impact of fission on r-process nucleosynthesis within the ... · impact of fission on r-process...
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Impact of fission on r-process nucleosynthesiswithin the energy density functional theory
Samuel A. Giuliani†, G. Martınez Pinedo†, L. M. Robledo‡,M.-R. Wu†
†Technische Universitat Darmstadt, Darmstadt, Germany‡Universidad Autonoma de Madrid, Madrid, Spain
January 17th, 2017
Neutron star mergers: from gravitational waves tonucleosynthesis
Hirschegg, Austria
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Outline
1. Introductionr-process in neutron star mergers (NSM)
2. Fission rates from the BCPM EDFFission within the Energy Density Functional approachFission properties of the BCPM EDFStellar reaction rates
3. r-process nucleosynthesis in NSMr-process abundances: BCPM vs FRDM-TFRadioactivityAveraged fission rates
4. Conclusions & Outlook
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Outline
1. Introductionr-process in neutron star mergers (NSM)
2. Fission rates from the BCPM EDFFission within the Energy Density Functional approachFission properties of the BCPM EDFStellar reaction rates
3. r-process nucleosynthesis in NSMr-process abundances: BCPM vs FRDM-TFRadioactivityAveraged fission rates
4. Conclusions & Outlook
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
r-process abundances distributions
Cowan & Sneden, Nature 440, 1151 (2006).
50 60 70 80 90
Atomic number
–12
–10
–8
–6
–4
–2
0
Rela
tive
log ε
CS22892–052HD115444BD+17°3248
CS31082–001HD221170SS r-process abundances
Site?
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
r-process in Neutron Star Mergers & fission10
0
150
200
250
10−310
−210−110
0101
r−process waiting point (ETFSI−Q)
Known massKnown half−life
N=126
N=82
Sola
r r a
bund
ance
s
r−process path
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 recycling
Mendoza-Tem
iset
al.,Phys.Rev.
C92,055805
(2015)
Condition for robust r-process abundances in NSM
Material in the fissioning region (A > 250) dominating at freeze-out→ systematic calculations of fission rates using different models are
required!
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Outline
1. Introductionr-process in neutron star mergers (NSM)
2. Fission rates from the BCPM EDFFission within the Energy Density Functional approachFission properties of the BCPM EDFStellar reaction rates
3. r-process nucleosynthesis in NSMr-process abundances: BCPM vs FRDM-TFRadioactivityAveraged fission rates
4. Conclusions & Outlook
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Fission within the Energy Density Functional approachPotential Energy Surface
Energy evolution from the groundstate to the scission point.
Relevant degrees of freedom?
SAG+ PRC90(2014); Sadhukhan+ PRC90(2014)
Collective inertias
Resistance of the nucleusagainst the deformation forces.
Hard to compute.
Baran+ PRC84 (2011).
0 10 20 30 40 50 60 70 80 90 100Q20 [b]-2060
-2056
-2052
-2048
-2044
-2040
E(Q
20) [
MeV
]
286114Fl
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Static fission propertiesI Parameters defining the potential energy surface:
- inner and outer fission barrier heights,- isomer excitation energy.
I Probability of tunneling under the fission barrier (WKB):
P =1
1 + exp(2S) ;
S =
∫ b
adQ20
[2 B(Q20)︸ ︷︷ ︸
inertias
(V (Q20)︸ ︷︷ ︸
PES
−E∗)] 1
2.
- Spontaneous fission lifetimes: tsf =ln 2
n×P ;- fission cross sections.
I Interaction:- Now: Barcelona-Catania-Paris-Madrid EDF Baldo et al., PRC87 (2013).- (next) Future: Gogny and BCPM∗ EDFs.
