four-body effects on be+208pb scattering around the ... · total fusion: comparisonwithexperiment...

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Four-body effects on 9Be+208Pb scattering around the Coulomb barrier

Mahir S. HusseinUniversidade de Sao Paulo-Brazil

In collaboration with

• P. Descouvemont and T. Druet (ULB)

• L.F. Canto (UFRJ Rio)

Phys. Rev. C 91, 024606 (2015)

1. Introduction

2. The CDCC method (scattering)

3. 9Be description (3-body ++n)

4. 9Be+208Pb elastic scattering

5. 9Be+208Pb breakup and fusion

6. Conclusion

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1. Introduction

Many CDCC studies (Continuum Discretzed Coupled Channel) Essentially• Three-body problems (two-body projectiles)• Elastic scattering• Fusion (few papers)

Problem for weakly bound projectiles: • very slow convergence (partial wave inside the projectile) of the cross sections

(essentially around the Coulomb barrier)• 11Be+64Zn: up to L=5 (in 11Be) is necessary around the Coulomb barrier

Ref. T. Druet and P.D., EPJA 48 (2012) 147

Here: 9Be+208Pb system (many experimental data are available)• 9Be described by a 3-body structure +n• Simultaneous description of:

• Elastic scattering• Breakup• Fusion

Example: 9Be=+n

• Pseudostates :𝐸 >0• Simulate breakup effects• Depend on the basis

• Ground state: 𝐸 <0• Independent of the basis

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2. The CDCC method

0

2

4

6

8

10

3/2 5/2 1/2

9Be

n

+n

9Be description (3-body ++n)

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Description of 9Be=++n• potential: Buck et al. NPA275 (1977) 246 (2 forbidden states for L=0, 1 fs for L=2)• +n potential: Kanada et al. Prog. Theor. Phys. 61 (1979) 1327 (1 forbidden state for L=0) supersymmetry transform to remove the forbidden states

Both reproduce the elastic phase shifts up to ~20 MeV (deep potentials)

3-body method: Hyperspherical coordinatesTests: different Kmax, number of basis functions

9Be

n

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3. 9Be +n description

Three-body ++n model

3/2

1/25/2

+n

Hyperspherical formalism for 3-body nuclei• essentially spectroscopy, continuum possible (more difficult!)• Hamiltonian: 𝐻 = 𝑇 + 𝑇 + 𝑇 + 𝑉 ( 𝒓 − 𝒓 ) + 𝑉 ( 𝒓 − 𝒓 ) + 𝑉 ( 𝒓 − 𝒓 )

𝒓𝟏 𝒓𝟑

𝒓𝟐

Jacobi coordinates(c.m. removed)

𝒙 = 𝒓𝟐 − 𝒓𝟏𝒚 = 𝒓𝟑 − 𝐴𝟏𝒓𝟏 + 𝑨𝟐𝒓𝟐 /𝐴

𝒚𝒙

Absolute coordinates Hyperspherical coordinates• Ω , Ω• 𝜌 = 𝑥 + 𝑦• tan𝛼 =

𝜌 =hyperradius𝛼= hyperangle

In hyperspherical coordinates: 𝐻 = 𝑇 + 𝑉 𝜌, 𝛼, Ω , ΩEigenstates of 𝑇 : hypersphercial functions 𝒴 𝛼, Ω , Ω = 𝒴 Ω

known functions (analytical)extension of spherical harmonics 𝑌 Ω in 2-body problemsK=hypermoment

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Discretization of the three-body +n continuum

• j=1/2, 3/2, 5/2• Kmax=20• Lagrange functions: N=20• Maximum Emax=12.5 MeV

many pseudostatesMany channels in the CDCC calculation


0

2

4

6

8

12

-2

10

3/2 5/2 1/2

𝐸,(M

eV)

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−ℏ2𝜇

𝑑𝑑𝑅

−𝐿 𝐿 + 1

𝑅+ 𝐸 − 𝐸 𝑢 𝑅 + 𝑉 𝑅 𝑢 𝑅 = 0

At large distances• Nuclear potential negligible, only Coulomb remains

• Wave function 𝑢 𝑅 → 𝐼 𝑘 𝑅 𝛿 − 𝑈 𝑂 𝑘 𝑅with 𝜔=entrance channel

𝐼 𝑥 , 𝑂 𝑥 = incoming and outgoing Coulomb functions

𝑈 =scattering matrix various cross sections (elastic, breakup, etc)

Procedure

• Calculation of the 3-body wave functions

• Calculation of the coupling potentials 𝑉 𝑅

• Solving the coupled-channel system (R matrix)

• Determining the scattering matrices 𝑈 and cross sections

Need to solve the coupled-channel equations

9Be+208Pb elastic scattering

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𝑹

208Pb

𝑟

9Be

𝛼

𝛼

𝑛𝑟

-208Pb potential: Goldring et al. Phys. Lett. B 32, 465 (1970).n-208Pb potential: Koning Delaroche, Nucl. Phys. A 713, 231 (2003).

