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

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

¬

Ò

X

 Ì Ì

¬ ; ¬ ;

Ò

i

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) (∇ − ∇ ) δ(r − r ) + h.c. 2 1 1 1 2 1 2 + t (1 + x Pσ)(∇ − ∇ ) δ(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 ) (∇ − ∇ ) δ(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 ∇ 3 t 3 3 = t =−t Xt 0 3X J J 2 + C∇ ρtt ∇ 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 · ∇ 3 t 3 · 3 t tt3 = t =−t Xt 0 3X j F s 2 + C∇ s ∇ j + C s F + C∇ (∇ 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 Closed shell nuclei are spherical.

0Ñω Shell Towards a unified Ab initio Model description of the nucleus few-body calculations No-Core Shell Model G-matrix

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] 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 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] 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. 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

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. Adjusting spin-isospin force

25

) 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 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.Adjusting proton-neutron spin-1 pairing Excitation energy (MeV)

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 spin-1 pairing force, with several functionals.

Hc.c. =Cs s2 + C∆s s 2s + CTs T + Cj j2 odd 1 11 1 11 · ∇ 11 1 11 · 11 1 11 F 2 + C∇j s ∇ j + C s F + C∇s (∇ s ) + V pn pair. 1 11 · × 11 1 11 · 11 1 · 11 0 × Rare-Earth Region Local fit of two parameters: strengths of spin-isospin force and proton-neutron spin-1 pairing force, with several functionals. Adjusting spin-isospin force 25c.c. s 2 ∆s 2 T j 2 ) s 3 H C =(SkOC′) =s 128.0+ MeVC fm s s + C s T + C j –1 odd 10 1 11 1 11 11 1 11 11 1 11 Cs (SkO -Nd) = 102.0 MeV fm3 · ∇ · 20 10 ′ Adjustment: proton-neutron pairingF 2 + C1∇j s11 ∇ j11 + C1 s11 F11 + C1∇s (∇ s11) + V0 pn pair. 15 · × · · × Extremely important for beta-decay half-lives, but not active in HFB (no 10 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.Adjusting proton-neutron spin-1 pairing Excitation energy (MeV)

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

Have J = 0 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 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' 4 60 0.154 � 0.576 MeV 40

points 20 3 0 Q values not given perfectly, -2 -1 0 1 2 5 Q -Q [MeV] 2 comp. exp. and β rate roughly Q . ∝ [MeV]

exp. 1 -Q comp. Q 0

-1

4 -2 10 0 2 4 6 8 10 12 14 uncertainty Qexp. [MeV] 103 SkO' bias Initially adjusted only 102 s 101

C , which moves GT res- exp.

1 /t 100

onance, and isoscalar comp. t pairing strength. 10-1 10-2

Uncertainty decreases 10-3 with increasing Q. 10-4 0 2 4 6 8 10 12 14

Qcomp. More comprehensive fit Additional adjustment

Fitting the Full Time-Odd Skyrme Functional Charge-Changing Part, That is ...

Hc.c. =Cs s2 + C∆s s 2s + CTs T + Cj j2 odd 1 11 1 11 · ∇ 11 1 11 · 11 1 11 j F s 2 + C∇ s ∇ j + C s F + C∇ (∇ s ) + V pn pair. 1 11 · × 11 1 11 · 11 1 · 11 0 ×

Initial two-parameter fit Additional adjustment

Fitting the Full Time-Odd Skyrme Functional Charge-Changing Part, That is ...

Hc.c. =Cs s2 + C∆s s 2s + CTs T + Cj j2 odd 1 11 1 11 · ∇ 11 1 11 · 11 1 11 j F s 2 + C∇ s ∇ j + C s F + C∇ (∇ s ) + V pn pair. 1 11 · × 11 1 11 · 11 1 · 11 0 ×

Initial two-parameter fit More comprehensive fit Fitting the Full Time-Odd Skyrme Functional Charge-Changing Part, That is ...

Hc.c. =Cs s2 + C∆s s 2s + CTs T + Cj j2 odd 1 11 1 11 · ∇ 11 1 11 · 11 1 11 j F s 2 + C∇ s ∇ j + C s F + C∇ (∇ s ) + V pn pair. 1 11 · × 11 1 11 · 11 1 · 11 0 ×

Initial two-parameter fit More comprehensive fit Additional adjustment Tried Lots of Things...

