Analytical model by B. McNamara

An analytical treatment of pileup has been performed byMcNamara (1997). He defines the pileup fraction as the ratio of number of frames with two or more events to the number of frames with one or more events, consistent with our definition above in eq. 6.14. The model does not include any grade selection effects, and we will thus compare its prediction with the measured pileup for All-255 grades in Table 6.22. In addition, the model predictions hold for the Al-K$\alpha$ energy only.

Figure 6.37 shows the plot of the difference between the predicted pileup fraction from McNamara's model and the observed pileup as a function of the source counts per frame, at the Al-K$\alpha$ energy and for All-255 grade selection. The predicted pileup fraction was extrapolated from the plot in Figure 2 ofMcNamara (1997). assuming an encircled energy fraction of 0.7, as appropriate for the XRCF data. Figure 6.37 shows that the analytical model always overstimates the measured pileup. The maximum discrepancy (30%) is obtained for the FI chip at large source fluxes. For our fiducial flux of 1 count/frame, the pileup is overestimated by 24% for the FI chip and 10% for the BI chip.
 

Figure 6.37:  Plot of the percent difference between the
pileup predicted by the analytical model of McNamara (1997) and the
pileup measured for phase H data (Table 2), versus incident source
flux. Only data for grade selection All--255 and for the Al-K-alpha
energy (where the model is defined) are used. It is apparent that the
model overestimates the pileup fraction, especially at high source fluxes.

 

We conclude that the analytical model provided byMcNamara (1997) can be used to obtain a conservative upper limit for the pileup for both FI and BI detectors, for a monoenergetic sources and no grade selection.