Analysis of low-energy scattering and weakly-bound states through - - PowerPoint PPT Presentation

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Analysis of low-energy scattering and weakly-bound states through effective-range functions. Application to 12C+α.

Jean-Marc Sparenberg, Oscar Leonardo Ram´ ırez Su´ arez1

Nuclear Physics and Quantum Physics, Universit´ e libre de Bruxelles (ULB), Brussels, Belgium, EU

June 2nd, 2015 Centre for Nuclear and Radiation Physics, University of Surrey, Guildford, England, UK

1PhD student 2011-2014 Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 1 / 29

ulbnorm

1

The 12C (α, γ)16O reaction in nuclear astrophysics

2

(Weakly-)bound states and asymptotic normalization constant

3

(Low-energy) elastic scattering and phase shifts

4

Effective-range expansion(s) for scattering and bound states

5

Analysis of the 12C + α experimental p-wave phase shifts

6

Analysis of the 12C + α experimental d-wave phase shifts

7

Conclusions and perspectives

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 2 / 29

ulbnorm The 12C (α, γ)16O reaction in nuclear astrophysics

1

The 12C (α, γ)16O reaction in nuclear astrophysics

2

(Weakly-)bound states and asymptotic normalization constant

3

(Low-energy) elastic scattering and phase shifts

4

Effective-range expansion(s) for scattering and bound states

5

Analysis of the 12C + α experimental p-wave phase shifts

6

Analysis of the 12C + α experimental d-wave phase shifts

7

Conclusions and perspectives

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 3 / 29

ulbnorm The 12C (α, γ)16O reaction in nuclear astrophysics

12C + α → γ + 16O radiative capture in red-giant stars

Importance: competes with 3α → 12C in red-giant helium burning ⇒ C/O in Universe Key quantity: cross section σ at “low” energy (Gamow peak) Ec.m. = 300 keV Dominant transitions: E1 (resp. E2)

◮ from p (resp. d) 12C + α scattering state ◮ to s 16O ground state (spin 0)

Experiment: Coulomb repulsion ⇒ very small cross section ⇒ data for E > 1 MeV only Betelgeuse [Dupree, Gilliland,

Hubble ST, NASA, ESA, 1996]

Theoretical extrapolation to low energy: necessary but difficult because p and d subthreshold bound states enhance cross section but are not well known

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 4 / 29

ulbnorm The 12C (α, γ)16O reaction in nuclear astrophysics

12C + α → γ + 16O

direct measurements (recent)

p-wave (E1) and d-wave (E2) capture to s-wave ground state Astrophysical S-factor S(E) = σ(E)E exp(2πη) dimensionless Sommerfeld parameter η = Z1Z2e2/v relative velocity v =

• 2Ec.m./2µ

[Plag et al., PRC 2012] [Kunz et al., PRL 2001; Hammer, Fey et al., NPA 2005]

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 5 / 29

ulbnorm The 12C (α, γ)16O reaction in nuclear astrophysics

Indirect information on subthreshold states

Well-known energies (experiment)

◮ 1−: Ec.m. = −45 keV ◮ 2+: Ec.m. = −245 keV

Structure rather-well understood (theory)

◮ 1−: 1 particule-1 hole shell-model state ◮ 2+: 12C+α cluster structure, part of

rotational band with 0+

2 and 4+

But asymptotic normalization constants (ANC ≈ width) not well known

• ✁

9.59 1- 8.87 2- 7.117 1- 6.917 2+ 12C+α S(E) 6.130 6.049 3- 9.85 2+ 7.162 + + 0.3 MeV E1,E2 π 16O

[Buchmann, private com. 2004]

Radiative cascade transitions ⇒ (difficult) access to 1− and 2+ states

[Kettner et al., ZPA 1982; Redder et al., NPA 1987; Plag et al., PRC 2012] 16N β-delayed α decay ⇒ access to 1− state [Azuma et al., PRC 1994; Tang et al., PRC 2010]

α transfer reactions: 12C(6Li, d)16O and 12C(7Li, t)16O ⇒ access to 1− and 2+ states [Brune et al., PRL 1999] SE1(300 keV) = 101 ± 17 keV b and SE2(300 keV) = 42+16

−23 keV b

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 6 / 29

ulbnorm The 12C (α, γ)16O reaction in nuclear astrophysics

Indirect information from 12C + α elastic scattering

Access to 1− and 2+ states? Motivated high-precision elastic scattering remeasurement

[Tischhauser et al., PRL 2002, PRC 2009]

Analysis with R-matrix formalism but questions raised

◮ partial-wave decomposition: contradicts scattering inverse problem? ◮ background description: channel radius + subthrehsold pole +

background pole

[Sparenberg, PRC 2004]

More reliable way?

