Pulse-shape calculations and applications using the AGATAGeFEM software package
EEPJ manuscript No. (will be inserted by the editor)
Pulse-shape calculations and applications using theAGATAGeFEM software package
J. Ljungvall Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
Abstract.
A software package for modeling segmented High-Purity Segmented Germanium detectors,AGATAGeFEM, is presented. The choices made for geometry implementation and the calculations ofthe electric and weighting fields are discussed. Models used for charge-carrier velocities are described. Nu-merical integration of the charge-carrier transport equation is explained. Impact of noise and crosstalk onthe achieved position resolution in AGATA detectors are investigated. The results suggest that crosstalkas seen in the AGATA detectors is of minor importance for the position resolution. The sensitivity of thepulse shapes to the parameters in the pulse-shape calculations is determined, this as a function of positionin the detectors. Finally, AGATAGeFEM has been used to produce pulse-shape data bases for pulse-shapeanalyses of experimental data. The results with the new data base indicate improvement with respect tothose with the standard AGATA data base.
PACS. γ -ray spectroscopy In nuclear-structure physics, as in many other fields of re-search, the development of better instrumentation and thediscoveries of new physics are closely linked. One of themost powerful techniques to study atomic nuclei is γ -rayspectroscopy, often combined with in-beam production ofthe species of interest. The development and constructionof new advanced radioactive-beam and stable-beam facili-ties has prompted development and construction of a newgeneration of γ -ray spectrometers. The last before presentgeneration of γ -ray spectrometers, such as EUROBALL[1] and GAMMASPHERE [2], have limitations in termsof efficiency, resolving power, and maximum count-ratecapabilities. A way to improve the performance is to usefully digital electronics combined with highly segmentedHigh-Purity Germanium detectors and perform so-called γ -ray tracking [3]. Two large projects have been developedand are presently in the construction phase. In Europethe Advanced GAmma-ray Tracking Array (AGATA) [4]and in the USA the Gamma-Ray Energy Tracking Array(GRETA) [5]. For a more complete discussion on γ -rayspectrometers and their development over time see, e.g.,Eberth and Simpson [6]. A recent discussion on the cur-rent performance of AGATA can be found in Ljungvall etal. [7].The main characteristic of γ -ray tracking spectrom-eters is the lack of Compton suppression shields. Theyhave been replaced by γ -ray tracking: The track of the γ -ray through the array is reconstructed using interactionpoints given by pulse-shape analysis (PSA). The trackingallows an increase of the solid angle covered by germa- nium, with an increase in efficiency. The γ -ray trackingalso gives a high peak-to-total. For this method to givehigh performance it is crucial that the PSA gives the in-teraction positions with high accuracy (i.e. less than 5 mmFWHM at 1 MeV). The accuracy of the PSA depends onhow well the detector response is known, and to a lesser de-gree, the algorithm used for the PSA. Characterisation ofthe HPGe detectors, and associated electronics, are henceof great importance.Extensive work has been done to model the responseof the highly segmented germanium detectors used byAGATA and GRETA. Over the years several softwarepackages have been developed within the AGATA col-laboration to model the pulse shapes from segmented de-tectors [8,9,10,11,12]. At the same time extensive effortshave been made to provide these codes with accurate inputin terms of impurity concentrations [13,14], crosstalk andelectronics response [15,16,17], and charge-carrier mobil-ity [18]. Within the AGATA community it is presentlythe ADL [11,12] package that is used to produce the databases of pulse shapes needed for pulse-shape analysis. Forwork related to GRETA and GRETINA see [19,20] andreferences therein.In this paper a software package written to model thesegmented HPGe germanium detectors used in AGATAis described. The main intended use of this package isPSA development. It can however also be used for detectorcharacterization or pulse-shape data bases production. Ashort introduction to pulse-shape calculations in semicon-ductor detectors is given in section 2. The AGATAGeFEMpackage together with models and assumptions made are a r X i v : . [ phy s i c s . i n s - d e t ] J a n J. Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package described in section 3. An effort to characterize wherein the detector volume the parameters entering in pulse-shape calculations have the largest influence on the pulseshapes is presented in 4. How a poorly known crystal ge-ometry might effect pulse-shape analysis is investigatedin section 5. The use of the software package to bench-mark the effect of crosstalk and noise on the two pulse-shape algorithms Extensive Grid Search (EGS) and Sin-gular Value Decomposition (SVD) is covered in section 6.To validate the pulse shapes calculated by the AGATAGe-FEM package pulse-shape data bases used for PSA havebeen calculated with it and used for PSA. Results usingthe AGATAGeFEM bases are compared to results pro-duced with bases calculated with the ADL [11,12]. This ispresented in section 7. Conclusions are given in section 8.
The signal (referred to as "pulse shapes" in this work)generation in all detectors based on the motion and col-lection of charge carriers is calculated using the Shockley-Ramo theorem [21,22]. The theorem states that the in-duced charge on an electrode due to moving charges is dQ ( t ) dt = e (cid:104) N h (cid:126)v h ( (cid:126)r h ) · (cid:126)W ( (cid:126)r h ) − N e (cid:126)v e ( (cid:126)r e ) · (cid:126)W ( (cid:126)r e ) (cid:105) , (1)where (cid:126)W ( (cid:126)r e,h ) = −∇ Φ W ( (cid:126)r e,h ) is the weighting field, N e,h the number of charge carriers for electrons and holes,respectively, and (cid:126)v e,h ( (cid:126)r e,h ) are the charge-carrier veloc-ities. The charge carrier velocities are functions of theelectric field (cid:126)E ( (cid:126)r ) . The electric field is calculated from theelectric potential as (cid:126)E ( (cid:126)r ) = −∇ Φ ( (cid:126)r ) .The calculation of pulse shapes for a semiconductordetector begins with solving the two partial differentialequations (PDE) ∇ Φ ( (cid:126)r ) = − ρ ( (cid:126)r ) (cid:15) Ge (2)and ∇ Φ W ( (cid:126)r ) = 0 (3)known