The figure linked here shows a typical event from run 000803g, where the x-ray collimator was positioned over the coordinate (0.,1.143) (mm) (ie. 0.3 mm below the red point in this figure). In this figure, the signals (volts vs. seconds) on all 8 channels of readout are shown. As expected, there are two distinct classes of pulses evident. For pads which collect some of the charge, a long fall time to 0 is observed. The pads that see an induced pulse quickly return to 0, once the charge is collected by the other pads. Because of the location of the x-ray collimator, the majority of the charge is collected by the central pad (seen as channels 1 and 5). Pads 2 and 8 also collect some charge. The remaining pads see only induced pulses. The induced pulses are smallest for pads 4 and 6, which might be expected, given that they are the furthest away. In all of the data analysed here, pads 3,4,6 and 7, collect essentially no charge, whereas the remaining pads have signals dominated by the collected charge.
The gemanal program performs quadratic fits to various regions of the pulses, to best estimate the pulse amplitudes and timings. Of all the quantities considered, the pulse amplitudes were found to be the most sensitive to the position of the primary x-rays. Interestingly, the pulse width of the induced pulses also show some sensitivity. In what follows, only the pulse peak amplitude (as determined by a quadratic fit near the peak) is used. Two separate analyses are performed for position estimation. The first uses the relative amplitudes of the peak signals in pads that collect charge. The second uses the amplitudes of the induced pulses.
The figure linked here, shows the charge fraction in pad 2, for two runs. The left figure is from run 000803a, with the x-rays centred over the vertex of the centre pad and pads 2,8 (at the red spot). Since the pulse amplitudes on the 3 pads are about equal, the charge fraction is about 0.33, with a resolution of about 0.04. The right figure is from run 000803g, where the x-rays are moved 0.3 mm in the negative y direction. The charge fraction reduces to about 0.23, reduced by some 2.5 sigma.
A weighting function, w(fi), is used to convert the charge fractions measured in an event into a position estimate of the absorbed x-ray, as follows:
r = Sum [ w(fi) ri ]where, r is the position estimate, the sum runs over each pad with direct charge, and ri is the position of the centre of pad i. To determine the form of the weighting function, the 15 data runs are used with the assumed central positions of the x-ray collimator. The figure linked here, shows estimates for the weighting function derived from the 15 runs, along with a 3rd order polynomial fit, that appears to describe the data quite well.
When the 3rd order polynomial is used as the weighting function and the formula, the data shows excellent resolution of individual x-ray events.
The figure linked here, shows the x and y estimates for run 000803c, where the x-rays are centred at (0.,1.343) (mm). The x and y coordinate estimates have standard deviations of 51 microns and 66 microns, respectively. Since this is part of the data used to define the weighting function, the bias in y should be exactly zero, but appears to be present at the 10 micron level.
The widths of these distributions are of the similar size as the expected size of the collimated x-ray beam and the expected range of the primary ionizing electrons. The intrinsic resolution of the detector may be much better than these values.
The figure linked here, shows the same for run 000803k, where the x-rays are centred at (0., 0.943) (mm). The width of the x distribution reduces to 35 microns, but the width for y increases to 86 microns.
The mean x and y estimates and their standard deviations for the 15 data runs are summarized in the figure linked here. Surprisingly, the resolution in x improves dramatically as the x-ray beam moves towards the centre of the middle pad, while the resolution in y degrades somewhat.
In summary, the weighting function method for charge sharing events provides a good estimate of the location of the x-ray beam. The difference between the true and average calculated y location is below 10 microns for all runs, except for the first run (20 microns). The standard deviations of the x and y coordinate estimates are found to be 30-60 microns, and 70-90 microns respectively.
g = induced peak amplitude / sum of all direct peak amplitudesis used to characterize the strength of the induced signal. An example of the ratio for pads 3 and 4 in run 000803c is shown the figure linked here. The induced signal in pad 4 is significantly smaller than that of pad 3.
The variation of g with distance between collimator position and centre of the pads as determined from the 15 runs is shown in the figure linked here. Note that each run allows a measurement of two points. A third order polynomial is used to parametrize the relation. One of the final four points (which were taken on the following day) show a slight deviation from the smooth curve.
To determine the position of the absorbed x-ray using the induced pulses alone, the distances from the charge pulse to the centres of the 4 pads with induced pulses are estimated by using r(g), given in the previous figure. These distances are used to define the radii of 4 circles centred at the induced pads. The location where these circles intersect is an estimate of the location of the absorbed x-ray. Since the 4 circles do not necessarily all intersect at a point, an analytic chi-squared minimization is performed, to define the best estimate.
As an example, the results from run 000803c are shown in the figure linked here. The position resolution is slightly worse than that determined from charge sharing.
The mean x and y x-ray location estimates, and their standard deviations, are summarized in the figure linked here. There is some indication of problem between the data samples taken on the two days. This was already noted by Jeff, before the analysis of the data, as there were problems with x-ray collimator stability. The collimator setup is being improved.
The position resolution using induced signals alone is about 110 - 150 microns. This is worse than that from charge sharing by 50 - 100%. Not surprisingly, the position resolution improves as that x-ray beam moves towards the centre of the middle pad. In the y direction, this nicely complements the behaviour of the position measurement using charge sharing, where the opposite relation is found.
For x-rays absorbed near the centre of a pad, which collects all the charge, the use of the induced signals will be critical to achieve optimal resolution. With the data sets used in this analysis, a direct measure of the resolution is not possible, but from extrapolation, it appears that sub 100 micron resolution can be achieved.
More interestingly, the two independent measurements might give an indication of the contribution to the widths of the position estimates coming from common sources. The x-ray beam profile and the range of the ionizing electrons are expected to contribute to the scatter in the position distributions, at the level of many 10s of microns. If these common sources of scatter dominated the measurement uncertainties, then the two measurements would show strong correlation with one another.
The figure linked here shows the location of x-ray hits in run 000803e using each method, and a correlation between the two is clearly visible.
A simple model of the situation would be to assume that the only source of correlation between the two measurements of x-ray position comes from the position of the x-ray itself. In this model, the covariance of the two x-position measurements would be of the form:
charge induced charge sig2c+ sig2beam sig2beam induced sig2beam sig2i+ sig2beam
The data for the two x-measurements and the
two y-measurements were fit according to this model for each run. If this
simple model to explain the correlation was correct, the width of the beam
should be constant for each run, and one would expect the width in the x
and y directions to be comparable. However, the data shows too strong correlation
between the measurements to be attributed to an external source alone. As
a result, the fit results usually return intrinsic error of the charge measurement
(sig_c) to be near zero. Another source of correlation is present, apart
from the beam size.
