Limitations (and merits) of PENELOPE
as a track-structure code
Jos´ e M. Fern´ andez-Varea1,∗, Gloria Gonz´ alez-Mu˜ noz2, Mariel E. Galassi3,
Kristin Wiklund4, Bengt K. Lind4, Anders Ahnesj¨ o2and Nina Tilly2
1Facultat de F´ ısica (ECM and ICC), Universitat de Barcelona. Diagonal 647, E-08028
2Department of Medical Physics, Uppsala University. Akademiska Sjukhuset, SE-751 85
3Instituto de F´ ısica de Rosario, CONICET-UNR. Av. Pellegrini 250, 2000 Rosario, Ar-
4Department of Oncology-Pathology, Medical Radiation Physics, Karolinska Institutet and
Stockholm University. PO Box 260, SE-171 76 Stockholm, Sweden
∗Corresponding author. E-mail email@example.com
Purpose: To outline the limitations of PENELOPE (acronym of PENetration and
Energy LOss of Positrons and Electrons) as a track-structure code, and to comment on
modifications that enable its fruitful use in certain microdosimetry and nanodosimetry
Methods: Attention is paid to the way in which inelastic collisions of electrons are
modelled and to the ensuing implications for microdosimetry analysis.
Results: Inelastic mean free paths and collision stopping powers calculated with PENE-
LOPE and two well-known optical-data models are compared. An ad hoc modification
of PENELOPE is summarized where ionization and excitation of liquid water by elec-
tron impact is simulated using tables of realistic differential and total cross sections.
Conclusions: PENELOPE can be employed advantageously in some track-structure
applications provided that the default model for inelastic interactions of electrons is
replaced by suitable tables of differential and total cross sections.
Monte Carlo (MC) techniques have become essential tools to describe radiation transport in
matter. In microdosimetry applications, liquid water plays a central role as a surrogate for
biological media. Hence, a number of track-structure MC codes has been developed over the
years to simulate the propagation and interaction of electrons and light ions in this substance
(Nikjoo et al. 2006). In principle, general-purpose MC codes may be used as well to simulate
electron tracks in liquid water. However, the generic cross sections currently implemented
in these programs are unable reproduce the finest details of the interactions in a condensed
medium like water in the liquid state.
PENELOPE (acronym of PENetration and Energy LOss of Positrons and Electrons) (Salvat
et al. 2008) is a general-purpose MC code for the coupled transport of electrons, photons and
positrons in arbitrary material systems. The databases of cross sections cover the energy
interval from 50 eV up to 1 GeV. PENELOPE offers the possibility to track electrons in
a “detailed” (i.e. event-by-event) mode. This feature prompted a number of researchers
to employ the program in various microdosimetry studies. Thus, Stewart et al. (2002),
Mainardi et al. (2004), Hugtenburg et al. (2007), Hsiao and Stewart (2008), Hugtenburg
(2008) and Bernal and Liendo (2009) have pointed out that PENELOPE performs quite
well for some microdosimetry problems when operated in an event-by-event mode. However,
Bernal and Liendo (2009) and others have reported on the appearance of artifacts in certain
simulation results. These shortcomings obviously arise because PENELOPE does not qualify
as a track-structure MC code.
In this context, the main purpose of the present article is to highlight the reasons that prevent
the successful use of PENELOPE in microdosimetry. In order to circumvent the most serious
of these limitations one needs to replace the default model for electron inelastic collisions by
a set of cross sections specifically tailored to describe the energy-loss interactions of electrons
in liquid water. The work done along these lines by Gonz´ alez-Mu˜ noz et al. (2011) is briefly
2 Limitations and merits of PENELOPE as a track-
Most of the interaction models in PENELOPE are accurate enough to describe reasonably
well the interactions of electrons in liquid water even at low energies around 100 eV. For
instance, the database of cross sections for elastic scattering was computed within the static-
exchange approximation using partial-wave methods (ICRU 2007)1. This formalism is basi-
cally the same as the one underlying the corresponding databases of various microdosimetry
codes, e.g. CELLDOSE (acronym of CELL DOSE) (Champion et al. 2008).
On the other hand, the features that limit the capability of PENELOPE (as distributed by
the OECD/NEA2) to simulate accurately electron tracks in liquid water are the following;
• Cross sections for the various interaction mechanisms pertain to isolated atoms, and
mean free paths are rescaled to the actual mass density of the traversed medium.
