Analysis

Before discussing how to reduce the BESSY data, we review the assumptions that go into generating the CCD response function. The spectral redistribution function (or matrix) contains all the information (e.g. gain, fluorescence and escape peak amplitudes, FWHM vs. energy) necessary to perform modeling of a pulseheight distribution. Refer to Section 4.1 for a full description of the relevant quantities and their determination. The tool that generates the redistribution matrix (hereafter RM) is rspgen, written by A. Rasmussen [Rasmussen1995b]. rspgen uses a description of the CCD's response to monochromatic X-ray sources over a range of energies. Table 4.25 lists the options used to create RMs for the analysis of BESSY data.

When invoked, the unity command normalizes all redistribution function to unity at each energy. This is equivalent to assuming the branching ratios are not a function of energy. Although this is generally incorrect, by limiting the analysis to a specific energy range (0.3-4 keV) and to a particular subset of events (ASCA grades 0,2,3,4,6), the assumption is valid. The threshold values are matched to those used in the extraction of events from the CCD output. Although the same CCD/electronics combination was used for the BESSY calibration, the energy scale and noise conditions can vary slightly between the two measurements. The gain, offset, and broadening options account for the differences in the operating conditions and modifies the RM accordingly.
 
 

Table 4.25: Summary of Adjustments to rspgen required for BESSY data

Option	Flag	Units	Remarks
unity	u	N/A	assumes unity branching ratios
split threshold	p	ADU	standard MIT values; depends on CCD electronics
event threshold	v	ADU	standard MIT values; depends on CCD electronics
grid spacing	E	keV	defines the energy grid that rspgen uses
gain	g	N/A	scales the ECD data
offset	o	ADU	offsets the ECD data
broadening	b	ADU	decreases or increases the ECD in quadrature
tailing	t	N/A	scales the low energy redistribution

The purpose of the BESSY experiments is to obtain an absolute calibration for the reference detectors. This goal is achieved by multiplying a model of the CCD quantum efficiency with the RM, convolving the model response with the input spectrum, and comparing the results to the data. The parameters of the CCD QE model are adjusted to the find the best fit to the data. The current CCD model is the Slab and Stop Model synthesized by K. Gendreau [Gendreau1995]. It approximates the gate structure as piecewise-uniform layers of Si, SiO₂, and Si₃N₄ and the channel stop as layers of Si and SiO₂ with finite width. The final two CCD parameters are the width of the pixel and the depletion depth. The model assumes that the gate and channel stop are dead, and only those photons interacting in the depleted silicon will be detected.

Fitting the BESSY data with this model does not constrain all the parameters. The channel stops occupy a small area compared to the gate structure. This reduces the role of the channel stop parameters on the overall fit and leads to degeneracy in these three best-fit parameters^4.3. Rather than determining these dimensions from BESSY data, we have measured the channel stops by other methods and frozen the parameters at these values. For the energy range of the White Light calibrations (E < 4 keV), the quantum efficiency is mainly determined by the gate parameters and is insensitive to the depletion depth. Therefore, this parameter is also frozen at a value determined by another method (see Section 4.6.2 for details). Table 4.26 gives the parameters names, describes it and states whether or not it is fixed.
 
 

Table 4.26: CCD Model Parameters for BESSY absolute measurements

Parameter Name	Description	Status
Dead	Si gate thickness	free
SiOx	SiO2 gate thickness	free
SiNit	Si3N4 gate thickness	free
Depl	Depletion depth	frozen-determined by branching ratio method$^{\dag }$
CSWidth	Channel Stop width	frozen-determined by SEM measurements$^{\ddag }$
CSSiOx	SiO2 stop thickness	frozen-determined by SEM measurements$^{\ddag }$
CSSi	Si p+ thickness	frozen-determined by mesh experiments$^{\ddag }$
Pix	Pixel width	frozen at 24 $\mu m$
$\dag$: see Section 4.6.2
$\ddag$: see Section 4.5

A total of eleven devices were characterized at BESSY during six separate 48 hour shifts^4.4. Table 4.27 shows when each CCD was calibrated, how many data sets were taken, and the number of CCD positions illuminated. A typical measurement consisted of acquiring 2000 frames. Integration times ranged from 0.83-1.53 seconds, depending on which readout electronics were used. Storage ring currents ranged from 2 to 50 electrons, but typically the ring current was adjusted to either 10 or 20 electrons (again, this depended on readout electronics) in order to provide 350 counts/frame/quadrant. For a typical measurement, this yields on order of $3 \times 10^{6}$ counts in the 0.3-4.0 keV band over the illuminated part of the CCD.

