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Here is an exemplary calculation of depletion depth for the ACIS flight device
*w*193*c*2. The dataset */ohno/di/database/w193c2/butthead/fe55/30nov96/1713/*
represents 416 frames of Fe^{55} 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.

In Fig. 4.53 is shown a
standard set of histograms of different grades. The number of counts in the
peak of combined grades 0 1 2 3 4 6 histogram within the
interval is (an error is calculated as a square root of
number of counts). The number of counts in grade 7 histogram below is (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. line induced error*

There is a source of systematic error associated with the nonmonochromatic
nature of the Fe^{55} radioactive source. Presence of the Kline in the spectrum leads to underestimation of the depletion depth
because the characteristic absorption length at 6.4 keV
( microns)
is bigger than at 5.89 keV ( microns) and, hence,
K line produces a larger share of events in the undepleted bulk.
To calculate the error introduced by the K line let us
rewrite equation (4.19) separately for both lines:

(32) |

(33) |

Quantities and can be determined from the experimental data as the number of counts in each of the corresponding peaks, whereas and cannot be separated. But, since the sum 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:

(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 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 2*E*_{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 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.

In this case charge

(35) |

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

(36) |

(37) |

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

(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 Fe^{55} frames are negligible (less than 0.3%).
The biggest sources of systematic errors are 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 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.

11/20/1997