2000(2817 nucl.) Proton (Atomic) number 20 -2004(2915 nucl.) stable N=28 N=50 N=82 N=126 0 0 20 40 60 80 100 120 140 160 Neutron number N

2 Z=114

100 Known nuclides SHE

Z=82 80

60 r-process Z=50 -1984(2204 nucl.) 40 -1992(2523 nucl.)

Z=28 -2000(2817 nucl.) Proton (Atomic) number 20 -2004(2915 nucl.) stable N=28 N=50 N=82 N=126 Known nuclides: taken from Chart of 0 the nuclides by JAERI 0 20 40 60 80 100 120 140 160 Neutron number N SHE

r-process

2 Importance of nuclear masses for the r-process nucleosynthesis

Neutron-rich nuclei(A>~ 80)=>quite diﬃcult to measure so far. Q , S , ...=>require to estimate λ ,λn, ... • β n β • Estimations of decay modes of α,βand SF =>important in the SHE region • Nuclear structure (magicity, isomer)=>gives abundance pattern • mass curvature=> reach 3rd peak (+U,Th) or not ．．． • 3 II. Phenomenological Mass Formulas

• First-principle calculation Theoretical approaches on nuclear masses (calculation from the realistic nuclear force) ﬁrst principle第一原理 To treat nuclear masses with the ﬁrst principle is not appropriate approxi- 現象論system- way from consideration on 近似 mation システマティックスatics properties of nuclear force, many- body force, computational time, modelモデル and accuracy for the present and WB in the near future.

Nuclear masses are estimated from various viewpoints for various purposes. 4 Systematics・Phenomenology Z ・Garvey-Kelson-like mass systematics focusing on relation between mass values and Z, N − + + + Comay-Kelson-Zidon, Jänecke-Masson (1988) ≈ 0 + − N ・Empirical shell term focusing on Bulk part (WB-like)+deviation（Shell term） Tachibana-Uno-Yamada-Yamada (1988) ・Phenomenological shell model calculation

Polynomials of particle and hole numbers, 100 obliged to assume magic numbers in advance. Liran-Zeldes (1976) 50

・... 0 0 50 100 150 Properties ・Good reproduction of masses for known nuclei + good prediction for unknown nuclei (quite) near mass-measured nuclides. (300-600 keV) ・No predictable power for superheavy nuclei (next magic number, etc.) ・No deformation is obtained. 5 Hilf-Groote-Takahashi(HGT,1976) Macroscopic part: deformed liquid-drop (Myers-Swiatecki-type) Microscopic part

6 S systematics 50 2n 50 N=20 N=8 Exp.(AW95) TUYY 40 N=20 40 N=28 N=28 N=50 N=82 30 30 N=50 N=126 N=82 N=126 (MeV) (MeV) 20

2n 20 2n S S 10 10 0 0 20 40 60 80 100 20 40 60 80 100 Proton number Z Proton number Z

50 N=20 N=50 GHT 40 N=28 N=82 30 N=126

(MeV) 20 2n S 10 0 20 40 60 80 100 Proton number Z 7 Mass Model, Approximation Recent mass formulas: ・ 310 are designed for nuclei with Z, N=8 to [126]184 or more ・have the RMS dev. from exp. masses. of 600-800 keV ・give deformation parameters β2, β4... and ﬁssion barriers

・Density functional theory <- recent project by M.Stoitsov, etc. ・Hartree-Fock method with Skyrme force micro Strong short-range force => δ-function => HF calc. (-like) ETFSI (1995), HFBCS (2001), HFB (2002-) ・Liquid-drop model by S.Goriely Deformed liquid-drop part+Micro. (folded Yukawa) macro+ FRDM (1995), FRLDM (2002), micro ・Mass formula with spherical-basis shell term by P.Möller Phenom. gross (WB-like)+spherical-basis shell part KUTY (2000), KTUY (2005) Koura, Uno, Tachibana, Yamada phenom.

