Crosstalk properties of 36-fold segmented symmetric hexagonal HPGe detectors

ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 599 (2009) 196–208

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Crosstalk properties of 36-fold segmented symmetric hexagonal HPGe detectors Bart Bruyneel ,1, Peter Reiter, Andreas Wiens, Ju¨rgen Eberth, Herbert Hess, Gheorghe Pascovici, Nigel Warr, Dirk Weisshaar 2 ¨t zu Ko ¨ r Kernphysik, Universita ¨ln Zu ¨ lpicherStr. 77, D-50937 Ko ¨ln, Germany Institut fu

for the AGATA Collaboration a r t i c l e in f o

a b s t r a c t

Article history: Received 24 July 2008 Received in revised form 24 October 2008 Accepted 3 November 2008 Available online 25 November 2008

Crosstalk properties of three 36-fold segmented, symmetric, large volume, HPGe detectors from the AGATA Collaboration were deduced from coincidence measurements performed with digitized segment and core signals after interaction of g rays with energies of 1.33 MeV. The mean energy values measured by the core signal fluctuate for g-ray interactions with energy deposited in two segments. A regular pattern is observed depending on the hit segment combinations. The core energy shifts deviate 0.03–0.06% from the average energy calibration. The segment-sum energy is reduced with respect to the core energy as a function of the decoupling capacitance and the segment multiplicity. The deviation of the segment-sum energies from multiplicity two events fluctuates within an interval of less than 0.1% depending on the different segment combinations. The energy shifts caused by crosstalk for the core and segment signals are comparable for all three detectors. A linear electronic model of the detector and preamplifier assembly was developed to evaluate the results. The fold-dependent energy shifts of the segment-sum energies are reproduced. The model yields a constant shift in all segments, proportional to the core signal. The measured crosstalk pattern and its intensity variation in the segments agree well with the calculated values. The regular variation observed in the core energies cannot be directly related to crosstalk and may be caused by other effects like electron trapping. & 2008 Elsevier B.V. All rights reserved.

Keywords: g-Ray instruments Segmented germanium detectors Pulse-shape analysis

1. Introduction With AGATA [1] and GRETA [2], a new generation of g-ray spectrometers is currently under construction which will employ the g-ray tracking technique as a novel position-sensitive detection method for reconstruction of the individual interaction points of a g ray within a 4p Ge shell [3]. Position sensitivity will be achieved by means of pulse-shape analysis (PSA) of signals from irregular shaped, highly segmented, encapsulated, n-type HPGe detectors. A very detailed understanding of the pulse shapes of these detectors is, therefore, an obvious necessity for the g-ray tracking technique. Pulses of HPGe detectors depend on electron and hole mobilities, crosstalk properties and response functions. The electron and hole mobilities are connected with the germanium

Corresponding author. Tel.: +49 221 470 3620.

E-mail address: [email protected] (B. Bruyneel). Supported by the German BMBF(06KY205 TP 8). 2 Present address: NSCL, Michigan State University. 1

0168-9002/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.11.011

band structure and, therefore, their contribution to the pulse shape is related to germanium crystal properties and the applied electrical field. Pulse shapes of direct and transient signals in segmented Ge detectors can be understood to high precision by weighting potential calculations. The measured position dependence of pulse shapes is caused by the anisotropic mobility, crystal geometry, changing field strength and space-charge effects. The response functions are determined by the experimental details of the analog front-end electronics and digitization. The crosstalk properties of the highly segmented HPGe detectors are caused by the Ge crystal and its coupling to the preamplifier front-end electronics. Crosstalk properties are described later within this paper by a quantitative electronic model of the segmented detector and its preamplifier electronics. The need for low noise and high amplification of the detector signals over a broad bandwidth in the high frequency domain has created demanding requirements for the construction of chargesensitive preamplifiers suited for position-sensitive HPGe detectors. The high segmentation of these detectors results in spatial limitations for the electronics and the inductance of the large number of wires provides an additional potential source of

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crosstalk. Therefore, a detailed investigation of crosstalk contributions in the first available symmetric AGATA detectors was carried out to quantify these effects for the first time in this novel type of detector. Although only AGATA detectors are discussed in this paper, similar crosstalk behavior can be expected in other segmented Ge detectors for spectrometers like e.g. EXOGAM [21,24] GRETA [5], MARS [27], MINIBALL [4], SEGA [6] and TIGRESS [7]. Moreover, highly segmented HPGe detectors are applied for g-ray detection in various related fields like nuclear structure research, nuclear astrophysics and g-ray imaging.

2. Motivation The most important property of HPGe detectors is the outstanding energy resolution for g-ray detection. For g-ray tracking with highly segmented detectors a similar energy resolution is required for the individual segments. A more detailed analysis of the g-ray energy deposition within the HPGe detector volume is performed by including the distribution of the energy into the available 36 detector segments. The number of firing segments—the segment multiplicity or segment fold—depends strongly on the g ray energy and causes e.g. for Eg ¼ 1:3 MeV in a large volume AGATA detector on average 2–3 firing segments, with a multiplicity distribution exceeding five segments for 1% of the events. The segmented detector allows the analysis of the energy registration by using either the information from the central core contact or the sum of energy deposited in the individual segments. One may expect the same result for the two independent energy measurements. However, as a function of the segment multiplicity, the comparison of the measured energies shows clear differences between the core and segment-sum energies for the new symmetric AGATA detectors, which is attributed to the inherent crosstalk properties of the segmented Ge detectors. The g-ray tracking method requires precise pulse shape information from the segments to deduce the positions of the interactions. A possible influence of crosstalk on the tracking performance is hiding in the potential distortion of the pulse shape. In highly segmented HPGe detectors crosstalk has been observed [4] to consist of different components: at high frequencies, it can be considerably larger, resulting in spurious transient signals. These signals, in practice often referred to as derivative crosstalk, can produce a significant fraction of the observed transient signals which are crucial for PSA [22]. In this paper, these high frequency effects will not be dealt with. The discussion here considers the low frequency phenomena, which yield the most relevant effects observable in energy spectra.

