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In the low flux limit x-rays incident on a CCD are assumed to be independently detectable. That is, the detection of any one x-ray is not affected by the x-ray flux incident on the detector. At sufficiently high flux this approximation breaks down as ``pileup'' occurs. In a CCD the piled-up x-rays do not directly interact with each other, but their electron charge clouds, which form in the depleted region of the CCD, merge or even overlap. All isolated x-rays produce charge clouds whose detection pattern varies strongly with (1) the x-ray energy, (2) the sub-pixel location of the initial photoelectric interaction, (3) the depth of the interaction, and (4) the detector electronics. All detections, or events, can be categorized by (1) the shape (grade) and (2) the magnitude (energy) of their charge cloud pattern. Figure 4.9 represents a matrix of the possible grade-energy combinations for which a single x-ray can be detected. The x-ray counts for a timed exposure will divide among the categories. In the low flux limit, the probability of a single x-ray to produce an event in any category approaches a ``pileup-free'' value characterisitic of the detector and the spectral content of the source. However, the probability of producing an event in any category is a function of the incident flux, and the the essential problem of pileup characterization is to determine this functional relationship. The effect of pileup is to redistribute the counts in each category, as exemplified by the arrows in the figure.
The largest effect of merging charge clouds is to reduce the number of detected ``good'' x-ray events. Here by ``good'' we mean that the event satisfies both of the following conditions: i) the event amplitude is in the spectral band of interest and ii) that the event grade (shape) is in the grade set of interest. The merged cloud resulting from pileup will most likely be detected as a single x-ray event with either a different grade (e.g. if the two x-rays landed in adjacent pixels) or a different energy (e.g. if the two x-rays landed in the same pixel). Many cases will appear as mixtures of these types of pileup. In either case, as a second good x-ray event is produced near a first good x-ray event, not only is the second event undetected, but the first x-ray is removed from detection as a good event. These major redistributions are represented by the heavy arrows in Fig. 4.9. The smaller redistributions shown by the light arrows occur very infrequently for quasi-monochromatic x-ray beams and are not considered here.
There are two complications in a general pileup analysis. The first concerns spatial distribution of the incident flux. We note that, in general, pileup effects are a function of fluence (number of events per unit area within a given CCD readout or exposure), and if the incident fluence varies with position on the detector, the pileup effects will also vary. Thus, in general, pileup can change the apparent spatial distribution of detected flux. In particular, the apparent HRMA/ACIS point-response function can be changed by pileup effects. The discussion in this section is based on experimental data with uniform illumination, and we do not specifically address effects of pileup on the point response function. We note, however, that the pileup ``cross-sections'' we report below are prerequisites for understanding the effects of pileup on spatial flux distribution measurements. A short discussion at the end of this section considers some aspects of pileup in the limit of a perfectly focussed point source.
The second complication for pileup analysis is the spectral distribution of the incident flux. Here there are two limits, those of a monochromatic source and a continuum source, respectively. The strategy followed in this discussion is to examine a monochromatic source first to understand the redistribution of events as a function of energy. Then, an approximation technique will be discussed to apply these results to a more general spectral distribution.
The remainder of this section is organized as follows. The next subsection details a simple pileup model for a quasi-monochromatic source. This approach only considers the effect of pileup on the number of detected counts at one major line in the spectrum. The strategy is to develop a theory based on fundamental pileup processes which can be applied for any CCD. The following subsection describes pileup experiments which when analyzed in the context of the preceeding theory provide generally applicable cross sections for pileup processe as a function of incident X-ray energy. The final section extends the simple model to include any arbitrary spectrum, specifically including spectral redistribution.