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 7 / 29

ulbnorm (Weakly-)bound states and asymptotic normalization constant

1

The 12C (α, γ)16O reaction in nuclear astrophysics

2

(Weakly-)bound states and asymptotic normalization constant

3

(Low-energy) elastic scattering and phase shifts

4

Effective-range expansion(s) for scattering and bound states

5

Analysis of the 12C + α experimental p-wave phase shifts

6

Analysis of the 12C + α experimental d-wave phase shifts

7

Conclusions and perspectives

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 8 / 29

ulbnorm (Weakly-)bound states and asymptotic normalization constant

Two-body potential model

Isolated two-nuclei system ⇒ center-of-mass-movement separation

◮ one-body Schr¨

• dinger equation

Hϕ(r) = Eϕ(r), H = p2

2µ + V (r),

E = 2

2µk2

◮ relative coordinate r, reduced mass µ, unit choice 2/2µ = 1 ◮ (complex) wave number k

Interacting through central potential (no spin), nuclear Bohr radius aB = 2Z1Z2e2

4πǫ0µ ,

η =

1 aBk

V (r) = Vnuclear(r) + VCoulomb(r) ∼

r→∞ 0,

VCoulomb(r) = 2ηk

r

Rotational invariance ⇒ CSCO {H, L2, Lz}

◮ angular separation, partial waves

ϕElm(r) = uEl(r)

r

Y m

l (θ, φ)

◮ one-dimensional radial Schr¨

• dinger equation, effective potential Vl(r)
• − d2

dr2 + l(l+1) r2

+ V (r)

• uEl(r) = EuEl(r)

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 9 / 29

ulbnorm (Weakly-)bound states and asymptotic normalization constant

Bound states and asymptotic normalization constants

Bound states: negative (discrete) energy Eb = −κ2

b

Potential model: normalized wave functions ub(r)

◮ asymptotic behaviour

ub(r) ≈

r→∞ Cb

e−κbr (2κbr)|ηb|

• Cbe−κbr for η = 0
• ◮ slowly decreasing for weakly bound states (κb ≪ 1)

◮ Asymptotic Normalization Constant (ANC) Cb ◮ related to nuclear vertex constant

More sophisticated models (coupled-channel, microscopic. . . ):

◮ unnormalized overlap functions

ub(r) ≡

• A
• ψ12C ⊗ ψα ⊗ Y m

l δ(r) r

• |ψ16O
• ◮ spectroscopic factor

∞ dr|ub(r)|2 = 1

◮ same definition for ANC Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 10 / 29

ulbnorm (Low-energy) elastic scattering and phase shifts

1

The 12C (α, γ)16O reaction in nuclear astrophysics

2

(Weakly-)bound states and asymptotic normalization constant

3

(Low-energy) elastic scattering and phase shifts

4

Effective-range expansion(s) for scattering and bound states

5

Analysis of the 12C + α experimental p-wave phase shifts

6

Analysis of the 12C + α experimental d-wave phase shifts

7

Conclusions and perspectives

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 11 / 29

ulbnorm (Low-energy) elastic scattering and phase shifts

Stationary-scattering-state partial-wave expansion

Stationary scattering states ϕk1z(r)

◮ solutions of 3D Schr¨

• dinger equation with positive (continuous) energy

E = k2

◮ rotationally invariant around z ◮ asymptotic behaviour: scattering amplitude fk(θ)

(2π)3/2ϕk1z(r) →

r→∞ eikz + fk(θ)eikr

r (η = 0)

◮ elastic-scattering differential cross section

dσ dΩ(E, θ) = |fk(θ)|2

Partial waves ϕklm(r) = ukl(r)

r

Y m

l (θ, ϕ)

