Role of fission in r-process nucleosynthesis100 2)Calculate stellar reaction rates from...

Role of fission in r-process nucleosynthesis

Samuel A. Giuliani

NSCL/FRIB, East Lansing

July 12th, 2018

FRIB and the GW170817 kilonova

NSCL/FRIB at MSU

Introduction Fission and r process Fission fragments distributions Conclusions & Outlook

Outline

1. Introduction

2. Impact of fission on r-process nucleosynthesis

3. Fission fragments distributions

4. Conclusions & Outlook


The r processr(apid neutron capture) process: τn � τβ−

β decayneutroncapture

neutron shell closureN

Z

unstablenucleistablenuclei

How far can the r process proceed? Number of free neutrons that seednuclei can capture (neutron-to-seed ratio).


r process and fission10

0

150

200

250

10−310

−210−110

0101

r−process waiting point (ETFSI−Q)

Known massKnown half−life

N=126

N=82

Solar

r ab

unda

nces

2830 32 34 36 38 40 42 44 46 48 50 52 54 56 58

6062

6466 68

7072

74 7678

8082

8486

88 90

92 9496

98100 102

104106

108110

112 114

116 118120

122124

126

128130 132

134 136138

140 142 144 146 148150

152154

156

158

160

162

164 166 168 170 172 174176

178180 182

184

186188 190

26

34

36

38

40

42

44

46

48

50

52

54

56

58

60

62

64

66

68

70

72

74

76

78

80

82

84

86

88

90

92

94

96

98

100

N=184

30

32

28

fission

For large neutron-to-seed ratiofission is unavoidable

- n-induced fission- β-delayed fission- spontaneous fission

I Where does fission occur?I How much material accumulates in fissioning region?I What are the fission yields?


1) Compute fission properties and binding energies using BCPM EDF.

120 140 160 180 200 220 240Neutron number

90

100

110

120

Proton

num

ber

Sn = 2 MeVSn = 0 MeV

-2 0 2 4 6 8 10 12 14

Bf−Sn (MeV)

2) Calculate stellar reaction rates from Hauser-Feshbach theory.

120 140 160 180 200 220 240Neutron number

90

100

110

120

Prot

on n

umbe

r

Dominating channel at nn =1028 cm−3

(n,γ)(n,fission)spont. fissionSn= 2 MeVSn= 0 MeV

3) Obtain r-process abundances using network calculations.

120 140 160 180 200 220 240Neutron number

90

100

110

120

Prot

on n

umbe

r

-25 -20 -15 -10 -5 0

log10(Y)


The fission process

0 10 20 30 40 50 60 70 80Q20 (b)

-2050

-2045

-2040

-2035

E HFB

(MeV

)

286Fl114

innerbarrier

fissionisomer

outer barrier

groundstate

E*

spontaneousfission

neutron-inducedbeta-delayed

photo-inducedfission

Potential Energy Surface

Energy evolution from the initialstate to the scission point.

SAG+ PRC90(2014); Sadhukhan+ PRC90(2014)

Collective inertias

Resistance of the nucleusagainst the deformation forces.

Baran+ PRC84 (2011)


The Hartree-Fock-Bogolyubov (HFB) formalismThe ground-state wavefunction is obtained by minimizing the total energy:

δE [|Ψ〉] = 0 ,

where |Ψ〉 is a quasiparticle (β) vacuum:

|Ψ〉 =∏µ

βµ|0〉 ⇒ βµ|Ψ〉 = 0 .

The energy landscape is constructed by constraining the deformation of thenucleus 〈Ψ(q)|Q̂|Ψ(q)〉 = q:

E [|Ψ(q)〉] = 〈Ψ(q)|Ĥ − λqQ̂|Ψ(q)〉 .

The energy density functionals (EDF) provide a phenomenological ansatz of theeffective nucleon-nucleon interaction:

- Barcelona-Catania-Paris-Madrid (BCPM);- Skyrme and Gogny interactions (UNEDF1, D1S);- relativistic EDF.


