Calculating β 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 8, 7

R-Process Abundances

Nuclear Landscape To convincingly locate the site(s) of the r process, we need to know reaction rates and properties in very neutron-rich nuclei. protons 8 8 8 Limits of nuclear existence 5 rp-process 8 A= A= 5 neutrons A~6 8 8 6 Density Functional Theory Selfconsistent Mean Field r-process β decay is particularly important. Responsible for increasing Z Ñω Shell Ab initio Model throughout r process, and its competition few-body with neutron capture calculations No-Core Shell Model during freeze-out can have large effect G-matrix on abundances. Towards a unified description of the nucleus

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 E 5 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.

What s Usually Been Used for β-decay in Simulations Èº ÅĐÓÐÐ Ö º È «Ö Ãº¹Äº ÃÖ ØÞ»ËÔ Ò ÙÔ Ø Ð Ð Ö¹ÔÖÓ º º º ½½ Ancient but Still in Some Ways Unsurpassed Technology Masses through finite-range droplet model with shell corrections. QRPA with simple space-independent interaction. First forbidden decay (correction due to finite nuclear size) added very crude way in 3. Shortens half lives. T β,calc /T β,exp 4 3 3 4 3 3 Möller, Pfeiffer, Kratz (3) β decay (Theory: GT) Total Error = 3.73 for 84 nuclei, T β,exp < s Total Error =.6 for 546 nuclei (3 clipped), T β,exp < s β decay (Theory: GT + ff) Total Error = 3.8 for 84 nuclei, T β,exp < s Total Error = 4.8 for 546 nuclei, T β,exp < s 3 3 Experimental β-decay Half-life T β,exp (s) ÙÖ Ê Ø Ó Ó ÐÙÐ Ø ØÓ ÜÔ Ö Ñ ÒØ Ð ¹ Ý Ð ¹Ð Ú ÓÖ ÒÙÐ ÖÓÑ ½ Ç ØÓ Ø Ú Ø ÒÓÛÒ

Modern Alternative: Skyrme Dens.-Func. Theory Started as zero-range effective potential, treated in mean-field theory: V Skyrme = t ( + x P σ ) δ(r r ) + [ ] t ( + x P σ ) ( ) δ(r r ) + h.c. + t ( + x P σ ) ( ) δ(r r )( ) + 6 t 3 ( + x 3 P σ ) δ(r r )ρ α ([r + r ]/) + iw (σ + σ ) ( ) δ(r r )( ) where P σ +σ σ. Re-framed as density functional, which can then be extended: ( ) E = d 3 r H even + H }{{ odd +H } kin. + H em H Skyrme H odd has no effect in mean-field 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: t { H even = C ρ t ρ tt 3 + C ρ t ρ tt3 ρ tt3 + Ct τ ρ tt3 τ tt3 H odd = t= t 3 = t t= t 3 = t t Time-even densities: } + C J t ρ tt3 J tt3 + C J t J tt 3 { Ct s s tt 3 + Ct s s tt3 s tt3 + Ct T s tt3 T tt3 + C j t j tt 3 } + C j t s tt3 j tt3 + Ct F s tt3 F tt3 + Ct s ( s tt3 ) 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-: normal density and pairing density HFB, -D lattice, SLy4 + volume pairing Ref: Artur Blazkiewicz, Vanderbilt, Ph.D. thesis (5) β β

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 protons Limits of nuclear existence 8 8 8 Ab initio few-body 5 rp-process 8 A= A= 5 neutrons A~6 Ñω Shell Model 8 8 6 Density Functional Theory Selfconsistent Mean Field Closed shell nuclei are spherical. r-process Towards a unified description of the nucleus

Initial Skyrme Application: Spherical QRPA In nuclei near waiting points, with no forbidden decay. 5 9 Zr Chose functional corresponding to Skyrme interaction SkO because did best with GT distributions. B(GT) 5 5 SkO SLy5 SkP Expt 4 6 8 4 6 8 excitation energy [MeV]

Initial Skyrme Application: Spherical QRPA In nuclei near waiting points, with no forbidden decay. Chose functional corresponding to Skyrme interaction SkO because did best with GT distributions. One free parameter: strength of proton-neutron spin- pairing (it s zero in schematic QRPA.) Adjusted in each of the three peak regions to reproduce measured lifetimes. calc expt T / / T/ B(GT) 5 5 5 9 Zr SkO SLy5 SkP Expt 4 6 8 4 6 8 excitation energy [MeV] - - 8 Zn 78 Zn 76 Zn 8 Ge 5 5 5 3 35 V [MeV]

