Breakup Reactions and Spectroscopic Factors: a Theoretical - - PowerPoint PPT Presentation

Breakup Reactions and Spectroscopic Factors: a Theoretical Viewpoint

Pierre Capel ULB, Belgium

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Breakup reaction

Breakup used to study exotic nuclear structures e.g. halo nuclei: large matter radius small Sn or S2n ⇒ seen as dense core with neutron halo Short lived ⇒ studied through reactions like breakup: halo dissociates from core by interaction with target Information sought through reactions: Binding energy (e.g. 19C) lj of halo neutron(s) (e.g. 31Ne) SF

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Introduction

Reaction models rely on single-particle model

• f a two-body projectile (core c + fragment f):

[Tr + V (r) − ǫ]φnlj(r) = 0, with ∞

0 |φnlj(r)|2dr = 1

In reality, there is admixture of configurations:

AY (Jπ) = A−1X(Jπ c ) ⊗ f(lj) + . . .

The overlap wave function is ψlj(r) = A−1X(Jπ

c )|alj(r)|AY (Jπ)

Spectroscopic Factor: Slj = ∞

0 |ψlj(r)|2dr

Single-particle approximation ≡ ψlj =

• Sljφnlj

⇒ usual idea: Slj = σexp

bu /σth bu

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11Be+Pb→10Be+n+Pb @69AMeV Experiment:

[Fukuda et al. PRC 70, 054606 (2004)]

They get Ss1/2 = 0.72 for 10Be(0+)⊗n(2s1/2) (our) Theory:

[Goldstein et al. PRC 73, 024602 (2006)]

• 1:3

• 6

[PC et al. PRC 70, 064605 (2004)]

Experiment Convoluted d5/2 Coulomb + Nuclear E (MeV) dσ/dE (b/MeV) 2.5 2 1.5 1 0.5 0.07 0.06 0.05 0.04 0.03 0.02 0.01

With Ss1/2=1

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Outline

Breakup models: CDCC, Time-Dependent, Dynamical Eikonal Approximation What do we probe in breakup ? Peripherality of breakup reactions (ANC vs SF) Description of the continuum Projectile-target interaction (VPT) Influence of couplings upon halo wave function Can we get SF from ANC? Ratio of angular distributions: a new way to remove VPT dependence Conclusion

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Framework

Projectile (P) modelled as a two-body system: core (c)+loosely bound fragment (f) described by H0 = Tr + Vcf(r) Vcf adjusted to reproduce bound state Φ0 and resonances Target T seen as structureless 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) with initial condition Ψ(r, R) − →

Z→−∞ eiKZ+···Φ0(r)

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CDCC

Solve the three-body scattering problem: [TR + H0 + VcT + VfT] Ψ(r, R) = ETΨ(r, R) by expanding Ψ on eigenstates of H0 Ψ(r, R) =

i χi(R)Φi(r)

with H0Φi = ǫiΦi Leads to set of coupled-channel equations (hence CC) [TR + ǫi + Vii] χi +

j=i Vijχj = ETχi,

with Vij = Φi|VcT + VfT|Φj The continuum has to be discretised (hence CD)

[Tostevin, Nunes, Thompson, PRC 63, 024617 (2001)]

Fully quantal approximation No approx. on P-T motion, no restriction on energy But expensive computationally (at high energies)

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Time-dependent model

P-T motion described by classical trajectory R(t)

[Esbensen, Bertsch and Bertulani, NPA 581, 107 (1995)] [Typel and Wolter, Z. Naturforsch. A54, 63 (1999)]

P structure described quantum-mechanically by H0 Time-dependent potentials simulate P-T interaction Leads to the resolution of time-dependent Schrödinger equation (TD) i ∂ ∂tΨ(r, b, t) = [H0 + VcT(t) + VfT(t)]Ψ(r, b, t) Solved for each b with initial condition Ψ − →

t→−∞ Φ0

Many programs have been written to solve TD Lacks quantum interferences between trajectories

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Dynamical Eikonal Approximation

Three-body scattering problem: [TR + H0 + VcT + VfT] Ψ(r, R) = ETΨ(r, R) with condition Ψ − →

Z→−∞ eiKZΦ0

Eikonal approximation: factorise Ψ = eiKZ Ψ TRΨ = eiKZ[TR + vPZ + µPT 2 v2] Ψ Neglecting TR vs PZ and using ET = 1