197

with Rfb ¼ 1:0 GO as the first stage of the preamplifier. This part of the analog electronics was located under vacuum in the cryostat and was kept at temperatures of typically 88–100 K maintained by the liquid nitrogen cooling system. The main part of the preamplification consisted of 37 fast low-noise preamplifiers [9], which were specifically designed and built for PSA applications with AGATA detectors. The core and 36 segment energy signals were digitized using 10 DGF modules. The DGF module for the core energy signal was configured as the master and provided a trigger signal for the read-out of the nine DGF modules for all 36 segment signals. For the final result, enough coincidences between all segments are needed. Especially Compton scattering of a 1.33 MeV g ray through the full crystal length of 9 cm requires long measuring periods of several days to collect sufficient events. For the measurements, three radioactive sources were used simultaneously (see Fig. 1): (i) a weak 60Co source of 16.4 kBq was positioned in front of the detector. This 60Co source was centered with respect to the detector axis at 1.5 cm distance in front of the detector crystal. (ii) A 57Co source (4.5 kBq) was placed at one side of the detector and (iii) a 241Am source (140 kBq) was located at the opposite side of the detector at a greater distance. In this way, all parts of the detector are irradiated by the 1332.5 keV line from 60 Co and either the 60 keV line from 241Am or the 122 keV line from 57Co. Calibration was performed using the full-energy photopeak events of the 1332.5 keV line and the stronger of the two low-energy lines (i.e. either 59.5 or 122 keV) within the individual segment volume. A linear calibration was applied. A non-linear behavior of the acquisition chain at energies between the calibration points was measured to be negligible. List mode data were typically acquired for 3–4 days, yielding a total of about 109 events. To account for possible fluctuations in the gain of the analog electronics, the data were divided up into small runs each containing 10 million events. Offline gain matching was performed for each of the 37 channels (core and 36 segments) and for each of the individual runs. A typical result is shown in Fig. 2 where uncorrected gain shifts are shown. Systematic effects in the form of a saw tooth, occurring on all the channels, are observed. The time intervals and oscillating behavior of the maxima is caused by the slight fluctuation in the temperature during the liquid nitrogen filling cycle. In all the subsequent analysis, these gain drifts were corrected to a level of better than 25 eV.

3. Experimental setup The experimental setup for the measurements consisted of (i) a cryostat housing the encapsulated, tapered, semi-hexagonal cut, symmetric AGATA detector, (ii) a 2-fold analog preamplifier stage for each signal within the cryostat and (iii) ten digitizers of the type DGF-4C Revision E (Digital Gamma Finder with four channels by XIA LLC [8]). The preamplification part was divided into two parts. For optimum noise performance, the detector pulses from the core signal and the 36 segments were connected with very short wires to the gate of a jFET (type BF862), which was operated together with a feedback impedance of C fb ¼ 1:1 pF3 in parallel 3 The feedback capacitance consists of a physical capacitance of 1.0 pF, in combination with a parasitic stray capacitance of up to 0.2 pF.

Fig. 1. Schematic drawing of the 36-fold segmented symmetric AGATA detector crystal: length 90 mm, diameter: back 80 mm, the radius varies at the hexagonal front side within 53–61 mm, core: inner diameter 10 mm, length 77 mm, distance of crystal to Al capsule: 0.4 mm. Segment labeling convention: the six sectors are labeled A–F, the rings are numbered 1–6. The standard positioning of the calibration sources is also indicated: a 60Co source 15 mm below the detector and 57 Co and 241Am on opposite sides.

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1335 C1 C2 C3 C4 C5 C6

1334

Energy [keV]

1333 1332 1331 1330 1329 0

10

20

30

40

50

60

70

80

90

100

Run number Fig. 2. The peak positions of the 60Co source for the six segments in sector C of detector S001 vary as a function of time which is shown for the uncorrected gain stability. The curves were measured over a period of three days. The saw-tooth structure corresponds to the irregular filling times.

The standard labeling convention for AGATA detectors is used: each of the 36 detector segments is labeled with a letter and a number. The labeling is illustrated in Fig. 1. The letters A to F correspond to the six sectors around the detector axis, while the numbers 1–6 correspond to the six rings, with 1 being at the front end (closest to the 60Co source) and 6 at the back (closest to the dewar). 3.1. Core-to-segment capacitance measurement Taking advantage of the on-board pulse generator, which is part of a new AGATA core preamplifier [10], the capacitance between the core electrode and the 36 individual segments was additionally measured (more details in Pullia et al. [11]). The charge pulse from the pulse generator is injected in the source of the core FET and is coupled to the segments by the core-tosegment capacitance. The signal amplitude registered by a single segment is proportional to the core-to-segment capacitance. After correction of small differences in the 36 segment preamplifier gains, the measured amplitude values are normalized to the total detector capacitance. The sum of the segment capacitances is equal to the bulk capacitance of the detector, for which data are provided by the manufacturer Canberra France; e.g. a value of 46.7 pF is provided at full depletion voltage for the measured symmetric AGATA crystal S001. This capacitance was measured before encapsulation. The measured capacitances for the symmetric AGATA crystal S001 are listed in Table 1. Due to the symmetry of the crystal, these values are only ring dependent. Deviations of the measured values from the listed averages are within a standard deviation of 3%. The considerable difference in the capacitance of the segments in ring 2 can be attributed to the low effective volume of this ring. This is due to the complex shape of the field lines in the noncoaxial part of the detector, especially affecting this ring of segments. Their distinct lower capacitance will represent a clear ‘‘finger print’’ in the energy shifts caused by crosstalk, as will be shown later.

