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 June 3, 26

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 28 2 2 8 2 8 Limits of nuclear existence 5 rp-process 28 2 A= A=2 5 neutrons A~6 82 82 26 Density Functional Theory Selfconsistent Mean Field r-process β decay is particularly important. It is responsible for increasing Z throughout the r process, and its competition with neutron capture during freeze-out can have a large effect on final abundances.

Sensitivity Studies Looked first at rare-earth peak formation and weak-r process peak formation.

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). These are differences between the masses that everyone has been talking about. matrix elements of the GT operator στ and forbidden operators rτ, rστ between the two states. The operator τ turns a neutron into a proton; the GT operator does that while flipping nucleon spin. Most of the time the GT operator leaves you above of is above threshold, by the way.

What s Usually Used for β Decay in Simulations 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 23. Shortens half lives. T β,calc /T β,exp 4 3 2 2 3 4 3 2 2 3 Möller, Pfeiffer, Kratz (23) β decay (Theory: GT) Total Error = 3.73 for 84 nuclei, T β,exp < s Total Error = 2.6 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 = 4.82 for 546 nuclei, T β,exp < s 3 2 2 3 Experimental β-decay Half-life T β,exp (s)

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 2 ) + 2 t ( + x P σ ) [ ( 2 ) 2 δ(r r 2 )+h.c. ] + t 2 ( + x 2 P σ )( 2 ) δ(r r 2 )( 2 ) + 6 t 3 ( + x 3 P σ ) δ(r r 2 )ρ α ([r + r 2 ]/2) + iw (σ + σ 2 ) ( 2 ) δ(r r 2 )( 2 ) where P σ +σ σ 2 2. 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 ρ2 tt 3 + C ρ t ρ tt3 2 ρ tt3 + C τ t ρ tt 3 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 J2 tt 3 { C s t s2 tt 3 + C s t s tt3 2 s tt3 + C T t s tt 3 T tt3 + C j t j2 tt 3 } + C j t s tt3 j tt3 + C F t s tt 3 F tt3 + C s t ( s tt3 ) 2 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-2: normal density and pairing density HFB, 2-D lattice, SLy4 + volume pairing Ref: Artur Blazkiewicz, Vanderbilt, Ph.D. thesis (25) β β

QRPA QRPA done properly is time-dependent HFB with small harmonic perturbation. Perturbing operator is β-decay transition operator. Matrix elements of transition operator between the initial state and final excited states obtained from response of nucleus to perturbation. Schematic QRPA of Möller et al. is very simplified. No fully self-consistent mean-field calculation to start. Nucleon-nucleon interaction (schematic) is used only in QRPA part of calculation.

Initial Skyrme Application: Spherical QRPA Example: late-9 s calculation, 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 T = pairing (it s zero in schematic QRPA.) Adjusted in each of the three peak regions to reproduce measured lifetimes. calc expt T /2 / T/2 B(GT) 25 2 5 5 2 - -2 9 Zr SkO SLy5 SkP Expt 2 4 6 8 2 4 6 8 2 excitation energy [MeV] 8 Zn 78 Zn 76 Zn 82 Ge 5 5 2 25 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 28 Cd K =, ± 6 5 4 3 2 7 46 Ba K = 6 5 4 3 2 7 46 Ba K = ± 6 5 4 3 2 GT strength [MeV ] GT strength [MeV ] GT strength [MeV ] 2 28 Pb K =, ± 9 6 3 7 66 Gd K = 6 5 4 3 2 7 66 Gd K = ± 6 5 4 3 2 Beta-decay rates obtained by integrating strength with phase-space weighting function in contour around excited states below threshold. 2 RPA energy [MeV] Re [S(ω)f(ω)] Im ω C Re ω 2 RPA energy [MeV] Im [S(ω)f(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 s2 + C s s 2 s + C T s T + C j j2 + C j s j + C F s F + C s ( s ) 2 + V isos. pair.

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. 25 Gamow-Teller strength (MeV ) Adjusting spin-isospin force odd =Cs s2 + C s s 2 s + C T s T + C j C j2 s (SkO )=28. MeVfm 3 2 5 5 C s (SkO -Nd) = 2. MeVfm 3 + C j s j + C F s F + C s ( s ) 2 + V isos. pair. 5 5 2 Excitation energy (MeV) 82 78 74 7 66 62 58 54 Adjusting proton-neutron spin- pairing 5 82 86 9 94 98 2 6 4 8 22 26 / 2

Odd Nuclei Have J Treat degeneracy as ensemble of state and angular-momentum-flipped partner (Equal Filling Approximation). GT strength (MeV ) EFA Even-even 2 2 4 6 8 QRPA energy (MeV)

How Do We Do? Open symbols mean used in fit T / 2 (calc) / T / 2 (expt) 4 SkO SkO -Nd SV-min 2 2 4 2 4 2 4 2 4 Experimental half-life (s) T / 2 (calc) / T / 2 (expt) 4 SLy5 unedf-hfb 2 2 4 2 4 2 4 Experimental half-life (s)

Predicted Half Lives Half-life (s) Z =58 Z =6 2 2 3 Z =58 Z =6 2 SV-min SLy5 SkO SkO -Nd UNEDF-HFB + global fit Fang x Möller Expt. 2 9 98 6 4 Neutron number 9 98 6 4

What s the Effect?

