Sources of Modelling Errors in the Absolute Calibration of Reference Detectors.

The CCD response model used to analyze the undispersed synchrotron radiation data suffers from a number of shortcomings. The magnitidue of the quantum efficiency errors resulting from these limitations is not well known at present. In principle, simulations comparing the accuracy of the simple models we have used to more realistic (and complicated) models could provide estimates of the magnitude of these errors. Indeed, we have already implemented more elaborate CCD models in our simulations (see section 4.14). The best approach, however, would be to fit these more sophisticated models to the calibration data. There is no technical reason why this could not be done in the near future.

For purposes of guiding such work, we list the major shortcomings of the model we used to establish the absolute response of ACIS reference detectors.

1.

The gate structure model is oversimplified. The slab and stop model ignores gate overlaps and phase-to-phase variations in gate thickness. The resulting error varies in lowest order as the square of the optical depth of the gate structure, so in the limit that the gates are optically thin, the error vanishes. Conversely, the magnitude of this error is largest just above the absorption edges of oxygen and silicon, and at very low energies. This error can readily be quantified via simulation.

2.

Absorption fine-structure was ignored in determing model parameters. The undispersed synchrotron radiation data were analyzed using standard Henke (1993) absorption coefficients; these omit edge structure which we have since measured (see section  4.6.4.) While the spectral resolution of the detector tends to smooth the fine structure, we have not yet established the magnitude of error introduced by neglect of fine structure. The relatively large residuals from the best-fit model (see section  4.6.1) near the absorption edges probably reflect this error. We hope to repeat the reference detector fits with the fine-structure included in the near future.

3.

The redistribution function is oversimplified. A phenomenological representation of the spectral redistribution function has been used in analysis of synchrotron radiation data. The response to a monochromatic input is modelled as the sum of 2 Gaussians plus a phenomenological low-energy tail. A better, physically-grounded model of the redistribution function is now available (see sections 4.3.2 and  4.14), but has not yet been used to analyze the PTB/BESSY data.

A more subtle but related difficulty is that aside from the fact that in our current model the channel stop is taken to be a dead volume, there is no allowance for variation of the redistribution function with position within a pixel. A more realistic picture is that the spectral redistribution function for photons absorbed in the channel stops is non-zero but differs markedly from that for the rest of the device. We are now modelling this effect.

4.

Channel stop parameter values have not been measured directly for the reference detectors. The most reliable measurements of channel stop dimensions are obtained using the mesh technique described in section 4.5, and from (destructive) scanning electron micrographs. Neither of these techniques has been applied to determine the channel stop parameters of the reference detectors themselves. It is hoped that mesh measurements can be made on at least one of the reference detectors.

5.

Spatial variations in quantum efficiency have not been physically modelled. While the method used to infer depletion depth is simple and apparently reasonbly accurate, it is subject to a number of systematic errors, as has been discussed in section 4.6.2. Of particular note are variations in depletion depth with position in each detector. The branching ratio data show small (several microns out of 65-75 microns) variations in depletion depth from quadrant to quadrant within a detector. Thus the use of spatially averaged quantum efficiency may lead to systematic errors at high energy. Indeed the quantum efficiency maps (see section 4.7) show some modest ($~\sim 2-3$) residual variation with detector position at 8 keV; little, if any, such variation is seen at lower energies in the FI devices. While we believe the best interpretation of these variations is that they reflect spatial variation in the depletion depth, this interpretation is speculative.