Calculating β Decay for the r Process J. Engel with M. Mustonen, T. Shafer C. Fröhlich, G. McLaughlin, M. Mumpower, R. Surman July 27, 2016 Nuclear Landscape 126 Limits of nuclear existence 82 To locate the site(s) of the r r-process process, need reaction rates and properties in 50 protons 82 very neutron-rich nuclei. 28 rp-process 20 50 8 28 neutrons Density Functional Theory 2 20 Selfconsistent Mean Field 2 8 A=10 A~60 A=12 β decay particularly important. Increases Z0Ñωthroughout Shell Towards the r a unified Ab initio Model description of the nucleus process, and competition withfew-body neutron capture during freeze-out calculations No-Core Shell Model can have large effect on abundances. G-matrix What’s Usually Used for β Decay in Simulations Ⱥ ÅÓÐ ÐÖ¸ º È«Ö¸ ú¹Äº ÃÖØÞ »ËÔÒ ÙÔ Ø Ð Ö¹ÔÖÓ ×× º º º ½½ Möller, Pfeiffer, Kratz (2003) 104 − 3 β decay (Theory: GT) Old but Still in Some Ways 10 102 Unsurpassed Technology 101 100 − Masses through finite-range 10 1 − droplet model with shell 2 10 Total Error = 3.73 for 184 nuclei, Tβ,exp < 1 s ,exp β Total Error = 21.16 for 546 nuclei (13 clipped), Tβ < 1000 s − 3 ,exp T corrections. / 10 4 10 ,calc − β 3 β decay (Theory: GT + ff) QRPA with simple space- T 10 2 independent interaction. 10 101 First forbidden decay added in 100 − crude way in 2003. Shortens 10 1 − 2 half lives. 10 Total Error = 3.08 for 184 nuclei, Tβ,exp < 1 s Total Error = 4.82 for 546 nuclei, Tβ,exp < 1000 s − 10 3 − − − 10 3 10 2 10 1 100 101 102 103 Experimental β-Decay Half-life Tβ,exp (s) ½6 ÙÖ ÊØÓ Ó ØÓ ÜÔ ÖÑÒØÐ ¬ Ý Ð¹ÐÚ× ÓÖ ÖÓÑ Ç ØÓ Ø Ú×Ø ÒÓÛÒ Ò ÓÙÖ ÔÖÚÓÙ× Ò ÑÓ Ð׺ ÖÖÓÖ× ØÖ Ö ÑÒÝ ÑÓÖ Ô ÓÒØ× ØÒ ÓÖ ÐÖ ÖÖÓÖ׺ Ì× × ÒÓØ ×Ò Ò Ø ¬ÙÖ׸ ÓÖ ×ÑÐÐ ÖÖÓÖ× ÑÒÝ Ô ÓÒØ× Ö ×ÙÔ ÖÑÔ Ó× ÓÒ ÓÒ ÒÓØÖº ÌÓ ÓØÒ ÑÓÖ ÙÒÖ×ØÒÒ Ó Ø ÖÖÓÖ Ò Ø Û ØÖÓÖ Ô ÖÓÖÑ ÑÓÖ ØÐ ÒÐÝ×׺ ÇÒ ÓØÒ ÒÐÝÞ× Ø ÖÖÓÖ Ò Ý ×ØÙÝÒ ÖÓ ÓعÑÒ¹×ÕÙÖ ´ÖÑ×µ ÚØÓÒ¸ Ò Ø× ÛÓÙÐ Ò X ½ ¾ ¾ ´Ì Ì µ ´½¿µ ÖÑ× ¬ ;ÜÔ ¬ ; Ò i½ ÀÓÛÚÖ¸ Ò ÖÖÓÖ ÒÐÝ×× × ÙÒ×ÙØÐ Ö¸ ÓÖ ØÛÓ Ö×ÓÒ׺ Ö×ظ Ø ÕÙÒØØ× ×ØÙ Modern Alternative: Skyrme Density-Functional Theory Started as zero-range effective potential, treated in mean-field theory. Later re-framed as density functional, which can then be extended: 3 E = d r Heven + Hodd +Hkin. + Hem Z HSkyrme | {z } Hodd has no effect in mean-field description of time-reversal even states (e.g. ground states), but large effect in β decay. Contains eight terms that depend on spin density, current density, spin-kinetic density, tensor-kinetic density. QRPA QRPA done properly is time-dependent mean-field theory with small harmonic perturbation by β-decay transition operator. Matrix elements of operator between the initial state and final excited states obtained from response of nucleus. QRPA of Möller et al. is simplified. No fully self-consistent mean-field calculation to start. Nucleon-nucleon interaction (schematic) is used only in QRPA part of calculation. Several groups have tried to do better. One free parameter: strength of isoscalar 102 pairing (zero in schematic QRPA.) 1 10 t 2 p / x 1 e T / c 0 2 l / a 10 1 Adjusted in each of the three peak regions c T to reproduce measured lifetimes. 