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 Results

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 So Far

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’d hoped. Is the reason the limited correlations in the QRPA? Or will better fitting and more data help? Comparison with Other Groups

(a) t1/2 ≤ 1000 s 1.5 1 )

exp. 0.5 /t 0 comp. -0.5 lg (t -1 -1.5 1A 1B 1C 1E 2 3A 3B 4 5 Mö Ho Na Co Ma

(b) t1/2 ≤ 100 s 1.5 1 )

exp. 0.5 /t 0 comp. -0.5 lg (t -1 -1.5 1A 1B 1C 1E 2 3A 3B 4 5 Mö Ho Na Co Ma

(c) t1/2 ≤ 10 s 1.5 1 )

exp. 0.5 /t 0 comp. -0.5 lg (t -1 -1.5 1A 1B 1C 1E 2 3A 3B 4 5 Mö Ho Na Co Ma

(d) t1/2 ≤ 1 s 1.5 1 )

exp. 0.5 /t 0 comp. -0.5 lg (t -1 -1.5 1A 1B 1C 1E 2 3A 3B 4 5 Mö Ho Na Co Ma (a) t1/2 ≤ 1000 s 1.5 1 )

exp. 0.5 /t 0 comp. -0.5 lg (t -1 -1.5 1A 1B 1C 1E 2 3A 3B 4 5 Mö Ho Na Co Ma

(b) t1/2 ≤ 100 s 1.5 1 )

exp. 0.5 /t 0 comp. -0.5 lg (t -1 -1.5 Comparison with Other1A 1B 1C 1E Groups2 3A 3B 4 5 Mö Ho Na Co Ma

(c) t1/2 ≤ 10 s 1.5 1 )

exp. 0.5 /t 0 comp. -0.5 lg (t -1 -1.5 1A 1B 1C 1E 2 3A 3B 4 5 Mö Ho Na Co Ma

(d) t1/2 ≤ 1 s 1.5 1 )

exp. 0.5 /t 0 comp. -0.5 lg (t -1 -1.5 1A 1B 1C 1E 2 3A 3B 4 5 Mö Ho Na Co Ma And they are easier to predict: RESULTS(∆E WITH+ δ)5 NO PAIRINGδ = 1 + 4 + ... (∆E)5 ∆E 208Pb Gamow-Teller Strength (SkO' Force) 60

Spherical Result 50 Modified Deformed Result

40

30 Strength

20

10

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

again, pretty much dead-on withCan more more interesting be resonance measured? 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: RESULTS(∆E WITH+ δ)5 NO PAIRINGδ = 1 + 4 + ... (∆E)5 ∆E 208Pb Gamow-Teller Strength (SkO' Force) 60

Spherical Result 50 Modified Deformed Result

40

30 Strength

20

10

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

again, pretty much dead-on withCan more more interesting be resonance measured? structure 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. 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... 0 6

) T=0 4 5 RPA

fm 4 SRPA 2 3 2

B(E0) (e 1 0 0 10 20 30 40 50 3

) Argghh! 4 2.5 T=1

RPA fm 2 gets location of resonances about right (spreading width 2 inserted1.5 by hand). Second RPA lowers them by several MeV. But problem1 turns out to be due to insconsistency with DFT...

B(E0) (e 0.5 0 0 10 20 30 40 50 Energy (MeV)

FIG. 3: (Color online) RPA, dashed (black) lines and SRPA, full (red) lines, for the isoscalar (upper panel) and isovector (lower panel) monopole strength distributions.

+ 2 200 T=0 ) 4 150 RPA

fm SRPA 2 100

50 B(E2) (e 0 0 10 20 30 40 50 50

) T=1 4 40 fm

2 30 20

B(E2) (e 10 0 0 10 20 30 40 50 Energy (Mev)

FIG. 4: (Color online) As in Fig. 3 but for the quadrupole case.

the transition amplitudes are easily calculated and they have the same expression both in RPA and SRPA, namely

0 [Q , F ] 0 HF [Q , F ] HF = h | ν | i ≈ h | ν | i

ν ν = X ∗ p F h + Y ∗ h F p . (11) ph h | | i ph h | | i Xph  

6 How can we do these calculations quickly? Usual procedure involves inversion of huge matrix. FAM hard to generalize in convenient way.

With Correction... 16O (a) 0.04

RPA 0.02

0 (b) 0.04 DFT-corrected SSRPA_F Second RPA 0.02

0

Fraction E0 EWSR/MeV (c) 0.04 Exp

0.02

0 5 10 15 20 25 30 35 40 E (MeV) Gambacurta, Grasso, JE With Correction... 16O (a) 0.04

RPA 0.02

0 (b) 0.04 How can we do these calculationsDFT-corrected quickly? Usual SSRPA_F procedure involves inversion of hugeSecond matrix. RPA FAM hard to generalize0.02 in convenient way.

0

Fraction E0 EWSR/MeV (c) 0.04 Exp

0.02

0 5 10 15 20 25 30 35 40 E (MeV) Gambacurta, Grasso, JE Finally...

TheTheThe End End End

Thanks for your kind attention.