- Institut für Kernchemie, Univ. Mainz, Germany - HGF VISTARS, Germany

1. Gross b–decay properties for astrophysical applications Karl-Ludwig Kratz - Institut für Kernchemie, Univ. Mainz, Germany - HGF VISTARS, Germany - Department of Physics, Univ. of Notre Dame, USA

2. Nuclear data in astrophysics What data are needed in nuclear astrophysics ? (B) Explosive nucleosynthesis e.g. rp-process, np-process; “weak” and “main” r-process • Quiescent nucleosynthesis • e.g. H-, He-burning; s-process • nuclear masses (reaction Q-values) • charged-particle reaction rates • (e.g. (p,g), (a,g), (a,n)) • neutron capture-rates • nuclear structure properties • (e.g. Esp, Jp, C2S) • nuclear-masses (Qb, Sp, Sn) • half-lives (T1/2,; g.s. , isomers) • b-delayed quantities (Pp, Pn, Pf) • neutron capture rates • neutrino reactions • nuclear-structure-properties • (e.g. e2, Esp, J p …) for 10’s to 100’s of isotopes NEARb-stability for 100’s to 1000’s of isotopes FAR-OFFb-stability

3. What are the nuclear data needed for? as input for astrophysical calculations star evolution, “chemical” evolution of Galaxy, specific nucleosynthesis processes WARNING ! Nuclear data (n.d.) are only ONE set of input parameters among SEVERALastrophysics parameter sets Depending on “mentality of the star-couturier”, nuclear data are considered important unimportant nuclear and astro-parameters of equal standing n.d. to constrain astro-parameters “learning” nucl. structure from astro-observables astro-parameters dominate n.d. just “telephone numbers” (too) many (free) parameters n.d. effects invisible mathematical nuclear astrophysics

4. Basic astronomical question: r-process • Historically, • nuclear astrophysics has always been • concerned with • interpretation of the • origin of the chemical elements • from astrophysical and cosmochemical • observations, • description in terms of specific • nucleosynthesis processes • (already B²FH, 1957). Solar system isotopic abundances, Nr, T9=1.35; nn=1020 - 1028 , Bi r-process observables CS 22892-052 abundances isotopic composition Ca, Ti, Cr, Zr, Mo, Ru, Nd, Sm, Dy ↷ r-enhanced scaled solar r-process ALLENDE INCLUSION EK-1-4-1 scaled theoretical solar r-process Zr Pt Os Pb Cd Ru Ba d [‰] Nd Sn Sr Ga Pd Dy Mo Gd Er Ge Sm Ce Yb Ir Hf Y Rh La Nb Ag Ho Eu Pr Tb Au Lu Th Tm U Mass number Elemental abundances in UMP halo stars “FUN-anomalies” in meteoritic samples

5. Nuclear models to calculate T1/2 and Pn – (I) Theoretically, the two gross/ integralb-decay quantities, T1/2 and Pn, are interrelated via their usual definiton in terms of the so-called b-strength function [Sb(E)] What is that? … a natural adoption of the strength function concept employed in other areas of nuclear physics, e.g.: single-particle strength functions, s-, p-wave neutron strength functions, multipole strength functions for photons. Slc refers to the behavior of the squares of overlap integrals (g²lc) between two sets of nuclear wave functions: l represents various states of excitation, classified by E, Jp, T; c refers to the different reaction / decay channels, classified by Epart, lpart,… rl is the density of levels l. Slc = <g²lc> rl

6. 106 103 1 Nuclear models to calculate T1/2 and Pn – (II) Application to b-decay: “Theoretical” definition (Yamada & Takahashi, 1972) “Experimental” definition (Duke et al., 1970) b(E) Sb(E) = D-1 ·M(E) ² · (E) [s-1MeV-1] [s-1MeV-1] Sb(E) = f(Z, Qb-E) · T1/2 b(E) absolute b-feeding per MeV, f(Z, Qb-E) Fermi function, T1/2 b-decay half-life. M(E)  average b-transition matrix element (E) level density D const., determines Fermi coupling constant gv² 1 T1/2 as reciprocal ft-value per MeV T1/2 = Sb(Ei) x f (Z,Qb-Ei) 0Ei Qb Sb(E) 6x105 Fermi function f(Z, Qb-Ei)  (Qb-Ei)5 3x103 T1/2 sensitive to lowest-lying resonances in Sb(Ei) Pn sensitive to resonances in Sb(Ei) just beyond Sn Qb same T1/2 ! 1 ↷ easily “correct” T1/2 with wrong Sb(E) 11 E*[MeV] 1 5 10

