Second analysis of GEM-TPC data

December 11, 2001

A study of the data taken with the 15 cm TPC with rectangular pads and Ar-CO₂ gas in late November and early December is summarized in this working document..
 

HV settings

Drift	5200?V
GEM	3150?V

 

Data runs

Run	Triggers	Filtered events
615
696

Triggers are provided by a cosmic telescope at a rate of about 0.4 Hz. A filter program selects only those with a signal in any channel exceeding 10 counts beyond pedestal level, which reduces the recorded event rate by a factor of 10. Events consist of 32 channels of 4000 8 bit samples in 5 ns time bins, and thus are each about 128 KB in size. A 24 hour run takes about 300 MB. Data spanning 550 hours has been recorded to date. The operating conditions of the GEM were chosen so as to roughly match the gain observed in the P10 data.

An interesting event from the first run is shown in this figure. The event in unusual in that it has two tracks and both are well contained within the fiducial volume of the pads. The vertical scale shows FADC counts from 0 to 256. For convenience, the plot shows the arrangement of pads rotated counterclockwise by 90 degrees. The event is due to 2 vertical charged particles, with the top being at the left.

In addition to regular data, some calibration data was recorded, as summarized below. The first 4 calibration files have charge applied to each channel through a single 22 pf capacitor, allowing the gain to be calculated for each channel. The last 2 calibration files have a double pulse applied through the test input on the Aleph preamps, which are fed to four channels simultaneously. This allows a check of the clock frequency and the consistency of timing between channels.
 

Calibration runs

Run	description
654	preamp #1, calibration pulse amplitude 3.34 mV
657	preamp #2, calibration pulse amplitude 3.34 mV
659	preamp #2, calibration pulse amplitude 2.66 mV
660	preamp #1, calibration pulse amplitude 2.66 mV
683	preamp #1, double pulse, dt = 12.628 us
684	preamp #2, double pulse, dt = 12.628 us

Analysis software

Data analysis is done using the program, tpcanal, version 0.5. See first analysis for a list of useful commands.

Calibration studies

Calibrated single pulses

An example of a calibrated pulse waveform is shown in this figure (the first event from run 654). Two characteristics of the pulse considered for gain calibration are vp, the peak value, and af, the fitted amplitude 500 ns after the peak. The quantity af is used in the tracking analysis to quantify the charge collected by the pad, so as to be unaffected by any induced signals. This figure shows the distribution of observed af and vp values for the two different calibration pulse heights. It appears that run 659 and 660 had different pulse amplitudes. This figure shows the means of the af and vp values for each channel, shown relative to channel 1. The vp values are quite consistent with one another. The larger variation in the af values may be a result of modified shaping times of the system. In the cosmic data, the fall time is found to be consistent for the two preamps. The observed variation of the gains in the preamp and FADC has a standard deviation of about 1.5%. This would result in systematics in estimated positions from fits of order 0.015 x 2.5 mm = 40 microns. We can safely ignore the variation in the gains at this time.

The peak value of 184 corresponds to the charge required to raise a 22 pF capacitor by 3.34 mV. This is 7.4 x 10^-14 C, or about 460,000 electrons. For P10 gas the average charge per row was about 50 FADC counts, corresponding to  125,000 electrons. Assuming an average of 45 primary electrons produced in the 5 mm track length implies a GEM gain of about 2800. The gain estimated independently using the Fe55 source is about 5000.

The distribution of leading edge times for the calibration pulses is quite broad, about 40 ns. This is studied in more detail in the next subsection.

Double pulse samples

An example of the double pulse waveform is shown in this figure. The leading edge (50% of maximum pulse height) for the two pulses are found using quadratic fit and the results are shown in this figure. The distribution of times with respect to the trigger time is seen to have a width of 40 ns. The difference between the two leading edge times, however, has a mean of 12631 ns with a standard deviation of only 1 ns. The small standard deviation confirms that the FADC has stable clocking within an event. The mean being close to the time difference measured by the digital oscilloscope (12628 ns) confirms that the clock is operating with a frequency very near 200 MHz. The 20% variation of maximum drift time observed in the P10 data is unlikely to be due to variations of the clock frequency, otherwise it would be unlikely to find this clock frequency within 0.03% of the design value for the calibration data set. The large width of the leading edge times (40 ns) indicates a inconsistent t0 as defined by the trigger to the FADC system. Note that the leading edge time distribution is the same for all channels separately, so it is not due to different t0 offsets in each channel.