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
BCPM: systematic of fission barriers
0
2
4
6
8
10
12
14
120 140 160 180 200 220 240Neutron number
90
100
110
120TF
2
2
2
2 2 2
2
4
4
4
4 4
6
6
6
6
6
8
8
10 12 14
90
100
110
120
Prot
on n
umbe
r HFB14
2 2 2
2
2
2
2
2
2
2
2
4 44
4 444
6 68810 12
Sn = 2 MeVSn = 0 MeV
90
100
110
120BCPM
2
2 2
4
4
4
4
4
66
6
6
8
8
10
12 12
14 14
Fission barrier (MeV)
I Mean-field models predict largerBf than mic-mac for Z ≤ 110;
I n-induced fission importantabove N = 184.
BCPM: SG, Martınez-Pinedo and Robledo (in
preparation); HFB14: S. Goriely et al., PRC75
(2007); TF: W. D. Myers and W. J. Swiatecki,
PRC60 (1999); FRDM: P. Moller et al., At. Data
Nucl. Data Tables 52 (1995).
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
BCPM: systematic of fission barriers
0
2
4
6
8
120 140 160 180 200 220 240Neutron number
90
100
110
120TF-FRDM
90
100
110
120
Prot
on n
umbe
r HFB14
Sn = 2 MeVSn = 0 MeV
90
100
110
120BCPM
Bf−Sn (MeV)
I Mean-field models predict largerBf than mic-mac for Z ≤ 110;
I n-induced fission importantabove N = 184.
BCPM: SG, Martınez-Pinedo and Robledo (in
preparation); HFB14: S. Goriely et al., PRC75
(2007); TF: W. D. Myers and W. J. Swiatecki,
PRC60 (1999); FRDM: P. Moller et al., At. Data
Nucl. Data Tables 52 (1995).
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Nuclear inputs for reaction rates
I Reaction rates require knowledge/modeling of nuclear level densities, γ-raystrengths, masses, fission barriers. . .
I The EDF theory useful framework for providing nuclear inputs:- neutron separation energies and barriers from the same interaction;
120 130 140 150 160 170 180 190 200N
0
2
4
6
8
10
12
14
16
18
20
S2n
Z =120
Z =84
BCPM
120 130 140 150 160 170 180 190 200N
0
2
4
6
8
10
12
14
16
18
20
S2n
Z =120
Z =84
FRDM
- tunneling probability from full fission barrier and microscopic collectiveinertias.
I Several optical models, level densities and γ-ray strengths implemented inTALYS.
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Cross sections: BCPM vs experimentI n-induced fission, γ-induced fission, spontaneous fission and n-capture rates
computed using the BCPM nuclear masses and fission barriers.
102
103
104
σ(n,fiss)
(mb)
238Pu 239Pu
10-2 10-1 100 101
Energy (MeV)
102
103
104
σ(n,fiss)
(mb)
240Pu
10-2 10-1 100 101
Energy (MeV)
241Pu
ExperimentConstant TemperatureBack-Shifted Fermi Gas
Goriely, Eur. Phys. J. A (2015) 51:22.
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Cross sections: BCPM vs experiment
I n-induced fission, γ-induced fission, spontaneous fission and n-capture ratescomputed using the BCPM nuclear masses and fission barriers.
0.01 0.1 1 10 0.01 0.1 1 10 0.01 0.1 1 10 0.01 0.1 1 10
102
103
0.01 0.1 1 10
238Pu(n,f)
n,f [m
b]
E [MeV]
102
103
0.01 0.1 1 10
239Pu(n,f)
E [MeV]
100
101
102
103
104
0.01 0.1 1 10
240Pu(n,f)
E [MeV]
102
103
104
0.01 0.1 1 10
E [MeV]
241Pu(n,f)
Goriely, Eur. Phys. J. A (2015) 51:22.