as the Poisson and Laplace equations, respec-tively. Together with appropriate boundary conditions theydescribe the electric (Poisson) and weighting (Laplace) po-tentials. In equation 2, ρ ( (cid:126)r ) is the free charge distributionin the detector and (cid:15) Ge the dielectric constant for ger-manium. Boundary conditions for the electric potentialare the applied detector bias or 0 V on the conductingsurfaces. For surfaces requiring passivation the boundarycondition should include charges and possible leakage cur-rents. As these are unknown the approximation of naturalboundary conditions is used in this work. The weighting potential Φ W ( (cid:126)r ) is calculated by setting the potential onthe electrode of interest to 1 and to 0 on all other con-ducting surfaces.One of the major difficulties when integrating equa-tion 1 is to find the correct function for the charge-carriervelocities (cid:126)v e,h ( (cid:126)r e,h ) . The physics and the models used inthis work is discussed in section 2.1.To produce realistic pulse shapes for a detector theeffects of the limited bandwidth of the electronics haveto be included. The effect of the bandwidth can in prin-ciple be measured for a system using a pulse generator.However, the approximation using an analytical functionseems sufficient and is more practical. The crosstalk be-tween different segments has to be modeled. Typically thedifference is made between so-called linear and differentialcrosstalks. The former has mainly the capacitive couplingof the electrodes as origin whereas the later has its ori-gin in the front-end electronics. The linear crosstalk cannot be avoided, whereas the differential crosstalk can bereduced using careful engineering. The AGATAGeFEMpackage models both types of crosstalk. For all pulse-shape calculations the models used to de-scribe the charge-carrier mobilities are of crucial impor-tance. A commonly used function [23] to describe thecharge-carrier velocity is (cid:126)v ( (cid:126)r ) = µ (cid:126)E ( (cid:126)r ) (cid:16) E ( (cid:126)r ) /E ) β (cid:17) /β − µ n (cid:126)E ( (cid:126)r ) , (4)where E , β , µ n , and µ are experimentally adjusted pa-rameters. This parameterization is valid when the electricfield is parallel to one of the symmetry axes of the Ger-manium crystal (<100>, <110>, or <111>). The charge-carrier velocity is only parallel to the electric field whenthe electric field is parallel to a symmetry axes. For germa-nium crystals cooled down to liquid nitrogen temperatures( ≈ − ◦ Celsius) several models have been developed forthe electron mobilities. For AGATAGeFEM the model ofNathan [24] is used. It treats the anisotropy of the electrondrift velocity observed in germanium with high accuracywith the formalism described in Mihailescu et al. [25].For the hole mobility B. Bruyneel et al. [18] have de-veloped a model based on the so-called "streaming mo-tion" concept. The holes are accelerated to a thresholdenergy, they emit an optical phonon losing most of theirenergy, are re-accelerated in the applied electric field tothe threshold energy, and so on. In this work a differ-ent approach has been used. Here it is assumed that thevariation in carrier velocity as a function of the electricfield can be described by the fraction of holes populat-ing the light-hole band and the heavy-hole band and afield dependent relaxation time. The anisotropy is givenby the effective masses the second derivative of the en-ergy of the hole bands. Despite the much higher the much . Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package 3 higher energy of the light-hole band as compared to theheavy-hole band the model reproduces experimental datafor hole drift velocities [26]. For the holes the surfaces ofequal energy in the conduction bands are not ellipsoids,which means that the reciprocal effective mass tensor willdepend on the direction of the wavevector (cid:126)k . Here the as-sumption is made that the wavevector is parallel to theapplied electric field. The hole energy functions read [27] (cid:15) h ( k ) = Ak ± (cid:2) B k + C (cid:0) k x k y + k y k z + k z k x (cid:1)(cid:3) / , (5)where the positive (negative) sign is for the light (heavy)-hole band. Using equation (cid:18) m ∗ (cid:19) µν = 1 (cid:126) d (cid:15) ( k ) dk µ dk ν ≡ ¯¯ Γ (6)to calculate the reciprocal effective mass tensor, we have (cid:126)v h = q T ( E ) (cid:104) F ( E ) ¯¯ Γ heavyh + (1 − F ( E )) ¯¯ Γ lighth (cid:105) (cid:126)E. (7)Comparing equation 7 with (cid:126)v = qt ¯¯ Γ (cid:126)E, (8)the factor T ( E ) in the latter equation corresponds to t inthe former and is considered an electric-field dependentrelaxation time. F ( E ) is the fraction of the holes mov-ing in the heavy-hole band and is also field dependent.As in the case of electrons, equation 4 can now be usedto calculate the hole drift velocities in the <100> and<111> directions. Using these velocities T ( E ) and F ( E ) are obtained at the electric-field strength in question. Abig difference for holes as compared to electrons is thatthe reciprocal effective mass tensor changes with the di-rection of the electric field. In figure 1 the anisotropy ofcharge carrier transport is illustrated. Without anisotropythe top row would show perfect spheres in one color andthe second and third rows would show zero velocities. Thedeficiency of the model can be noted from the v hϕ shown inthe bottom right corner. There should be no anisotropy inany of the <100>, <110>, or <111> directions as theseare symmetry axes in germanium. This is different for theelectrons shown in the left column.Neither the drift model for electrons nor for holes takesinto account the effects of crystal temperature or impu-rity concentrations on the charge-carrier drift velocitiesalthough these effects modify the drift velocities [28]. Theeffect of varying the hole-drift velocity was studied withinthe GRETINA [20] and AGATA collaborations [29]. Inthese works it is concluded that the position resolution isnot limited by the hole mobility models. When the holemobility is varied within reasonable limits it is difficultto separate the effects from those coming from the front-end electronics. Numerical values used in this work for themobility parameters are presented in table 2. Several software packages have been developed to calcu-late pulse shapes from the High-Purity Germanium detec- v er x yz -120000 -90000 v hr x yz 70000 77000 84000v e θ x yz -30000 0 30000 v h θ x yz -15000 0 15000v e φ x yz -30000 0 30000 v h φ x yz -10000 0 10000 Fig. 1.