8 (Koura, Uno, Tachibana, Yamada, NPA674, 2000) III. KTUY (KUTY) mass formula (Koura, Tachibana, Uno, Yamada, PTP113, 2005) M(Z, N)=M (Z, N)+M (Z, N)+M (Z, N) gross eo shell M (Z, N): gross term gross 2 2.39 M gross(Z,N)=M HZ+M n N+a(A)A+b(A) N–Z +c( A)(N–Z) / A + EC(Z,N) – k el Z 1/3 2/3 1 a(A)=a1 + a2A− + a3A− + a4(A + αa)− Volume, Surface, etc... 1/3 2/3 1 b(A)=b1 + b2A− + b3A− + b4(A + αb)− Wigner term  1/3 2/3 1  c(A)=c1 + c2A− + c3A− + c4(A + αc)− Asymmetry term Z2 1 E = a (Z,N) 1 0.76C  C C A1/3 − x Z2/3 Coulomb with an exchange term ! " M (Z, N): shell term shell Spherical nuclei Calculated from Spherical single-particle potential for any nuclei (includes the BCS paring, reduction) Deformed nuclei (Spherical-basis condieration) Obtained by an appropriate mixture of the above spherical shell energies + liquid-drop deform. energies

9 RMS dev. of masses and separation energies from Exp. Neutron sep. Proton sep. Mass formula Masses Sn S2n Sp S2p

Z, N≥2 (2219 nuclei) (2054 nuclei) (1997 nuclei) (2016 nuclei) (1897 nuclei) KTUY03(05) 666.7 keV 352.9 keV 442.0 keV 389.9 keV 532.2 keV KUTY(00) 689.8 keV 389.1 keV 462.4 keV 473.6 keV 558.2 keV

Z, N≥8 (2149 nuclei) (1988 nuclei) (1937 nuclei) (1948 nuclei) (1835 nuclei) KTUY03(05) 652.8 keV 316.2 keV 379.1 keV 353.0 keV 490.1 keV KUTY(00) 670.7 keV 339.5 keV 386.5 keV 379.2 keV 499.7 keV FRDM(95) 655.5 keV 399.3 keV 511.7 keV 395.2 keV 493.6 keV HFBCS-1(01) 770.7 keV 451.8 keV 477.3 keV 496.5 keV 603.8 keV

1000 652.8 keV KTUY05 KUTY00 800 379.1 keV FRDM95 HFBCS01 600 316.2 keV 400 200

RMS deviation (keV) 0 Mass S1n S2n S1p S2p 10 S systematics 2n 50 50 N=20 Exp.(AWT03) KUTY02KTUY03KTUY05 N=28 N=50 40 N=8 (even-N) 40 N=20 N=82

30 N=28 N=50 30 N=126 N=126 N=82 (MeV) (MeV)

2n 20 20 2n S S 10 10

0 0 20 40 60 80 100 Proton number Z 20 40 60 80 100 Proton number Z 50 50 N=20 FRDM N=20 N=28 HFBCS 40 N=28 40 N=50 N=82 N=50 N=82 30 N=126 30 N=126

(MeV) 20 (MeV) 20 2n 2n S S 10 10

0 0

20 40 60 80 100 20 40 60 80 100 Proton number Z 11 Proton number Z S systematics 2n 50 50 N=20 Exp.(AWT03) KUTY02KTUY03KTUY05 N=28 N=50 40 N=8 (even-N) 40 N=20 N=82

30 N=28 N=50 30 N=126 N=126 N=82 (MeV) (MeV)

2n 20 20 2n S S 10 10

0 0 Smoothness 20 40 60 80 100 Proton number Z 20 40 60 80 100 Proton number Z 50 50 N=20 FRDM N=20 N=28 HFBCS 40 N=28 40 N=50 N=82 N=50 N=82 30 N=126 30 N=126

(MeV) 20 (MeV) 20 2n 2n S S 10 10 Anomalous 0 Kink 0 Zigzag

N=22,30 line 20 40 60 80 100 20 40 60 80 100 Proton number Z N=20,28 gap Proton number Z ->crossing 11->already weak S systematics 2n 50 50 N=20 Exp.(AWT03) KUTY02KTUY03KTUY05 N=28 N=50 40 N=8 (even-N) 40 N=20 N=82

