Axes determination for segmented true-coaxial HPGe detectors

A fast method to determine the crystallographic axes of segmented true-coaxial high-purity germanium detectors is presented. It is based on the analysis of segment-occupancy patterns obtained by irradiation with radioactive sources. The measured patterns are compared to predictions for different axes orientations. The predictions require a simulation of the trajectories of the charge carriers taking the transverse anisotropy of their drift into account.


Introduction
High purity germanium detectors, HPGeDs, are used in a wide variety of applications in particle and nuclear physics [1,2,3,4]. Segmented detectors can be used to disentangle event topologies [5,6,7]. The detectors considered here are segmented cylindrical true-coaxial detectors. The segmentation in azimuth angle 1 φ is the decisive feature for the proposed method.
The orientation of the crystallographic axes of a germanium detector is important for many analyses where pulse shapes are used. The difference between the charge-carrier mobilities along the crystallographic axes 110 and 100 is significant. It is called longitudinal anisotropy. Experimental data on hole mobility exist for the 111 and 100 axes [8]. Relevant here is the plane containing the 110 and 100 axes. Appropriate calculations [9] imply that the difference of the velocities of the holes along the 110 and 100 axes is about 10 % for realistic values of the electric field. The different charge carrier velocities result in different rise times of pulses originating at equal distances from the electrode. The differences in rise times between pulses along the 110 and 100 is typically around 10% [9,10] depending on the relative location of the charge deposits to the crystal axes. In addition, the paths of charge carriers not drifting along the crystallographic axes are bent; this is called transverse anisotropy. The anisotropies have to be taken into account in so called pulse shape analyses [11,12] and whenever simulated pulses [9] are compared to data.
The crystallographic 001 -axis of a cylindrical germanium detector is usually aligned with the z axis. The position of the 110 axis, φ 110 , is, however, a priori not known after the detector is processed. Therefore the orientation of the 110 axis with respect to the segment boundaries has to be determined during the characterization of the detector. This paper presents a comparison of two methods to determine the axes. The first method is based on scanning the detector. It is widely used, well proven and based on data only. However, it requires that the detector is mounted in a special test setup with a movable source and it is time consuming. The second method involves the comparison to Monte Carlo simulation. Its advantages are that it is fast and can be performed in any configuration, even if the detector is part of a complex detector system. The determination of the axes is required to have an accuracy of about 5 • . A simple application using the orientation of the axes is the calculation of the probability that a neutrinoless double beta decay event actually produces a signal in two segments of a detector, even though it is a so called single-site event. This requires that the knowledge about the axes orientation is sufficient to calculate whether a drifting charge cloud gets separated across segment boundaries. The charge cloud representing a neutrinoless double beta decay event has a diameter of about 2 mm [13], and thus covers about 5 • at a radius of 25 mm in a cylindrical detector.

Detector and test environment
The detector used for this study was an n-type true-coaxial 18-fold segmented detector with a diameter of 75 mm, a height of 70 mm and an inner bore of 10 mm. It was produced by Canberra, France. The segmentation scheme is three in z and six in φ. The layout is depicted in Fig. 1. The first such detector was characterized in detail [10] including the determination of the orientation of the crystallographic axes. The second detector of the series, used for this study, was previously used to test the performance of such a device while submerged in a cryogenic liquid [14] and the temperature dependence of pulse lengths [15]. The density of electrically active impurities, ρ imp , was given by the manufacturer as 0.35 · 10 10 cm −3 at the top and 0.55 · 10 10 cm −3 at the bottom of the detector. The change in impurities is assumed to be linear with height, z. The operational voltage was 2000 V. At the time of the measurements, the detector drew a leakage current of up to 100 nA, all through segment 9. It was, however, fully functional. The only effect was that the energy resolutions in segment 9 and the core had deteriorated.
For all measurements, the detector was mounted inside the vacuum cryostat K1 [15]. The detector was scanned in φ at z = 0 with a 152 Eu source placed at r=9.6 cm. It was also irradiated from the top (z = 18.5 cm, r=0 cm) with a 228 Th and a 60 Co source. The 228 Th source was also placed at the side (z = −2.3 cm, r = 17.6 cm, φ = 155 • ), in front of segment 16.