Experimental half-life (s) A 8 Region Two coupling constants fit again in same way. Even Nuclei Odd Nuclei 4 This work 2 Möller has clear bias here in even-even nuclei. T / 2 (calc) / T / 2 (expt) 2 4 4 Möller et al. (23) 2 2 4 2 2 4 6 2 2 4 6

A 8 Region Predictions Predictions with SV-min: Z =24 Z =25 Z =26 Z =27 Z =29 Half-life (s) 2 3 44 46 48 46 48 5 46 48 5 49 5 5 52 Z =3 Z =3 Z =32 Z =33 Z =34 Half-life (s) 2 3 54 56 55 57 54 56 58 6 56 58 6 62 Neutron number 58 6 62 64 SV-min x Möller

r-process in A 8 Region Us Möller Marketin et al. REACLIB This figure shows results of using new rates only for individual isotope chains, but is not terribly enlightening.

Fast QRPA Now Allows Global Skyrme Fit Fit to 7 GT resonance energies, 2 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 2 points 6 4 2-2 - 2 Q comp.-q exp. [MeV].54.576 MeV Qvalues not given perfectly, and β rate roughly Q 5. - -2 2 4 6 8 2 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. 4 3 2 - -2-3 -4 2 4 6 8 2 4 Q comp. uncertainty SkO' bias

Fitting the Full Time-Odd Skyrme Functional Charge-Changing Part, That is Hodd c.c. =Cs s2 + C s s 2 s + C T s T + C j j2 + C j s j + C F s F + C s ( s ) 2 + V isos. pair. Initial two-parameter fit

Fitting the Full Time-Odd Skyrme Functional Charge-Changing Part, That is Hodd c.c. =Cs s2 + C s s 2 s + C T s T + C j j2 + C j s j + C F s F + C s ( s ) 2 + V isos. pair. Initial two-parameter fit More comprehensive fit

Tried Lots of Things GT resonances SD resonances half-lives 8 2 6 4 E computed - E exp. 4 2-2 -4-6 -8 48 Ca 76 Ge 9 Zr 2 Sn 3 Te 5 Nd 28 Pb E computed - E exp. 2-2 -4 9 Zr 28 Pb T comp. / T exp. - -2 E D 3A 3B 4 5 58 Ti 78 Zn 98 Kr 26 Cd 52 Ce 66 Gd 226 Rn All SkO E = Experimental Q values, 2 parameters D = Computed Q values, 2 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 Four-parameter fit 4 3 2 uncertainty SkO' refit, exp. Q values bias t comp. /t exp. - -2-3 -4 2 4 6 8 2 4 Q comp. [MeV] Not significantly better than restricted two-parameter fit.

Summary of Fitting So Far t /2 s t /2 s t /2 s t /2 s 2 2 2 2.5.5.5.5 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 -.5-2 -2-2 -2 t /2.5 s t /2.2 s t /2. s 2 2 2.5.5.5 log (t comp./texp.).5 -.5 - -.5-2 log (t comp./texp.).5 -.5 - -.5-2 log (t comp./texp.).5 -.5 - -.5-2 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 2 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?

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 ( E + δ) 5 ( E) 5 = + 4 δ E +... 28Pb Gamow-Teller Strength (SkO' Force) 5 Spherical Result Modified Deformed Result 4 Strength 3 2 4 6 8 2 4 6 8 2 22 24 26 28 3 QRPA Energy/MeV Can more be measured?

Role of Forbidden Decay 4 Forbidden contribution to beta-decay rate proton number Z 2 8 6 4 2 8 6 4 2 forbidden contribution [%] 5 5 2 25 neutron number N Message: the neutron-rich nuclei invariably have significant contributions from both allowed and forbidden operators.

Possible Measurement of Allowed vs. Forbidden? This would really help as well, since short-lifetime nuclei nearly always have significant contributions to both.

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 2 fm 4 ) 6 5 4 3 2 T= RPA SRPA 2 3 4 5 Aaagghhhh! RPA gets location of resonances about right (spreading width inserted by hand). Second RPA lowers them by several MeV. But problem turns out to be due to insconsistency with DFT

Higher RPA and Density Functional Theory Hohenberg-Kohn theorem: E[ρ] is universal, i.e. in the presence of an additional local operator λq(r), E[ρ] becomes E[ρ] E λ [ρ] =E[ρ]+λ dr Q(r)ρ(r)

Higher RPA and Density Functional Theory Hohenberg-Kohn theorem: E[ρ] is universal, i.e. in the presence of an additional local operator λq(r), E[ρ] becomes E[ρ] E λ [ρ] =E[ρ]+λ dr Q(r)ρ(r) RPA-like equation If λ small, density changes: for exact response ρ λ (r) =ρ (r)+λ dr R(ω =, r, r )Q(r ) = R(ω = ) =R RPA KS (ω = ) Runge-Gross (TDDFT) theorem implies R(ω,r, r )=R KS(ω, r, r )+ dr dr 2 R KS(ω, r, r )V(ω, r, r 2 )R(ω, r 2, r ), V(ω, r, r ) is frequency-dependent effective interaction given by higher RPA. But R(ω = ) =R RPA KS so V(ω =, r, r 2 )=V RPA δ 2 E[ρ] (r, r 2 )= δρ(r )δρ(r 2 ) ρ

Subtraction Results 6 O.4 (a).2 RPA Fraction E EWSR/MeV.4.2.4 (b) (c) DFTcorrected SSRPA_F Second RPA Exp.2 5 5 2 25 3 35 4 E (MeV) Gambacurta, Grasso, JE

Finally The End Thanks for your kind attention.