2µPTv2 + ǫ0

iv ∂ ∂Z

• Ψ(r, b, Z) = [H0 − ǫ0 + VcT + VfT]

Ψ(r, b, Z) solved for each b with condition Ψ − →

Z→−∞ Φ0(r)

This is the dynamical eikonal approximation (DEA)

[Baye, P. C., Goldstein, PRL 95, 082502 (2005)]

Same equation as TD with straight line trajectories

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15C + Pb @ 68AMeV Comparison of CDCC, TD, and DEA

[PC, Esbensen, and Nunes, PRC 85, 044604 (2012)]

dσbu/dE

Exp. dea td cdcc E (MeV) dσbu/dE (mb/MeV)

5 4 3 2 1 400 300 200 100

All models agree

Data: [Nakamura et al. PRC 79, 035805 (2009)]

dσbu/dΩ

dea td cdcc θ (deg) dσbu/dΩ (b/sr)

5 4 3 2 1 140 120 100 80 60 40 20

DEA agrees with CDCC TD reproduces trend but lacks oscillations

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ANC vs SF

Is Slj = σexp

bu /σth bu ?

Is breakup really sensitive to SF ? i.e. do we probe the whole overlap wave function ? Isn’t breakup peripheral? i.e. sensitive only to asymptotics ? ψlj(r) − →

r→∞ Clj e−κr

Asymptotic Normalisation Coefficient: Clj Test this with two descriptions of projectile with different interiors but same asymptotics.

[PC and Nunes, PRC 75, 054609 (2007)]

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SuSy transformations

Use 2 Vcf with different interior but same asymptotics

• btained by SuSy transfo. [D. Baye PRL 58, 2738 (1987)]

SuSy Deep r (fm) Veff (MeV) 10 8 6 4 2 20

• 20
• 40
• 60
• 80
• 100

SuSy Deep r (fm) up3/2 (fm−1/2) 10 8 6 4 2 0.6 0.4 0.2

• 0.2
• 0.4

Deep potential ⇒ spurious deep bound state ⇒ node in physical bound state Remove deep state by SuSy ⇒ remove node but keep same asymptotics (ANC and phase shift) Analyse difference in σth

bu between deep vs SuSy

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Peripherality of breakup reactions

8B+58Ni @ 26MeV

20 40 60 80 100 120 140 θ (degrees) 20 40 60 80 100 120 140 (dσ/dΩ)

Deep SuSy

8B+208Pb @ 44AMeV

SuSy Deep E (MeV) dσbu/dE (b/MeV) 3 2.5 2 1.5 1 0.5 0.5 0.4 0.3 0.2 0.1

No difference between deep and SuSy potentials

at low and intermediate energies,

• n light and heavy targets,

for energy and angular distributions

⇒ breakup probes only ANC ⇒ SF extracted from measurements are questionable?

[PC, Nunes, PRC 75, 054609 (2007)]

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Similar study

Garcia-Camacho et al. NPA 776, 118 (2006)

11Be+Pb @70AMeV

0.5 1 1.5 2

ε (MeV)

500 1000 1500 2000 2500 3000

dσ/dε (mb/MeV)

Using either single particle wave function (solid)

• r its asymptotic expansion (dashed)

⇒ same conclusion with SF=1

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Asymptotic version

ψlj and φnlj exhibit same asymptotics: ψlj(r) − →

r→∞ Clj e−κr

φnlj(r) − →

r→∞ bnlj e−κr

⇒ Asymptotic version of the single-particle approx.: ψlj − →

r→∞ Clj bnljφnlj ⇒ Slj = C2

lj

b2

nlj

Since ANC accessible to breakup reactions, can we still extract SF from reaction data? What effects of couplings between configurations ? ψlj compared to φnlj SF Slj ANC Clj

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c-f system with couplings

We use a model where core can be in different states Φi(ξ) described as levels of deformed rotor ΨJπ =

i ψi(r)Yi(Ω)Φi(ξ)

The c-f Hamiltonian reads

[Nunes NPA 596, 171 (1996)]

