Effects Of Target Resonances In Low-energy

Nucleon Scattering from Weakly-bound Nuclei

Ken Amos

The University of Johannesburg

South Africa

and

The University of Melbourne,

Australia

1

Colleagues (5 nations – 5 continents)

S. KarataglidisThe University of Johannesburg, South Africa

P. R. Fraser, K. Massen-Hane, I. Bray, A. S. KadyrovCurtin University, Perth, Australia

L. CantonI.N.F.N., Padova, Italy

R. Fossi onU.N.A.M., Mexico City, Mexico

J. P. SvenneThe University of Manitoba, Winnipeg, Canada

and

D. van der KnijffThe University of Melbourne, Australia

2

Why this study?

• Weakly bound radioactive nuclei now formed as RIBs

• Their low excitation spectra have particle emission resonant states.

Questions: with weakly bound nuclei in scattering (beam/ta rget)

• What effect do such particle-emitting states have on cross sections?

• What physical considerations

need be taken in modelling these resonances?

We seek answers using:

• The MCAS method is used to solve Lippmann-Schwinger equations

Currently built for n+A and α+A coupled-channel systems.

• Compound system spectra found:

all bound and (low excitation) resonant states of the cluste r.

• The S-matrix is formed −→ elastic and reaction cross sections.

3

The Lippmann-Schwinger equations:The coupled-channel problem of a two-nucleus system (no resonant states)

• In momentum space , with an interaction matrix, VJπ

cc′(p,q),

(c: unique quantum number sets), the multi-channel T -matrix is

TJπ

cc′(p,q;E) = VJπ

cc′(p,q)

+µ

[

open∑

c′′=1

∫

∞

0

VJπ

cc′′(p,x)x2

k2

c′′ − x2 + iεTJ

π

c′′c′(x,q;E)dx

−closed∑

c′′=1

∫

∞

0

VJπ

cc′′(p,x)x2

h2

c′′ + x2TJ

π

c′′c′(x,q;E)dx

]

• For an n+A system , c = (ℓ 1

2) jI;Jπ; and with Ec: target state energies

k2

c = µ (E−Ec) ; h2

c = µ (Ec − E) .

4

Some details of the MCAS approach:References: 1. K. Amos et al., Nucl. Phys. A728, 65 (2003).

2. L. Canton et al. Phys. Rev. C 83, 04763 (2011).

• Pauli principle: coupled-channels and local interactions — can violate P.P.

• Overcome: use orthogonalizing pseudo-potentials ( OPPs)

Vcc′(p,q) −→ Vcc′(p,q) + λAc(p)Ac′(q)

Ac(p) : bound state functions for occupied orbits (in Vc,c)

• Effect: New Hamiltonian has no solutions involving occupied states

• If a target state is a resonance: Replace excitation energy ( Ec) with

Ec −→ Ec +∆c(E) + i12U(E)Γc

where the shift that restores causality is given by

∆c(E) =Γc

2

1

πP∫

∞

0

U(E′)

(E′ −E)dE′

5

Why the scale factor, U(E)?

• Lorentzian forms: can lead to non-physical features in results:

1. Reaction cross sections

E → 0 (scattering threshold) may diverge, and

2. bound states in the compound system spectrum

may have spurious widths.

• Cause of problems: Lorentzians are non-zero below the threshold.

• Remedy: use a scaling function on the Lorentzian form.

• Restore causality: Add the energy-dependent shift, ∆c(E)

• Scale function conditions:

1. U(E) = 0 and dU(E)dE

= 0 for E ≤ 0

2. U(E) → 0 as E → ∞3. U(Ec) = 1

6

Case: n+12C scattering – 3 states, 0+g.s.

