Current Status of Nuclear Mass Formulae 1 RIBF-ULIC-Symposium: Physics of Rare-RI Ring, RIKEN, Nov , 2011 Hiroyuki KOURA Advanced Science Research Center, Japan Atomic Energy Agency (JAEA) Bulk properties of atomic masses Phenomenological mass formulas Atomic mass model: deviation from masses Application for the r-process and the superheavy nuclei Summary

taken from Chart of the nuclides by JAERI and JAEA r-process SHE 2 amdc.in2p3.fr/mastables/filel.ht ml wwwndc.jaea.go.jp/CN10/index.html Identified Mass- measured ~3000 nuclei ~2400 nuclei

RMS dev. from AWT03: 2.93 MeV (Z, N ≥ 8) M exp. − M WB Mass data : 2003 Atomic mass evaluation (Audi, Wapstra & Thibault) Existence of magic number N=28,50,82,126 Z=28,50, Pb 132 Sn aVaV asas aIaI aCaC a eo (MeV) 3 M exp  M WB (MeV) N=Z ridge Wigner energy N=Z ridge Depression due to the deform. rare-earth, actinide N  Z=24 Next figure M(Z, N)=Z m H +N m n −B(Z,N) =Z m H +N m n −a V A+a s A 2/3 +a I (N−Z) 2 /A+a C Z 2 /A 1/3 +  eo Weizsäcker-Bethe semi-empirical atomic mass formula Bulk properties of atomic mass Shell energy =

- Experiment - Shape transition and shell energies  from B(E2) : S. Raman et al., ADNDT78 (2001) 4 Notable feature on discontinuity of derivative of mass values Z=50, N=82 and Z=82 discontinuity of derivative: Spherical single-particle shell closure N=88-90 discontinuity: Shape transition N= Schematic - Fig. (a) N-Z plane Fig. (b) cross section along dashed line in (a)

A consideration of cancellation of core + valence nucleons (based on the shell model) Assumption: Cores among related (six) nuclei are the same. Mass relation: Garvey-Kelson systematics 5 Regionnum.AverageRMS dev. All (keV)341.6(keV) A> A≤ N  Z=24 Light region larger Example of mass formula: Comay-Kelson-Zidon, Jänecke-Masson,... (ADNDT39, 1988)

Shell gaps:  N, Z=20, 28, 50, 82,126(only N) and a change of magicities (ex. N=14 to 16) Transition of sphere to deformation:  Discontinuity of derivatives at N=88 to 90 near the β-stable region. Wigner term:  Discontinuity at N=Z. Averaged even-odd effect:  Staggering change of masses alternates even and odd-N/Z. Bulk properties of mass surface:  In macro.-micro. models, it is explicitly introduced. In full microscopic calculation, this is one of the most difficult points. Some points on parametrization of a mass formula 6

・ Garvey-Kelson-type mass systematics focusing on relation between mass values and Z, N Comay-Kelson-Zidon, Jänecke-Masson (1988) ・ 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, obliged to assume magic numbers in advance. Liran-Zeldes (1976), Duflo-Zuker (1995) ・... Properties ・ Good reproduction of masses for known nuclei + good prediction for unknown nuclei (quite) near mass-measured nuclides. ( keV) ・ No predictable power for superheavy nuclei (next magic number, etc.) ・ No deformation is obtained. 7 Systematics ・ Phenomenology

Comay-Kelson-Zidon88 8 Janecke-Masson88    ≈0 Garvey-Kelson-type mass formula (1988) Good reproduction in the whole region, but worse in the light n-rich region. Referred mass data:AME88 Mass Sn Mass Sn

Gives the best RMS dev. (380 keV for AME11) among global mass formulae (without GK type). Nether deformation nor fission barrier are obtained. How about the superheavy mass region? Duflo-Zuker mass formula (1995) 9 by Zuker Referred mass data:AME93 Mass Sn

