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Locating events in a segment of a segmented HPGe detector at KRISS
Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162680
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research, A journal homepage: www.elsevier.com/locate/nima
Locating events in a segment of a segmented HPGe detector at KRISS Jubong Han, K.B. Lee β, Jong-Man Lee, S.H. Lee, Tae Soon Park, J.S. Oh KRISS, Daejeon, Korea
ARTICLE
INFO
Keywords: Segmented HPGe detector Gamma ray tracking system Finite element method Shockley-Ramo theorem Signal simulation
ABSTRACT A specific segment of the 24 fold segmented HPGe detector was studied for improving the interaction position resolution from several cm to mm. The segmented HPGe detector was modeled by CAD program with their electrodes which was isolated by the Boolean function. The Shockley-Ramo theorem changed the Gauss law to Laplace equation problem for calculating the induced charge on each electrode. Finite Element Method solved the Poisson and Laplace equation with their boundary conditions for Electric field and potential. The drift velocities of charge carriers were compared between both along the < 100 > and one between < 100 > and < 110 > Ge crystal axis at a specific segment and used to simulate the pulse shapes with 1 mm position resolution. Coincidence system between a specific segment and 3 Γ 3 HPGe detector verified the simulated pulse shapes by comparing these with the experimental ones which are generated at the specific target region with a fixed scattering angle. The experimental and simulated ones were π 2 fitted to find the least π 2 value and locate the interaction positions. The interaction positions of experimental pulse shapes were analyzed statistically after the rise time correction along < 100 > and < 110 > Ge crystal axis. The position resolution as FWHM on x, y, and z axis is the 3.12, 4.38 and 8.69 mm with the collimator size 3, 6, and 6 mm, respectively.
1. Introduction 1.1. Nuclear structure and πΎ-ray tracking Nuclear structure has been studied by the πΎ-ray energy spectrum of the radioisotopes. This study has been improved by the Compton suppression which is done by the passive and active shield. The passive shield used the lead block to isolate the radioisotopes from the outside radio active source. The active shield used the coincidence system between the main and guard detector. The signals which were detected simultaneously at the main and guard detectors were considered as the out side radio active source or the partly deposited energy of the interesting radioisotopes. The πΎ-ray tracking is one of the Compton suppression technique by tracking the πΎ-ray to its originating site. It is elementary to locate the πΎ-ray interaction positions in a segment for πΎ-ray tracking. The position resolution of the segmented detector is increased from its segment size to several mm by locating the interaction positions in a segment. When a πΎ-ray interacts with a specific segment. The signals with characteristic pulse shapes are generated at the electrodes of this segment and the core electrode and all other segments show the transient signals.
The pulse shapes at a specific electrode and all other segments during its rise time contains the information of where it is created. The pulse shapes during its rise time were simulated by using the Shockley-Ramo theorem which changed the Gauss law problem to Laplace equation problem for calculating the induced charge on each electrode [2]. 1.2. CSDA, pre-amplifier rise time, anisotropy and cross-talk The hot electron which is generated by the πΎ-ray collides with the Germanium atoms. These Germanium atoms are ionized by this hot electron within a certain range. This electron range is calculated as Continuous Slowing Down Approximation(CSDA)range and opened by National Institute of Standards and Technology(NIST). The pre-amplifier has its own rise time and affects the signal shapes during signal processing. The ionized electrons drift to the electrodes with a certain drift velocities depending on the crystal axis. The segmented Ge detector itself is a complex network of capacitors which are Capacitances of the segments and between the segments. The cross talk is created by the coupling capacitor of the core signal and the capacitive network of the detector [3].
β Corresponding author. E-mail address: [email protected] (K.B. Lee).
https://doi.org/10.1016/j.nima.2019.162680 Received 24 October 2018; Received in revised form 29 August 2019; Accepted 1 September 2019 Available online 12 September 2019 0168-9002/Β© 2019 Elsevier B.V. All rights reserved.
J. Han, K.B. Lee, J.-M. Lee et al.
Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162680
Fig. 1. Drawing of the electrically segmented detector β 6 layers Γ 4 azimuthal pies [1].
2. Materials and methods
The PSC 823C pre-amplifier has its rise time as 20 ns/0pF and slope as 0.8 ns/pF. The digital system consists of a PXIe-1082 chassis, 8135 controller, flex RIO-7966R, 7965R Γ 2, 7975R and digitizer 5751(5 Γ 107 samplings/s, 16 channels) Γ 2, 5761(2.5 Γ 108 samplings/s, 6 channels). Signals are digitized by the 5751 digitizer and memorized into First-In, First-Out(FIFO) memory in PXIe-7965R. This Field Programmable Gate Array(FPGA) programmed in the PXIe-7965R used the 6 state machines for data taking [4].
2.1. Segmented and 3 Γ 3 inch HPGe detector Coincidence system was set up for both segmented and 3 Γ 3 coaxial HPGe detector using the program based on Laboratory Virtual Instrument Engineering Workbench (LabVIEW). Coaxial HPGe crystal is segmented electrically by applying separated electrodes in longitude and azimuthal along coaxial axis. It has 1 anode biased at 5000 V at the inner hole and 24 cathodes biased at ground through the gate of Field Effect Transistor (FET) on the cylinder outer surfaces which are connected by pre-amplifiers for each electrode. It has 90 mm longitudinal length and 80 mm diameter with 10 mm inner hole diameter. It is segmented to the six 15 mm layers along the longitudinal axis with each containing 4 equal pie-shaped segments. Fig. 1 shows a drawing of the segmented HPGe detector which is a N-type semiconductor detector. Segmentation is ordered from A to D and 1 to 6 which is 1 at the closed end of cylinder and 6 at the open end of it. The inner hole is depicted in a dashed line in Fig. 1 and the top segments are indicated with the A1, B1, C1 and D1 segments for convenience. The origin of Cartesian coordinates for simulating pulse shapes is placed at the bottom center of cylinder as shown in the left of Fig. 1. The 3 Γ 3 coaxial HPGe detector(GEM-25185-p) used for coincidence system was manufactured by ORTEC which is p-type and unsegmented HPGe detector.