H0 = Hc + Tr + Vcf(r, β, ξ) with Vcf(r, β, ξ) = V0

• 1 + exp
• r−R0[1+βY 0

2 (Ω)]

a

−1 ⇒ set of coupled equations [Tr + Vii(r) + Ei − ǫ]ψi(r) = −

i′=i Vii′(r)ψi′(r),

with Vii′(r) = Φi(ξ)Yi(Ω)|Vcf(r, β, ξ)|Φi′(ξ)Yi′(Ω) We analyse the validity of single-particle approx. for one-neutron halo nucleus 11Be

[PC, Danielewicz, Nunes, PRC 82, 054612 (2010)]

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Influence of coupling (ψ vs. φ)

11Be ≡ 10Be+n has two bound states

ε1/2+ = −0.504 MeV Ψ1/2+ = ψs1/2Φ0+ +ψd3/2Φ2+ + ψd5/2Φ2+

β = 0.8 β = 0.6 β = 0.4 β = 0.2 β = 0 r (fm) |rψrot

2s1/2|/

• Srot

2s1/2 (fm−1/2)

14 12 10 8 6 4 2 0.4 0.3 0.2 0.1

ε1/2− = −0.184 MeV Ψ1/2− = ψp1/2Φ0+ +ψp3/2Φ2+ + ψf5/2Φ2+

β = 0.8 β = 0.6 β = 0.4 β = 0.2 β = 0 r (fm) |rψrot

1p1/2|/

• Srot

1p1/2 (fm−1/2)

14 12 10 8 6 4 2 0.5 0.4 0.3 0.2 0.1

⇒ single-particle approx. fails: ψlj(r) =

• Sljφnlj(r)

But, for the ground state, ψs1/2 − →

r→∞

Ss1/2φ2s1/2 ∀β

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Comparing S and C2/b2

We find ψs1/2 − →

r→∞

Ss1/2φ2s1/2 ∀β ⇒ Asymptotic version of single particle approx.? i.e. is C2

lj/b2 nlj a good approx. of Slj ?

C2

p1/2/b2 1p1/2

S1p1/2 C2

s1/2/b2 2s1/2

S2s1/2 β Snlj or C2

lj/b2 nlj

1 0.8 0.6 0.4 0.2 1 0.9 0.8 0.7 0.6 0.5

g.s.: Small admixture,

• approx. OK

e.s.: Large admixture,

• approx. fails β > 0.2

⇒ Approx. breaks at large admixture and/or coupling?

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Exploring the model

To understand this, we push the model to its limits

β = 0.8 β = 0.4 β = 0.1 S2s1/2 C2

s1/2/b2 2s1/2

1 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2

General trend validates S ∼ C2/b2 Very large admixture

• btained even for small β
• Approx. breaks down at

large couplings for large admixtures ⇒ S ∼ C2/b2 for small coupling strength β and/or when component is dominant (i.e. large S) i.e. when coupling term in equations is small

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Sensitivity to the c-f continuum

Is breakup sensitive only to bound-state properties? Influence of c-f continuum

11Be on Pb @ 69AMeV

V6 V5 V4 V3 V2 V1 E (MeV) (dσbu/dE)/b2

1s1/2 (fm b/MeV)

2 1.5 1 0.5 3 2.5 2 1.5 1 0.5

[PC, Nunes, PRC 73, 014615 (2006)]

Sensitivity to continuum of projectile I can get what you want for SF... (PC 2006)

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Role of continuum

Where does it come from? p-wave contributions

⇒ need contraints on c-f continuum e.g. from microscopic structure calculations Wave functions

• 0.1
• 0.2
• 0.3

• 0.2
• 0.4
• 0.6

• 0.2
• 0.4
• 0.6

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Influence of VPT

11Be on C @ 67AMeV

Sensitivity to P-T optical potentials NB: Coulomb breakup less sensitive to VPT ⇒ phenomenological inputs not free from uncertainty ⇒ cautious when extracting SF/ANC from data Can we remove/reduce the sensitivity to VPT? Maybe using the Ratio technique...

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Recoil Excitation and Breakup

Assumes

[R. Johnson et al. PRL 79, 2771 (1997)]

adiabatic approximation VnT = 0 ⇒ excitation and breakup due to recoil of the core Elastic scattering: dσel

dΩ = |F00|2( dσ dΩ)pt

F00 =

• |Φ0|2eiQ · rdr

Q ∝ (K − K′) ⇒ scattering of compound nucleus ≡ form factor × scattering of pointlike nucleus Similarly for breakup:

dσbu dEdΩ = |FE0|2( dσ dΩ)pt

|FE0|2 =

ljm

• Φljm(E)Φ0eiQ · rdr
• 2

⇒ explains similarities in angular distributions provides the idea for the ratio technique...

