The R-matrix method and 12C()16O

Pierre DescouvemontUniversité Libre de Bruxelles, Brussels, Belgium

1. Introduction

2. The R-matrix formulation: elastic scattering and capture

3. Application to 12C()16O

4. Conclusions and outlook

Introduction

• Many applications of the R-matrix theory in various fields

• “Common denominator” to all models and analyses

• Can mix theoretical and experimental information

• Two types of applications: data fittingvariational calculations

• Application to 12C()16O: nearly all recent papers

References:A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257F.C. Barker, many papers

R-matrix formulation

• Main idea: to divide the space into 2 regions (radius a)

• Internal: r ≤ a : Nuclear + coulomb interactions

• External: r > a : Coulomb only

Internal region

16O

Entrance channel12C+

Exit channels

12C(2+)+

15N+p, 15O+n

12C+

CoulombNuclear+Coulomb:R-matrix parametersCoulomb

In practice limited to low energies (each J must be considered individually). well adapted to nuclear astrophysics

Example: 12C+

Physical parameters = “observed” parameters

Resonances:

R-matrix parameters = “formal” parameters

Poles:

Similar but not equal

16O

1- , ER=2.42 MeV, =0.42 MeV

Reduced width 2 :=22 P(ER), with P = penetration factor

12C+

R-matrix parameters = poles

Background pole

Isolated resonances:

Treated individually

High-energy states with the same J

Simulated by a single pole = background

Energies of interest

Non resonant calculations possible:only a background pole

a. Hamiltonian: H =E

With, for r large:

Il, Ol= Coulomb functions

Ul = collision matrix (→ cross sections)

= exp(2il) for single-channel calculations

• Total wave function

b. Wave functions

• Set of N basis functions u(r) with

Derivation of the R matrix (elastic scattering)

c. Bloch-Schrödinger equation:

With L = Bloch operator (restore the hermiticity of H over the internal region)

Replacing int(r) and ext(r) by their definition:

Solving the system, one has:

R-matrix parameters

R matrix

=reduced width

P=penetration factorS=shift factor

Reduced width: proportional to the wave function in a ”measurement of clustering”

Dimensionless reduced width

“first guess”: 2=0.1

Total width:

Depend on a

1.E-10

1.E-081.E-06

1.E-04

1.E-021.E+00

1.E+02

-1 0 1 2

l=2l=0

-4

-3

-2

-1

0-1 0 1 2

l=2

l=0

-1

0

1

2

-1 0 1 2

l=2l=0

-3

-2

-1

0

1

-1 0 1 2

l=2

l=0Sl

Pl

E (MeV)

+ n+ 3He

Penetration and shift factors P(E) and S(E)

Two approaches:

1. Fit: The number of poles N is determined from the physics of the problem

In general, N=1 but NOT in12C()16O : N=3 or 4 (or more) are fitted

Phase shift:

2. Variational calculations (ex: microscopic calculations):

• N= number of basis functions

• are calculated (depend on a, but should not)

Breit-Wigner approximation: peculiar case where N=1

One-pole approximation: N=1

Resonance energy:

Thomas approximation:

Then R-matrix parameters(calculated)

Observed parameters(=data)

Capture cross sections in the R-matrix formalism

New parameters: = gamma width of the poles = interference sign between the poles

is equivalent to the Breit-Wigner approximation if N=1

Relative phase between Mint and Mext : ±1Mint and Mext are NOT independent of each other:

a must be commonU in Mext should be derived from R in Mint

Sometimes in the literature:

exp(-Kr)

Extension to 12C()16O: N>1

• Problem: many experimental constraints (energies, and widths)→ how to include them in the R-matrix fit?

• Previous techniques: fit of the R-matrix parameters2+11.52

3 poles + background →

• 12 R-matrix parameters to be fitted

• + constraints (experimental energies, widths)

New technique: start from experimental parameters (most are known) and derive R matrix parameters strong reduction of the number of parameters!

• Generalization of the Breit-Wigner formalism: link between observed and formal parameters when N>1

C. Angulo, P.D., Phys. Rev. C 61, 064611 (2000) C. Brune, Phys. Rev. C 66, 044611 (2002)

• idea:

• Information for E2:• 2+ phase shift• E2 S-factor• spectroscopy of 2+ states in 16O: energy and widths

2+11.52Three 2+ states + background

Energy(MeV)

width(MeV)

width (eV)

-0.24 ? 0.097

2.68 3.68 x 10-4 0.0057

4.36 1.39 x 10-2 0.61

Backg. 10 ? ?