This simplification is implicit in most, if not all, general-purpose MC codes, and is
tantamount to the complete neglect of aggregation effects, which are appreciable for
low-energy electrons moving in molecular or condensed media.
• Ionization by electron impact is modelled with a simple generalized oscillator strength,
and excitations are disregarded.
• The emission of fluorescent radiation (characteristic x-rays and/or Auger electrons)
is decoupled from the ionization of atomic inner shells; this simulation strategy may
induce negative energy imparted in small scoring volumes.
1International Commission on Radiation Units and Measurements, URL http://www.icru.org/
• The smallest absorption (i.e. cut-off) energies of 50 eV allowed in PENELOPE pre-
cludes transport of radiations down to the lowest excitation threshold of the water
molecule in the liquid phase, which is around 7 eV. Thus, the simulation of electron
track-ends is incomplete.
The second of these items deserves a more detailed explanation owing to its relevance for
the topic at hand. To this end, let us consider an electron with linear momentum ⃗ p and
kinetic energy E that undergoes an inelastic collision with a target atom or molecule. The
corresponding quantities after the collision will be denoted as ⃗ p′and E′. The plane-wave
Born approximation constitutes the conventional framework to describe inelastic interactions
of charged particles (Inokuti 1971). For the sake of simplicity, formulas are displayed in a non-
relativistic fashion, although the actual implementation in PENELOPE is fully relativistic.
Moreover, proportionality factors that are not essential for the present discussion are omitted
from the equations altogether; the complete formulas can be found in (Salvat et al. 2008). In
the plane-wave Born approximation, the doubly differential cross section may be expressed
in terms of the momentum transfer ⃗ q ≡ ⃗ p − ⃗ p′(whose modulus is q = |⃗ q|) and the energy
transfer W ≡ E − E′as
where the generalized oscillator strength of the active shell i, which has Nielectrons and
binding energy Ui, is given by
∝??⟨ψf|exp(i⃗ q·⃗ r/?)|ψi⟩??2. (2)
In the matrix element, ψiand ψfare the initial (bound) state and final (bound or free) state
of the active electron, respectively. The numerical calculation of dfi(q,W)/dW is straight-
forward (albeit cumbersome) for an atom but far from trivial if the target is a molecule. In
addition, handling of bidimensional tables of generalized oscillator strengths is impractical
for MC simulation purposes, e.g. for the random sampling of q and W, even with present-day
computing resources. Hence the need for simplifications that nevertheless retain the main
advantages of the plane-wave Born formalism. The asymptotic behaviours of the generalized
oscillator strength provide useful information to devise a simplified form for dfi(Q,W)/dW,
where the recoil energy Q ≡ q2/2me (me is the electron mass) has been introduced as a
convenient variable. In the case of distant interactions, i.e. those with Q ≪ Ui,
dfi(Q,W)/dW ∝ σi,ph(W),
where σi,phis the photoelectric cross section (in the dipole approximation). For close colli-
sions, namely those with Q ≫ Ui, it is
dfi(Q,W)/dW ≈ Niδ(W − Q).
Inspired by the asymptotic behaviours of dfi(Q,W)/dW, the Sternheimer–Liljequist model
adopted in PENELOPE (Salvat et al. 2008) represents the generalized oscillator strength as
[δ(W − Wi)Θ(Ui− Q) + δ(W − Q)Θ(Q − Ui)],dfi(Q,W)/dW = fi
where Θ is the Heaviside step function. The first and second terms in this expression cor-
respond to distant and close interactions, respectively. Notice that the excitation spectrum
of distant interactions is collapsed into a single resonance energy Wiand that the δ(W −Q)
factor in the second term appears because in close (binary) collisions the target electrons
react as if they were free and at rest (W = Q). Besides, all inelastic interactions are treated
as ionizations. Furthermore, it is assumed that fi = Ni and, for an insulator like liquid
water, Wi≈ aUi
of the atom at Q = 0, df(0,W)/dW =∑
Z lnI =
3. The parameter a is set imposing that the generalized oscillator strength
idfi(0,W)/dW, leads to the currently accepted
mean excitation energy I (ICRU 1984) through the relation
being Z =∑
As an illustration, Figure 1 shows the generalized oscillator strength of liquid water in
the optical limit Q = 0 as parameterized by Dingfelder et al. (1998, 2008). The partial
contributions of the five molecular orbitals are plotted as well. PENELOPE places the
resonance energies Wiof equation (5) at 31.4, 64.0 and 1200 eV with oscillator strengths fi
equal to 6, 2 and 2, respectively (the three outermost molecular orbitals are grouped into
a single resonance); these values of Wiare obtained when binding energies of 538, 28.5 and
13.6 eV are employed and I = 75 eV (ICRU 1984) is inserted in equation (6).
iNithe number of electrons in the target atom or molecule.
