Breakup models Pierre Capel 17 July 2015 1 / 28 Summary of - - PowerPoint PPT Presentation

Breakup models

Pierre Capel 17 July 2015

1 / 28

Summary of Lecture 1

In a quantum collision various reactions can take place each corresponds to one channel that can be open or closed Elastic scattering corresponds to

a + b → a + b

always open ; described by stationary scattering states

Ψ

K ˆ

Z(R) −→

R→∞(2π)−3/2

• eiKZ+... + f(θ)eiKR+...

R

• Cross section obtained from scattering amplitude f(θ)

dσ dΩ = | f(θ)|2

Other channels affect elastic scattering

⇒ use of optical potential Uopt(R) = V(R) + iW(R)

imaginary part W simulates absorption from elastic channel We now see how to include the breakup channel

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Halo nuclei

Halo nuclei

Light, neutron-rich nuclei small S n or S 2n low-ℓ orbital One-neutron halo

11Be ≡ 10Be + n 15C ≡ 14C + n

Two-neutron halo

6He ≡ 4He + n + n 11Li ≡ 9Li + n + n

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 ✲ N ✻ Z

n 1H 2H 3H 3He 4He 6He 8He 6Li 7Li 8Li 9Li 11Li 7Be 9Be 10Be 11Be 12Be 14Be 8B 10B 11B 12B 13B 14B 15B 17B 19B 9C 10C 11C 12C 13C 14C 15C 16C 17C 18C 19C 20C 22C 12N 13N 14N 15N 16N 17N 18N 19N 20N 21N 22N 23N 13O 14O 15O 16O 17O 18O 19O 20O 21O 22O 23O 24O

Proton halœs are possible but less probable : 8B, 17F Two-neutron halo nuclei are Borromean. . .

c+n+n is bound but not two-body subsystems

e.g. 6He bound but not 5He or 2n

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Halo nuclei

Borromean nuclei

Named after the Borromean rings. . . [M. V. Zhukov et al. Phys. Rep. 231, 151 (1993)]

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Breakup reaction

Breakup reaction

Breakup ≡ dissociation of projectile in constituent clusters by interaction with target

11Be + 12C

→

10Be + n + 12C 8B + 208Pb

→

7Be + p + 208Pb

Used to study cluster structure in nuclei e.g. halo nuclei infer reaction rates of astrophysical interest Need a good understanding of the reaction mechanism i.e. an accurate theoretical description of reaction coupled to a realistic model of projectile Elastic breakup ≡ all clusters measured in coincidence target not excited

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Breakup reaction

Framework

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

H0 = Tr + Vc f(r) Vc f adjusted to reproduce P spectrum

Target T seen as structureless particle

R r 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)

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Breakup reaction

Projectile Hamiltonian H0

H0 = −2∆r 2µc f + Vc f(r) Vc f has usually a Woods-Saxon form factor Vc f(r) = V0 1 + e(r−R0)/a c- f relative motion described by H0 eigenstates ǫnl < 0 : discrete set of bound states H0 φnlm(r) = ǫnl φnlm(r) ǫ > 0 : c-f continuum ≡ broken up projectile H0 φklm(r) = ǫ φklm(r) where ǫ = 2k2/2µc f

Breakup ≡ transition from bound state to continuum through interaction with target (Coulomb and nuclear) Breakup can take place in one or more steps will be sensitive to both bound and continuum states (for simplicity, spin will be ignored)

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Breakup reaction

Example : 11Be

1/2+

• 0.504 1s1/2

1/2−

• 0.184 0p1/2

10Be + n

5/2+ 1.274 d5/2

11Be spectrum 11Be ≡ 10Be(0+) + n 10Be cluster assumed in 0+ ground state

(extreme shell model)

⇒ spin and parity of 11Be states

fixed by angular momenta l and j of n :

1/2+ ground state in s1/2 1/2− excited state in p1/2 5/2+ resonance in d5/2 ⇒ fit V0 in s1/2, p1/2 and d5/2 waves

but not in p3/2. . .

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Breakup reaction

Projectile-target interaction : VcT and V fT

The breakup channel is now included in the collision description However other channels not included : absorption of the fragment by the target breakup of the core . . .

c-T and f -T interactions described by optical potentials VcT and VfT

Their imaginary parts simulate the other channels Usually chosen in the literature

VcT : problematic if c-T scattering not measured ⇒ extrapolate what exists VfT : many N-T global potentials exist

[Becchetti and Greenlees, Phys. Rev. 182, 1190 (1969)] [Koning and Delaroche NPA 713, 231 (2003) ]

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Reaction models

Three-body Scattering Problem

Within this framework breakup reduces to three-body problem

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

with the initial condition

Ψ(r, R) −→

Z→−∞ eiKZ+···φl0m0(r)

