Extending the Eikonal Approximation to Low Energy
Pierre Capel, Tokuro Fukui & Kazuyuki Ogata

2nd June 2014

1

Halo nuclei

2

Reaction modelling
CDCC
Eikonal approximation

3Coulomb breakup of 15 C
68AMeV
20AMeV

4

SummaryHalo nuclei

Halo nuclei
Exotic nuclear structures are found far from stability
In particular halo nuclei withpeculiar quantal structure :Light, n-rich nuclei
Low Sn or S2nExhibit large matter radius
due to strongly clusterised structure :neutrons tunnel far from the core and form a halo
One-neutron halo
11Be ≡ 10 Be + n15C ≡ 14 C + n
Two-neutron halo
6He ≡ 4 He + n + n
11Li ≡ 9 Li + n + n

13 O 14 O 15 O 16 O 17 O 18 O 19 O 20 O 21 O 22 O 23 O 24 O
12 N 13 N 14 N 15 N 16 N 17 N 18 N 19 N 20 N 21 N 22 N 23 N
9C6Z

10 C 11 C 12 C 13 C 14 C 15 C 16 C 17 C 18 C 19 C 20 C

8B

10 B 11 B 12 B 13 B 14 B 15 B

7 Be
6 Li
3 He 4 He
1H2Hn

9 Be 10 Be 11 Be 12 Be
7 Li8 Li6 He

9 Li8 He

3HN-11 Li17 B22 C

19 B

14 Be

Noyau stable
Noyau riche en neutrons
Noyau riche en protons
Noyau halo d’un neutron
Noyau halo de deux neutrons
Noyau halo d’un proton

Proton halœs are also possible, but less probableHalo nuclei

Reactions with halo nuclei
Halo nuclei are fascinating objects
but difficult to study [τ1/2 (6 He)= 0.8 s]⇒ require indirect techniques, like reactions
Elastic scattering[see A. Di Pietro on Wednesday]

Breakup ≡ dissociation of halo from coreby interaction with target [see A. Bonaccorso on Thursday]

Need an good understanding of the reaction mechanism
i.e. an accurate theoretical description of reaction
coupled to a realistic model of projectileReaction modelling

Framework
Projectile (P) modelled as a two-body system :core (c)+loosely bound nucleon (f ) described by

H0 = Tr + Vcf (r)P

fr

Vcf adjusted to reproduce
bound state Φ0cband resonances

R

Target T seen as
structureless particle

ZT

P-T interaction simulated by optical potentials⇒ breakup reduces to three-body scattering problem :hiTR + H0 + VcT + VfT Ψ(r, R) = ET Ψ(r, R)
with initial condition Ψ(r, R) −→ eiKZ+··· Φ0 (r)Z→−∞Reaction modelling

CDCC

Continuum Discretised Coupled Channel (CDCC)
Solve the three-body scattering problem :hi
TR + H0 + VcT + VfT Ψ(r, R) = ET Ψ(r, R)

by expanding Ψ on eigenstates of H0Ψ(r, R) =Pi χi (R)Φi (r)with H0 Φi = i Φi

Leads to set of coupled-channel equations (hence CC) :[TR + i + Vii ] χi +XVij χj = ET χi ,j,iwith Vij = hΦi |VcT + VfT |Φj i
The continuum has to be discretised (hence CD)[Austern et al. , Phys. Rep. 154, 125 (1987)][Tostevin, Nunes, Thompson, PRC 63, 024617 (2001)]

Fully quantal approximation
No approximation on P-T motion, nor restriction on energy
But expensive computationally (at high energies)

Reaction modelling

Eikonal approximation

Eikonal approximation
Three-body scattering problem :hi
TR + H0 + VcT + VfT Ψ(r, R) = ET Ψ(r, R)

with condition Ψ −→ eiKZ Φ0Z→−∞b
Eikonal approximation : factorise Ψ = eiKZ ΨµPT 2 bv ]Ψ
2Neglecting TR vs PZ and using ET = 12 µPT v2 + 0∂ bb b, Z)
i~v Ψ(r,b, Z) = [H0 − 0 + VcT + VfT ]Ψ(r,∂Zb −→ Φ0 (r)solved for each b with condition ΨTR Ψ = eiKZ [TR + vPZ +Z→−∞

This is the dynamical eikonal approximation (DEA)[Baye, P. C., Goldstein, PRL 95, 082502 (2005)]

