12
a
R
RICHARD DEBOER
TRENTO, ITALY 2013

OUTLINE
What data is considered?
The R-matrix Fit
Successes and remaining problems
Uncertainty analysis
Monte Carlo

SOME BACKGROUND
15 N(p
,g
) 16 O
15 N(p,p
0
) 15 N
15 N(p, a
) 12 C
15 N(p
,a
1
) 12 C
12 C( a,g
) 16 O
12 C( a,a
12 C( a
,p) 15 N
12 C( a,a
1
0
) 12 C
) 12 C
???

R -matrix theory: reaction framework for low energy nuclear reactions
Based on the algorithms developed for AZURE FORTRAN by R.E. Azuma →
Written in C++, Graphical Interface created with Qt
Utilizes currently maintained pubic libraries
• MINUIT2
• GNU Scientific Library
Open Source soon
Multiple entrance/exit channels
• Particle, particle
• Particle, gamma
• Beta delayed particle emission
Full external capture calculation with E1, E2, and M1 as well as contributions to the gamma widths of resonances
AZURE2

SCHÜRMANN
(2012)
Global analysis of 12 C( a,g
) 16 , 16 N( ba
) 12 C, and 12 C( a,a
) 12 C phase shifts
Systematic Uncertainties considered
Monte Carlo uncertainty analysis
S(300 keV) = 161  19 (stat) +8
-2
(syst)
Independent analysis
Consider broader set of data
Extend uncertainty analysis to entire cross section curve and to reaction rate over a broad temperature range

SYSTEMATIC
UNCERTAINTY
Schürmann et al. (2012)

HIGHER ENERGY
DATA

HIGH ENERGY DATA
1 - 1 -

LOW ENERGY DATA
Total data points to date: 12600

FULL ANALYSIS
REGION
Tischhauser et al. (2009)
Azuma et al. (1994)
Schürmann et al. (2005)

WHAT MAKES 12 C( a,g ) 16 O
SO TOUGH FOR
R -MATRIX?
Subthreshold states (E x
= 7.12 and 6.92 MeV)
Broad resonances with strong interferences
Direct Capture
E1 and E2 components

12 C( a , g
0
) 16 O – 12 LOW
ENERGY MEASUREMENTS
Brochard et al. (1973)
Dyer and Barns (1974)
Kettner et al. (1982)
Redder et al. (1987)
Kremer et al. (1988)
Ouellet et al. (1996)
Roters et al. (1999)
Gialanella et al. (2001)
Kunz et al. (2001)
Fey Thesis (2004)
Assuncao et al. (2006)
Makii et al. (2009)
Schürmann et al. (2011)

GROUND STATE
ANGULAR
DISTRIBUTIONS

MORE GROUND STATE
ANGULAR
DISTRIBUTIONS

E2 INTERFERENCE

E1 INTERFERENCE:
“TWO SOLUTIONS”
Michael Fey Thesis (2004) systematic study of interferences c
2 /n = 2.5 c
2 /n = 9.0 c
2 /n = 8.6
Current analysis n = 146 c
2 /n = 9.6

16 N( ba ) 12 C a) Azuma et al. (1994) b) Tang et al. (2010)
Also seen recently in
Schürmann et al. (2012)
Data sets have some dissagreements

CASCADE
TRANSITIONS
Redder et al. (1987)
Kunz (priv. comm.)
Schürmann et al. (2011)
Matei et al. (2006)
Kettner et al. (1982)

CASCADE
TRANSITION ISSUES
For previous fit, ANC of 6.05 MeV state was about 1600 fm -1/2
Fairouz Hammache, private communication

12 C( a,a
0
) 12 C –
TISCHHAUSER
(2002) & (2009)
ET AL.

FIT SUMMARY
Generally good fit for all data
Issues
Beta delayed particle emission data
Cascade transition data – 6.13 and 6.05 MeV transitions
The data constrain the interferences, transfer measurements constrain the low energy cross section and uncertainty
Background poles for 12 C( a,g
0
) 16 O may not be necessary

ASYMPTOTIC
NORMALIZATION
COEFFICIENTS
Contribution of the subthreshold states is determined by
ANCs and g
widths
ANCs can be constrained by (see Mukhamedzhanov and
Tribble (1998) ) scattering data -- BGP
(  ) beta delayed emission -- BGP
(  )
12 C( a,g
) 16 O - 7.12 and 6.92 transitions – need data transfer reactions
(  )

THE SUBTHRESHOLD
STATES: ANCs
Ground State
F. Hammache, private communication

SUBTHRESHOLD
STATES: g WIDTHS

MONTE CARLO
UNCERTAINTY ANALYSIS
Multiparameter fit function uncertainties tough to extract parameter correlations reaction rate
Monte Carlo method assume data point uncertainties are Gaussian assume systematic uncertainties are Gaussian or
Linear
Best fit is an accurate description of the true cross section
Create new data sets from random variations and do analysis again

PROBABILITY
DENSITY FUNCTION
Only 1000 throws

RECENT ESTIMATES
OF S(300 KEV)
Woosley, Heger, and Weaver (2002) want
D
S(300 keV) < 10%

PRELIMINARY UNCERTAINTY –
S(300 KEV) = 160 (10% + MODEL) KEV B

FUTURE
Model uncertainties must be estimated radius parameter background poles
What is going on with the a
radius parameter?
Higher angular momentum terms necessary?
Narrow resonance contributions to the rate

CONCLUSIONS
ANCs dominate the low energy uncertainty
Including higher energy data decreases the dependence of the R -matrix fit on background poles
12 C( a,g
) 16 O cross section may be approaching the 10% level of uncertainty
Cascade transitions are worth measuring don’t contribute much to the total but are very helpful for the R -matrix analysis of the total cross section data

COLLABORATORS
J. G ö rres
K. Smith
E. Uberseder
M. Wiescher
THANKS TO
R.E. Azuma
C. Brune
D. Schürmann
G. Imbriani
F. Strieder and many others