Outline

1. Introduction





Nuclear inputs from the BCPM EDFWe study the impact of fission in the r process by comparing BCPM withprevious calculations based on Thomas-Fermi (TF) barriers and Finite RangeDroplet Model (FRDM) masses.

0 2 4 6 8 10 12 14

Fission barrier (MeV)

120 140 160 180 200 220 240Neutron number

90

100

110

120

Prot

on n

umbe

r TF

90

100

110

120

Prot

on n

umbe

r BCPM


2

6

10

14

18

S2n (M

eV)

184

126

174

BCPM

90 100 110 120Proton number

2

6

10

14

18S2n

(MeV

)

184

126

174

FRDM

BCPM: Giuliani et al. (2018); TF: Myers and Świaţecky (1999); FRDM: Möller et al. (1995).


Compound reactionsReaction rates computed within the Hauser-Feshbach statistical model.

compoundnucleus

target

γ gammadecay

particleemission

fission

- Based on the Bohr independence hypothesis: the decay of the compoundnucleus is independent from its formation dynamics.

- BCPM nuclear inputs implemented in TALYS reaction code to computen-induced fission and n-capture rates.


Cross sections from BCPM

Energy (MeV)

101

102

103

104

σ(n

,fiss) (m

b)

235U(n,fis)ExperimentBCPM

Energy (MeV)

238U(n,fis)

Energy (MeV)

238Pu(n,fis)

10-2 10-1 100 101

Energy (MeV)

101

102

103

104

σ(n

,γ) (mb)

235U(n,g)

10-2 10-1 100 101

Energy (MeV)

238U(n,g)

10-2 10-1 100 101

Energy (MeV)

238Pu(n,g)


Stellar reaction rates - impact of collective inertias?

120 140 160 180 200 220 240Neutron number

90

100

110

120

SEMP-r

90

100

110

120

Proton

num

ber

GCM-r


90

100

110

120

ATDHFB-r

spont. fis.α-decay

(n,γ)(n,α)

(n,fis)(n,2n)

SAG, Mart́ınez-Pinedo and Robledo, Phys. Rev. C 97, 034323 (2018)


The dynamical ejecta in neutron mergersTrajectory from 3D relativistic simulations of 1.35 M�-1.35 M� NS mergers.

x [km]

y [km

]

30 20 10 0 10 20 3030

20

10

0

10

20

30

9

9.5

10

10.5

11

11.5

12

12.5

13

13.5

14

14.5

x [km]

y [km

]

30 20 10 0 10 20 3030

20

10

0

10

20

30

9

9.5

10

10.5

11

11.5

12

12.5

13

13.5

14

14.5

13.0056 ms 13.4824 ms

x [km]

y [km

]

13.8024 ms

50 0 5050

40

30

20

10

0

10

20

30

40

50

9

9.5

10

10.5

11

11.5

12

12.5

13

13.5

14

14.5

x [km]

y [km

]

15.167 ms

50 0 5050

40

30

20

10

0

10

20

30

40

50

9

9.5

10

10.5

11

11.5

12

12.5

13

13.5

14

14.5

Bauswein et al., ApJ 773, 78 (2013).

- Large amount of ejecta (0.001-0.01 M�).- Material extremely neutron rich (Rn/s & 600).- Role of weak interactions?


r-process abundances: BCPM vs FRDM+TF

I Trajectory: 3D relativistic simulations from1.35 M�-1.35 M� NS mergers [Bauswein+(2013)].

I BCPM Giuliani+(2017) vs TF+FRDM Panov+(2010).I We changed the rates of nuclei with Z ≥ 84.I Same β-decay rates [Möller et al. PRC67(2003)].

I BCPM barriers larger than TF:

- nuclei around A > 280 longer lifetimes ,- accumulation above 2nd peak.

I BCPM shell gap smaller than FRDM at N = 174:

- FRDM-TF peak at A ∼ 257,- impact on final abundances at A ∼ 110.

I Same 232Th/238U ratio: progenitors of actinideshave Z < 84 ⇒ can initial nuclei with Z ≥ 84survive to fission?