7. Nuclear models to calculate T1/2 and Pn – (III) Before any theoretical approach is applied, its significance and sophistication should be clear ! In general, 2 groups of models: (2) Models that use an effective nuclear interaction and solve the microscopic, quantum-mechanical Schrödinger or Dirac equation. • “Models” where the physical quantity of interest • is given by a polynomial or some other algebraic • expression. • parameters adjusted to exp. data • describes only a single nucl. property • no nuclear wave functions • no insight into underlying SP structure • provides nuclear wave functions • within the same framework, • describes a number of nucl. properties • (e.g. g.s.-shape; Esp, Jp , log(ft), T1/2 … ) Examples: Examples: Kratz-Herrmann Formula (1973) Gross Theory (1973) · · New exponential law for T1/2(b+) (Zhang & Ren; 2006) T1/2(b-) from GTGR + known log(ft)’s (Kar, Chakravarti & Manfredi; 2006) FRDM+QRPA (1997; 2006) Self-consist. Skyrme-HFB + QRPA (Engel et al.; 1999) Large-Scale Shell Model (Martinez-P. & Langanke; 1999, 2003) Density-Functional + Finite-Fermi System (Borzov et al.; 2003) PN-Relativistic QRPA (Niksic et al.; 2005) ·

8. Nuclear models to calculate T1/2 and Pn – (IV) • Simple “statistical” approaches • assumptions: • b-decay energy is large (Qb≳ 5 MeV) • high level density • Sb(E) is a smooth function of E (e.g. Sb=const.; Sb r (E)); • is insensitive to nature of final states; • does not vary significantly for different types of nuclei (ee, o-mass, oo). The Kratz-Herrmann Formula, applied to Pn values Sb(Ei) x f (Z,Qb-Ei) with Sb=const. b SnEi Qb (Qb – Sn) Pn = Pn≃ a Sb(Ei) x f (Z,Qb-Ei) (Qb – C) CEi Qb a, b as “free parameters”, to be determined by a log-log fit to known Pn-values C is a “cut-off parameter” (↷ pairing-gap in b-decay daughter)

9. Region Lin. regression Least-squares fit a [%] b red. ² a [%] b r² 106 5.5 81 40 0.6 29  Z  43 88.2 4.1 0.81 123 4.7 57 41 0.5 47  Z  57 84.4 3.9 0.86 81 4.7 78 21 0.3 85.2 4.0 0.83 29  Z  57 Lin. regression Least-squares fit a [ms] b red. ² a [ms] b r² 7.07E05 4.0 1.1E04 5.33E05 0.4 2.74E06 4.5 0.72 Nuclear models to calculate T1/2 and Pn – (V) From Pfeiffer, Kratz & Möller, Prog. Nuclear Energy 41 (2002) 39-62 Parameters from fits to known Pn-values full line dashed line … as a kind of “joke”: -b T1/2≃ a (Qb-C) Parameters from fit to known T1/2 of n-rich nuclei

10. Nuclear models to calculate T1/2 and Pn – (VI) … NO joke ! in 2006, two examples for big steps BACKWARDS : • K. Kar, S. Chakravarti & V.R. Manfredi; • arXiv: astro-ph/0603517 v1 • “Beta-decay rates (115 < A < 140) for r-process • nucleosynthesis” • X. Zhang & Z. Ren; PRC73, 014305 • “New exponential law for b+ decay half-lives • of nuclei far from b-stable line” “…we have discovered a new exponential law for T1/2(b+)…as a function of neutron number…” … the xth re-invention of the Gross Theory ! log10 T1/2 = a x N + b “… shell model results… indicate that the GT strength distribution.. can be taken as a Gaussian.” authors give fit parameters for a and b, for (I) different Z-regions (II) allowed b+-decay (III) first-forbidden b+-decay (IV) second-forbidden b+-decay • “…GT strength distributes among 3 different types of • final states: • discrete low-lying states with known log ft’s; • discrete states above with unknown strengths; • a part of the GT giant resonance (GTGR).” ↷ finally “a simple and accurate formula” emerges: admitted “problems”: centroid of GTGR ↷ from Bertsch & Esbensen (1987) width of GTGR ↷ free parameter ! log10 T1/2 = (c1Z + c2) N + c3Z + c4 “…useful to experimental physicists for analyzing b+-decay data.”