30 N=28 N=50 30 N=126 N=126 N=82 (MeV) (MeV)

2n 20 20 2n S S 10 10

0 0 20 40 60 80 100 Proton number Z 20 40 60 80 100 Proton number Z 50 50 N=20 FRDM N=20 N=28 HFBCS 40 N=28 40 N=50 N=82 N=50 N=82 30 N=126 30 N=126

(MeV) 20 (MeV) 20 2n 2n S S 10 10

0 0

20 40 60 80 100 20 40 60 80 100 Proton number Z 11 Proton number Z S systematics 2n 50 50 N=20 Exp.(AWT03) KUTY02KTUY03KTUY05 N=28 N=50 40 N=8 (even-N) 40 N=20 N=82

30 N=28 N=50 30 N=126 N=126 N=82 (MeV) (MeV)

2n 20 20 2n S S 10 10

0 0 20 40 60 80 100 Proton number Z 20 40 60 80 100 Proton number Z 50 50 30 KTUY03 N=28 N=20 FRDM N=20 N=28 HFBCS 40 N=28 40 20 N=50 N=82 N=50 N=82 N=50

30 N=126 30 (MeV) 10 N=126 2n S

(MeV) 20 (MeV) 20

2n 2n 0 S S 10 10N=8 4 8 12 16 20 24 28 (N=16) N=20 (N=32) (N=58) 0 0 N=14,20,28,50 gap 20 40 60 80 100 20 40 60 80 100 Proton number Z 11 ->decreasingProton number Z Light neutron-drip line

16 KTUY05 known RMS dev. : 0.35MeV for Sn nuclides(-2000) 0.44MeV for S2n 12 Notani (02)

Z Baumann (07) 42Al 8 S =−0.05 31 n 26O F 4 S =+0.21 Fail to predict 8 nuclei whether bound/unbound 2n S2n=−0.47 22 0 C 31 All the n-sep. energy of these nuclei are within 1.5 19 Ne

Proton number C S = 0.31 2n − times of RMS dev. -4 16B S =−0.04 Sn=−0.58 n 19B Sn=+0.02 -8 S2n=−0.51

4 8 12 16 20 24 28 32 36 40 Neutron number N

16 FRDM Notani(02) 44 RMS dev. : 0.40MeV for Sn Baumann(07) P 12 Sn=−0.38 0.51MeV for S2n Z 44Si 8 S =−0.16 2n Fail to predict 10 nuclei whether bound/unbound 4 43Al 37Mg The n-sep. energy of the most nuclei considered Sn=−0.30 0 26 S = 0.33 are within 1.5 times of RMS dev, while O n − 42

Proton number Al 37 43 26 43 42 -4 S2n=1.53 Na Si S =−2.61 those of three nuclei, O, Si, Al, exceed 3.0 40 2n 34Ne S = 0.07 Al n − S2n=−1.68 times of its RMS dev. S = 0.10 -8 2n − Sn=−0.02

4 8 12 16 20 24 28 32 36 40 Neutron number N 12 Light neutron-drip line

16 KTUY05 16 TUYY known Z Notani(02) nuclides(-2000) Baumann(07) 12 Notani (02) 12 Z Baumann (07) 42 Al 31 8 Ne S =−0.05 8 31 n 28 26O F F 4 31 S =+0.21 4 F 2n S2n=−0.47 19 26 Proton number 22 O 22 14 C C 0 C 31 0 Be 19 Ne Proton number C S2n=−0.31 -4 16B S =−0.04 Sn=−0.58 4 8 12 16 20 24 28 32 n 19B Sn=+0.02 Neutron number N -8 S2n=−0.51

4 8 12 16 20 24 28 32 36 40 Neutron number N 16 HFBCS 16 FRDM Notani(02) Notani(02) 44 Baumann(07) Baumann(07) P 12 12 Sn=−0.38 Z Z 44Si 8 8 40Al S2n=−0.16 4 Sn=−0.1 4 43Al 37 31 Mg S =−0.30 0 26 28 F n O F 31 0 26 S = 0.33 Proton number Ne O n − 42

Proton number Al 37 43 -4 -4 S2n=1.53 Na Si S =−2.61 34 40 2n Ne S =−0.07 Al n S2n=−1.68 -8 S = 0.10 -8 2n − Sn=−0.02 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 36 40 Neutron number N Neutron number N 13 α-decay Q-values of heavy and superheavy nuclei