10-910-810-710-610-510-410-310-2

abundances at n/s=1

10-910-810-710-610-510-410-310-2 abundances at τ(n,γ) = τβ

100 150 200 250 300A

10-910-810-710-610-510-410-310-2 abundances at 1 Gyr

solarBCPMFRDM+ TF


r-process abundances: BCPM vs FRDM+TF

I Trajectory: 3D relativistic simulations from1.35 M�-1.35 M� NS mergers [Bauswein+(2013)].

I BCPM Giuliani+(2017) vs TF+FRDM Panov+(2010).I We changed the rates of nuclei with Z ≥ 84.I Same β-decay rates [Möller et al. PRC67(2003)].

I BCPM barriers larger than TF:- nuclei around A > 280 longer lifetimes ,- accumulation above 2nd peak.

I BCPM shell gap smaller than FRDM at N = 174:- FRDM-TF peak at A ∼ 257,- impact on final abundances at A ∼ 110.

I Same 232Th/238U ratio: progenitors of actinideshave Z < 84 ⇒ can initial nuclei with Z ≥ 84survive to fission?

10-910-810-710-610-510-410-310-2

abundances at n/s=1

10-910-810-710-610-510-410-310-2 abundances at τ(n,γ) = τβ

100 150 200 250 300A

10-910-810-710-610-510-410-310-2 abundances at 1 Gyr

solarBCPMFRDM+ TF


Averaged fission rates

10−1

100

101

102

103

104

Time (s)

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Rat

e (s−1

)

BCPMFRDM+TFn-induced fissionspontan. fissionβ-delayed fission

10−6

10−4

10−2

100

102

neut

ron-

to-s

eed

ratio

n/s

I n-induced dominates until freeze-out and revived by β-delayed neutrons⇒ β-delayed fission rates from BCPM barriers required!

I decay of material to stability triggers spontaneous fission.


Emitted radioactive energyEnergy emitted by radioactive products in NSM crucial for predicting kilonovalight curves [J. Barnes et al., ApJ 829 110 (2016)].

10-5 10-4 10-3 10-2 10-1 100 101 102

Days

10-2

10-1

100

Frac

tion

of ra

dioa

ctiv

e en

ergy

BCPMFRDM+TFβ-decayα-decaytotal fission

I Minor impact in the radioactive energy production ⇒ progenitors of actinidesfrom Z < 84 [Mendoza-Temis et al., Phys. Rev. C92, 055805 (2015)].

I Fission subdominant → impact of multi-chance bdf [Mumpower et al., arXiv:1802.04398]?


Outline

1. Introduction





Impact of fission yields on r process

Figure 1. Final abundances of the integrated ejecta around the second and third peak for an NSM Korobkin et al. 2012; Rosswog et al. 2013 at a simulation time

10 s, employing the FRDM mass model combined with four different ssion fragment distribution models see the text . For reasons of clarity the results arepresented in two graphs. The abundances for Th and U are indicated by crosses. In the left-hand panel the lower crosses belong to the Panov et al. 2008 modeldashed line , while the lower crosses in the right-hand panel belong to the ABLA07 distribution model dashed line . The dots represent the solar -processabundance pattern Sneden et al. 2008

Figure 2. Fission rates at 1 s in s for -delayed and neutron-induced ssion at freeze-out from equilibrium for one representative trajectorywhen utilizing the FRDM mass model and Panov et al. 2010 ssion rates. : Corresponding ssion fragment production. The distribution model here is ABLA07.

The Astrophysical Journal, 808:30 13pp , 2015 July 20 Eichler et al.

M. Eichler et al., Astrophys. J. 808, 30 (2015).

• Final abundances strongly affected by fragments distributions[see also B. Côté et al., Astrophys. J. 855, 99 (2018)].

• Most of the models are parametrizations/phenomenological → validity farfrom stability?

• This talk: compute fission yields (FY) using DFT+Langevin.


