Determination of $\hat{\epsilon}$ and $\epsilon$

An example of experimental pileup data is shown in Figs. 4.23 - 4.26. The CCD w203c2 was used with a 7 second exposure on all 12 x-ray targets. The data for each target is presented with two graphs forming two columns . The left hand column plots the number of detected counts in the $K_{\alpha}$ per frame per quadrant and is normalized to the x-ray tube current (in units of $\mu$A) versus the x-ray tube current. The counts are determined by fitting a three parameter gaussian curve to the $K_{\alpha}$ peak and using the fitted coefficients to determine the total counts. The four grade selections displayed are G0 (triangles), G0234 (stars), G02346 (squares), and G01234567 = All (diamonds). The value for each quadrant is plotted independently. The lines result from a least squares polynomial fit to Eqn. 4.16. The fit for each quadrant is plotted independently. The zero current intercept averaged for all four quadrants is listed in the first column within each figure, and the zero current slope is listed in the second column. The third column is the slope normalized by the square of the intercept and multiplied by 10⁵, which is called the rate. This number provides a configuration-independent slope which only depends on the number of detected x-rays, not on the x-ray tube current. The units for the rate are (Counts/Frame/Qd)^-1. The fourth column displays the error for the rate.

In an ideal detector with no pileup effects, a horizontal lines would result. The non-zero slope of all the lines clearly indicates the presence of pileup. The presence of a non-linear slope for high Z indicates higher order pileup events, for which the quadratic correction of Eqn. 4.16 becomes important.

In the righthand column are similar plots for the number of pixels above the threshold (diamonds) and the total charge collected (triangles). Printed underneath the values of intercept and slope are the respective errors. The fact that the normalized charge doesn't vary significantly with x-ray tube current is good evidence for linearity of the x-ray flux with tube current.

Figures 4.27 and 4.28 are parallel to Figs. 4.23 - 4.26, except the analysis uses all x-rays in the spectrum, not just those in the $K_{\alpha}$ peak. A useful number from these plots is the intercept value from a linear fit to the x-ray fluxes from all grades. From this number the pileup-corrected branching ratios can be computed.

For our pileup analysis, the most significant number presented in each figure is the rate and error for the G02346 grade set, the set used most commonly for the ACIS calibration. These figures were produced for all 11 test configurations, although the other 10 configuration's figures are not included for space reasons. A summary for the G02346 rate values for all configurations is presented in Figs. 4.29 and 4.30. Figure 4.29 shows the rate for the first targets elements. The graph for each target shows the rate value with errors for each test configuration. Not every configuration has a value for every element since due to some experimental problems and time constraints. A mean value has been determined for the front sided data (i.e. not including w140c4r), excluding any suspicious data points. The mean value is plotted as the horizontal dashed line and is printed in the upper right of each figure. Figure 4.30 corresponds to Fig. 4.29, but shows results for the other six targets.

Figure 4.31 presents a summary of the mean rate values, both as a function of the $K_{\alpha}$ energy (top) and the x-ray penetration length (bottom) in silicon. With the exception of the Cl line at 2621 eV, the rate is roughly a constant at low energies, and increases quickly at high energies. The Cl line is a known exception since the target source is actually KCl and the K line competes with the Cl, causing more relative pileup than the other targets.

An example of pileup corrections for an ACIS CCD is shown in Table  4.22. Typical detected flux rates for the CCD w215c2r (which presently is located in the I3 position of the ACIS flight focal plane) are listed in the second column for the 13 different $K_{\alpha}$ x-rays listed in the first column (Mn $K_{\alpha}$ has now been included, whose source is radioactive Fe⁵⁵). The third column lists the mean values for $\hat{\epsilon}$ presented in Figs. 4.29 and  4.30. The fourth column lists the corresponding incident fluxes as determined by Eqn. 4.16, and the next column is the ratio N_i/N_d. These measurements were conducted with a 3.28 s exposure time. The last column lists the corresponding ratio if a 7.15 second exposure time had been used for the same incident flux.
 
 

Table 4.22: Example of pileup correction factors for typical detection fluxes in w215c2r

 

$K_{\alpha}$ X-ray	Detected Flux	$\hat{\epsilon}$	Incident Flux	Correction Factor	Correction Factor
	(G02346)	(105)	(G02346)	(3.28 sec)	(7.15 sec)
Al	1145	6.69	1246$\pm$15	1.088
Si	1310	4.88	1404$\pm$17	1.072	1.186
P	1120	7.15	1225$\pm$13	1.094	1.257
Cl	605	11.1	651 $\pm$6	1.076
Ti	1285	4.62	1370$\pm$8	1.066	1.169
V	1180	4.57	1251$\pm$8	1.059
Mn	880	7	938$\pm$6	1.060	1.174
Fe	1040	8.07	1143$\pm$10	1.099
Co	950	10.3	1062 $\pm$11	1.118
Ni	840	13.5	959 $\pm$19	1.142
Cu	700	18.2	815 $\pm$22	1.164	1.643
Zn	660	26.4	829 $\pm$46	1.256
Ge	480	46.9	662 $\pm$87	1.379