2.3. Program and language used for position resolution improvement 2.3.1. Finite element method for electric field and potentials The geometry of the coaxial HPGe detector which is segmented on the outer surface was modeled by using FreeCAD. The meshed geometry was saved as an Standard Triangle Language (STL) file in ASCII-based format. This STL file was imported into Elmer, a multiphysics program, especially to find the electric field and potential using the Finite Element Method (FEM). The Dirichlet boundary condition was used for a 5000 V biased voltage on the inner hole of the cylinder and for a 0 V biased voltage on the 24 segments of the cylinder surface. The Neumann boundary condition was used for the space charge in the HPGe crystal of the detector. The space charge was denoted as π and the Neumann boundary condition was ππ/ππ§ = constant. This derivative was integrated over the z axis which was satisfied by the space charges at the specific points, net impurity concentration β front (closed end):1.60 β 1010 cmβ3 and rear (open end):0.47 β 1010 cmβ3 . Poissonβs equation was solved by Elmer to calculate the potential with the boundary conditions and dielectric constant [1]. The geometry and coordinate are shown in Fig. 2. The solutions of Poissonβs equation are depicted as the electric field line in the left of Fig. 2. The solution of Laplaceβs equation is shown as a weighting potential at the A3 segment in the right of Fig. 2. The weighting potentials at the A3 segment were
2.2. Data acquisition system The EGC100-SEG24 segmented detector manufactured by Canberra is equipped with 1 resistive feed-back charge sensitive pre-amplifier card PSC 823C with cold FET in AC coupling for Full Volume and the external contacts are equipped with 24 cold FET in DC coupling. Polarity of the outputs is negative excepted the Full Volume signal [1]. 2
J. Han, K.B. Lee, J.-M. Lee et al.
Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162680
Fig. 2. Geometry, coordinates, electric field and weighting potential of segmented HPGe crystal.
Fig. 3. Electric field strength and drift velocity: weighting potentials.
the highest as unit less 1 on the outer electrode and decreased to unit less 0 at the inner electrode. The weighting potentials for the transient pulse shapes can be specified by biasing a unit less one on the outer electrode of the A2, A4, B3 and D3 segments, respectively.
respectively. Drift velocity v is defined as Eq. (1). π£=
π0 πΈ (1 + (πΈβπΈ0 )πΎ )1βπΎ
(1)
where the π0 is the mobility of charge carrier at a low electric field. Both πΎ and E0 are treated as adjustable parameters fitted to the experimental drift velocity measurements [5]. The E is the electric field strength at a charge carrier position. The π0 , πΎ and E0 for the electron and hole are 40180 cm2 /Vs, 0.72, 493 V/cm [6] and 6633 cm2 /Vs, 0.744, 181 V/cm [7] along the β¨100β© crystal axis, respectively. The positions of the charge carriers were calculated by using the electric field line and their drift velocity at a specific time. The weighting potentials of the charge carriers can be specified at these positions and can be collected to simulate a pulse shape which is called the ββmain pulse shapeββ. There are four segments next to the A3 segment: the up-A4, down-A2, right-B3 and left-D3 segments. Their weighting potentials can be specified at the positions of the charge carriers and
2.3.2. Drift velocity of the charge carriers Elmer solved Poissonβs and Laplaceβs equations for the potential and this potential can be saved as a solution to the VTU file which can be imported to ParaVIEW. The electric field and weighting potential were used to find the path of the charge carriers and to simulate a pulse shape, respectively. The A3 segment was located in the middle of the cylindrical crystal and its field was simple because of its simple geometry. The charge carriers are electron and hole which are created at interacting positions with the πΎ ray in the semiconductor and move along the electric field to the + and β electrodes with drift velocity, 3
J. Han, K.B. Lee, J.-M. Lee et al.
Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162680
Fig. 4. Simulated main and transient pulse shapes at p1, p2, p3, p4 and p5.
Fig. 5. Coincidence system of segmented and 3 Γ 3 coaxial HPGe for studying the pulse shapes at the specific positions.
can be collected to simulate a pulse shape which is called ββtransient pulse shapeββ.
π§-axis. As shown in Fig. 3, the electric field strength and drift velocity were higher as 4.5 kV/cm and 0.15 mm/ns at the inner electrode and lower as 0.5 kV/cm and 0.07 mm/ns at the outer electrode. The drift velocities of the hole and electron were different at the starting point as 0.073 and 0.081 mm/ns, respectively. The charge carriers which were the electrons and holes were separately moved from their creation site to the outer and inner electrodes. The holes and electrons were moved for 40 and 300 ns until they were absorbed by each electrode. The moving distances during a specific time increased when the holes and electrons were close to the inner electrode because of the high drift velocity. The weighting potentials of the holes and electrons were added to simulate the pulse shape depending on the time in Fig. 3. Unit less 1 were appended to the weighting potentials of the holes after 40 ns. The weighting potentials
2.3.3. Simulated pulse shapes with 1 mm position resolution CSDA range of hot electron in Germanium was calculated as 0.79 mm by using the stopping-power and range tables for electrons(estar), The hot electrons loose their energies and change to the cold electron which starts to drift to the electrode. thus the maximum position resolution is larger than 0.79 mm. Pulse shapes are different from each other depending on where the charge carriers are created in a A3 segment. The pulse shapes are simulated for a 1 mm grid and the total points were 20,257 in a A3 segment. The electric field strength, drift velocity and weighting potential starting at x37y5z55 can be calculated along the electric field line over the position from the 4
J. Han, K.B. Lee, J.-M. Lee et al.
Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162680
Fig. 6. Re-sampled pulse shapes by the linear interpolation for main and transient pulse shapes.
Fig. 7. Relative change along β¨100β© crystal axis and drift velocities using Eq. (4) along β¨100β©, β¨111β© and one between β¨100β© and β¨110β© for electron and hole.