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• Experimental data (Elab~38-60 MeV, Ecm ~36-58 MeV, Vcoul~38 MeV) • R. J. Woolliscroft et al. Phys. Rev. C 69, 044612 (2004).• N. Yu et al., J. Phys. G37, 075108 (2010).

• Various convergence tests• Emax, number of basis functions: specific to 9Be• Partial waves in 9Be 𝑗 : 3/2±,5/2±,1/2±

• Calculation (with the same conditions) of• Elastic scattering• Breakup• Fusion

• Known problematic issues • from 11Be+64Zn, T. Druet and P.D., EPJA 48 (2012) 147, Di Pietro et al., PRC 85,

054607 (2012)• Convergence (11Be partial waves) slower at low energies• Improved when the binding energy is larger (11Be: 0.5 MeV, 9Be: 1.57 MeV)

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Convergence with Emax (cut off energy in 9Be)

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0.5

0.6

0.7

0.8

0.9

1.0

1.1

0 30 60 90 120 150 180

Emax=2.5

Emax=5

Emax=7.5

Emax=10

Emax=12.5

𝐸_𝑙𝑎𝑏=38 MeV

0

2

4

6

8

12

-2

10

3/2 5/2 1/2

𝜃 (deg)

𝜎/𝜎

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 30 60 90 120

Emax=2.5

Emax=5

Emax=7.5

Emax=10

Emax=12.5



Convergence with j values (9Be partial waves)

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𝜃 (deg)

𝜎/𝜎

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0 30 60 90 120 150 180

j=g.s.

j=3/2

j=3/2 ,5/2

j=3/2 ,5/2 ,1/2


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 30 60 90 120

j=g.s.

j=3/2

j=3/2 ,5/2

j=3/2 ,5/2 ,1/2


𝜎/𝜎

𝜃 (deg)


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0.4

0.6

0.8

1

1.2

0 30 60 90 120 150 180


0

0.2

0.4

0.6

0.8

1

1.2

0 30 60 90 120 150 180


0 30 60 90 120 150 180

𝜃 (deg)

𝜎/𝜎

𝜎/𝜎

Comparison withexperiment


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0

0.2

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0.6

0.8

1

1.2

1.4

0 30 60 90 120 150 180


0 30 60 90 120 150 180

10-1

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10-4

10-2

1

10-3

𝜃 (deg)

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0.2

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1

1.2

1.4

0 30 60 90


0 30 60 90

10-1

10

10-3

10-2

1

𝜎/𝜎

𝜎/𝜎

9Be+208Pb breakup and

fusion

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0

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4

6

8

10

3/2 5/2 1/2

Breakup cross sections: simulated by transitions to pseudostates Equivalent to « inelastic channels »

Breakup cross section: 𝜎 𝐸 = ∑ ℓ𝓁 ℓ𝓁 2𝐽 + 1 𝑈 ℓ𝓁, ℓ𝓁 (𝐸)𝑓=final states

𝑈 ℓ𝓁, ℓ𝓁 (𝐸)=scattering matrix (non diagonal terms gs pseudo states 𝑓)

gs

pseudostates

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Data: R. J. Woolliscroft et al., Phys. Rev. C 68, 014611 (2003).

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0

50

100

150

30 35 40 45 50

Emax=2.5 Emax=5

Emax=7.5 Emax=10

Emax=12.5

0

50

100

150

30 35 40 45 50

j=3/2

j=3/2, 5/2

j=3/2, 5/2, 1/2

𝐸 (MeV)

𝜎(m

b)

𝜎(m

b)𝐸 (MeV)


Fusion cross section: 2 processes• Complete fusion CF: the full projectile is captured• Incomplete fusion IF: the projectile breaks up, and one constituent is captured

Total fusion: sum of both processes: 𝜎 = 𝜎 + 𝜎

Previous CDCC calculations: Many calculations with two-body projectiles• A. Diaz-Torres and I. J. Thompson, PRC65 (2002) 024606: 11Be+208Pb• V. Jha et al., PRC89 (2014) 034605 : 9Be +heavy targets• Many others

Present work: • CDCC with three-body projectiles• Total fusion only

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Calculation from scattering matrices

