Transcript Slide 1

Error bars for reaction rates in astrophysics:
the R-matrix theory context.
Claudio Ugalde
University of North Carolina
Outline
Introduction: The problem
The theory
The experiments
Mix and match: the extraction of astro info
What does my number mean? Error bars.
Conclusion
The problem
Equations of stellar evolution
r
1

P
GM

 L2 r T
GM rT    4r
M r 4r   nP
  GM
CM
T

P
4

r

p
r
4
Y
a
b

r
1
r 

M

M
4

r
P
  
Y Yr N 
Y Y N 
  2
4 T1   GM T
Lt
1   M
rr   

   C
T  r P 44

M

M
4r P
r1
M
i
i
i
r
r
j
ij
p
j
A

k i
ij
k ,l
kl
A
r
4
kl
abundance
YiY j Qij N A  ij
  radiative
n  
 ij
j 1
(by number)
N A  reaction rate
n
r

i

r
Yi
Yi
ai i
bi

of
species
convective
    Hydrostatic
YiMass
Yj N 

 
Yk Yi N A 
A distribution
equilibrium
ij
t
1   kl
j 1   Energy
ij
generation k ,l
Composition
change
Energy transport
kl
The reaction rate

8
 E 
 
 E E exp   dE
3 0
kT 
 kT 
T is the temperature of the plasma
E is the energy of the particle pair
(E) is the integrated cross section
But...
Problems : Coulomb barrier prevents us from
measuring the reaction cross section at small energies.
Therefore, the main goal here becomes to extrapolate the
cross section into the Gamow window.
Are there more resonances inside the Gamow window?
(We may get an idea if we look into the nuclear structure
of the compound) What are their properties?
Are there non-resonant contributions to the cross section?
Also, sometimes the number of parameters (energies of
resonances + reduced width amplitudes) is huge.
The theory
The 2-step model for low energy nuclear reactions
Compound
19
F
Exit channel
Entrance channel
Step 1
V(r)
23
Step 2
Na
p
22
Ne
Coulomb+centrifugal
Point
Potential
r
Nuclear
Direct reactions
As opposed to resonant reactions, the model for the direct reaction corresponds
to a one-step process.
19
F
Entrance channel
Exit channel
One step
p
22
Ne
It is thought that during a direct reaction, only some of the nucleons may be involved.
This means that these reactions are fast and peripheral. Therefore, not all nucleons
share the energy of the collision.
Some examples are transfer reactions, radiative capture, stripping, pick-up,
knock-out, etc.
The compound
22
Ne
Compound
?
In fact, we don't know what happens to the nucleons
during the formation of the compound.
The energy of the system is distributed among all
the nucleons.
26
Mg
The compound “looses memory” of the way in which it was
formed.
Basic rules still apply: conservation of energy,
angular momentum, charge, etc. Whatever happens
to the compound forward in time needs to follow the
rules.
Most interesting is that the process of formation of the
compound is time reversal symmetric !
Formation
Destruction
The Wigner hypersurface
Compound
26
Mg
The surface splits space in two:
a) Inside- where ALL nuclear
reactions between the pair of
nuclei take place
R
b) Outside-everything else
R can have any size as long as all reactions take place inside the surface.
The model restricts R to be finite. A very large R (say the size of a “finite” universe) is
possible but computations get extremely complex. In practice R < 10 fm.
Wigner chose a truncated octahedron to describe the
boundary (for historical reasons, irrelevant to the theory).
In general, the boundary is an hypersurface in a 3A
dimensional space, such that A is the number of
nucleons in the projectile+target system.
Each dimension corresponds to a spatial coordinate.
Each face of the hypersurface is called a channel.
A channel is one of the many ways the compound can be formed (or destroyed).
A channel c is defined by c = c{a(I1I2)slm}
a is the particle pair
I1 and I2 are the spins of the 2 particles
s is the channel spin s=I1+I2 and  its projection
l is the orbital angular momentum of the 2 particles
and m its projection
The experiments
19
22
Example: F(a,p) Ne
792 < Elab/keV< 1993
1471 data points
Finding an initial set of R-matrix parameters
(needs to be done by hand)
1) Try to restrict the N space as much as possible. (Basically, answer the question
“How much we know about the compound?”)
2) Select the levels that should have a strong influence in the measured curves.
3) Set by hand the energies of these levels. Get peaks at the right position.
4) Turn off all resonances but the ones for a single J.
5) Within a single J, work in pairs trying to figure out how one resonance affects the others
in the group. Try to figure out what are the strongest conditions in the group (signs of
reduced width amplitudes + their absolute value) governing a “reasonable trend”
6) Once the signs of the reduced width amplitudes are set, turn on 2 groups
of J's. Work for all possible pairs of J's.
7) Turn on all J's, changing one of the N parameters + signs, one at a time.
8) A small variation in one of the N parameters affects all the curves at the same time
(this is independent of the method).
9) The method is iterative and therefore very time-consuming. This means that all steps
in the fitting process need to retraced over and over again (3 to 5 times, as average).
19F(a,p )22Ne
0
19F(a,p )22Ne
1
The meaning of numbers
Formal parameter error bars
Determined in a two-step process:
a) Quantify the sensitivity of the experimental data set to the R-matrix fit.
(via bootstrap)
b) Compute the contribution of individual parameters to the quality of the fit.
(via Monte Carlo)
experimental
data set
c2
R-matrix
formal
parameter
set
Formal parameter error bars
Determined in a two-step process:
a) Quantify the sensitivity of the experimental data set to the R-matrix fit.