4. Experimental results All the measurements with the first three symmetric AGATA detectors were performed utilizing digital electronics. However,

Table 1 Measured capacitances: between core and a single segment and between a single segment and the grounded encapsulation for a symmetric AGATA detector. Ring number (pF)

1

2

3

4

5

6

Core-to-segment capacitance C 0;j Segment-to-ground capacitance C g;i

1.20 20

0.84 15

1.22 20

1.53 20

1.45 20

1.55 25

The capacitances between core and segment were normalized to the nominal value for the total bulk detector capacitance. For segment labeling see Fig. 1. Due to the symmetry of symmetry of the crystal these values are only ring dependent.

Table 2 Energy resolutions (full width half maximum in keV, measured with analog electronics) for the detector crystals S001, S002 and S003.

S001 S002 S003

Core FWHM

Seg. FWHM @59.5 keV

Seg. FWHM @1.3 MeV

@122 keV

@1.3 MeV

Min.

Mean

Max.

Min.

Mean

Max.

1.16 1.19 1.10

2.20 2.08 2.13

1.07 0.98 0.96

1.14 1.07 1.03

1.27 1.18 1.11

1.87 1.87 1.83

1.99 1.98 2.01

2.19 2.12 2.16

For the segments, the minimum, mean and maximum values are given for Eg ¼ 59:5 and 1332.5 keV.

these detectors were also carefully tested with standard analog electronics employing spectroscopy amplifiers and 13-bit resolution ADCs. The energy resolution of the analog measurements of the core signals were 1.16 and 2.20 keV at Eg ¼ 122 and 1332.5 keV, respectively, for detector S001. The energy resolution of the segments was in the range Eg ¼ 1:0721:27 keV with a mean value of Eg ¼ 1:14 keV at Eg ¼ 59:5 keV. For the higher energy Eg ¼ 1332:5 keV, the energy resolution of the segments was in the range Eg ¼ 1:8722:19 keV with a mean value of Eg ¼ 1:99 keV. See also Table 2, where additionally the results for the crystals S002 and S003 are summarized. The energy measurement with segmented detectors provides double information from the anode (core) and cathode (segment) of the energy deposited in the Ge detector volume. The core electrode collects all the electrons after g-ray interaction and electron-hole creation in the n-type Ge material, while the holes are collected at the outer segment contacts. Mainly through

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counts

15e5

1−fold 2−fold 3−fold 4−fold 5−fold

199

20% 38% 27% 11% 3%

1e6

5e5

0 1320

1322

1324

1326

1328

1330

1332

1334

1336

Energy [keV] Fig. 3. The segment-sum energies for Eg ¼ 1332:5 keV plotted for different segment multiplicities. The centroids of the energy peaks are lowered as a function of the segment fold. The multiplicity distribution is also given e.g. only 20% of the detected g rays are contributing to the 1-fold photo peak. The energy calibration can be adjusted only for known full-energy interactions in the single segment.

Compton scattering process, energy can be deposited simultaneously at several positions in the Ge crystal, causing energy signals in more than one segment. Ideally the energy from the single signal of the core equals the sum of energies of all the firing segments. However, measurements with the new 36-fold segmented detectors show a clear difference between the core energy and the sum of all segment energies. The effect for the segmentsum energy is shown in Fig. 3 for Eg ¼ 1332:5 keV as a function of the segment multiplicity. It reveals that the sum or add-back energies lack a considerable amount of energy, which is linearly proportional to the number of hit segments (see also Fig. 4(a)). A classification according to the number of hits, therefore, becomes natural: events in which exactly n segments are hit will be designated as n-folds. This experimental result clearly motivates the need for a careful investigation of crosstalk in highly segmented detectors: a detailed understanding—especially for the crosstalk observed in the segments—is of crucial importance for g-ray tracking. The tracking method reconstructs the original g energies by adding the individual segment energies assigned to a single g ray, to prevent summing effects when more than one g ray is present in the detector at a given time. Uncorrected spectra produced by the tracking algorithms would be severely limited in resolution as can be deduced from Fig. 4(a) and (b). The final performance of the spectrometer will depend crucially on the ability to perform a proper add-back correction including crosstalk contributions. The calibration of the segment energies is done with photopeak events in single segments and is synonymous to an alignment of 1-fold events. The first and simple choice to observe crosstalk in segments is thus provided by the 2-fold events. Higher folds have much more combinations, and the statistics for each combination is drastically reduced, so that finally, the analysis of higher folds becomes much more involved. For the core, the standard calibration is performed with the photo peak events as seen by the entire crystal, which does not distinguish between different folds in the segments. The resulting core calibration reference is a complex superposition of the various segment folds. A more refined inspection of the peak position of only the 2-fold events from Fig. 3 is shown in Fig. 5. The peak shifts of these 2-fold events reveal a regular pattern as a function of

pairwise segment combinations. The fitted centroid energies of the 1332.5 keV line is shown for all possible two segment combinations. The energies are shown for the core energy signal (þ, red symbol) and the sum of the two segment energy signals (, green symbol). All 2-fold data obtained on three similar symmetric AGATA crystals (S001, S002 and S003) are shown in Fig. 5. It is noteworthy to emphasize that the energy differences are not caused by an inadequate energy calibration of the core or segment signals. These measurements were performed with the three crystals in three different cryostats. The similar pattern with a 6-fold periodicity demonstrates that the observed energy shifts are reproducible and show an intrinsic property of the detectors. The shifts are only slightly affected by possible small differences in the mechanical and electrical environment of the detector crystals. The regular structure is related to the pair of chargecollecting segments involved, which is repeated six times due to the symmetry of the crystal. In contrast to the segment-sum energy one observes no large energy offset for the core energy. Although a clear indication for a hit pattern is observed.

