Download Diapositive 1 - Theoretical Astro-, Nuclear

THEORETICAL PREDICTIONS
OF THERMONUCLEAR RATES
P. Descouvemont
1.
2.
3.
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6.
Reactions in astrophysics
Overview of different models
The R-matrix method
Application to NACRE/SBBN compilations
Typical problems/questions
Conclusions
Low level densities:
Light nuclei (typically A < 20, pp chain, CNO)
or close to the drip lines (hot burning)
Then:
• In general data are available:
problems for compilations:
* extrapolation
* coming up with “recommended” cross sections
* computing the rate from the cross section
• Specific models can be used
• Each reaction is different: no systematics
High level densities: Hauser-Feshbach
* accuracy?
* data with high energy resolution are required
Types of reactions:
1. Capture (p,g), (a,g): electromagnetic interaction
• Non resonant
• Isolated resonance(s)
• Multi resonance
2. Transfer: (p,a), (a,n), etc: nuclear process
• Non resonant
• Isolated resonance(s)
• Multi resonance
Transfer cross sections always larger than capture cross sections
1.0
3
S (keV-b)
0.8
He(a,g)7Be
5/25/2-
6Li+p
7/2-
0.6
Non resonant
a+3He
0.4
1/23/2-
0.2
7Be
0.0
0
1
2
Ecm (MeV)
3+
S-factor (eV-b)
200
7
150
Be(p,g)8B
100
1+
7Be+p
50
0
2+
8B
0
1
2
3
5/2+
3/2-
1000
S-factor (keV-b)
Resonant
lmin=0
lR=1
12
100
C(p,g)13N
1/2+
12C+p
10
1/2-
1
0
0.2
0.4
0.6
13N
Resonant
lmin=0
lR=0
S-factor (keV-b)
100
12
2+
12-
C(a,g)16O
10
a+12C
12+
30+
1
0
1
2
3
4
0+
16O
S-factor (MeV-b)
E (MeV)
10
Subthreshlod
states 2+, 1-
1
22
10
1
10
8
Ne(a,n)25Mg
n+25Mg
20 states
a+22Ne
125 states
0
1
2
E (MeV)
 Many different situations
0
0+
26Mg
multiresonant
Theoretical models
Model
Applicable to Comments
Potential model
Capture
• Internal structure neglected
• Antisymmetrization approximated
R-matrix
Capture
Transfer
• No explicit wave functions
• Physics simulated by some parameters
Light
systems
DWBA
Transfer
• Perturbation method
• Wave functions in the entrance and exit
channels
Low level
densities
Microscopic
models
Capture
Transfer
• Based on a nucleon-nucleon interaction
• A-nucleon problems
• Predictive power
Hauser-Feshbach
Capture
Transfer
Capture
• Statistical model
Shell model
• Only gamma widths
Heavy
systems
Question: which model is suitable for a compilation?
• Potential model: limited to non resonant reactions (or some specific
resonances):
– NO
•
DWBA: limited to transfer reactions, too many parameters:
– NO
•
Microscopic: too complicated, not able to reproduce all resonances:
– NO
•
R-matrix: only realistic common procedure:
– If enough data are available
– If you have much (“unlimited”) time
– MAYBE
Conclusion:
• For a broad compilation (Caltech, NACRE): no common method!
• For a limited compilation (BBN): R-matrix possible
Problems for a compilations:
• Data evaluation
• Providing accurate results (and uncertainties)
• Having a method as “common” as possible
• “Transparency”
• Using realistic durations and manpower
This system has no solution
 a compromise is necessary
The R-matrix method
Goal: treatment of long-range behaviour
Internal region
External region
E<0
E>0
 The R matrix
● Applications essentially in:
● atomic physics
● nuclear physics
● Broad field of applications
● Resonant AND non-resonant calculations
● Scattering states AND bound states
● 2-body, 3-body calculations
● Elastic scattering, capture, transfer (Nuclear astrophysics)
beta decay, spectroscopy, etc….
● 2 ways of using the R matrix
1. Complement a variational calculation with long-range wave functions
2. Fit data (nuclear astrophysics)
● Main reference: Lane and Thomas, Rev. Mod. Phys. 30 (1958) 257.
• Main idea of the R matrix: 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
Basic ideas (elastic scattering)
High-energy states with the same Jp
Simulated by a single pole = background
Isolated resonances:
Energies of interest
Treated individually
• Phenomenological R matrix: El, gl are free parameters
• Non-resonant calculations are possible: only a
background pole
Transfer reactions
Threshold 2
Elastic scattering
Inelastic scattering,
transfer
Poles
El>0 or
El<0
Threshold 1
Pole properties:
energy
reduced width in different channels ( more parameters)
gamma width  capture reactions
R matrix  collision matrix  transfer cross section
Capture reactions: more complicated
 ( E ) ~| M |2 , with M  M int  M ext
Internal contribution:
New parameter (g width)
elastic
Elastic:
El, gl: pole energy and particle width
Capture:
+ Ggl : pole gamma width
 3 parameters for each pole (2 common with elastic)
External contribution:

1.
