Review of recent work using the simulation code PENELOPE
F. Salvat∗, X. Llovet† , C. Campos∗, S. Segui∗ , E. Acosta∗, J.M. Fernández-Varea∗
and J. Sempau∗∗
∗
Facultat de Física (ECM). Universitat de Barcelona. Societat Catalana de Física (IEC).
Diagonal 647. 08028 Barcelona. Spain.
†
Serveis Cientificotècnics. Universitat de Barcelona. Lluís Solé i Sabarís 1-3. 08028 Barcelona. Spain.
∗∗
INTE. Universitat Politècnica de Catalunya. Diagonal 647. 08028 Barcelona. Spain.
Abstract.
The general-purpose Monte Carlo code PENELOPE for the simulation of coupled electron and photon transport is used
to generate x-ray emission spectra from targets irradiated by electrons. This code provides a realistic description of the
penetration and slowing down of electrons with energies in the range from ∼ 1 keV to 1 GeV. The simulation of bremsstrahlung
emission is based on numerical partial-wave cross sections, differential in both the photon energy and the direction of emission.
The ionization of K and L shells by electron impact is simulated by using total ionization cross sections calculated from the
distorted-wave first Born approximation; these cross sections, which are different from those in the public version 2001
of PENELOPE, provide a better description of the ionization process. The relaxation of ionized atoms is accounted for
by combining transition probabilities from the LLNL Evaluated Atomic Data Library with experimental x-ray energies. A
systematic comparison of simulated x-ray spectra with absolute spectra measured with an electron microprobe, for various
elemental solid targets and 20 keV electron beams will be presented. The discrepancies between Monte Carlo results and
experiment will be discussed.
INTRODUCTION
PENELOPE [1] is a general-purpose Monte Carlo code
for the simulation of coupled electron-photon transport
in homogeneous material media of arbitrary composition. It generates random histories of electrons, positrons
and photons in the energy range from about 1 keV
up to 1 GeV. The code system includes a geometry package that allows automatic tracking of particles
in material systems consisting of homogeneous bodies
limited by quadric surfaces. PENELOPE is distributed
by the OECD Nuclear Energy Agency Data Bank
(www.nea.fr). The document [1] (http://www.
nea.fr/lists/penelope.html) provides a very
detailed description of the code system, with emphasis on the physical interaction models, electron transport
mechanics and numerical algorithms.
The simulation of electron (and positron) transport is
performed by means of a “mixed” class II algorithm [2]
in which “hard” interactions, with energy loss W and/or
scattering angle θ larger than certain cutoff values Wc
and θc, are simulated individually. “Soft” interactions,
with energy loss and/or angular deflections smaller than
the cutoff values, are described by means of multiple
scattering approximations. With this strategy, the global
effect of the multiple soft interactions that occur between
a pair of consecutive hard interactions is simulated in a
single computational step, thus minimizing the numerical work needed to generate each charged particle history. In PENELOPE the cutoff θc, which separates hard
and soft elastic events, is determined dynamically and
varied continuously with the kinetic energy E of the
transported particle. The cutoff value decreases for decreasing E. At moderately low energies, of the order of
50 keV or smaller (depending on the composition of the
sample), its value goes to zero. The mixed simulation
then becomes detailed (i.e. all interactions are simulated
individually) and the results are “exact”.
The electron interaction models adopted in PENELOPE were selected to allow simulations in a wide energy range and, hence, their complexity had to be kept
below reasonable limits. These models were devised to
yield accurate results under multiple scattering conditions, i.e. when each transported particle undergoes a
minimum of ∼ 10 interactions. In this case, repeated interactions tend to smear out the details of the underlying
single scattering differential cross sections (DCS) and the
results are essentially determined by a few integral properties (moments) of the DCS. PENELOPE uses either
simple analytical DCSs, with parameters determined to
reproduce the values of the relevant moments obtained
from the most reliable information or theory available,
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
129
or numerical databases, combined with physically motivated interpolation schemes.
In this communication we describe recent development of PENELOPE aimed at the simulation of x-ray
generation by keV electron beams. For this application,
the public version 2001 of the code was modified by considering ionization of inner shells by electron impact as
a separate process. The code system has been validated
by comparison of simulated x-ray spectra with measurements performed on an electron microprobe for several
elemental solids [3].