5. Linear electronic model In order to describe the experimental results, the following electronic model of the detector and preamplifier assembly was employed. In Fig. 6(a) the coupling between the detector and the preamplifier is shown. For simplicity, only a 2-fold segmented encapsulated detector is shown, but its extension to a 36-fold segmented AGATA detector is trivial. The core electrode is connected to high voltage over a load resistor Rl. The core preamplifier (open-loop gain Ac ¼ 8  104 ) is AC coupled with C ac . The segment preamplifiers are DC coupled and have an open-loop gain of As ¼ 1  104 [10]. All feedback loops are implemented with the same impedance Z fb consisting of a feedback capacitance in parallel with a large resistor. The impedances and open-loop gains for the AGATA detectors under discussion are summarized in Table 3. Using Miller equivalence theorem [12], the small signal equivalent scheme (or AC equivalent scheme) of the circuit in Fig. 6(a) is deduced and shown in Fig. 6(b). As documented by Gatti et al. [13], Pellegrini [18] and Pullia et al. [11], the AC

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1336 1334 1332

Energy [keV]

1330 1328 1326 1324 1322 1320 1318 1316 1

2

3

4

5

6

7

Number of segments hit

5

4.5

FWHM [keV]

4

3.5

3

2.5

2 1

2

3

4

5

6

7

Number of segments hit Fig. 4. Peak-energy shift and variation of the peak width as a function of segment multiplicity is shown for the segment sum. The corresponding energy spectra are shown in Fig. 3. (a) Peak shift. (b) Line width.

equivalent scheme of a depleted detector can be described by a network of capacitances: each electrode i is coupled to the other electrodes j by a capacitance C ij . The most important amongst them, the capacitances C 0j between core and segments, were determined experimentally (see Table 1) and are all of the order of 1 pF depending on the size of the segments and the crystal geometry. The capacitances C ij between segments were not directly accessible to a measurement. These values are determined by the segment areas, the distance between the two segments i and j and the relative orientation of their surfaces. Most of the capacitances C ij are estimated considerably smaller due to large distances and smaller effective areas in comparison with the measured C 0j values. For neighboring segments the capacitance uncertainties are larger due to the unknown details of the segmentation: the distance between segments, the segmenta-

tion profile and the thickness of the electrical contacts. We have varied these C ij values in the reasonable range between 0 and 3.5 pF to improve the agreement between the model and the experimental results. Moreover, one may expect an additional contribution to the crosstalk coming from the cabling inside the capsule. Unfortunately, these details are not provided by the detector manufacturer. For encapsulated detectors, there is also a capacitive coupling C gi between segments and the grounded encapsulation. Values for C gi are also measured and listed in Table 1. The current induced on electrode i by movement of free charge carriers inside the detector is given by Shockley–Ramo theorem [4,14–16] and is incorporated in the model by an AC current source ii generated between the ground and the electrode i. Within this model, the feedback currents ~ifb depend linearly on the currents ~i created in the charge

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1335

core sum of segments

1334

1334

1333

1333

Energy [keV]

Energy [keV]

1335

1332

1331

1331

1330

1329

1329

1328

A1-A2 A3-F6 B1-A1 B3-F6 C1-A1 C3-F6 D1-A1 D3-F6 E1-A1 E3-F6 F1-A1 F3-F6 F6-F5

core sum of segments

1332

1330

1328

201

A1-A2 A3-F6 B1-A1 B3-F6 C1-A1 C3-F6 D1-A1 D3-F6 E1-A1 E3-F6 F1-A1 F3-F6 F6-F5

1335

core sum of segments

1334

Energy [keV]

1333

1332

1331

1330

1329

1328

A1-A2 A3-F6 B1-A1 B3-F6 C1-A1 C3-F6 D1-A1 D3-F6 E1-A1 E3-F6 F1-A1 F3-F6 F6-F5

Fig. 5. Energy shifts observed in 2-fold events as function of hit pattern. Three different symmetric AGATA crystals (S001, S002 and S003) were investigated, each built into a different cryostat. The centroid shifts, as observed in the core energy, are marked with ‘‘þ, red’’. The corresponding shift in the segment-sum energy with ‘‘, green’’. The order used on each x-axis to enumerate the 2-fold segment combinations is: first all combinations with A1: A1–A2,A1–A3,. . .,A1–A6,A1–B1,. . .,A1–F6, then all combinations with A2: A2–A1,A2–A3,. . .,A2–F6, etc. (a) S001. (b) S002. (c) S003.

a

+Bias

Rfb

Rl

b

Zfb

Cfb

Ac

Rl

Zfb Ac

Cg,0

Vout,0

Ac

Vout,0

ifb,0

0

C

0

02

Core

2

Cac

ifb,0 i0

Cac

i*0 Z 00

core

2

1

Zfb

Cfb Rfb

As

C 01 1

i*1 Z 11

As

C12

ifb,1

i1

Cg,1

Zfb As

Vout,1

ifb,1

Vout,1

Fig. 6. (a) Schematic layout of the coupling of the preamplifier to a 2-fold segmented detector. Shown are an AC-coupled core preamplifier and a DC-coupled segment. (b) The small signal equivalent scheme of the circuit also including the capacitance from segment electrode to ground C g;i and Miller equivalent of the feed back impedance Z fb =As . (a) DC scheme. (b) AC equivalent scheme.