For spin and parity 𝐽 : matrix 𝑈 𝐸 =

𝑈 𝑈 … 𝑈𝑈 𝑈 … 𝑈⋮ ⋮ ⋮ ⋮

𝑈 𝑈 … 𝑈

Entrance channel: 1Peudostates (breakup) channels: 2N

• Elastic scattering cross section: 𝜎 𝐸 ← 𝑈

• Reaction cross section: 𝜎 𝐸 ← 1 − |𝑈 |

• Breakup cross section: 𝜎 𝐸 ← ∑ |𝑈 |

• Total fusion cross section: 𝜎 𝐸 = 𝜎 𝐸 − 𝜎 𝐸

Note: if real potential: 𝜎 𝐸 = 0 (from unitarity of the scattering matrix ∑ |𝑈 | = 1)


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From the scattering matrices (sum over 𝐽 ) 𝜎 𝐸 = 𝜎 𝐸 − 𝜎 𝐸

Alternative definition of the fusion cross section

𝜎 𝐸 ←4𝑘𝐸

,

∫ 𝑢 𝑟 𝑊 , 𝑟 𝑢 𝑟 𝑑𝑟

• 𝑊 , 𝑟 =imaginary potential (computed from +208Pb and n+208Pb potentials)

• 𝑢 𝑟 =wave function in channel 𝑐

• 𝜎 𝐸 = 0 if 𝑊 , 𝑟 = 0 (as expected)

• Stricly equivalent ( strong numerical test!) more difficult to use (requires the wave function 𝑢 𝑟 ) More accurate at low energies (𝜎 𝐸 ≈ 𝜎 𝐸 ) Allows a decomposition into the different channels

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30 35 40 45 50

Total fusion

gsEmax=2.5Emax=5Emax=7.5Emax=10Emax=12.5

103

100

101

102

30 35 40 45 50Ecm (MeV)

Total fusion

j=g.s.

j=3/2

j=3/2, 5/2

j=3/2, 5/2, 1/2

103

100

101

102

𝑉_𝐵

𝜎(m

b)0

2

4

6

8

12

-2

10

3/2 5/2 1/2

Total fusion: convergence with Emax and j

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Total fusion: comparison with experimentData from M. Dasgupta et al., PRC 70, 024606 (2004), PRL82 (1999) 1395

• Neglecting neutron capture is necessary• Good agreement with the data except at the lowest energies


30 35 40 45 50

Ecm (MeV)

Total fusion

𝜎_𝑇𝐹

(mb)

103

100

101

102

0

200

400

600

800

1000

1200

30 35 40 45 50

• Red lines: capture of and n

• Black lines: capture of only𝜎 𝐸

←4𝑘𝐸

,

∫ 𝑢 𝑟 𝑊 , 𝑟 𝑢 𝑟 𝑑𝑟

n-208Pb imaginary potential set to 0

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𝑁 = 20

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0

2

4

6

8

-2

𝑁 = 30

6. Numerical issues

Role of the 9Be basis: calculation limited to j=3/2- with N=20 and N=30

0.9

0.95

1

1.05

1.1

30 35 40 45 50 55 60

elastic 90 deg.

elastic 60 deg.

breakup

fusion

𝐸 (MeV)

9Be pseudostates (j=3/2-)

Ratio of the cross sections

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6. Numerical issues

Role of the -208Pb potential1. G. Goldring et al., Phys. Lett. B 32, 465 (1970).2. A. R. Barnett and J. S. Lilley, Phys. Rev. C 9, 2010 (1974).

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

30 35 40 45 50 55 60

elastic 60 deg.

elastic 90 deg.

breakup

fusion

𝐸 (MeV)

Effect more important in breakup and fusion

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6. Numerical issues

Main problem: large number of channels high accuracy is required Very long computer times

j=1/2+: 36, j=1/2-: 37, j=3/2+: 59, j=3/2-: 63, j=5/2+: 71, j=5/2-: 71

+ different L values, 𝐽 − 𝑗 ≤ 𝐿 ≤ 𝐽 + 𝑗+ 𝐽 large (~301/2)

Statistical CDCC??M.S. HusseinC. Bertulani,P.D.B. Carlsson

• Explicit treatment of some « important » channels• Statistical treatment of the other pseudostates

• C. A. Bertulani, P. D., M. S. HusseinEPJ Web of Conferences 69 (2014) 00020

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3/2 5/2 1/2

Conclusion

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7. Conclusion

• Spectroscopy of 9Be• Described by a 3-body +n model• Hyperspherical coordinates• and +n potentials reproduced the elastic phase shifts• A small 3-body force is necessary to adjust the energy of the ground state• Fair energy spectrum, spectroscopic properties

• 9Be+208Pb elastic scattering and breakup• +208Pb and n+208Pb taken from literature no fit• Slow convergence with 9Be states (known effect for 2-body nuclei)• Good agreement with experiment

• Fusion• Total fusion: fast convergence and good agreement with the data (except at very low

energies: theory is larger than experiment)

• To be done:• Separation between CF and IF?• Many pseudostates Statistical CDCC??

four-body effects on be+208pb scattering around the ... · total fusion: comparisonwithexperiment...

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