KAHIBARO
45.7. Adding Energy Resolution
Table of Contents
Why Add Energy Resolution?
In a real scintillation detector, the measured energy of a gamma ray is not a single exact value. Even if all gamma rays have exactly the same true energy, the detector output shows a spread of measured energies around that value. This spread is called energy resolution.
In your basic gamma detector simulation, you first record the deposited energy without any detector imperfections. This gives you a spectrum with very sharp lines, which is not realistic. Adding energy resolution is the step where you convert these ideal deposited energies into realistic, blurred energies that mimic what a physical detector would measure.
Energy resolution is usually described by the full width at half maximum, or FWHM, of a photopeak in the energy spectrum. For a peak at energy $E_0$, the energy resolution $R$ (in percent) is
$$
R = \frac{\text{FWHM}}{E_0} \times 100 \,\%.
$$
Good scintillation detectors have smaller $R$ values, which means narrower peaks and better ability to distinguish nearby energies.
Energy resolution is modeled by replacing the ideal deposited energy with a random value drawn from a Gaussian distribution centered on the true energy, with a standard deviation $\sigma$ related to the detector FWHM.
From Ideal Energy to Blurred Energy
The key idea is to apply a statistical fluctuation to each recorded deposited energy value. For every ideal energy deposit $E$ in the detector, you generate a new value $E_{\text{blur}}$:
$$
E_{\text{blur}} \sim \mathcal{N}\left(E,\ \sigma^2\right),
$$
where $\mathcal{N}$ is a normal (Gaussian) distribution. The parameter $\sigma$ is the width of the energy resolution and is related to the FWHM by
$$
\text{FWHM} = 2.355 \, \sigma.
$$
If the detector has a fractional FWHM resolution $R$ at energy $E$ (for example $R = 0.1$ for $10\%$ resolution), then
$$
\text{FWHM} = R \, E,
$$
and therefore
$$
\sigma = \frac{\text{FWHM}}{2.355} = \frac{R \, E}{2.355}.
$$
Putting this together, the blurred energy for a given deposit $E$ is drawn from
$$
E_{\text{blur}} \sim \mathcal{N}\!\left(E,\ \left(\frac{R E}{2.355}\right)^2\right).
$$
In practice this means that higher deposited energies have larger absolute spread in their measured values. Many detector models also use energy‑dependent formulas where $R$ is not simply a constant, but for a first practical example a constant resolution is enough.
To convert a desired FWHM energy resolution into the Gaussian width used for blurring, always use
$$
\sigma = \frac{\text{FWHM}}{2.355}.
$$
If your resolution is given in percent, first compute $\text{FWHM} = \left(R_{\%}/100\right) \, E$, then apply the formula above.
Configuring Energy Blurring in the Example
In your gamma detector project, you already have a world, a detector crystal, a gamma source, physics, and a way to record energy deposition. At that stage, the output contains one or more ideal deposited energies for each detected event, before any detector response effects.
To include energy resolution, you add an energy blurring step at the appropriate place in the readout chain. Earlier in the course, the digitizer chapter explains the general concept of digitizer chains and where energy blurring fits. Here you focus on the concrete choices for this particular detector.
For a simple scintillation detector, you typically want one blurred energy per detected event. Depending on how your example is organized, this is usually done after summing all energy deposits that belong to a single detector signal. The sum gives a single ideal energy value for the event. Then you apply Gaussian smearing to that summed energy to obtain the measured energy.
In the code of your example, you will usually specify:
- A target energy resolution, for example a FWHM at a reference energy such as 662 keV.
- A functional form that converts from the true energy to the Gaussian width $\sigma$ used for blurring each event energy.
A common choice for a simple example is to assume that the fractional resolution $R$ is constant over the energy range of interest. If you choose, for instance, $R = 10\%$ at all energies, then for each event:
- Compute $\sigma = (0.10 \times E)/2.355$.
- Draw $E_{\text{blur}}$ from a normal distribution with mean $E$ and standard deviation $\sigma$.
- Store $E_{\text{blur}}$ as the energy value that will appear in the final spectrum.
If you prefer to match a real detector more closely, you can specify $R$ at a particular calibration energy, then use the same relative resolution at all energies near that value. For the purposes of the beginner example, it is enough to set a single percentage and keep it fixed.
Table 1 summarizes the relationship between resolution, FWHM, and $\sigma$ at a reference energy:
| Reference energy $E_0$ | Resolution $R$ | FWHM $= R E_0$ | $\sigma = \text{FWHM}/2.355$ |
|---|---|---|---|
| 662 keV | 10 % | 66.2 keV | 28.1 keV |
| 662 keV | 7 % | 46.3 keV | 19.7 keV |
| 140 keV | 12 % | 16.8 keV | 7.13 keV |
When you configure the value in your simulation script, be careful to use consistent units. If energies are in keV, then your FWHM and $\sigma$ must also be in keV.
Effects on the Energy Spectrum
Once energy resolution is added, the ideal, sharp features in your spectrum change into realistic peaks with finite width:
- A monoenergetic gamma source that previously produced a narrow spike now gives a broadened photopeak.
- Compton continua become smoother and broader.
- Very narrow structures are hidden if they fall inside the detector resolution.
After applying energy blurring, you can proceed to the next chapter of the project, where you build an energy spectrum from your simulated data and analyze the resulting peaks. With realistic energy resolution applied, that analysis will be directly comparable to spectra measured with a physical scintillation detector.
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