Gain Determination & Uncertainty

The value of the simulator can be demonstrated by the following example. Allyn Tennant of the MSFC Project Science team analyzed XRCF data from ACIS which was taken with a continuum source dispersed across the ACIS-S array. These data provided an excellent check of the ACIS intrinsic energy resolution, because by separating the image into separate spectra, one can get effectively an incident monochromatic beam, with the incident energy slowly increasing with position.

The surprise that appeared was that the `gain', as measured by fitting a Gaussian peak to the ACIS spectra at each point, showed a jump at the Silicon K edge energy. (See Figure 7.6.) This is surprising because the CCD should have no intrinsic gain within the detector, and the analog and digital electronics should have no special sensitivity to the Si K edge (as the electronics are only seeing a charge packet).
 

Figure 7.6:  Peak of a Gaussian fit to CCD spectra extracted
from  a grating dispersed incident spectrum.  The abscissa gives the
energy of the incident photons inferred from the grating dispersion
equation.  Note the apparent `break' in slope near 1.84 keV.
[Data & analysis by MSFC Project Science].

While the CCD should have no significant gain change at the Si K edge, the energy resolution function shows an abrupt change across the edge. First, in FI devices, photons above the K edge can create fluorescent Si K$\alpha$ photons at 1.74 keV due to absorptions in the gates, which cause reabsorbed photon detections of Si K$\alpha$ in the depletion region. Second, photons just above the K edge have very shallow penetration depths into the CCD, so, proportionally, a much high proportion of these photons interact in the channel stops than photons with energy just below the K edge. (In this case the lower energies penetrate more, and so many can pass right through the stop and interact normally in the depletion level below.)

The combined effect of Si K fluorescent photons and more channel stop photons is to build up an apparent low energy wing onto the energy response function. We have run our CCD simulation program and find about a factor of ten higher soft shoulder for photons above the K edge, than for photons below it.

To see how much this shoulder affects fits, we created simulated datasets at 1830 eV (just below the K edge) and 1860 eV (just above the edge). (See Fig. 7.7.) Then, simple Gaussian models were fit to the data. In the first case only a single Gaussian was used (which is what we believe was done by MSFC Project Science). Next, two or three Gaussians were used (one for the main peak, one broad and lower energy for the channel stop events, and one [for the 1860 eV case] for the fluorescent peak at 1740 eV). The results were:

Figure 7.7:  ACIS simulations of 1830 eV (left) and 1860 eV (right)
incident photons.  Solid lines are model fits of 1, 2, or 3 Gaussian
components.  Note that the scale is logarithmic, and the soft shoulder
is ten times higher in 1860 eV simulation than 1830 eV.

 

Note that for the single Gaussian fit the mean of the primary peak `jumps' upward by 1.003-0.9956 = 0.74%. This is very similar to the jump that MSFC Project Science found of 2.5/415 channels = 0.6%. We believe that the discontinuity in `gain' results from a discontinuity in the instrument energy response profile at the Si K edge, and not in either a gain jump in the Si of the CCD or the CCD camera electronics and processing. Approximating the CCD response by a single Gaussian is inexact and the effect of this inaccuracy is to induce this non-linearity at the edge when the energy response changes rapidly.

It is interesting to observe how adding a low energy shoulder onto a Gaussian peak can raise the mean of the peak. What seems to happen is that the fit results in a larger sigma to accomodate the soft shoulder, which results in a higher mean because the high energy tail of the Gaussian model simply ignores the data. (There are more channels better fit by the single Gaussian model on the soft side of the peak, so the model chooses that over fitting the hard side and ignoring the soft.)

Note that using 2 or 3 Gaussians to approximate the response is better (the jump decreases from 0.74% to 0.21%), but it is not perfect. To reach accuracies significantly better than 1% will require treating the energy response by using the full simulation distribution or an accurate high order empirical approximation. Alternatively we can make approximations using simpler functions (single or multiple Gaussians) but then we need to calibrate them using the full simulator (i.e. fit the simulation results to the simple functions, and determine a calibration relation between the simple functions and the true values).

Currently, the determination of X-ray energies from ACIS data is based on laboratory calibration measurements done at MIT, as described in Chap. 4. These calibrations apply multi-Gaussian fits to the lab data, and so are presumably no more accurate than the results shown in Table 7.2. Thus we suggest that the knowledge of the ACIS energy is accurate to a level better than 1%, but that these systematic effects must be limiting the accuracy to no better than 0.2%.
 

Table 7.2: Comparison of apparent peak shifts when fitting one, two or three Gaussians to ACIS spectra. 

Ratio of fitted energy in main peak to true energy
Energy	1 Gaussian	2 Gaussians	3 Gaussians
1830 eV	0.9956	0.9967	-
1860 eV	1.003	0.9989	0.9989