To check if there is inconsistency in the t0 between separate FADC channels, the leading edge of the first pulse in one channel is compared to that of the other 3 channels that received the signals at the same time. This is summarized in this figure for pads 1-16 and this figure for pads 17-32. Apart from channel 4, there is no evidence that the different channels in the FADC have a random t0 offset. That problematic channel was excluded from the figures in the previous paragraph. The t0 is quite consistent across the combinations of channels studied. The origin for the width of the leading time distribution of these events appears to be due to a random offset common across the FADC channels.

General properties of the data samples

A scatter plot of rising edge times versus time, shown in this figure, indicates that the drift velocity is not constant, just as was observed in the P10 data. The drift velocity appears to vary by as much as 10%.

The FWHM pulse distribution (ie. tf - tr) is shown in this figure.  Compared to the P10 data, induced pulses are less predominant.  A fit to the short (induced) pulses, gives a mean of 160 ns. This corresponds to the time required for the charge clouds to completely drift through the final 5.7 mm GEM gap.  The induced pulses are observed with greater efficiency for tracks with small drift distance. This is demonstrated in this figure, where only events with tr < 3000 ns are shown (ie. drift distance before GEM is less than about 9 mm). The amplitude of induced pulses reduce as the charge cloud diffuses in the z direction because induced pulses do not combine for charges separated in time by more than the drift time in the final GEM gap. Because the drift velocity in the drift region is much slower for Ar-CO2 than for the P10, the induced pulses are much less predominant in this data.

A comparison of the various channels is shown in these figures. There is a significant problem with the readout channel for pad #4. Fortunately the pad will not be used directly for tracking studies, since it is an outer pad.

Gain studies

The total charge collected on each row can be used as a measure of the gas gain. To ensure that the active pads in a row collect all the charge in that row, it is required that the pads on the two ends of the row have no charge themselves. The average total charge is shown vs time in this figure. The gas gain is stable within 5% over a period of 550 hours. The variations in the gain appear to be somewhat correlated to the variation in the apparent drift velocity, as demonstrated in this figure. The total charge seen in each row is shown in this figure.

Tracking studies

Track fitting to the data is performed to understand the resolution properties of the TPC. This figure shows the coordinate system for the track parameters. The track is defined by the coordinate (x₀,0,z₀), the azimuthal angle, phi, and the angle in the y-z plane, psi. It is useful to define a local offset from the center of each pad, b. Tracking studies are performed on events with contain signals in all rows, and in which the edge pads on all rows contain less than 10% of the total charge collected by that row. Track fitting is performed in y-z and x-y separately. No channel to channel calibrations have yet been performed.

y-z tracking

Pads that have a signal peak of 5 or more ADC channels below the pedestal are used as hit candidates. The pulse arrival time is defined as the time that the rising edge reaches 50% of the peak pulse height, (tr). Since the cosmic telescope triggers the FADC in common stop mode a fixed delay after the trigger, the arrival time is linearly related to the z coordinate. Pulses in a given row (common y coordinate) are combined to form a single z coordinate for that row.

There is an unexplained systematic relation between time and amplitude of the pulses, which is corrected for. As usual, the pulse amplitude is estimated using a fit to the pulse 500 ns after the peak of the pulse (an). This figure shows the difference between a row average tr using large pulses only (those with 50 < -an < 200) and the tr for all pulses as a function of the average pulse amplitude an. Pulses with amplitudes less than about 30 counts arrive earlier than larger amplitude pulses by as much as 90 ns. An exponential fit to this data is used to define a corrected rising edge time, trc. This effect, also seen in the P10 data, needs to be better understood, and perhaps another algorithm be used to define the pulse arrival time.

