For an optimal operation of a Timepix based detector, each pixel should ideally have the same threshold. While all pixels are theoretically identical, imperfections in the production process will always lead to slight differences, either locally (transistor threshold voltage or current mismatches (Llopart et al. 2007; Pelgrom, Duinmaijer, and Welbers 1989) ) or global effects like small supply voltage instabilities. Therefore, each pixel has 4 independently selectable current sources to minimize the spread of threshold values (Llopart and Poikela 2006; Llopart et al. 2007). Together all 4 sources act as an effective 4-bit DAC to slightly adjust the threshold. The absolute current for the 4 sources is dependent on the global THS DAC, allowing currents in the range of \(\SIrange{0}{40}{nA}\).

To achieve a good calibration for a homogeneous threshold, first the THS DAC has to be set correctly. This is referred to as the THS optimization. Once the correct value is found, the 4-bit DAC on each pixel can be adjusted to minimize the spread of threshold values of all pixels together.

If the THS DAC is set too high, the 4-bit DAC on each pixel will be too coarse for a fine adjustment (as the 'current steps' will be too large). If it is too low, not enough range will be available to adjust each pixel to an equal noise / sensitivity level (not enough current available via the 4 current sources). The goal of the THS optimization is therefore to find just the right value, as to provide a range of values such that all values can be shifted to the same threshold of the threshold DAC THL.

The algorithm scans a range of THL values through a subset of \(\num{4096}\) pixels using different 4-bit DAC values. First 0 for all pixels and then the maximum value of 15. At each THL and 4-bit value the number of hits due to pure noise is recorded for each pixel. The weighted mean of the THL values, using the number of hits as weight, is the value of interest for each pixel and each 4-bit DAC value: the effective THL noise value for that pixel. For each of the two cases (4-bit value 0 and 15) we can then compute a histogram of the number of pixels at each THL value. The resulting histogram will be a normal distribution around a specific THL value. The stopping criterion, which defines the final THS value, is such that these two distributions overlap at the 3 RMS level. This is performed by comparing the means of the 0 and 15 value distribution at a starting THS value and again at half of that THS value. Using linear regression of the two differences, the optimal THS value is computed.

With a suitable THS value set, the actual threshold equalization can start. The algorithm used is fundamentally very similar to the logic of the THS optimization. Each pixel of the chip is scanned for a range of THL values and the weighted THL noise mean is computed both at a 4-bit DAC value of 0 and at 15. The normalized deviation of each pixel's THL value to the mean THL value of all pixels is computed. Using a linear regression the optimal required shift (in units of the 4-bit DAC) yields the final 4-bit DAC value for each pixel.

An example of the 0 and 15 value distributions as well as the distribution using the final 4-bit DAC values for each pixel is shown in fig. 1(a). ¹ Each of the distributions represent different 4-bit DAC settings of all pixels of the chip. Orange ("min") represents all pixels using a 4-bit DAC value of 0, purple ("max") of 15. In green is the same distribution for the case where every pixel uses its optimal 4-bit DAC value. The threshold equalization thus yields a very strong reduction in the THL spread of all pixels. Fig. 1(b) shows how all pixels are spread in the values of the 4-bit DAC. The narrow equalized line of fig. 1(a) is achieved by a normal distribution around \(\num{8}\) of the 4-bit DAC values, with only very few at the edges of the DAC (\(\num{0}\) and \(\num{15}\)). Finally, fig. 2 shows a heatmap of an entire chip with its 4-bit DAC values after equalization.

Similar plots for all other chips during both run periods can be found in the extended thesis.

Once the detector is THS optimized and threshold equalized, the final threshold value of the THL DAC can be determined for the data taking. While measurements like an S-Curve scan (see sec. 19.1.3) can be used to understand where the noise level of the chip is in terms of THL values, it is typically not a reliable measure as the real noise depends strongly on the shutter length. If an experiment – like a low rate experiment as CAST – requires long shutter lengths, the best way to determine the lowest possible noise-free THL value is to perform a simple scan through all THL values using the shutter length in use for the experiment.