11. Nuclear models to calculate T1/2 and Pn – (VII) VGT = GT : st_· st+ (2) QRPA – type, “microscopic” models Recent review by J. Engel; Proc. Workshop on The r-Process… ; Seattle (2004); World Scientific Among “recent theoretical schemes”… “Some methods emphasize global applicability, others self-consistency, and still others the comprehensive inclusion of nuclear correlations. None of the methods includes all important correlations, however.” (2.2) Self-consistent Skyrme-HFB + QRPA (2.1) FRDM + QRPA Macroscopic-microscopic mass model FRDM; Schrödinger equation solved in QRPA: GT force with “standard choice” for GT interaction latest version includes ff-strength from Gross Theory. Skyrme interaction SKO ↷ reasonable reproduction of energies and strengths of GT resonances; strength of T=0 np pairing “adjusted” to fit known T1/2 • disadvantages: only spherical shape; only GT; only n-magic (N=50, 82, 128); Skyrme interaction not good enough to make…decisive improvement • advantage: self-consistency GT = 23 MeV/A • disadvantage: not “self consistent” • advantages: globalmodel for all shapes and types of nuclei; large model space ↷ T1/2 shorter than those from FRDM + QRPA

12. Nuclear models to calculate T1/2 and Pn – (VIII) (2.3) Large-scale Shell Model (2.4) Density Functional HFB + QRPA shell-model code ANTOINE; restricted, but sufficiently large SP model space, with residual interaction split into: (I) monopole part (II) renormalized G-matrix component monopole interaction tuned to reproduce exp. spectra; admitted, that truncated space may still miss some correlations. density-functional / Greens-function-based model + finite-Fermi-systems theory; not quite selfconsistent, but with well-developed phenomenology. • disadvantage: only spherical nuclei • disadvantages: only n-magic nuclei (N=50, 82, 126); only GT-decay; only spherical. • advantages: all types of nuclei (ee, o-mass, oo); includes ff-strength microscopically. • advantages: several essential correlations included; treatment of ee and odd-p isotopes. ↷ T1/2 even shorter than those of SC-HFB + QRPA ↷ T1/2 (in particular with ff) short

13. Nuclear models to calculate T1/2 and Pn – (IX) (2.5) Fully consistent relativistic pn-QRPA use of new density-dependent interaction in relativistic Hartree-Bogoliubov calculations of g.s. and particle-hole channels; finite-range Gogny D1S interaction for T=1 pairing channel; inclusion of pn particle-particle interaction. Conclusions J. Engel “… it is argued on the basis of a measurement of a strength distribution(i.e. N=82 130Cd) that the transitions at N=82 calculated by the shell model, HFB + QRPA and Density-functional + FFS are too fast. …this will force the other groups to go back and examine their calculated strength distributions.” • disadvantages: only spherical ee nuclei; Ni half-lives overestimated by factor ∼ 10 (spherical QRPA “normalized” to deformed 66Fe40 …! ); “… our model predicts that 132Sn is stable against b-decay…” (exp.: T1/2=40 s ; Qb=3.12 MeV). P. Möller “…there is no “correct” model in nuclear physics. Any modeling of nuclear-structure properties involves approximations … to obtain a formulation that can be solved…, but that “retains the essential features” of the true system.” • advantages: “…theoretical T1/2 reproduce the exp. data for Fe, Zn, Cd, and Te…”; sufficiently large model space.

14. The r-process “waiting-point“ nucleus 130Cd ...obtain a physically consistent picture! T1/2, Q, E(1+), I(1+), log ft Q Sn 7.0 8.9 J=1+ {g7/2, g9/2} 2QP 4QP 1.2 2.9 “free choice” of combinations: T1/2(GT) 233 ms 1130 ms 76 ms 246 ms low E(1+) with low Qb high E(1+) with low Qb low E(1+) with high Qb high E(1+) with high Qb

