G. Blanchon, M. Dupuis, H. F. Arellano · PDF fileMicroscopic potential with Gogny interaction G. Blanchon, M. Dupuis, H. F. Arellano CEA, DAM, DIF P(ND)2-2, Bruy`eres-le-Chˆatel,

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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