The differential cross section for distant interactions, which are the most frequent ones, is
dσi,dist/dW ∝ δ(W − Wi). (7)
As a consequence, the kinetic energy of the primary electron after the ionizing collision is
assigned the discrete value E′= E − Wi. The secondary electron gets Es= Wi− Uiexcept
that the approximation Es= Wiis made in PENELOPE when Uiis smaller than the absorp-
tion (cut-off) energy. These kinetic energies of the outgoing electrons are unrealistic since,
as already mentioned, they are not distributed following the actual (continuous) excitation
spectrum of the medium. Notice that the energy deposit E − (E′+ Es) = Uiis only correct
when Es= Wi− Ui.
The Sternheimer–Liljequist model outlined above yields collision stopping powers of electrons
in liquid water which are in good agreement with those tabulated in ICRU Report 37 (ICRU
1984) if the I value recommended in that publication is chosen. This is because the Bethe
formula for the collision stopping power, which was used to prepare the tables in that ICRU
report, can be derived from the model in the limit E ≫ I (Salvat et al. 2008). At lower
energies, below a few hundred eV, both the collision stopping powers and the inelastic mean
free paths resulting from PENELOPE’s algorithm depart from the values calculated with
more elaborate approaches such as the optical-data models of Dingfelder et al. (1998, 2008)
and Emfietzoglou et al. (2005), which were used to generate the cross-section databases
included in some track-structure codes. Figure 2 depicts inelastic mean free paths λ and
collision stopping powers S computed with these optical-data models and with PENELOPE.
3The actual relationship between Wiand Uiincludes the Lorentz–Lorenz correction (Salvat et al. 2008).
Ad hoc modification of PENELOPE/penEasy
In an attempt to reduce somewhat the aforementioned artifacts caused by the δ-oscillators,
Tilly et al. (2002) increased the number of discrete resonances Wiso as to mimic the shape
of the df(0,W)/dW distribution. A much better way to remedy for this oversimplification
is to replace PENELOPE’s algorithm for the simulation of ionizing collisions by a set of
tabulated differential and total cross sections pertinent to liquid water, as recently done
by Gonz´ alez-Mu˜ noz et al. (2011) adapting the structured main program penEasy (Sempau
2008). In particular, the cross sections for the various ionization and excitation channels
were computed from [cf equation (1)]
employing Dingfelder et al.’s (1998, 2008) parameterization of the complex dielectric function
ϵ(q,W) = ϵ′(q,W)+iϵ′′(q,W) of liquid water and adding exchange and Coulomb corrections.
It is worth recalling that these cross sections are incorporated in the PARTRAC (acronym
of PARticle TRACk) (Dingfelder et al. 2008) and PITS04 (acronym of Positive Ion Track
Structure) (Wilson et al. 2004) track-structure codes. Besides, in order to enforce energy
conservation in each energy deposit, either a KLL Auger electron or a Kα characteristic
x-ray is released isotropically whenever an ionization occurs in the oxygen K shell.
Incidentally, in the course of a MC investigation on the sensitivity of the spatial extent of elec-
tron tracks to the adopted physics models, Wiklund et al. (2011) found that depth-dose dis-
tributions of keV electrons simulated using their track-structure code with Dingfelder et al.’s
(1998, 2008) cross sections differ from those generated by PENELOPE even though the two
codes rely on the ICRU (2007) database for elastic scattering. PENELOPE’s (on average)
longer electron tracks came as a surprise because the selection of I = 75 eV for liquid water
should yield for E ≫ I a larger collision stopping power than that predicted by Dingfelder
et al.’s (1998, 2008) formalism which leads to I = 81.8 eV (see Figure 2b). The inconsis-
tency was traced to an anomalous behaviour of their energy-loss function Im[−1/ϵ(q,W)]
at intermediate and large values of q, namely that the Bethe sum rule is too high by up
to 34%, which may be attributed to the use of truncated Drude functions in ϵ′′(q = 0,W)
to describe the ionization of the molecular orbitals of H2O and to the adopted dispersion
algorithm. The inelastic mean free path is less sensitive than the collision stopping power
to this drawback of the model. Then, the relation ⟨W⟩ = S λ implies that, on average,
energy losses in ionizing collisions will be slightly too large although energy deposits are
correctly predicted. In this respect, the analytical representation of ϵ(q,W) by Emfietzoglou
and co-workers (2005) might be more consistent because it has recourse to a more realistic
3.1 Ongoing work with the modified PENELOPE/penEasy code
A simple MC code, LIonTrack (acronym of Light ION TRACK), has been developed to
simulate the slowing down of swift light ions in liquid water (Gonz´ alez-Mu˜ noz et al. 2011).