Reduction of the data begins by extracting events from the raw data and saving the location, pulse-height value, and frame number of each event in an event list. The storage ring current is monitored by the BESSY facility, and this data stream is searched for the loss of electrons from the storage ring during our measurements. If such a loss occurred, the event list is temporal filtered. Pileup effects are very dependent on the initial flux rate, and limiting the data to a single ring current allows the most accurate correction. Finally, the event list is further filtered by event grade selection and a XSPEC compatible PHA file is produced. At this time, the data from an entire quadrant is averaged together in a 5mm $\times$ 6mm spatial bin.
 
 

Table 4.27: Summary of synchrotron measurements made with PTB beamlines at BESSY.

Chip	Date	Data Sets	Chip Positions	Electronics
w34c3	Mar 95	7	4	acis 3
w34c3	Apr 95	17	4	acis 3
w34c3	May 95	6	5	acis 6
w34c3	Aug 95	3	1	acis 3
w103c1	Mar 95	7	4	acis 3
w102c3	May 95	6	5	acis 6
w103c2	May 95	5	5	acis 6
w103c4	May 95	6	5	acis 6
w103c4	Aug 95	5	5	acis 3
w147c3 (BI)	Aug 95	15	5	acis 3
w148c4 (BI)$^{\ddag }$	Aug 95	6	5	acis 3
w190c1	Jun 96	10	5	dea-sn014
w190c1	Dec 96$^{\dag }$	6	5	dea-sn017
w190c3	Jun 96	10	5	dea-sn014
w168c2	Dec 96$^{\dag }$	5	5	dea-sn017
w203c2	Dec 96$^{\dag }$	19	5	dea-sn017
$\dag$: Wavelength Shifter Measurement
$\ddag$: ACIS-2C device

The rspgen-generated RM, an ``atable'' model of the incident synchrotron radiation spectrum, the CCD model, and the PHA file are loaded into XSPEC. The atable model (/usr/acis/atable/bessy_new.mod) has two parameters, an overall normalization factor and another parameter that exists for historical reasons but is no longer used. The data is rebinned, the energy range limited to 0.3-4.0 keV and the model parameters are left free or frozen according to Table 4.26. If pileup is significant, two techniques are available to correct the data. One method developed by A. Rasmussen utilizes a mtable model that is compatable with XSPEC. This empirical model simulates the effects of pileup from BESSY data and relies on the incident flux levels being sufficiently low [Rasmussen1995a]. It has one phenomenological parameter that corresponds to the incident flux rate. XSPEC then fits for gate thicknesses, the source normalization and the pileup parameter. The other method developed by S. Jones (see Section 4.4) utilizes extensive laboratory data from incident flux-count rate linearity studies. The pulse-height spectrum is directly corrected, and this corrected PHA file is then read directly into XSPEC.

Figure 4.59, Figure 4.60, and Figure 4.61 show the best fit models with the data for individual quadrants of detectors w190c3, w190c1, and w103c4. The RMs used for the fit can be found in:
/ohno/d9/mjp/BESSY/XSPEC/w190c3/t0852/c1.rsp
/ohno/d9/mjp/BESSY/XSPEC/w190c1/t2347/tmp_memo.rsp
/ohno/d9/mjp/BESSY/XSPEC/w103c4/t1414/tmp_c1.rsp
Pileup corrections have been applied to the data for all three devices. The Rasmussen method was used for devices w190c1 and w190c3, while the Jones method was used for w103c4. To check the consistency of each method, w190c3 was also corrected using the Jones method. Normalization values from the two techniques differed by $\sim 0.1$%, indicating excellent agreement between the two correction methods. Table 4.28 shows the best-fit parameters, the RMS error, and the normalization accuracy for each reference detector as well as listing the values of the frozen parameters.
 
 

Figure 4.59: w190c3, quad B: BESSY data vs. model

 

Figure 4.60: w190c1,quad A: BESSY data vs. model

 

Figure 4.61: w103c4, quad B: BESSY data vs. model

 

Table 4.28: CCD model parameter fit results from synchrotron radiation measurements

Chip	CCD Parameters (in $\mu m$)							(0.3-4 keV) RMS Deviation of Fit	Best Fit Normalization
	Free			Fixed
	Si$^{\dag }$	SiO$_{2}^{\ddag }$	Si3N4	CS Si	CS SiO2	CS Width	Depletion Depth
w190c3	0.259	0.354	0.031	0.35	0.45	4.1	71.3	2.54%	$1.000 \pm .003$
w190c1	0.261	0.358	0.029	0.35	0.45	4.1	70.6	2.24%	$0.994 \pm .003$
w103c4	0.291	0.202	0.030	0.35	0.45	4.1	57.9	3.88%	$0.956 \pm .003$
$\dag$: typical 90% confidence error is $\pm 0.008$$\mu$m
$\ddag$: typical 90% confidence error is $\pm 0.011$$\mu$m

