A typical example and possible sources of errors

a. Statistical errors

Here is an exemplary calculation of depletion depth for the ACIS flight device w193c2. The dataset /ohno/di/database/w193c2/butthead/fe55/30nov96/1713/ represents 416 frames of Fe⁵⁵ X-rays taken in Butthead at -120^o C with standard imaging array voltages of +1 and +11 Volts. The average count rate is 1110 counts/sector/frame. The calculation is made for the chip sector 2.

  

Figure 4.53: Histograms of different grades for Fe⁵⁵ X-rays. Eight panels (from top to bottom) correspond to ASCA grades 0 through 7.

In Fig. 4.53 is shown a standard set of histograms of different grades. The number of counts in the $K_{\alpha}$ peak of combined grades 0 1 2 3 4 6 histogram within the $\pm 3\sigma$ interval is $355719 \pm 596$ (an error is calculated as a square root of number of counts). The number of counts in grade 7 histogram below $E_c-3\sigma$is $33092 \pm 182$ (again assuming an error being the square root although, strictly speaking, it may not be the right estimate because this is not the gaussian distribution). This results in the depletion depth of 70.9 microns. Assuming the worst case of statistical errors going in the opposite directions we get the limits for d_d from 70.69 to 71.07 microns, which means that the statistical accuracy for the standard dataset is better than 0.3%.

b. $K_{\beta}$ line induced error

There is a source of systematic error associated with the nonmonochromatic nature of the Fe⁵⁵ radioactive source. Presence of the K$_{\beta}$line in the spectrum leads to underestimation of the depletion depth because the characteristic absorption length at 6.4 keV ($\lambda_{\beta}=38.0$ microns) is bigger than at 5.89 keV ($\lambda_{\alpha}=28.77$ microns) and, hence, K$_{\beta}$ line produces a larger share of events in the undepleted bulk. To calculate the error introduced by the K$_{\beta}$ line let us rewrite equation (4.19) separately for both lines:

 

$$N_{und \alpha}=(N_{d \alpha}+N_{und \alpha}) \exp(- \frac {d_d}{\lambda_{\alpha}})$$ (32)

 

$$N_{und \beta}=(N_{d \beta}+N_{und \beta}) \exp(- \frac {d_d}{\lambda_{\beta}})$$ (33)

Quantities $N_{d \alpha}$ and $N_{d \beta}$ can be determined from the experimental data as the number of counts in each of the corresponding peaks, whereas $N_{und \alpha}$ and $N_{und \beta}$ cannot be separated. But, since the sum $N_{und \alpha} + N_{und \beta}$ is known (it is number of counts in the grade 7 tail), equations (4.21) and (4.22) can be solved numerically after transforming them into the following:

 

$$\frac{N_{d \alpha}}{exp(\frac{d_d} {\lambda_{\alpha}})-1} + ... ...ac {d_d} {\lambda_{\beta}})-1}-(N_{und \alpha}+N_{und \beta})=0$$ (34)

The program I<>gyp/het/dep_Kbeta_corrected implements the Newton-Rafson method to solve (4.23). For the above example (number of counts in the $K_{\beta}$ peak equals 45709) the corrected solution gives d_d=73.8 microns, or, 4% increase of d_d.

c. Pileup

One other source of error can be pileup. It throws events out of the main peak into events with higher amplitude. The most simple first order correction would be to assume that all the events in the pileup peak (located at the energy 2E_c) originates from two main peak photons landed on the top of each other, and that the ``high energy tail'' of grade 7 distribution (events to the right from the $K_{\beta}$ peak) comes from one main peak photon and one grade 7 event piled up together. After adding the corresponding number of counts to the main peak and to the grade 7, the corrected depletion depth (using formula (4.20)) becomes 69.64 microns.

d. Interactions beyond detection limit

Some photons interact so deep in the bulk of silicon that the portion of charge collected in one pixel is too small to be detected. An event will not be registered by the data analysis software if the charge in each pixel falls below the event threshold. To determine the distance from the depletion region border beyond which the charge will stay undetected. S. Jones' code was used to implement the solution of diffusion equation as described in the Hopkinson's paper [G.R.Hopkinson1987]. The solid line on Fig. 4.54 shows charge collected in the pixel for 5.89 keV photon stopped in the field free region as a function of distance from the cloud center to the depletion region boundary. A dotted line in the plot shows the same thing calculated on the simple assumption that charge diffuses uniformly in all directions. This approach is attractive since the result can be expressed by a single analytical formula.

  

Figure 4.54: Charge collected in the center pixel from the Fe⁵⁵ photon landing in the neutral bulk as a function of distance from depletion region boundary.

In this case charge Q_pix collected in one pixel is proportional to the solid angle under which the pixel is seen from the center of the cloud. Solving the stereometry quiz in the case of cloud center located at the pixel center line results in the following:

$$Q_{pix}=\frac {Q_{tot}}{4\pi} (8 \arcsin{\sqrt{ \frac {2{d_c}^2 +a^2}{4{d_c}^2+a^2}}} -2 \pi)$$ (35)

where a is a pixel size, d_c is the distance from the center of the charge cloud to the depletion region border, Q_tot is total electron charge created by the photon. As expected, both curves in Fig. 4.54 converge at large distances, when the influence of the boundary becomes negligible. At small d_c simplified approach results in the collected charge approximately factor of 2 smaller than the full solution, which also makes sense. In the middle range of distances smaller than 140 microns, though, the difference is surprisingly big to justify the use of the simple approach.

For the data analyzed above the event threshold is 38 ADU, or, multiplied by gain, 35 electrons. From the above plot (solid line) one can determine critical distance d_c = 59 microns beyond which a photon is not detected. It means that 1.1% of all the events stay undetected (the ones that are deeper than 70.9 + 59 microns from surface) instead of being counted as grade 7 events.

In order to calculate the correction for the depletion depth introduced by the undetected photons the equation (4.19) should be modified into

$$N_{und}+N_{lost}=(N_d+N_{und}+N_{lost}) \exp(- \frac {d_d}{\lambda})$$ (36)

and supplemented by a similar equation for the region beyond the depletion boundary:

$$N_{lost}=(N_{und}+N_{lost}) \exp(- \frac {d_c}{\lambda})$$ (37)

where we introduced N_lost - number of interactions beyond the detection border located at the distance d_d+d_c from the silicon surface.

These two equations can be converted into the corrected formula for the depletion depth  

$$d_d = \lambda \ln(\frac{N_d}{N_{und}(\frac {1}{1- \exp(- \frac {d_c} {\lambda})})} + 1)$$ (38)

For the above analyzed example the corrected depletion depth according to (4.27) is 67.3 microns, which constitutes a 5% correction. It is clear from this analysis that reducing the event threshold will significantly improve the accuracy of the original formula (4.20).

e. Summary of error estimates

The conclusion is that statistical errors for the depletion depth determination using standard set of Fe⁵⁵ frames are negligible (less than 0.3%). The biggest sources of systematic errors are $K_{\beta}$ line photons and photons interacting deep in the bulk of the silicon. Corrections from those two factors go in the opposite directions and tend to compensate each other because the $K_{\beta}$ photons increase number of events counted as grade 7, whereas undetected ones diminish this number. For the analyzed example they almost entirely cancelled each other, although, undetected photons seem to pull d_d down a little stronger. As a result, a calculation made with the simple formula (4.20) seems to overestimate depletion depth by about 2% due to pileup and the detection limit. To get a more accurate result a very detailed modeling is necessary.