Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
Accounting for incomplete charge collection in Monte Carlo
simulations of the efficiency of well-type Ge-detectors
F. Herna! ndez*, F. El-Daoushy
Department of Physics, Uppsala University, Box 530, Uppsala 75121, Sweden
Received 30 July 2002; received in revised form 22 November 2002; accepted 25 November 2002
Abstract
The influence of the process of incomplete charge collection on the performance properties of HPGe well-type
detectors is studied. A simplified sub-routine, that takes into account the total collection time of charges within the Gecrystal, has been developed and coupled to the Monte Carlo photopeak efficiency. Only the manufacturer’s data of the
detector’s geometry and the detector’s operating voltage was used in the sub-routine. The corrected photopeak
efficiency simulations, as based on this new sub-routine, showed much better agreement with the experimental data,
especially for energies above 40 keV. The ratio between the experimental and simulated efficiency at 1460 keV were, for
example, improved from 0.73 to 0.96 after performing these corrections on our largest well-type Ge-detector.
r 2002 Elsevier Science B.V. All rights reserved.
PACS: 29
Keywords: Monte carlo simulations; HPGe detectors; Efficiency; Incomplete charge collection
1. Introduction
Environmental studies often involve the measurement of very small amounts of material. This
is partly due to sampling restrictions and partly
because the sampled material is generally shared
by multiple users that require different type of
measurements. Well-type Ge-detectors, with
their high efficiency, is the type of detectors most
often used for the measurement of small samples
(e.g. Refs. [1–2]).
*Corresponding author. Tel.: +46 -(0 1 8)-4713564; fax:
+46-(0 1 8)-4713524.
E-mail address: francisco.hernandez@fysik.uu.se
(F. Hern!andez).
The calculation of absolute activities of natural
and artificial radionuclides by gamma spectrometry requires reliable and accurate determination
of the detector’s photopeak efficiencies. Although
the experimental determination of the efficiency of
Ge-detectors is still the most accurate approach
[1,3], this often requires extensive and delicate
laboratory work, both in term of source preparations and measuring time.
A complete evaluation of the photopeak efficiency of Ge detectors using only Monte Carlo
simulations has been evolving since the seventies
(e.g. Refs. [4–6]). Different approaches and codes
can be found in literature [2,5,6–15] where various
sample-to-detector geometries and photon energy
ranges are treated. However, efficiency results are
0168-9002/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0168-9002(02)02080-6
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F. El-Daoushy / Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
341
mostly given for planar and coaxial type detectors
since these are the most commonly used types of
detectors. Scarce data are available for well-type
Ge-detectors [2,11,15].
The Monte Carlo code used to simulate the
efficiency of the different Ge-detectors, in this
paper, was the well-established MCNP Monte
Carlo code [16]. Simulations of the performance of
the planar-type Ge-detectors, available in our
laboratory, showed that this code yields good
agreement between the simulated and experimental efficiency [17]. However, this code failed to
reproduce the experimental data of two well-type
Ge-detectors available also in our laboratory. The
largest differences between the experimental and
the simulated data were observed for the largest of
the two well-type Ge-detectors (Fig. 1).
The discrepancies commonly existing in literature between experimental and simulated data of
the efficiency calibration of Ge-detectors have
been attributed to the uncertain knowledge of
the geometric characteristics of the detectors.
These discrepancies have been dealt with by
manipulating the dimensions of the detector’s
crystal and the contact layers [e.g. Refs [2,14]).
In this paper, we show that incomplete
charge collection is a more likely cause of the
discrepancies in the simulation of the efficiency of
well-type Ge-detectors than inaccurate detector
specifications supplied by manufactures. Our
approach consists in two steps: we use a collimated
source to demonstrate that the efficiency in
different sections of the well-type crystal is not
what one would expect from simple geometric
considerations, and we show that Monte Carlo
results improve dramatically when incomplete
charge collection is accounted for with the method
we developed.