Each row z coordinate is defined by an amplitude weighted average of the corrected pad times in that row. The row y coordinate is simply defined by the centre of the pad row. An unweighted linear fit of (z,y) coordinates is performed. The distribution of z₀ and psi are shown in this figure. (A time offset of 2000 ns and drift velocity of 9 micron/ns is assumed for these plots). The angle psi has an unexplained asymmetrical shape.

To estimate the resolution of a single row z measurement, the linear fit is repeated without the row, and the z coordinate from the fit is compared to the z coordinate of the row. The difference between the row and fit z coordinate is shown in this figure, for four different ranges of drift distances. Accounting for the fact that the track fit is determined from 4 measurements, the estimated z resolution for these events is 11% less than the standard deviations of the fits in the figure. The z resolution for drift distances up to 10 mm is about 130 microns. This is much better than observed in P10, simply because the of the slow drift velocity in Ar-CO2.

x-y tracking

As in the y-z track fit, each row is used to define a single hit on a track. For linear collider TPC designs, it is important to correctly deal with situation where only one or two pads within a row collect charge. For this case it was shown in our space point resolution studies that it is incorrect to use a linear charge weighted centroid estimate. In those studies, a charge cloud shape of a 2D Gaussian was integrated over the pad areas to estimate the centroid of the charge cloud, given the observed charge fractions in each pad.

A logical extension for the tracking analysis is to use a model with a uniform line of charge with the transverse distribution in the x-y plane given by a Gaussian. The integration of such a line charge over a rectangular pad is shown in this figure. The charge fluctuations along the length of the track are not included in the model. The observed charge fractions in the pads are unaffected by such fluctuations for track angles near phi=0.

For each row, charge fractions in each pad are calculated and compared to the expected charge fractions given by the integrals, I. The track fit is performed by minimizing the chi**2 difference of the observed and expected charge fractions in the 5 rows, while varying the track parameters, x₀ and phi, and the transverse standard deviation of charge, sigma. Since the tracks are mostly vertical, a single choice for sigma for all rows is a reasonable approximation. The chi**2 is summed only over pads that are observed to have at least 2% of the row charge. The absolute uncertainty in the charge measurement from the pads is assumed to be the same for all pads.

This figure shows the result for x₀ and phi. These track parameters are limited in range due to the fiducial volume cut described above. There appears to be a systematic effect in the fitting that moves some events towards x₀ = +/- 2.5 mm and phi = +/- 0.25 rad.

This figure compares the distribution of the fitted values for sigma, the transverse scale of the line charge, for Ar-CO2 data and P10 data. The transverse size of the line charge is smaller for the Ar-CO2 data, as expected, since the diffusion is less. There is a large enhancement for sigma < 0.2 mm and a small peak for sigma between 0.2 and 0.35 mm, not present in the P10 data. These are an artifact of the fitting algorithm, and are studied in more detail below.

This figure shows sigma (for sigma greater than 0.35) as a function of drift time. It follows the expected form of diffusion, in that the square of sigma is a linear function of the drift distance. Assuming  a drift velocity of 9 microns/ns, the transverse diffusion constant is 0.059 sqrt(mm) or 0.019 sqrt(cm). The diffusion constant calculated by Magboltz is about 0.025 sqrt(cm). The diffusion for tracks at the edge of the GEM is about 0.38 mm. The GEM system has an 8 mm gap which, with diffusion constant of 0.020 sqrt(cm) over this higher field region, would provide a diffusion at this point of 0.18 mm. The diffusion due to the gas does not properly account for the observed width of the line charge for tracks with very small drift distances.

