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U.P.B. Sci. Bull., Series A, Vol. 71, Iss. 2, 2009
ISSN 1223-7027
PULSE SHAPE CALCULATIONS FOR THE MARS
SEGMENTED GAMMA RAY DETECTOR
Raluca MARGINEAN 1
Lucrarea descrie pe scurt principiile care stau la baza funcţionării
detectorilor segmentaţi HPGe şi a modelării formei semnalelor care se obţin pentru
aceşti detectori. Au fost facute calcule de forme de semnal pentru detectorul
segmentat MARS folosind două programe diferite; în lucrare se exemplifică si se
discută rezultatele obtinute.
The paper shortly describes the principles of HPGe segmented detectors and
of the pulse shape calculations for this type of detectors. The results obtained in
pulse shape calculations for the MARS segmented detector are discussed in detail.
A comparison was made between the results obtained with two pulse shape
calculations software
Keywords: pulse shape calculations, segmented gamma ray detectors
1. Introduction
A new concept was required to further increase the efficiency and
granularity of detector arrays used in γ spectroscopy, and in this respect the idea of
building a shell purely made of Ge was approached in the frame of the European
project AGATA [1] and US project GRETA [2].
The present gamma detector arrays don't cover a “pure” 4π angle, due to
the fact that a part of the solid angle was covered not by the HPGe detectors, but
by the anti-Compton shields. In order to reach solid angle values closer to 4π, no
anti-Compton shields will be used in AGATA and GRETA and new techniques
will be used in order to deal with the Compton background and improve the
array's detecting parameters.
The geometry of MARS [3], one of the first European segmented
detectors, is presented in Fig. 1. It has a cylindrical (tapered) shape, 9.0 cm in
length and 7.2 cm in diameter, and 25 segments (6 radial sectors × 4 slices + 1
front). Each sector has 60º.
AGATA will be made up of 180 HPGe high-fold position sensitivity
detectors called segmented detectors, and for reconstructing the incident gamma
rays will use the concept of γ-tracking combined with pulse shape analysis.
1
Researcher, Horia Hulubei National Institute of Physics and Nuclear Engineering - IFIN-HH
Bucharest, PhD Student, University POLITEHNICA of Bucharest, Romania, e-mail:
[email protected]
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Raluca Marginean
The HPGe segmented detectors have their outer electrical contact
bidimensionally (radial and perpendicular to Oz) divided in segments (see as
example the segmentation of MARS presented in Fig.1). When a γ-ray interacts
with the HPGe crystal, it may suffer a photoelectric absorption or a series of
Compton scatterings or pair production events followed, eventually, by a final
photoelectric absorption (see Fig.1, left). A so-called “net” electrical signal is then
obtained on the contact corresponding to the area where the hit point was located,
and transient signals are collected on the neighboring contacts.
5 6
1
4
3 2
A
B
C
D
Fig. 1 In the right, the segmentation of MARS is presented, with the segments
denomination. In the left, a possible path (2 Compton scatterings and a photoelectric
absorption) of a γ ray is illustrated.
It was observed that, by analyzing the shape of the transient signals, the 3D
position of the incidence point could be obtained [4]. Using as input the positions
of the γ incidence points, all the individual interaction points belonging to a certain
γ event will be identified and used to reconstruct the tracks of the γ ray in the Ge
material using a γ-ray tracking algorithm.
2. Segmented detector signal characteristics
The observation of a net charge signal on one of the charge-collecting
electrode allows to identify the segment where the interaction γ - Ge material took
place. Depending on the radius where the charge carriers are produced, they will
have different drift distances to the electrodes, and accordingly, the shapes of the
transient image (induced) current signals are different for different interaction
radii. The shape of the signals contains the information on the three-dimensional
position of each individual interaction within the Ge volume and the energy
released at each interaction. To extract the position information from the obtained