2.
3.
3 steps:
Elastic scattering  R matrix, phase shift d
Introduction of C  , external contribution Mext
Introduction of gamma widths  Calculation of Mint
If external capture [7Be(p,g)8B, 3He(a,g)7Be]:
 A single parameter: ANC
M ~ M ext
R matrix fit: P.D. et al., At. Data Nucl. Data Tables 88 (2004) 203
Only the ANC is fitted
1.2
3
Parker 63
Nagatani 69
7
He(a,g) Be
S-factor (keV-b)
1.0
Kräwinkel 82 (x1.4)
Osborne 84
Hilgemeier 88
Singh 04
0.8
0.6
S(0)=0.51±0.04 keV-b
0.4
0.2
0.0
0.01
Cyburt04: 4th order polynomial:
S(0)=0.386 keV-b
 danger of polynomial
extrapolations!
0.1
Ecm (MeV)
1
Comparison of 2 compilations:
• NACRE (87 reactions): C. Angulo et al., Nucl. Phys. A656 (1999) 3
 previous: Fowler et al. (1967, 1975, 1985, 1988)
• SBBN (11 reactions): P.D. et al., At. Data Nucl. Data Tables 88 (2004) 203
 previous: M. Smith et al., ApJ Supp. 85 (1993) 219
K.M. Nollett, S. Burles, PRD61 (2000) 123505.
NACRE
• fits or calculations taken from
literature
• Polynomial fits
• Multiresonance (if possible)
• Hauser-Feshbach rates
• Rough estimate of errors
SBBN
• R matrix for all reactions
• Statistical treatment of errors
Example: 3He(a,g)7Be
NACRE
SBBN
• RGM calculation by Kajino
• Independent R-matrix fits of all
experiment
• Scaled by a constant factor
• Determination of averaged S(0)
1
3
S-factor (keV-b)
T9
10
3
HO59
PA63
NA69
KR82, RO83a
OS84
AL84
HI88
adopted
He(a,g)7Be
1
1.0
He(a,g)7Be
Parker 63
0.8
Nagatani 69
Kräwinkel 82 (x1.4)
Osborne 84
S (keV-b)
0.1
1.5
0.5
0.6
Hilgemeier 88
0.4
0.2
0
0
0.5
1
1.5
2
2.5
E (MeV)
0.0
0
0.5
1
1.5
Ecm (MeV)
• S(0)=0.54±0.09 MeV-b
• S(0)=0.51± 0.04 MeV-b
2
2.5
Typical problems for compilations:
• Difficulty to have a “common” theory for all reactions
• Data inconsistent with each other  how to choose?
1000
E2 S-factor (keV b)
12
C(a,g )16O
100
Stuttgart 2001
Orsay 2006
Orsay 2006
10
1
0.1
0
1
2
Ec.m. (MeV)
3
• In resonant reactions, how important is the non-resonant term?
• Properties of important resonances?
•
15O(a,g)19Ne
• Very little is known exp.
• 3/2+ resonance not described
by a+15O models
• Level density
0.1
10
1
T9
1000
19F+p
S-factor (MeV-b)
19
CL57 (norm.)
IS58 (norm.)
BR59
WA63b
MO66 (norm.)
CA74
CU80
non-resonant
F(p,a0)16O
100
10
1
0.1
0.1
1
E (MeV)
• How to relate the peaks in the S-factor with the 20Ne levels?
• How to evaluate (reasonably) the uncertainties?
10
Error treatment
• Assume n parameters pi, N experimental points
• Define
• Find optimal values pi(min) and c2(min)
• Define the range
• Sample the parameter space (Monte-Carlo, regular grid)
p2
p2(min)
p1(min)
• Keep parameters inside the limit
• Determine limits on the S factor
p1
Common problems:
• c2(min)>1 : then statistical methods cannot be applied
• different experiments may have very different data points ( overweight of some
experiments)
• Giving the parameters with error bars
p2
p2
Dp1
p1
p1
No correlation:
Strong correlation between p1 and p2
p1 given as p1(min)± Dp1
 Need of the covariance matrix
Analytical fits  tables with rates
• “Traditional” in the Caltech compilations
• Useful to understand the physical origin of the rates
• Difficult to derive with a good precision (~5%) in the full temperature range
• Question for astrophysicists:
• Tables only?
• Fits only?
• Tables and fits?
Conclusion
• Compilations are important in astrophysics
But
• Having a high standard is quite difficult (impossible?)
– Large amount of data (sometimes inconsistent and/or not
sufficient)
– No systematics
– No common model
• Ideally: should be regularly updated
Then
• Long-term efforts
• Small groups: difficult to find time
• Big groups: difficult to find agreements
• Compromises are necessary