KAHIBARO
27.2. Energy Resolution
Table of Contents
Gaussian energy smearing
In Geant4 you typically obtain an ideal energy deposit for each event, for example the total energy deposited in a scintillator or semiconductor detector. Real detectors, however, never measure this ideal value exactly. Instead, many effects in the detector and the readout electronics introduce fluctuations. The result is that a monoenergetic line, such as a 662 keV gamma from cesium 137, appears as a broadened peak in the measured energy spectrum.
Energy resolution describes how broad that peak is relative to its energy. A common way to quote it is the full width at half maximum (FWHM) divided by the peak energy, often written as a percentage. In this chapter the goal is not to model the detailed physics that causes the resolution, but to apply a simple mathematical model to convert ideal Geant4 energy deposits into realistic smeared energies.
The most frequently used model for detector energy resolution is a Gaussian distribution. You start from the ideal deposited energy from Geant4 and replace it by a random value drawn from a Gaussian centered on that ideal value, with a standard deviation that encodes your detector resolution. You typically apply this smearing when filling histograms or writing output, not inside the transport itself.
Gaussian energy smearing
A Gaussian distribution is fully determined by its mean and its standard deviation. For energy smearing, the mean is the ideal deposited energy from Geant4, and the standard deviation depends on the detector resolution at that energy.
If you know the resolution as a FWHM at a given energy, you convert between FWHM and standard deviation using the fixed factor for a Gaussian:
For a Gaussian distribution, the relation between FWHM and standard deviation $\sigma$ is
$$\text{FWHM} = 2\sqrt{2\ln 2}\,\sigma \approx 2.355\,\sigma.$$
Hence,
$$\sigma = \frac{\text{FWHM}}{2.355}.$$
If the detector resolution is specified as a percentage at some energy $E$, for example 7 % FWHM at 662 keV, the FWHM at that energy is
$$\text{FWHM}(E) = \frac{\text{Resolution}[\%]}{100}\,E,$$
and the corresponding $\sigma(E)$ is obtained from the formula above.
In practice, to smear an ideal deposited energy $E_{\text{dep}}$ you perform two steps. First, determine the standard deviation $\sigma(E_{\text{dep}})$ from your resolution model, which may be constant or energy dependent. Second, draw a random number $E_{\text{meas}}$ from a Gaussian with mean $E_{\text{dep}}$ and width $\sigma(E_{\text{dep}})$. This sampled value represents what the detector would actually measure in one event.
In a Geant4 application, the smearing is usually implemented on the analysis side. Inside an event or stepping action, or more commonly in RunAction or EventAction where you fill histograms, you take the accumulated ideal energy for a detector and transform it into a smeared value. Geant4 provides a random number interface, for example through G4RandGauss::shoot(mean, sigma), which directly returns a Gaussian distributed number. You then fill your energy histogram with the smeared energy instead of the ideal one.
It is also common to prevent unphysical negative energies after smearing, especially at low energies where the Gaussian tail can extend below zero. A simple way is to check the smeared value and, if it is negative, either set it to zero or redraw a new value. This choice depends on how strictly you want to preserve the Gaussian shape near threshold.
Different detector types have different resolution behaviors as a function of energy. For many scintillation detectors, an approximate dependence of the form
$$\frac{\text{FWHM}(E)}{E} = \sqrt{\left(\frac{a}{\sqrt{E}}\right)^2 + b^2}$$
is often used, where $E$ is in suitable units and $a$ and $b$ are detector specific constants that describe statistical and non statistical contributions. For semiconductor detectors, the resolution is often better and may be closer to a constant term plus a small energy dependent term. Whatever parameterization you choose, the key step is always to convert the chosen FWHM into $\sigma(E)$ and use it in the Gaussian sampling.
By incorporating Gaussian energy smearing into your Geant4 analysis you turn an ideal energy spectrum into one that resembles real experimental data, with finite peak widths and realistic line shapes. This is essential when you want to compare simulation results directly to measured spectra or to study how detector resolution affects peak separation, background, and the visibility of spectral features.
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