Fig. 8. Probability density of rise time β experimental and evenly simulated pulse shapes.
for the transient pulse shapes can be calculated by finding the potentials at the same positions but the biased unit less one was not on the A3 segment but was on the A2, A4, B3 and D3 segments. The pulse shapes at five different starting points were plotted over the time(ns) in Fig. 4. p1, p2, p3, p4 and p5 were at x37y5z55, x4y8z53, x9y13z48, x7y25z51 and x25y21z46 as shown in Fig. 5. The pulse shapes which starting from close to the inner and outer electrodes such as p2 and p1 have longer rise times than that which starting from the radial middle line such as p4. The amplitudes of the transient pulse shapes were higher at the A2 and A4 segments than at the B3 and D3 segments because electrodes were close vertically and
far horizontally. The pulse shape at p1 close to the D3 segment had the higher amplitude at the D3 segment than that at the B3 segment but that at p4 did not because its starting position was close to the radial middle line. The weighting potentials of the hole and electron starting at the point close to the radial middle line were canceled out because their rise times were approximately the same. The pulse shapes starting at the inner position like p2 and p3 had the minus transient pulse shapes because their major moving charges were the holes which inducing a negative image charge. Otherwise, the pulse shapes at the outer like p1 and p5 had plus transient pulse shapes because their moving charges were the electrons which inducing a positive image 5
J. Han, K.B. Lee, J.-M. Lee et al.
Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162680
Fig. 9. Experimental main and transient signals at p1.
Fig. 10. π 2 values for main(top left) and transient(top right) of the located events; Position distribution on the three axis(bottom left) and 3D(bottom right) without any correction along β¨100β© crystal axis.
charge. Here, the unit less minus one was multiplied for the π 2 fitting with the experimental ones.
collimator was made of two lead blocks with a 3 mm gap between them. Target region p1 had the volume from 34 to 37 mm for the π₯-axis, from 1 to 7 mm for the π¦-axis and from 52 to 58 mm for the π§-axis.
2.4. Coincidence system 2.5. π 2 Fitting The pulse shapes were studied by the coincidence system of the segmented and 3 Γ 3 coaxial HPGe detectors. A3 segment was targeted by the πΎ rays from the 137 Cs source which was collimated by the lead block with a hole β6 mm in diameter. The 137 Cs πΎ rays were scattered and only those scattered at 90 degree reached the 3 Γ 3 coaxial HPGe detector through the second collimator as shown in Fig. 5 [4]. This
The Maxima code compares the pulse shapes from the experimental and simulated ones during their rise times. The experimental data which were acquired by DAQ system are divided into three parts: the acquired data before the charge carrier creation, the acquired data during the charge carriers drift from their creation position to each 6
J. Han, K.B. Lee, J.-M. Lee et al.
Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162680
Fig. 11. π 2 values and Position distribution after the rise time correction along β¨100β© crystal axis.
Fig. 12. Probability in percentage over the difference between the located positions without any and rise time correction along β¨100β© crystal axis(left); That between after rise time correction along β¨100β© and β¨110β© crystal axis(right).
electrode and the acquired data after the charge carrier absorption. The pulse shape data during its rise time were separated from the experimental data by cutting out them under and over the specific thresholds. These taken data were considered as the data before creation and after absorption, respectively. The best matched pulse shape by the least π 2 fitting were saved as a png file for main and transient pulse shapes with its interaction position on x, y and z axis. The π 2 fitting equation is modified for escaping the infinity and undefined error in Maxima code as shown in Eqs. (2) and (3). 2 πππππ =
2 ππ‘πππ =
πβ1 β (πΈππππ,π β πππππ,π )2 π=2 πβ1 β 4 β
πππππ,π (πΈπ‘ππππ ,π β ππ‘ππππ ,π )2
indexed as the 1, 2, 3 and 4 in the j index. The first and last data points in the i index are neglected for escaping ββundefined errorββ of Maxima code in both main and transient π 2 fitting equations. The denominator of π 2 fitting equation is removed in the transient π 2 fitting equation for escaping ββinfinity errorββ in Maxima code. 2.6. Correction of experimental pulse shapes by rise time The pre-amplifier intrinsic rise time was evaluated by inputting the testing square pulse to test input node and measuring the pulse shape of A3 output node. The pre-amplifier was connected to biased 12 voltage but the detector was not. The input square pulse shape had its rise time 9.6 ns and the pre-amplifier output pulse shape showed its rise time 38.4 ns. The pre-amplifier intrinsic rise time was calculated by subtracting quadratically the input rise time from the output rise time as 37.2 ns. The pre-amplifier intrinsic rise time 37.2 ns were applied to the experimental main and transient ones by re-sampling these after considering the extended rise time as shown in Fig. 6.
(2)
(3)
π=2 π=1
Here, πΈππππ and πππππ are the experimental and simulated main pulse shape data points. πΈπ‘πππ and ππ‘πππ are the transient pulse shape data 2 2 points. πππππ and ππ‘πππ are π 2 fitting value for the main and transient pulse shape data points. The 4 different transient pulse shape data points for up, down, left and right segments next to main segment are 7
J. Han, K.B. Lee, J.-M. Lee et al.
Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162680
Fig. 13. Locating test46 pulse shape at A3 for non, rise time correction along β¨100β© and β¨110β© (top left); non correction(top right), rise time correction along β¨100β©(bottom left) and β¨110β©(bottom right) for transient pulse shapes. Table 1 Parameter values of equation 4 for electron(E) and hole(H) along each crystal axis.
a b c d
E@β¨100β©
H@β¨100β©
E@β¨100β©-β¨110β©
H@β¨100β©-β¨110β©
3 212 421.866 β0.1055632 646.842869 β12 488 348.55
2 550 414.084 0.04107491 β728.99177 β8 335 162.455
3 212 421.866 0 0 12 488 348.55
2 550 414.084 0.085 β1050 β8 335 162.455
3.2. Experimental signals The experimental main and transient signals at p1 were plotted in Fig. 9. The signals from DAQ system had a minus amplitude because of its electronics. The experimental and simulated pulse shapes can be compared by multiplying a minus one to the experimental signals. 3.3. Locating events
2.7. Correction of simulated pulse shapes by anisotropy
Experimental and simulated pulse shapes at p1 were compared by the modified π 2 equation in Eqs. (2) and (3) as shown in Fig. 13. The experimental pulse shape Test46 without any correction and rise time correction along β¨100β© and β¨110β© was fitted with the simulated ones as shown in Fig. 13. The main and transient pulse shapes without any correction were located at x33y10z52 position. Two ones after rise time correction along β¨100β© and β¨110β© were located at x33y9z53 and x32y10z53 position, respectively.
