Microscopic potential with Gogny interaction
G. Blanchon, M. Dupuis, H. F. Arellano
CEA, DAM, DIF
P(ND)2-2, Bruyeres-le-Chatel, 14-17 octobre 2014
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Phenomenology/Microscopy
Phenomenological opticalpotentials
Very good in theparametrization range
Very useful for applications
Lack of predictive power whenexperiments are missing
Local/Non-local potential
Microscopic approaches
Link with NN interaction Predictive power
Computer cost Non-local, energy-dependent,
complex potential
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Microscopic approaches
Bare NN interaction: Nuclear matter method
H. F. Arellano et al., PRC 66,024602 (2002).
SCRPAH. Dussan et al., PRC 84, 044319 (2011).
Coupled ClusterG. Hagen et al., PRC 86, 021602(R) (2012).
NCSMS. Quaglioni et al., PRL 101, 092501 (2008).
Effective NN interaction: Nuclear structure method
N. Vinh Mau, Theory of nuclear structure (IAEA, Vienna 1970) p. 931.
Y. Xu et al., JPG 41, 015101 (2014).
cPVCK. Mizuyama et al., PRC 86, 041603 (2012).
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NUCLEAR STRUCTURE METHOD
N. Vinh Mau, Theory of nuclear structure (IAEA, Vienna 1970)N. Vinh Mau, A. Bouyssy. NPA 257 (1976) 189-220V. Bernard and N.V. Giai, NPA 327, 397 (1979)
F. Osterfeld, et al. PRC 23, 179 (1981)
V = VHF +∆VRPA
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Nuclear structure method
V = VHF + ∆VRPA
Elastic scattering off a mean field Elastic scattering with excitationof the target
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Nuclear structure approach
Optical potential
V = V HF + V PP + V RPA − 2V (2)
Use of EDF (Gogny interaction)Particle-particle correlations already contained in Hartree-Fock
Im[V PP
]≈ Im
[V (2)
]
V = V HF + Im
[V (2)
]+ V RPA − 2V (2)
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Self-consistency
Schrodinger
VHF (ρ)ρ
NN interaction
SCHF
Schrodinger
VHF (ρ) + ∆V RPA(ρ)ρ
NN interaction
SCRPA
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Schrodinger integro-differential equation
n+A complex, non-local and energy-dependent potential
[d2
dr2+
l(l + 1)
r2− k2
]fjl (r) + r
∫νjl (r , r
′;E )fjl (r′)r ′dr ′ = 0
V (r, r’;E ) =∑
ljm
Yljm (r)νlj(r , r′;E )Y†
ljm (r′)
No localization of the potential
Solved in a 15 fm box
Bound statesR. H. Hooverman, NPA 189, 155 (1972).
ContinuumJ. Raynal, DWBA98, 1998, (NEA 1209/05).
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HARTREE-FOCK APPROXIMATION
V (r, r’,E ) = VHF (r, r’) + ∆VRPA(r, r’,E )
C. B. Dover and N. V. Giai, NPA 190 (1972) 373C. B. Dover and N. V. Giai, NPA 177 (1971) 559
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HF approximation to the n+A potential
VHF (r, r’) =
∫dr1v(r, r1)ρ(r1)δ(r− r’)− v(r, r’)ρ(r, r’)
ρ(r) =∑
i
ni |φi (r)|2,
ρ(r, r’) =∑
i
niφ∗i (r)φi (r’)
v: Gogny forceFinite range NN interaction → VHF non-local.
v is real and energy independentVHF is real and energy independent.
HF in coordinate space→ Good asymptotic behavior of the wave functions(not the case with HO basis).→ Correct treatment of the continuum(Distorted Wave φλ, Resonances).