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Table 3 Overview of impedances shown in Fig. 6. Symbol

Description

Value

C 0;j C i;j Rl C ac Rfb C fb C g0 C gi Ac

Capacitance between core and segment j C 0j Capacitance between segment i and segment j C ij High voltage load resistor High voltage decoupling capacitance Feedback resistor Feedback capacitance Capacitance between encapsulation and core Capacitance between encapsulation and segment i Core preamplifier open-loop gain

Table 1 0–3.5 pF 1:0 GO 1.0 nF 1:0 GO 1.1(1) pF 8(1) pF Table 1

As

Segment preamplifier open-loop gain

104

Z fb

Rfb kC fb

Z 00 Z ii r0

Approximation

Norm at 1 MHz

ðsC fb Þ1

Rl kC g0 kðZ fb =Ac þ ðsC ac Þ C gi kðZ fb =As Þ

1

Þ

ðsC ac Þ1 ðsAs C fb Þ1 ðC ac þ C g 0 Þ=C ac

1.008

ðAs C fb þ C gi Þ=As C fb C ac =r 0 ðC ac þ C det Þ P C fb =r i ðC fb þ j C ij =As Þ

1.002 0.956 0:998

ri T 00 T ii

ðZ fb =Ac þ ðsC ac Þ1 Þ=Z 00 Z fb =As Z ii ðC det kZ 00 Þ=r 0 Z 00 P ðZ ii k j C ij Þ=ri Z ii

D0i

sr i C 0i Z ii

r i C 0i =As C fb

8:3  105

Di0

sr 0 C 0i Z 00

r 0 C 0i =C ac

1:01  103

Dij

sr j C ij Z jj

r j C ij =As C fb

8:3  105 ðpF1 Þ  C ij

The evaluation is made on the basis of Table 3, at a frequency of 1 MHz with i; j40, C 0i ¼ 1 pF, C ij ¼ 0 pF for non neighboring segments.

collection process 0 1 0 P 1 i0 i Z 0i C B Z 1 ~i ¼ B @ i1 A ¼ B 01 @ i2 Z 1 02

Z 1 01 P 1 i Z 1i Z 1 12

1 0 1 r 0 Z 00  ifb0 B 1 C 1 Z 12 C  @ r 1 Z 11  ifb1 C A ¼ T  ~ifb P 1 A r 2 Z 22  ifb2 Z Z 1 02

i 2i

(1) 1

with the impedance Z ij ¼ ðsC ij Þ between nodes i and jai (see Fig. 6 and Table 4). The notation Z ii is used for the impedance between node i and ground. The quantity r i refers to the ratio ii =ifbi in Fig. 6. Using Eq. (1), the transfer function T can be expressed, exploiting the principle of a virtual ground in Fig. 6, as ~ vout ¼ Z fb  ~ifb ¼ Z fb  T ~i

r 1 Z 11 Z 12

i 2i

r 2 Z 22

(4) 3

Table 4 Overview of impedances and matrix elements of matrix Eq. (4). Full formula

r 0 Z 00 Z 02

8  104

The given numbers correspond to correspond to standard values used in AGATA detectors.

Symbol

diagonal part T1 and a small mixing term D, we get for the d inverse, to first order4: T ¼ Td  Td  D  Td . For this particular type of network, as the electronics is designed to register all the current produced, we have T ffi Td ffi 1 (see also Table 4) which makes it possible to simplify the inverse further to T ¼ Td  D or 0 P 1 1 1 ½ Z  r 1 Z 11 r 2 Z 22 i 0i 0 1 r Z Z Z B C 0 00 01 02 T 00 D01 D02 P 1 1 B C ½ Z 1i  B D C B C r Z r Z 0 00 i 2 22 T 11 D12 A ¼ B Z T@ 10 C r Z Z 01 1 11 B P 121 1 C @ D20 D21 T 22 ½ Z  A

(2)

The first factor Z fb takes care of the integration of the current signals. In the time domain, it represents a convolution with the transfer function 8 > < 1 et=t if ðtX0Þ 1 L ðZ fb Þ ¼ C fb (3) > :0 elsewhere The exponential decay has a time constant t ¼ Rfb C fb ffi 1 ms which corresponds to the characteristic decay time of the first stage of the preamplifier. The actual crosstalk is described by the off-diagonal matrix elements in T. Numerically, the inversion of the frequencydependent matrix T1 ðsÞ poses no problem. However, explicit analytical expressions for the matrix inverse become rapidly obscure. A simple approximation can be made by assuming that crosstalk is a small effect within a well-behaved circuit and, therefore, the matrix T1 is quasi diagonal. After separation into a

6

The bandwidth of the energy filter is limited to f ¼ 10 210 Hz. In this frequency range, the matrix elements in Eq. (4) are real and constant. Explicit values, based on Table 3, are listed in Table 4. The influence of the resistors within the bandwidth of the energy filter can be neglected, which allows more simple expressions for the matrix elements (also summarized in Table 4). The core has a slightly smaller gain than the segments: T 00 oT ii P due to the detector capacitance C det ¼ i C 0i . As a consequence the signal-to-noise ratio is reduced. This effect may not be observed directly due to the rather large uncertainty in C fb and consequently a large spread in gain. The crosstalk matrix elements obtained in Eq. (4) have a rather simple interpretation: the current generated on electrode i will not entirely go to preamplifier i, where it sees an impedance Z ii . Part of the signal will also be split off by the capacitive coupling Z ij between electrode i and any other electrode j. The current picked up by segment j, as calculated if only segment j would be present, would amount Z ii =ðZ ij þ Z ii Þ ffi Z ii =Z ij . This yields basically the same crosstalk matrix elements as Eq. (4), taken into account that Z ij have to be much larger than Z ii for a well performing detector. The dominating observable crosstalk effect is a reduced energy signal of the segment energy which is caused by the AC coupled core preamplifier. This is described by the matrix elements Di0 ffi C 0i =C ac in the first column of matrix Eq. (4). The values from Table 4 cause approximately a 0.1% shift of the deposited energy for an AGATA detector. Conversely, the influence on the core energy signal by the DC coupled segments is described by the segment-to-core matrix elements D0i ffi C 0i =As C fb in the first row of matrix Eq. (4). Here the results are calculated to be at least one order of magnitude smaller than for the segments due to the much higher value of As C fb with respect to C ac . The other matrix elements of Eq. (4) are relevant for the segment-to-segment crosstalk; their contribution is calculated to be negligible in most cases for the same reason. However, exceptions exist in adjacent segments, where the segment-tosegment capacitance can surmount the core-to-segment capacitance.