(via bootstrap)
b) Compute the contribution of individual parameters to the quality of the fit.
(via Monte Carlo)
experimental
data set
c2
R-matrix
formal
parameter
set
The bootstrap method
Bootstrap (verb): To help oneself, often through improvised means.
The idea is to "improvise" a population out of a single sample.
single = 5 marbles
sample
Rules of the game:
1) A marble can not change color.
2) You pick a marble randomly. (You
can't look into the hat).
3) Only one marble can be drawn
at a time. (You need to return
the marble to the hat before taking a
new one)
4) A new, "synthetic" sample, is
the same size as the original
The bootstrap method II
Valid synthetic samples:
Synthetic
Population
Invalid:
Bootstrapping the data set
From the original data set, create a synthetic population of datasets
Tip: dY includes both systematic and statistical error bars
For each synthetic data set, compute c2 by leaving fixed all formal parameters
1471 points
(E,Y,dY)
N=40,000
Formal parameter error bars
Determined in a two-step process:
a) Quantify the sensitivity of the experimental data set to the R-matrix fit.
(via bootstrap)
b) Compute the contribution of individual parameters to the quality of the fit.
(via Monte Carlo)
experimental
data set
c2
R-matrix
formal
parameter
set
Individual parameter contribution to the fit
Vary each formal parameter around the central value (Monte Carlo).
Compute c2 using only the original experimental data set.
Individual parameter contribution to the fit
Upper limits come out naturally !
Error bars for the reaction rate I
experiment
Region measured in experiments
The space defined by the 201 formal parameters is sampled with Monte Carlo
All parameters are sampled simultaneously within their individual 95% confidence
interval
THE SINGLE PARAMETER DISTRIBUTION IS ASSUMED FLAT.
With the R-matrix, compute the "T-collision" matrix. Integrate the cross section.
The integrated cross section
Measured region
The cross section is computed
for every set of parameters.
The reaction rate is calculated for
every cross section.
All resulting cross sections (reaction
rates) in the population are compared
with each other at every energy
(temperature).
The error bands are defined by the
upper and lower values found from
the sample population.
Error bars for the reaction rate II
not measured
(need to extrapolate)
Extrapolations
So far, we have discussed how to treat reaction rates in the R-matrix context for
experiment-MEASURED energy regions.
However, the astrophysical interesting regions are far from our current technological
reach (with maybe a couple of exceptions).
Therefore, almost all charged-particle nuclear reactions need to be extrapolated.
Possible solutions:
a) keep pushing direct measurements to the limit. (Be patient here!)
b) use the R-matrix as a tool to compile reaction information that has been
measured indirectly. For example, energies of states in compound,
spin-parities, widths (spectroscopic factors).
Fast, one-step processes need to be understood and incorporated in the
formalism as well.
22Ne(p,p)22Ne
and 22Ne(p,p')22Ne*
Extrapolation to lower energies
From proton scattering experiments we got information about the compound nucleus
structure and proton widths.
But, what about a-widths?
g2
a(J,)
= 10
<log(g2a)>
Interference between resonances
In the future, probably the most important sources of uncertainty in reaction
rates important to hydrogen and helium burning will be:
a) Fast, one step
processes (such
as direct captures)
b) Interference between
resonances
The effects of this
kind of uncertainty
needs to be simulated
with Monte Carlo
Error bars for the reaction rate III
not measured
Extrapolation to higher energies
There are various experimental works at higher energies:
direct measurements of 19F(a,p)22Ne
studies of the nuclear structure of 23Na
Spins & parities of states mostly unknown!
However, density of states is high enough (Rauscher et al. 1997) to apply
Hauser-Feshbach.
With the matching temperature T=1x109 K, extend our experimental rate to
higher temperatures following the statistical model energy dependence.
A lot of work is still needed here!!
Reaction rate
Other sources of error
(swept under the carpet in this work)
The R-matrix radius
The target features
Target integration
In the laboratory, most common is to measure the yield of a
reaction instead of the differential cross section.
If one needs to describe the experimental data (yield) with
the output of the R-matrix theory (aka, fit data), a differential
cross section to yield transformation needs to be performed.
The basic idea is to simulate the effects of particle energy
loss in the target.
Conclusions
The R-matrix theory is so far the best theory available for extrapolating cross
sections into the astrophysically relevant temperature regimes.
No! It is not O.K. to ignore error bars when using the R-matrix to compute
reaction rates.
Our method does not yield the shape of the
statistical distribution (yet!). Only confidence
intervals are provided.
One must be careful when computing rates
with statistical models or narrow resonance,
non-interfering formalisms. The R-matrix estimates
may fall in-between.
We must be advocates trying to remind people
(specially nuclear astrophysicists) that the R-matrix
will be the ultimate tool for understanding the massive
amount of upcoming radioactive beam data sets.
Thanks!
R. Azuma
A. Couture
J. Goerres
H. Y. Lee
E. Stech
E. Strandberg
W. Tan
M. Wiescher