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Outline
1. Introductionr-process in neutron star mergers (NSM)
2. Fission rates from the BCPM EDFFission within the Energy Density Functional approachFission properties of the BCPM EDFStellar reaction rates
3. r-process nucleosynthesis in NSMr-process abundances: BCPM vs FRDM-TFRadioactivityAveraged fission rates
4. Conclusions & Outlook
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
BCPM rates
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)
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
I Narrow (n, γ) path through A ∼ 300; r-path up to A ∼ 320.I Fission dominates at A & 350: heavier nuclei cannot be produced.I SF relevant when Bf < 2 MeV.
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
r-process abundances: BCPM vs FRDM-TF Panov+, A&A 513 (2010)
I Trajectory: 3D relativistic simulations from1.35 M�-1.35 M� NS mergers [Bauswein+(2013)].
I Same rates for Z ≤ 83 and same τβ .
I BBCPMf > BTF
f and ∆BCPMn > ∆FRDM
n atN = 184: larger production of nuclei withA > 280.
I Accumulation above the 2nd peak:
- formed by fission fragments of nuclei withA ∼ 283− 286.
I FRDM-TF peak at A ∼ 257:
- jump in Sn at N = 172 (not present inBCPM).
I Similar 232Th/238U ratio: relevant physicsbelow Z = 84 or close to stability.
10-910-810-710-610-510-410-310-2
abun
danc
es a
t n/s
=1
10-910-810-710-610-510-410-310-2
abun
danc
es a
t τ(n,γ
)=τ β
100 150 200 250 300A
10-910-810-710-610-510-410-310-2
abun
danc
es a
t 1 G
y
PANOVBCPMsolar
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
r-process abundances: BCPM vs FRDM-TF Panov+, A&A 513 (2010)
I Trajectory: 3D relativistic simulations from1.35 M�-1.35 M� NS mergers [Bauswein+(2013)].
I Same rates for Z ≤ 83 and same τβ .
I BBCPMf > BTF
f and ∆BCPMn > ∆FRDM
n atN = 184: larger production of nuclei withA > 280.
I Accumulation above the 2nd peak:
- formed by fission fragments of nuclei withA ∼ 283− 286.
I FRDM-TF peak at A ∼ 257:
- jump in Sn at N = 172 (not present inBCPM).
I Similar 232Th/238U ratio: relevant physicsbelow Z = 84 or close to stability.
10-910-810-710-610-510-410-310-2
abun
danc
es a
t n/s
=1
10-910-810-710-610-510-410-310-2
abun
danc
es a
t τ(n,γ
)=τ β
100 150 200 250 300A
10-910-810-710-610-510-410-310-2
abun
danc
es a
t 1 G
y
PANOVBCPMsolar
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
r-process abundances: BCPM vs FRDM-TF Panov+, A&A 513 (2010)
I Trajectory: 3D relativistic simulations from1.35 M�-1.35 M� NS mergers [Bauswein+(2013)].
I Same rates for Z ≤ 83 and same τβ .
I BBCPMf > BTF
f and ∆BCPMn > ∆FRDM
n atN = 184: larger production of nuclei withA > 280.
I Accumulation above the 2nd peak:
- formed by fission fragments of nuclei withA ∼ 283− 286.
I FRDM-TF peak at A ∼ 257:
- jump in Sn at N = 172 (not present inBCPM).
I Similar 232Th/238U ratio: relevant physicsbelow Z = 84 or close to stability.
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at
n/s
=1
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at τ (n,γ)=τ β
100 150 200 250 300A
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at
1 G
y
PANOV
BCPM
solar
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
r-process abundances: BCPM vs FRDM-TF Panov+, A&A 513 (2010)
I Trajectory: 3D relativistic simulations from1.35 M�-1.35 M� NS mergers [Bauswein+(2013)].
I Same rates for Z ≤ 83 and same τβ .
I BBCPMf > BTF
f and ∆BCPMn > ∆FRDM
n atN = 184: larger production of nuclei withA > 280.
I Accumulation above the 2nd peak:- formed by fission fragments of nuclei with
A ∼ 283− 286.