The drift velocity π£100 and π£111 along the β¨100β© and β¨111β© crystal axis was defined as the parametric equation in Eq. (1). The modified drift velocity π£β²100 along the β¨100β© crystal axis was defined by the combination of the log and quadratic equation as shown in Eq. (4). The relative change between π£100 and π£β²100 was defined in Eq. (5). π£β²100 = ππππ(πΈ) + ππΈ 2 + ππΈ + π π .πΆ =
π£β²100 β π£100 π£100
(4)
3.4. Statistical analysis of located events
(5)
The 88 experimental main and transient signal sets were compared with the 20,257 simulated ones which are generated in the A3 segment. The Maxima code located 85 signals but the other 3 signals were not. These 3 signals were undershot at the beginning of its rise time or overshot at the end of it. The maxima code selects the pulse shape data during its rise time using the base line data but these over and undershot signals are failed to be selected as the proper ones. The π 2 values for main and transient pulse shapes without any correction were distributed by their frequencies at the top right and left of Fig. 10. These located signals were distributed on the x-, y- and π§-axis with their positions with their frequencies in the bottom left of Fig. 10. Gaussian fitted mean and standard deviation values were 34.23 Β± 1.76, 6.12 Β± 2.95 and 52.93 Β± 3.22 mm. The Full Width at Half Maximum (FWHM) was defined as 2.355 multiplied by the standard deviation of the Gaussian distribution and evaluated as the 4.14, 6.95 and 7.58 mm with 3, 6 and 6 mm collimator sizes. FWHMs after the rise time correction along β¨100β© crystal axis were 3.11, 4.38 and 8.69 mm on x, y and z axis with 3, 6 and 6 mm collimator sizes. The located events were plotted on three-dimensional(3D) x-y-z spaces with the target region as shown in a 7.8 Γ 8.4 Γ 8.4 mm cube in the bottom right of Figs. 10 and 11. The original target region was 3 Γ 6 Γ 6 mm cube by the collimator size but this region was spread by the distance between the target region and the source and collimator. The target region and located events were plotted on the axis origin for simplicity.
Here, a is the parameter for natural log equation. The b, c and d are the parameters of the quadratic equation. The drift velocity along the β¨100β© and one between β¨100β© and β¨110β© axis was expressed using the modified parametric equation with parameter values for electron and hole in Table 1. R.C represents Relative Change between the π£100 and π£β²100 drift velocities along β¨100β© crystal axis. The relative change were plotted in percentage over the electric field strength from 500 to 5000 V/cm for electron and hole as shown in the left of Fig. 7. The drift velocity along one between β¨100β© and β¨110β© crystal axis was calculated by the π£β²100,110 with parameters in table as shown in the right of Fig. 7, respectively. 3. Result 3.1. Rise time distribution of experimental and simulated pulse shapes in A3 segment Rise time distribution of experimental pulse shapes can be compared with that of simulated ones at A3 segment. Probability density of simulated pulse shape rise times can be calculated over whole region of A3 segment assuming evenly distributed interaction positions. That of experimental ones was obtained by locating the 137 Cs source at the A3 segment surface. Probability densities of experimental and simulated ones were depicted in Fig. 8. That of experimental ones were more probable at the short rise time than that of long one. 8
J. Han, K.B. Lee, J.-M. Lee et al.
Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162680
4. Conclusion
References
It was about 90% within 2 mm on each axis that the probability of difference between without any and rise time correction along β¨100β© crystal axis as shown in the left of Fig. 12. The rise time correction improves the position resolution about 2 mm on each axis. The difference between along β¨100β© and β¨110β© was about 1 mm within 90% probability on x and z axis as shown in the right of Fig. 12. The position resolution between them was improved about 1 mm on x and z axis by choosing β¨100β© than β¨110β© crystal axis. The pulse shape along β¨110β© crystal axis extended its rise time than that of β¨100β© but π 2 value increased, also. The A3 segment has simple geometry and is affected by two β¨100β© and β¨110β© but β¨111β© crystal axis. The difference of drift velocities was 0.01 mm/ns at 5000 V/cm for electron along β¨100β© and β¨110β© crystal axis. That was 0.06 mm/ns at the same condition along β¨100β© and β¨111β© crystal axis.. These collimated pulse shapes can be assumed to be generated along one of two. The cross talk was neglected here because All triggered events were hit once and only the segment signals were used.
[1] Benoit Pirard, Segmented coaxial HPGe detector EGC100-SEG24 userβs manual, CANBERRA, 2014. [2] Zhong He, Review of the Shockley-Ramo theorem and its application in semiconductor πΎ-ray detectors, Nucl. Instrum. Methods A 463 (2001) 250β267. [3] Bart Bruyneel, Peter Reiter, Gheorghe Pascovici, Characterization of large volume HPGe detectors. Part I: Electron and hole mobility parameterization, Nucl. Instrum. Methods A 569 (2006) 764β773. [4] Jubong Han, K.B. Lee, Jong-Man Lee, S.H. Lee, Tae Soon Park, J.S. Oh, Performance of a segmented HPGe detector at KRISS, Appl. Raidat. Isot 134 (2018) 177β181. [5] G.F. Knoll, Radiation of Detection and Measurement, third ed., Wiley, New York, 1999. [6] L. Mihailescu, W. Gast, R. Lieder, H. Brands, H. Jager, The influence of anisotropic electron drift velocity on the signal shapes of closed-end HPGe detectors, Nucl. Instrum. Methods A 447 (2000) 350β360. [7] L. Reggiani, C. Canali, F. Nava, G. Ottaviani, Hole drift velocity in germanium, Phys. Rev. B 16 (1977) 2781.
