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

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

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

Ì Ì

¬ ; ¬ ;

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] 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 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] 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] 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 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. 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

60 Protons 55

50 70 cold inc cold dec

60 Protons 55

50 70 nsm inc nsm dec

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

) s 3 C (SkO′) = 128.0 MeV fm

–1 10 s 3 20 C10(SkO′-Nd) = 102.0 MeV fm

10 150Nd 5

Gamow-Teller strength (MeV 0 0 5 10 15 20 Excitation energy (MeV) 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

) s 3 C (SkO′) = 128.0 MeV fm

–1 10 s 3 20 C10(SkO′-Nd)Adjustment: proton-neutron pairing = 102.0 MeV fm

15 Extremely important for beta-decay half-lives, but not active in HFB (no 10 150Nd proton-neutron mixing) ⟶ free parameter. 5 Adjust to approximately reproduce experimental lifetimes as close as Gamow-Teller strength (MeV 0 0possible 5 10 to the region 15 of 20interest. Excitation energy (MeV) 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 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. 10–1

10–2

90 98 106 114 90 98 106 114 Neutron number What’s the Effect?

x 1.0e-04 0.35 0.30 0.25 0.20 Y(A) 0.15 0.10 0.05 hot

0.35 0.30 0.25 0.20 Y(A) 0.15 0.10 0.05 cold

0.35 0.30 FRDM 0.25 SkO'-Nd SkO' 0.20

Y(A) Sly5 0.15 SV-min UNEDF1 0.10 0.05 nsm 145 150 155 160 165 170 175 180 A 1. Initial two-parameter fit (spin-isospin, isoscalar pairing interactions) 2. Four-parameter fit (add terms with kinetic spin density, kinetic tensor density) 3. Adjust three additional terms with current density and divergence of spin density on top of four-parameter fit

Fast QRPA 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. Fast QRPA 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.

1. Initial two-parameter fit (spin-isospin, isoscalar pairing interactions) 2. Four-parameter fit (add terms with kinetic spin density, kinetic tensor density) 3. Adjust three additional terms with current density and divergence of spin density on top of four-parameter fit Tried Lots of Combinations of Data/Parameters...

GT resonances SD resonances half-lives

8 102

4 6

4 101 2 2 exp. exp. exp. - E - E 0 0 / T 100 comp. T computed computed

E -2 E 1E -2 1D -4 -1 10 3A 3B -6 -4 4 5 -8 10-2 48Ca 76Ge 90Zr 112Sn 130Te 150Nd 208Pb 90Zr 208Pb 58Ti 78Zn 98Kr 126Cd 152Ce 166Gd 226Rn All SkO’ 1E = Experimental Q values, 2 parameters 1D = 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 Initial Step: Two Parameters Again Accuracy of the computed Q values with SkO' 4 60 0.154 � 0.576 MeV 40

points 20 3 0 -2 -1 0 1 2 Q values not given perfectly, Qcomp.-Qexp. [MeV] 2 5

[MeV] and β rate roughly Q .

exp. 1 -Q ∝ comp. Q 0

-1

-2 0 2 4 6 8 10 12 14

Qexp. [MeV] 104 uncertainty 103 SkO' bias Adjusted only spin- 102 isospin interaction and 101 exp. /t isoscalar pairing. 100 comp. t -1 Uncertainty decreases 10 -2 with increasing Q. 10 10-3

10-4 0 2 4 6 8 10 12 14

Qcomp. More Parameters

Four-parameter fit

104 uncertainty 103 SkO' refit, exp. Q values bias 102

101 exp. /t 100 comp. t 10-1

10-2

10-3

10-4 0 2 4 6 8 10 12 14

Qcomp. [MeV]

Not significantly better than restricted two-parameter fit. Summary of Fitting

t1/2 ≤ 1000 s t1/2 ≤ 100 s t1/2 ≤ 10 s t1/2 ≤ 1 s 2 2 2 2

1.5 1.5 1.5 1.5

1 1 1 1 ) ) ) )

exp. 0.5 exp. 0.5 exp. 0.5 exp. 0.5 /t /t /t /t 0 0 0 0 comp. comp. comp. comp. -0.5 -0.5 -0.5 -0.5 log (t log (t log (t log (t -1 -1 -1 -1

-1.5 -1.5 -1.5 -1.5

-2 -2 -2 -2

t1/2 ≤ 0.5 s t1/2 ≤ 0.2 s t1/2 ≤ 0.1 s 2 2 2

1.5 1.5 1.5

1 1 1

) ) ) Möller et al.

exp. 0.5 exp. 0.5 exp. 0.5 SV-min baseline /t /t /t SkO' baseline (short-lived) 0 0 0 SkO' baseline (long-lived) comp. comp. comp. SkO' baseline (mixed) -0.5 -0.5 -0.5

log (t log (t log (t SkO' baseline (new nuclei) -1 -1 -1 SkO' refit, step 1 SkO' refit, step 2 -1.5 -1.5 -1.5 Refit with exp. Q values -2 -2 -2

Meh.... Not doing as well as we had hoped.

Is the QRPA near its limits? We think so. And they are easier to predict: (∆E + δ)5 δ RESULTS WITH= NO1 + 4 PAIRING+ ... (∆E)5 ∆E 208Pb Gamow-Teller Strength (SkO' Force) 60

Spherical Result 50 Modiﬁed Deformed Result

30 Strength

0 4 6 8 10 12 14 16 18 20 22 24 26 28 30 QRPA Energy/MeV

again, pretty much dead-on with more interesting resonance structure

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: (∆E + δ)5 δ RESULTS WITH= NO1 + 4 PAIRING+ ... (∆E)5 ∆E 208Pb Gamow-Teller Strength (SkO' Force) 60

Spherical Result 50 Modiﬁed Deformed Result

30 Strength

0 4 6 8 10 12 14 16 18 20 22 24 26 28 30 QRPA Energy/MeV

again, pretty much dead-on with more interesting resonance structure Role of Forbidden Decay

Forbidden contribution to beta-decay rate 140 100

120 80 100

80 60

60 40

proton number Z 40 20 20 forbidden contribution [%]

0 0 0 50 100 150 200 250 neutron number N

Message: the neutron-rich nuclei usually have significant contributions from both allowed and forbidden operators. Real uncertainty is larger, though.

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. TheTheThe End End End

Thanks for your kind attention.

Finally...

Need to go beyond QRPA. With density functionals some conceptual problems arise but they are solvable.

See, e.g., work of E. Litvinova, V. Tselyaev, D. Gambacurta, M. Grasso, etc. Finally...

Need to go beyond QRPA. With density functionals some conceptual problems arise but they are solvable.

See, e.g., work of E. Litvinova, V. Tselyaev, D. Gambacurta, M. Grasso, etc.

TheTheThe End End End

Thanks for your kind attention.