The fission process

J. Sadhukan et al. Phys. Rev. C 93, 011304(R) (2016)


The stochastic Langevin framework

Path from outer turning point to scission given by dissipative Langevin:

dpidt = −

pjpk2

∂

∂xi(M−1)jk −

∂V∂xi− ηij︸︷︷︸

friction

(M−1)jkpk + gijΓj(t)︸︷︷︸random forcedxi

dt = (M−1)ijpj

J. Sadhukan et al. Phys. Rev. C 96, 061301(R) (2017)20

220 270 320 37020

40

60 1110

987

6

54 3 2 1

Q20(b)

Q 30(b

)


240Pu: Fission yields

J. Sadhukan et al. Phys. Rev. C 96, 061301(R) (2017)

RAPID COMMUNICATIONS

SADHUKHAN, NAZAREWICZ, AND SCHUNCK PHYSICAL REVIEW C 93, 011304(R) (2016)

50 60

0.1

1

10

120 140 160

0.1

1

10

240Pu

chargeyield(%)

fragment charge

massyield(%)

fragment mass

FIG. 5. Mass (left) and charge (right) distributions of heavier SF

yields of 240Pu. The symbols are the same as in Fig. . The shaded

regions are uncertainties in the distributions due to variations in

(narrow red band), dissipation tensor (wider cyan band), and scission

configuration (linear hatch pattern).

restrict the dynamical space in the classically allowed region

to the surface defined by 20,Q30 . In the following, we

calculate the fission paths on this surface for a collection of

900 outer turning points around the most probable outThe Langevin propagation is studied in three different

scenarios. In the first variant, the mass and charge distributions

of fission fragments are computed without invoking dissipation

and fluctuation by setting ij 0 (thus ij 0). Under such

conditions, the Langevin equations resemble the deterministic

Newtonian equations of motion with a one-to-one correspon-

dence between outer turning points and scission points. By

computing 900 trajectories to scission, we obtain mass and

charge yield distributions marked by the red dashed line in

Fig. . The most probable values of the fission yields are

consistent with the data but the distribution tails are clearly

off. In the second variant, we incorporate a constant collective

dissipation tensor ij with reasonable values 11 50 2240 , and 12 , but take a diagonal unit mass tensor

and obtain the green dashed-dotted line. In this case, fission

dynamics is dominated by the static features of the PES.

However, since the excitation energy is small, dissipation

effects are weak. As a result, the distribution width is even

narrower than in the first variant. It is only by combining a

constant dissipation tensor with the nonperturbative cranking

inertia that we obtain the solid blue lines, which nicely agree

with experiment over the whole range of mass-charge splits.

The results shown in Fig. correspond to 100 different runs

per each outer turning point, hence the distributions contain

contribution from 90 000 trajectories.

To illustrate the sensitivity of yield distributions to the

initial collective energy , the narrow red band in Fig.

shows the distribution uncertainty when taking a sample of 11

different values of within the range 0 2 MeV.

While such a variation in changes the SF half-life by

over two orders of magnitude, its impact on fission yield

distributions is minimal. The wider cyan band shows the spread

in predicted distributions when sampling the dissipation tensor

in the range of 0 12 30 and ( 11,η22 [30 400

with the constraint 1 11/η22 25. Note that we consider

a very broad range of variations in order to account for the

uncertainties in the theoretical determination the dissipation

tensor. Finally, the linear pattern in Fig. indicates the

uncertainty related to the definition of scission configurations

and corresponds to 0 0. It is very encouraging

to see that the predicted yield distributions vary relatively

little, even for nonphysically large values of ij and . We

have also found that the distributions are practically indistin-

guishable when the level density parameter varies from A/

to A/13.

Conclusions. In this work, we propose a microscopic

approach rooted in nuclear DFT to calculate mass and charge

distributions of SF yields. The SF penetrabilities, obtained

by minimizing the collective action in large multidimensional

PESs with realistic collective inertia, are used as inputs to

solve the time-dependent dissipative Langevin equations. By

combining many trajectories connecting the hypersurface of

outer turning points with the scission hypersurface, we predict

SF yield distributions. The results of our pilot calculations for240Pu are in excellent agreement with experiment and remain

reasonably stable under large variations of input parameters.