◮ solutions of radial Schr¨

• dinger equation ukl(r)

◮ asymptotic behaviour: scattering phase shifts δl(k)

ukl(r) →

r→∞ 1 k

• 2

πeiδl(k) sin

• kr − l π

2 + δl(k)

• (η = 0)

Expansion (few terms at low E) (2π)3/2ϕk1z(r) = ∞

l=0 clϕkl0(r)

⇒ fk(θ) =

1 2ik

∞

l=0(2l + 1)(e2iδl − 1)Pl(cos θ)

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 12 / 29

ulbnorm Effective-range expansion(s) for scattering and bound states

1

The 12C (α, γ)16O reaction in nuclear astrophysics

2

(Weakly-)bound states and asymptotic normalization constant

3

(Low-energy) elastic scattering and phase shifts

4

Effective-range expansion(s) for scattering and bound states

5

Analysis of the 12C + α experimental p-wave phase shifts

6

Analysis of the 12C + α experimental d-wave phase shifts

7

Conclusions and perspectives

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 13 / 29

ulbnorm Effective-range expansion(s) for scattering and bound states

Effective-range function(s) and expansion(s)

Scattering matrix Sl(k) ≡ e2iσl × e2iδl ≡ Γ(l+1+iη)

Γ(l+1−iη) × cot δl(k)+i cot δl(k)−i

Effective-range function, analytical at k2 = 0, for η = 0 K0(k2) ≡ k cot δ0(k) ≈

k2≈0 − 1 a0 + r0 2 k2 − P0r3 0k4 . . .

(l = 0) Kl(k2) ≡ k2l+1 cot δl(k) ≈ − 1

al + rl 2 k2 − Plr3 l k4 . . .

(l = 0)

◮ scattering length a0, effective range r0, shape parameter P0. . . ◮ few-term Taylor expansion at low energy but poles for E ∈ R+ ◮ Pad´

e expansion more useful: few terms for all energies, corresponding exactly-solvable potential from SUSYQM, e.g. neutron-proton 1D2, 3 parameters [Midya et al., arXiv:1501.04011, PRC 2015]

Charged case: Kl(k2) ≡ 2wl(η2)

l!2a2l+1

B

• π cot δl(k)−i

e2πη−1

+ h(η2)

• ≈ − 1

al + . . .

◮ wl(η2) = l

n=0

• 1 + n2/η2

◮ h(η2) = ψ(iη) − ln(iη) + (2iη)−1,

ψ = digamma function

◮ historically introduced to get analytic function, real for E ∈ R+

[Breit et al. PR 1936, Landau & Smorodinsky JPAS USSR 1944, Schwinger, Bethe PR 1949]

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 14 / 29

ulbnorm Effective-range expansion(s) for scattering and bound states

Simplified effective-range function for charged case

∆l(k2) = π cot δl(k)

e2πη−1

Also analytic at E = 0, also real for E ∈ R+ Zeros E0,j = k2

0,j and poles E∞,j = k2 ∞,j, in particular for resonances

δl(k) π/2 =

• even,

for k = k∞,j

• dd,

for k = k0,j Low energy resonance → weakly bound-state study ⇒ conjecture (OLRS): ∆(−κ2

b) = 0 (single zero)

Actually related to modified K-matrix K−1

l

(k2) = − 2wl(η2)

l!2a2l+1

B

∆l(k2)

[Humblet et al. NPA 1976, PRC 1990], which confirms conjecture

Pad´ e expansion expected to be efficient ∆l(k2) ≈ l!2a2l+1

B

−2al N

j=1[1−(k2/k2 0,j)]

M

j=1[1−(k2/k2 ∞,j)] ◮ N + M + 1 parameters: scattering length, zeros, poles

(real or mutually-conjugated complex in energy plane)

◮ optional high-energy constraint: δl(k) ∼k→∞ k−1 ⇐

⇒ N = M + 1

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 15 / 29

ulbnorm Effective-range expansion(s) for scattering and bound states

Bound-state properties from scattering matrix

Bound state = scattering-matrix pole in complex wave-number plane

◮ position: energy ◮ residue: ANC

Sl(k) ∼

k→iκb

(−1)l+1ie−πηb

C2

b

k−iκb

(1)