2. Method
2.1. Measurements
Four ultra low-level gamma spectrometry systems consisting of seven different detectors are
available at our laboratory [1,17]. Two of these
detectors, that are discussed here, are HPGe well-
Fig. 1. Description of the physical dimensions, contact layers
and operating voltages of the two well-type Ge-detectors as well
as the physical dimensions of the samples. (a) The smallest welltype Ge-detector. (b) The largest well-type Ge-detector. The
Figures are not drawn to scale.
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F. El-Daoushy / Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
type detectors. Fig. 1(a) and (b) specify the
physical dimensions and operating voltages of
these two detectors. A total of five different
radioactive standards were used for the preparation of numerous calibration sources for the
experimental calibration of the two well-type Gedetectors. These radioactive standards included
mixed sources containig the well-known single
gamma ray emitters, single Pb-210, single Am-241,
single K-40 and Ra-226 + daughters. Radionuclides such as Ce-139, Y-88 and Co-60 were
discarded from the final data-sets because of
coincidence summing. Multiple runs were performed for each sample-to-detector geometry to
reduce the random uncertainties related to the
preparation of the different calibration sources.
Further description of the Ge-detection systems
and the experimental calibration of the efficiencies
is given elsewhere [1,15,18].
A collimated spherical source with radioactive
material at its centre was used to measure the
efficiency response of the largest well-type Gedetector at two different energies, 59.5 and
186 keV. This source allowed scanning the active
parts of the Ge-crystal by moving inside the well
cavity of the detector. The source could be placed
at different positions on the z-axis, which is
passing through the centre of the well cavity. It
could, also, be rotated around the z-axis [0–2p]
and around the radial-axis [0–p] of the horizontal
plane which is perpendicular to the z-axis. These
movements allowed directing a collimated beam,
from the pin hole of the source, into different
active zones within the Ge-crystal. Ultra-pure lead
(1 mBq/g), electrolytic copper and styrene were
used for shielding and sealing the source. The
spherical source had an inner and outer radius of
1.5 and 6.5 mm, respectively. The radioactive
material was concentrated in a low absorption
spherical solid matrix and placed in the inner
central part of the source. The pin hole opening
of the source provided the collimated photons with
a radius of 1 mm. Several plastic frames of
different lengths allowed holding the source in
stable positions at various heights inside the well
cavity.
The photon emission rate of the collimated
source was determined for the two different
photon energies using three different HPGe planar
detectors. Description of these detectors is given
elsewhere [15]. The source was always placed on
the symmetry axis of the planar-type detectors
with the collimated beam perpendicular to the
detectors’ windows. No measurements were performed at other angles of photon incidence.
Measurements at various distances ranging from
8 to 20 mm, from the hole of the source to the
surface of two of the planar Ge-crystals, were
performed to study the influence of the source-todetector distance on the measured count rate.
Calculation of the source’s expected count rate in
the well-type detectors involved minor corrections
to consider the absorption of photons in the Al
windows of these detectors. Two of the planartype Ge-detectors available in our laboratory
constitute a sandwich Ge-system. This system
allowed calculating the percentage of the 186 keV
photons that escaped isotropically the shield
of the source due to its small dimensions and the
relative long range of these photons. All the
subsequent measurements of the 186 keV photons
count rate were corrected for this small percentage
(o 2%).
2.2. Simulations
The simulations of the efficiency were done,
using MCNP [16], for both well-type Ge-detectors,
and for energies ranging from 20 to 1460 keV.
Only the dimensions given by the detector’s
manufacturer were used in the simulation. The
simulation results comprise the average of several
runs with variable numbers of initial photons
ranging from 105 to 1010. The largest number of
the initial photons was used for the simulations
involving photon energies corresponding to the
lowest efficiency in order to improve the statistical
uncertainty. Simulations of the collimated source
were also made for the largest well-type Gedetector to allow comparisons with the experimental data. A simplified diagram of the collimated source-to-detector geometry is shown in
Fig. 2. These simulations were done for different
beam angles around the radial-axis [0–p] and
different distances from the bottom of the well
cavity.