This is an important outcome, as SF yield distributions are

important observables for benchmarking theoretical models

of SF [52]. This finding is reminiscent of the analysis of

Ref. [17] for low-energy neutron- and -induced fission,

which found that the yield distributions predicted in the

Brownian-motion approach are insensitive to large variations

of dissipation tensor. On the other hand, according to our

analysis, the collective inertia tensor impacts both tunneling

and the Langevin dynamics.

The results of our study confirm that the PESs is the most

important ingredient when it comes to the maxima of yield

distributions. This is consistent with the previous DFT studies

of most probable SF splits [31 53 56], which indicate that the

topology of the PES in the prescission region is the crucial

factor. On the other hand, both dissipative collective dynamics

and collective inertia are essential when it comes to the shape

of the yield distributions. The fact that the predictions are fairly

robust with respect to the details of dissipative aspects of the

model is most encouraging.

Acknowledgments. Discussions with A. Baran, J.

Dobaczewski, J. A. Sheikh, and S. Pal are gratefully acknowl-

edged. This work was supported by the U.S. Department of

Energy, Office of Science, Office of Nuclear Physics under

Awards No. DOE-DE-NA0002574 (the Stewardship Science

Academic Alliances program) and No. DE-SC0008511 (NU-

CLEI SciDAC-3 collaboration). Part of this research was per-

formed under the auspices of the U.S. Department of Energy

by Lawrence Livermore National Laboratory under Contract

No. DE-AC52-07NA27344. Computational resources were

provided through an INCITE award, “Computational Nuclear

Structure,” by the National Center for Computational Sciences

(NCCS) and by the National Institute for Computational

Sciences (NICS). Computing resources were also provided

through an award by the Livermore Computing Resource

Center at Lawrence Livermore National Laboratory.

011304-4

• Good agreement with experimental data (circles).• Results are robust against variations in theoretical quantities (ηij , E0,. . . ).• Random force responsible for the tails of the distribution.


Fission yields of 294Og

How robust is the method against:

• Choice of collective variables?• Choice of collective inertias?• Choice of functional?

Testground: 294118Og176 [Oganessian et al., PRC 74 (2006)]• Heaviest element produced on Earth (2005-2010 JINR, Dubna).• τ ∼ 0.7 ms.• Very few events (1-2 fission?).

Very exotic nucleus → “blind” EDF calculation. . .


294Og: potential energy surface

• Two competing fission modes: symmetric (Q30 = 0) vs asymmetric (Q30 6= 0).

• From localization functions: 294118Og176 −−→ 20882Pb126 + 8636Kr50 .• 294Og decays via cluster emission.



• Two competing fission modes: symmetric (Q30 = 0) vs asymmetric (Q30 6= 0).• From localization functions: 294118Og176 −−→ 20882Pb126 + 8636Kr50 .

• 294Og decays via cluster emission.



• Two competing fission modes: symmetric (Q30 = 0) vs asymmetric (Q30 6= 0).• From localization functions: 294118Og176 −−→ 20882Pb126 + 8636Kr50 .• 294Og decays via cluster emission.


294Og barriers: UNEDF1 vs D1S

10203040

Q30(b

3 2)

UNEDF1HF B

0 50 100 150Q20 (b)

10203040

Q30(b

3 2) D1S

0

3

6

9Energy(M

eV)

clusterfission

cluster

fission

Matheson et al. (in preparation)

UNEDF1 and D1S predict similar evolution of the potential energy surface, butD1S has larger barrier → impact on yields?


294Og fission yields: UNEDF1 vs D1S

160 180 200 220Heavy fragmen mass

10−1

100

101%

yie

ld

UNEDF1HFBD1S

60 70 80 90Heavy fragmen charge

10−1

100

101294Og

Matheson et al. (in preparation)


Outline

1. Introduction





Conclusions & Outlook

I HFB + Hauser-Feshbach are valuable tools for studying the role of fissionin the r-process nucleosynthesis.