The left (right) column shows the charge-carrier ve-locity as a function of the direction of the electric field forelectrons (holes). The three rows show the ˆ r, ˆ θ, ˆ φ componentsof the velocity, respectively. tors used in AGATA. Examples from the AGATA collab-oration are MGS [30,10], JASS [9,9,17] and ADL [11,18,12]. Although differing in details they all have in commonthe use of finite difference PDEs solvers for the electricfield and the weighting potentials in the detector. How-ever, the complex shapes of the AGATA crystals are notwell reproduced using a finite difference scheme with rect-angular grids. This is a problem that can be circumventedusing finite element methods (FEM). Another strong pointof FEM is that the solution is an approximation of a func-tion describing the electric field and not the electric fieldat certain points. This removes the need to interpolatebetween grid points as the solution is defined on the en-tire volume of the detector. It is beyond the scope of thepresent work to describe FEM and the reader is referredto [31] and references therein.The AGATAGeFEM package written in C++, useshigh-quality open-source FEM software to calculate the J. Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package electric and weighting potentials of AGATA type germa-nium detectors. For the charge-carrier transport the or-dinary differential equation solvers of the Gnu ScientificLibrary [32] are used. The geometry is described to ma-chine precision for charge transport and mesh generation.Earlier versions of the program used mainly the FEM li-brary dealii [33,34]. This is a very flexible code that al-lows an iterative refinement of the FEM mesh in a verysimple way. However, the mesh cell geometry is limitedto quadrilaterals and hexahedra. From the point of viewof solving the partial differential equations this is a goodchoice. However, in AGATAGeFEM the solutions of thePoisson and Laplace equations are not projected down toa regular grid when used in the charge carrier-transportprocess. This is also the case for calculations of the inducedsignals via the Shockley-Ramo theorem. The idea Behindis that the mesh refinement procedure tells where a highgranularity is needed and all projection to a regular griddeteriorates this information. The problem is then to findthe correct cell in an irregular mesh. The curved bound-aries of hexahedra cells make these calculations compli-cated. As a result the first version of AGATAGeFEM wascapable of calculating about 2-3 pulse shapes/s. Sufficientto calculate a basis for PSA it is far from enough for usingAGATAGeFEM in the fitting of parameters in the pulse-shape calculations or to use it in a complete Monte Carlosimulation chain. The FEM part of the program was there-fore changed to the libmesh library [35]. It uses tetrahedrawith each side defined by three points, speeding up find-ing the correct mesh cell. The AGATAGeFEM is furthermore restrained to the use of only linear bases functionsin the solution. This way the calculations are a factor ofalmost 100 faster while reproducing while reproducing theresults using dealii.AGATAGeFEM is fully parallelized with threads andthe MPI interface. This applies to the field calculationsand the pulse-shape calculations. Parallelism is also usedwhile fitting the pulse-shape calculations parameters. Tominimize the χ Minuit and Minuit2 [36] are employed.AGATAGeFEM further has interfaces allowing calculat-ing and displaying fields and pulses from the ROOT [37]interpreter interface. This inside the chosen detector ge-ometry if wanted. It has further a very simple server clientmechanism allowing other programs to ask the server tocalculate pulse shapes for it.The AGATAGeFEM package also contains miscella-neous codes for – applying pre-amplifier response – crosstalk – re-sample pulse shapes – compare pulse shapes – calculate pulse shapes from the output of the AGATAgeant4 MC [38] – create data bases for PSA. The AGATA crystals are mm long and have an diame-ter of mm. They were produced in four different shapes.A symmetric hexagonal shape for three prototypes andthree different non-symmetric hexagonal shapes for usein the AGATA [4]. For the generation of the FEM meshOpenCASCADE models of the detectors are generated.For the charge transportation the detector geometries areimplemented in C++ as the union of a cylinder and sixplanes or using the CSG geometry of geant4 [39]. The holecorresponding to the central contact has several parame-ters allowing it shape and orientation to be varied. Theseare the radius of the hole, the radius at the bottom of thehole joining the side of the bore hole with the bottom ofit, and translation and rotation of the axes of the hole.The two different geometrical models of the detectors areequivalent. Examples of the geometries are shown in figure2. XYZ XYZ
XY Z
Fig. 2.
Top row, three views of an "A" type AGATA crystal.Bottom left shows a symmetric crystal. Bottom right, a "C"type AGATA crystal with half the volume hidden to show thecentral contact.
AGATAGeFEM uses a total of 40 fields for calculatingthe pulse shapes. Thirty seven of these are the weightingfields for the 36 segments and the central contact. Exceptfor the central contact which is trivial, these are definedeither using the limiting depth values and start and stopangles of a segment or using the intersection between thedetector surface and four planes. The segments do nothave to cover the entire surface of the detector. This is . Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package 5 intended for modeling the gaps between the segment con-tacts on the outer surface [40]. Presently no implementa-tion of suitable boundary conditions for the electric fieldexists in AGATAGeFEM limiting the value of this option.When solving the charge transport equations, AGATAGe-FEM uses three fields to calculate the electric field. Thefirst one is the solution to the Poisson equation with V onthe surface of the detector and V bias V on the central con-tact, and the nominal impurity concentration. The choiceto include charge impurities here was made to maximizethe benefits of mesh refinement. Representative values forthe charge impurities are presented in table 3 in appendixA. The second field is the solution of the Poisson equationassuming V on both the surface and the central contactbut with an impurity contribution of at the front of thedetector that decreases linearly as a function of depth tothe back of the detector where it is . The third and fi-nal field is like the second one but reversing the slope ofthe impurity concentration. The use of three fields allowsvarying the effective impurity concentration and its effecton the electric field in the detector without recalculat-ing the electric field. Solving the Poisson equation whenvarying the impurity concentration is too computation-ally intensive to allow the fitting of detector parametersto experimental signals.For solving the Laplace equation and the Poisson equa-tion AGATAGeFEM uses the libmesh library [35], withthe meshes generated with gmsh [41]. As a first step theLaplace or the Poisson equation is solved using a uniformmesh with an average cell size of 2 mm. This solution isthen used to estimate the largest acceptable geometricalapproximation of the mesh in order to ensure an error onthe field of less than one per mil of the maximum valueof the field. This step is then followed by repeated stepsof refinement of the mesh based on an estimate of the lo-cal error of the solution [42] until the field is described byat least . ∗ degrees of freedom. A limit that gives agood approximation of the fields (see section 7). The finalstep for each of the 40 fields is to create look-up tables ona 2x2x2 mm grid over the detector volume to allow fastaccess to the correct mesh cell when evaluating the fields. The detector pulses are calculated by first transportingthe point representing electrons and the point representingholes from the point of the γ -ray interaction until theyreach the boundary of the detector volume using d (cid:126)r e,h dt = (cid:126)v e,h (cid:16) (cid:126)E (cid:17) . (9)The equations are solved separately for the holes and theelectrons using an solver algorithm with an adaptive timestep. AGATAGeFEM allows the user to choose betweenany of the possible algorithms provided by GSL, but thedefault choice it the embedded Runge-Kutta Prince-Dormandmethod [43]. The paths of the charge carriers are calcu-lated with an adaptive time step and sampled at a cho-sen frequency, by default 100 MHz. As the charge carriers approach the boundary of the detector the sampling fre-quency is adapted to allow an accurate description of thepulse shape as a ns time step typically gives a paththat ends outside the detector.In the next step the charge on electrode i is calculatedusing Q i ( t ) = q (cid:0) Φ iW ( (cid:126)r e ( t )) − Φ iW ( (cid:126)r h ( t )) (cid:1) (10)for all signals. The signals can be convoluted with the response of theelectronics. This is not done when generating a basis forPSA. In this work the response of the electronics havebeen modeled by the function [8,9] shown in figure 3. Aconvolution (in time-domain) is made for all 37 calculatedsignals. A possible improvement is to use a function witha slower rise time for the central contact. However, theconvolution is mainly used for PSA development whereAGATAGeFEM is used both for generating the bases andthe signals that are analyzed so the exact form of thefunction is of minor importance. A m p lit ud e Time [ µ s]Transfer function Fig. 3.
Transfer function of the electronics shown in timedomain.
The effect of both linear and differential crosstalk canbe included in the transfer function. For in-depth discus-sion concerning crosstalk in segmented germanium detec-tors, see [15,44]. An example of the signals calculated withand without response function is given in figure 4. The ef-fects of linear and derivative crosstalks are also shown.