10-1 80Zn 78Zn 76Zn -2 10 82Ge 0 50 100 150 200 250 300 350 V0 [MeV] Initial Skyrme Application: Spherical QRPA 25 90 Example: late-90’s calculation, in nuclei Zr near “waiting points;” no forbidden decay. 20 ) 15 T G ( Chose functional SkO because did best B 0 10 with GT distributions. 5 SkO' SLy5 SkP Expt 0 0 2 4 6 8 10 12 14 16 18 20 excitation energy [MeV] 102 101 t 2 p / x 1 e T / c 0 2 l / a 10 1 c T 10-1 80Zn 78Zn 76Zn -2 10 82Ge 0 50 100 150 200 250 300 350 V0 [MeV] Initial Skyrme Application: Spherical QRPA 25 90 Example: late-90’s calculation, in nuclei Zr near “waiting points;” no forbidden decay. 20 ) 15 T G ( Chose functional SkO because did best B 0 10 with GT distributions. 5 SkO' SLy5 SkP Expt 0 0 2 4 6 8 10 12 14 16 18 20 excitation energy [MeV] One free parameter: strength of isoscalar pairing (zero in schematic QRPA.) Adjusted in each of the three peak regions to reproduce measured lifetimes. Re [S(ω)f(ω)] Im [S(ω)f(ω)] Im ω Im ω Beta-decay rates obtained C C by integrating strength with phase-space Re ω Re ω weighting function in contour around excited states below threshold. New: Fast Skyrme QRPA in Deformed Nuclei Finite-Amplitude Method (Nakatsukasa et al.) 7 12 128 208 ] K , Cd =0 1 ] Pb K =0, 1 1 ± 1 ± − 6 − 9 5 4 6 3 Strength functions 2 3 GT strength [MeV 1 GT strength [MeV 0 0 computed directly, in 7 146 7 166 K K ] Ba =0 ] Gd =0 1 1 − 6 − 6 orders of magnitude less 5 5 4 4 3 3 time than with matrix 2 2 GT strength [MeV 1 GT strength [MeV 1 0 0 ] 7 146 ] 7 166 QRPA. 1 Ba K = 1 1 Gd K = 1 − 6 ± − 6 ± 5 5 4 4 3 3 2 2 GT strength [MeV 1 GT strength [MeV 1 0 0 0 10 20 0 10 20 RPA energy [MeV] RPA energy [MeV] Re [S(ω)f(ω)] Im [S(ω)f(ω)] Im ω Im ω C C Re ω Re ω New: Fast Skyrme QRPA in Deformed Nuclei Finite-Amplitude Method (Nakatsukasa et al.) 7 12 128 208 ] K , Cd =0 1 ] Pb K =0, 1 1 ± 1 ± − 6 − 9 5 4 6 3 Strength functions 2 3 GT strength [MeV 1 GT strength [MeV 0 0 computed directly, in 7 146 7 166 K K ] Ba =0 ] Gd =0 1 1 − 6 − 6 orders of magnitude less 5 5 4 4 3 3 time than with matrix 2 2 GT strength [MeV 1 GT strength [MeV 1 0 0 ] 7 146 ] 7 166 QRPA. 1 Ba K = 1 1 Gd K = 1 − 6 ± − 6 ± 5 5 4 4 3 3 2 2 GT strength [MeV 1 GT strength [MeV 1 0 0 0 10 20 0 10 20 RPA energy [MeV] RPA energy [MeV] Beta-decay rates obtained by integrating strength with phase-space weighting function in contour around excited states below threshold. Two Projects Described Here 1. Local focus on rare-earth