Detector scan
A well established way to find the crystallographic axes is to perform an azimuthal scan, where a source with a low energy gamma line, like 152 Eu with its 122 keV line, is moved around the detector in small steps in φ. Low energy gamma rays predominantly deposit their energy close to the surface. The rise times of the resulting pulses vary with φ, reaching a maximum for the drift along the 110 axes.
A scan was performed with a collimated 40 kBq 152 Eu source with a 1σ beam spot diameter of about 5 mm and a step size of 10 • in φ. The events were selected using an energy window of 10 keV. The rise time of a given pulse was determined by fitting a simulated pulse to the measured one. This is described in detail elsewhere [9,15]. This method uses all the available information about a measured pulse while suppressing the influence of noise. For every φ position, the average rise time, t r , was computed for all pulses, for which χ 2 /ndof < 1.5 for the fit with the simulated pulse. Figure 2 shows the dependence of t r on the angle of the source position, φ . The data were fitted with the function where A, B and φ 110 are the free parameters. The coordinate system was chosen such that the 4-15 segment boundary was at 0 • .
A study of the systematic uncertainties was performed. The energy window, in which events were selected, was changed to 5 keV and 20 keV. In addition, the χ 2 used for the pulse selection was modified from 1.5 to 1.2 and 1.8. In both cases, a systematic uncertainty of 0.5 • was deduced. The dominating systematic uncertainty is connected to the placement of the detector within the cryostat. This could only be controlled to 3 • . The total systematic uncertainty was determined by adding the individual contributions in quadrature. The statistical uncertainty, as determined by the fit, is 0.4 • . The final result is:

Occupancy method
The drift of the charge carriers within a germanium detector only follows radial lines along the crystallographic axes. The transverse anisotropy causes bent trajectories elsewhere in the crystal [9, Fig. 3]. As a result, the effective segment boundaries do not coincide with the geometrical boundaries and the segment electrodes collect charge carriers from volumina of different sizes. This reflects in the occupancies of the segments when the detector is irradiated.

Measured Occupancies
A 28 kBq 228 Th and a 40 kBq 60 Co source were used. The occupancies were measured for the 0.58 MeV and 2.61 MeV lines from 208 Tl and the 1.17 MeV and 1.33 MeV lines from 60 Co. Only events, in which one segment registered the same energy, E s , as the core within 50 keV, and the energy seen in any other segment was less than 150 keV were used. As the separately measured background spectra did not show lines associated to 60 Co, the background was not subtracted for the analysis of these lines. The background did show lines associated to 208 Tl decays. Therefore the background was subtracted for the analysis of the corresponding spectra. Figure 3(a) shows the spectrum measured in segment 3 for the irradiation from top and the corresponding normalized background spectrum. The background contribution for this relatively low-energy line is maximal at the bottom of the detector. The resulting segment-spectra were fitted using the log-likelihood method. The fit function was a combination of a Gauss function of width σ and a sigmoid: where E γ is the energy of the line under study and A, B, C are the free parameters. The sigmoid represents the background shape. It drops for energies E s > E γ . The occupancy, D i , of each segment i was taken as the fitted number of events under the peak, D i = A · binwidth −1 .  the analysis. Figure 4 shows the spectra of all 18 segments for the irradiation with 60 Co from the top 2 . Also shown are the fit results.