234 277 271 275 related to Bk [112] [110] ( [112]) RIKEN(02) RIKEN1(04) 11 RIKEN(03) GSI 2(04) GSI(98) ToI 12 GSI1(96) 6) 12 KTUY 6) 2(00) 11) KTUY 6) FRDM 10 11) KTUY FRDM 19) 11 11) 11 HFBCS 19) FRDM 12) HFBCS 19) HFB8 12) HFBCS

(MeV) 9 HFB8 12) (MeV) (MeV) α α 10 HFB8 α 10 Q Q oo Q 8 9 9

7 ee odd-A odd-A 8 8 128 130 132 134 136 138 N 155 157 159 161 163 165 N 153 155 157 159 161 163 N (86) (88) (90) (92) (94) (96) (97) (Z ) (102) (104) (106) (108) (110) (112) Z (102) (104) (106) (108) (110) (112) Z 234 214 218 222 226 230 277 255 259 263 267 271 Rn Ra Th U Pu Cm 257No 261RF 265Sg 269Hs 273Ds [112] No Rf Sg Hs Ds 234Bk Exp.: gradually decreasing to N=136 Exp.: steep increasing at N=162 Exp.: gradually increasing suppressed at N=164

KTUY O or △ KTUY X KTUY O FRDM X FRDM △ FRDM △ HFBCS X HFBCS O HFBCS X HFB8 X HFB8 X HFB8 X or △

*Measured Qα of odd-A nuclei is not surely represented the g.s.-to g.s. decay. Black mark: exp. data 14 R-process nucleosynthesis -Check the mass formulas as astrophysical data- • Canonical model Steady ﬂow +Waiting point Approximation Neutron-number density (N ) and temperature (T ) are constants n 9 (n,γ)-(γ,n) equilibrium is established over an irradiation time τ • Saha equation (gives the r-process path) Y (Z, A + 1) G(Z, A + 1) A 5.04 log = log 34.075 + log N + 1.5 log T + S (Z, A + 1) Y (Z, A) G(Z, A) − n A + 1 9 T n ! " 9

S (Z, A + 2)/2 S0 (34.075 log N + 1.5 log T )T /5.04 S (Z, A)/2 2n ≤ a ≡ − n 9 9 ≤ 2n S2n: 2-neutron separation energy

Time evolution dY/dt = λZ YZ (t) + λZ 1YZ 1(t) • − − − YZ: abundance, λZ: β-deacy constant (gross theory 2nd) Nn,T9,τ: chosen to reproduce the abundance peak at A=130 (obs.) S ,Q : estimated from mass formulas (TUYY, KUTY, FRDM) 2n β 15 120 New measurement ('96-) Z 100 Mass measured nucleus (-'95) 80

40 r-path TUYY Proton number 20 r-path KUTY r-path FRDM 0 0 20 40 60 80 100 120 140 160 180 200 Neutron number N

16 Nr=N(Solar abund.)-N s

Nr r-only nuclei

2 TUYY KUTY FRDM 10 logNn=25.10 logNn=24.26 logNn=23.89 T =1.5 T =1.5 T9=1.5 1 9 9 )

6 10 τ=1.5 τ=0.9 τ=2.0

0 10

-1 10

-2 10

-3 10

-4 10 Solar abundance (Si=10 -5 10

-6 10 80 120 160 200 80 120 160 200 80 120 160 200 Mass number Mass number Mass number TUYY+GT2 KUTY+GT2 FRDM+GT2

17 S2n systematics

A=130peak A=130peak A=130peak A=130peak

Experiment TUYY KUTY FRDM 15 (1995 Audi & Wapstra) Z=50 Z=50 Z=60 Z=60 10

/2 [MeV] Z=40 Z=40 2n

S 5 S 0=2.75 0 a Sa =3.0

0 Z=36 Sa =3.1 MeV Z=38 Z=36 Z=36 0 Z=36 Z=38 Z=38 Z=38 40 50 60 70 80 9040 50 60 70 80 9040 50 60 70 80 9040 50 60 70 80 90 Neutron number N Neutron number N Neutron number N Neutron number N