9
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research, A journal homepage: www.elsevier.com/locate/nima
Locating events in a segment of a segmented HPGe detector at KRISS Jubong Han, K.B. Lee β, Jong-Man Lee, S.H. Lee, Tae Soon Park, J.S. Oh KRISS, Daejeon, Korea
ARTICLE
INFO
Keywords: Segmented HPGe detector Gamma ray tracking system Finite element method Shockley-Ramo theorem Signal simulation
ABSTRACT A specific segment of the 24 fold segmented HPGe detector was studied for improving the interaction position resolution from several cm to mm. The segmented HPGe detector was modeled by CAD program with their electrodes which was isolated by the Boolean function. The Shockley-Ramo theorem changed the Gauss law to Laplace equation problem for calculating the induced charge on each electrode. Finite Element Method solved the Poisson and Laplace equation with their boundary conditions for Electric field and potential. The drift velocities of charge carriers were compared between both along the < 100 > and one between < 100 > and < 110 > Ge crystal axis at a specific segment and used to simulate the pulse shapes with 1 mm position resolution. Coincidence system between a specific segment and 3 Γ 3 HPGe detector verified the simulated pulse shapes by comparing these with the experimental ones which are generated at the specific target region with a fixed scattering angle. The experimental and simulated ones were π 2 fitted to find the least π 2 value and locate the interaction positions. The interaction positions of experimental pulse shapes were analyzed statistically after the rise time correction along < 100 > and < 110 > Ge crystal axis. The position resolution as FWHM on x, y, and z axis is the 3.12, 4.38 and 8.69 mm with the collimator size 3, 6, and 6 mm, respectively.
1. Introduction 1.1. Nuclear structure and πΎ-ray tracking Nuclear structure has been studied by the πΎ-ray energy spectrum of the radioisotopes. This study has been improved by the Compton suppression which is done by the passive and active shield. The passive shield used the lead block to isolate the radioisotopes from the outside radio active source. The active shield used the coincidence system between the main and guard detector. The signals which were detected simultaneously at the main and guard detectors were considered as the out side radio active source or the partly deposited energy of the interesting radioisotopes. The πΎ-ray tracking is one of the Compton suppression technique by tracking the πΎ-ray to its originating site. It is elementary to locate the πΎ-ray interaction positions in a segment for πΎ-ray tracking. The position resolution of the segmented detector is increased from its segment size to several mm by locating the interaction positions in a segment. When a πΎ-ray interacts with a specific segment. The signals with characteristic pulse shapes are generated at the electrodes of this segment and the core electrode and all other segments show the transient signals.
The pulse shapes at a specific electrode and all other segments during its rise time contains the information of where it is created. The pulse shapes during its rise time were simulated by using the Shockley-Ramo theorem which changed the Gauss law problem to Laplace equation problem for calculating the induced charge on each electrode [2]. 1.2. CSDA, pre-amplifier rise time, anisotropy and cross-talk The hot electron which is generated by the πΎ-ray collides with the Germanium atoms. These Germanium atoms are ionized by this hot electron within a certain range. This electron range is calculated as Continuous Slowing Down Approximation(CSDA)range and opened by National Institute of Standards and Technology(NIST). The pre-amplifier has its own rise time and affects the signal shapes during signal processing. The ionized electrons drift to the electrodes with a certain drift velocities depending on the crystal axis. The segmented Ge detector itself is a complex network of capacitors which are Capacitances of the segments and between the segments. The cross talk is created by the coupling capacitor of the core signal and the capacitive network of the detector [3].
β Corresponding author. E-mail address: [email protected] (K.B. Lee).
https://doi.org/10.1016/j.nima.2019.162680 Received 24 October 2018; Received in revised form 29 August 2019; Accepted 1 September 2019 Available online 12 September 2019 0168-9002/Β© 2019 Elsevier B.V. All rights reserved.
J. Han, K.B. Lee, J.-M. Lee et al.
Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162680
Fig. 1. Drawing of the electrically segmented detector β 6 layers Γ 4 azimuthal pies [1].
2. Materials and methods
The PSC 823C pre-amplifier has its rise time as 20 ns/0pF and slope as 0.8 ns/pF. The digital system consists of a PXIe-1082 chassis, 8135 controller, flex RIO-7966R, 7965R Γ 2, 7975R and digitizer 5751(5 Γ 107 samplings/s, 16 channels) Γ 2, 5761(2.5 Γ 108 samplings/s, 6 channels). Signals are digitized by the 5751 digitizer and memorized into First-In, First-Out(FIFO) memory in PXIe-7965R. This Field Programmable Gate Array(FPGA) programmed in the PXIe-7965R used the 6 state machines for data taking [4].
2.1. Segmented and 3 Γ 3 inch HPGe detector Coincidence system was set up for both segmented and 3 Γ 3 coaxial HPGe detector using the program based on Laboratory Virtual Instrument Engineering Workbench (LabVIEW). Coaxial HPGe crystal is segmented electrically by applying separated electrodes in longitude and azimuthal along coaxial axis. It has 1 anode biased at 5000 V at the inner hole and 24 cathodes biased at ground through the gate of Field Effect Transistor (FET) on the cylinder outer surfaces which are connected by pre-amplifiers for each electrode. It has 90 mm longitudinal length and 80 mm diameter with 10 mm inner hole diameter. It is segmented to the six 15 mm layers along the longitudinal axis with each containing 4 equal pie-shaped segments. Fig. 1 shows a drawing of the segmented HPGe detector which is a N-type semiconductor detector. Segmentation is ordered from A to D and 1 to 6 which is 1 at the closed end of cylinder and 6 at the open end of it. The inner hole is depicted in a dashed line in Fig. 1 and the top segments are indicated with the A1, B1, C1 and D1 segments for convenience. The origin of Cartesian coordinates for simulating pulse shapes is placed at the bottom center of cylinder as shown in the left of Fig. 1. The 3 Γ 3 coaxial HPGe detector(GEM-25185-p) used for coincidence system was manufactured by ORTEC which is p-type and unsegmented HPGe detector.