Expected Occupancies
Events for the experimental setup as described in Section 2 were simulated with the MaGe [16] package and trajectories were computed with the pulse shape simulation package [9]. The package provides calculations of the electrical and weighting fields and uses experimental input and model based calculations to provide the mobilities, and thus the velocities, of holes and electrons at any given point in the detector. The orientation of the 110 axis, φ sim 110 , was an input parameter to the simulation of the drift. The endpoints of the resulting trajectories were used to assign each energy deposit of an event to a segment. The energies were summed for each segment. The event selection was performed as for the measurements.
The resulting expected occupancies for φ sim 110 = −20 • and φ sim 110 = 30 • are shown in Fig. 6 for the irradiation from the top and the side. The occupancies depicted in Fig. 6 were computed for the middle layer, for which an average ρ imp = 0.45 · 10 10 cm −3 was assumed. For the irradiation from the top, the result for the 1. For the irradiation from the side, a clear difference between the predictions for φ sim 110 = −20 • and φ sim 110 = 30 • is visible. There are no apparent symmetries, but the relative occupancies significantly change for varying φ 110 .
The amplitudes of the observed patterns depend on the amount of transverse anisotropy in the crystal. This is influenced by the hole mobility and by ρ imp . The dependence on ρ imp was investigated by comparing expectations for the nominal detector, i.e. the one supposed to be close to the real device with a ρ imp of 0.52 · 10 10 cm −3 , 0.45 · 10 10 cm −3 and 0.38 · 10 10 cm −3 , to predictions for a lower impurity test detector with a ρ imp of 0.29 · 10 10 cm −3 , 0.18 · 10 10 cm −3 and 0.06 · 10 10 cm −3 for the top, middle and and bottom layers.  The sensitivity of the predictions on the exact location of the source is demonstrated in Fig. 8 for the case of (a) the 1.33 MeV line of 60 Co and (b) the 2.61 MeV line of 208 Tl. Shown are the normalized predicted occupancies for a perfectly aligned source and the effects on the pattern for two source misalignments of 5 mm, one along φ = 45 • and one along φ = 90 • . The misalignments cause changes in the pattern for the middle layer breaking the 15-14-13 6-5-4 degeneracy. The effect on a single occupancy is as much as 0.005, which is a 3 % effect on the occupancy, i.e. as much as 15 % on the amplitude.

Extraction of the axes orientation
The occupancies were computed for φ sim 110 varying in 1 • steps. The resulting occupancies were analyzed by computing the test statistic ǫ, defined as where D i and S i denote the measured and simulated occupancies in segment i. The simulation was normalized to the total number of events in a given layer and the sum runs over all segments in this layer.
The resulting function ǫ(φ sim 110 ) was fitted in a 20 • window with a second order polynomial. The φ sim 110 corresponding to the minimum of the polynomial, ǫ min , was taken as the result of the procedure, φ meas . Figure 9(a) shows ǫ(φ sim 110 ) plus the result of the polynomial fit for the case of the 1.33 MeV line in the middle layer for irradiation from the top. The shape of the pattern observed in the data is best reproduced by the prediction for φ sim 110 ≈ 4 • . Figure 9(b) depicts the corresponding measured and predicted occupancies.  The comparison between the predicted and measured pattern in Fig. 9 shows that the shape of the pattern is very well reproduced. However, the amplitude of the predicted pattern is larger than the amplitude of the measured pattern. As was shown in Section 4.2, this cannot be explained by the choice of ρ imp in the simulation or by any misalignment of the source. As the hole mobility is not well known [8], its simulation could be the source of the discrepancy.   Figure 10 shows the situation for the 2.61 MeV line in the middle layer for the irradiation from the side. The value of ǫ min is significantly higher than for the top case depicted in Fig. 9, but the minimum is as distinct. The pattern observed in the data is well reproduced by the prediction for φ 110 ≈ 164 • = − 16 • .

