From 8fcc9a0b44ebde54a1ae3679c572833772823060 Mon Sep 17 00:00:00 2001 From: tlatorre Date: Tue, 24 Sep 2019 12:40:50 -0500 Subject: update shower position distribution parameters for muons 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. --- src/muon.c | 26 ++++++++++++++------------ 1 file changed, 14 insertions(+), 12 deletions(-) (limited to 'src') diff --git a/src/muon.c b/src/muon.c index 9d6c91a..14574d3 100644 --- a/src/muon.c +++ b/src/muon.c @@ -95,15 +95,20 @@ void muon_get_position_distribution_parameters(double T0, double *a, double *b) * * f(x) = x**(a-1)*exp(-x/b)/(Gamma(a)*b**a) * - * I determined the b parameter by simulating high energy muons using - * RAT-PAC and determined that it's roughly equal to the radiation length. - * To calculate the a parameter we use the formula from the PDG, i.e. + * I determined a and b by simulating high energy muons using + * RAT-PAC and fitting the histogram of the position of all photons as a + * function of the distance along the track length. * - * tmax = (a-1)/b = ln(E/E_C) - 0.5 + * Note: Unlike the case of a shower produced by an electron, the + * distribution of photons from high energy muons does not seem to follow a + * gamma distribution very well. In addition, the numbers I use here are + * really approximate. The b parameter was obtained by a single degree + * polynomial fit because it looked pretty good, but for the a parameter, I + * couldn't find any functional form that would describe it well as a + * function of energy and so I decided to just approximate it by a + * constant. * - * Therefore, we calculate a as: - * - * a = tmax*b+1. + * FIXME: Should update this in the future. * * `T` should be in units of MeV. * @@ -116,11 +121,8 @@ void muon_get_position_distribution_parameters(double T0, double *a, double *b) * See http://pdg.lbl.gov/2014/reviews/rpp2014-rev-passage-particles-matter.pdf. * * FIXME: Double check that this is correct for muons. */ - double tmax; - - *b = RADIATION_LENGTH; - tmax = log(T0/MUON_CRITICAL_ENERGY_D2O) - 0.5; - *a = fmax(1.1,tmax*(*b)/RADIATION_LENGTH + 1); + *b = -7.8 + 0.118928*T0; + *a = 1.5; } double muon_get_angular_distribution_alpha(double T0) -- cgit