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Current Status of Nuclear Mass Formulae

RIBF-ULIC-Symposium: Physics of Rare-RI Ring, RIKEN, Nov. 10-12, 2011. Current Status of Nuclear Mass Formulae. Hiroyuki KOURA Advanced Science Research Center, Japan Atomic Energy Agency (JAEA). Bulk properties of atomic masses Phenomenological mass formulas

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Current Status of Nuclear Mass Formulae

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  1. RIBF-ULIC-Symposium: Physics of Rare-RI Ring, RIKEN, Nov. 10-12, 2011 Current Status of Nuclear Mass Formulae 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 1

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

  3. = • Wigner energy • N=Z ridge N-Z=24 N=Z ridge Next figure Shell energy 208Pb 132Sn • Depression due to the deform. • rare-earth, actinide Bulk properties of atomic mass Weizsäcker-Bethe semi-empirical atomic mass formula M(Z, N)=Z mH+N mn−B(Z,N) =Z mH+N mn−aVA+asA2/3+aI(N−Z)2/A+aCZ2/A1/3+δeo (MeV) Mexp.−MWB • Existence of magic number • N=28,50,82,126 • Z=28,50,82 RMS dev. from AWT03: 2.93 MeV (Z, N ≥ 8) Mexp-MWB (MeV) Mass data : 2003 Atomic mass evaluation (Audi, Wapstra & Thibault) 3

  4. Shape transition and shell energies - Experiment - - Schematic - N=88-90 • 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 Fig. (b) cross section along dashed line in (a) Fig. (a) N-Z plane β from B(E2) : S. Raman et al., ADNDT78 (2001) 4

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

  6. Some points on parametrization of a mass formula • 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. 6

  7. Systematics・Phenomenology ・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. (300-600 keV) ・No predictable power for superheavy nuclei (next magic number, etc.) ・No deformation is obtained. 7

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

  9. Duflo-Zuker mass formula (1995) by Zuker Mass • 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? Sn Referred mass data:AME93 9

  10. ・Density functional theory <- recent project by Dobaczewski et al. Mass model, Approximation 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 600-800 keV ・give deformation parametersβ2, β4... and fission barriers ・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 micro (-like) by S.Goriely et al. macro+ micro by P.Möller et al. phenom. 10 by H. Koura et al.

  11. Skyrme-Hartree-Fock-Bogoliubov mass formula (2002-2010) by S. Goriely et al. Etot = EHFB+ Ewigner BSk21 force parameter set: t0=-3961.39 MeV fm3, t1=396.131 MeV fm5 t2=0 MeV fm5, t3=22588.2 MeV fm3+3α t4=-100.000 MeV fm5+3β, t5=-150.000 MeVfm5+3γ x0=0.885231, x1=-0.0648452, t2x2=1390.38 MeV fm5 x3=1.03928, x4=2.00000, x5=-11.0000 W0=109.622 MeV fm5, α=1/12, β=1/2, γ=1/12 f+n=1.00, f+p=1.07, f-n=1.05, fp=1.13 VW=-1.80 MeV, λ=280, V'W=0.96, A0=24 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 Sn. Referred mass data:AME03 11

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

  13. Spherical-Basis (KTUY) mass formula (2005) by H. Koura et al. M(Z, N)=Mgross(Z, N)+Meo(Z, N)+Mshell(Z, N) • Mgross smooth function of N and Z. (same as the TUYY formula) • Mshell: modified Woods-Saxon pot.+BCS+deform. config. • 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. Mass Sn Derivatives of mass like Sn ,Qα, Qβ, gives a good reproduction. Referred Mass data:AME03 13

  14. Atomic mass formula competition • S. Maripuu, Special ed., 1975 Mass Predictions, Atomic Data and Nuclear Data Tables 17, 411(1976) • P.E. Haustein., Special ed., 1988-89 Atomic Mass Predictions, Atomic Data and Nuclear Data Tables 39, 185 (1988) With the use of AME11, various mass models are compared and estimated. 14

  15. RMS deviation of mass formulae: masses • Janecke-Masson(1988) is the best, Duflo-Zuker(1995) is the second best. • Among the macro-micro or HFB mass formulae, HFB21(2010) gives a best RMS dev. in masses. 15

  16. RMS deviation of mass formulae: Derivatives: Sn, S2n Sn,S2n: required for the r-process nucleosynthesis study. • JM(1995) is the best. Among the macro-micro or HFB mass formulae, KTUY(2005) gives the best RMS in both Sn and S2n. 16

  17. 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 Sn, Sp, Qα, Qβ, 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

  18. 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 • Mass table part: Dobaczewski • goal: RMS dev. less than 500 keV. Scientific strategy • Bertsch • Nazarewicz • Dobaczewski • Thompson • Furnstahl • ... Mass difference masses Deformation 18

  19. Mass formula on the RMF (1998-) First: D. Hirata, et al., NPA616 438 (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 • Deformation is obtained • Overestimated at closed shell region and deformation region (Result from Geng’s paper) Lagrangian density: 19

  20. Change of shell closure far from the stable nuclei S2n vs Z S2p vs N p-rich n-rich N=20 gap decreases, while N=16 gap increases in the n-rich region. N=56 gap evolves in the n-rich region. Z=82 gap decreases in the p-rich region (even penetrating the p-drip line). S2p lines parallels each other in most cases. 20

  21. N=16 Smoothness Zigzag Anomalous Kink N=20,28 gap -> already weak N=22,30 line -> crossing shell gap and smoothness S2n systematics 21

  22. dip R-process nucleosynthesis -Check the mass formulae as astrophysical data- • Canonical model Steady flow +Waiting point Approximation Neutron-number density (Nn) and temperature (T9) are constants (n,γ)-(γ,n) equilibrium is established over an irradiation time τ Nn,T9,τ: chosen to reproduce the abundance peak at A=130 (obs.) S2nfor equilibrium eq. (determine the path) andQβ for λβ: estimated from mass formulae (TUYY, KUTY, FRDM) • 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

  23. A=130peak To measure S2n of 108,110Sr, 110,112Zr, etc. gives an answer. Bunched (Twisted) A=130peak A=130peak A=130peak T1/2 of 110Zr was measured by Nishimura et al. 112Zr 112Zr 112Zr 112Zr S2n systematics Experiment TUYY KUTY FRDM 23

  24. Change of shape? Garvey-Kelson mass relationship 24

  25. Deformation parameter α2 A=130 KUTY Proton number Z dip of S2n <=> change of α2 (FRDM) Proton number Z FRDM 25

  26. Closed-shell Closed shell Long-lived Proton number Z Proton number Z Proton number Z Long-lived Shell energy in the superheavy mass region ΔM(Z, N)=MFRDM(Z, N)−(Mgross(Z, N)+Meo(Z, N)) ETFSI Mgross(Z, N):KUTY gross term Meo(Z, N):KUTY average even-odd term ΔM of FRDM Esh of KUTY ΔM of ETFSI 26

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

  28. 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 (Sn, Sp, Qα, Qβ...). 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. Conclusion 28

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