Systematic Uncertainties
The systematic uncertainties fall into two categories: (1) parameters used in the treatment of the data (2) imperfections of the simulation The first category is dominated by uncertainties arising through the definition of ǫ and the fit to the resulting ǫ-function. The test statistic ǫ deviates from a classical χ 2 by putting extra emphasis on bins with lower occupancies. For the irradiation from the top, the differences in the results are small and the resulting systematic uncertainty is 1 • . For the irradiation from the side, the differences in the results are significant. This is intended. However, a study using different test statistics yielded a systematic uncertainty of 4 • . The effect of a variation of the width of the fit window for the ǫ-function was also studied. The window was widened to 30 • and narrowed to 10 • and the resulting uncertainty is 1 • . Using other functions then a second order polynomial to fit the ǫ-function yielded a systematic uncertainty of 2 • .
The total systematic uncertainty connected with ǫ is evaluated by adding the individual uncertainties in quadrature. It is 2.5 • and 4.6 • for irradiation from the top and the side, respectively.
In the second category, the influence of the combination of hits before drifting the charge carriers was studied. Per default, hits within a distance of 1 mm were clustered. That was changed to 2 mm and the uncertainty due to this was found to be negligible. Any diffusion of the charge cloud is not simulated; the effect is expected to be small compared to the influences of charge clustering.
The main uncertainties are connected with the assumptions on the detector parameters and the source location.
As shown in Section 4.2, a mismatch of simulated and real ρ imp slightly changes the amplitude of the pattern, but not the shape. Therefore, the absolute values of ǫ will change, but the result will actually not. This was confirmed by simulation.
Also discussed in Section 4.2 was the influence of any misalignment of the source. The source position was controlled to about 5 mm. Different layers are affected slightly differently due to the difference in angular coverage. The simulations show that the systematic uncertainty due to possible misalignments of up to 5 mm is 5 • for irradiation from the top.
The systematic uncertainty for the irradiation from the side directly reflects the uncertainty in the φ-position of the source with respect to the detector. The detector was remounted for the side measurement and its relative rotation within the cryostat could, as mentioned before in Section 3, only be controlled to 3 • . In addition, the position of the source relative to the cryostat could only be controlled to 5 • . Added in quadrature, this results in a total systematic uncertainty of 6 • .
For the analysis of lines which are also present in the background, possible changes in the background over time are another source of systematic uncertainty. Such changes can occur, if the cryostat or any other equipment in the lab is moved. Unfortunately, this was the case. The resulting patterns are very sensitive, especially for the irradiation from the side. The uncertainty was evaluated conservatively by performing the complete analysis for the 208 Tl lines with and without background subtraction. The result is a systematic uncertainty of 1 • and 7 • for the top and side cases, respectively.
In total, the systematic uncertainties for the irradiation from the top and the side were evaluated to be 6 • and 10 • , respectively. They are dominated by the uncertainty in the source position and, for the irradiation from the side, possible changes in the background.

