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Calculating Beta Decay for the R Process

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Calculating Beta Decay for the R Process Calculating β Decay for the r Process J. Engel with M. Mustonen, T. Shafer C. Fröhlich, G. McLaughlin, M. Mumpower, R. Surman D. Gambacurta, M. Grasso January 18, 2017 R-Process Abundances β decay is particularly important. Responsible for increasing Z throughout r process, and its competition with neutron capture during freeze-out can have large effect on abundances. Nuclear Landscape 126 Limits of nuclear existence To convincingly locate the 82 site(s) of the r process, we r-process need to know reaction 50 rates and properties in protons 82 28 rp-process very neutron-rich nuclei. 20 50 8 28 neutrons Density Functional Theory 2 20 Selfconsistent Mean Field 2 8 A=10 A~60 A=12 0Ñω Shell Towards a unified Ab initio Model description of the nucleus few-body calculations No-Core Shell Model G-matrix Nuclear Landscape 126 Limits of nuclear existence To convincingly locate the 82 site(s) of the r process, we r-process need to know reaction 50 rates and properties in protons 82 28 rp-process very neutron-rich nuclei. 20 50 8 28 neutrons Density Functional Theory 2 20 Selfconsistent Mean Field 2 8 A=10 A~60 A=12 β decay is particularly important. Responsible0Ñω Shell for increasingTowards aZ unified Ab initio Model description of the nucleus throughout r process, and its competitionfew-body with neutron capture calculations No-Core Shell Model during freeze-out can have large effectG-matrix on abundances. So nuclear structure model must do good job with masses, spectra, and wave functions, in many isotopes. Calculating β Decay is Hard Though, As We’ll See, It Gets a Bit Easier in Neutron-Rich Nuclei To calculate β decay between two states, you need: an accurate value for the decay energy ∆E (since contribution to rate ∆E5 for “allowed” decay). / matrix elements of the decay operator στ- and “forbidden" operators rτ-; rστ- between the two states. The operator τ- turns a neutron into a proton; the allowed decay operator does that while flipping spin. Most of the time the decay operator leaves you above threshold, by the way. Calculating β Decay is Hard Though, As We’ll See, It Gets a Bit Easier in Neutron-Rich Nuclei To calculate β decay between two states, you need: an accurate value for the decay energy ∆E (since contribution to rate ∆E5 for “allowed” decay). / matrix elements of the decay operator στ- and “forbidden" operators rτ-; rστ- between the two states. The operator τ- turns a neutron into a proton; the allowed decay operator does that while flipping spin. Most of the time the decay operator leaves you above threshold, by the way. So nuclear structure model must do good job with masses, spectra, and wave functions, in many isotopes. What’s Usually Been 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 β− Ancient but Still in Some Ways 103 decay (Theory: GT) Unsurpassed Technology 102 101 Masses through “finite-range 100 − droplet model with shell 10 1 − 2 10 Total Error = 3.73 for 184 nuclei, Tβ,exp < 1 s ,exp corrections.” β Total Error = 21.16 for 546 nuclei (13 clipped), Tβ < 1000 s − 3 ,exp T / 10 4 10 “QRPA” with simple ,calc − β 3 β decay (Theory: GT + ff) space-independent interaction. T 10 102 First forbidden decay 101 (correction due to finite nuclear 100 − size) added very crude way in 10 1 − 2 2003. Shortens 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 Dens.-Func. Theory Started as zero-range effective potential, treated in mean-field theory: VSkyrme = t0 (1 + x0Pσ) δ(r1 - r2) 1 h 2 i + t (1 + x Pσ) (r - r ) δ(r - r ) + h:c: 2 1 1 1 2 1 2 + t (1 + x Pσ)(r - r ) δ(r - r )(r - r ) 2 2 1 2 · 1 2 1 2 1 α + t (1 + x Pσ) δ(r - r )ρ ([r + r ]=2) 6 3 3 1 2 1 2 + iW (σ + σ ) (r - r ) δ(r - r )(r - r ) 0 1 2 · 1 2 × 1 2 1 2 1+σ1 σ2 where Pσ · . ≡ 2 Re-framed as density functional, which can then be extended: 3 E = d r Heven + Hodd +Hkin. + Hem Z HSkyrme Hodd has no effect in mean-field| {z description} of time-reversal even states (e.g. ground states), but large effect in β decay. Time-Even and Time-Odd Parts of Functional Not including pairing: 1 t ρ 2 ∆ρ 2 τ Heven = C ρ + C ρtt ρtt + C ρtt τtt t tt3 t 3 r 3 t 3 3 = t =-t Xt 0 3X J J 2 + Cr ρtt r Jtt + C J t 3 · 3 t tt3 1 t s 2 ∆s 2 T j 2 H = C s + C stt stt + C stt Ttt + C j odd t tt3 t 3 · r 3 t 3 · 3 t tt3 = t =-t Xt 0 3X j F s 2 + Cr s r j + C s F + Cr (r s ) t tt3 · × tt3 t tt3 · tt3 t · tt3 Time-even densities: ρ = usual density τ = kinetic density J = spin-orbit current Time-odd densities: s = spin current T = kinetic spin current j = usual current Couplings are connected by “Skyrme interaction,” but can be set independently if working directly with density-functional. Starting Point: Mean-Field-Like Calculation (HFB) Gives you ground state density, etc. This is where Skyrme functionals have made their living. Zr-102: normal density and pairing density HFB, 2-D lattice, SLy4 + volume pairing Ref: Artur Blazkiewicz, Vanderbilt, Ph.D. thesis (2005) G3H G3H G=HD !βT bREVU! #@3D !βT bREVTGWH!C! PEE!G91%!#6!)EC!&A5E!#8E!U!GTRRXH! TFTXFSR! 2)(#4! #4!(#4C!1"#4 ')6! `! QRPA QRPA done properly is time-dependent HFB with small harmonic perturbation. Perturbing operator is β-decay transition operator. Decay matrix elements obtained from response of nucleus to perturbation. Schematic QRPA of Möller et al. is very simplified version of this. No fully self-consistent mean-field calculation to start. Nucleon-nucleon interaction is schematic. Initial Skyrme Application: Spherical QRPA 126 Limits of nuclear existence 82 r-process 50 protons 82 28 rp-process 20 50 8 28 neutrons Density Functional Theory 2 20 Selfconsistent Mean Field 2 8 A=10 A~60 A=12 0Ñω Shell Towards a unified Ab initio Model description of the nucleus few-body calculations No-Core Shell Model G-matrix Closed shell nuclei are spherical. One free parameter: strength of proton-neutron spin-1 pairing (it’s zero in 102 schematic QRPA.) Adjusted in each of the 101 t 2 p / x 1 three peak regions to reproduce measured e T / c 0 2 l / a 10 1 lifetimes. 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 In nuclei near “waiting points,” with no Zr forbidden decay. 20 ) 15 T G ( Chose functional corresponding to B 10 Skyrme interaction SkO0 because did best 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 In nuclei near “waiting points,” with no Zr forbidden decay. 20 ) 15 T G ( Chose functional corresponding to B 10 Skyrme interaction SkO0 because did best 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 proton-neutron spin-1 pairing (it’s 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 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.
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