・ Hartree-Fock method with Skyrme force Strong short-range force => δ-function => HF calc. ETFSI (1995), HFBCS (2001), HFB (2002-) ・ Liquid-drop model Deformed liquid-drop part+Micro. (folded Yukawa) FRDM (1995), FRLDM (2002), ・ Mass formula with spherical-basis shell term Phenom. gross (WB-like)+spherical-basis shell part KUTY (2000), KTUY (2005) Koura, Uno, Tachibana, Yamada Recent mass formulas: ・ are designed for nuclei with Z, N=8 to 310 [126] 184 or more ・ have the RMS dev. from exp. masses. of keV ・ give deformation parameters  2,  4... and fission barriers micro (-like) macro+ micro phenom. 10 by S.Goriely et al. by P.Möller et al. ・ Density functional theory <- recent project by Dobaczewski et al. by H. Koura et al. Mass model, Approximation

Skyrme-Hartree-Fock-Bogoliubov mass formula ( ) BSk21 force parameter set: t 0 = MeV fm 3, t 1 = MeV fm 5 t 2 =0 MeV fm 5, t 3 = MeV fm 3+3 α t 4 = MeV fm 5+3 β, t 5 = MeVfm 5+3 γ x 0 = , x 1 = , t 2 x 2 = MeV fm 5 x 3 = , x 4 = , x 5 = W 0 = MeV fm 5, α=1/12, β=1/2, γ=1/12 f + n =1.00, f + p =1.07, f - n =1.05, f p =1.13 V W =-1.80 MeV, λ=280, V ' W =0.96, A 0 =24 11 E tot = E HFB+ E wigner Current version: HFB-21 (2010) Mass Sn HFB21 gives a less than 600 keV of the RMS dev. In the light region there is some discrepancy in derivatives as S n. Referred mass data:AME03 by S. Goriely et al.

Deformation, fission barrier is obtained Good prediction on fission properties. Finite-Range-Droplet Model (FRDM) mass formula (1995) 12 Mass Sn E(Z, N, shape)=E macro (Z, N, shape)+E micro (Z, N, shape) E macro : Droplet part as a function of Z and N E micro : Folded Yukawa-type potential + Nilsson-Strutinsky method Current version is FRLDM (2003-) Good for the heavier mass region. Some large discrepancies appear in the light region. Referred mass data:AME93 by P. Möller et al.

M gross smooth function of N and Z. (same as the TUYY formula) M shell : modified Woods-Saxon pot.+BCS+deform. config. Spherical-Basis (KTUY) mass formula (2005) 13 M(Z, N)=M gross (Z, N)+M eo (Z, N)+M shell (Z, N) Derivatives of mass like S n,Q α, Q β, gives a good reproduction. Mass Sn Referred Mass data:AME03 Deformation, fission barrier is obtained Change of magicties in the n-rich nuclei is predicted. (N=20 -> 16, etc.) Topic: decay modes for superheavy nuclei are applied for. by H. Koura et al.

S. Maripuu, Special ed., 1975 Mass Predictions, Atomic Data and Nuclear Data Tables 17, 411(1976) P.E. Haustein., Special ed., Atomic Mass Predictions, Atomic Data and Nuclear Data Tables 39, 185 (1988) 14 With the use of AME11, various mass models are compared and estimated. Atomic mass formula competition

Janecke- Masson(19 88) is the best, Duflo- Zuker(1995 ) is the second best. Among the macro- micro or HFB mass formulae, HFB21(201 0) gives a best RMS dev. in masses. RMS deviation of mass formulae: masses 15

RMS deviation of mass formulae: Derivatives: S n, S 2n JM(1995) is the best. Among the macro-micro or HFB mass formulae, KTUY(2005) gives the best RMS in both S n and S 2n. S n,S 2n : required for the r-process nucleosynthesis study. 16

Conclusion in the RMS deviation of mass formulae In current status (as I evaluated ) by 2011: Jänecke-Masson formula (Garvey-Kelson Consideration) gives the best RMS deviation in any mass-related quantities. Among the macro-micro or microscopic mass formulae, HFB21(2010) gives the best RMS deviation in absolute mass values. Regarding the derivatives as S n, S p, Q α, Q β, KTUY(2005) gives the best RMS. FRDM(1995): between HFB21 and KTUY, or comparable. Other mass formulae: Duflo-Zuker (1995): without the GK formula, DZ has still good predictable power. GHT, HGT(1976): RMS dev. diverge for recent exp. mass values. especially lighter and/or neutron-rich mass region. Satpathy-Nayak(1988): RMS dev. remarkably diverge for recent exp. mass values. (insufficient parameter choice in 1988?) 17

