Landau distribution

Landau distribution function

This module computes the Landau distribution, which governs the fluctuations in energy loss of particles travelling through a relatively thin layer of matter.

Currently, this module only contains functions to calculate the exact function using two integral representations of the defining complex integral. This should be extended by approximations when the need for doing serious work arises.

References are made to Fokkema2012, DOI: 10.3990/1.9789036534383.

sapphire.analysis.landau.pdf_kernel(y, sf)
sapphire.analysis.landau.pdf_kernel2(u, lf)

The Landau kernel

Fokkema2012, eq 2.13.

class sapphire.analysis.landau.Scintillator
thickness = 0.02
xi = 0.172018
epsilon = 3.10756e-11
delta = 2.97663
Euler = 0.577215665
mev_scale = 1
gauss_scale = 1
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pdf_values = None
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landau_pdf(energy_loss)

The Landau energy loss distribution function

Fokkema2012, eq 2.9, where lf is eq 2.10.

Parameters:energy_loss – Energy loss in the scintillator.
Returns:energy loss probability.
lf(energy_loss)

Calculate the lambda parameter

Fokkema2012, eq 2.10. With additional shift by delta.

Parameters:energy_loss – Energy loss in the scintillator.
Returns:lambda parameter.
pdf(lf)

The Landau probability density function

Fokkema2012, eq 2.13.

Parameters:lf – lambda parameter.
Returns:probability.
conv_landau_for_x(x, count_scale=1, mev_scale=None, gauss_scale=None)

Landau convolved with Gaussian

Fokkema2012, eq 5.4.

Parameters:
  • x – energy loss(es) for which to get the probability.
  • count_scale – total number of counts.
  • mev_scale – number of MeV per unit of x.
  • gauss_scale – width of the normal distribution.
Returns:

probability.

conv_landau(x, count_scale=1, mev_scale=None, gauss_scale=None)

Bare-bones convoluted landau function

This thing is fragile. Use with great care! First and foremost, x must be symmetrical around zero. Second, x must contain most of the Landau function (including a significant part of the tail). If not, the results cannot be trusted!

Better use conv_landau_for_x, which better handles this.

residuals(p, xdata, ydata, a, b)
constrained_residuals(p, xdata, ydata, a, b)
sapphire.analysis.landau.discrete_convolution(f, g, t)

Discrete convolution

Parameters:
  • f,g – two functions that take one argument (t).
  • t – values for which the functions will be evaluated, and the along which the convolution will be performed.
Returns:

convolution of the two functions.