, 2+1 , 0+2 , no widths

10-3

10-2

10-1

100

Elab

(MeV)

0

2

4

6

8σ el

astic

(b)

0 1 2 3 4 5

5/2_

5/2+

7/2+

1/2_

3/2+

3/2+

9/2+

5/2+

7

n+12C elastic scattering — adding widths to 2+1 , 0+2 states

8

n+12C reaction cross sections — small widths

2 3 4 5E

lab (MeV)

0

0.1

0.2

0.3

0.4σ re

actio

n(b)

0.06 MeV0.05 MeV0.04 MeV0.03 MeV0.02 MeV0.01 MeV0.00 MeV

Γc: 2

+ & 4

+

9

Target nuclei:

8Be

6He

0+

2+

4+

2+

0+

2+

(2+, 1

_, 0

+)

(1_, 2

_)

[5.57 eV]

[1.51 MeV]

[~3.5 MeV]

[108 keV]

[113 keV]

[12.1 MeV]

[7.4 MeV]

0

2

4

6

8

10

12

14

16

Ex(MeV)

10

Case: 7He (n+6He) — g.s. 32

−

: E = 0.445 MeV (0.52 lab.); Γ = 0.15 MeV

1

2

3

σ reac

t(b)

2

4

6

8

σ elas

tic(b

)

g.s.

CausalityLorentzianNo widths

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2E

lab(MeV)

2

4

6

8

σ tota

l(b)

7He g.s. E; Γ = 0.449; 0.145 MeV

11

Case: Spectra of 12C (α+8Be)

0

2

4

6

8

10

12

14

Ex(M

eV)

12C exp. causality

0+

2+

0+

3_

1_

(0+)

(2+)

1+

4+

2_

2_

3_

(8.5x10-3

) keV

(2+)

2+

Lorentzian

(α+8Be)

7.366

3.8 keV 700 keV

12

Case: Spectra of 10Be (α+6He)

-2

0

2

4

6

8

10

Ex(M

eV)

0+

2+

2+ 1

_0+ 2

_

3_2

+

(4_)

2+

3_

0+

2+

1_0

+ 2_2

+2+ 1

_3

_

1+

3+ 1

_4+2+ 2

_3

_

0+

2+

2+

0+

1_2_2

+ 4+3

_

1_3

+1

_

2_2

+

1+

10Be exp. Lorentziancausality

13

Case: α+6He scattering

2 3 4 5 6 7E

cm(MeV)

100

101

102

dσ/d

Ω (

mb/

sr)

95o

105o

95 - 105

40 60 80 100 120 140θ

cm(deg)

100

101

102

103

Ecm

= 4.4

Ecm

= 3.8 (x10)

Data: D. Suzuki et al. Phys. Rev. C 87, 054301 (2013)

14

Conclusions:MCAS: developed to solve coupled-channel two-nucleus cluster pr oblems

This method:

• accounts for target state particle instability

• is free of any unphysical behaviour at the scattering threshold

• satifies the Pauli principle (via OPP)

• conserves causality .

• allows scaling the usual Lorentz form of a resonant target state

(to meet physical requirements while preserving causality )

• yields compund spectra and scattering cross sections

(Results are sensitive to target state particle instabilit y)

Structure and scattering involving weakly-bound RIBs will be influenced.

15

References:

• Basic MCAS: K. Amos et al. Nucl. Phys. A728 , 65 (2003).

• Pauli principle violation and coupled-channel calculatio ns:

L. Canton et al. Phys. Rev. Lett. 94 , 122503 (2005).

K. Amos et al. Phys. Rev. C. 72 , 064604 (2005).

• Predicting resonances in drip-line spectra:

L. Canton et al. Phys. Rev. Lett. 96 , 072502 (2006).

• Non-local effective potentials from MCAS:

P. Fraser et al. Euro. Phys. J. A 35 , 69 (2008).

• MCAS with particle unstable target states:

P. Fraser et al. Phys. Rev. Letts. 101 , 242501 (2008).

L. Canton et al. Phys. Rev. C 83 , 047603 (2011)

16

MCAS details: Optimal functions, χcn(q), from Sturmians,

|χcn〉 =∑

c′ Vcc′ |Φc′n〉 ;∑

c′ G(0)c Vcc′ |Φc′n〉 = −ηn |Φcn〉

• Expand: the potential matrix: Vcc′(p,q) ∼ ∑

nχcn(p) η

−1n

χc′n(q)

• Leads to: a multi-channel S-matrix: Scc′ = δcc′ − iπµ√kckc′ Tcc′

where

Tcc′ =∑

n,n′

χcn(p)(

[η −G0]−1

)

nn′χc′n′(q)

• Involves: a resolvent matrix with

[G0]nn′ = µ

[

open∑

c′′=1

∫

∞

0

χc′′n(x)x2

k2c′′ − x2 + iǫχc′′n′(x) dx

−closed∑

c′′=1

∫

∞

0

χc′′n(x)x2

h2c′′ + x2

χc′′n′(x) dx

]

; [η ]nn′ = ηn δnn′

17

What are Sturmians?