K. Davies, S. Krieger, and M. Baranger, Nuclear Physics 84, 545 (1966).10 / 23
HF phase shift n/p+40Ca
0
1/2
1
3/2
2
5/2
1 10 100 1000
δ (
rad/
π)
E (MeV)
40Ca(n,n) Phaseshift
j=1/2 l=0j=1/2 l=1j=3/2 l=1j=3/2 l=2j=5/2 l=2j=5/2 l=3j=7/2 l=3j=7/2 l=4j=9/2 l=4j=9/2 l=5
0
1/2
1
3/2
2
5/2
1 10 100 1000
δ (
rad/
π)
E (MeV)
40Ca(p,p) Phaseshift
j=1/2 l=0j=1/2 l=1j=3/2 l=1j=3/2 l=2j=5/2 l=2j=5/2 l=3j=7/2 l=3j=7/2 l=4j=9/2 l=4j=9/2 l=5
Resonances when δ = nπ/2 (n odd).
Correct DW treatment of the intermediate wave φλ.
Impact on ∆VRPA
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VHF vs. Re(Vpheno)
Total cross section n+40Ca
1 10 100E (MeV)
1000
10000
σ Tot
(m
b)
Exp.KDCHE
0,1 1 10 100
E (MeV)
0
1000
2000
3000
4000
5000
6000
σ El
(mb)
HFRe(KD)Re(CHE)
Bound states HF/D1S Exp. CHE
V HF gives the main contribution to the real part of the potential
(B. Morillon and P. Romain, Phys. Rev. C 70, 014601 (2004).) → dispersive potential
(A. J. Koning and J. P. Delaroche, Nuclear Physics A 713, 231 (2003).)12 / 23
Hartree-Fock volume integral
0
100
200
300
400
500
600
700
800
0 5 10 15 20 25
J V
(MeV
fm3 )
PW
40 MeV
17 MeV
10 MeV
5 MeV
0.5 MeV
VHF
HartreePB
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Elastic cross section n+40Ca
0 20 40 60 80 100 120 140 160 180
θc.m.
(deg.)
1
10
100
1000
10000
dσ/d
Ω
(mb/
sr)
Exp.HF
n + 40
Ca @ 30.3 MeV
0 20 40 60 80 100 120 140 160 180
θc.m.
(deg.)
0,01
1
100
10000
dσ/d
Ω
(mb/
sr)
Exp.HF
n + 40
Ca @ 40 MeV
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RANDOM-PHASE APPROXIMATION
V (r, r’,E ) = VHF (r, r’) + ∆VRPA(r, r’,E )
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RPA potential∆VRPA = Im
[V (2)
]+ V RPA − 2V (2)
V RPA(r, r′,E) = limη→0+
∑
N 6=0,ijkl
∫
∑
λ
χ(N)ij χ
(N)kl
×
(
nλ
E − ǫλ + EN − iη+
1− nλ
E − ǫλ − EN + iη
)
Fijλ(r)F∗klλ(r
′)
withFijλ(r) =
∫
d3r1φ
∗i (r1)v(r, r1)[1− P]φλ(r)φj (r1)
φ’s are HF wave functions.
We include both bound and continuumparticles in constructing our intermediatestate φλ.
Excitations of the target described withRPA/D1S
Blaizot, et al., NPA 265, 315 (1976).
Berger, et al., Comp. Phys. Com. 63, 365(1991).
occ
unocc
HF
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Effect of HF intermediate propagator
p+40Ca
VHF + Im(VRPA)
Coupling to the first 1− E1− = 9.7MeV
10 12 14 16 18 20
Einc
(MeV)0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
σ R
(mb)
DWCoul
Eλ = 2.15 MeV
j=3/2 l=1
Eλ = 5.65 MeV
j=5/2 l=3
Eλ = 9.55 MeV
j=9/2 l=4
Eλ = 3.70 MeV
j=1/2 l=1
Effect of resonances of the intermediate HFpropagator.
Enhancement of σR compared as with aCoulomb wave.
