Transcript The R-matrix method and 12C(a,g)16O Pierre Descouvemont

The R-matrix method and 12C(a,g)16O
Pierre Descouvemont
Université Libre de Bruxelles, Brussels, Belgium
1. Introduction
2. The R-matrix formulation: elastic scattering and capture
3. Application to 12C(a,g)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 fitting
variational calculations
• Application to 12C(a,g)16O: nearly all recent papers
References:
A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257
F.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
Exit channels
12C(2+)+a
Entrance channel
12C+a
Internal region
12C+a
16O
15N+p, 15O+n
Coulomb
Nuclear+Coulomb:
R-matrix parameters
Coulomb
In practice limited to low energies (each Jp must be considered individually).
 well adapted to nuclear astrophysics
R-matrix parameters = poles
Example: 12C+a
1- , ER=2.42 MeV, Ga=0.42 MeV
12C+a
Reduced width g2 :
Ga=2 g2 P(ER), with P = penetration factor
16O
Physical parameters
= “observed” parameters
R-matrix parameters
= “formal” parameters
Resonances:
Poles:
Similar but not equal
Background pole
High-energy states with the same Jp
Simulated by a single pole = background
Isolated resonances:
Energies of interest
Treated individually
Non resonant calculations possible:
only a background pole
Derivation of the R matrix (elastic scattering)
a. Hamiltonian: H Y=E Y
With, for r large:
Il, Ol= Coulomb functions
Ul = collision matrix (→ cross sections)
= exp(2idl) for single-channel calculations
b. Wave functions
•
Set of N basis functions ul(r) with
•
Total wave function
c. Bloch-Schrödinger equation:
With L = Bloch operator (restore the hermiticity of H over the internal region)
Replacing Yint(r) and Yext(r) by their definition:
Solving the system, one has:
R matrix
P=penetration factor
S=shift factor
R-matrix parameters
Depend on a
=reduced width
Reduced width: proportional to the wave function in a  ”measurement of clustering”
Dimensionless reduced width
“first guess”: q2=0.1
Total width:
Penetration and shift factors P(E) and S(E)
a + 3He
a+n
1.E+02
2
1.E+00
1.E-02 -1
Pl
0
l= 0
1.E-04
1.E-06
1
2
l= 2
1
1.E-10
0
1
l= 0
-2
-4
0
1
2
-1
0
1
2
1
-1
-3
-1
-1
0
Sl
l= 2
0
1.E-08
-1
l= 0
2
0
-1
l= 2
-2
-3
E (MeV)
l= 0
l= 2
Phase shift:
Two approaches:
1. Fit:
The number of poles N is determined from the physics of the problem
In general, N=1 but NOT in12C(a,g)16O : N=3 or 4 (or more)
are fitted
2. Variational calculations (ex: microscopic calculations):
•
•
N= number of basis functions
are calculated (depend on a, but d 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
exp(-Kr)
 New parameters: Gg = gamma width of the poles
el = interference sign between the poles

is equivalent to the Breit-Wigner approximation if N=1
 Relative phase between Mint and Mext : ±1
Mint and Mext are NOT independent of each other:
a must be common
U in Mext should be derived from R in Mint
 Sometimes in the literature:
Extension to 12C(a,g)16O: N>1
• Problem: many experimental constraints (energies, a and g widths)
→ how to include them in the R-matrix fit?
• Previous techniques: fit of the R-matrix parameters
11.52
2+
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 a and g widths
Application to 12C(a,g)16O: E2 contribution
Main goal: to reduce the number of free parameters
11.52
Three 2+ states + background
2+
Energy
(MeV)
a width
(MeV)
g width
(eV)
21
-0.24
?
0.097
22
2.68
3.68 x 10-4
0.0057
23
4.36
1.39 x 10-2
0.61
Backg.
10
?
?
From phase
shift
From S factor
3 parameters + interference signs in capture
 2 steps: 1) phase shifts: a widths
2) S factor: g width of the background
 the S-factor is fitted with a single free parameter
First step: fit of the 2+ phase shift
2 parameters:
g 12 andg 42
Phase shift:
2+
10
total
phase shift (deg)
11.52
0
0
1
2
3
4
5
-10
HS + 1
HS
-20
HS +1 + 3
-30
Ec.m. (MeV)
Strong influence of the background!
6
Second step: fit of the E2 S-factor
1 remaining parameter:
Gg4
4 poles→4 signs e1, e2, e3, e4,
e1=+1 (global sign)
e4=+1 (very poor fits with e4=-1)
1000
50
40
+/+
-/+
c2
30
E2 S-factor (keV b)
100
20
-/-
10
-/+
10
+/+
-/+
+/-
1
-/+/+
+/-
+/-
0
-/-
0.1
0
10
20
30
G4g (eV)
40
50
0
1
2
3
4
Ec.m. (MeV)
SE2(300 keV)=190-220 keV-b
5
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 MeV
Kunz et al.: background at 7.2 MeV
R matrix:
“well” known
S factor at 300 keV
Between 1~3 MeV, terms 1 and 4:
background
have opposite signs
are large and nearly constant
4
Several equivalent possibilities
3
R matrix
2
1
a-scattering does not provide
without ambiguities!
pole 4
0
pole 1
-1
pole 3
Consistent with a recent work by
J.M. Sparenberg
-2
-3
0
1
2
3
g 12
4
5
6
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
Original potential
Transformed potential
V
V
r
r
Supersymmetric
transformation
Both potentials have exactly the same phase shifts (different wave functions)
 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 fm
No bound state
V(r)
Supersymmetric
partners
Identical phase shifts!
Conclusion:
 It is possible to define different potentials giving the same phase shifts but
different g 12
 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!
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
2
proportional to g 1
“Final” conclusions
What do we know?

12C(a,g)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 b-decay)
 Elastic scattering is a useful constraint, but not a precise way to derive g 12
 Possible constraints from astrophysics?
 New project 16O+g→a12C (Triangle, North-Carolina)
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
E2 S-factor (keV b)
1000
Angulo 2000
100
Kunz 2001
10
Please
avoid
this!
1
0.1
0
0.5
1
Ec.m. (MeV)
1.5
2