I FRDM-TF peak at A ∼ 257:
- jump in Sn at N = 172 (not present inBCPM).
I Similar 232Th/238U ratio: relevant physicsbelow Z = 84 or close to stability.
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at
n/s
=1
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at τ (n,γ)=τ β
100 150 200 250 300A
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at
1 G
y
PANOV
BCPM
solar
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
r-process abundances: BCPM vs FRDM-TF Panov+, A&A 513 (2010)
I Trajectory: 3D relativistic simulations from1.35 M�-1.35 M� NS mergers [Bauswein+(2013)].
I Same rates for Z ≤ 83 and same τβ .
I BBCPMf > BTF
f and ∆BCPMn > ∆FRDM
n atN = 184: larger production of nuclei withA > 280.
I Accumulation above the 2nd peak:- formed by fission fragments of nuclei with
A ∼ 283− 286.
I FRDM-TF peak at A ∼ 257:- jump in Sn at N = 172 (not present in
BCPM).
I Similar 232Th/238U ratio: relevant physicsbelow Z = 84 or close to stability.
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at
n/s
=1
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at τ (n,γ)=τ β
100 150 200 250 300A
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at
1 G
y
PANOV
BCPM
solar
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
r-process abundances: BCPM vs FRDM-TF Panov+, A&A 513 (2010)
I Trajectory: 3D relativistic simulations from1.35 M�-1.35 M� NS mergers [Bauswein+(2013)].
I Same rates for Z ≤ 83 and same τβ .
I BBCPMf > BTF
f and ∆BCPMn > ∆FRDM
n atN = 184: larger production of nuclei withA > 280.
I Accumulation above the 2nd peak:- formed by fission fragments of nuclei with
A ∼ 283− 286.
I FRDM-TF peak at A ∼ 257:- jump in Sn at N = 172 (not present in
BCPM).
I Similar 232Th/238U ratio: relevant physicsbelow Z = 84 or close to stability.
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at
n/s
=1
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at τ (n,γ)=τ β
100 150 200 250 300A
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
ab
und
ance
s at
1 G
y
PANOV
BCPM
solar
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
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
BCPMPanovβ-decayα-decaytotal fission
I Minor impact in the radioactive energy production ⇒ physics Z < 84 and/orβ-decay more relevant.
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Averaged fission rates
10-1 100 101 102
Time (s)
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Rat
e (s−
1)
BCPMPanovn-induced fissionβ-delayed fissionspontan. fissionγ-induced fission
I n-induced dominates until freeze-out and revived by β-delayed neutrons⇒ β-delayed fission rates from BCPM barriers required!
I photo-induced fission always subdominant;I decay of material to stability triggers spontaneous fission.
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
What is next? Gogny interaction
0 2 4 6 8 10 12 14
120 140 160 180 200 220 240Neutron number
90
100
110
120D1N
2
2
4
4
4
6
6
6
8 8
8
8
88
8
1010
10
12 12 14
90
100
110
120
Prot
on n
umbe
r D1S
2
2
2
4
4
4
6
6
6
6
66 6
8
8
8
8
8
8 8
10 10
10
10
10
10
12
12
1414 14
Sn = 2 MeVSn = 0 MeV
90
100
110
120D1M
2
2
4
4
4
6
6
6
8 8
8
8
8
8
8
1010
10
12 12 14
Fission barrier (MeV)
-5 -3 -1 1 3 5
120 140 160 180 200 220 240Neutron number
90
100
110
120D1S-BCPM
90
100
110
120
Prot
on n
umbe
rD1M-BCPM
Sn = 2 MeVSn = 0 MeV
90
100
110
120D1M-BCPM
Fission barrier: GOGNY vs BCPM (MeV)
I Three Gogny parametrizations in the market: D1S, D1M, D1N.I BCPM and Gogny differ up to 5 MeV for r-process nuclei: impact in
rates/abundances?