New: Fast Skyrme QRPA in Deformed Nuclei Finite-Amplitude Method Strength functions computed directly, in orders of magnitude less time than with matrix QRPA. GT strength [MeV ] GT strength [MeV ] GT strength [MeV ] 7 8 Cd K =, ± 6 5 4 3 7 46 Ba K = 6 5 4 3 7 46 Ba K = ± 6 5 4 3 GT strength [MeV ] GT strength [MeV ] GT strength [MeV ] 8 Pb K =, ± 9 6 3 7 66 Gd K = 6 5 4 3 7 66 Gd K = ± 6 5 4 3 RPA energy [MeV] Re [S(ω)f(ω)] RPA energy [MeV] Im [S(ω)f(ω)] Beta-decay rates obtained by integrating strength with phase-space weighting function in contour around excited states below threshold. Im ω C Re ω Im ω C Re ω

Rare-Earth Region Local fit of two parameters: strengths of spin-isospin force and proton-neutron spin- pairing force, with several functionals. H c.c. odd =Cs s + C s s s + C T s T + C j j + C j s j + C F s F + C s ( s ) + V pn pair.

Rare-Earth Region Local fit of two parameters: strengths of spin-isospin force and proton-neutron spin- pairing force, with several functionals. Adjusting spin-isospin force Gamow-Teller strength (MeV ) 5 H c.c. odd =Cs s + C s s s + C T s T + C j C j s (SkO ) = 8. MeV fm 3 5 5 Adjustment: proton- neutron pairing C s (SkO -Nd) =. MeV fm 3 + C j s j + C F s F + C s ( s ) + V pn pair. 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 proton-neutron spin- pairing 5 5 Excitation energy (MeV) 8 78 74 7 66 6 58 54 5 8 86 9 94 98 6 4 8 6 /

Odd Nuclei Have J Treat degeneracy as ensemble of state and angular-momentum-flipped partner (Equal Filling Approximation). GT strength (MeV ) EFA Even-even 4 6 8 QRPA energy (MeV)

How Do We Do? Open symbols mean used in fit T / (calc) / T / (expt) 4 SkO SkO -Nd SV-min 4 4 4 4 Experimental half-life (s) T / (calc) / T / (expt) 4 SLy5 unedf-hfb 4 4 4 Experimental half-life (s)

Predicted Half Lives Half-life (s) Z = 58 Z = 6 3 Z = 58 Z = 6 SV-min SLy5 SkO SkO -Nd UNEDF-HFB + global fit Fang x Möller Expt. 9 98 6 4 Neutron number 9 98 6 4

What s the Effect? x.e-4.35.3.5..5..5 Y(A) Y(A).35.3.5..5..5 hot cold Y(A).35.3.5..5..5 nsm FRDM SkO'-Nd SkO' Sly5 SV-min UNEDF 45 5 55 6 65 7 75 8 A

Fast QRPA Now Allows Global Skyrme Fit Fit to 7 GT resonance energies, spin-dipole resonance energies, 7 β-decay rates in selected spherical and well-deformed nuclei from light to heavy.

Initial Step: Two Parameters Again Accuracy of the computed Q values with SkO' Q comp.-qexp. [MeV] 4 3 points 6 4 - - Q comp.-q exp. [MeV].54.576 MeV Q values not given perfectly, and β rate roughly Q 5. - - 4 4 6 8 4 Q exp. [MeV] Initially adjusted only C s, which moves GT resonance, and isoscalar pairing strength. Uncertainty decreases with increasing Q. t comp. /t exp. 3 - - -3-4 4 6 8 4 Q comp. uncertainty SkO' bias

Fitting the Full Time-Odd Skyrme Functional Charge-Changing Part, That is... Hodd c.c. =Cs s + C s s s + C T s T + C j j + C j s j + C F s F + C s ( s ) + V pn pair. Initial two-parameter fit

Fitting the Full Time-Odd Skyrme Functional Charge-Changing Part, That is... Hodd c.c. =Cs s + C s s s + C T s T + C j j + C j s j + C F s F + C s ( s ) + V pn pair. Initial two-parameter fit More comprehensive fit