pulse shapes, they have to be compared to shapes already known for each point of
the detector. In principle, the shapes can be experimentally registered, using
tightly collimated γ-ray sources and imposing for a Compton scattered γ to have a
Pulse shape calculations for the MARS segmented gamma ray detector
77
coincidence from an external collimated detector. It has however been shown [5]
that these are extremely lengthy measurements if the required position definition
of the scattering is of 1 mm3. The only viable way is then to calculate these pulse
shapes from the electric field inside the crystal and the drift velocities of the
charge carriers and by taking into account the fact that the conductivity in Ge is
anisotropic with respect to the crystallographic axis directions.
Detector
geometry
Electrical field
Electrons/Holes
mobility
Electrons/Holes drift
velocities
Weighting fields
Impurities
concentration
and distribution
Electrons/Holes
trajectories
Pulse shapes
Interaction point
coordinates
Fig.2 Logic scheme of a pulse shape calculation software
Net and transient image charge signals are obtained by calculating the
charge carrier's pathway to the electrode, for a given interaction position. The
motion of the charge carriers is determined by the electric field, which depends on
the detector geometry, the applied voltage, the intrinsic space charge density and
charge carrier mobility. The additional effect of the crystal lattice orientation of
the detector has also to be taken into consideration. A pulse shape calculation
software has the logic scheme presented in Fig.2.
Two different software were employed in order to calculate pulse shapes
for the MARS detector. The first one, called MGS [6], is a MatLAB based
software, and uses the MatLAB commercial routines for calculating the fields; the
other one, called APP [7], is a C++ software and uses the Diffack routines for
fields calculations.
3. Electrical field and weighting potentials
The electric field intensity influences the charge carriers drift velocity. Therefore,
its distribution in the Ge volume is important in defining detectors' response
characteristics like rising time, signal forming, etc.
The electric field is calculated by solving the Poisson equation with the
boundary conditions Φ(r1) = V and Φ(r2) = 0, where V is the applied potential on
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Raluca Marginean
the anode (4000 V in our case) and by taking into account a constant impurity
distribution. For complex geometries (as in our case), the Poisson equation is
solved numerically, using FEM (Finite Elements Method) and a grid adaptive
method to define the finite elements. The adaptive grid will increase the number of
elements in the more sensitive areas like the area near the anode's curved surface
or near the tapered outer surface [6].
The weighting field describes the capacitive coupling between the
segmented electrodes and the charge inside the detector. Its potential is calculated
for every segment S, using the boundary conditions VS = 1 V for the S segment
and Vi = 0 for the other segments. The lines of the electric field and of the
weighting field are very different and only rarely coincide.
For calculating the fields, the detector geometry has to be described inside the
software. In MGS the MARS geometry was described using a cubic grid with a
1mm step. This gave a slightly rough approximation in describing the segments'
edges, with consequences that will be presented later. On the other hand, being
user-developed software, APP has the possibility to have a thinner grid, leading to
an adequate description of MARS geometry. Implicitly, although the fields were
calculated using an adaptive grid, the results were saved for pulse shape
calculations using a fixed grid with a 1 mm step in MGS. On the other hand, in
APP, the fields were saved using the adaptive grid elements' positions, so a higher
accuracy is provided.
4. Drift velocities and pulse shapes calculations
The drift velocities, necessary in order to obtain the drift trajectories, have
been calculated using a model developed by Mihailescu et al. [8], briefly
presented below. In this model, the drift velocity is given by:



n
γ
E
j
j
0


v
E
=
A
E

Σ
d
1
/
2
n


E
γ
E
0
j
0

(1)
where E0 is the normalized electric field vector. The dependence of repopulation
of the jth valley on the orientation of the applied electric field nj/n and γj are:

1
/
2

1



n
E
γ
E
1
j
0
j
0

1


γ
=
R
γ
R
,
γ
=
m
0
0
=
R
E

+

j
j 0j
0
t

1
/
2

4

n
4


ΣE
γ
E
0
j
0




(2)
0m
l 0




00m
t



where ml = 1.64 and mt = 0.00819, Rj = Rz(α)Rx(β) is the rotation matrix around
the Ox axis with the β = acos(√2/3) angle, and Rz(αj) is the rotation matrix around
Pulse shape calculations for the MARS segmented gamma ray detector
79
the Oz axis with the αj = jπ/4 angle. The parameters A(E) and R(E) are obtained
from the empirically deduced drift velocities along the <111> and <110>
crystallographic axes.
Due to the fact that the lines of the weighting field for a given segment
expands over the segment's border in the Ge volume, the charge carriers will
“feel” in a given point inside the Ge crystal the sum of all weighting potentials.
The drift movement of the charge carriers between the points described by the


position vectors r0 and r will produce at the electrode k a current:
n


n 
e

h e




I
=
φ
r

φ
r

v

v

ε






k
i
i
0
d
d
i
V
i
=
1
i
=
1


0


 
(3)


where V0 is the applied potential, φ k r , φ k r0  are weighting potentials in the
h
e
initial and final points, vd , vd are the drift velocities for holes and electrons,
respectively, and n is the number of segments.
After calculating the electric field and the weighting potentials, they were
saved in separate files. In the pulse shapes calculating procedures we need the

values of these variables in a point r that can be different from any of the points
where the fields were calculated and saved. In this respect, a separate function is

used for calculating the field at r as the average value of the 8 values from the

nearest points that define a cube around r , and to smooth the obtained values.
In the pulse shapes calculations, a step-by-step approach is used. Taking
as input the incidence point coordinates, the next position is calculated using a
fixed time-step (1 ns) and the drift velocities separately calculated; then, for the
movement between the two positions, the charge produce on every electrode k is:


dQ k = ±e  φ k r   φ k r0

(4)
The procedure is done for electrons and holes separately and is repeated
until the charge carrier arrives onto the outer surfaces of the Ge crystal.
5. Results
A database with calculated charge signals was created and a comparison
was made between the results obtained with MGS and APP respectively.
The comparison between the pulse shapes calculated with MGS and APP
lead to conclusion that both software generally identify correctly the segment
where the incidence was located – the segment with the net charge signal. Also,
they correctly identify the segments where higher transient signals appear – the
segments located near the one where net signal was produced.
126
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Raluca Marginean
Examples of the obtained results are presented in Fig. 3 – 5. In these
figures, in the case of APP, only the transient signals with high enough amplitude
are presented. The interaction points for these examples were selected to be
located in the geometrical center of a segment and of an inner slice – best case
possible, and near a border (radial border and/or the Z border) between segments
or near the anode – worst case possible.
C1
B1
B2
B6
0
0
-5
-2 .5
-1 0
-1 5
-2 0
A1
-5
-7 .5
-1 0
0
0
-1 0
-1 5
-2 0
0
-1 0 0
-5
B2
A1
B6
-2 .5
-5
-2 0 0
B1
-3 0 0
-4 0 0
-1 2 .5
-2 5
-1 5
-3 0
-1 7 .5
-3 5
-2 0
-4 0
-2 2 .5
-2 5
-3 0
-2 5
-2 7 .5
-1 2.5
-5 0 0
-1 5
-6 0 0
-1 7.5
C1
-2 0
-7 0 0
-4 0
-2 2.5
-8 0 0
-2 5
-4 5
-9 0 0
-2 7.5
-1 0 0 0
-3 0
-5 0
-5 5
-3 0
-1 0
-3 5
-4 5
-5 0
-7 .5
-5 5
Segment
A1
B1
B2
B6
C1
Relative Amplitude MGS
-5.2
-1000
-2.9
-3.0
6.0
Relative Amplitude APP
-5.2
-1000
-2.9
-3.0
5.9
Fig.3 Pulse shapes calculated using MGS (upper) and APP (lower) for an interaction point located
in the geometrical center of the B1 segment. The values of the transient signals' amplitudes,
normalized to the net signal, are presented.
81
Pulse shape calculations for the MARS segmented gamma ray detector
D3
D4
C3
D6
C5
B3
B5
10
B6
55
55
9
50
50
8
45
45
7
5
B3
B4
40
35
30
35
30
20
20
15
15
2
10
10
5
0
0
0
18
-2 0 0
C3
14
1 75
-4 0 0
150
-5 0 0
1 25
12
C6
10
-7 0 0
C5
100
C4
-6 0 0
-8 0 0
75
50
8
6
25
-9 0 0
4
0
2
-2 5
-1 0 0 0
0
10
9
8
7
65
65
12
60
60
11
55
55
10
50
50
45
45
8
40
40
7
35
35
30
30
D4
25
D3
1
9
6
5
6
2
16
20 0
-3 0 0
11
3
20
25 0
22 5
12
5
0
27 5
-1 0 0
13
4
3.5
3
2.5
2
1 .5
1
0 .5
5
0
B6
7
6 .5
6
5 .5
5
4.5
4
25
25
3
1
10
9 .5
9
8 .5
8
7.5
B5
40
4
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
C4
C6
B4
6
D5
20
15
10
25
D5
20
15
10
5
5
0
0
4
D6
3
2
1
0
0
Segment
B3 B4 B5 B6
Relative Amplitude MGS 6.7
Relative Amplitude APP
C3
C4
C5
C6
20 167 367 -120 -834 -1000 8.3
1.4 5.2
5
11
21
-1000
262
2.2
D3
D4
D5
D6
5
20
20
5
1.3
6.3
6
1.5
Fig.4 Pulse shapes obtained using MGS (upper) and APP (lower) for an interaction point
located in segment C4, near the border between C4 and C5 and between slices C and B. The
values of the transient signals' amplitudes, normalized to the net signal' maximum, are presented
82
Raluca Marginean
D3
D6
C5
C6
B4
B5
D4
D5
C3
C4
0
0
0
-1
-1
-2
-3
-1
-2
-3
-2 0
-2
-3
-4
-5
-4
-5
0
-6
-7
-4
-5
-6
-7
-6
-7
-8
-9
-8
-9
-1 0
-1 1
-1 0
-1 1
-8
-9
-1 0
-1 1
-1 2
-1 3
-1 4
-1 2
-1 3
-1 2
-1 3
-1 4
-1 5
-1 4
-1 5
-1 6
-1 7
-1 6
-1 7
-1 5
-1 6
-1 7
-1 8
0
0
-1
-2
-3
-4
0
-2 0
-4 0
-6 0
-8 0
-1 0 0
-1 20
-1 40
-1 6 0
-1 8 0
-2 0 0
-2 20
-2
-4
-5
-6
-7
-6
-8
-4 0
-6 0
-8 0
-1 0 0
-1 20
-1 40
-1 6 0
-1 8 0
-2 0 0
-2 20
0
-2 5
-5 0
-7 5
-1 0 0
-1 25
-1 0
-8
-9
-1 0
-1 1
-1 5 0
-1 2
-1 75
-1 4
-1 2
-1 3
-1 4
-1 5
-1 6
-1 7
-1 8
-2 0 0
-1 6
-1 8
-2 75
-3 0 0
0
-2 5
-2
-5 0
-7 5
-2 5 0
-2 2
0
-1 0 0
-2 25
-2 0
-4
-6
-8
-1 25
-1 0
-1 5 0
-1 2
-1 75
-1 4
-2 0 0
-2 25
-2 5 0
-1 6
-1 8
-2 75
-2 0
-3 0 0
-2 2
Fig.5 Pulse shapes calculated using MGS (upper) and APP (lower) for an interaction point
located in the C5 segment, near the anode and near the border between slices C and D.
The APP transient signals' amplitudes are normalized to the net signal
In Fig.3 the case of the interaction point located in the center of B1
segment is presented. It can be seen that the net signal (signed in red for APP) has
Pulse shape calculations for the MARS segmented gamma ray detector
83
a distinct shape, different from the transient signals shapes. APP normalizes the
transient signals to the net signal (normalized by taking its maximum equal to 1000). By doing the same procedure with the signals obtained with MGS
