J. Engel with M. Mustonen, T. Shafer C. Fröhlich, G. McLaughlin, M. Mumpower, R. Surman D. Gambacurta, M. Grasso January 18, 2017
To convincingly locate the site(s) of the r process, we need to know reaction rates and properties in very neutron-rich nuclei.
To convincingly locate the site(s) of the r process, we need to know reaction rates and properties in very neutron-rich nuclei. 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.
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 ΔE (since contribution to rate ∞ ΔE 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. 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 ΔE (since contribution to rate ∞ ΔE 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. So nuclear structure model must do good job with masses, spectra, and wave functions, in many isotopes.
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 2003. Shortens half lives. Möller, Pfeiffer, Kratz (2003) Modern Alternative: Skyrme Dens.-Func. Theory Started as zero-range effective potential, treated in mean-field theory:
Hodd has no effect in mean-field description of time-reversal even states (e.g. ground states), but large effect in β decay.
Not including pairing:
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.
Gives you ground state density, etc. This is where Skyrme functionals have made their living. 2/26/10 Volues Oberacoer, Vanderb h 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.
In nuclei near “‘waiting points,” with no forbidden decay. Chose functional corresponding to Skyrme interaction SkO’ because did best with GT distributions.
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-1 pairing (it’s zero in schematic QRPA.) Adjusted in each of the three peak regions to reproduce measured lifetimes. New: Fast Skyrme QRPA in Deformed Nuclei Finite-Amplitude Method Strength functions computed directly, in orders of magnitude less time than with matrix QRPA. New: Fast Skyrme QRPA in Deformed Nuclei Finite-Amplitude Method Strength functions computed directly, in orders of magnitude less time than with matrix QRPA. Beta-decay rates obtained by integrating strength with phase-space weighting function in contour around excited states below threshold. Rare-Earth Region Local fit of two parameters: strengths of spin-isospin force and proton-neutron spin-1 pairing force, with several functionals.
Local fit of two parameters: strengths of spin-isospin force and proton-neutron spin-1 pairing force, with several functionals. Adjusting spin-isospin force Adjusting proton-neutron spin-1 pairing
Have / ≠ 0 Treat degeneracy as ensemble of state and angular-momentum-flipped partner (Equal Filling Approximation).
� SV-min □ SLy5 ● SkO' ▼ SkO’-Nd △ UNEDF1-HFB + global fit * Fang x Möller ● Expt.
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.
Fitting the Full Time-Odd Skyrme Functional Charge-Changing Part, That is ….
▶ Initial two-parameter fit Fitting the Full Time-Odd Skyrme Functional Charge-Changing Part, That is ….
▶ Initial two-parameter fit ▶ More comprehensive fit Fitting the Full Time-Odd Skyrme Functional Charge-Changing Part, That is ….
▶ Initial two-parameter fit ▶ More comprehensive fit ▶ Additional adjustment
All SkO' 1E = Experimental O values, 2 parameters 3A = Computed Q values, 4 parameters 4   = Start with 3A, 3 more parameters 1D = Computed Q values, 2 parameters 3A = Experimental Q values, 4 parameters 5   = Computed Q values, three more parameters
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?
(3) t 1/2ミ1000 s (b) t 1/2 ≤ 100 s 1A 1B 1C 1E 2 3A 3B 4 5  Mö  Ho  Na  Co  Ma
(c) t 1/2ミ10 s 1A 1B 1C 1E 2 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.
These are the most important for the r process. . . And they are easier to predict: ( ΔE + δ) 5 (ΔΕ)5
Significance of Factor-of-Two Uncertainty
Significance of Factor-of-Two Uncertainty
Add two-phonon states to RPA’s one-phonon states; should describe spreading width of resonances and low-lying strength much better. But… Argghh! 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…
160
160 How can we do these calculations quickly?  Usual procedure involves inversion of huge matrix.  FAM hard to generalize in convenient way. Finally… Thanks for your kind attention.