INTERACTION PHYSICS
The considered interactions of electrons with the
medium are elastic scattering, inelastic scattering and
bremsstrahlung emission. Elastic collisions are described
by means of a modified Wentzel model [1], with a simple
analytical DCS defined by three parameters. These are
determined to reproduce the total cross section and the
first and second transport cross sections obtained from
accurate Dirac partial-wave calculations for free atoms,
using Dirac-Hartree-Fock electron densities [4, 5]. Thus,
the modified Wentzel model yields the “correct” values
of the mean free path between elastic collisions and the
first and second moments of the angular deflection in
each collision. This suffices to ensure reliable results
under multiple scattering conditions. These conditions
are not met, for instance, in simulations of multilayered
specimens, which require a more elaborate description
of elastic events. We have recently developed a new
algorithm for efficient simulation of elastic collisions using numerical partial-wave DCSs, which is free from this
drawback. However, it implies a considerable increase
in the volume of numerical information to be handled by
the code, with a gain in accuracy that is appreciable only
for very thin media.
Inelastic collisions are described by means of the
DCS, differential in energy and scattering angle, obtained from the relativistic first Born approximation [6].
The generalized oscillator strength (GOS) is modelled as
a set of discrete δ -oscillators [7] with resonance energies chosen so as to reproduce the empirical values of
the mean ionization energy recommended by Berger and
Seltzer [8]. This ensures the reliability of the predicted
stopping power, even for moderately low energies where
the so-called shell corrections become appreciable. The
GOS model also predicts approximately the observed energy dependence of the inelastic mean free path. However, the distribution of energy losses in single soft inelastic collisions differs substantially from the distribution obtained from more realistic GOS models. Again,
this does not affect the global transport results when the
130
number of inelastic collisions of an electron within the
specimen is of the order of 10 or larger.
The default model for inelastic collisions is too simple to describe the ionization of inner atomic electron
shells, a process of primary importance for the generation of x rays. To describe this process, it is sufficient
to know the ionization cross section of each individual
subshell as a function of the energy E of the projectile.
To allow the use of more accurate cross sections without altering the structure of the code, we consider that
the projectile’s energy and direction of flight remain unaltered in the ionizing event and, therefore, the transport
mechanics of PENELOPE is not affected by the introduction of these events. The current public version of
PENELOPE simulates ionization in the K-shell and Lsubshells; the corresponding cross sections are obtained
from the Born approximation, using partial approximate
GOSs constructed from the photoelectric cross section of
the shell [1]. For quantitative purposes, we have recently
incorporated ionization cross sections calculated from
the distorted-wave Born approximation [9]. The accuracy of this theoretical approach has been demonstrated
by direct comparison with measurements performed in
our laboratory [10].
After an ionizing event, the target atom is left in a
highly excited state with a vacancy in an inner shell that
decays by emission of characteristic x rays and Auger
electrons. The transition rates used to simulate the relaxation cascade were taken from the LLNL Evaluated
Atomic Data Library [11]. The energies of the released
x rays are set equal to their experimental values [12].
It is worth recalling that PENELOPE also simulates the
transport of photons. Photons can also produce ionizations in inner shells by photoelectric absorption and, to a
lesser extent, by Compton scattering [13]. Photoelectric
events are simulated by means of numerical cross sections extracted from the LLNL Evaluated Photon Data
Library [14]. The mean free path for photoelectric absorption is obtained from the atomic cross section; when
a photon is absorbed, the active atomic electron shell is
selected randomly according to the partial photoelectric
cross sections of the K-shell, the L-subshells and the set
of outer shells. If the vacancy is produced in the K- or
L-shells, the relaxation of the residual ion is simulated
as described above and, hence, the code automatically
accounts for (photoelectric and Compton) fluorescence
corrections.
Bremsstrahlung events are described by means of
numerical partial-wave DCSs, differential in the photon energy and direction of emission. The energy of
the emitted photon is sampled according to the scaled
bremsstrahlung cross sections tabulated by Seltzer and
Berger [15]. The direction of the photon is generated
from an analytical interpolation of the shape functions
given by Kissel et al. [16]. The algorithm, which has
−
-7
detector
C
-8
10
-9
10
-1
-1
photons eV sr electron
-1
e beam
10
Interaction volume
-10
10
-11
10
-12
10 0
FIGURE 1. Geometry of a typical electron microprobe measurement.
5
10
20
15
E (keV)
X-RAY GENERATION
Ni
-6
10
-7
10
-1
-1
photons eV sr electron
been described in detail by Acosta et al. [17], is based
on the fact that the intrinsic angular distribution of
bremsstrahlung photons (relative to the direction of the
emitting electron) can be represented as a boosted dipole
distribution. This representation faithfully reproduces the
correlation between the energy of the photon and the direction of emission and, moreover, it allows the sampling
of photon direction by a fast analytical method.