[Baz’, Zel’dovich & Perelomov 1969, Blokhintsev et al. SJPN 1984, Mukhamedzhanov & Tribble, PRC 1999]

⇒ bound-state energy and ANC (vertex constant) seem to be extractable form scattering-matrix properties, hence from phase shifts Apparent contradiction with scattering-inverse-problem result: phase-shifts determine number of bound states Nb, through Levinson theorem: δl(0) − δl(∞) = Nbπ but binding energy and ANC arbitrary

[Chadan & Sabatier, 1989]

Explanation: (1) only valid for potentials decreasing faster than exp(−2κbr) ⇒ useful for weakly-bound states only

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 16 / 29

ulbnorm Effective-range expansion(s) for scattering and bound states

Example: 12C+α d-wave phase-equivalent potentials

• 0.2
• 0.15
• 0.1
• 0.05

0.05 1 2 3 4 5 δ2 (rad) Ec.m. (MeV) background phase shift potential models

• 20
• 10

10 20 30 40 2 4 6 8 10 12 Veff (MeV) r (fm) no bound state Cb = 20 x 103 fm−1/2 Cb = 200 x 103 fm−1/2 Cb = 2000 x 103 fm−1/2

Built with arbitrary ANC by SUSYQM, only some of them reproduce rotational band ⇒ Cb ≈ 144.5 ± 8.5 × 103 fm−1/2 [Sparenberg, PRC 2004] Only some of them are short-ranged [Sparenberg et al., INPC 2010]

• 14
• 12
• 10
• 8
• 6
• 4
• 2

2 2 4 6 8 10 12 14 Vnuc (MeV) r (fm) Cb = 20 x 103 fm−1/2 180 x 103 fm−1/2 192 x 103 fm−1/2 200 x 103 fm−1/2 2000 x 103 fm−1/2 1e-15 1e-10 1e-05 1 100000 1e+10 5 10 15 20 25 30 35 40 |Vnuc| (MeV) r (fm) Cb = 20 x 103 fm−1/2 180 x 103 fm−1/2 192 x 103 fm−1/2 200 x 103 fm−1/2 2000 x 103 fm−1/2

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 17 / 29

ulbnorm Effective-range expansion(s) for scattering and bound states

Bound-state properties from effective-range function

Standard effective-range function: cot δl(iκb) = i F −1

l

≡ Kl(−κ2

b) − 2wl(η2

b)h(η2 b)

l!2a2l+1

B

= 0, Cb = κl

b Γ(l+1+|ηb|) l!

• dF −1

l

dk2

• − 1

2

k2=−κ2

b

⇒ Pad´ e expansion of ERF gives easy access to Eb and Cb

[Iwinski et al. PRC 1984, Blokhintsev et al. PRC 1993]

Same idea with pole expansion of modified K-matrix

[Humblet et al. NPA 1976, Mukhamedzhanov et al. PRC 1999]

Simplified effective-range function: ∆(−κ2

b) = 0

Cb = Γ(l+1+|ηb|)

|ηb|l

• aB

2|wl(ηb)|

• d∆l

dk2

• − 1

2

k2=−κ2

b

Weakly-bound state ⇒ ∆l(E) ≈E∈[Eb,0]

l!2a2l+1

B

−2al

• 1 − E

Eb

• Cb ≈ Γ(l+1+|ηb|)

l!

κl+1

b

• al

wl(ηb)

• ⇒ Cb error bar from al error bar

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 18 / 29

ulbnorm Effective-range expansion(s) for scattering and bound states

Test: 12C+α d-wave Gaussian potential

∆2(k2) has a simpler energy dependence than ∆2(k2)w2(η2) ≡ −l!2a2l+1

B

K−1

2 (k2)/2

Linear extrapolation to negative energies (a2 = 58 910 fm5, r2 = 0.1580 fm−3)

◮ Eb = −213 keV

[Eexact

b

= −245 keV]

◮ Cb ≈ 130 540 fm−1/2 for linear ∆2

[Cexact

b

= 138 400 fm−1/2]