15. Shape of Nr, abundance peak rising wing 122<A<130 solar r abundances “short“ T1/2 Deficiencies explained by : • neutrino induced reactions ? • Qian,Haxton et al. (1997) • waiting-point concept breaks down ? • Martinez-P. & Langanke (1999) • nuclear structure below 132Sn not understood ? • Kratz et al. (since 1993) • importance of ng7/2 pg9/2 GT • position of ng7/2 SP state nd3/2 rel. to nh11/2 • spin-orbit splittingn3p3/2 - n3p1/2 n2f7/2 - n2f5/2 p2p3/2 - p2p1/2 p1f7/2 - p1f5/2 • N=82 shell quenching QRPA (Nilsson, Woods-Saxon, Folded Yukawa) OXBASH

16. Level systematics of the lowest 1+ state in neutron-rich even-mass In isotopes OXBASH (B.A. Brown, Oct. 2003) Experimental 1+ 2181 1+ 2120 (new) 1+ 1173 1731 keV 1+ 1382 1+ 688 (old) Reduction of the TBME (1+) by 800 keV 1+ 243 3+ 473 3+ 0 3+ 0 3+ 0 3+ 389 124In75 126In77 128In79 1- 0 1- 0 130In81 130In81 Dillmann et al., 2003 Configuration 3+ : nd3/2 pg9/2 Configuration 1+ : ng7/2 pg9/2 Configuration 1- : nh11/2 pg9/2

17. Beta-decay odd-mass, N=82 isotones nSP states in N=81 isotones S1n=3.98MeV S1n=3.59MeV S1n=5.246MeV S1n=2.84MeV E*[MeV] 2648 2643 ng7/2 7/2+ 2637 7/2+ 67% 4.1 45% 4.25 24% 4.5 2607 7/2+ ng7/2 88% 4.0 2565 89% 4.0 P1n=25% P2n=45% P3n=11% P1n=39% P2n=11% P3n= 4.5% P1n=29% P2n= 2% S1n=1.81MeV Pn=4.4% Pn=9.3% P4n= 8.5% P5n= 1% 908 1/2+ 1/2+ 814 1/2+ 728 601 1/2+ 536 ns1/2 524 3/2+ 472 3/2+ 414 3/2+ 331 3/2+ 282 nd3/2 0 Ib log(ft) nh11/2 11/2- 11/2- 11/2- 11/2- 1.2% 6.3 0.6% 6.4 2.3% 6.3 0.9% 6.4 0.5% 6.45 123Mo81 125Ru81 131Sn81 127Pd81 129Cd81 42 44 46 50 48

18. high-j orbitals  (e.g. nh11/2) • low-j orbitals  (e.g. nd3/2) • evtl. crossing of orbitals • new “magic” numbers / shell gaps • (e.g. 110Zr70, 170Ce112) N/Z ) 0 w g 9/2 h 58 40 126 g 9/2 of p 1/2 i 7.0 f 13/2 112 5/2 p Units h ;f 3/2 i 9/2 5/2 13/2 p h 1/2 ( 6.5 9/2 f p 7/2 3/2 f 7/2 Energies h 11/2 70 6.0 h 11/2 g d 7/2 3/2 g d 7/2 s 3/2 1/2 5.5 s d 1/2 5/2 50 d 5/2 Single – Neutron g g 9/2 5.0 9/2 40 p 1/2 f 5/2 f p 5/2 1/2 70% 10% 100% 40% Strength of ℓ -Term 2 82 B. Pfeiffer et al., Acta Phys. Polon. B27 (1996) Effects of N=82 „shell quenching“ change of T1/2 ?

19. Possible effect of “shell quenching” Nilsson potential; gradual reduction of l2-term E*[MeV] L2 standard 20% red. 40% red. 60% red. 10% red. ng7/2 3549 ng7/2 3327 ng7/2 3027 912keV 684keV ng7/2 2806 379keV 2648 2643 2637 7/2+ ng7/2 7/2+ 2607 7/2+ ng7/2 2565 nd3/2 2497 199keV T1/2=4.6/6.15ms T1/2=14.4/17.3ms T1/2=2.0/2.85ms 1771 3/2+ T1/2=41.4/48.4ms T1/2=157ms 1.96MeV 1057 1299keV 3/2+ 643keV 650 3/2+ 3/2+ 536 472 3/2+ 319keV 414 3/2+ 331 3/2+ nd3/2 282 nh11/2 11/2- 0 11/2- 11/2- 11/2- 131Sn81 123Mo81 125Ru81 50 127Pd81 129Cd81 42 44 46 48

