5 Single Compton scatter events: Single scatter discrimination

CKD will only work for N > 2 since there are no independent scattering angle measurements for a single Compton scatter followed by a photoabsorption. It turns out that the ordering of two-site photopeak events can still be determined with a high probability; however, the ability to reject background events is lost. As is discussed further in Sect. 6, the loss of background rejection, coupled with low peak-to-Compton ratios, and a larger fraction of backscatter events mean that the inclusion of two-site events will likely hurt the sensitivity of a GCT; however, discussion of event ordering is still included for completeness.

Given a two-site event, the first test one can perform is to determine whether both possible orderings of the interaction sites are energetically compatible with a single Compton scatter, i.e. are compatible with the requirement that $\cos{\phi_{1}} < 1$ (Eq. 2). In Fig. 5, the fraction of spatially-resolved, photopeak events with unique orderings are plotted versus energy. Also plotted are the fraction with ambiguous orderings. At energies below $\sim$0.4 MeV the majority of resolved photopeak events have a unique ordering, while at higher energies most events are ambiguous.

As an empirical test of the ambiguous events, the relative magnitude of the energy lost in the initial scatter (E₁) compared to the photoabsorption (E₂) can be compared. The fraction of resolved two-site photopeak events which have ambiguous orderings with E₁ > E₂ is plotted in Fig. 5. At higher energies, nearly all of the resolved photopeak events with ambiguous interaction orders have E₁ > E₂, which can be used to determine the most likely interaction order. This empirical result can be easily understood, in hindsight, by the fact that photons which deposit most of their energy in the initial Compton scatter are much more likely to be photoabsorbed in the second interaction.

Figure 5: Empirical distributions for fully-resolved, two-site photopeak events. The fraction of events with only one physically possible ordering ($\Box$), as well as events with both orderings physically possible ($\triangle$) are shown. Also shown are the events with both orderings possible, but with the larger energy deposit in the initial scatter rather than the photoabsorption, E1 > E2 ($\diamond$)

Therefore, a simple Single Scatter Discrimination (SSD) technique to determine the most likely interaction ordering of two-site events follows. First one determines whether a physically unique ordering exists; if not, the larger energy deposit is assumed to be the initial Compton scatter. Only at the lowest energies are two-site events possible in which neither ordering is acceptable (unresolved events, Compton continuum), and some background rejection is possible.

The fraction of two-site photopeak events which are spatially resolved is shown in Fig. 6 as a function of energy, for the instrument model in Appendix A. Roughly $80\%$ of all events from 0.2-20 MeV are resolved. This number is about $10\%$ lower than the 3+ site events, due to the smaller path lengths of the lower energy scattered photons in two-site photopeak events. Also shown in Fig. 6 is the fraction of events which SSD has properly reconstructed. SSD allows proper reconstruction (hence imaging) of $\sim 60-80\%$ of the photopeak events, while improperly imaging the remaining $\sim 20-40\%$ into the off-source background. Only a relatively small number of low energy events can be rejected outright. SSD is least effective around 0.5 MeV, where the unique/ambiguous ordering signatures are not as clear. For comparison, if the order of the interaction sites were randomly chosen $\sim 40\%$ would be properly imaged, while the remaining $\sim 60\%$ would be improperly imaged into the background.

Figure 6: Photopeak distributions for two-site events. The fraction of events with the first/second interaction sites spatially resolved are presented ($\diamond$), along with the fraction of events which have been properly ordered (hence imaged) using SSD ($\triangle$). The fraction of events rejected due to no physically possible orderings ($\Box$), as well as the fraction incorrectly imaged ($\times$) are also shown