Experiment TUYY 18 KUTY FRDM Kink S2n systematics

A=130peak A=130peak A=130peak A=130peak

Experiment TUYY KUTY FRDM 15 (1995 Audi & Wapstra) Z=50 Z=50 Z=60 Z=60 10

/2 [MeV] Z=40 Z=40 2n

S 5 S 0=2.75 0 a Sa =3.0

0 Z=36 Sa =3.1 MeV Z=38 Z=36 Z=36 0 Z=36 Z=38 Z=38 Z=38 40 50 60 70 80 9040 50 60 70 80 9040 50 60 70 80 9040 50 60 70 80 90 Neutron number N Neutron number N Neutron number N Neutron number N

Experiment TUYY 18 KUTY FRDM Garvey-Kelson mass relationship 100 Experiment (Audi-Wapstra95) Garvey-Kelson eq. Ave =−13. 8 keV GK Z 80 RMSGK=434 keV − + ≡ΔGK 60 + − − + 40 N 20 ≈ 0

20 40 60 80 100 120 140 0 250 500 750 1000 Neutron number N |ΔGK| [keV] 100 KUTY 100 AveGK =−0. 47 keV FRDM Ave = . 75 keV RMSGK=270 keV GK −2 80 80 RMSGK=599 keV

60 60

40 40

20 20

20 40 60 80 100 120 140 20 40 60 80 100 120 140 19 Neutron number N Neutron number N Deformation parameter α 40 50 60 70 80 290 Z 60 60 A=130 r e b m 50 50 u n KUTY n ot 40 40 ro Z=38

P Proton number Z 40 50 60 70 80 90 Neutorn number N 40 50 60 70 80 90 Z 60 60 r e b m 50 50 u dip of S <=> change of α n 2n 2 FRDM n ot 40 40 (FRDM) ro Z=38

P Proton number Z 40 50 60 70 80 90 Neutron number N

-0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3 20 α2 Z N=81 N=83

SkP

S :FRDM+SkP n Sn

shell-quenching => decreasing of dips around the peaks (Chen et al. 1995)

Kink of S2n, or shell quenching? 21 Deformation parameter alpha2 of KUTY Potential energy surface 120 Z sph 100 Esh(Z, N) = (< E (Z, N) >def. +∆ES(Z, N) ∆EC(Z, N) ∆Epro(Z, N)) sh − − def. 80 ! number number 60 Sn Isotopes 40 Possibility of Proton Proton 20 shape isomer 20 40 60 80 100120140160180200 Neutron number N sph. min sph. min sph. min prolate min 112Sn62 132Sn82 142Sn92 152Sn102 0.10 -0.2 -0.1 -0.00.10 0.1 0.2 0.10 0.10 0.05 0.05 0.05 0.05 0.00 alpha20.00 0.00 0.00 alpha4 alpha4 alpha4 -0.05 -0.05 alpha4 -0.05 -0.05 -0.10 -0.10 -0.10 -0.10 -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4 alpha2 alpha2 alpha2 alpha2

0.0 5.0 10.0 15.0 20.0 -10.0 -5.0 0.0 5.0 10.0 0.0 5.0 10.0 15.0 20.0 5.0 10.0 15.0 20.0 Esh (MeV) Esh (MeV) Esh (MeV) Esh (MeV)

112Sn62 132Sn82 142Sn92 152Sn102 Systematical Study of Tin deformation (Shape coexistence) r-process mass region

N=126 N=184 298 [114]184 140 278 [113] Z=126 165

120 Z=114

Z 100

Z=82 80

235 208 U 60 Pb

Proton (Atomic) number 40 an example of r-process path (Canonical, Waiting point app.) nuclei with 1ns or more 20 Proton-drip line β-stable nuclei known nucl.(2004) 0 0 20 40 60 80 100 120 140 160 180 200 Neutron number N Calculated from the KTUY mass formula r-process mass region

N=126 N=184 298 [114]184 140 278 [113] Z=126 165

120 Z=114 superheavy,

Z neutron-rich 100 (n-capture, β- Z=82 80 decay, ﬁssion)

235 208 U 60 Pb Endpoint?