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F. El-Daoushy / Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
343
Fig. 2. Simplified geometry used in the simulation of the efficiency of the collimated source. In this case, the source is placed on the
symmetry axis at a distance of 47 mm from the base of the detector. Theta (y) represents the rotation angle of the collimated beam
around a radial axis situated in the horizontal plane. This horizontal plane is perpendicular to the symmetry axis z.
2.3. Modelling the charge collection process
In order to model the charge collection process,
it was necessary to calculate the electric field inside
the active regions of the different detectors.
Analytical models exist for the calculation of the
field inside planar and true-coaxial Ge-detectors
[19]. However, the complicated geometry of the
well-type Ge-detector had to be solved numerically. The FlexPDE software [20], a programme
designed for solving differential equations for
given boundary conditions, was used to perform
the calculations of the electric field for the planar,
true-coaxial and well-type Ge-detectors. The
results were compared to the available analytical
solutions. The good agreement of the magnitude
of the electric field between the analytical and
simulated data indicated the reliability of the used
software. Fig. 3 shows a simplified version of the
cross-section of the calculated electric field for
the largest well-type detector. A more detailed and
full colour version of this figure is available
elsewhere [17, http://www.fysik.uu.se/isotopgeo/
research/compmaterial/]. The boundary conditions used in the calculations of the field corresponded to the operating voltages given by the
detectors’ specification sheet. The voltage drop in
the detector’s load resistor, of the bias circuit, was
neglected since it was found to be very small. The
variations in the uniformity of the electric field, i.e.
the strength and the direction, are more significant
in the well-type Ge-detector compared to the
planar-type Ge-detector. Strong variations in the
strength and direction of the field were observed at
the bottom corners of the well cavity, and the
bottom corners of the Ge-crystal. The lowest
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F. El-Daoushy / Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
Fig. 3. Cross-section of the largest well-type Ge-detector with a simplified version of the electric field distribution within the active
detection region. The arrows indicate the approximate field direction.
values of the field are located at the bottom
corners of the Ge-crystal of the well-type detector.
3D maps of the drift velocities of charges due to
the electric field for various detectors were
calculated using the empirical expression [21]:
mEðx; y; zÞ
Vdrift ðx; y; zÞ ¼ h
b i1=b
1 þ Eðx; y; z=e
ð1Þ
where E(x,y,z) is the magnitude of the electric field
at position (x,y,z), m is the mobility of the charge at
low electric field, and e as well as b are constants.
Values for e and b for both electrons and holes are
available elsewhere [21]. A computer code was
built to simulate the motion of the different
charges inside the Ge crystals, and to simulate
the output pulse of the detector as a function of
time. This code was, also, used to calculate the
total collection time of all the charges as a function
of their initial position. The initial positions for the
electron–hole pairs were fed from the outcome of
the Monte Carlo simulations of the photopeak
efficiency. Only the photons that transferred all the
energy inside the detector, i.e. that belonged to the
photopeak, were considered. Secondary photons,
which deposited energy in several locations inside
the detector, were initially treated as independent
events and their contributions were recombined
and compiled in the final analysis of the data.
Trapping of the charges inside the detector due to
impurities was not considered in our model. This
process is expected to have little influence in high
purity Ge detectors and should only be considered
if heavy radiation damage is suspected. Fig. 4(a)
displays a cross-section of the largest well-type Gedetector with the total collection time distribution
as a function of position. Fig. 4(b) shows the
development of the output pulse versus time due to
the drift of electrons and holes for 9 different
initial positions as marked in Fig. 4(a). The
magnitude of the total charge collection time
agrees well with the data given by Ref. [22]. These
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F. El-Daoushy / Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
345
Fig. 4. (a) Cross-section of the largest well-type Ge-detector showing the total charge collection time at the recommended operating
voltage of 3000 V. (b) Calculated output pulse for 9 given positions in the largest well-type Ge-detector as marked in Fig. 3(a). The
pulses are due to the electron’s and the hole’s motion in the detector.