6. Comparing data with the model The results of the electronic model are compared with the different experimental findings on crosstalk properties. Firstly, the fold-dependent energy shifts of the segment-sum energies, which were shown in Figs. 3 and 4(a), are reproduced: the model showed that for each event a constant shift, which is proportional to the core signal, is observed in every hit segment. For a n-fold event, n such terms contribute to a reduction of the segment-sum energy. Moreover, the overall energy shifts are expected to be inversely proportional to the core decoupling capacitance. This behavior 4 2 Proof: T  T 1 ¼ ðT d  T d DT d Þ  ðT 1 d þ DÞ ¼ 1  ðT d  DÞ . Using the max-norm, 2 kðT d  DÞ2 k is bounded by kðT d  DÞ2 k1 p½kDk1 =minðjT 1 di;i jÞ , which is small compared to 1 by assumption.

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7 6

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800

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Coupling Capacitor [pF] Fig. 7. The non-linear shift in 2-fold segment-sum energies as function of the decoupling capacitance is explained by crosstalk contributions. The fit function corresponds to y½% ¼ 750ð17Þ% pF=ðC ac ½pF þ 31ð3Þ pFÞ.

was already experimentally verified on a 6-fold segmented MINIBALL detector [17,19] and the result is shown in Fig. 7 where the observed energy shifts are in very good agreement with the 1=C ac fit function as indicated by the line. In the following, the measured crosstalk properties of three symmetric AGATA detectors were compared with the results of the linear electronic model, summarized by Eq. (4). For the comparison, measured events were taken as a vector ~iðsÞ ¼ ð1jx x   Þ 1 2

(5)

with the restriction that the sum of energies deposited in the P segments yields i xi ¼ 1, while the core signal equals 1. Of importance is the calibration procedure and its impact on the comparison between experiment and calculation. The energy deposited within the single segment is calibrated selecting 1-fold full-energy events with xi ¼ di;j which is completely defined. The standard core calibration is different; here no restrictions to the individual segments and its coefficients xi are required or meaningful. The ensemble average of firing segments x¯ i defines the calibration reference value. 6.1. Crosstalk observed in the segments In Fig. 8 the variation of the centroid energies for two-segment events from g rays with Eg ¼ 1:3 MeV is shown as calculated using the model (Eq. (4)). The results are divided into three parts and corresponding figures in order to decompose the three main components, which are included in the simulation. The first picture in Fig 8(a) shows results of the most basic and simplified simulation: it was assumed (i) that the energy for the 2-fold combinations is shared equally between the two participating segments (2-folds of the type ~i ¼ ð1j1212ÞT in Eq. (5)), (ii) that no segment-to-segment crosstalk was considered. At this point the variation in the pattern is clearly depending only on the variation of the core-to-segment capacitances C 0i (Table 1), which are strictly ring dependent. The weakest effect is showing up in combinations involving ring 2, due to its distinct lower capacitance. In Fig. 8(b) the calculation takes into account the different energy splitting between two segments which is governed by the angular dependence of Compton scattering cross-section and the

attenuation of the g rays after Compton scattering: ~i ¼ ð1jhx ihx iÞT . Experimental values (hx i and hx i ¼ 1  hx i) i j i j i were derived for all the 2-fold combinations i, j from data taken with detector S001 (for the standard source positioning shown in Fig. 1). Although the effect on the energy is rather small—only about a quarter of all 2-fold combinations show an energy difference which exceeds 20%—this effect is large enough to be observed because it behaves very systematically. For the standard source position, it causes an additional diagonal pattern. In the last picture, Fig. 8(c), the segment-to-segment crosstalk was included between direct neighboring segments. This contribution created four extra diagonals: two diagonals are caused by crosstalk between segments belonging to the same ring. These diagonals lie directly besides the main diagonal. Two additional strong secondary diagonals are caused by crosstalk between adjacent segments in neighboring rings. In the graph, they produce the diagonal line which is shifted six segments away from the main diagonal. The simulation was, in this case, performed with a constant capacitance of 3.5 pF between direct neighboring segments in the same sector and 1.0 pF between direct neighboring segments in the same ring. The last picture is in good agreement with the measured data for the three symmetric AGATA detectors as shown in Fig. 9. To fully agree with the experimental values, the theoretical values had to be multiplied by a constant value of 1.2; the theory underestimates the measurement by 20%. Data from all three measurements showed a similar small discrepancy. During the measurement with detector S002, Fig. 9(b), an untypical crosstalk behavior was observed between segment D3 and all other segments. This was traced back to a malfunctioning channel of the digitizer electronics (DGF) itself. The core-to-segment capacities were measured only for detector S001. As these values do depend on crystal properties and depletion voltage one may expect differences between the three detectors. Also the capacitances between segments seem to vary between the three detectors. This can be observed on the data in Fig. 9 by comparing the amplitudes in the diagonals which originates from crosstalk between neighboring segments. Indeed the crosstalk contribution is largest in the S001 detector (Fig. 9(a)), it becomes smaller in S002 (Fig. 9(b)) and it is barely visible in S003 (Fig. 9(c)). To demonstrate the very good quantitative agreement between model and simulation, the column averages of both plots,