◮ Cb ≈ 123 000 fm−1/2 for linear ∆2w2

[Sparenberg et al., PRC 2010; Mukhamedzhanov et al., PRC 1999]

Linear on small energy range only ⇒ extrapolation from experimental data range (Ec.m. ≥ 2 MeV) probably difficult

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 19 / 29

ulbnorm Analysis of the 12C + α experimental p-wave phase shifts

1

The 12C (α, γ)16O reaction in nuclear astrophysics

2

(Weakly-)bound states and asymptotic normalization constant

3

(Low-energy) elastic scattering and phase shifts

4

Effective-range expansion(s) for scattering and bound states

5

Analysis of the 12C + α experimental p-wave phase shifts

6

Analysis of the 12C + α experimental d-wave phase shifts

7

Conclusions and perspectives

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 20 / 29

ulbnorm Analysis of the 12C + α experimental p-wave phase shifts

Simplified effective-range function ∆1(k2) = π cot δ1(k)

e2πη−1

1 2 3 4 5 Ec.m. [MeV] 0.0 0.5 1.0 1.5 2.0 2.5 δ1 [rad] 1 2 3 4 5 Ec.m. [MeV] 0.00020 0.00015 0.00010 0.00005 0.00000 0.00005 0.00010 0.00015 0.00020 ∆1

raw no res. no res. no bs

Very precise experimental data [Tischhauser et al., PRL 2001, PRC 2009] Based on R-matrix fit, with Cb = 166+63

−166 × 1012 fm−1/2

Well-known resonance at Eres = 2.44 MeV, removed ⇒ approximately linear ∆1(1 − E/Eres) Extrapolation from first two points ⇒ bound state at Eb = −130 keV [Eexp

b

= −45 keV] ⇒ promising!

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 21 / 29

ulbnorm Analysis of the 12C + α experimental p-wave phase shifts

ANC extraction from ∆1(1 − E/Eres)(1 − E/Eexp

b

)

1 2 3 4 5 Ec.m. [MeV] 0.0 0.5 1.0 1.5 2.0 2.5 δ1 [rad] 1 2 3 4 5 Ec.m. [MeV] 0.0000010 0.0000011 0.0000012 0.0000013 0.0000014 0.0000015 0.0000016 ∆1

no res. no bs Brune et al. 99

Structure (threshold effect?) above 4.1 MeV ⇒ data not used Structure on [1.96 − 4.1] MeV, plateau below 2.3 MeV (if no R-matrix bias and if realistic error bars!) Excellent agreement with Cb = 208(19) × 1012 fm−1/2 [Brune et al., PRL

1999] but accuracy potentially 10 times better

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 22 / 29

ulbnorm Analysis of the 12C + α experimental p-wave phase shifts

Numerical fits of ∆1 with [N/M] Pad´ e approximants

• rder

Ecm (MeV) Nfit χ2/Nfit ANC (×1012 fm−1/2) [2/0] 6.35 212(2) [2/0] [1.96-2.48] 27 0.98 215(1) [3/0] → [2/0] [2/1] 1.77 221(1) [2/1] [1.96-3.88] 176 0.99 226(1) [4/0]∗ [3/1] → [2/0] [2/2] 0.57 258(2) [5/0] 0.51 194(8) [4/1]∗ [3/2]∗ 0.52 238.9(2.5) [2/3] 0.51 203(12) [6/0] 0.51 156(24) [5/1] → [4/0]∗ [4/2] 0.51 223(4) [3/3] → [2/2] [2/4]∗

Ecm ∈ [1.96-4.1] MeV, Nfit = 249, χ2 on randomized data

∗: unphysical bound or resonant states; →: zero/pole merging

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 23 / 29

ulbnorm Analysis of the 12C + α experimental p-wave phase shifts

Most plausible and precise fits

2 4 6 8 10 Ec.m. [MeV] 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 δ1 [rad] 2 2 4 6 8 10 Ec.m. [MeV] 0.0000008 0.0000010 0.0000012 0.0000014 0.0000016 0.0000018 ∆1 no res. no bs

exp. Brune [2/0] [2/1] [2/2] [3/2] [4/2]