20. Beta-decay of 129Ag isomers 127Ag pg9/2 pp1/2 pp1/2 129mAg82 pg9/2 129gAg82 T1/2(g)=(46 ) ms +5 -9 T1/2(m)=(15860) ms Separation of isomers by fine-tuning of laser frequency pp1/2 30% pg9/2 158ms 70% 46ms

21. Terrestrial and stellar half-lives of odd-mass N=82 waiting-point isotopes Isotope Experiment QRPA(GT+ff)*) T1/2(stellar) T1/2(stellar) T1/2(pg9/2) T1/2(pp1/2) T1/2(pg9/2) T1/2(pp1/2) 131In 280ms 350ms 300ms 157ms 477ms 253ms 49 129Ag 46ms 158ms 80ms 43ms 140ms 72ms 47 127Rh ------ ----- ------ 14.4ms 25.4ms 17.7ms 45 125Tc ------ ----- ------ 4.60ms 4.45ms 4.5ms 43 123Nb ------ ----- ------ 2.01ms 1.91ms 1.98ms 41 *) Nuclear masses: ADMC,2003 & ETFSI-Q

22. Astrophysical consequences • ...mainly resulting from new nuclear structure information: • better understanding of formation and shape of, as well as r-process matter flow • through the A»130 Nr, peak • no justification to question waiting-point concept • (Langanke et al., PRL 83, 199; Nucl. Phys. News 10, 2000) • no need to request sizeable effects from n-induced reactions • (Qian et al., PRC 55, 1997)  r-process abundances in the Solar System and in UMP Halo stars... ...are governed by nuclear structure! „short“ T1/2 „long“ T1/2 Nuclear masses from AMDC, 2003 ETFSI-Q Normalized to Nr,(130Te)

23. Let’s come back to global calculations of gross b-decay properties… … only model that can calculate on a macroscopic-microscopic basis all types of nuclei (nearly) all nuclear shapes g.s. and odd-particle excited-states decays: mass models: FRDM (ADNDT 59, 1995) ETFSI-Q (PLB 387, 1996) QRPA model: pure GT (ADNDT 66, 1997) GT + ff (see above; URL: http://t16web/moeller/publications/rspeed2002.html; ADNDT, to be submitted; KCh Mainz Report (unpubl.), URL: www.kernchemie.uni-mainz.de)

24. T1/2 and Pn calculations in 3 steps – (I) “Typical example”: • FRDM /ETFSI-Q • ↷ Qb, Sn, e2 • Folded-Yukawa wave fcts. • QRPA pure GT • with input from mass model • potential: Folded Yukawa • Nilsson (different , m ) • Woods-Saxon • pairing-model: Lipkin-Nogami • BCS Sn Qb (2) as in (1) with empirical spreading of SP transition strength, as shown in experimental Sb(E) (3) as in (2) with addition of first-forbidden strength from Gross Theory note: effect on Pn !

25. T1/2 and Pn calculations in 3 steps – (II) Another“spherical”case: …and a typical“deformed”case: Note: low-lying GT-strength; ff-strength unimportant! note : effect on T1/2 !

26. Experimental vs.theoretical b-decay properties T1/2, Pn gross b-strength properties from FRDM + QRPA Requests: (I) prediction / reproduction of correct experimental “number” (II) detailed nuclear-structure understanding ↷full spectroscopy of “key” isotopes, like 80Zn50 , 130Cd82. Total Error = 3.73 Total Error = 5.54 Pn-Values Half-lives QRPA (GT) QRPA (GT) QRPA (GT+ff) QRPA (GT+ff) (P. Möller et al., PR C67, 055802 (2003)) Total Error = 3.52 Total Error = 3.08

27. T1/2x 3 T1/2: 3 Effects of T1/2 on r-process matter flow T1/2 (GT + ff) • Mass model: ETFSI-Q • all astro-parameters kept constant • r-process model: • “waiting-point approximation“ r-matter flow too slow r-matter flow too fast

28. Conclusion Despite impressive experimentaland theoretical progress, situation of nuclear-physics data for explosive nucleosynthesis calculations still unsatisfactory ! • better globalmodels with sufficiently large SP model space, for all nuclear shapes (spherical, prolate, oblate, triaxial, tetrahedral,…) and all nuclear types (even-even, odd-particle, odd-odd) • more measurements masses gross b-decay properties level systematics full spectroscopy of selected “key“ waiting-point isotopes