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I found when simulating high energy muons that the expected charge for some
PMTs which should be getting hit was zero. The reason for this is that the
integrand was very sharply peaked at the Cerenkov angle which makes it
difficult to integrate for numerical integration routines like cquad. To solve
this I split up the integral at the point when the track was at the Cerenkov
angle from the PMT to make sure that cquad didn't miss the peak. However,
calling cquad twice takes a lot of time so it's not necessarily good to do this
for all fits. Also, it's not obvious if it is necessary any more now that the
angular distribution calculation was fixed.
I think the real reason that cquad was missing those integrals was that for a
high energy muon the range is going to be very large (approximately 40 meters
for a 10 GeV muon). In this case, I should really only integrate up to the edge
of the cavity or PSUP and hopefully cquad picks enough points in there to get a
non zero value.
I also added a check to only compute tmean when at least one PMT has a valid
time. This prevents a divide by zero which causes the likelihood function to
return nan.
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spaced
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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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This commit contains code to fit for the energy, position, and direction of
muons in the SNO detector. Currently, we read events from SNOMAN zebra files
and fill an event struct containing the PMT hits and fit it with the Nelder
Mead simplex algorithm from GSL.
I've also added code to read in ZEBRA title bank files to read in the DQXX
files for a specific run. Any problems with channels in the DQCH and DQCR banks
are flagged in the event struct by masking in a bit in the flags variable and
these PMT hits are not included in the likelihood calculation.
The likelihood for an event is calculated by integrating along the particle
track for each PMT and computing the expected number of PE. The charge
likelihood is then calculated by looping over all possible number of PE and
computing:
P(q|n)*P(n|mu)
where q is the calibrated QHS charge, n is the number of PE, and mu is the
expected number of photoelectrons. The latter is calculated assuming the
distribution of PE at a given PMT follows a Poisson distribution (which I think
should be correct given the track, but is probably not perfect for tracks which
scatter a lot).
The time part of the likelihood is calculated by integrating over the track for
each PMT and calculating the average time at which the PMT is hit. We then
assume the PDF for the photons to arrive is approximately a delta function and
compute the first order statistic for a given time to compute the probability
that the first photon arrived at a given time. So far I've only tested this
with single tracks but the method was designed to be easy to use when you are
fitting for multiple particles.
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temperature and pressure to density in his tables
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wavelength
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