Error bars for reaction rates in astrophysics: the R-matrix theory context.

Claudio UgaldeUniversity of North Carolina

Outline

Introduction: The problemThe theoryThe experimentsMix and match: the extraction of astro infoWhat does my number mean? Error bars. Conclusion

The problem

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

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

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?

The theory

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

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

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

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.

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

The experiments

Example: 19F(,p)22Ne

1471 data points

792 < Elab/keV< 1993

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).

19F(,p0)22Ne

19F(,p1)22Ne

The meaning of numbers

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

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

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).

The bootstrap method II

Valid synthetic samples:

Invalid:

SyntheticPopulation

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

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

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.

Individual parameter contribution to the fit

Upper limits come out naturally !

Error bars for the reaction rate I

experiment

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.

The integrated cross section

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.

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 theformalism 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 nucleusstructure and proton widths.

But, what about -widths?

2(J,) = 10 <log(2

)>

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

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(,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 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.

Thanks!

R. AzumaA. CoutureJ. GoerresH. Y. LeeE. StechE. StrandbergW. TanM. Wiescher