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Ratio technique

dσbu/dσel = |FE0(Q)|2/|F00(Q)|2 completely independent of reaction process not affected by VPT; i.e. the same for all targets probes only projectile structure no need to normalise exp. cross sections Test this using Dynamical Eikonal Approximation,

[B. Baye, P.C., G. Goldstein, PRL 95, 082502 (2005)]

without adiabatic approximation including VnT Alternative: dσbu/dσsum = |FE0|2 =

ljm

• Φljm(E)Φ0eiQ · rdr
• 2

with dσsum

dΩ

= dσel

dΩ + dσinel dΩ +

• dσbu

dEdΩdE

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Testing with DEA

11Be+Pb @ 69AMeV [P. C., R. Johnson, F. Nunes, PLB 705, 112 (2011)]

|FE,0|2 Ratio dσsum/dσR dσbu/dEdΩ θ (deg)

10 8 6 4 2 103 102 10 1 0.1 10−2 10−3 10−4

removes most of the angular dependence REB predicts ratio = |FE0|2 confirmed by DEA calculations ⇒ probe structure with little dependence on reaction

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(In)sensitivity to VPT

11Be+C @ 67AMeV 11Be+Pb @ 69AMeV

|FE,0|2 Q (fm−1) (dσbu/dEdΩ)/(dσsum/dΩ) (MeV−1)

0.3 0.25 0.2 0.15 0.1 0.05 0.1 10−2 10−3 10−4

Similar for Coulomb and nuclear dominated collisions ⇒ nearly independent of the reaction process

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Sensitivity to projectile description

Study sensitivity to binding energy bound-state orbital

E0 = 5 MeV E0 = 0.5 MeV E0 = 50 keV θ (deg) (dσbu/dEdΩ)/(dσsum/dΩ) (MeV−1)

10 8 6 4 2 1 0.1 10−2 10−3 10−4 10−5 10−6

0d5/2 0p1/2 1s1/2 θ (deg) (dσbu/dEdΩ)/(dσsum/dΩ) (MeV−1)

10 8 6 4 2 0.1 10−2 10−3 10−4 10−5

Sensitive to both binding energy and orbital in both shape and magnitude Works better for loosely-bound projectile (adiabatic approximation ?)

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Sensitivity to radial wave function

0s1/2 R = 4 fm R = 1 fm R = 2.585 fm r (fm) u (fm−1/2) (a)

12 10 8 6 4 2 0.4 0.2

• 0.2
• 0.4

0s1/2 R = 4 fm R = 1 fm R = 2.585 fm θ (deg) |FE0|2 (MeV−1) (b)

10 8 6 4 2 0.06 0.05 0.04 0.03 0.02 0.01

Changes in |FE0|2 similar to those in ulj Forward angles probe asymptotics of ulj Large angles probe the interior of ulj may be difficult to distinguish experimentally ⇒ Ratio scans radial wave function ⇒ maybe can get SF

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Conclusion and outlook

Good understanding of reaction process Breakup models agree with each other (@70AMeV) SF extracted from σexp

bu /σth bu BUT:

Probes only ANC but maybe link with SF? Sensitive to description of continuum to be constrained by structure models? Sensitive to VPT can be reduced using ratio Next step: improve projectile description core excitation, e.g. XCDCC microscopic description

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Thanks to my collaborators

Filomena Nunes Daniel Baye Mahir Hussein

Universidade de São Paulo

Ron Johnson Henning Esbensen Ian Thompson

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15C + Pb @ 20AMeV dσbu/dE

dea td cdcc E (MeV) dσbu/dE (mb/MeV)

5 4 3 2 1 500 400 300 200 100

TD ≡ CDCC DEA too high dσbu/dΩ

td (str. lines) dea td cdcc θ (deg) dσbu/dΩ (b/sr)

18 16 14 12 10 8 6 4 2 12 10 8 6 4 2

TD gives trend of CDCC (lacks oscillations) DEA peaks too early DEA=CDCC due to Coulomb deflection (TD straight lines)

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