To understand the influence of the different detector pa-rameters on the shape of the signals the sensitivity of thepulse shapes to each parameter was calculated for a largenumber of points inside a detector of symmetric type. As
J. Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package P u l s e h e i gh t [ a . u . ] Time index [10 ns]Hit segment-1400-1000-600-200 0 10 20 30 40 50 P u l s e h e i gh t [ a . u . ] Time index [10 ns]Nearest neighbour
Fig. 4.
Left: Example of a front segment weighting field.Shown is also the mesh used for solving the Poisson equation.Right: Examples of net-charge signals and transient signalswith and without convolution with the transfer function. Themodulating effect of the response of the electronics is clearlyseen (red shape). The effect of linear (green shape) and deriva-tive (blue shape) crosstalk is also shown. the absolute value of the different parameters vary overmany orders of magnitude normalized dimensionless pa-rameters was used to estimate this sensitivity. The sensi-tivity was evaluated as the second derivative of a χ atzero fractional variation of the parameter in question. Itis extracted by fitting a second degree polynomial and ishence the curvature of the χ function. The χ is calcu-lated using the original pulse shape and a pulse shapecalculated using the changed parameter. In figure 5 theextraction of the sensitivity is shown for the hole mobilityin the <111> direction. " χ " Δ µ h<111> / µ h<111> Second degree polynomial fi t Fig. 5.
Example of how the sensitivity of the pulse shapes toa parameter (in this case µ < >h ) is extracted. In figure 6 is shown for how many positions in thedetector the pulse shapes are most sensitive to each pa-rameter. It can be seen that at most points it is the param-eters that control the velocity of the charge carriers in the<100> direction that are dominating. It is not surprisingas in the coaxial part of the detector the charge transportis never in a <111> direction. Worth noting is that thehole mobility parameters for the <111> direction have a large impact at more positions than the <111> directionparameters for the electrons. This can also be understoodgeometrically as the paths close to the <111> directionwith an overlapping large weighting potential are domi-nated by hole transport occurring at the corners of thefront face of the detector. Crystal orientation only domi-nates at one position, a consequence of the definition andits evaluation at zero change. µ e < > β e < > µ h < > β h < > E e < > E < > µ h < > β h < > E < > µ e < > E e < > β e < > C r y s t a l N u m b e r o f po i n t s Fig. 6.
Number of points in the crystal where a parameterhas the largest influence on the pulse shapes.
Figure 7 shows the average sensitivity of the pulseshapes to a parameter at the positions dominated by theparameter. One can notice that the highest average sensi-tivity is found for the crystal orientation followed by theparameterization of the hole mobility in the <111> di-rection. It can be understood as these parameters are themost likely to change in which segment the charges arecollected. C r y s t a l E < > µ h < > β h < > E < > β h < > µ h < > β e < > E e < > E e < > β e < > µ e < > µ e < > A vg . c u r v a t u r e o f s qu a r e s u m Fig. 7.
The sensitivity of pulse shapes to the parameters usedin pulse-shape calculations averaged over the points where thepulse shapes are most sensitive to the respective parameter.
In figure 8 the relative sensitivity as a function of po-sition in the detector volume is shown for the µ < >e pa-rameter. It is homogeneous inside the volume althoughthe projection on the XY plane shows that, apart fromthe volume effect, there is an increase in sensitivity closeto the <100> directions. This is expected as the charge . Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package 7 carrier velocity mainly depends on parameters for thatcrystal axes at these positions. Fig. 8.
The sensitivity of the pulse shapes to the µ < >,< >e,h parameters as a function of position. The size of the cubes areproportional to the sensitivity. Note that the cube sizes are notcomparable between the figures. Top left: µ < >e Top right: µ < >h Bottom row left: µ < >e Bottom right: µ < >h A similar pattern can be seen for the µ < >h parameterin figure 8, but with the maximum shifted towards lowerradii corresponding to pulses in which the hole drift con-tributes more to the pulse shapes. The situation is differ-ent for the parameters µ < >e and µ < >h , also shown infigure 8. For the electrons the pattern is easily understood,i.e. parameters concerning the <111> direction show sen-sitivity in the region where charge transport is parallel tothe <111> direction.For the holes the pattern is not reflecting the <111>direction in the crystal. This is not imposed by the model,unlike for the model of the electrons. When the electricfield is parallel to a symmetry axis of the crystal thecharge carriers move, by symmetry arguments, parallel tothe field and the axis. This can also be seen in figure 1where the ϕ component of the hole velocity is non zeroin the xy-plane <100> directions. Imposing this symme-try on the hole velocity model is planned for future work, but as shown in section 7 this deficiency does not seem todegrade the results. As the exact geometry (here considering contact thicknessdead layers etc as a part of the geometry) is imperfectlyknown it is interesting to investigate its possible impacton the pulse shapes. A different geometry was used togenerate the basis for PSA. The influence of an imperfectgeometry has been investigated for three different cases,ranging from an extreme (unrealistic) case to a small erroron the used front-face segmentation. a, b, c,
Fig. 9.
The three different geometries used for investigatingthe impact of an imperfect geometry on the pulse-shape databases. In picture a, the nominal geometry of a capsule type Ais shown. In b the bore hole for the central contact has beendisplace 5 mm and with an angle of φ = . radians and θ = . radians, respectively. In figure c the bore hole has been enlargedwith 3 mm, this as an (extreme) example of the lithium driftedcentral contact. Correct 0 0.2 0.4 0.6 0.8 1 W e i gh ti ng po t e n ti a l [ V ] Wrong 0 0.2 0.4 0.6 0.8 1 W e i gh ti ng po t e n ti a l [ V ] Fig. 10.
The difference between the correct (left) and "naive"(right) segmentation.
Using the nominal geometry of an A type crystal with arepresentative impurity concentration a pulse-shape basisfor PSA with a grid size of 1x1x1 mm and a sample rate of100 MHz was first calculated. Using three differently mod-ified geometries the same 1x1x1 mm grid of points were J. Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package used to calculate pulse shapes for each of those. The fol-lowing geometry modifications were used: 1) An incorrectfront-face segmentation. 2) A displaced and tilted boreholefor the central contact. 3) An enlarged bore hole for thecentral contact. The nominal geometry and geometries 2and 3 are shown in figure 9 whereas the difference in frontsegmentation is shown in figure 10. The pulse-shape ba-sis of the nominal geometry was then used for PSA onthe three different pulse-shape sets corresponding to eachgeometry variation and on itself as a reference. The usedPSA is a simple extensive grid search with 5 keV Gaussiannoise added and an interaction energy of 1 MeV. In figure11 the χ for the four different cases are shown. It can beseen that for none of the geometries it is possible to usethe χ distribution to make a statement on whether thegeometry used to produce the basis is good or bad - allfour distributions are reasonable. C oun t s χ On it selfWrong segmentationCore R=8 mmCore not correct
Fig. 11.