Tried Lots of Things... GT resonances SD resonances half-lives 8 6 4 E computed - E exp. 4 - -4-6 -8 48 Ca 76 Ge 9 Zr Sn 3 Te 5 Nd 8 Pb E computed - E exp. - -4 9 Zr 8 Pb T comp. / T exp. - - E D 3A 3B 4 5 58 Ti 78 Zn 98 Kr 6 Cd 5 Ce 66 Gd 6 Rn All SkO E = Experimental Q values, parameters D = Computed Q values, parameters 3A = Computed Q values, 4 parameters 3A = Experimental Q values, 4 parameters 4 = Start with 3A, 3 more parameters 5 = Computed Q values, three more parameters

Results t comp. /t exp. 4 3 - - Four-parameter fit uncertainty SkO' refit, exp. Q values bias -3-4 4 6 8 4 Q comp. [MeV] Not significantly better than restricted two-parameter fit.

Summary of Fitting So Far t / s t / s t / s t / s.5.5.5.5 log (t comp./texp.).5 -.5 - log (t comp./texp.).5 -.5 - log (t comp./texp.).5 -.5 - log (t comp./texp.).5 -.5 - -.5 -.5 -.5 -.5 - - - - t /.5 s t /. s t /. s.5.5.5 log (t comp./texp.).5 -.5 - -.5 - log (t comp./texp.).5 -.5 - -.5 - log (t comp./texp.).5 -.5 - -.5 - Möller et al. SV-min baseline SkO' baseline (short-lived) SkO' baseline (long-lived) SkO' baseline (mixed) SkO' baseline (new nuclei) SkO' refit, step SkO' refit, step Refit with exp. Q values Meh... Not doing as well as we d hoped. Is the reason the limited correlations in the QRPA? Or will better fitting and more data help?

Comparison with Other Groups lg (t comp. /t exp. ) (a) t / s.5.5 -.5 - -.5 A B C E 3A 3B 4 5 Mö Ho Na Co Ma lg (t comp. /t exp. ).5.5 -.5 - -.5 (b) t / s A B C E 3A 3B 4 5 Mö Ho Na Co Ma (c) t / s

-.5 Comparison with Other A B C E Groups 3A 3B 4 5 Mö Ho Na Co Ma lg (t comp. /t exp. ) (c) t / s.5.5 -.5 - -.5 A B C E 3A 3B 4 5 Mö Ho Na Co Ma lg (t comp. /t exp. ).5.5 -.5 - -.5 (d) t / s A B C E 3A 3B 4 5 Mö Ho Na Co Ma

But We Really Care About High-Q/Fast Decays These are the most important for the r process.

But We Really Care About High-Q/Fast Decays These are the most important for the r process. And they are easier to predict: 6 RESULTS WITH NO PAIRING ( E + δ) 5 ( E) 5 = + 4 δ E +... 8Pb Gamow-Teller Strength (SkO' Force) 5 Spherical Result Modified Deformed Result 4 Strength 3 4 6 8 4 6 8 4 6 8 3 QRPA Energy/MeV again, pretty much dead-on with Can more more interesting beresonance measured? structure

What s at Stake Here? Significance of Factor-of-Two Uncertainty

What s at Stake Here? Significance of Factor-of-Two Uncertainty Real uncertainty is larger, though.

The Future: Second (Q)RPA Add two-phonon states to RPA s one-phonon states; should describe spreading width of resonances and low-lying strength much better. But... B(E) (e fm 4 ) 6 5 4 3 T= 3 4 5 3.5 T= Argghh! RPA gets location of resonances about right (spreading width inserted.5 by hand). Second RPA lowers them by several MeV. But problem turns out to be due to insconsistency with DFT... E) (e fm 4 ) + RPA SRPA

With Correction... 6 O.4 (a). RPA Fraction E EWSR/MeV.4..4 (b) (c) DFT-corrected SSRPA_F Second RPA Exp. 5 5 5 3 35 4 E (MeV) Gambacurta, Grasso, JE

With Correction... 6 O.4 (a). RPA Fraction E EWSR/MeV.4 (b) How can we do these calculations DFT-corrected quickly? Usual SSRPA_F procedure involves inversion of huge Secondmatrix. RPA FAM. hard to generalize in convenient way. (c).4 Exp. 5 5 5 3 35 4 E (MeV) Gambacurta, Grasso, JE

Finally... The End Thanks for your kind attention.