As a whole, the model fits for detectors w190c1 and w190c3 are quite good. In both cases the data:model ratio shows two deviation from unity: a narrow feature around 1.8 keV and a systematic underestimation of the flux above 2.2 keV. An underestimation of the Si K$\alpha$ fluorescence could help contribute to the narrow feature. Analysis of the response function data indicates that the fluorescence yields in the current response matrices are too low (see Section 4.14). Another probable source of error is the use of the Henke optical constants. Our EXAFS measurements (see Section 4.6.4) show large deviations from the tabulated Henke values around both the O K$\alpha$ and Si K$\alpha$ absorption edges. In the future we will incorporate these optical constant measurements in our model. The excess above 2.2 keV results from the good, but not exact, correction for pileup. The best-fit normalizations are within 1% of the calculated value, but could change with an improved pileup algorithm and inclusion of more accurate fluorescence data. Another measure of the goodness of fit for these two devices is the comparison of the derived gate thicknesses. w190c1 and w190c3 came from the same wafer, and hence, underwent the exact same fabrication process. The differences between the derived thickness for the Si, SiO₂, and Si₃N₄ layers are well within the errors.

The fit for reference detector w103c4 is noticeably worse than the other two devices. The RMS error is higher, and the best-fit normalization is nearly 6% too low. Above 2 keV, the model underestimates the data by almost 10%. The deviation from unity illustrates the importance of correcting the data for pileup. At the same time, however, the best-fit values for the three gate layers are reliable. Studies with devices w190c1 and w190c3 indicate that neglecting pileup influences the RMS error and normalization but has only small effects on the best-fit gate thicknesses. This behavior is consistent with the low level pileup model discussed in Section 4.4. To first order, pileup shifts events out of the acceptable grades (this explains the low best-fit normalization) and deposits some small fraction of these events' charge into non-physical, higher energy events (this accounts for the excess of counts above 2 keV).

Extensive measurements at MIT CSR have shown that the QE of front-illuminated CCDs have little spatial variation over bin sizes of 0.77 mm². Because the intensity and shape of the BESSY spectrum changes over the illumination pattern, similar measures of fine spatial uniformity are non-trivial. Our current spectral fitting procedure is to integrate all the data into 5mm $\times$ 6mm spatial bins. Table 4.29 shows the average counting rate (cts/sec/e^- ring current) normalized to the y001 position for each quadrant of w103c4 at five different CCD positions. Data is from the August 95 shift. The counting rates for y768 are expected to be higher than those for any other quadrants since the beam current was lower for this measurement and pileup effects should be smaller. Statistical errors for the data sets are less than 0.002. Finally, on average there is one bad column per quadrant that does not extend the length of the CCD. This analysis does not account for the reduction of the active area caused by the bad column, and this introduces an uncertainty on order of 0.004 to the ratios.
 
 

Table 4.29: Average counting rate (cts/sec/e^- ring current) for w103c4 normalized to CCD position y001. Statistical errors for all measurements are below .002. No pileup corrections have been applied.

CCD Position	Ring Current	Quad A	Quad B	Quad C	Quad D
y001	19 e-	1.0000	1.0000	1.0000	1.0000
y209	19 e-	1.0069	1.0087	1.0056	1.0049
y417	19 e-	1.0069	1.0062	1.0080	1.0025
y625	19 e-	1.0031	1.0050	1.0043	0.99938
y768	18 e-	1.0056	1.0019	1.0025	1.0012

 

A final check of the BESSY data is a comparison of the relative quantum efficiency measurements made at MIT CSR. Reference standards w190c3 and w103c4 were calibrated with respect to one another at discrete energies using the HEXS (see Section 4.7.1 for details). The model fitting to the BESSY data yields absolute efficiencies for both devices. Dividing these continuous curves by one another provides an independent check of both the MIT CSR measurements and the quality of the CCD model. The upper panel of Figure 4.62 shows the absolute quantum efficiencies determined from the synchrotron data. The higher efficiency of w103c4 is easily understood as it has a thinner gate oxide layer than w190c3. The bottom panel of Figure 4.62 shows the discrete relative measurements made at MIT CSR vs. the continuous BESSY-derived relative quantum efficiencies. The errors associated with the MIT CSR values are systematic and currently estimated at 2%^4.5.
 

Figure 4.62: BESSY absolute efficiencies vs. MIT relative efficiencies for reference detectors w190c3 and w103c4.