⇔ P in its ground state φl0m0 impinging on T

R r T P c f

Various methods developed to solve that equation Coupled-channel method with discretised continuum (CDCC) Time-dependent approach (TD) (semiclassical) Eikonal approximation

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Reaction models CDCC

Coupled-Channel method

The eigenstates of H0 {|φi} are a basis in r :

H0|φi = ǫi|φi

Idea : expand Ψ on that basis :

Ψ(r, R) =

i χi(R)|φi

• TR + H0 + VcT + VfT
• Ψ(r, R)

= ETΨ(r, R) ⇔

• i

TRχi(R)|φi + χi(R)H0|φi + (VcT + VfT)χi(R)|φi =

• i

ETχi(R)|φi φ j| ⇓ TR χj(R) + ǫ j χ j(R) +

• i

φ j|VcT + VfT|φi χi(R) = ET χ j(R)

This is a set of coupled equations in χi(R) where the coupling terms are φ j|VcT + VfT|φi i.e. connect the various projectile states through the P-T interaction Problem : continuum states φklm are not discrete. . .

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Reaction models CDCC

Discretising the Continuum

Model of breakup requires description of continuum must be tractable in computations, i.e. discrete

[Rawitscher, PRC 9, 2210 (1974)]

φklm

with k ∈ R+ → φilm with i ∈ N Various methods exist : mid-point : divide continuum in bins [ǫi − ∆ǫi/2, ǫi + ∆ǫi/2] and choose φilm(r) = φkilm(r) to describe bin i average the wave function over the bin

φilm(r) = 1 Wi ǫi+ ∆ǫi

2

ǫi− ∆ǫi

2

fi(ǫ) φklm(r) dǫ

with Wi =

ǫi+ ∆ǫi

2

ǫi− ∆ǫi

2

fi(ǫ)dǫ ⇒ square-integrable wave functions φilm

pseudo-states : solve H0 φilm = ǫ φilm on a finite basis

⇒ square-integrable wave functions φilm but ǫi not chosen

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Reaction models CDCC

CDCC

Continuum Discretised Coupled-Channel : CDCC

[Austern et al. , Phys. Rep. 154, 125 (1987)] [Tostevin, Nunes, Thompson, PRC 63, 024617 (2001)] Recent review : [Yahiro et al. , Prog. Th. Phys. Supp. 196, 87 (2012)]

Fully quantal approximation No approximation on P-T motion, nor restriction on energy But expensive computationally (at high energies) Various codes have been written to solve these coupled equations fresco written by Ian Thompson is free on www.fresco.org.uk

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Reaction models CDCC

CDCC breakup cross sections

Expanding χ into spherical harmonics

χj(R) = 1 R

• L

iLujL(R) Y0

L(Ω)

. . . and coupling l and L into J, the coupled equations read

− 2 2µPT d2 dR2 − L(L + 1) R2

• uJ

c(R) +

• c′

V J

cc′(R)uJ c′(R)

= (ET − ǫ j)uJ

c(R)

with c ≡ {j, L} and JT = L + l These equations are solved assuming the asymptotic behaviour

uJ

c(R) −→ R→∞

• δc0 IL(η, KR) − S J

c0 OL(η, KR)

• where

IL = GL − iFL

incoming Coulomb function

= O∗

L

• utgoing Coulomb function

The S matrix is used to compute the breakup cross sections

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Reaction models CDCC

Example : 8B breakup

8B + 58Ni → 7Be + p + 58Ni

@26MeV

• Exp. :[V. Guimar˜

aes et al. PRL 84, 1862 (2000)]

20 40 60 80 θlab(

7Be) (degrees)

20 40 60 80 100 120 dσ/dΩc (mb/sr)

• Th. :[Tostevin et al. PRC 63, 024617 (2001)]

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Reaction models Time-dependent approach

Time-dependent model

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

T c f P b R(t) r RcT (t) RfT (t)

P structure described quantum-mechanically by H0

Time-dependent potentials simulate P-T interaction

⇒ time-dependent (TD) Schr¨

• dinger equation

i ∂ ∂tΨ(r, b, t) = [H0 + VcT(t) + VfT(t) − Vtraj(t)]Ψ(r, b, t)

Solved for each b with initial condition Ψ(m0) −→

t→−∞ φl0m0

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Reaction models Time-dependent approach

Numerical resolution of the TD Schr¨

• dinger equation

Time-step evolution approximating the evolution operator

Ψ(m0)(r, b, t + ∆t) = U(t + ∆t, t) Ψ(m0)(r, b, t)

with U(t′, t) = exp[ i

• t′

t H(τ)dτ] and Ψ(m0)(r, b, t → −∞) = φl0m0(r)

Faster computation compared to CDCC because each trajectory treated separately Lacks quantum interferences between trajectories Many programs developed to solve TD Partial-wave expansion of Ψ :

