the r -matrix method and 12 c( a,g ) 16 o pierre descouvemont université libre de bruxelles,...
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The R -matrix method and 12 C( a,g ) 16 O Pierre Descouvemont Université Libre de Bruxelles, Brussels, Belgium. Introduction The R -matrix formulation: elastic scattering and capture Application to 12 C( a,g ) 16 O Conclusions and outlook. Introduction. - PowerPoint PPT PresentationTRANSCRIPT
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!