## 2 Materials and Methods  
### The PiXirad Detector  
The PiXirad detector system (PiXirad IC srl c/o INFN Pisa, Italy) was used for these measurements. The detector consists of a CdTe solid state sensor connected to a complementary metal-oxide semiconductor (CMOS) Application-Specific Integrated Circuit (ASIC) readout [16]. The CMOS ASIC has 512 \(\times\) 476 pixels arranged on a hexagonal pitch of 60 \(\upmu\)m \(\times\) 52 \(\upmu\)m. The sensor thickness for the PiXirad detector is 650 \(\upmu\)m with a corresponding detection efficiency of 100% at 10 keV and 98% at 50 keV [48]. Each pixel includes an electrode, connected to a charge amplifier, which provides two discriminators and two 15-bit counters per register. This enables each pixel to discriminate photons in terms of their energy. The X-ray photons are converted to electric charges at the semiconductor, which are collected and transferred to the ASIC where the signal per pixel is electronically treated. This technology enables the PiXirad detector to generate a maximum of four colour images per single exposure, by setting four individual energy thresholds. For each energy threshold, the PiXirad detector counts photons with energies greater than the threshold energy up to the cut off X-ray energy of the spectrum. The four-energy thresholds produced by the PiXirad detector (shown in Figure 1 as \(E_{1}\), \(E_{2}\), \(E_{3}\) and \(E_{4}\)) are used to produce two discrete energy bandwidths (\(\Delta E_{a}\) and \(\Delta E_{b}\)), which only contain photon energies within the specified energy range. The discrete energy bandwidths, \(\Delta E_{a}\) and \(\Delta E_{b}\) are approximated here as two quasi-monochromatic beams each having a bandwidth (\(\Delta E/E\)) less than 34%. This bandwidth can readily be defined using the PiXirad whose spectral energy resolution is shown in Figure S2 in the Supplementary Material.  
A key consideration here is that the performance of the PiXirad detector can be affected by charge-sharing. Charge-sharing is an artefact which arises when a single photon is absorbed and produces a signal that is'shared' across two or more detector pixels. The  
Figure 1: The measured X-ray source spectrum for a tungsten target at a 40 kVp tube voltage, with the four chosen threshold energies \(E_{1}\), \(E_{2}\), \(E_{3}\) and \(E_{4}\) (indicated by dashed vertical blue lines) which together, following image subtraction, create two discrete energy bands shown by the blue shaded areas (\(\Delta E_{a}\) and \(\Delta E_{b}\)).  
spread of the charge cloud during drift or cloud splitting arising from X-ray fluorescence generally leads to a reduction in the measured energy of the detected photons [14]. In pixelated energy-resolving X-ray detectors, charge-sharing can impact the majority of detection events and thus needed to be taken into account in this study. The characteristics of charge-sharing include the appearance of low energy tails in the observed photopeaks when using monochromatic X-rays which, particularly for CdTe and CdZnTe detectors, can be due to carrier trapping, which results in 'hole tailing' [49]. However, we note that in the PiXirad (and other high-resolution energy-resolving X-ray detectors) the effect of hole tailing is ameliorated due to the small pixel size of the detector. There are several interconnected phenomena that determine the degree and characteristics of charge sharing when using polychromatic X-ray sources [41]. The most obvious of these are the apparent increase in the number of detected counts within the low energy region of the spectrum and concomitant reduction in the number of counts within the higher energy region [12].