[Kido, Yabana, and Suzuki, PRC 50, R1276 (1994)] [Esbensen, Bertsch and Bertulani, NPA 581, 107 (1995)] [Typel and Wolter, Z. Naturforsch.A 54, 63 (1999)]

Expansion on a 3D spherical mesh :

[P . C., Melezhik and Baye, PRC 68, 014612 (2003)]

Expansion on 3D cubic lattice : [Fallot et al. NPA700, 70 (2002)]

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Reaction models Time-dependent approach

Semicassical breakup cross sections

For each trajectory (b) a breakup probability can be computed

dPbu(b) dE = µc f 2k 1 2l0 + 1

• m0
• l,m

|φklm|Ψ(m0)(b, t → ∞)|2

We can build an angular distribution from b ↔ θ

dσbu dEdΩ = dσel dΩ dPbu[b(θ)] dE ,

where dσel/dΩ is obtained from Vtraj And an energy distribution

dσbu dE = 2π ∞ dPbu(b) dE b db

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Reaction models Time-dependent approach

Example : 15C Coulomb breakup

15C ≡ 14C(0+) + n 15C + 208Pb → 14C + n + 208Pb

@68AMeV

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

0.5 0.4 0.3 0.2 0.1 4 3.5 3 2.5 2 1.5 1 0.5 d/dErel (b MeV-1) Erel (MeV) (B) < 6o < 2.1o Folded + scaled Dynamic calc.

• Th. :[Esbensen, PRC 80, 024608 (2009)]

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Reaction models Eikonal approximation

Eikonal approximation

Three-body scattering problem :

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

with condition Ψ(m0) −→

Z→−∞ eiKZφl0m0

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(m0) −→

Z→−∞ φl0m0(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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Reaction models Eikonal approximation

Eikonal cross section

After some mathematical developments. . .

[Goldstein, Baye, P .C. PRC 73, 024602 (2006)]

dσbu dEdΩ ∝ 1 2l0 + 1

• m0
• lm
• ∞

J|m0−m|(qb) S (m0)

klm (b) bdb

• 2

, S (m0)

klm (b) = φklm|

Ψ(m0)(Z → ∞) are breakup amplitudes dσbu dEdΩ

• dΩ

−→ dσbu dE ⇒ Dynamical eikonal extends TD

takes into account interferences between trajectories (sum of breakup amplitudes)

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Reaction models Eikonal approximation

Usual Eikonal

iv ∂ ∂Z

• Ψ(r, b, Z) = [H0 − ǫ0 + VcT + VfT]

Ψ(r, b, Z)

The usual eikonal uses adiabatic approx. H0 − ǫ0 ∼ 0

• Ψ(m0)

eik (r, b, Z) = exp

• − i

v Z

−∞

dZ′ VcT(r, b, Z′) + V fT(r, b, Z′)

• φl0m0(r)

Easy to interpret and implement Neglects internal dynamics of projectile

⇒ dynamical eikonal generalises eikonal

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Reaction models Eikonal approximation

Example : 11Be Coulomb breakup

11Be + 208Pb → 10Be + n + 208Pb

@69AMeV

• Exp. :[Fukuda et al. PRC 68, 054606 (2004)]

• E
• 1

• (degrees)

• Th. :[Goldstein, Baye, P

.C. PRC 73, 024602 (2006)]

DEA exhibits interferences (oscillations) Usual eikonal diverges at forward angles (adiabatic approx.)

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Comparison : Coulomb breakup of 15C

15C+Pb @ 68AMeV : energy distribution

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

5 4 3 2 1 400 300 200 100

Excellent agreement between all three models

[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 15C

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Comparison : Coulomb breakup of 15C

15C+Pb @ 68AMeV : angular distribution dea td cdcc θ (deg) dσbu/dΩ (b/sr)

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

TD lacks quantum interferences but reproduces the general trend at small θ DEA exhibits quantum interferences though much less time consuming than CDCC

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Comparison : Coulomb breakup of 15C

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 DEACDCC due to Coulomb deflection Eikonal is a high-energy approximation

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Conclusion

Conclusion

Breakup can be included assuming cluster structure of P Two-body structure leads to a three-body scattering probem CDCC : Ψ expanded over H0 eigenstates

◮ fully quantal model ⇒ valid at all energies ◮ requires continuum discretisation ◮ heavy computationally

Time-dependent : collision simulated by trajectory

◮ semiclassical approximation ⇒ no quantal interferences ◮ simple interpretation and light numerically

Eikonal : high-energy approximation

◮ DEA : includes interferences and dynamic

• nly valid at high energy

◮ Usual eikonal : add adiabatic approximation

⇒ not valid for Coulomb ⇒ important to know the range of validity of the models

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