For fitted sigmas below 0.35 mm, the situation is more complex. This figure shows the track parameters (x0,phi, and z0) for the events with sigma below 0.2 mm and between 0.2 and 0.35 mm. The excess for small sigma is due to events with phi near +/- 0.25 rad and x0 near 0 and +/- 2.5 mm. Note that tracks with these track parameters cross each row in such a way that the charge is shared equally between the the pads in a row, independent of the transverse diffusion sigma. From a visual scan of such events (example), it is evident that this results from tracks with phi near 0.25 rad, but with one or both of the outer rows having only 1 pad with significant charge. The fit finds a solution to the problem by setting the value for sigma to near zero.

The small excess of sigmas between 0.2 and 0.35 mm arises from the fit for nearly vertical tracks which leave significant charge in only one pad for rows where it passes near the centre of the pad. Larger values of sigma would result in charge deposited in neighbouring pads for all rows, so the track fit finds this erroneous solution. Here is one such example.

The track fit needs to be improved to avoid these cases where the track parameters are biased due to one or more rows having just one pad hit. One simple way to reduce the effect is to fix the value of sigma in the fit to a value determined from the drift distance. In the current track fit, the chi**2 is calculated assuming the charge fractions all have equal uncertainties, and the correlations are not taken into account.

Rows with only single pads hit are much more prevalent in the Ar-CO2 data because of the low transverse diffusion of the gas. This figure compares the fraction of rows having only one pad hit, for fully contain events, for Ar-CO2 and P10 data. The upper figures are for all events, the lower figures are for vertical tracks (|phi| < 0.05).

To estimate the transverse track resolution from a single row, events are fit by excluding the row to determine x₀, phi, and sigma. The data from the single row is then fit with only x₀ free, while phi and sigma are fixed to the values determined from the fit to the four other rows. Events with a large chi**2 value from the 4 row fit are discarded so as to exclude events with poorly measured reference tracks. This figure shows the difference between the single row and 4-row fit x₀ track parameter for different drift distances. The residuals do not follow a Gaussian distribution.

When sigma is fixed to the value determined from the drift distance, the residuals more closely follow Gaussian distributions as shown in this figure.  Note that the width of the residual distribution increases as a function of drift distance. Diffusion would cause the width to increase. The reduction in the fraction of pads would cause the width to decrease. Perhaps these two competing effects are playing a role. For drift distances of less than 10 mm, the resolution is about 230 microns (remembering to divide the fitted standard deviations by sqrt(1.25)). This is actually worse than the resolution seen for P10 gas for the same drift length, where the value was 200 microns.

This figure shows the resolution when the single row x₀ coordinate is estimated by the simple linear centroid method. The resolution is significantly worse for the smallest drift distances. Once again, this demonstrates that the linear method of defining hit centroids should not be used for precision track measurements. The resolution slightly improves for larger drift distances likely due to the reduced fraction of pads with one row hit.

The sensitivity to non-uniform ionization along the length of the track is greater for tracks that are further from phi=0 (the so-called track angle effect). This figure compares the resolution for tracks with abs(phi) < 0.05, with 0.05 < abs(phi) < 0.15,  and with abs(phi) > 0.15 (all within the first 10 mm of the drift volume). There is a strong track angle dependence on the resolution. For the first phi bin, a resolution of about 150 +/- 14 microns is seen. The other two bins in phi have resolutions consistent with each other of about 225 +/- 15 and 260 +/- 21 microns. These results are from fits with sigma fixed to values calculated by the drift distance.

Note that for the fits to the P10 data within the first 10 mm of drift, the resolution was found to be 100 +/- 15 microns for abs(phi)<0.05. These track fits were done with sigma as a free parameter. When the P10 data is fit with fixed sigma (as is done in the previous paragraph), resolutions are found to be 175 +/- 45, 210 +/- 45, and 242 +/- 125 microns.

This figure shows the Ar-CO2 resolutions for the different phi bins for drift distances between 10 and 30 mm. The derived resolutions are 216 +/- 11, 220 +/- 8, and 296 +/- 10 microns.

The resolution depends on the number of primary electrons produced. This figure compares the resolution for events in for bins of total row charge for all drift distances. As expected the resolution is better for the rows with greater collected charges. The resolution degrades for very large row charge, presumably because such events include delta rays.