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Outline
1. Introductionr-process in neutron star mergers (NSM)
2. Fission rates from the BCPM EDFFission within the Energy Density Functional approachFission properties of the BCPM EDFStellar reaction rates
3. r-process nucleosynthesis in NSMr-process abundances: BCPM vs FRDM-TFRadioactivityAveraged fission rates
4. Conclusions & Outlook
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Conclusions
I Fission plays a crucial role in r-process nucleosynthesis.I EDFs valuable tool for a microscopic description of the fission process.I New set of stellar rates suited for r-process calculations using the BCPM
EDF:- n-induced, γ-induced fission and spontaneous fission stellar rates.
I BCPM fission barriers higher than TF: r-process towards A ∼ 320.I Small impact on radioactive energy generation and 232Th/238U ratio.
I Future work:- computation of fragments distributions;- β-delayed fission rates;- explore sensitivity to BCPM and different EDF’s (Gogny, BCPM∗).
Introduction Fission rates from the BCPM EDF r-process nucleosynthesis in NSM Conclusions & Outlook
Thank you!
Minimizing the action: B(∆N 2) vs E(∆N 2) - 234U
S =
∫ b
ads√
2× B(s)[E(s)− E0
]
5 10 15 20 25 30 35
〈∆N2〉
-1777
-1776
-1775
-1774
-1773
-1772
-1771
-1770
-1769
-1768Erot(M
eV)
Erot ∼ 〈∆N2〉2
2.00e-03
4.00e-03
6.00e-03
8.00e-03
1.00e-02
1.20e-02
1.40e-02
1.60e-02
1.80e-02
2.00e-02
B(h
2/M
eVb2)
B∼ 1/〈∆N2〉2
4.09e-03
4.29e-03
4.49e-03
4.69e-03
4.90e-03
5.10e-03
5.30e-03
5.50e-03
5.70e-03
S(h)
action S
The least action path
I The least action path (black) strongly differ from the leastenergy one (red)!
Fission barriers and E0 - 244U
S =
∫ b
ads√
2× B(s)[E(s)− E0
]244U
10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
Q2 (b)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
EHFBE
GS (
MeV
)
DYNAMIC
STATIC
Spontaneous fission lifetimes (BCPM results)
• Exp: Xiaojun Bao et al., Nucl. Phys. A906, 1–13 (2013).
35 36 37 38 39 40 4121110
17
50
70
232T
h
232U
234U
235U
236U
238U
240C
m24
2Cm
243C
m24
4Cm
245C
m24
6Cm
248C
m25
0Cm
250N
o25
2No
254N
o25
6No
258N
o26
0No
262N
o
37 38 39 40 41 42 43
241A
m24
3Am
237C
f23
8Cf
240C
f246C
f24
8Cf
249C
f25
0Cf
252C
f25
4Cf
256C
f
245M
d24
7Md
255D
b25
7Db
260D
b26
3Db
36 37 38 39 40 41ZZ2 /AA
236P
u23
8Pu
239P
u24
0Pu
242P
u24
4Pu
249B
k
241F
m24
3Fm
244F
m24
6Fm
248F
m25
0Fm
252F
m25
4Fm
255F
m25
6Fm
257F
m25
8Fm
259F
m26
0Fm
260F
m
39 40 41 42 43 44ZZ2 /AA
253E
s25
5Es
258L
r25
9Lr
261L
r26
2Lr
253R
f25
4Rf
255R
f25
6Rf
257R
f25
8Rf
259R
f26
0Rf
262R
f
258S
g26
0Sg
264H
s
ATDHFBGOA-GCMSEMPEXP
log10(tSF[s])
BCPM barrier heights and isomer energy
• Exp:R. Capote et al., Nucl. Data Sheets 110, 3107 (2009);
S. Bjørnholm and J. E. Lynn, Rev. Mod. Phys. 52, 725 (1980).
125 130 135 140 145 150 155N
4
2
0
2
4
BC
PM
-EX
PE
RIM
EN
T (
MeV
)
BI
140 145 150 155N
BII
Bjørnholm Capote
140 142 144 146 148 150N
4
2
0
2
4EII
BCPM barrier heights and isomer energy BCPM: Baldo et al., PRC87 (2013)
Exp: Sing et al., Nucl. Data Sheets 97, 241 (2002); R. Capote et al., Nucl. Data Sheets 110, 3107 (2009).