Coulomb breakup of 15 C

1568AMeV

C+Pb @ 68AMeV : energy distribution
dσbu /dE (mb/MeV)

400cdccdea300Exp.
200

1000012345E (MeV)

Excellent agreement between cdcc and dea[P.C., Esbensen and Nunes, PRC 85, 044604 (2012)]

Excellent agreement with experiment[Nakamura et al. PRC 79, 035805 (2009)]⇒ Confirms the validity of the approximations. . . and the two-body structure of 15 C

Coulomb breakup of 15 C

C+Pb @ 68AMeV : angular distribution
140cdccdea

120

dσbu /dΩ (b/sr)

1568AMeV

100
8060
40
2000123θ (deg)dea agrees well with cdcc
Though a slight shift compared to cdcc. . .

45

Coulomb breakup of 15 C

1520AMeV

C+Pb @ 20AMeV
12CDCCDEA

dσbu /dΩ (b/sr)

10864202468θ(◦ )101214dea too high and too forward due to lack of Coulomb deflection[P.C., Esbensen and Nunes, PRC 85, 044604 (2012)]

Can E-CDCC solve the problem ?

Coulomb breakup of 15 C

20AMeV

Eikonal-CDCC (E-CDCC)
Solving the eikonal problem expanding Ψ upon H0 eigenstates Φi (r)assuming discretised continuum [Ogata et al. PRC 68, 064609 (2003)]

Ψ(r, R) =X

ξi (b, Z)Φi (r)ei{Ki Z+ηi ln[Ki R−Ki Z]}i⇒ set of coupled equations
1 X∂ξi (b, Z) =Fii0 (b, Z)ξi0 (b, Z)ei(Ki0 −Ki )Z Rii0 (b, Z),∂Z
i~vi (R) i0
with coupling potential

D
E
Fii0 (b, Z) = Φi VcT + VfT − VC Φi0 .
r

E-CDCC takes proper account of energy conservation : vi (R)

Rii0 (b, Z) =(Ki0 R−Ki0 Z )iηi0(Ki R−Ki Z)iηiaccounts for part of the Coulomb distortion

Coulomb breakup of 15 CC+Pb @ 20AMeV
12CDCCDEA
E-CDCC10dσbu /dΩ (b/sr)

1520AMeV

864202468θ(◦ )e-cdcc also too high and too forward

101214

Coulomb breakup of 15 C

C+Pb @ 20AMeV

σbu (L) (mb)1012CDCCDEA
E-CDCCb′

CDCC
DEA
E-CDCC10dσbu /dΩ (b/sr)

1520AMeV

1864
20.1

0100

200

300

400 500
L(b)

600

700

800

02468θ(◦ )1012e-cdcc also too high and too forward
Shift in L ⇒ correction b → b0 (classical closest approach)14

Coulomb breakup of 15 C

C+Pb @ 20AMeV

σbu (L) (mb)1012CDCCDEA
E-CDCCb′CDCC
DEA
E-CDCChybrid10dσbu /dΩ (b/sr)

1520AMeV

1864
20.1

0100

200

300

400 500L(b)

600

700

800

02468θ(◦ )1012e-cdcc also too high and too forward
Shift in L ⇒ correction b → b0 (classical closest approach)hybrid solution : cdcc at low L (b) and eikonal at large L (b)⇒ excellent agreement with full cdcc

14

Coulomb breakup of 15 C

C+Pb @ 20AMeV

σbu (L) (mb)1012CDCCDEA
E-CDCCb′

CDCC
DEA
E-CDCChybridb′

10dσbu /dΩ (b/sr)

1520AMeV

186420.1

0100200

300400 500L(b)

600

700

80002468θ(◦ )1012e-cdcc also too high and too forward
Shift in L ⇒ correction b → b0 (classical closest approach)hybrid solution : cdcc at low L (b) and eikonal at large L (b)⇒ excellent agreement with full cdcc
Improve eikonal using Coulomb correction : b → b014Summary

Summary and prospect
Halo nuclei exhibit very exotic structure : core + halo
Studied mostly through reactions
I elastic scattering
I breakup
Mechanism of reactions with halo nuclei understoodcdcc & eikonal agree at 70AMeV
At 20AMeV, eikonal fails, due to Coulomb deflection
But :I e-cdcc can be extended to hybrid version⇒ agreement with full cdcc
I Simple Coulomb correction works fine
Can it be improved ?
How far down can we go in energy ? hie-isolde ?