2.3. Program and language used for position resolution improvement 2.3.1. Finite element method for electric field and potentials The geometry of the coaxial HPGe detector which is segmented on the outer surface was modeled by using FreeCAD. The meshed geometry was saved as an Standard Triangle Language (STL) file in ASCII-based format. This STL file was imported into Elmer, a multiphysics program, especially to find the electric field and potential using the Finite Element Method (FEM). The Dirichlet boundary condition was used for a 5000 V biased voltage on the inner hole of the cylinder and for a 0 V biased voltage on the 24 segments of the cylinder surface. The Neumann boundary condition was used for the space charge in the HPGe crystal of the detector. The space charge was denoted as π and the Neumann boundary condition was ππ/ππ§ = constant. This derivative was integrated over the z axis which was satisfied by the space charges at the specific points, net impurity concentration β front (closed end):1.60 β 1010 cmβ3 and rear (open end):0.47 β 1010 cmβ3 . Poissonβs equation was solved by Elmer to calculate the potential with the boundary conditions and dielectric constant [1]. The geometry and coordinate are shown in Fig. 2. The solutions of Poissonβs equation are depicted as the electric field line in the left of Fig. 2. The solution of Laplaceβs equation is shown as a weighting potential at the A3 segment in the right of Fig. 2. The weighting potentials at the A3 segment were
2.2. Data acquisition system The EGC100-SEG24 segmented detector manufactured by Canberra is equipped with 1 resistive feed-back charge sensitive pre-amplifier card PSC 823C with cold FET in AC coupling for Full Volume and the external contacts are equipped with 24 cold FET in DC coupling. Polarity of the outputs is negative excepted the Full Volume signal [1]. 2
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Fig. 2. Geometry, coordinates, electric field and weighting potential of segmented HPGe crystal.
Fig. 3. Electric field strength and drift velocity: weighting potentials.
the highest as unit less 1 on the outer electrode and decreased to unit less 0 at the inner electrode. The weighting potentials for the transient pulse shapes can be specified by biasing a unit less one on the outer electrode of the A2, A4, B3 and D3 segments, respectively.
respectively. Drift velocity v is defined as Eq. (1). π£=
π0 πΈ (1 + (πΈβπΈ0 )πΎ )1βπΎ
(1)
where the π0 is the mobility of charge carrier at a low electric field. Both πΎ and E0 are treated as adjustable parameters fitted to the experimental drift velocity measurements [5]. The E is the electric field strength at a charge carrier position. The π0 , πΎ and E0 for the electron and hole are 40180 cm2 /Vs, 0.72, 493 V/cm [6] and 6633 cm2 /Vs, 0.744, 181 V/cm [7] along the β¨100β© crystal axis, respectively. The positions of the charge carriers were calculated by using the electric field line and their drift velocity at a specific time. The weighting potentials of the charge carriers can be specified at these positions and can be collected to simulate a pulse shape which is called the ββmain pulse shapeββ. There are four segments next to the A3 segment: the up-A4, down-A2, right-B3 and left-D3 segments. Their weighting potentials can be specified at the positions of the charge carriers and
2.3.2. Drift velocity of the charge carriers Elmer solved Poissonβs and Laplaceβs equations for the potential and this potential can be saved as a solution to the VTU file which can be imported to ParaVIEW. The electric field and weighting potential were used to find the path of the charge carriers and to simulate a pulse shape, respectively. The A3 segment was located in the middle of the cylindrical crystal and its field was simple because of its simple geometry. The charge carriers are electron and hole which are created at interacting positions with the πΎ ray in the semiconductor and move along the electric field to the + and β electrodes with drift velocity, 3
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Fig. 4. Simulated main and transient pulse shapes at p1, p2, p3, p4 and p5.
Fig. 5. Coincidence system of segmented and 3 Γ 3 coaxial HPGe for studying the pulse shapes at the specific positions.
can be collected to simulate a pulse shape which is called ββtransient pulse shapeββ.
π§-axis. As shown in Fig. 3, the electric field strength and drift velocity were higher as 4.5 kV/cm and 0.15 mm/ns at the inner electrode and lower as 0.5 kV/cm and 0.07 mm/ns at the outer electrode. The drift velocities of the hole and electron were different at the starting point as 0.073 and 0.081 mm/ns, respectively. The charge carriers which were the electrons and holes were separately moved from their creation site to the outer and inner electrodes. The holes and electrons were moved for 40 and 300 ns until they were absorbed by each electrode. The moving distances during a specific time increased when the holes and electrons were close to the inner electrode because of the high drift velocity. The weighting potentials of the holes and electrons were added to simulate the pulse shape depending on the time in Fig. 3. Unit less 1 were appended to the weighting potentials of the holes after 40 ns. The weighting potentials
2.3.3. Simulated pulse shapes with 1 mm position resolution CSDA range of hot electron in Germanium was calculated as 0.79 mm by using the stopping-power and range tables for electrons(estar), The hot electrons loose their energies and change to the cold electron which starts to drift to the electrode. thus the maximum position resolution is larger than 0.79 mm. Pulse shapes are different from each other depending on where the charge carriers are created in a A3 segment. The pulse shapes are simulated for a 1 mm grid and the total points were 20,257 in a A3 segment. The electric field strength, drift velocity and weighting potential starting at x37y5z55 can be calculated along the electric field line over the position from the 4
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Fig. 6. Re-sampled pulse shapes by the linear interpolation for main and transient pulse shapes.
Fig. 7. Relative change along β¨100β© crystal axis and drift velocities using Eq. (4) along β¨100β©, β¨111β© and one between β¨100β© and β¨110β© for electron and hole.