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F. El-Daoushy / Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
results were, also, tested for the other well-type
and planar-type Ge-detectors.
Though the corrections of the efficiency simulation for the incomplete charge collection should
have been based on the initial slope of the pulses,
this proved to be time consuming and inconvenient. Multi-parameterisation was required to add
each of the charge functions produced by the
motion of both the electrons and holes to their
respective collection contacts. In addition, no
information is available from the manufacturer
about the discrimination settings of the different
preamplifiers. It was, therefore, considered to be
more advantageous to base the corrections on the
total charge collection time, where a cut-off time
would be used as a discriminating limit for the
rejection of pulses. There were two main reasons
for this. Firstly, this parameter (the total collection
time) depends solely on the detector’s operating
voltage and the geometrical data provided by the
manufacturer, which eases the extension of the
model to other well-type Ge-detectors. Secondly,
there exists a good correlation between the regions
with high probability for shallow initial slope of
the pulses and the longest collection times as
observed in Figs. 4(a) and (b). This model saves
considerable computing time. However, it has
some drawbacks, especially when applied to
photon energies below 40 keV. In general, pulses
with shallow initial slope would give the longest
collection times. However, pulses originating from
low energy photons (detected very close to the
surface of the active volume) may have rather
steep initial slopes and long total collection
times. This is due to the fast collection of the
holes in the contact nearest to the surface of the
Ge-crystal facing the source, and the drift of
electrons all the way to the other side of the
depletion layer (see pulse from position 6). The
collection cut-off time in the model was set equal
to the total collection time required for an
electron–hole pair produced at position 9 (or
equivalent for the other well-type Ge-detector).
This time was observed to be compatible with
the collection times calculated for the planar
detectors and with the experimental results
obtained from the scanning of the largest welltype Ge-detector.
3. Results and discussion
Fig. 5 displays the experimental results and the
MCNP simulations of the relative efficiency of the
largest well-type Ge-detector at two different
energies, 59.5 and 186 keV. This data corresponds
to the case where the collimated source is being
centred on the symmetry axis at a distance of
47 mm from the bottom of the detector and
rotating around the radial-axis in the horizontal
plane perpendicular to the bottom of the detector.
Zero degrees correspond to the collimated beam
perpendicular to the bottom of the detector.
Normalisation and conversion of the data to
relative efficiencies was done by comparing the
different counting rates measured in the well-type
detector at different incident angles to the counting rate measured in one of the planar-type
detectors (2.5 cm thick) at zero incidence angle.
Therefore, 100% relative efficiency corresponds to
the same measured counting rate. The counting
rates measured by the well-type Ge-detector were
corrected for the different thicknesses of the Al
window seen by the photons. Comparison of the
counting rates between the detectors was possible
since the thickness of the planar-type detector is
almost equal to that of the side of the well-type
Ge-detector. Because of the geometrical aspect of
the field distribution, the planar-type detector was
assumed to have complete charge collection. The
simulated data was included in Fig. 5 to indicate
what the expected efficiency would have been if
this detector had complete charge collection. The
error bars given in Fig. 5 represent one-sigma
uncertainties. The uncertainty in the angle was
estimated to be below 7%. The high attenuation
coefficient of the 59.5 keV photons in germanium
limits their range in the Ge-crystal. The mean free
path of these photons is of the order of 1 mm. This
means that the efficiency at this energy is
independent of the direction of the collimated
beam as long as the beam hits the depletion layer.
In the case of 186 keV photons, the mean free path
is of the order of 1 cm and thus the efficiency
depends on the thickness of the detector as well.
