Analysis of 8B breakupexperiments with a Dynamical

Eikonal ApproximationPierre Capel, Gerald Goldstein, Daniel Baye

ULB, Brussels, Belgium

– p.1/19

What makes 8B so interesting?One-p halo candidate (Sp = 137 keV);seen as 8B ≡ 7Be + pBreakup used to study halo structure

Coulomb breakup of astrophysical interest:Inverse reaction of 7Be(p, γ)8B,important for solar-neutrino studiesIdea: extract σcapture from σbu

We (re)analyse several 8B-breakup experiments

within one model: Dynamical Eikonal Approximation

– p.2/19

OutlineDynamical Eikonal Approximation8B breakup experiments:

44 & 81 AMeV p‖ distributions (MSU)83 AMeV energy distribution (MSU)52 AMeV angular distribution (RIKEN)

Analysis of dynamics: role of nuclear interaction,multipole contributions, higher-order effects...

Conclusion and prospects

– p.3/19

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

H0 = Tr + Vcf(r)

Vcf adjusted to reproducebound state

Target T seen asstructureless particle

R

b

r

Z T

P

c

f

P -T interaction simulated by optical potentials

⇒ breakup reduces to three-body scattering problem:

[TR + H0 + VcT + VfT ] Ψ(R, r) = ETΨ(R, r)

– p.4/19

Dynamical eikonal approximation (1)Three-body scattering problem

[TR + H0 + VcT + VfT ] Ψ(r, R) = ETΨ(r, R)

with condition Ψ(r, R) −→Z→−∞

eiK0ZΦ0(r)

To remove the rapid variation in R we factoriseΨ(r, R) = eiK0ZΨ(r, R):

HΨ = eiK0Z [TR + vPZ +1

2µPTv2

+ (H0 + VcT + VfT )]Ψ

Neglecting TR vs vPZ and using ET = 12µPTv2 + E0

i~v∂

∂ZΨ(r, b, Z) = [H0 − E0 + VcT + VfT ]Ψ(r, b, Z)

– p.5/19

Dynamical eikonal approximation (2)

i~v∂

∂ZΨ(r, b, Z) = [H0 − E0 + VcT + VfT ]Ψ(r, b, Z)

is equivalent to a TDSE with straight line trajectoriesBUT here b and Z are quantal ⇒ no trajectory

The usual eikonal uses adiabatic approx. H0 −E0 ∼ 0⇒ neglects internal dynamical effects of projectile

Ψeik(r, b, Z) = e−i

~v

∫ Z

−∞dZ ′[VcT (r,b,Z ′)+VfT (r,b,Z ′)]Φ0(r)

⇒ dynamical eikonal generalises TDSE and eikonal

improves TDSE by including interferences

improves eikonal by including dynamical effects

we know how to solve accurately TDSE– p.6/19

Breakup cross sectionBreakup transition matrix element:

T bufi = 〈eiK ·Rχ

(−)

k|VcT + VfT |e

iK0ZΨ〉,

where H0χ(−)

k= Eχ

(−)

k(ingoing scattering wave)

At forward angle, assuming q = K −K0Z transverse

T bufi ≈ i~v

∫dbeiq·b〈χ

(−)

k|Ψ(Z → ∞)〉

∝∑

lm

Y ml (Ωk)e

−imϕ

∫ ∞

0

J|m|(qb)〈Φklm|Ψ(Z → ∞)〉bdb,

Cross sections:dσbu

dkdΩ∝ |T bu

fi |2

∫dΩk,

∫dΩ

−→ dσbu

dEdΩ and dσbu

dE

⇒DEA includes interferences between trajectories– p.7/19

8B8B has only one 2+ loosely-bound state Sp = 137 keV

seen as |8B(2+)〉 = | 7Be(32

−)⊗p(p3/2)〉

Description of Esbensen & Bertsch [NPA 600, 37 (96)]7Be assumed spherical, its spin is neglected7Be-p potential is WS+S0

Coulomb breakup can be used to infer 7Be(p, γ)8Bradiative capture cross section if

purely Coulomb (ie nuclear interaction negligible)

purely E1 (ie no E2 contribution)

occurs at first-order (ie no higher-order effects)⇒We study 8B breakup addressing these issues[G. Goldstein, P.C., D. Baye PRC 76, 024608 (2007)]

– p.8/19

Parallel-momentum distributions8B + Pb @ 44AMeV (MSU) [Davids PRL 81, 2209 (98)]

θc = 1.5

θc = 2.4

θc = 3.5

44AMeV

pc‖ (MeV/c)

dσ

bu/d

p c‖

[mb/(

MeV

/c)]

2100205020001950

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

Excellent agreement with exp.(no fitting parameter)in particular:

magnitude (no scaling factor)

width

asymmetry due to E1-E2interferences(no need to renormalise E2)

Validates the DEA for breakup calculations

Indicates that a simple description of 8B is sufficient

– p.9/19

Analysis of the dynamics8B + Pb @ 44 AMeV (MSU) [Davids PRL 81, 2209 (98)]

Coul.