Fig. 8. Probability density of rise time β experimental and evenly simulated pulse shapes.
for the transient pulse shapes can be calculated by finding the potentials at the same positions but the biased unit less one was not on the A3 segment but was on the A2, A4, B3 and D3 segments. The pulse shapes at five different starting points were plotted over the time(ns) in Fig. 4. p1, p2, p3, p4 and p5 were at x37y5z55, x4y8z53, x9y13z48, x7y25z51 and x25y21z46 as shown in Fig. 5. The pulse shapes which starting from close to the inner and outer electrodes such as p2 and p1 have longer rise times than that which starting from the radial middle line such as p4. The amplitudes of the transient pulse shapes were higher at the A2 and A4 segments than at the B3 and D3 segments because electrodes were close vertically and
far horizontally. The pulse shape at p1 close to the D3 segment had the higher amplitude at the D3 segment than that at the B3 segment but that at p4 did not because its starting position was close to the radial middle line. The weighting potentials of the hole and electron starting at the point close to the radial middle line were canceled out because their rise times were approximately the same. The pulse shapes starting at the inner position like p2 and p3 had the minus transient pulse shapes because their major moving charges were the holes which inducing a negative image charge. Otherwise, the pulse shapes at the outer like p1 and p5 had plus transient pulse shapes because their moving charges were the electrons which inducing a positive image 5
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Fig. 9. Experimental main and transient signals at p1.
Fig. 10. π 2 values for main(top left) and transient(top right) of the located events; Position distribution on the three axis(bottom left) and 3D(bottom right) without any correction along β¨100β© crystal axis.
charge. Here, the unit less minus one was multiplied for the π 2 fitting with the experimental ones.
collimator was made of two lead blocks with a 3 mm gap between them. Target region p1 had the volume from 34 to 37 mm for the π₯-axis, from 1 to 7 mm for the π¦-axis and from 52 to 58 mm for the π§-axis.
2.4. Coincidence system 2.5. π 2 Fitting The pulse shapes were studied by the coincidence system of the segmented and 3 Γ 3 coaxial HPGe detectors. A3 segment was targeted by the πΎ rays from the 137 Cs source which was collimated by the lead block with a hole β6 mm in diameter. The 137 Cs πΎ rays were scattered and only those scattered at 90 degree reached the 3 Γ 3 coaxial HPGe detector through the second collimator as shown in Fig. 5 [4]. This
The Maxima code compares the pulse shapes from the experimental and simulated ones during their rise times. The experimental data which were acquired by DAQ system are divided into three parts: the acquired data before the charge carrier creation, the acquired data during the charge carriers drift from their creation position to each 6
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Fig. 11. π 2 values and Position distribution after the rise time correction along β¨100β© crystal axis.
Fig. 12. Probability in percentage over the difference between the located positions without any and rise time correction along β¨100β© crystal axis(left); That between after rise time correction along β¨100β© and β¨110β© crystal axis(right).
electrode and the acquired data after the charge carrier absorption. The pulse shape data during its rise time were separated from the experimental data by cutting out them under and over the specific thresholds. These taken data were considered as the data before creation and after absorption, respectively. The best matched pulse shape by the least π 2 fitting were saved as a png file for main and transient pulse shapes with its interaction position on x, y and z axis. The π 2 fitting equation is modified for escaping the infinity and undefined error in Maxima code as shown in Eqs. (2) and (3). 2 πππππ =
2 ππ‘πππ =
πβ1 β (πΈππππ,π β πππππ,π )2 π=2 πβ1 β 4 β
πππππ,π (πΈπ‘ππππ ,π β ππ‘ππππ ,π )2
indexed as the 1, 2, 3 and 4 in the j index. The first and last data points in the i index are neglected for escaping ββundefined errorββ of Maxima code in both main and transient π 2 fitting equations. The denominator of π 2 fitting equation is removed in the transient π 2 fitting equation for escaping ββinfinity errorββ in Maxima code. 2.6. Correction of experimental pulse shapes by rise time The pre-amplifier intrinsic rise time was evaluated by inputting the testing square pulse to test input node and measuring the pulse shape of A3 output node. The pre-amplifier was connected to biased 12 voltage but the detector was not. The input square pulse shape had its rise time 9.6 ns and the pre-amplifier output pulse shape showed its rise time 38.4 ns. The pre-amplifier intrinsic rise time was calculated by subtracting quadratically the input rise time from the output rise time as 37.2 ns. The pre-amplifier intrinsic rise time 37.2 ns were applied to the experimental main and transient ones by re-sampling these after considering the extended rise time as shown in Fig. 6.
(2)
(3)
π=2 π=1
Here, πΈππππ and πππππ are the experimental and simulated main pulse shape data points. πΈπ‘πππ and ππ‘πππ are the transient pulse shape data 2 2 points. πππππ and ππ‘πππ are π 2 fitting value for the main and transient pulse shape data points. The 4 different transient pulse shape data points for up, down, left and right segments next to main segment are 7
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Fig. 13. Locating test46 pulse shape at A3 for non, rise time correction along β¨100β© and β¨110β© (top left); non correction(top right), rise time correction along β¨100β©(bottom left) and β¨110β©(bottom right) for transient pulse shapes. Table 1 Parameter values of equation 4 for electron(E) and hole(H) along each crystal axis.
a b c d
E@β¨100β©
H@β¨100β©
E@β¨100β©-β¨110β©
H@β¨100β©-β¨110β©
3 212 421.866 β0.1055632 646.842869 β12 488 348.55
2 550 414.084 0.04107491 β728.99177 β8 335 162.455
3 212 421.866 0 0 12 488 348.55
2 550 414.084 0.085 β1050 β8 335 162.455
3.2. Experimental signals The experimental main and transient signals at p1 were plotted in Fig. 9. The signals from DAQ system had a minus amplitude because of its electronics. The experimental and simulated pulse shapes can be compared by multiplying a minus one to the experimental signals. 3.3. Locating events
2.7. Correction of simulated pulse shapes by anisotropy
Experimental and simulated pulse shapes at p1 were compared by the modified π 2 equation in Eqs. (2) and (3) as shown in Fig. 13. The experimental pulse shape Test46 without any correction and rise time correction along β¨100β© and β¨110β© was fitted with the simulated ones as shown in Fig. 13. The main and transient pulse shapes without any correction were located at x33y10z52 position. Two ones after rise time correction along β¨100β© and β¨110β© were located at x33y9z53 and x32y10z53 position, respectively.