For a correctly equalized chip a sharp drop off of noisy pixels should be visible at a certain threshold. In principle the THL value at which no more pixels are noisy is the ideal THL value.

TOS first performs a quick scan in a THL range given by the user, using short shutter lengths. The determined drop values that still see some noise to a noise-free range is used as a basis for a long shutter length scan using a shutter length given by the user. For safe noise free operation one should choose a THL value 2 or 3 above the first noise-free THL value at the target shutter length. Especially for long shutter lengths it is important to perform this calibration without any high voltage applied to the detector as otherwise cosmic background starts to affect the data.

The S-Curve scan is one of 2 different ways to determine the optimal THL value.

The purpose of the S-curve scan is to understand the relationship between injected charges in electron and the THL DAC values by providing a \({\text{\# } e^-}/\mathtt{THL}\text{ step}\) number (or without a ToT calibration \(\mathtt{ToT}/\mathtt{THL}\)).

It works by injecting charges onto each pixel and checking the pixel response of each pixel at different THL values. Below a certain THL value all pixels will respond to the injected charge. At some point certain pixels will be insensitive to the induced charge and a 90° rotated "S" will form. By fitting an error function to this S an ideal THL value can be deduced.

By calculating THL value at which half of all test pulses are recorded, we can compute the number of electrons corresponding to that THL DAC value, as we know the amplitude of the test pulse and thus number of injected electrons.

Fig. 3 shows an S-Curve scan of chip 0 of the Septemboard using the calibration from July 2018. The center peak in the middle is the noise peak of the detector at the shutter length used for the S-Curve scan. The symmetrical shape is due to specific implementation details of how the pixels function, the upper side is the one of interest. The center point (half way between both plateaus) corresponds to the effective threshold of the detector at that injected charge. The falling edge of each curve can be fit by the cumulative distribution function of a normal distribution, eq. \eqref{eq:daq:s_curve_fit_function}.

where the parameter \(N\) is simply a scaling factor and \(μ\) represents the x value of the half-amplitude point. \(σ\) is the spread of the drop and \(\text{erfc}\) is the complementary error function \(\text{erfc}(x) = 1 - \text{erf}(x)\). The error function is of course just the integral over a normal distribution up to the evaluation point \(x\):

\[ \text{erf}(x) = \frac{2}{\sqrt{π}} ∫_0^{x} e^{-t²} \dd t. \]

Given that the number of injected electrons is known for each test pulse amplitude (see sec. 8.1.1), we can compute the relationship of the number of electrons per THL value step. This is called the THL calibration and an example corresponding to fig. 3 is shown in fig. 4, where the THL values used correspond to the \(μ\) parameters of eq. \eqref{eq:daq:s_curve_fit_function}. The resulting fit is useful, as it allows to easily convert a given THL DAC value into an effective number of electrons, which then corresponds to the effective threshold in electrons required to activate a pixel on average. When looking at the distribution of charges in a dataset, that cutoff in electrons is of interest (see sec. 19.1.4).

Table tab. 29 shows the fit parameters for the fits of fig. 3. See the extended thesis for all fit parameters and plots.

In a gaseous detector the gas amplification (see sec. 6.3.6) allows to easily exceed the minimum detectable charge of \(\mathcal{O}(\SIrange{500}{1000}{e^-})\). The typically used THL threshold will be quite a bit higher than the 'theoretical limit' however for multiple reasons. One can either compute the effective threshold in use based on the THL calibration as explained in sec. 19.1.3, or use an experimental approach by utilizing the Pólya distribution as introduced in sec. 6.3.6 and 8.1.2. By taking data over a certain period of time and computing a histogram of the charge values recorded by each pixel, a Pólya distribution naturally arises.

