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This commit updates the likelihood fit to use the KL path expansion. Currently,
I'm just using one coefficient for the path in both x and y.
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refraction
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Previously I had been assuming that a particle undergoing many small angle
Coulomb scatters had a track direction whose polar angle was a Gaussian.
However, this was just due to a misunderstanding of the PDG section "Multiple
scattering through small angles" in the "Passage of particles through matter"
article. In fact, what is described by a Gaussian is the polar angle projected
onto a plane. Therefore the distribution of the polar angle is actually:
(1/(sqrt(2*pi)*theta0**2))*theta*exp(-theta**2/(2*theta0))
This commit updates the code in scattering.c to correctly calculate the
probability that a photon is emitted at a particular angle. I also updated
test-likelihood.c to simulate a track correctly.
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To fit the path of muons and electrons I use the Karhunen-Loeve expansion of a
random 2D walk in the polar angle in x and y. This allows you to decompose the
path into a sum over sine functions whose coefficients become random variables.
The nice thing about fitting the path in this way is that you can capture
*most* of the variation in the path using a small number of variables by only
summing over the first N terms in the expansion and it is easy to calculate the
probability of the coefficients since they are all uncorrelated.
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photons
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The GSL library only has the Nelder Mead Simplex algorithm for doing
multidimensional minimization without gradient information. The nlopt library
has lots of different minimization algorithms so it's easier to switch between
them to see which one works best.
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The RMS scattering angle calculation comes from Equation 33.15 in the PDG
article on the passage of particles through matter. It's not entirely obvious
if this equation is correct for a long track. It seems like it should be
integrated along the track to add up the contributions at different energies,
but it's not obvious how to do that with the log term.
In any case, the way I was previously calculating it (by using the momentum and
velocity at each point along the track) was definitely wrong.
I will try this out and perhaps try to integrate it later.
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There was no entry for heavy water at
http://pdg.lbl.gov/2018/AtomicNuclearProperties/index.html, so I emailed Don
Groom who maintains the website. Amazingly he agreed to rerun the code to add
an entry for D2O.
Apparently all of the old Fortran code was not set up to deal with isotopes,
but he updated everything and reran the code for heavy water! The new results
are at http://pdgprod.lbl.gov/~deg/AtomicNuclearProperties/ but they should
make it to the main page soon.
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