C.+N.

pc‖ (MeV/c)

dσ

bu/d

p c‖

[mb/(

MeV

/c)]

2100205020001950

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

E1

C.+N.

pc‖ (MeV/c)

2100205020001950

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

1st

C.+N.

pc‖ (MeV/c)

2100205020001950

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

Nuclear interactionnegligibleat forward angles

Significant E1-E2interference(asymmetry)

First-order:more asymmetric⇒ higher-order

– p.10/19

Parallel-momentum @ 81AMeV8B + Pb @ 81AMeV (MSU) [Davids PRL 81, 2209 (98)]

θc = 1.0

θc = 1.5

θc = 2.5

81AMeV

pc‖ (MeV/c)

dσ

bu/d

p c‖

[mb/(

MeV

/c)]

2850280027502700

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

Less good agreement with exp.(no fitting parameter)in particular:

too high 10–25%

a bit too wide

but correct slope

Representative of the actual exp./th. uncertainty ?

Larger energy ⇒ larger relativistic effects ?

– p.11/19

Analysis of the dynamics8B + Pb @ 81 AMeV (MSU) [Davids PRL 81, 2209 (98)]

C.C.+N.

pc‖ (MeV/c)

dσ

bu/d

p c‖

[mb/(

MeV

/c)]

2850280027502700

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

E1C.+N.

pc‖ (MeV/c)

2850280027502700

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

FOC.+N.

pc‖ (MeV/c)

2850280027502700

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

Same conclusions about the dynamics as @ 44AMeV

– p.12/19

Energy distribution8B + Pb @ 83AMeV (MSU) [Davids PRL 83, 2750 (01)]

Exp.F.O.

E1

Coul.

C.+N.

E (MeV)

83AMeV

s

p

d

dσ

bu/d

E(m

b/M

eV)

21.510.50

140

120

100

80

60

40

20

0

Fair agreement with exp. (need better 8B model?)

No influence of nuclear interaction

Small influence of E2 ⇒ no study of E2

Higher-order effects (s and d , while p )– p.13/19

Angular distribution8B + Pb @ 52AMeV (RIKEN) [Kikushi PLB 391, 261 (97)]

Exp.F.O.

E1

Coul.

C.+N.

θ (deg)

E = 1.25–1.5 MeV

dσ

bu/d

θ(b

/rad

)

1086420

0.1

0.01

0.001

Good agreement with experiment

Nuclear interaction influent only at large angle

Small influence of E2 ⇒ no study of E2

First-order too large ⇒ higher-order effects– p.14/19

No effect of nuclear interaction?Actually there is an effect of nuclear interaction,even at low scattering angle:

C.

C.+N.

θ (deg)

dσ/d

θ(b

/rad

)

109876543210

1.4

1.2

1

0.8

0.6

0.4

0.2

0

E = 0.5–0.75 MeV

C.+N. oscillates around C. [Ogata PRC 73, 024605 (2006)]⇒ unseen for too low angular resolution

What affects these oscillations?– p.15/19

S17

Using this 8B description, the 7Be(p, γ)8B S17 is

Junghans 03

Baby 03Hammache 01

E (MeV)

S(e

Vb)

1.51.2510.750.50.250

30

25

20

15

10

5

0

We obtain S17 = 19.2 b eV at E = 0Good agreement with Hammache [PRL 86, 3985 (01)]Too low but good shape compared to Junghans

[PRC 68, 065803 (03)]

– p.16/19

ConclusionStudy of several 8B Coulomb breakup experiments

New reaction model: Dynamical Eikonal Approx.

Good agreement for various observables withonly one reaction model, and one 8B description⇒ physics of Coulomb breakup well understood

Nuclear interaction negligible at forward angleLarge E1-E2 interference in breakupSignificant higher-order effects

⇒Uncertainties in extracting σcapture from σbu

S17 from 8B description in agreement with direct data

Future: influence of 8B descriptionreasons of the oscillations in angular distr.

– p.17/19

Relativistic correction

PiR→ mPvi = ~K0

NR→ DEA calcul.

NR→ mcvc

R→ pc

avant correction +90 MeV/cavant correctionapres correction

pc‖ (MeV/c)

dσ/d

p c‖

(mb/(

MeV

/c))

210020502000195019001850

5

4

3

2

1

0

Distribution shifted to the right momentum,

and increased in magnitude and width

– p.18/19

Influence of the projectile descriptionFirst test: different scattering lengths (still Ic = 0)

Exp.a0 = −7 fm

a0 = −2.8 fma0 = 5.9 fm

E (MeV)

dσ/d

E(m

b/M

eV)

totals

p

d

f

21.510.50

140

120

100

80

60

40

20

0

Difference only in s wave

in magnitude (no distortion)⇒ small influence of projectile description ?

A proper test should include Ic

– p.19/19