+
HF
1
Intermediate HF propagator
RPA excitation
0
1/2
1
3/2
2
5/2
1 10 100 1000
δ (
rad/
π)
E (MeV)
40Ca(p,p) Phaseshift
j=1/2 l=0j=1/2 l=1j=3/2 l=1j=3/2 l=2j=5/2 l=2j=5/2 l=3j=7/2 l=3j=7/2 l=4j=9/2 l=4j=9/2 l=5
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Effect of HF intermediate propagator
σR from VHF + Im(VRPA)
σR from VHF + Im(VPH)
0 10 20 30 40 50E (MeV)
0
500
1000
1500
2000
σ R
(mb)
KDHF+Im(RPA) Γ= 0 MeVHF+Im(PH) Γ= 0 MeV
First excited stateE = 3.1 MeV
n + 40
CaΓ= 0 MeV
→ Effect of the HF resonanceson Im(VRPA)
Zero width calculation:
σR = 0 for incident energies below
the energy of the first excited state
of the target nucleus
40Ca RPA states J = 0 → 8
0 50 100 150 200 250E
N (MeV)
0
1000
2000
3000
4000
5000
Num
ber
of R
PA s
tate
s
RPA states J = 0 to 1440
Ca
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Averaged potential
p + 40Ca
5 10 15 20 25 30 35 40E (MeV)
0
200
400
600
800
1000
σ R
(mb)
Exp.KDV
HF + ∆V
n + 40Ca
0 10 20 30 40E (MeV)
1500
2000
2500
3000
3500
4000
σ Tot
(m
b)
Exp.KDV
HF + ∆V
Physical origin of width
Self-consistent scheme η 6= 0 when HF propagator gets
dressed by RPA EN → EN + iΓN(EN)
Damping (doorway state) &
continuum
0 50 100 150 200 250E
N (MeV)
0
10
20
30
40
Γ N
(MeV
)
Phenomenological width for RPA states
Use of a phenomenological width
(Harakeh and van der Woude)
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Averaging...
S = 〈S〉+ S
Averaged cross section
〈σE 〉 =π
k2〈|1− S |2〉
〈σR〉 =π
k2〈1− |S |2〉
〈σT 〉 =π
k2〈1− Re[S ]〉
Averaged potential
σE =π
k2|1− 〈S〉|2
σR =π
k2(1− |〈S〉|2)
σT =π
k2(1− Re[〈S〉])
〈σE 〉 = σE + σCE
〈σR〉 = σR − σCE
〈σT 〉 = σT
Compound elastic
σCE =π
k2〈|S |2〉
TALYS: Hauser-Feshbach/ Koning-Delaroche
particularly relevant for neutron scatteringbelow 10 MeV
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Elastic cross section n/p+40Ca
0 20 40 60 80 100 120 140 160 180
θc.m.
(deg.)
10-4
10-2
100
102
104
106
108
1010
1012
1014
dσ/d
Ω
(mb/
sr)
9.9113.9
16.9
25.5
30.3
40.
19.
21.7
3.29
5.3
6.5
2.06
5.88
7.91
0 20 40 60 80 100 120 140 160 180
θc.m.
(deg.)10
-10
10-8
10-6
10-4
10-2
100
102
104
σ(θ)
/σR
uth
9.86
10.37
13.49
14.51
15.97
18.57
30.3
40.
19.57
21.
23.5
25.
26.3
27.5
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Analyzing powers n/p+40Ca
-1-0.5
00.5
1
-1-0.5
00.5
1
-1-0.5
00.5
1
0 20 40 60 80 100 120 140 160 180
θc.m.
(deg.)
-1-0.5
00.5
1
9.91
11.
13.9
16.9
Ay(θ
)
-1-0.5
00.5
1
-1-0.5
00.5
1
-1-0.5
00.5
1
0 20 40 60 80 100 120 140 160 180
θc.m.
(deg.)
-1-0.5
00.5
1
14.51
15.97
18.57
40.
Ay(θ
)
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Conclusion
Summary: We take into account absorption coming from the coupling to
RPA states. Consistent scheme. Tools to deal with non-local potentials (bound and continuum
states, HF in coordinate space). Exact treatment of the intermediate state with resonances. Good agreement with experiment (cross section, analyzing
power) for 40Ca up to 30 MeV.
48Ca, 90Zr, 132Sn and 208Pb in production
Outlooks:
Bound single particle dressing. Consistent width. Consistent Compound elastic. QRPA potential, deformed nuclei. Inelastic scattering.
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