The larger the active Ge-volume ‘seen’ by the
collimated beam, the higher the efficiency. Irregularities in the uniformity of the P+ contact layer in
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F. El-Daoushy / Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
347
Fig. 5. Scanning of the efficiency of the largest well-type Ge-detector using a collimated beam directed, at different angles y, towards
the surface of the Ge-crystal. y=0 corresponds to the beam perpendicular to the base of the detector. The collimated source is centred
on the axis of symmetry at a distance of 47 mm from the base of the detector. Two different photon energies are given. Information
about the normalisation of the data is given in the text.
the detector should not make substantial contributions to the efficiency measurements at any of
these energies.
The biggest differences between experimental
and simulated data appear at the extreme angles
for both energies, i.e. at 01 and 1351. The
difference at 1351 can partly be explained by the
escape of photons from the top of the detector.
This is most likely due to the spread of the
collimated beam with distance from the pin hole.
For the 59.5 keV photons the relative efficiency
should have been the same as for any of the other
angles. However a lower efficiency was, also,
measured. For the 186 keV photons, 01 would
have corresponded to the position where the
detector size, as seen by the collimated beam, is
largest. However, the efficiency was much lower
than expected. These results give clear evidence
that the whole germanium crystal is not equally
efficient in detecting the photons or translating the
deposited energies to electric pulses. As shown in
Fig. 4(a), the longest charge collection times are
calculated for photon interactions that occur at the
bottom of the well cavity (which correspond to an
angle of 01) and the bottom of the Ge-crystal
(0–301) due to ‘irregularities’ in the field. Due to
the field characteristics, a larger number of pulses
would be rejected in these zones explaining the
lower efficiency that is seen experimentally. An
estimation of the differences between the experimental and the simulated data for a cylindrical
source, as based on the measurements described
above, gave values of 10–20% for the 186 keV
photons and 1–5% for the 59.5 keV photons. The
experimental results of the collimated beam at
different distances from the base of the detector
showed the same relation between the 186 keV
photons and the volume of Ge-crystal ‘seen’ by the
beam. The efficiency of the collimated beam at
different angles around the symmetry axis did not
indicate variations in magnitude.
Calculations of the charge collection cut-off
times, as described in the ‘modelling of the charge
collection process’ section, provided values of
250 ns for the largest well-type detector and
190 ns for the smallest well-type detector. These
values can also be estimated directly from Figs. 4(a)
and 6. The smallest well-type Ge-detector showed
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F. El-Daoushy / Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
a faster and more uniform collection time map
(Fig. 6). This was not only due to the smaller
dimensions of this detector but also because it runs
at a higher operating voltage. The smaller dimensions of this detector give the better field homogeneity while the dimensions and the operating
voltage determine the field strength.
Figs. 7(a) and 8(a) show the experimental and
MCNP simulated (with and without correction)
efficiency for the two well-type Ge-detectors. The
fitting of the experimental data was done using
multiple polynomial functions of the logarithm of
the efficiency and the logarithm of the energy. The
calculation of the uncertainties of the simulated
efficiency where based on the following assumptions: (1) the uncertainties in the dimensions of the
detector are small and (2) the uncertainty in the
charge collection cut-off time is 10 ns for both
well-type Ge-detectors. Analysis of the efficiency
corrections for small variations in the cut-off time
showed that intermediate energies (80–800 keV)
are not especially sensitive. However, low and high
energies showed variations in magnitude of up to
10%. These variations were more substantial in
the smaller well-type detector probably due to the
smaller collection time differences across the whole
detector.
It can be observed from Figs. 7(a), (b), 8(a) and
(b) that a better agreement between the experimental and the simulated data is obtained for both
detectors after charge collection correction. Above
40–50 keV, the statistical agreement between the
experimental and simulated data after these corrections is, in fact, rather good. Further improvements
are, however, still needed at energies below 40 keV.
