KAHIBARO
11.4. Energy Distributions
Table of Contents
Monoenergetic
An energy distribution describes how the primary particles created by the General Particle Source (GPS) are distributed in energy. The simplest case is a monoenergetic source, where every generated particle has exactly the same kinetic energy.
With GPS, a monoenergetic source is the default behavior if you only set a single energy value. Conceptually, you tell GPS: "Always generate this particle with energy $E_0$."
In a macro, you typically configure this in two steps. First choose the distribution type, then set the value. For a monoenergetic distribution, the type is "Mono". After that, you provide the energy:
- Select the energy distribution type as monoenergetic.
- Set the energy value and its unit.
Most often, the configuration is done through commands like /gps/energy with a unit, for example MeV. Once this is set, all primaries created by GPS will have that kinetic energy until you change the distribution again.
Monoenergetic sources are useful when you want to study detector response or physics processes at a fixed energy, for example a 511 keV annihilation photon or a 1 MeV gamma beam.
Important rule: A monoenergetic GPS configuration always produces particles with exactly one specified kinetic energy. Change the energy distribution type before setting multiple values or more complex shapes.
Gaussian
Real beams often have some spread around a central energy. A Gaussian energy distribution models this situation with a mean energy and a standard deviation $\sigma$ that describes the width of the distribution. The probability density for the energy $E$ is then
$$
f(E) = \frac{1}{\sqrt{2\pi}\,\sigma} \exp\left(-\frac{(E - E_0)^2}{2\sigma^2}\right),
$$
where $E_0$ is the mean energy of the beam.
In GPS, you tell the source to use a Gaussian distribution and then provide at least two essential parameters:
- The mean energy.
- The standard deviation (spread) of the energy.
You also need to provide the units consistently for both values, for example MeV. The mean controls where the peak of the distribution sits, and the standard deviation controls how wide the distribution is. A small $\sigma$ gives a narrow, almost monoenergetic beam. A large $\sigma$ gives a broad range of energies.
Gaussian energy distributions are common when simulating realistic accelerator beams or detector responses where an incoming nominal energy is smeared by upstream materials or beam optics. When you use such a distribution, remember that low probability tails can extend quite far in energy. If your detector or geometry is sensitive to very high or very low energies, you may need enough statistics to sample those tails.
Key formula: In a Gaussian energy distribution, about 68 percent of particles have energy in the interval $[E_0 - \sigma, E_0 + \sigma]$, and about 95 percent in $[E_0 - 2\sigma, E_0 + 2\sigma]$.
User-defined spectra
Sometimes neither monoenergetic nor simple Gaussian shapes are realistic. For example, radioactive sources, cosmic rays, or complex beamlines can produce arbitrary spectra. In these cases, GPS supports user-defined energy spectra.
Conceptually, a user-defined spectrum specifies a probability distribution in energy by a list of energy points and associated weights. GPS then samples from this distribution when generating events. You can think of this as a histogram or a tabulated function. GPS internally normalizes the weights, for example the total probability is 1 after normalization, and uses interpolation between points when needed.
A typical user-defined spectrum has the following components:
- A set of energies, for example $E_1, E_2, \dots, E_n$.
- A set of weights or relative probabilities, for example $w_1, w_2, \dots, w_n$.
- An interpolation scheme that defines how probabilities behave between the tabulated energies.
For a discrete spectrum such as a few gamma lines from a radionuclide, you would define several discrete energy points with weights proportional to their branching ratios. For a continuous spectrum, such as a beta spectrum or a broad X ray distribution, you provide many points covering the whole energy range.
When you configure such a spectrum in GPS with macro commands, you must keep units consistent for all energy points. The relative weights do not need a unit, but they should reflect the shape you want. The absolute normalization does not matter, because GPS rescales the weights so that they define a proper probability distribution.
User-defined spectra are powerful when you want to reproduce measured or published spectra in your simulation. For instance, if you have an energy distribution measured in an experiment, you can digitize it into energy bin centers and counts, then use those as the energies and weights for GPS. This allows your primary source in Geant4 to mimic the actual physical spectrum very closely.
Important rule: In a user-defined GPS energy spectrum, only the relative values of the weights matter. GPS automatically normalizes them, so doubling all weights does not change the distribution shape.
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