Mass table part: Dobaczewski goal: RMS dev. less than 500 keV. Mass formula on the UNEDF project (2006-) UNDEF(Universal Nuclear Energy Density Functional) 2001 SciDAC(Scientific Discovery through Advanced Computing) program started 2006 Dec. - : starting Scientific strategy Bertsch Nazarewicz Dobaczewski Thompson Furnstahl... Deformation Mass difference 18 masses

Deformation is obtained Overestimated at closed shell region and deformation region Mass formula on the RMF (1998-) First: D. Hirata, et al., NPA (1998): TMA parameter, no pairing, 8≤Z≤120 e-e nuclei: RMS dev.=2.71 MeV Later: G. A. Lalazissis, et al., ADNDT 71, 1 (1999), NL3 parameter+BSC, 10≤Z≤98 e-e nuclei: RMS dev.=2.6 MeV Recent: L.S. Geng, et al., PTP113 (2005): TMA parameter, state dependent BCS, 8≤Z≤100, RMS dev.=2.1 MeV Lagrangian density: (Result from Geng’s paper) 19

Change of shell closure far from the stable nuclei N=20 gap decreases, while N=16 gap increases in the n-rich region. N=56 gap evolves in the n-rich region. n-rich p-rich Z=82 gap decreases in the p-rich region (even penetrating the p-drip line). S 2p lines parallels each other in most cases. 20 S 2n vs ZS 2p vs N

N=20,28 gap -> already weak Zigzag Smoothness N=22,30 line -> crossing Anomalous Kink 21 shell gap and smoothness S 2n systematics N=16

TUYY: gross term (WB-like with higher expansion) + empirical shell term. KTUY: TUYY gross term + deformed shell with a modified Woods-Saxon pot. FRDM: Macroscopic Droplet + microscopic deformed shell with a folded Yukawa pot. 22 dip -Check the mass formulae as astrophysical data- S 2n for equilibrium eq. (determine the path) and Q  for λ β : estimated from mass formulae (TUYY, KUTY, FRDM) Steady flow +Waiting point Approximation Neutron-number density (N n ) and temperature (T 9 ) are constants (n,γ)-(γ,n) equilibrium is established over an irradiation time τ Canonical model N n, T 9,τ: chosen to reproduce the abundance peak at A=130 (obs.) R-process nucleosynthesis

A=130peak ExperimentTUYYKUTYFRDM 23 A=130peak S 2n systematics Bunched (Twisted) To measure S 2n of 108,110 Sr, 110,112 Zr, etc. gives an answer. 112 Zr T 1/2 of 110 Zr was measured by Nishimura et al.

Garvey-Kelson mass relationship 24 Change of shape?

Deformation parameter  2 KUTY FRDM Proton number Z dip of S 2n change of  2 (FRDM) A=130 25

E sh of KUTY ΔM of ETFSI ΔM of FRDM Δ M(Z, N)=M FRDM (Z, N) − (M gross (Z, N)+M eo (Z, N)) ETFSI M gross (Z, N):KUTY gross term M eo (Z, N):KUTY average even-odd term Proton number Z Long-lived Closed-shell Closed shell Long-lived 26 Shell energy in the superheavy mass region

α-decay Q-value of superheavy nuclei (unphysical) zigzag waving no shell closure rather geometrical Z=114 magic near N=184 deformed shell N=184 magic => Prediction of structure for SHE

There are various mass formulae in the history of the nuclear physics study. Each mass formula has its specific property, therefore we need to understand the different properties when we use. Only mass values are required: Among global mass formulae, JM and CKZ, Garvey-Kelson type mass model, gives the best RMS dev., DZ (phenomenological shell model) also good though there is no information on the nuclear structure as nuclear shapes and fission properties. Regarding mass models capable to calculate nuclear shapes and deformations, the HFB-21 mass formula gives the best RMS dev., besides KTUY gives good properties on derivatives of mass (S n, S p, Q α, Q β...). RMF, UNEDF are in progress. The current RMS is over one MeV. To explore unknown mass regions as the n-rich region relevant to the r- process or the superheavy mass region, the mass formulae are still important tools. 28 Conclusion