Proton (Atomic) number 40 an example of r-process path Recycling? (Canonical, Waiting point app.) nuclei with 1ns or more 20 Proton-drip line β-stable nuclei known nucl.(2004) 0 0 20 40 60 80 100 120 140 160 180 200 Neutron number N Calculated from the KTUY mass formula 3. β-delayed fission (β-df) probability Region of β-delayed and n-induced fission Nuclear masses and fission barrier: KTUY (Koura-Tachibana-Uno-Yamada) mass formula

240 n-induced ﬁssion 120 Pu • Z 234 236 239 S >B (Z, N), n 110 U U U n ﬁss fission 100 Sn(Z, N) γ 90 Bfiss 80 Sn and S2n>0 Proton number (Z, N+1) 70 Sn>Bfiss Sn>Bfiss+1 MeV neutron-induced fission 60 140 160 180 200 220 240 Neutron number N 278 298 120 [113] [114]184

β-delayed fission Z Q >B • 235 β fiss 110 U (Z, N) neutron emission 100 Sn fission 90 Q (Z, N) 80 β S and S >0 Bfiss Proton number n 2n 70 Qβ>Bfiss (Z+1, N-1) 60 -delayed neutron emission and -delayed fission 140 160 180 200 220 240 β β Koura 24 Neutron number N FissionFission Barrier barrier height height of KUTY 298 [114] 184 proton-drip line β-stable nuclei Fission Barrier height of 278[113] 130 Howard &Möller (1980): 354 [126] 120 228 Used for calc. of β-delayed fission 110 prob. from QRPA (Klapdor, 88)

100 Z 100

90 90

number number neutron-drip line

130 140 150 160 170 180 190 200 80210 220 230 240 250 208 238 Neutron number Proton N Pb U the neutron-rich nuclides 140 150 from160 this170 line may180 have large Neutron number N outer barriers over α2 > 0.5 0 1 2 3 4 5 6 7 8 9 10 Bfiss (MeV) 0 these1 2 nuclides3 4 5 must6 7have8 large9 10 BFiss(MeV)outer barriersfrom Haward over α &2 Moeller(1980)> 0.5 RMS dev. from exp. :~1MeV 25 Koura FissionFission Barrier barrier height height of KUTY 298 [114] 184 proton-drip line β-stable nuclei Fission Barrier height of 278[113] 130 Myers &Swiatecki (1996): 354 [126] 120 228

110

100 Z 100

90 90

number number neutron-drip line 130 140 150 160 170 180 190 200 210 220 230 240 250 80 208 238 Neutron number Proton N Pb U the neutron-rich nuclides 140 150 from160 this170 line may180 have large Neutronouter number barriers N over α2 > 0.5 0 1 2 3 4 5 6 7 8 9 10 Bfiss (MeV) 0 these1 2 nuclides3 4 6 must7 have8 9 large10 outer barriers over α > 0.5 PnPfMS2_transform_x2 RMS dev. from exp. :~1MeV 26 Koura Fission barrier height of KUTY Fission Barrier height of KUTY 298 [114] *) 184 proton-drip line -stable nuclei ETFSI (β2, β4, β6β deformation) 278[113] 130 278[113] Dubna(1999-) Dubna(1999-)354 [126] 120 228

110

100 Proton number Z 90 neutron-drip line 130 140 150 160 170 180 190 200140210150220160230170240180250190 200 208 238 Neutron numberNeutron N number N Neutron number N Pb U the neutron-rich nuclides *)Mamdouh,etfrom this al line NPA679 may have (2001) large outer barriers over α2 > 0.5 0 1 2 3 4 5 6 7 8 9 10 Bfiss (MeV) these nuclides must have large outer barriers over α2 > 0.5Koura 27 Summary Bulk properties of nuclear masses: •Notable differences in the n-rich region among mass formulae -Kink of the S2n systematics , Garvey-Kelson discrepancy, etc. Relation between the r-process nucleosynthesis cal. and mass formulae: •R-process nucleosynthesis calculation with a canonical model -Abundance deﬁcient both sides of A=130 <= kink of S2n sys.

Effect of beta-delayed ﬁssion to the r-process nucleosynthesis: •Reaction Network calculation -Fission fragment tends to compensate a discrepancy near A=110 and 140 region (without adopting an N=82 quenching) Qualitatively!