The drift velocity π£100 and π£111 along the β¨100β© and β¨111β© crystal axis was defined as the parametric equation in Eq. (1). The modified drift velocity π£β²100 along the β¨100β© crystal axis was defined by the combination of the log and quadratic equation as shown in Eq. (4). The relative change between π£100 and π£β²100 was defined in Eq. (5). π£β²100 = ππππ(πΈ) + ππΈ 2 + ππΈ + π π .πΆ =
π£β²100 β π£100 π£100
(4)
3.4. Statistical analysis of located events
(5)
The 88 experimental main and transient signal sets were compared with the 20,257 simulated ones which are generated in the A3 segment. The Maxima code located 85 signals but the other 3 signals were not. These 3 signals were undershot at the beginning of its rise time or overshot at the end of it. The maxima code selects the pulse shape data during its rise time using the base line data but these over and undershot signals are failed to be selected as the proper ones. The π 2 values for main and transient pulse shapes without any correction were distributed by their frequencies at the top right and left of Fig. 10. These located signals were distributed on the x-, y- and π§-axis with their positions with their frequencies in the bottom left of Fig. 10. Gaussian fitted mean and standard deviation values were 34.23 Β± 1.76, 6.12 Β± 2.95 and 52.93 Β± 3.22 mm. The Full Width at Half Maximum (FWHM) was defined as 2.355 multiplied by the standard deviation of the Gaussian distribution and evaluated as the 4.14, 6.95 and 7.58 mm with 3, 6 and 6 mm collimator sizes. FWHMs after the rise time correction along β¨100β© crystal axis were 3.11, 4.38 and 8.69 mm on x, y and z axis with 3, 6 and 6 mm collimator sizes. The located events were plotted on three-dimensional(3D) x-y-z spaces with the target region as shown in a 7.8 Γ 8.4 Γ 8.4 mm cube in the bottom right of Figs. 10 and 11. The original target region was 3 Γ 6 Γ 6 mm cube by the collimator size but this region was spread by the distance between the target region and the source and collimator. The target region and located events were plotted on the axis origin for simplicity.
Here, a is the parameter for natural log equation. The b, c and d are the parameters of the quadratic equation. The drift velocity along the β¨100β© and one between β¨100β© and β¨110β© axis was expressed using the modified parametric equation with parameter values for electron and hole in Table 1. R.C represents Relative Change between the π£100 and π£β²100 drift velocities along β¨100β© crystal axis. The relative change were plotted in percentage over the electric field strength from 500 to 5000 V/cm for electron and hole as shown in the left of Fig. 7. The drift velocity along one between β¨100β© and β¨110β© crystal axis was calculated by the π£β²100,110 with parameters in table as shown in the right of Fig. 7, respectively. 3. Result 3.1. Rise time distribution of experimental and simulated pulse shapes in A3 segment Rise time distribution of experimental pulse shapes can be compared with that of simulated ones at A3 segment. Probability density of simulated pulse shape rise times can be calculated over whole region of A3 segment assuming evenly distributed interaction positions. That of experimental ones was obtained by locating the 137 Cs source at the A3 segment surface. Probability densities of experimental and simulated ones were depicted in Fig. 8. That of experimental ones were more probable at the short rise time than that of long one. 8
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4. Conclusion
References
It was about 90% within 2 mm on each axis that the probability of difference between without any and rise time correction along β¨100β© crystal axis as shown in the left of Fig. 12. The rise time correction improves the position resolution about 2 mm on each axis. The difference between along β¨100β© and β¨110β© was about 1 mm within 90% probability on x and z axis as shown in the right of Fig. 12. The position resolution between them was improved about 1 mm on x and z axis by choosing β¨100β© than β¨110β© crystal axis. The pulse shape along β¨110β© crystal axis extended its rise time than that of β¨100β© but π 2 value increased, also. The A3 segment has simple geometry and is affected by two β¨100β© and β¨110β© but β¨111β© crystal axis. The difference of drift velocities was 0.01 mm/ns at 5000 V/cm for electron along β¨100β© and β¨110β© crystal axis. That was 0.06 mm/ns at the same condition along β¨100β© and β¨111β© crystal axis.. These collimated pulse shapes can be assumed to be generated along one of two. The cross talk was neglected here because All triggered events were hit once and only the segment signals were used.
[1] Benoit Pirard, Segmented coaxial HPGe detector EGC100-SEG24 userβs manual, CANBERRA, 2014. [2] Zhong He, Review of the Shockley-Ramo theorem and its application in semiconductor πΎ-ray detectors, Nucl. Instrum. Methods A 463 (2001) 250β267. [3] Bart Bruyneel, Peter Reiter, Gheorghe Pascovici, Characterization of large volume HPGe detectors. Part I: Electron and hole mobility parameterization, Nucl. Instrum. Methods A 569 (2006) 764β773. [4] Jubong Han, K.B. Lee, Jong-Man Lee, S.H. Lee, Tae Soon Park, J.S. Oh, Performance of a segmented HPGe detector at KRISS, Appl. Raidat. Isot 134 (2018) 177β181. [5] G.F. Knoll, Radiation of Detection and Measurement, third ed., Wiley, New York, 1999. [6] L. Mihailescu, W. Gast, R. Lieder, H. Brands, H. Jager, The influence of anisotropic electron drift velocity on the signal shapes of closed-end HPGe detectors, Nucl. Instrum. Methods A 447 (2000) 350β360. [7] L. Reggiani, C. Canali, F. Nava, G. Ottaviani, Hole drift velocity in germanium, Phys. Rev. B 16 (1977) 2781.
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