![Page 1: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/1.jpg)
Error bars for reaction rates in astrophysics: the R-matrix theory context.
Claudio UgaldeUniversity of North Carolina
![Page 2: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/2.jpg)
Outline
Introduction: The problemThe theoryThe experimentsMix and match: the extraction of astro infoWhat does my number mean? Error bars. Conclusion
![Page 3: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/3.jpg)
The problem
![Page 4: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/4.jpg)
Equations of stellar evolution
24
1
rM
r
r
44 r
GM
M
P r
r
PTCM
Lpn
r
r
Pr
TGM
M
T r
r44
klAiklk kl
iijAji
j ij
ii NYYb
NYYa
t
Y
, 11
24
1
rM
r
r
Mass distribution
Energy generation
PTCM
Lpn
r
r
ijAijjij ij
n NQYY
1
1
Hydrostatic equilibrium
44 r
GM
M
P r
r
Energy transport
Pr
TGM
M
T r
r44
convective
radiativereaction rate
klAiklk kl
iijAji
j ij
ii NYYb
NYYa
t
Y
, 11
AN
Composition change
iYabundance (by number) of species i
![Page 5: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/5.jpg)
The reaction rate
dE
kT
EEE
kT
exp
803
T is the temperature of the plasma
E is the energy of the particle pair
(E) is the integrated cross section
![Page 6: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/6.jpg)
But...
Also, sometimes the number of parameters (energies ofresonances + reduced width amplitudes) is huge.
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?
![Page 7: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/7.jpg)
The theory
![Page 8: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/8.jpg)
The 2-step model for low energy nuclear reactions
PointPotential
V(r)
r
Coulomb+centrifugal
Nuclear
Entrance channel
F19
Compound
Na23
Exit channel
Ne22
pStep 1
Step 2
![Page 9: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/9.jpg)
Direct reactionsAs opposed to resonant reactions, the model for the direct reaction correspondsto a one-step process.
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.
Entrance channel
F19
Exit channel
Ne22
pOne step
![Page 10: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/10.jpg)
Ne22
Compound
Mg26
The compound
?In fact, we don't know what happens to the nucleonsduring the formation of the compound.
The energy of the system is distributed among all the nucleons.
The compound “looses memory” of the way in which it was formed.
Basic rules still apply: conservation of energy,angular momentum, charge, etc. Whatever happensto the compound forward in time needs to follow the rules.
Most interesting is that the process of formation of thecompound is time reversal symmetric !
Formation Destruction
![Page 11: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/11.jpg)
The Wigner hypersurface
Compound
Mg26
R
The surface splits space in two:
a) Inside- where ALL nuclear reactions between the pair of nuclei take place
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.
![Page 12: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/12.jpg)
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 3Adimensional space, such that A is the number ofnucleons 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{(I1I2)slm}
is the particle pair
I1 and I
2 are the spins of the 2 particles
s is the channel spin s=I1+I
2 and its projection
l is the orbital angular momentum of the 2 particlesand m its projection
![Page 13: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/13.jpg)
The experiments
![Page 14: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/14.jpg)
Example: 19F(,p)22Ne
![Page 15: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/15.jpg)
1471 data points
792 < Elab/keV< 1993
![Page 16: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/16.jpg)
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 groupsof 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).
![Page 17: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/17.jpg)
19F(,p0)22Ne
19F(,p1)22Ne
![Page 18: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/18.jpg)
![Page 19: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/19.jpg)
The meaning of numbers
![Page 20: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/20.jpg)
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)
experimentaldata set
formalparameter
set
R-matrix
Formal parameter error bars
![Page 21: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/21.jpg)
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)
experimentaldata set
formalparameter
set
R-matrix
![Page 22: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/22.jpg)
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 sample Rules of the game:
1) A marble can not change color.
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
= 5 marbles
2) You pick a marble randomly. (You can't look into the hat).
![Page 23: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/23.jpg)
The bootstrap method II
Valid synthetic samples:
Invalid:
SyntheticPopulation
![Page 24: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/24.jpg)
Bootstrapping the data set
From the original data set, create a synthetic population of datasets
For each synthetic data set, compute 2 by leaving fixed all formal parameters
1471 points
(E,Y,dY)
Tip: dY includes both systematic and statistical error bars
N=40,000
![Page 25: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/25.jpg)
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)
experimentaldata set
formalparameter
set
R-matrix
![Page 26: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/26.jpg)
Individual parameter contribution to the fit
Vary each formal parameter around the central value (Monte Carlo).
Compute 2 using only the original experimental data set.
![Page 27: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/27.jpg)
Individual parameter contribution to the fit
Upper limits come out naturally !
![Page 28: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/28.jpg)
Error bars for the reaction rate I
experiment
![Page 29: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/29.jpg)
Region measured in experiments
With the R-matrix, compute the "T-collision" matrix. Integrate the cross section.
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.
![Page 30: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/30.jpg)
The integrated cross section
![Page 31: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/31.jpg)
Measured region
The cross section is computedfor every set of parameters.
All resulting cross sections (reaction rates) in the population are compared with each other at every energy (temperature).
The reaction rate is calculated forevery cross section.
The error bands are defined by the upper and lower values found from the sample population.
![Page 32: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/32.jpg)
Error bars for the reaction rate II
not measured(need to extrapolate)
![Page 33: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/33.jpg)
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 theformalism as well.
![Page 34: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/34.jpg)
22Ne(p,p)22Ne and 22Ne(p,p')22Ne*
![Page 35: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/35.jpg)
Extrapolation to lower energies
From proton scattering experiments we got information about the compound nucleusstructure and proton widths.
But, what about -widths?
2(J,) = 10 <log(2
)>
![Page 36: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/36.jpg)
Interference between resonances
In the future, probably the most important sources of uncertainty in reactionrates important to hydrogen and helium burning will be:
a) Fast, one step processes (such as direct captures)
b) Interference between resonances
The effects of thiskind of uncertainty needs to be simulated with Monte Carlo
![Page 37: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/37.jpg)
Error bars for the reaction rate III
not measured
![Page 38: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/38.jpg)
Extrapolation to higher energies
There are various experimental works at higher energies:
direct measurements of 19F(,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!!
![Page 39: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/39.jpg)
Reaction rate
![Page 40: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/40.jpg)
Other sources of error(swept under the carpet in this work)
The R-matrix radius
The target features
![Page 41: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/41.jpg)
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.
![Page 42: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/42.jpg)
![Page 43: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/43.jpg)
![Page 44: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/44.jpg)
![Page 45: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/45.jpg)
Conclusions
The R-matrix theory is so far the best theory available for extrapolating crosssections 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 rateswith statistical models or narrow resonance,non-interfering formalisms. The R-matrix estimatesmay 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 massiveamount of upcoming radioactive beam data sets.
![Page 46: Error bars for reaction rates in astrophysics: the R-matrix theory context. Claudio Ugalde University of North Carolina](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ed35503460f94be3583/html5/thumbnails/46.jpg)
Thanks!
R. AzumaA. CoutureJ. GoerresH. Y. LeeE. StechE. StrandbergW. TanM. Wiescher