4 Multiple Compton scatter events: Compton kinematic discrimination

One method has been suggested to overcome these complications in the context of liquid xenon time projection chambers (Aprile et al. 1993). Here, this method is formalized as Compton Kinematic Discrimination (CKD) and examined in more detail. This technique allows the order of the photon interactions to be determined with high probability, as well as providing the basis of a powerful tool for background suppression in GCTs.

CKD takes advantage of redundant measurement information in an event to determine the most likely interaction sequence. A photon of initial energy E (using the notation in Sect. 2) interacts in the instrument at N sites, depositing an energy of E_i at each location $\vec{r}_{i}$. It is assumed that the interactions $1, \ldots ,N-1$ are Compton scatters, and interaction N is the final photoabsorption. Given the correct ordering of the interactions, there are two independent ways of measuring N-2 of the scattering angles, $\cos{\phi_{2}}, \ldots , \cos{\phi_{N-1}}$.

Geometrical measurement of $\cos{\phi_{i}}$. From simple vector analysis, given the correct ordering of the interaction sites one can derive the scatter angles

$$\cos{\phi_{i}} = \vec{\hat{r}}_{i}' \cdot \vec{\hat{r}}_{i-1}', i = 2, \ldots,N-1 ~,$$ (9)

where the uncertainties in the scattering angles, $\delta \cos{\phi_{i}}$, can be estimated from the spatial uncertainty in the scattering angles (Eq. 8), yielding

$$\delta (\cos{\phi_{i}}) = \delta \phi_{i,r} \sin{\phi_{i}} ~.$$ (10)

Compton kinematics measurement of $\cos{\phi_{i}'}$. Given the correct ordering, the measured values of W_i can be derived, which were defined earlier as the energy of the photon after each scattering i, in units of $m_{\rm e}c^{2}$. The Compton scatter formula (Eq. 2) gives:

$$\cos{\phi_{i}'} = 1+\frac{1}{W_{i-1}}-\frac{1}{W_{i}}, i = 1, \ldots,N-1 ~,$$ (11)

$$\delta (\cos{\phi_{i}'}) = \left[\left(\frac{\delta W_{i-1}^{2}}{W_{i-1}^{4}}\right)\right.$$

$$\hspace*{1cm}\left.+ \delta W_{i}^{2}\left(\left(\frac{1}{W_{... ...-1}^{2}}\right)^2-\frac{1}{W_{i-1}^{4}}\right)\right]^{1/2} ~.$$ (12)

Given the N-2 independent measurements of $\cos{\phi_{2}}, \ldots , \cos{\phi_{N-1}}$, a trial ordering of the interaction sites can be tested for consistency. If the assumed ordering is incorrect $\cos{\phi_{i}}$ will not equal $\cos{\phi_{i}'}$ in general. Every possible permutation of orderings can be tested to determine the one most consistent with $\cos{\phi_{i}} = \cos{\phi_{i}'}$. Given a trial ordering, the two angle cosines for sites $i = 2, \ldots ,N-2$ are relabelled for convenience $\eta_{i} = \cos{\phi_{i}}$, $\eta_{i}' = \cos{\phi_{i}'}$.

As a first test, trial orderings that produce values of $\mid \eta_{i}' \mid ~ \geq 1$ are ruled out, since $\cos{\phi_{i}'} < 1$ for any scattering angle $\phi_{i}'$. This condition will eliminate many orderings which cannot physically be due to multiple Compton scatters followed by photoabsorption. Next a least-squares statistic measuring the agreement between the redundant scatter angle measurements is defined:

$$\chi^{2} = \frac{1}{N-2} \sum_{i=2}^{N-1} \frac{(\eta_{i}-\eta_{i}')^{2}}{(\delta \eta_{i}^{2} + \delta \eta_{i}'^{2})} ~\cdot$$ (13)

In general, $\chi^{2}$ will be minimized when the interactions are properly ordered (i.e. the order in which they occurred). Therefore, all possible permutations can be tested for their value of $\chi^{2}$, and the ordering corresponding to the minimum value, $\chi ^{2}_{\rm min}$, is taken as the most likely ordering.

This consistency statistic also provides a powerful tool for rejecting background events. If the event is truly a multiple Compton scatter event followed by a photoabsorption then $\chi^{2}_{\rm min} \sim 1$. By setting a maximum acceptable level for $\chi ^{2}_{\rm min}$, events that do not fit this scenario can be rejected. Such events include partially-deposited photons which scatter out of the instrument (Compton continuum), photon interactions with spatially unresolved interaction sites, events with interactions below the detector threshold, pair-production events, and similarly $\beta ^{+}$ decays. These events frequently have $\chi^{2}_{\rm min} \gg 1$, allowing a strong rejection statistic that is not very sensitive on the level set on $\chi ^{2}_{\rm min}$. Here, $\chi ^{2}_{\rm min}$ has been treated as a normal least-squares statistic with N-2 degrees of freedom, and events are rejected which have probabilities of $\chi^{2}_{\rm min} < 5\%$. Variations in the level between $1\%$ and $10\%$ do not strongly affect CKD rejection capabilities. For example, varying this level from $5\%$ to $1\%$ shifted the CKD efficiency curves in Fig. 4 by 1-2%.

Figure 4: Photopeak distributions for 3+ site events. The fraction of events with the first and second interaction sites spatially resolved are presented ($\diamond$), along with the fraction of events which have been properly ordered (hence imaged) using CKD ($\triangle$). The fraction of events rejected by the CKD statistic ($\Box$) as well as the fraction incorrectly imaged ($\times$) are also shown

The fraction of 3+ site photopeak events which have the first and second interaction sites spatially resolved - and hence could be imaged to the proper direction - is shown in Fig. 4 as a function of energy, for the instrument model discussed in Appendix A. Roughly $90\%$ of all events from 0.2-20 MeV have their first and second sites spatially resolved from each other. (Some of these events do not have their second, third, etc., interactions spatially resolved from each other, and will be rejected by the limits on $\chi ^{2}_{\rm min}$.) This figure also shows the fraction of the photopeak events which CKD properly orders (correctly reconstructed), as well as the fractions improperly ordered (hence incorrectly imaged to off-source background), and the fraction completely rejected. For energies below $\sim$10 MeV, CKD allows proper reconstruction (hence imaging) of $\sim 60-70\%$ of the photopeak events, while rejecting $10-20\%$. The remaining $10-20\%$ are incorrectly imaged into the off-source background. For comparison, if the order of the interaction sites were randomly chosen $<15\%$ would be correctly imaged, while the remaining $>85\%$ would be incorrectly imaged into the background.