Chi-square distributions for the four different casesof PSA using one exact and three inexact geometries. The χ rest close to 1 even if the basis used is calculated for a geometrythat does not coincide with the correct one. The average error on the determined positions (figure12 shows this for the x coordinate) is the most importantparameter evaluating the performance of a pulse-shapebasis. The incorrect segmentation lines do not produce anerror that is significant as compared to the experimen-tally determined value ( σ ~1.7 mm [45]). The two othergeometries give an error in the determination of the ac-tual position that is larger than the experimental result.A tentative conclusion is that the geometries of actualAGATA crystals are better known than the two ratherextreme cases used for this test.Looking at scatter plots of the determined interactionpositions, see figure 13, no clustering effects are seen forthe case where PSA is done on pulses belonging to thereference basis (i.e. on itself) nor with a small error onthe front segmentation. However, when the basis is madewith a geometry that deviates from the geometry of thedetector, one of the effects is clustering of events. Theorigin of this clustering is that the rise times contain mostof the information in the pulse shapes and all pulses withextreme rise times will be clustered towards the positionin the basis with the closest rise time. The empty voids area combination of this rise-time mismatch and the union of
100 1000 10000 100000 1x10 -10 -5 0 5 10 C oun t s / mm Δ XOn itself, σ =1.3 mmWrong segmentation, σ =1.4 mmCore R=8 mm, σ =2.9 mmCore not correct, σ =3.2 mm Fig. 12.
The difference between the position given by PSAas compared to the position in the crystal for the calculatedsignal. This for the X coordinate. The other coordinates aresimilar. In all four cases the PSA used the "nominal" geometrybasis. For details see text. the central contacts of the nominal geometry and that ofthe two variations of the bore hole geometry.
Using the AGATAGeFEM package the resolution for Ex-tensive Grid Search (EGS) and the Singular Value Decom-position matrix inversion (SVD) [46] as a function of noiselevel and the inclusion of differential and linear crosstalkhas been investigated. The amount of linear crosstalk usedfor this investigation is typical for AGATA crystals whenmounted in AGATA triple cryostats [15,16] and is aboutone per mil. The differential crosstalk is assumed propor-tional to the linear crosstalk with the proportionality fac-tor taken as the one used in the AGATA online PSA (afactor of 10).Assuming that the physics of a segmented germaniumdetector is well known, the possibility to determine thecoordinates of a γ -ray interaction in a large volume HPGedetector is limited by the knowledge of the response ofthe electronics and by the signal-to-noise ratio. These twoaspects have been studied by performing PSA on a dataset of pulse shapes calculated using the AGATAGeFEMwith exactly the same parameters as the basis used by thePSA code. Each pulse shape in the data set was analysedusing 6 different levels of noise and using EGS and SVD.This with or without linear and differential crosstalk fora total of 48 different combinations. Each pulse was ana-lyzed 20 times with noise regenerated for each time. Forboth PSA methods linear and differential crosstalks wereadded to the analyzed pulses but not to the pulse-shapebasis used for the PSA. The results are summarized in ta-ble 1. According to this work the crosstalks have a verylimited influence on the resolution, both for the averagereconstructed position and by not introducing systematicerrors. In figures 14 and 15 two-dimensional projectionsof interaction positions as determined by the two differentPSA algorithms are shown. Looking at figure 14 and 15a striking difference shows up for large noises. The EGS . Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package 9 Fig. 13.
Scatter plots of positions determined with PSA usinga basis calculated with the nominal AGATA A type detector.From left, PSA performed with the basis on itself with noiseadded, PSA performed on signals calculated with an incor-rect front-face segmentation, PSA performed with signals cal-culated using a too large central contact diameter, and finally,to the extreme right, PSA performed using signals calculatedwith a bore hole for the central contact off center and tilted. tends to cluster points towards the segment boundarieswhereas the SVD seems to move the points towards thebarycenter of the segment.Data from Söderström et al. [45] is presented togetherwith the result from present work in figure 17. One notesthat the EGS is better on simulated data than what hasbeen experimentally measured for energies above about50 keV. The Matrix inversion using SVD decompositionto increase the signal-to-noise ratio is better at very lowinteraction energies. It is of interest to try SVD on low-energy experimental data.
One of the objectives of the AGATAGeFEM package isto produce pulse-shapes bases used for the PSA of exper-imental data. In the AGATA collaboration an adaptive
Fig. 14.
Two dimensional projections of positions determinedby EGS on calculated pulse shapes. The level of noise havebeen varied in the interval .6%→12%. Note clustering close tosegment boundaries for low signal-to-noise ratio.
Fig. 15.
Two dimensional projections of positions determinedby SVD on calculated pulse shapes. The level of noise havebeen varied in the interval .6%→12%. Note how the largernoise drives the results towards the center of the segments. grid search algorithm is presently used [49]. The validationof bases calculated with AGATAGeFEM for the AGATAPSA is presented in this section. This step also validatesthe AGATAGeFEM package for use in the development ofpulse-shape analyses by proving that the pulse shapes arerealistic. Pulse-shape data bases have been calculated for6 AGATA crystals. These crystals were previously used toestimate the achieved position resolution [50,7] employingbases calculated with ADL [11,12].An important parameter when calculating the electricfields inside the in fully depleted detectors is the spacecharge. The space charges come from impurities in theGermanium crystals. For the AGATA crystals they havebeen measured by the community using techniques basedon the capacity of the crystal [13,14] and are used as in-put here with numerical values presented in table 3 in ap-pendix A. The parameters for the electron and hole mobil-ity are taken from Ljungvall et al. [26]. They differ slightlyfrom values used in ADL [11,12] and are presented in table2 appendix A.The addition of electronics transfer function and crosstalkto the pulse-shape data bases were performed by the stan-dard AGATA PSA codes. For the crosstalk values mea-
Table 1.
Full width at half maximum of the distributionsinteraction positions for different levels of noise for EGS andSVD PSA on a 1x1x1 mm basis.No crosstalkNoise GS [mm][% rms] ∆ x ∆ y ∆ z0.6 1.3 1.4 1.33.1 2.4 2.5 2.46.1 4.2 4.7 4.212 8.8 10 8.918 13 14 1337 20 19 17SVD [mm] ∆ x ∆ y ∆ z0.6 2.4 2.5 1.83.1 5.5 5.7 4.26.1 7.5 7.8 5.912 10 10 8.418 12 12 1037 16 16 13Full crosstalkNoise GS [mm][% rms] ∆ x ∆ y ∆ z0.6 1.4 1.4 1.33.1 2.4 2.5 2.46.1 4.3 4.7 4.212 8.8 10 8.918 13 14 1337 20 20 17SVD [mm] ∆ x ∆ y ∆ z0.6 2.8 2.8 2.03.1 5.5 5.7 4.36.1 7.4 7.7 5.912 10 10 8.418 12 12 1037 16 16 13 sured during each experiment for each crystal are used (fortypical values see Bruyneel at al. [15,16]). The responsefunction of the electronics is modeled as an exponentialwith a rise time of 35 ns, the default used for PSA inAGATA.To optimize the calculated bases, the parameters con-trolling the direction of the <100> crystal axis, the as-sumed radius of the central contact, and scaling of thehole and electron velocities, were varied. The best resultsfor the PSA was sought in this parameter space for eachcrystal. The orientation of crystal lattice was varied by ro-tating the lattice around the <100> crystal axis assumedparallel to the bore hole for the central contact. Rotationsin steps of 5 degrees until 90 degrees were performed. Forthe bore hole, three different radii were used: 5 mm (nom-inal), 6 mm, and 7mm. The charge carrier velocities werescaled from . to . is steps of . of their nominal val-ues. In figure 18 and 19 the variation of the pulse shapesover the used parameter space is illustrated. As can beseen in figure 18 the lattice orientation has a noticeableimpact on the pulse shapes. The varied parameters with R M S [ mm ] Δ r Δ x Δ y Δ z Fig. 16.
