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This commit updates guess_energy() which is used to seed the energy for the
likelihood fit. Previously we estimated the energy by summing up the charge in
a 42 degree cone around the proposed direction and then dividing that by 6
(since electrons in SNO and SNO+ produce approximately 6 hits/MeV). Now,
guess_energy() estimates the energy by calculating the expected number of
photons produced from Cerenkov light, EM showers, and delta rays for a given
particle at a given energy. The most likely energy is found by bisecting the
difference between the expected number of photons and the observed charge to
find when they are equal.
This improves things dramatically for cosmic muons which have energies of ~200
GeV. Previously the initial guess was always very low (~1 GeV) and the fit
could take > 1 hour to increase the energy.
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This commit updates the a and b parameters for the gamma distribution used to
describe the position distribution of shower photons produced along the
direction of the muon. Previously I had been assuming b was equal to the
radiation length and using a formula from the PDG to calculate a from that.
However, this formula doesn't seem to be valid for muons (the formula comes
from a section describing the shower profile of electrons and gammas, so it's
not surprising). Therefore, now we don't assume any relationship between a and
b.
Now, the value of a is approximated by a constant since I couldn't find any
functional relationship as a function of energy to describe a very well (and
it's approximately constant), and b is approximated by a single degree
polynomial fit to the values I got from simulating muons in RAT-PAC as a
function of energy.
Note that looking at the simulation data it seems like the position
distribution of shower photons from muons isn't even very well described by a
gamma distribution, so in the future it might be a good idea to come up with a
better parameterization.
Even if I stick with the gamma distribution, it would be good to revisit this
in the future and fit for a and b over a wider range of energies.
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The range and energy loss tables have different maximum values for electrons,
muons, and protons so we have to dynamically set the maximum energy of the fit
in order to avoid a GSL interpolation error.
This commit adds {electron,muon,proton}_get_max_energy() functions to return
the maximum energy in the tables and that is then used to set the maximum value
in the fit.
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Similarly to electrons, I fit an analytic form to the ratio of the number of
photons produced via shower particles over the radiative energy loss. In this
case, I chose the functional form:
ratio = a*(1-exp(-T/b))
since the ratio seemed to reach a constant value after a certain energy. I then
simulated a 10 GeV muon and it appears that the ratio might actually decrease
after that, so for higher energies I may have to come up with a different fit
function.
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To enable the fitter to run outside of the src directory, I created a new
function open_file() which works exactly like fopen() except that it searches
for the file in both the current working directory and the path specified by an
environment variable.
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This commit updates the likelihood function to take into account Cerenkov light
produced from delta rays produced by muons. The angular distribution of this
light is currently assumed to be constant along the track and parameterized in
the same way as the Cerenkov light from an electromagnetic shower. Currently I
assume the light is produced uniformly along the track which isn't exactly
correct, but should be good enough.
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To characterize the angular distribution of photons from an electromagnetic
shower I came up with the following functional form:
f(cos_theta) ~ exp(-abs(cos_theta-mu)^alpha/beta)
and fit this to data simulated using RAT-PAC at several different energies. I
then fit the alpha and beta coefficients as a function of energy to the
functional form:
alpha = c0 + c1/log(c2*T0 + c3)
beta = c0 + c1/log(c2*T0 + c3).
where T0 is the initial energy of the electron in MeV and c0, c1, c2, and c3
are parameters which I fit.
The longitudinal distribution of the photons generated from an electromagnetic
shower is described by a gamma distribution:
f(x) = x**(a-1)*exp(-x/b)/(Gamma(a)*b**a).
This parameterization comes from the PDG "Passage of particles through matter"
section 32.5. I also fit the data from my RAT-PAC simulation, but currently I
am not using it, and instead using a simpler form to calculate the coefficients
from the PDG (although I estimated the b parameter from the RAT-PAC data).
I also sped up the calculation of the solid angle by making a lookup table
since it was taking a significant fraction of the time to compute the
likelihood function.
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To calculate the expected number of photons from reflected light we now
integrate over the track and use the PMT response table to calculate what
fraction of the light is reflected. Previously we were just using a constant
fraction of the total detected light which was faster since we only had to
integrate over the track once, but this should be more accurate.
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the quantum efficiency
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This commit updates the calculation of the muon kinetic energy as a function of
distance along the track. Previously I was using an approximation from the PDG,
but it doesn't seem to be very accurate and won't generalize to the case of
electrons. The kinetic energy is now calculated using the tabulated values of
dE/dx as a function of energy.
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This commit adds the absorption length to the likelihood calculation. For now
I'm just using a single number independent of wavelength. I should update this
in the future to actually use the absorption lengths as measured by SNO and
then calculate an overall absorption length weighted by the Cerenkov spectrum
and the PMT quantum efficiency.
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This commit adds code to read in the PMT response from the PMTR bank from
SNOMAN. This file was used for the grey disk model in SNOMAN and was created
using a full 3D simulation of the PMT and concentrator. Since the PMT response
in SNOMAN included the quantum efficiency of the PMT, we have to divide that
out to get just the PMT response independent of the quantum efficiency.
I also updated the likelihood calculation to use the pmt response. Currently
the energy is being fit too high which I think will improve when we update the
solid angle calculation to use the radius of the concentrator instead of the
PMT.
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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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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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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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