The previous analysis is directly specific to the spectra emitted by HEXS. That is, $\hat{\epsilon}$(E) includes all the effects of spectral impurities particular to the HEXS source. While this is useful for computing pileup effects on HEXS data, a more general analysis requires $\epsilon$, which is independent of the spectra. Equation  4.13 gives the relation between $\hat{\epsilon}$ and $\epsilon$in terms of $\alpha$ (the number of bad events per good event) and $\epsilon ^{'}$ (the average $\epsilon$ associated with the ``out-of-band'' portion of the spectrum). Mathematically, a direct solution for $\epsilon$(E) from the 12 different energies is complicated since $\epsilon ^{'}$ depends on $\epsilon$, and we use an iterative approximation described as follows. Using interept values for G02346 and G01234567 from the left hand column in Figs. 4.23 - 4.28, we compute f_B or $\alpha$. Plots of$\alpha$ for each target from every data set are presented in Figs. 4.32 and  4.33. Using initial guesses for $\epsilon$, spectrally averaged values of $\epsilon ^{'}$ are computed. Together with $\alpha$, new values of $\epsilon$ are generated according to Eqn. 4.13, and the process is repeated until the values become self-consistent. Plots of $\epsilon ^{'}$ for all targets and data sets are shown in Figs. 4.34 and 4.35. Figures 4.36 and  4.37 display the final values of $\epsilon$, as computed by Eqn. 4.13. These values of the pileup cross section as a function of energy for a monochromatic incident beam, which are summarized in Fig. 4.38, represent the major result of the pileup experiments.

Notice that the low energy values are close to the theoretical minimum of $\epsilon = 3.4\times10^{-5}$. Recall this value corresponds to a cross-sectional area of 9 pixels in one quadrant (viz, 9/262144). We do not yet understand why the value of $\epsilon$ has a slight increase at the lowest energies. The discrepancy is no more than about one standard deviation. A comparison with Fig. 4.32 suggests that spectral impurities may be the cause since the 1740 eV Si point (which is most spectrally pure) lies near the minimum while the 2622 eV Cl point (which is most spectrally impure due to a potassium line) lies far from the minimum. The 1487 eV Al source is also known to be very spectrally impure.

Figure 4.23: Raw HEXS pileup data for detector w203c2 with a 7 second exposure for characteristic K lines from Al, Si, and P targets, with best-fit model for various ASCA event grade sets. triangles:G0; stars:G0234; squares:G02346; diamonds:G01234567. Points for each detector quadrant are shown. Best-fit model parameters are shown in each panel. See text for details.

 

Figure 4.24: Raw HEXS pileup data for K  detector w203c2 with a 7 second exposure for characteristic K lines from Cl, Ti, and V targets, with best-fit model for various ASCA event grade sets. triangles:G0; stars:G0234; squares:G02346; diamonds:G01234567. Points for each detector quadrant are shown. Best-fit model parameters are shown in each panel. See text for details.

 
  Figure 4.25: Raw HEXS pileup data for K  for w203c2 with a 7 second exposure for Fe, Co, and Ni targets.

 
  Figure 4.26: Raw HEXS pileup data for K  for w203c2 with a 7 second exposure for Cu, Zn, and Ge targets.

 
  Figure 4.27: Raw HEXS pileup data for entire spectrum for w203c2 with a 7 second exposure for Al, Si, P, Cl,Ti, and V targets.

 
  Figure 4.28: Raw HEXS pileup data for entire spectrum for w203c2 with a 7 second exposure for Fe, Co, Ni, Cu, Zn, and Ge targets.

 
  Figure 4.29: Mean g02346 epsilon for all data sets with no spectral correction (Al - V)

 
  Figure 4.30: Mean g02346 epsilon for all data sets with no spectral correction (Fe - Ge)

 
 
Figure 4.31: Variation of epsilon with Si penetration depth for raw HEXS data

 
 
Figure 4.32: Variation of alpha, the ratio of bad to good x-ray events, for Al to V.

 
 
Figure 4.33: Variation of alpha, the ratio of bad to good x-ray events for Fe to Ge.

 
 

Figure 4.34: Variation of spectrally averaged epsilon ($\epsilon ^{'}$) for Al to V

 
 

Figure 4.35: Variation of spectrally averaged epsilon ($\epsilon ^{'}$) for Fe to Ge

 
 

Figure 4.36: Mean g02346 epsilon for all data sets including a spectral correction (Al - V)

 
 

Figure 4.37: Mean g02346 epsilon for all data sets including a spectral correction (Fe - Ge)

 
 

Figure 4.38: Variation of epsilon including corrections for spectral impurities