RMS values for different amounts of noise forEGS and SVD PSA on a 1x1x1 mm basis. Results withoutcrosstalk, with only linear crosstalk, or with linear and differ-ential crosstalk added to the test signals are shown. the largest impact are the 10 % step scaling of the charge-carrier velocities, clearly seen in figure 19. This is coherentwith what was shown in section 4. Evaluation of the per-formance of the PSA was done using the peak width of the1221 keV γ -ray transition in Zr, as illustrated in figure20. Doppler correction was performed using first interac-tion point as defined by the γ -ray tracking and the recoilvelocity of the nucleus given by the VAMOS spectrome-ter (for details see Li et al. [50]). An automatic fit pro-cedure was chosen to exclude biases for one or the otherset of bases. The experimental data was processed, withthe exception of choice of pulse-shape data bases, exactlyas described in Ljungvall et al. [7]. For the detectors usedin this work neutron-damage correction was not needed.Similar efforts to optimize the results of PSA using ADLhave recently been published by Lewandowski et al. [29].The reader is cushioned that due to strong correlations be-tween different parameters entering in the calculation ofpulse shapes and in the PSA it is difficult to compare indi-vidual parameters, especially as different figures of meritsare used.A first optimisation varying only the lattice orienta-tion and the central contact hole radius was performed.Pulse-shape analyses were done using a total of 72 differ-ent bases for each crystal followed by γ -ray tracking. Theresulting spectra were fitted by an automatic routine andthe FWHM were extracted for each crystal and for the to-tal spectrum. The results, for each crystal and for the sum . Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package 11 P o s iti on r e s o l u ti on F W H M [ mm ] Energy interaction point [keV]Gridsearch 5keV rms noiseGridsearch 3keV rms noiseMatrix inversion 5keV rms noiseMatrix inversion 3keV rms noise γ -ray 246 keV γ -ray 770 keV γ -ray 1352 keV γ -ray 1826 keV γ -ray 2333 keV γ -ray 3905 keVScan data GSScan data MI Fig. 17.
Position resolution as a function of γ -ray interactionenergy for simulated data, for different γ -ray energies [45], andfor data from the 3D-scanning of the S002 at Liverpool [47,48]using extensive grid search (GS) or SVD (MI). of the crystals, are presented in figure 21. The minimumFWHM is achieved close to the nominal orientation formost of the crystals. However, the minimum of the sum isclose to a lattice rotation of 30 ◦ and with an assumed coreradius of 6 mm. This seems to be driven by crystal A007.The nominal direction of the lattice is 45 ◦ . In figure 21the results obtained with the data bases calculated usingADL are also shown as a reference.A second minimisation was performed on a parameterspace including the three different central contact radii, 5different crystal lattice orientations and 16 different scal-ings of the charge carrier velocities. In figure 22 the varia-tion of the FWHM for crystal A002 is shown for the threedifferent central contact radii and as a function of the scaleof the electron and hole velocities. For each data point acrystal lattice orientation of 45 ◦ was chosen. As can beseen there is a correlation between the central contact ra-dius and the best scale factor for the electron velocity. It isreasonable that it is the electron velocity that can be usedto compensate for changes in the central contact radius asthey are (mainly) responsible for the generation of the sig-nal on the central contact. Furthermore, the change of thecentral contact radius generates the strongest change inthe electric field in the regions where the largest contribu-tion from the electrons to the signal is created. From thisoptimisation it is also clear that different parameters forthe pulse-shape calculations are strongly correlated andthat it is difficult, if not impossible, to determine individ-ual parameters using only the width of a γ -ray peak (orany other single figure of merit).From the second minimisation the bases that producethe smallest FWHM of the 1221 keV peak were chosen foreach crystal. In figure 23 the lowest FWHM for each crys-tal is marked with a circle. It seems as if a slight increase -7 A m p lit ud e [ a . u . ] Time [s]Central contact -1000-800-600-400-200 0 200 400 600 800 1000 1200 0 4x10 -7 A m p lit ud e [ a . u . ] Time [s]Segment F4-1500-1000-500 0 500 1000 1500 0 4x10 -7 A m p lit ud e [ a . u . ] Time [s]Segment A3 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0 4x10 -7 A m p lit ud e [ a . u . ] Time [s]Segment A4 -2500-2000-1500-1000-500 0 500 1000 1500 2000 2500 0 4x10 -7 A m p lit ud e [ a . u . ] Time [s]Segment A5-1000-800-600-400-200 0 200 400 600 800 1000 1200 0 4x10 -7 A m p lit ud e [ a . u . ] Time [s]Segment B4
Fig. 18.
Variation in rise time and shape of transient signalsas a function of crystal lattice angle. The figure shows changesfor the pulse shapes as the crystal lattice is rotated over ◦ for three different radii , R=8 mm (black), 20 mm (grey), and36 mm (dark grey), respectively. of the hole mobilities, reflected by the scaling factor S h ,improves the performance on average. With an exceptionof the C001 crystal. This is not a general statement abouthole mobilities when modeling HPGe detectors but onlyapplies to this work. As for the bore hole radius, 6 mmseems to be preferred with, again, C001 in disagreement.Possible reasons for the different behaviour of the C001crystal will be discussed later in this section.The most important parameter in the adaptive gridsearch PSA algorithm used within the AGATA collabora-tion [49] is the power used to calculate the figure of merit(FOM) for a pulse shape in the basis when compared tothe experimental signal. A minimum is sought for the ex-pression (cid:88) i | y expi − y basei | p (11) -7 A m p lit ud e [ a . u . ] Time [s]Central contact -1000-800-600-400-200 0 200 400 600 800 1000 0 4x10 -7 A m p lit ud e [ a . u . ] Time [s]Segment F4-1500-1000-500 0 500 1000 1500 0 4x10 -7 A m p lit ud e [ a . u . ] Time [s]Segment A3 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0 4x10 -7 A m p lit ud e [ a . u . ] Time [s]Segment A4 -2500-2000-1500-1000-500 0 500 1000 1500 2000 2500 0 4x10 -7 A m p lit ud e [ a . u . ] Time [s]Segment A5-1000-800-600-400-200 0 200 400 600 800 1000 0 4x10 -7 A m p lit ud e [ a . u . ] Time [s]Segment B4
Fig. 19.