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A F

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A F A F A ABCDE F ABCDE F ABCDE FABCDE F ABCDE F ABCDE F Ring 1 Ring 2 Ring 3 Ring 4 Ring 5 Ring 6

Fig. 8. Theoretically expected crosstalk patterns for the centroid position of the segment-sum energy for 2-fold events. The color indicates the measured energy of the 1332.5 keV peak. (a) Including only core-to-segment capacitances C 0i ðhxi i ¼ 1=2Þ. (b) Folding in real observed energy splitting hxi i between segments. (c) Additional crosstalk between nearest neighbor segments.

Figs. 8(c) and 9(a) of detector S001 (for which the simulation was optimized), are directly compared in Fig. 10(a). To achieve this consistency the theoretical values were slightly rescaled by 20% and the capacitances between direct neighboring segments were

adjusted. To illustrate the effect of the last modification the difference between Figs. 9(a) and 8(b) is plotted in Fig. 10(b). For Fig. 8(b) the simulation excluded a contribution from segment-tosegment crosstalk. To illustrate the additional crosstalk between

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A F A F A ABCDE F ABCDE F ABCDE FABCDE F ABCDE F ABCDE F Ring 1 Ring 2 Ring 3 Ring 4 Ring 5 Ring 6

Fig. 9. Observed crosstalk patterns for the 2-folds segment sum for—from top to bottom—the detectors S001, S002 and S003 (see also Fig. 5). All data were taken with the standard source positioning. The color indicates the measured energy of the 1332.5 keV peak. (a) S001. (b) S002. (c) S003.

direct neighboring segments, the values were shown as function of the difference between row and column numbers. The enlarged crosstalk between neighboring segments shows up clearly at numbers (jrow  colj ¼ 1 and jrow  colj ¼ 6). As discussed before, the encapsulation technology does not allow for a direct

measurement of the segment-to-segment capacities and a large uncertainty on these capacities is expected. Obviously, adjacent segments have a pronounced capacitive coupling, compared to other segment combinations, which makes these segment combinations more vulnerable for crosstalk. However, in all three

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Column index –Row index Fig. 10. Comparison between theory (from Fig. 8) and experiment (Fig. 9) for the 2-fold segment-sum energies. (The values are given relative to 1332.5 keV.) (a) The column averages of Fig. 8(c) (-cross) and Fig. 9(a) (line) are shown for detector for detector S001. (b) The differences between Figs. 8(b) and 9(a) are shown as function of the column–row index difference. In Fig. 8(b) the segment-to-segment crosstalk was excluded from the simulation. This causes the non zero residues for index differences 6,1,1 and 6.

detectors, simulation needs only a capacitive coupling of less than 3.5 pF to match the observed crosstalk patterns. There is no need to assume any further crosstalk contribution present in the segments. All other crosstalk effects are described within the model down to a level of 104 .

6.2. Core crosstalk The electronic model also yields results related to the core energies, which are affected by a crosstalk contribution coming from the segments. The energy shift is calculated to be at least one order of magnitude smaller than the crosstalk shift visible in the segments due to the much higher value of As C fb in the denominator of the crosstalk matrix elements D0i with respect to C ac . The basic information on this small change in core energy can be deduced from events with grays fully absorbed in only one segment. In these cases, the core energy shifts of the 1-fold events directly yield the segment-to-core crosstalk matrix elements D0i (first row in matrix Eq. (4)) of the individual segment i. The calculated values are shown in Fig. 11(c) and are compared to various experimental values which exceed the calculated crosstalk effects obviously. The experimental centroid energy shifts of the 1332.5 keV lines are plotted in Fig. 11(b) depending on the single segment which fired. A shift of the energy position of nearly 0.5 keV is observed between the segments in the front part and the back part of the detector. This unexpected characteristic was observed in all three detectors and turned out to be always more dominant than the theoretical crosstalk effect. To elucidate the surprising behavior of the three detectors, the measurement and calibration procedure was repeated by changing the 60Co source positions. The three different source positions with respect to the AGATA detector are given in Fig. 11(a). The relative shifts between the front and back part of the detector did not change for the three measurements (see different curves of Fig. 11(b)). However a change in offset is

observed due to the change in source position. In the standard measurement, the front rings are more hit than the back rings. Therefore, these rings are weighted more in the calibration procedure relative to the back rings, which causes the calibration energy to lie close to the 1-fold energies of the front rings. If the calibration sources are moved to the back, one observes that the pattern is systematically shifted upwards, such that the calibration comes to lie closer to the energy shifts observed in the back rings. The calibration is clearly influenced by the position of the calibration source: as introduced before, the standard core calibration is determined by the ensemble average ~i ¼ ð1j¯x1 x¯ 2    ÞT of full absorption events in the core. The averages x¯ i which through Eq. (4) define the calibration reference value, are therefore, dependent on the position of the source and on the energy of the selected calibration line. The measured changes in peak energy position which are shown in Fig. 11(c) are clearly not in accordance with the crosstalk calculation and other effects may be responsible for the depth dependent energy shifts. Effects caused by ballistic deficit can be excluded as digital electronics was used. The trapezoidal filter with a long 1:2 ms flat top was chosen to exclude effects coming from different signal rise times. One intriguing possible explanation is related to the effect of incomplete charge collection especially by a small reduction of the measured core signals due to electron trapping. The AGATA detectors are built out of n-type HPGe crystals and collect after electron-hole pair production the electron part at the positive high voltage of the core contact. In previous work by Raudorf and Pehl [26] it was demonstrated that electron trapping causes energy shifts in a large HPGe crystal which are substantially larger than the expected crosstalk in the core while its influence on the segments is less pronounced, such that it is not observed there. In order to quantify this explanation we started with the following consideration: within a new HPGe crystal, the electron mean free path is given to be of the order of 2500 cm [25]. The left tailing of the energy peaks which one typically associates with trapping is only pronounced at smaller