region, consequences of new rates for rare-earth “bump.” 2. Global calculation, attempt at comprehensive fit of relevant part of Skyrme functional. Sensitivity Studies For Rare-Earth Peak: 70 hot inc hot dec 65 60 Protons 55 50 70 cold inc cold dec 65 60 Protons 55 50 70 nsm inc nsm dec 65 60 Protons 55 50 90 95 100 105 110 115 120 90 95 100 105 110 115 120 Neutrons Neutrons Adjustment: proton-neutron pairing Extremely important for beta-decay half-lives, but not active in HFB (no proton-neutron mixing) ⟶ free parameter. Adjust to approximately reproduce experimental lifetimes as close as possible to the region of interest. Adjusting isoscalar pairing 82 82 78 78 74 74 70 70 66 66 62 62 58 58 54 54 50 50 82 86 90 94 98 102 106 110 114 118 122 126 82 86 90 94 98 102 106 110 114 118 122 126 82 86 90 94 98 102 106 110 114 118 122 126 / 2 Rare-Earth Region Local fit of two parameters: strengths of spin-isospin force and proton-neutron isoscalar pairing force, with several functionals. Adjusting spin-isospin force 25 ) s 3 C (SkO′) = 128.0 MeV fm –1 10 s 3 20 C10(SkO′-Nd) = 102.0 MeV fm 15 10 150Nd 5 Gamow-Teller strength (MeV 0 0 5 10 15 20 Excitation energy (MeV) Adjustment: proton-neutron pairing Extremely important for beta-decay half-lives, but not active in HFB (no proton-neutron mixing) ⟶ free parameter. Adjust to approximately reproduce experimental lifetimes as close as possible to the region of interest. 82 82 78 78 74 74 70 70 66 66 62 62 58 58 54 54 50 50 82 86 90 94 98 102 106 110 114 118 122 126 82 86 90 94 98 102 106 110 114 118 122 126 82 86 90 94 98 102 106 110 114 118 122 126 / 2 Rare-Earth Region Local fit of two parameters: strengths of spin-isospin force and proton-neutron isoscalar pairing force, with several functionals. Adjusting spin-isospin force 25 ) s 3 C (SkO′) = 128.0 MeV fm –1 10 s 3 20 C10(SkO′-Nd) = 102.0 MeV fm 15 10 150Nd 5 Gamow-Teller strength (MeV 0 0 5 10 15 20 Excitation energy (MeV) Adjusting isoscalar pairing Odd Nuclei J = 0, degenerate ground state 6 Treat degeneracy as ensemble of state and angular-momentum- flipped partner (equal filling approximation). 10 EFA Even-even ) –1 1 10–1 GT strength (MeV 10–2 0 2 4 6 8 10 QRPA energy (MeV) 10 Borzov et al. (1995) ) –1 1 10–1 10–2 GT strength (MeV 10–3 0 2 4 6 8 10 Excitation energy (MeV) How Do We Do? Open symbols mean “used in fit” 104 SkO′ SkO′-Nd SV-min (expt) 2 2 10 / 1 T 1 10–2 (calc) / 2 / –4 1 10 T 1 102 104 1 102 104 1 102 104 Experimental half-life (s) 4 10 SLy5 unedf1-hfb (expt) 2 2 / 10 1 T 1 –2 (calc) / 10 2 / 1 10–4 T 1 102 104 1 102 104 Experimental half-life (s) Predicted Half Lives 103 Z = 58 Z = 60 102 SV-min 10 SLy5 1 SkO’ SkO’-Nd –1 10 UNEDF1-HFB 10–2 103 Z = 58 Z = 60 2 Half-life (s) 10 + global fit 10 Fang ∗ 1 x Möller Expt.