Variation in rise time and shape of transient signalsas a function of central contact radius (black=5mm, red=6mm,blue=7mm), and scaled charge-carrier velocities (thin=0.8,thicker=1.0, thickest=1.1), respectively. This is shown for in-teraction at three different radii (solid=8 mm, dotted=20mm,dashed-dotted=36 mm). C oun t s / . k e V Gamma-ray energy [keV]ADLFit 0 200 400 600 800 1000 1200 1200 1220 1240 C oun t s / . k e V Gamma-ray energy [keV]AGATAGeFEMFit
Fig. 20.
Examples of how the Full Width at Half Maximumis extracted from the data. An automatic procedure has beenchosen to minimize biases. The fit has been done on the upperhalf of the peak because of the tail to the left that is a resultof the nucleus slowing down in the target. F W H M [ k e V ] Crystal B010 R=5 mmR=6 mmR=7 mmADL 4 5 6 7 Crystal C001 F W H M [ k e V ] Crystal A007 4 5 6 7 0 10 20 30 40 50 60 70 80 90Crystal B007 F W H M [ k e V ] Lattice rotation [deg] 10 20 30 40 50 60 70 80 90Crystal C007Lattice rotation [deg] 5 6 7 0 10 20 30 40 50 60 70 80 90 F W H M [ k e V ] Lattice rotation [deg]
Fig. 21.
Extracted Full Width at Half Maximum for the1221 keV γ -ray peak in Sr as a function of lattice rotationand assumed radius on the central contact. Results using thestandard AGATA ADL pulse-shape data bases are also shown. which for p = 2 is the typical square sum FOM and i is theindex of the samples points. Using the ADL bases it hasbeen shown that an p of . give the best result [51] (fora recent in-depth discussion of the impact of the distancemetric on PSA, see Lewandowski et al. [29]). A scan of p ’s were performed using the six selected AGATAGeFEMbases, and the resulting FWHM are presented in figure 24.For reference the FWHM achieved using the ADL basesand the AGATAGeFEM are also shown. The AGATAGe-FEM bases were calculated with a central contact radiusof 6 mm and a lattice orientation of 45 ◦ . This is the fi-nal result of the optimisations made in this work. On thelower panel in figure 24 an approximate conversion to po- . Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package 13 E l ec t r on m ob ilit y s ca l e f ac t o r Hole mobility scale factorCore R=5 mminput matrix 5 5.5 6 6.5 70.80.91.01.1 0.8 0.9 1.0 1.1 E l ec t r on m ob ilit y s ca l e f ac t o r Hole mobility scale factorCore R=6 mminputCoreR6 matrix 5 5.5 6 6.5 70.80.91.01.1 0.8 0.9 1.0 1.1 E l ec t r on m ob ilit y s ca l e f ac t o r Hole mobility scale factorCore R=7 mminputCoreR7 matrix 5 5.5 6 6.5 7
Fig. 22.
Full Width at Half Maximum of the 1221 keV γ -raypeak in Sr for the A002 crystal using pulse-shape bases cal-culated with different central contact radii and charge-carriervelocity scaling. The crystal lattice was kept fixed to 45 ◦ . sition resolution has been added (for details see Li et al.[50]) allowing to estimate the improvements made. Look-ing at the lower panel in figure 24 an improved positionresolution of a few tens of millimeters is suggested, thisby comparing the points using the ADL bases and the op-timised AGATAGeFEM bases (at 0.3 on the x-axis). Thecorresponding γ -ray spectra are shown in figure 25. Vi-sually the difference in FWHM of 0.35 keV for the peak1221 keV peak is not obvious, and no manual fit has beenperformed to verify the difference. This in the spirit ofminimizing biases when analysing the data. One can how-ever state that the AGATAGeFEM produced bases per- e,h =0.9,1.1 F W H M [ k e V ] Crystal B010, S e,h =1.0,1.1R=5 mmR=6 mmR=7 mmADL 4 5 6 7 Crystal C001, S e,h =1.1,1.0 F W H M [ k e V ] Crystal A007, S e,h =1.0,1.1 4 5 6 7 35 40 45 50 55Crystal B007, S e,h =1.0,1.1 F W H M [ k e V ] Lattice rotation [deg] 35 40 45 50 55Crystal C007, S e,h =1.0,1.1Lattice rotation [deg]
Fig. 23.
Full Width at Half Maximum of the 1221 keV γ -ray peak in Sr as a function of crystal lattice orientation forthe set of charge carrier velocity scaling giving the smallestFWHM for each crystal used for evaluating the PSA perfor-mance. The best combination for each crystal is marked witha circle. Results using the ADL bases are given as reference. form as well as the ADL bases presently used for analysingAGATA data.An homogeneous γ -ray flux over the solid angle cov-ered by one AGATA crystal is an excellent approximation,and as a consequence the γ -ray interaction points shouldbe homogeneously distributed on planes parallel to frontface of the detector (here the small effect coming from thatstrictly speaking we should refer to spherical surfaces isignored). Looking at the projections of γ -ray interactionpoints onto the plane parallel to the front face is thereforean indicator of how well the PSA is performing. In the fig-ures 26, 27, and 28 projections of γ -ray interaction pointsare shown for the six crystals and three different sets ofbases, respectively. In all figures similar features can beseen. With the exception of crystal C001 the intensity of γ -ray interactions is clearly seen to be lower at segmentboundaries. This is interpreted as coming from the one-interaction per segment approximation used in the PSAand supported by the results of performing PSA on cal-culated signals as previously shown in figures 14 and 15.Based on the number of counts in the maximum bin the F W H M [ k e V ] Crystal B010 4 5 6 7 Crystal C001 F W H M [ k e V ] Crystal A007 4 5 6 7 0.1 0.3 0.5 0.7 0.9 1.1 1.3Crystal B007 F W H M [ k e V ] Metric exponetial p 0.1 0.3 0.5 0.7 0.9 1.1 1.3Crystal C007Metric exponetial p 5 6 7 0.1 0.3 0.5 0.7 0.9 1.1 1.3 4.5 5 5.5 6 F W H M [ k e V ] Δ r i [ mm ] Metric exponetial p
Fig. 24.