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1331.5 hit segment Fig. 11. The centroid energy of the 1.3 MeV peaks are measured for different source positions and compared in Fig. 11(c) to the expected crosstalk pattern. Only events with full-energy deposition in one segment are required. The detected energy is decreasing by more than 0.5 keV by moving the interaction points from the front to the back part of the detector. Moreover the calibration is depending on the different source positions (see Fig 11(a)) which is causing different offset values (see Fig. 11(b)). For segment labeling see see Fig. 1. (a) Three source positions used for the measurements summarized in Fig. 11(b). (b) The 1.3 MeV peak energy position measured with different source positions. (c) The results from Fig. 11(b) are corrected for a constant offset constant offset and compared to a calculated crosstalk contribution.

mean free paths. At such large path lengths, the influence of trapping is mainly a constant energy shift, combined with a broadened line width. These energy shifts are strongly dependent on the diameter of the crystal. For an electron mean free path of 2500 cm, the energy shift of the 1332.5 keV line for a crystal with a 4 cm radius is about 300 eV more than for a crystal with a radius of 2 cm. As the AGATA crystals have a tapered shape (see Fig. 1), one basically selects a selects a particular crystal radius by gating on 1-fold events in a specific segment. Therefore, the smooth shifts shown in Fig. 11(c) can be attributed to the change in crystal radius with depth. Indeed for all three observed in the core (see Fig. 5). The amplitude seems to vary within a factor of less than two which can readily understood by the variation in mean free path length from crystal to crystal. Also the energy resolution has been observed to decrease with radius within the expected range. Unfortunately, this effect cannot be disentangled from the true

crosstalk. Its magnitude is estimated to be higher than the crosstalk contribution. As a consequence the observed energy shifts in the core are inconclusive to validate the crosstalk model for segment-to-core crosstalk.

7. Discussion and conclusion Detailed investigations of peak energy shifts in tapered large volume 36-fold segmented HPGe detectors were performed as a function of the segment multiplicity and the 2-fold segment combination for the core energy and the segment-sum energy. The major crosstalk effects of the segmented Ge detectors are related to the coupling from the core signal to the segments and are observable in the sum energy of the segments. The measured deviations from a constant and full-energy detection are very well

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reproduced by the results of an electronic detector model for the segment-sum energy. The main crosstalk contribution is induced by the core signal and causes about 0.1% of crosstalk per fold in the segment-sum energy. Its main dependence on the ring dependent, different core-to-segment capacitance values is perfectly reproduced in comparison with the experimental result. The segment-to-segment crosstalk has only been observed between adjacent segments with a maximum amplitude of 0.04%. Within the uncertainties on the unknown physical coupling between segments, this is in good agreement with the detector model. Energy variations of 0.03–0.06% around the calibration reference are observed for the core signal as a function of the segment ring or detector depth. In comparison with the crosstalk model these values are one order of magnitude higher than calculated and other effects cover the expected small energy shifts. In particular, electron trapping is put forward as an explanation for the major effects observed in the core. The energy differences vary systematically as a function of the radius of the Ge crystal, and the observed energy shifts are in agreement with calculations based on literature values. Future investigations have to demonstrate the suggested trapping effects e.g. by measuring radius dependent point like interactions. Upgrades of the electronic detector model were also considered. Calculations of more realistic preamplifier models were started e.g. using the cold resistor model [20]. Here the gate-tosource capacitances of the FET input stage were taken into account [9]. However, this does not yield an improvement, because such models primarily describe more accurately the rise time of the transfer function (Eq. (3)), which has no major influence on the detected energy. Such models form the basis for the observed differential crosstalk [22]. Another approximation of the model, which may influence the result, is the assumption of an ideal common ground. In reality, the grounding in HPGe detectors operated at low, liquid nitrogen temperature is challenging: a perfect ground cannot be realized as the thermal conductivity of the wiring to the detector must be strictly limited. The trade-off that has to be made here is individual to each detector and, therefore, no general model can be given at this level. The effect is expected to increase with frequency. Here the impedances of the wires become important. In extreme cases, it can lead to detector resonances, which show up in more detailed models including wire impedances. This has been worked out in more detail by Pascovici et al. [23], and motivated e.g. the development of compensated preamplifiers. In summary, detailed information on crosstalk contributions in energy pulses from g-ray interaction in highly segmented HPGe detectors were obtained for the first three symmetric AGATA detectors. A linear electronic model was developed to evaluate the measured results. The linear model proves that basically every segmented detector will be affected by crosstalk due to the capacitive coupling between the core and the segments. These crosstalk contributions behaved in all three detectors very similarly, implying that the development of the AGATA cryostats

and the preamplifier electronics has progressed in such a way that these fundamental constraints are reached and detectable.

Acknowledgment This research was supported by the German BMBF under Grant 06 KY 205 TP 8. References [1] [2] [3] [4] [5]

[6] [7]

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

[18] [19] [20] [21]

[22]

[23]

[24] [25]

[26] [27]

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