2
1
0
1
2
3
4
5
6
7
8
9
10
11
E [M
eV
]
Inner barrier
Isomer excitation
Outer barrierEXP
BCPM
Potential energy surface parameters
234U
236U
238U
238P
u
240P
u
242P
u
244P
u
240C
m
242C
m
244C
m
246C
m
248C
m
250C
f
252C
f
SAG and Robledo, Phys. Rev. C88, 054325 (2013)
- Outer barriers and isomer energies quite well reproduced for all nuclei.- Inner barriers are reduced when triaxiality is allowed (Erler+(2012), Guzman+(2014)).
Uranium fission barrier heights: theoretical predictionsBSk14: S. Goriely et al., Phys. Rev. C75, 064312 (2007). Moller: P. Moller et al., Phys. Rev. C91, 024310 (2015).
138 144 150 156 162 168 174 180 186 192 198 204
N
2
3
4
5
6
7
8
9
10
11
12
13
Fis
sio
n b
arr
iers
he
igh
ts [
Me
V]
BCPM
BSk14
Moeller
Experimental
Z=92
138 144 150 156 162 168 174 180 186 192 198 204
N
20
40
60
80
100
120
140
160
180
log
10(tsf (s
)) BCPM
BSk14
Experimental
Z=92
Collective inertias
BSk14: B = 0.054A5/3 MeV−1
BCPM: B =12
(M−2)2
(M−1)3 with M(−n) =∑α>β
|〈αβ|Q20|0〉|2
(Eα + Eβ)n
The BCPM functional
The energy of a finite nucleus is given by
E = T0 + E∞int + EFRint + Es.o. + EC + Epair
E∞int[ρp, ρn] =
∫d~r[Ps(ρ)(1− β2) + Pn(ρ)β2]ρ
with ρ(~r) = ρn(~r) + ρp(~r) and β(~r) = (ρn(~r)− ρp(~r))/ρ(~r).
Ps and Pn are polynomial fits to reproduce microscopic EoS in nuclear matter.
I Phenomenological surface contribution
EFRint [ρn , ρp] =
12∑t,t′
∫∫d~rd~r ′ρt(~r)vt,t′ (~r − ~r ′)ρt′ (~r ′)
with vt,t′ (r) = Vt,t′ e−r2/r0 tt2; Vn,n = Vp,p = VL = 2b1/(π3/2r3
0Lρ0);Vn,p = Vp,n = VU = (4a1 − 2b1)/(π3/2r3
0Uρ0) .M.Baldo et al. Phys. Lett. B663 (2008) 390; Phys. Rev. C87 064305 (2013)
Remaining contributions to the EDF
I Coulomb Direct EHC = (1/2)
∫∫d~rd ~r ′ρp(~r)|~r − ~r ′|−1ρp(~r ′) Exchange:
EexC = −(3/4)(3/π)1/3 ∫ d~rρp(~r)4/3
I Spin-Orbitvso
ij = iWLS(~σi + ~σj) · [~k′ × δ(~ri − ~rj)~k]
Free parameters
WLS and r0L, r0U
I Pairing Correlations (E. Garrido et al. Phys. Rev. C 60, 064312 (1999)) Zero-rangeinteraction,
vpp(ρ(~r)) = η × v0
2
[1− γ
(ρ(~r)
ρ0
)α], ρ0 =
23π2 k3
F .
η ≡multiplicative parameter setting the pairing strength. . .