The improvements are more dramatic for the
largest well-type Ge-detector than for the smallest
one. The ratios between the experimental and
simulated data displayed in Figs. 7(b) and 8(b)
increase, for example, from 0.73 to 0.96 after
correction for the largest well-type detector at
1460 keV, and from 0.85 to 0.92 for the smallest
well-type detector at the same photon energy.
Well-type Ge-detectors are difficult to use
because of summing effects arising from low
energy photons. Lining of these detectors with a
high-Z absorber has been recommended by some
authors to prevent coincidence summing. In those
cases the presented method would work perfectly.
A more detailed model considering the rising
slope of the output pulse would, however, be
required to improve the data at energies below
40 keV. Furthermore, experimental analysis of the
dimensions of the detector, the end-cup window
Fig. 6. Cross-section of the smallest well-type Ge-detector showing the total charge collection time at the recommended operation
voltage of 3500 V.
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349
The largest Well-type Ge-detector
Photopeak Efficiency
1
0.1
Experimental data
Experimental data fit
Original MCNP simulation
Corrected MCNP simulation
10
(a)
100
1000
Energy (keV)
The largest well-type Ge-detector
Ratio Experimental / Simulated Efficiencies
1.2
1.0
0.8
0.6
Uncorrected simulation
0.4
Corrected Simulation
0.2
10
(b)
100
1000
Energy (keV)
Fig. 7. (a) Experimental and simulated efficiency of the largest well-type Ge-detector for a cylindrical source (radius = 6.5 mm and
height = 40 mm). (b) Ratio between the experimental (or fitted curve of the experimental data) and simulated efficiency before and
after charge collection correction.
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F. El-Daoushy / Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
The smallest well-type Ge-detector
Photopeak Efficiency
1
0.1
Experimental data
Original MCNP simulation
Corrected MCNP simulation
Experimental data fit
0.01
10
100
(a)
1000
Energy (keV)
The smallest well-type Ge-detector
Ratio Experimental / Simulated Efficiencies
1.2
1.0
0.8
0.6
Uncorrected simulation
Corrected Simulation
0.4
0.2
10
(b)
100
Energy (keV)
1000
Fig. 8. (a) Experimental and simulated efficiency of the smallest well-type Ge-detector for a cylindrical source (radius = 4 mm and
height = 40 mm). (b) Ratio between the experimental (or fitted curve of the experimental data) and simulated efficiency before and
after charge collection correction.
and the dead contact layers is also required to
unfold the different uncertainty contributions due
to the input parameters in the simulations at such
low photon energies. Currently, semi-empirical
Monte Carlo approaches still provide a more
accurate (within 5%) alternative for the simulation
of the performance of well-type Ge-detectors at
energies below 40 keV [15].
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F. El-Daoushy / Nuclear Instruments and Methods in Physics Research A 498 (2003) 340–351
4. Conclusions
Corrections for the incomplete collection of
charges in Ge-crystals are necessary for improving
the simulation of the efficiency of well-type Gedetectors. The scanning experiment of the efficiency of Ge-detectors presented here shows clear
differences in the efficiency in different parts of the
Ge-crystal. The lower part of the Ge-crystal shows
the largest discrepancies in efficiency, most likely
due to the lack of uniformity in direction and
strength of the electric field. A simple sub-routine
that simulates the total collection time of charges
produced by the interaction of photons in the Gecrystal can be used to correct the Monte Carlo
efficiency simulation. Good statistical agreement is
obtained between the experimental and simulated
efficiencies after charge collection correction for
both well-type Ge-detectors and for photon
energies above 40 keV. A more detailed model
considering the initial slope of the produced pulse
from the detector is required to correct the
efficiency simulation at energies below 40 keV.
Further experiments, using collimated sources, are
necessary for detailed studies of the different parts
of other Ge-crystals.
Acknowledgements
We would like to thank Cecilia Johansson for
her introductory lesson in MCNP Monte Carlo
Coding and for helping us to set the first geometry.
We would, also, like to thank the referee for his
positive criticisms.
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