I New set of stellar rates suited for r-process calculations:I Abundances sensitive to height of fission barriers and local changes in

neutron separation energies around A = 257 and A > 280.I No impact on radioactive energy generation and 232Th/238U ratio:

progenitors of actinides have Z < 84 ⇒ no nuclei with Z ≥ 84 survive tofission?

I EDF + Langevin is a useful method to compute fission yields → smallsensitivity on choice of the functional.

I Future work:- β-delayed fission rates from BCPM barriers;- calculation of fission fragments distributions using EDFs;- explore different initial astrophysical conditions;- extend calculations using different EDF.


Some questions

• Which observables could prove the production of actinides/SHE during ther process? (see Y. Zhu et al., arXiv:1806.09724 and Nicole’s talk)

• How shall we conciliate consistency and accuracy in the calculations ofnuclear inputs? (Nicolas’ talk)

• Is it time for new sensitivity studies of r-process abundances? (seeL. Neufcourt et al., arXiv:1806.00552 Witek’s talk)


Collaborators

- G. Mart́ınez Pinedo (TUD/GSI, Darmstadt)- Z. Matheson and W. Nazarewicz (NSCL/FRIB, East Lansing)- L. Robledo (UAM, Madrid)- J. Sadhukhan (VECC, Kulkata)- N. Schunck (LLNL, Livermore)- M.-R. Wu (Sinica, Taiwai)

Thank you!

The dynamic description of spontaneous fission

tSF ∼ exp(2S) ⇐ S(L) =∫ b

ads√

2× B(s)[E(s)− E0

]Expand the multidimensional PES: relevant d.o.f. in s?

I Deformation multipoles: Q20,Q22,Q30, . . .I Pairing correlations ∆ (Babinet and Moretto, PLB 49 (1974)).

How to determine the fission path L(s)?I Minimizing the energy E(s): static approximation.I Minimizing the action S(L): dynamic approach.

State-of-the-art SF calculations:Sadhukhan et al, PRC88(2013) and PRC90(2014); SAG et al, PRC90(2014); Zhao et al, PRC92(2015) andPRC93(2016).

Static vs dynamic fission: 240Pu and 234U

Triaxial case: 240Pu - SkM* interaction

Q20 [b]

Q22

[b]

E(s) [MeV]

from Shadukhan et al., PRC90(2014),

see also Zhao et al., PRC93(2016).

dynamic paths:2D: s = {Q20,Q22}3D: s = {Q20,Q22,∆N 2}

Pairing fluctuations restore the axial symmetry! Artifact?

Static vs dynamic fission: 240Pu and 234UAxial case: 234U - BCPM interaction

Method tsf (s)Emin (static) 0.81× 1043Smin(Q20,Q30) 0.44× 1042Smin(Q20,Q40) 0.12× 1043Smin(Q20,∆N 2) 0.18× 1023Experiment 7.8× 1023

SAG, Robledo and Guzmán-Rodriguez

PRC90(2014).

- Pairing correlations reduce collective inertias → spontaneous fissionlifetimes decrease when pairing is included as d.o.f.

Conclusion

Spontaneous fission dynamics strongly modified by pairing fluctuations!

Static vs dynamic fission: 240Pu and 234UAxial case: 234U - BCPM interaction

Method tsf (s)Emin (static) 0.81× 1043Smin(Q20,Q30) 0.44× 1042Smin(Q20,Q40) 0.12× 1043Smin(Q20,∆N 2) 0.18× 1023Experiment 7.8× 1023

SAG, Robledo and Guzmán-Rodriguez

PRC90(2014).

- Pairing correlations reduce collective inertias → spontaneous fissionlifetimes decrease when pairing is included as d.o.f.

Conclusion

Spontaneous fission dynamics strongly modified by pairing fluctuations!

IntroductionImpact of fission on r-process nucleosynthesisFission fragments distributionsConclusions & OutlookAppendix

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