presented in the upper part of Fig.3, it can be seen that the amplitudes have
coincident or much closed values. Obviously, the relative amplitude of a transient
image signals decrease as the distance to the interaction point decreases – for
example, the signal obtained in B3 has lower relative amplitude than that obtained
for B2. The values of these normalized relative amplitudes are presented in the
table under the figure.
On the other hand, there are some cases when MGS and, rarely, APP don't
identify correctly the segment where the incidence was located. In the case of
MGS, this happens for incidence points located near the segments' borders: the
border between two segments and/or between two slices (see Fig.4) or near the
anode (see Fig.5). As it can be seen in Fig.4, for an interaction point located in
segment C4, near the border between C4 and C5 and near the border between
slices C and B, MGS calculates two (C4 and C5) pulses of a shape typically
encountered for net signals – proving in this way that the fixed step grid used by
MGS for describing detector's geometry and fields doesn't allow a fine geometric
discrimination between segments. Looking at the normalized amplitudes, MGS
incorrectly gives a higher charge collected in C5 than in C4; also the charge signal
calculated for the neighboring signals from slice B – a prove that neither in the
case of the slices the geometric discrimination works. In the same case, APP
identified correctly the segment where the incidence was located and the segments
where higher transient signals appeared.
Near the anode (Fig. 5), a fine geometric discrimination is very important.
For the incidence points located here, the charge drift pathway tends to its
minimum for holes and maximum for the electrons. Also, in this region, the
electric fields have high variations over very small distances. All these reasons can
explain why both the software sometimes fail to correctly calculate pulse shapes
for incidence points located in this region. An example in this respect is presented
in Fig.5, where it can be seen that APP and MGS fail to identify the segment
where the incidence was located, although APP calculates correctly the transient
signals relative amplitudes.
6. Conclusions
The comparison between the pulse shapes calculated with MGS and APP
lead to conclusion that both software generally identify correctly the segment
where the incidence was located. An exception occurs for incidence points located
near the border between two segments and/or between two slices in the case of
MGS. The two software correctly identify the segments where higher transient
signals appear – the neighboring segments. Near the anode APP and MGS
sometime failed to identify the segment where the incidence was located, although
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Raluca Marginean
APP calculates correctly the transient signals relative amplitudes. The differences
in calculating the charge near borders or anode may arrive from the different way
of building the geometry of the detector. Modifications in MGS were already been
done to correct the problems presented in this paper.
BIBLIOGRAPHY
[1] Dino Bazzacco, “The Advanced Gamma Ray Tracking Array AGATA”, in Nuclear Physics A
Volume 746, 27 December 2004, Pages 248-254
[2] Web site: greta.lbl.gov
[3]. D. Bazzacco for the MARS collaboration, Workshop on GRETA Physics, Berkeley, LBNL41700 CONF-980228, 1998
[4] Th.Kroll et al, “Analysis of simulated and measured pulse shapes of closed-ended HPGe
detectors”, in Nuclear Instruments and Methods in Physics Research Section A, vol 371,
Issues 3, 1996, pp. 489-496
[5] K. Vetter, et. al,, “Three-dimensional position sensitivity in two-dimensionally segmented
detectors”, in Nuclear Instruments and Methods in Physics Research Section A, vol 452,
Issues 1-2, 2000, pp. 223-238
[6] C. Santos, C.Parisel, “MGS Study of performances”, oral communication at MGS2005
WorkShop, 19/20 April 2005, Strasbourg, France
[7]. Th Kroll and D.Bazzacco, “Simulation and analysis of pulse shapes from highly segmented
HPGe detectors dor the gamma-ray tracking array MARS”, in Nuclear Instruments and
Methods in Physics Research Section A, vol 463, Issues 1-2, 2001, pp. 227-249
[8]. L. Mihailescu, W. Gast, R.M. Lieder, H. Brands, H. Jager, “The influence of anisotropic
electron drift velocity on the signal shapes of closed-end HPGe detectors”, in Nuclear
Instruments and Methods in Physics Research Section A, vol. 447, 2000, pp.350-360