122232

3 parameters + interference signs in capture

2 steps: 1) phase shifts: widths2) S factor: width of the background

the S-factor is fitted with a single free parameter

From phase shift

From S factor

Application to 12C()16O: E2 contribution

Main goal: to reduce the number of free parameters

First step: fit of the 2+ phase shift

0

1

2

3

4

5

0.0 0.1 0.2 0.3 0.4 0.5 0.6

2

Emax=3.4 MeV

Emax=4.2 MeV

)MeV(21

2 parameters: 24

21 and

-30

-20

-10

0

10

0 1 2 3 4 5 6

Ec.m. (MeV)

phas

e sh

ift (d

eg)

total

HS

HS +1 + 3

HS + 1

Phase shift:

Strong influence of the background!

2+11.52

Second step: fit of the E2 S-factor

1 remaining parameter:

4 poles→4 signs 1, 2, 3, 4, 1=+1 (global sign)4=+1 (very poor fits with 4=-1)

4

0.1

1

10

100

1000

0 1 2 3 4 5

Ec.m. (MeV)

E2

S-fa

ctor

(keV

b)

+/-

+/-

-/+

-/++/+

+/+

-/-

-/-

0

10

20

30

40

50

0 10 20 30 40 50

c2

-/++/+

-/-

+/-

)eV(4

SE2(300 keV)=190-220 keV-b

Paper by Kunz et al., Astrophy. J. 567 (2002) 643

Similar analysis (with new data)

SE2(300 keV)=85 ± 30 keV-b

very different result

Origin: difference in the background treatment

Here: background at 10 MeVKunz et al.: background at 7.2 MeV

R matrix:

-3

-2

-1

0

1

2

3

4

0 1 2 3 4 5 6

R m

atrix pole 4

pole 1pole 3

S factor at 300 keV “well” known background

Between 1~3 MeV, terms 1 and 4: have opposite signsare large and nearly constant

Several equivalent possibilities

-scattering does not provide without ambiguities!

21

Consistent with a recent work by J.M. Sparenberg

Recent work by J.-M. Sparenberg: Phys. Rev. C69 (2004) 034601

Based on supersymmetry (D. Baye, Phys. Rev. Lett. 58 (1987) 2738)

acts on bound states of a given potential without changing the phase shifts

V V

Supersymmetric transformation

Both potentials have exactly the same phase shifts (different wave functions)

r r

Original potential Transformed potential

With this method: different potentials with Same phase shifts Different bound-state properties

Example: V(r)=V0 exp(-(r/r0)2)/r2, with V0=43.4 MeV, r0=5.09 fmNo bound state

V(r)

Supersymmetric partners

Identical phase shifts!

Conclusion: It is possible to define different potentials giving the same phase shifts but

different No direct link between the phase shifts and the bound-state properties Consistent with the disagreement obtained for R-matrix analyses using

different background properties (~ potential) the background problem should be reconsidered!

21

One indirect method: cascade transitions to the 2+ state

F.C. Barker and T. Kajino, Aust. J. Phys. 44 (1991) 369

L. Buchmann, Phys. Rev. C64 (2001) 022801

•Weakly bound: -0.24 MeV

•Capture to 2+ is essentially external

•Mint negligible

The cross section to the 2+ state is proportional to 2

1-0.5

0

0.5

1

1.5

0 20 40 60 80 100

r (fm)

I ()

7Be(p)8B

3He()7Be

12C(p)13N

12C()16O is probably the best example where the interplay between experimentalists, theoreticians and astrophysicists is the most important

Required precision level too high for theory alone we essentially rely on experiment

E1 probably better known than E2 (16N -decay)

Elastic scattering is a useful constraint, but not a precise way to derive

Possible constraints from astrophysics?

New project 16O+→12C (Triangle, North-Carolina)

21

“Final” conclusions

What do we know?

0.1

1

10

100

1000

0 0.5 1 1.5 2

Ec.m. (MeV)

E2

S-fa

ctor

(keV

b) Angulo 2000

Kunz 2001

What do we need?

• Theory: reconsider background effects

• Precise E1/E2 separation (improvement on E2)

• Capture to the 2+ state

• Data with lower error bars:precise data near 1.5 MeV are more useful than data near 1 MeV with a huge error

Please avoid this!