• Two body Hamiltonian H = H0 +V

• Standard Schr odinger equation (E−H0)ΨE = VΨE

(Here E is the spectral variable, and ΨE is the eigenstate)

• Sturmians are the eigensolutions of (E−H0)Φi(E) = V

ηi(E)Φi(E)

(Here E is a parameter, and ηi(E) is a potential scale)

• Spectrum consists of all rescalings that give solution to th e equation

(for a given energy and well-defined boundary conditions).

– The energy E here can be chosen as any useable value

– But of practical importance: use a negative value.

– A single value suffices: the objective is to define a basis exp ansion set

(for use with an energy independent set of interactions)

18

Finding resonances• Rapid determination of all resonances (no matter how narrow)

1− Sel = iπµ

M∑

nn′=1

k χ1n(k)[

(η −G0)−1]

nn′χ1n′(k)

= iπµ

M∑

nn′=1

kχ1n(k)√ηn

[

(

1− η−

1

2G0η−

1

2

)

−1]

nn′

χ1n′(k)√ηn′

• To find resonances: diagonalize the complex, symmetric matrix

N∑

n′=1

[

η−

1

2

]

nn[G0]nn′

[

η−

1

2

]

n′n′

Qn′r = ζrQnr ,

• Resonances: occur at energies for which ℜ(ζr) = 1

• Resonance widths relate to the imaginary part of ζr

19

Choice of scaling function:

• Conditions: on a scaling of resonance widths

1. U(E) and dU(E)dE

= 0 for E ≤ 0

2. U(E) → 0 as E → ∞3. U(Ec) = 1

4. Example of Wigner function form −→0 1 2 3 4

E (MeV)

0

0.5

1

scal

e

εc = 1.0

• A model choice for U(E): a modified Wigner distribution

U(E) = em(

E

Ec

)n

e−m( E

Ec)n

H(E)

m,n are positive integers

H(E) is the Heaviside function (No resonance effect for E < 0)

20

The energy shift ∆c(E):Case of Wigner scale function with m = 1, n = 2, Ec = 1.0,Γc = 2.0

1 2 3E (MeV)

-0.5

-0.25

0

0.25

0.5

0.75

∆ c(Ε)

5 10 15

-0.7

0

0.7

21

n+12C reaction cross section with no Uc(E)

22

n+12C reaction cross sections — using Uc(E)

23

The low excitation (resonant) states of 9Be (relative to the n+8Be threshold)

Energy centroids [widths] are in MeV

Jπ expt. Causality Lorentz

32

− −1.665 −1.669 −1.668

12

+0.018 [0.214] −0.399 −0.399

52

−

0.764 [0.078] 2.499 [0.983] 1.991 [2.302]

12

−

1.11 [1.08] 0.773 [0.54] 0.445 [1.216]

52

+1.384 [0.282] 1.337 [0.406] 1.184 [0.73]

Results shown in black lie within 0.5 MeV of measured ones.

24

Case: 9Be (n+ 8Be )

0 1 2 3 4E

lab(MeV)

0

2

4

6

8

10

12

14

σ elas

tic(b

)

No widthsLorentzianCausality

0 1 2 3 4E

lab(MeV)

0

0.5

1

1.5

2

2.5

σ reac

tion(b

)

25

The low excitation (resonant) states of 7He (relative to the n+6He threshold)

Energy centroids [widths] are in MeV

Jπ = 32

− 52

−

( 12−

)

Experiment 0.445 3.365 ± 0.09 6.425 ± 0.3

[0.150± 0.020] [1.990± 0.170] [4 ±1]

Causality 0.449 [0.145] 3.229 [0.073] 6.294 [10.17]

Lorentz 0.489 [0.177] 3.452 [0.219] 6.764 [0.096]

Results shown in black lie within 0.5 MeV of measured ones.

26

Case: α+8Be scattering

0 1 2 3 4 5 6 7E

lab (MeV)

0

1

2

3

4σ/

σ ruth

θc.m.

60o

90o

120o

60 90 120 150θ

c.m.(deg)

100

101

102

103

dσ/d

Ω (

mb/

sr)

3.0 MeV4.4 MeV6.0 MeV

Elab

27