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**Up:** Pileup Measurements and Modelling
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The simple model starts by defining any x-ray event detected with the
desired energy and grade to be ``good''. All other events (i.e., those
with different energy or grade) are defined as ``bad''.
Most practical x-ray line
sources are not monochromatic, and are typically a mixture of a
monochromatic line, other spectral features, and some continuum. The
pileup model quantifies this aspect by defining that for every
``good'' x-ray event there are ``bad'' events. Alternatively, if,
in the absence of pileup, the number of good incident x-rays is
N_{i} out of a total number of events N_{T}, then the fraction
f_{B} of good x-ray events is
. This model of a quasi-monochromatic beam
is a good representation of the x-ray sources used in the ACIS quantum
efficiency calibration.

Although for any CCD exposure the time history of event interactions
is unknown, for the purposes of analysis we can picture the
x-rays incident in one exposure as striking the CCD serially.
The goal is to describe a function *N*_{d}(*N*_{i}) which
represents the number of detected good x-rays as a function of the
number of incident good x-ray events. By inverting this
function, we can determine *N*_{i} from an experimental measurement of
*N*_{d}. We begin construction of this function by examining the effect
of a single x-ray.

Let be the effective area of the CCD affected by the occurrence of a ``good'' x-ray event (typically the desired energy is that of a K line the desired shapes are ASCA grades 0,2,3,4,and 6). In general, will be a function of energy; the energy dependence is described in detail later. Similarly, is the average effective area corresponding to any other x-rays, i.e. those with different energies and grades. The physical meaning of is that if a second x-ray is absorbed near a prior x-ray event such that the center of the second photoelectric absorption occurs within the area centered on the first x-ray, then an interaction occurs. Specific interaction effects are described mathematically below. We can derive the minimum size for commensurate with our event detection criteria. All (standard) event discrimination is based on the 3x3 pixel subarray surrounding a local maximum of detected charge. Thus, any second x-ray landing within the subarray invokes an interaction, and the area of 9 pixels forms a lower limit for . In all that follows, we express and in units of the area of one quadrant of an ACIS CCID17 detector. In these units, a nine-pixel ``island'' has an area of .

The mathematical model begins by schematically dividing the area of a
CCD as shown in Fig. 4.10. In our units, the total surface area of
the detector (one quadrant of a CCID17) is normalized to unity.
Let A_{1} be the total area occupied by all good
x-ray events (A_{1} can also be interpreted as a probability or cross section
for interaction; however the interpretation as an area is useful for
developing a model). The number of detected events is taken to be . This assumption is approximate since two good x-ray events
could lie close enough together so that their s overlap
while they do not interact. Let A_{2} be the total area occupied by
charge produced by all other events. Then 1-*A _{1}*-

(18) |

(19) |

The solution is obtained by combining the two equations to separate variables. This results in a second order differential equation with the following solution,

(20) |

(21) |

(22) |

(23) |

and

(24) |

Equation 4.9 has the desired asymtotic limit of *N*_{i} = *N*_{d} for low
flux, with the following useful expansion for the logarithm of the ratio:

(25) |

In the limit where and then

Solutions to Eqn. 4.9 are plotted in Fig. 4.11
for two different flux
ranges and for =0,2,4,...40 (x10^{5}) using
. Specifically, the cross-section
equals the fractional area of a CCD region of
interest, and *N*_{d} and *N*_{i} are the corresponding counts in that
region.
Note that corresponds to a area of 26.2
pixels. The deviation from the line *N*_{i} = *N*_{d} increases as
increases. All the curves bend over for
significantly high pileup and should assymtote to 0 for high enough
*N*_{i}. This high pileup limit corresponds to severe charge cloud
overlap so that very few x-rays satisfy the event selection criteria.

11/20/1997