Weakness of the method: order choice delicate! Order [2/2]: not compatible with [Brune et al.] value Order [3/2]: unphysical “anti-resonance” Orders [2/0], [2/1], [4/2] ⇒ Cb = 220.5(6.5) × 1012 fm−1/2

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 24 / 29

ulbnorm Analysis of the 12C + α experimental d-wave phase shifts

1

The 12C (α, γ)16O reaction in nuclear astrophysics

2

(Weakly-)bound states and asymptotic normalization constant

3

(Low-energy) elastic scattering and phase shifts

4

Effective-range expansion(s) for scattering and bound states

5

Analysis of the 12C + α experimental p-wave phase shifts

6

Analysis of the 12C + α experimental d-wave phase shifts

7

Conclusions and perspectives

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 25 / 29

ulbnorm Analysis of the 12C + α experimental d-wave phase shifts

Simplified effective-range function ∆2(k2) = π cot δ2(k)

e2πη−1

1 2 3 4 5 Ec.m. [MeV] 1.5 1.0 0.5 0.0 0.5 1.0 1.5 δ2 [rad] 1 2 3 4 5 Ec.m. [MeV] 0.004 0.003 0.002 0.001 0.000 0.001 0.002 ∆2

raw no resonance no res. no bs

Experimental phase shifts 10 times less precise than p-wave (except above 4.1 MeV) [Tischhauser et al., PRL 2001, PRC 2009] Based on R-matrix fit, with Cb = 154(18) × 103 fm−1/2 Well-known resonances at Eres = 2.68 and 4.39 MeV, removed ⇒ ∆no res.

2

still not linear Extrapolation [3/3] from first three points ⇒ zero at 170 keV [Eexp

b

= −245 keV] ⇒ not promising!

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 26 / 29

ulbnorm Analysis of the 12C + α experimental d-wave phase shifts

ANC extraction from ∆no res.

2

(1 − E/Eexp

b

)?

1 2 3 4 5 Ec.m. [MeV] 1.5 1.0 0.5 0.0 0.5 1.0 1.5 δ2 [rad] 1 2 3 4 5 Ec.m. [MeV] 0.00010 0.00008 0.00006 0.00004 0.00002 0.00000 ∆2

no res. no bs Brune et al. 99

Not constant, linear fit crosses E-axis ⇒ unphysical resonance or bound state often appears Large error bars ⇒ no improvement expected wrt Cb = 114(10) × 103 fm−1/2 [Brune et al., PRL 1999] Preliminary results (error estimates to be improved): [3/3] Cb = 144(18) × 1012 fm−1/2, [4/3] Cb = 112(8) × 103 fm−1/2

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 27 / 29

ulbnorm Conclusions and perspectives

1

The 12C (α, γ)16O reaction in nuclear astrophysics

2

(Weakly-)bound states and asymptotic normalization constant

3

(Low-energy) elastic scattering and phase shifts

4

Effective-range expansion(s) for scattering and bound states

5

Analysis of the 12C + α experimental p-wave phase shifts

6

Analysis of the 12C + α experimental d-wave phase shifts

7

Conclusions and perspectives

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 28 / 29

ulbnorm Conclusions and perspectives

Conclusions and perspectives

Elastic-scattering phase shifts can be used to check/evaluate bound-state binding energy and asymptotic normalization constant (ANC), but only for weakly-bound states A simplified effective-range function (related to but simpler than K-matrix for the charged case) seems sufficient to do the job Pad´ e approximants of effective-range functions are useful

◮ to fit data on large energy intervals (not a low-energy method anymore) ◮ with minimal number of parameters (scattering length, poles and zeros) ◮ to solve the inverse-scattering problem with exactly-solvable potentials

(for the neutral case)

◮ to extrapolate positive-energy data to weakly negative energies

but choosing their order can be delicate

12C + α: factor-3 (at least?) improvement for p-wave ANC,

not promising for d-wave Similar simplification for radiative capture (cf K matrix) ⇒ consistent calculation of astrophysical S-factor? Exactly-solvable potentials for the charged case?

Jean-Marc Sparenberg (ULB) Analysis of low-energy 12C+α scattering University of Surrey 2015 29 / 29