-1
-5
10
-8
10
-9
10
-10
10
-11
10 0
131
5
10
20
15
E (keV)
-1
-5
10
W
-6
10
-7
10
-1
-1
photons eV sr electron
The straight simulation of x-ray generation by keV electron beams is intrinsically very inefficient due to the
fact that production events (i.e. inner-shell ionization,
bremsstrahlung emission) occur very seldom. To get results with low statistical uncertainties with reasonable
computation times, variance-reduction techniques have
to be applied. PENELOPE permits the use of the socalled interaction forcing (also known as the method
of weights), which consists of artificially increasing the
probability of occurrence of a given process; to keep the
simulation unbiased, the extra interactions are assumed
to keep the projectile state unaltered and weights smaller
than unity are assigned to the produced particles. The result is a net gain in efficiency, i.e. a reduction of the statistical uncertainty for a given calculation time.
Measurements of x-ray spectra were performed on a
Cameca SX-50 electron microprobe, with the geometry
displayed in fig. 1, using a Si(Li) energy dispersive xray spectrometer. The detector covers a small solid angle
about a direction corresponding to a take-off angle of 40
degrees. The recorded spectra were transformed into absolute units (i.e. probability of emission of a photon per
unit energy, unit solid angle and incident electron) using
the procedure described by Llovet et al. [3], assuming
that the efficiency of the detector is unity. The estimated
overall uncertainties of the absolute spectra are about 4–
-8
10
-9
10
-10
10 0
5
10
15
20
E (keV)
FIGURE 2. Measured (dots) and simulated (solid lines) xray spectra from bulk samples of C, Ni and W generated by 20
keV electron beams.
8%.
Fig. 2 compares results from PENELOPE simulations for 20 keV electron beams impinging normally on
bulk samples of carbon, nickel and tungsten with abso-
lute spectra measured under the same conditions. Results from the Monte Carlo simulations were convolved
with a Gaussian distribution, with an empirical energydependent width, to account for the response of the detector. Notice that the simulated spectra are properly normalized (i.e. in absolute units). The experimental spectra
were found to differ systematically from the simulations
by a constant factor, slightly larger than unity and that
increases with the atomic number of the material. The
“measured” spectra plotted in fig. 1 for C, Ni and W are
the absolute spectra multiplied by 1, 1.06 and 1.10, respectively.
In the energy interval from ∼ 3 keV to ∼ 15 keV,
where the intrinsic efficiency of the Si(Li) detector is
nearly constant with energy, the results of the simulation are found to be in close agreement with measured
absolute spectra. The difference, which reduces to a constant factor, may be partially due to the uncertainty in the
assumed intrinsic efficiency. Another possible cause of
this discrepancy is stray radiation produced by backscattered electrons, which contributes as a continuous background to the measured spectrum and increases in magnitude with the atomic number. Outside this intermediate energy range simulation and experiment are seen to
differ in a systematic way. At low energies, E < 3 keV,
the observed discrepancies are mainly due to the loss of
efficiency caused by absorption in the detector window
and passive layers. This effect has not been accounted
for due to the limited information available on the internal structure of the detector. Small differences are also
found between the characteristic peak intensities of measured and simulated spectra. These differences probably
originate from approximations used in the calculations
of the transition rates and the ionization cross sections.
They could be minimized by empirically adjusting the
transition rates.
For heavy elements the high-energy part of the simulated spectrum is found to be slightly lower than the measurement. This difference cannot be due to the estimated
efficiency of the detector, because it is not observed for
low-Z elements. We have speculated [3] that it originates
from the adopted bremsstrahlung shape function. Since
photons with energies close to that of the incident beam
can only be emitted in the early stages of the electron
penetration, when the direction of the emitting electron
is still close to the direction of incidence, the tip of the
measured spectrum is more sensitive to the angular dependence of the bremsstrahlung DCS.
In conclusion, PENELOPE has been shown to yield
results in close agreement with electron microprobe measurements. The simulation scheme is robust and efficient,
and most of the observed discrepancies between calculation and experiment seem to arise from an incomplete
description of the geometry of the instrument and from
the lack of knowledge of the internal structure of the de-
132
tector. With an ideal detector, or one with a well-known
response function, simulation could be effectively used
for probing the accuracy of the adopted cross sections
and relaxation data and, ultimately, for improving our
knowledge of these fundamental physical parameters.
ACKNOWLEDGMENTS
This work has been partially supported by the Spanish
Fondo de Investigación Sanitaria, project no. 01/0093,
and by the PICASSO Program (“Acciones integradas
entre España y Francia”), grant no. HF2000-0053.
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