## 4 Energy Resolution  
The energy resolution represents the broadening of a gamma-ray line due to photoelectron statistics and detector effects, it quantifies how well the detector can define a gamma-ray line for a specific energy. The energy resolution, \(R\), of a detector is defined as  
\[R(\%)\equiv\frac{\text{FWHM}}{\mu}\times 100\,, \tag{3}\]  
where the full-width at half-maximum (FWHM) of the energy pulse is divided by the location of the peak's centre, \(\mu\)[38]. The FWHM is calculated using the standard deviation, \(\sigma\), of the Gaussian fit function, as follows:  
\[FWHM=2\sqrt{2 ln2}\sigma\,. \tag{4}\]  
There are a number of contributing factors resulting in the increase of a detector's energy resolution such as the aforementioned photoelectron statistics, changes to the configuration of GMOD and non-uniformity of the scintillator's response or the SiPMs' performance. The sources used in this testing were only a sample of the full range of energies that could be emitted by astrophysical sources. The resolution of the detector for this larger energy range can be extrapolated by fitting the following function to the physical measurements [38]:  
\[\sigma=\sqrt{a^{2}+b^{2}E+c^{2}E^{2}}\,. \tag{5}\]  
The constant, \(a\), is negligible because SiPMs are high-gain devices and the noise is not an issue for the measurement of gamma rays above 50 keV. The \(\sqrt{b^{2}E}\) term accounts for the fluctuations in the number of scintillated photons and photoelectrons produced by the SiPMs. The final term, \(\sqrt{c^{2}E^{2}}\), is attributed to the non-ideal transfer of the scintillated photon energy from their creation locations to the SiPMs. Figure 7: Energy–channel calibration relation for GMOD FM. The relationship was fit with a quadratic (dashed line) with the coefficients \(a=3.56\times 10^{-5}\), \(b=0.42\) and \(c=-17.30\). The real points representing the mixed source were not used in the final fit due to the lab temperature being different for these measurements. Measurement errors for gamma-ray line centres are on the order of 0.3 ADC channel. The energy calibration was first applied to the spectra produced by GMOD for the sources listed in Table 2, such that the spectral counts/keV/hour were plotted against energy in keV, rather than channel number, as in Figure 8.  
The photopeaks were then re-fitted with a combination of a Gaussian(s) and a polynomial function to account for other contributions to the photopeak to determine \(\sigma\) and, using Equation (4), the FWHM in terms of energy. Figure 9 presents four examples of fitting the photopeaks of the sources in energy space. Figure 10 presents the \(\sigma\) values for the photopeaks as a function of energy. Equation (5) was then fitted to these points with coefficients \(b\) = 0.6761 and \(c\) = 8.9754 \(\times\) 10\({}^{-3}\). Figure 8: GMOD FM-calibrated spectra (teal) in counts/keV/hour for the six sources in Table 2.  
Figure 9: A sample of four fits to GMOD spectra in terms of energy (keV). Each panel shows the fit (dashed grey line) to the energy-calibrated spectrum. The composite functions of the fit are plotted as solid lines. The energy resolution of the photopeaks were determined using Equation (3). This energy resolution was plotted as a function of energy, as in Figure 11. Equation (5) can be expressed in terms of the FWHM (%), as follows:  
\[\text{FWHM}\left(\%\right)=235.5\times\sqrt{\frac{b^{2}}{E}+c^{2}}\,. \tag{6}\]  
and can be fitted to the data in Figure 11 to determine the resolution of the GMOD FM over a wide range of energies. The obtained coefficients of the fit are \(b\) = 0.6294 and \(c\) = 6.8994 \(\times\) 10\({}^{-3}\). The energy resolution of GMOD FM improves for incoming photons with higher energies and was found to be 7.07% at 662 keV. \[FWHM=2\sqrt{2 ln2}\sigma\,. \tag{4}\]0 Energy resolution in terms of FWHM (%) fit with Equation (6) and coefficients \(a\) = 0, \(b\) = 0.6294 and \(c\) = 6.8994 \(\times\) 10\({}^{-3}\). Relative statistical errors in the resolution values are under 4%.