Results
The axes orientation was determined for all energies and all layers. The results, φ meas , are given in Table 1. Also given are weights, w = ǫ −1 min , for all measurements. The statistical uncertainty of the fit to the ǫ-function was always negligible compared to the systematic uncertainty.  Table 1 Extracted 110 -axis positions, φ meas , for different gamma lines and layers. Also listed are the weights, w, as defined in the text.
For the irradiation from the top, the results for the 208 Tl lines show a spread which is slightly larger than expected from the evaluation of the systematic uncertainties. This could be due to the effect of statistical fluctuations in individual bins. An ensemble test was used to determine the statistical uncertainties. The test was performed by randomly varying the content of each bin according to its statistical uncertainty and extracting φ meas for the modified occupancy pattern. The resulting φ meas distribution was fitted with a Gaussian function and the result used to determine the statistical uncertainty. The largest uncertainties were observed for the 2.61 MeV line for the irradiation from the top. They were 4 • for all layers.
The statistical and systematic uncertainties are different for different energies and layers. Therefore, weighted averages were computed: where the sums run over a combination of layers and line energies. The results for the 60 Co and 208 Tl lines in the three layers for irradiation from the top are: φ Tl,top 110 = −3.6 • ± 4 • (stat) ± 6 • (syst) .
(7) Figure 11 shows the measured occupancies together with the best predictions for the middle and bottom layer for the 60 Co irradiation. The high statistics 60 Co distributions show that the predictions reproduce the shapes of all four measured patterns well, but that the predicted amplitudes are consistently too large. The measured amplitudes are about 40 % smaller than the predicted ones. As discussed previously, this cannot be explained by an impurity level assumed to be too high in the simulation. The reproduction of the shapes of the measured distribution also indicates that the alignment of the source was well within the 5 mm uncertainty.
The simulation uses the hole mobilities as calculated by the pulse shape simulation package [9] using measured input parameters [8]. There is a non-negligible uncertainty in these calculations and the input parameters. The observation of less pronounced patterns than predicted could eventually be used to test the assumptions about hole mobilities. A simulation confirmed that the discrepancy between the predicted and observed amplitudes could be explained by an underestimate of the ratio of the mobilities along 111 and 100 . The trajectories were recalculated for mobilities, for which the component along the 111 axis was globally, i.e. independent of the electric field, increased by 10 %. This resulted in amplitudes about the size observed. However, as the calculation of mobilities along the 110 axis and between the axes depend on several approximations, this will need further systematic studies. Figure 12 shows the measured occupancies together with the best predictions for the middle and bottom layer for the 208 Tl irradiation from the top. The statistics is much lower than for the irradiation with 60 Co, such that single bins can be affected by significant statistical fluctuations. The 2.61 MeV line does not have a large probability to get fully absorbed in the volume of a single segment. The bottom layer has fewer events by design, because of the absorption in the crystal. It should be noted that the spread of the single measurements is still reasonable, even though single bins are fluctuating.
The individual values obtained for the irradiation from the side deviate consistently by more than 10 • from the expectation. The ensemble test for the 2.6 MeV line, as discussed previously in this chapter, yielded the statistical uncertainty of 4 • and 6 • for the middle and bottom layers, respectively. Segment 9 was not used in the measurements using the top layer. Even though this segment was directly facing the source, the resulting measurements are still consistent with the results for the other layers. The top layer was, however, This result is barely consistent with the one obtained with the scan. Figure 13 depicts the measured occupancies together with the best predictions for the middle and bottom layer for the 208 Tl irradiation from the side. An irradiation directly from the side is a priori not favorable for this method. However, the patterns are reproduced quite well, indicating that probably the misalignment caused by the remounting of the detector and the changes in the background due to the movement of the equipment were larger than hoped for. The usage of this data for the analysis presented here was originally not planned. Therefore the systematic uncertainties relevant for this analysis were not really well controlled. With some care, they certainly could be reduced significantly.
The overall results confirm the ability to extract the axes orientation from the occupancy patterns in a φ-segmented detector, even if the detector is not in a perfect condition. The precision depends on the care taken to avoid systematic uncertainties. The source position has to be well controlled under all circumstances. A symmetric configuration reduces the systematic uncertain- ties. The background has to be well known, if lines are used that also occur in the background. In general, a source should be chosen which has lines with a reasonably high probability of containment in a segment and high enough energy not to be disturbed too much by the surrounding material. The selection will depend on the setup and the detector. In the case considered here, the 60 Co lines were found to be best suited.

Conclusions
A new method is presented to determine the axes orientation of φ-segmented germanium detectors. The method relies on the ability to predict occupancy patterns caused by the transverse anisotropy of the drift of the relevant charge carriers. The axes orientation of a test detector was determined with a precision of about 6 • . The precision was limited by the knowledge about the source locations. In general, a precision of 5 • should be attainable. This is compatible to the precision achieved with the well known scan method and is the precision needed for the usages of pulse shapes as currently envisioned. The new method can be used in any setup, also with multiple detectors, using any calibration source as long as the setup can be simulated with sufficient accuracy and background, if present, is properly taken into account. Therefore, it becomes unnecessary to perform time consuming scans to characterize detectors before they are integrated in a larger device. The method was developed and tested for true-coaxial detectors. However, it can be applied to any detector with an appropriate segmentation scheme. The modifications for a φ-segmented closed-end cylindrical detector will be minimal.
The method has systematic uncertainties depending on the precision of the simulation of the setup and of the transverse anisotropy in the detector crystal. The dependence of the patterns on the transverse anisotropy provides the potential to determine charge carrier mobilities.

Acknowledgements
We would like to thank the members of the MaGe Monte Carlo group for their kind support.