Full Width at Half Maximum of the 1221 keV γ -raypeak in Sr as a function of the exponential p in the metricused to compare entries in the pulse-shape bases with exper-imental pulse shapes. The test was performed with the basesselected to be the best according to the FWHM for each crys-tal. The best value of . is in accordance with earlier results.A range of 0.1-3 in steps of 0.1 was scanned but as the resultsgrow worse as the metric increases beyond . results are onlyshown up to . . Results from using the ADL bases and withAGATAGeFEM bases assuming a central contact radius of 6mm and a crystal lattice rotation of 45 ◦ are given as reference(shown as grey circles and blue rectangles, respectively). Onthe lower panel an approximate conversion to position resolu-tion (FWHM) is given (see [50]). C t s / . k e V Gamma-ray energy [keV]ADLFinal bases setR=6 mm, <100> 45 deg 700 800 900 1000 1100 1200 1300 1400 1500 238 240 242 244 C t s / . k e V Gamma-ray energy [keV] 1212 1217 1222 1227 0 200 400 600 800 1000 1200 1400 C t s / . k e V Gamma-ray energy [keV]
Fig. 25.
Histograms made using three different set of pulse-shape data bases, the standard ADL data base, an AGATAGe-FEM data base, and the optimized AGATAGeFEM data base. quality of the PSA improves when moving from ADL basesto optimized AGATAGeFEM bases for four out of six de-tectors. This is seen when comparing the scales in figure26 and figure 28. However, the "hot spots" seen for crystalC001 remains pronounced for all bases. They are locatedat the corners of the front face and for depths in the crystalthat is smaller than 4 mm (as determined using the ADLbases). To investigate these events closer an average of alltraces belonging to the hot spot close to x~-30 mm, y~0mm, z<4 mm was produced together with an average ofevents that gave x and y coordinates next to the hot spot(the two regions are marked in figure 26). On the eventsused for averaging the condition of a net-charge in only onesegment was also enforced. In figure 29 the resulting av-erages are shown together with the pulse shapes from theADL and optimized AGATAGeFEM bases coming frompositions in the bases corresponding to the hot spot. Ascan be seen the average trace for events ending up at thehot spot does not reach its full value. This is most likelyrelated to an incorrect determination of the start time forthese events (in figure 29 the traces have been aligned forclarity). When trying to fit the time-misaligned traces thePSA algorithm always finds the same best position as thetime alignment can be compensated by choosing an ex-treme rise time in the pulse-shape data basis. When com-paring the traces from the ADL bases and the optimizedAGATAGeFEM bases an explanation to why the C001 hotspots are "less hot" for the AGATAGeFEM bases is given.The start of the signal for the AGATAGeFEM bases is dif-ferent allowing for more positions to reproduce the "false"rise time of the time-misaligned traces. It is however diffi-cult to state if one basis is more realistic than the other asthe "hot spot" is related to the preprocessing of the tracesperformed before the PSA or to the time-pickup made in . Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package 15
Fig. 26.
Gamma-ray interaction points as determined withthe ADL pulse-shape data base. For crystal C001 the regionsused to create averaged traces when investigating the originsof "hot spots" are found inside the red circle and are markedin white for events belonging to the "hot spot" and in greenfor reference events, respectively (for details see text). the digitizers of AGATA. This underlines the importanceof preprocessing for successful PSA.
The C++ based software package AGATAGeFEM aimingat modeling segmented High-Purity Germanium detectorshas been described. It allows the implementation of the de-tector geometry and segmentation schemes to within ma-chine precision and uses Finite Element Methods to solvethe Laplace and Poisson equations. The resulting fields arecalculated using the basis functions and support points ofthe actual FEM grid, i.e. using function evaluation ratherthan interpolation.The charge-transport equations are solved using timeadaptive Runge-Kutta methods from the GNU ScientificLibrary. To the induced charge signals linear and differen-tial crosstalk is added together with the transfer functionof the electronics. Convolution is made in the time do-main. The model used in AGATAGeFEM for hole-charge
Fig. 27.
Gamma-ray interaction points as determined withthe AGATAGeFEM pulse-shape data base with a central con-tact hole radius of 6 mm. carrier velocity has proven to give good results for PSAbut still has room for improvements.In this work AGATAGeFEM has been used to inves-tigate the impact of crosstalk and noise for the EGS PSAand for the SVD PSA. The result suggests that crosstalkat the level of what is found in AGATA has a small impacton the resolution of the PSA. Furthermore the influenceof an imperfectly known crystal geometry has been in-vestigated. It was found that a χ figure-of-merit statingon how good the experimental signals could be fitted us-ing the pulse-shape basis is not a good indicator for theprecision of the geometry. In extreme cases the measuredposition resolution using in-beam methods can give indi-cations. These results point to the importance to have welldefined crystal geometries when modeling pulse shapes.As a validation of the pulse-shapes calculated usingAGATAGeFEM pulse-shape data bases for PSA on AGATAdata have been produced and optimized. The resultingbases allow for analysing the data with results that are asgood as the other state-of-the-art pulse-shape data bases,showing that the concepts and models used in AGATAGe-FEM are producing pulse shapes as realistic as the ADLpresently used within the AGATA collaboration. It hasalso been shown that "hot spots" seen in the distribution Fig. 28.
Gamma-ray interaction points as determined withthe best performing AGATAGeFEM pulse-shape data base. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 2.5e-07 5e-07 A m p lit ud e [ a . u . ] Time [s]Segment D1 "Hot spot"Next to "Hot spot"Basis AGATAGeFEMBasis ADL
Fig. 29.
Signals from the net-charge segment for one-segmentevents coming from the "hot spot" seen for crystal C001 (forthe ADL bases, see figure 26). of γ -ray interaction points from the AGATA PSA can belinked to problems in the data treatment prior to PSA,e.g. the time alignment of the traces. Acknowledgement
The author would like to thank the AGATA collabora-tion and the GANIL technical staff. Gilbert Duchêne isthanked for providing the in-beam data set used to ex-tract the position resolution of the pulse-shape analysis.R.M. Pérez-Vidal, A. Lopez-Martens, C. Michelagnoli, E.Clément, J. Dudouet, and H. J. Li are thanked for theircontribution given via the work performed in the scopeof earlier publications. The excellent performance of theAGATA detectors is assured by the AGATA Detector Work-ing group. The AGATA project is supported in France bythe CNRS and the CEA. This work has been supportedby the OASIS project no. ANR-17-CE31-0026.
A Parameters used for pulse-shapecalculations
Table 2.
Charge carrier mobility parameters used in this workParameter Value µ < >e cm V s β < >e E < >e Vcm µ < >en -167 cm V s µ < >e cm V s β < >e E < >e Vcm µ e< >en -133 cm V s µ < >h cm V s β < >h E < >h Vcm µ < >hn cm V s µ < >h cm V s β < >h E < >h Vcm µ < >hn cm V s
Table 3.
Space charge densities used for field calculationsCrystal Front [ /cm ] Back [ /cm ]A002 0.50 1.18B010 0.54 1.55C001 0.93 0.67A007 0.42 1.59B007 0.52 1.76C007 0.60 1.39. Ljungvall: Pulse-shape calculations and applications using the AGATAGeFEM software package 17 References
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