The Pólya distribution can be used to determine the actual activation threshold by simply checking what the lowest charge is that sees significant statistics.

An example of such a Pólya distribution with a very obvious cutoff at low charges is seen in fig. 5. We see chip 0 using the same calibration from July 2018 as in the figures in the previous section 19.1.3. The data is a \(\SI{90}{min}\) interval of background data at CAST. The pink line represents the fit of the Pólya distribution to the data. The dashed part of the line was not used for the fit and is only an extension using the final fit parameters. The cutoff at the lower end due to the chip's threshold is clearly visible. The fit determines a gas gain of about \(\num{2700}\), compared to the mean of the data yielding about \(\num{2430}\).

Based on the data a threshold value of – very roughly – \(\num{1000}\) can be estimated. Using the THL calibration of the chip as shown in fig. 4 yields a value of

\[ Q(\text{THL} = 419) = 26.8 · 419 - 10300 = 929.2 \]

where we used the fit parameters as printed on the plot (\(1/m\) and \(f⁻¹(y=0)\)) and the THL DAC value of \(\num{419}\) as used during the data taking for this chip. Indeed, the real threshold is in the same range, but clearly a bit higher than the theoretical limit for this chip. This matches our expectation.

See sec. 8.1.1.1 for the section producing the single plot used in the thesis.

# To generate fig:septem:tot_calibration_example
for run in 2 3; do
    for chip in {0..6}; do
        plotCalibration --tot --chip $chip --runPeriod Run$run --useTeX --file ~/CastData/ExternCode/TimepixAnalysis/resources/ChipCalibrations/Run$run/chip$chip/TOTCalib$chip.txt --outpath ~/phd/Figs/detector/calibration --quiet &;
    done
done

All plots without caption, too much repetition and the run period & chip number are in the title. Looking at these plots one of the things I've long thought is that the fit function is way too overspecified, given that usually one parameter (most of the time \(t\)) is just zero. The errors go crazy in Run-2 for chip 0, because the fit produces crazily large errors. Not quite sure why, but I guess it's just another reason of the function having too many parameters. :)

Again, all plots without caption or anything. That's why they have titles, you know.

Excuse the mismatched order of the tables… I don't really expect anyone to be particularly interested in this, hence I didn't want to spend time sorting them by hand.

All figures:

All tables:

Figure 6 shows the optimized THL distributions of all chips after threshold equalization for run 2 (left) and 3 (right).

Note: The text should be placed in relative coordinates based on the data range, but that's currently not done. Sorry about tcho "hat.

This document contains the data for the calibration of the MM veto scintillator. It is a 'report' created while data taking and thus may contain conflicting information. Not to be understood as a simple reference protocol.

The scintillator has a Canberra 2007 base, which accepts positive HV. The PMT is a Bicron Corp. 31.49x15.74M2BC408/2-X, where the first two numbers are the scintillators dimensions in inch.

For calibration we're using $\SI{1400}{\volt}$, while Juanan mentioned in his mail to use $\SI{1200}{\volt}$ during data taking.

For calibration we're using an Ortec 9302 amplifier after the PMT with a gain of 20. This is fed into an LRS 621CL discriminator. The PMT and base are used at a HV of \(+\SI{1200}{\volt}\).

Scintillator is of size \(\SI{42}{\cm}\) times \(\SI{82}{\cm}\). Which is an area of

let x = 0.42
let y = 0.82
echo x * y 

• [ ] TODO: CROSS CHECK THESE NUMBERS HERE

At a cosmic muon rate of \(\sim\SI{100}{\hertz \per \meter \squared \steradian}\), the expected signal rate of munons is thus \(\sim \SI{33}{\hertz}\).

let area = 0.34
let total_muons = 60000.0
echo area * total_muons

• [ ] UPDATE: Muon rate about \(\SI{1}{cm^{-2}.min^{-1}} \approx \SI{166.67}{m^{-2}.s^{-1}}\)