The code permits the transport of bare ions H+, He2+, C6+, etc with specific energies between
0.5 and 300 MeV/u. Ionization differential and total cross sections are precalculated using the
continuum distorted wave with eikonal initial state formalism (Galassi et al. 2000). In turn,
excitation is described within the first Born approximation resorting to velocity scaling from
corresponding electron cross sections as well as scaling with the square of the projectile charge
(Dingfelder et al. 2000). Furthermore, LIonTrack assumes rectilinear motion of the ions and
disregards nuclear interactions. In this way, the code is suitable for track-segment conditions.
LIonTrack generates a “phase-space file” containing the position, direction and kinetic energy
of all electrons ejected by ion-impact ionization along the simulated track segment. These
secondary electrons are subsequently transported by the modified PENELOPE/penEasy
program down to 50 eV.
The LIonTrack and modified PENELOPE/penEasy codes have been employed to simulate
the 3D distribution of energy deposits by ion tracks in liquid water. The clustering capa-
bilities of single ion tracks were then analyzed by looking into the frequencies of distances
between energy deposits for several neighbouring orders (1st, 2nd, 3rd, etc nearest neigh-
bours). Other microdosimetry applications include the generation of radial absorbed dose
distributions and the calculation of the mean energy imparted by ions tracks in spherical
volumes with diametres between 2 and 30 nm (i.e. typical DNA sizes). These simulations
and results constitute the subject of a forthcoming publication (Gonz´ alez-Mu˜ noz et al. 2011)
where they are explained at length.
PENELOPE is a general-purpose MC code whose chief limitation for accurate track-structure
applications lies in the manner it simulates inelastic interactions of electrons. The perfor-
mance of PENELOPE can be improved if the default model is replaced by tables of differen-
tial and total ionization and excitation cross sections pertaining to the medium of interest.
In particular, cross-section tables for liquid water have been linked to PENELOPE/penEasy
by Gonz´ alez-Mu˜ noz et al. (2011). The modified code permits taking advantage of the rest of
PENELOPE’s characteristics, such as the coupled electron/photon transport, the flexibility
of the geometry package and the possibility of simulation in materials other than liquid wa-
ter. It can be used in conjunction with a program that stores the initial state of electrons
ejected by ion impact, e.g. LIonTrack, to simulate the patterns of energy deposits caused by
light ions (Gonz´ alez-Mu˜ noz et al. 2011).
We are indebted to Prof. F. Salvat (Universitat de Barcelona), Dr. M. Dingfelder (East
Carolina University) and Dr. D. Emfietzoglou (University of Ioannina) for enlightening dis-
cussions. JMFV acknowledges financial support from the Spanish Ministerio de Ciencia
e Innovaci´ on (project no. FPA2009-14091-C02-01) and FEDER as well as the Generalitat
de Catalunya (project no. 2009 SGR 276). NT thanks partial funding from the Swedish
Declaration of interest
The authors report no declarations of interest.
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W / eV
df(0,W)/dW / eV-1
Figure 1: Generalized oscillator strength at Q = 0 of liquid water (Dingfelder et al. 1998).
The continuous curve is the oscillator strength distribution of one H2O molecule, encom-
passing both excitation and ionization, whereas the thin dashed curves are the contributions
due to the ionization of the five molecular orbitals. The vertical lines indicate the positions
of the resonances Wi, see equation (5) and the text, and the associated numbers are the
corresponding oscillator strengths fi.
101 Download full-text
E / eV
λ / nm
E / eV
S / eV nm-1
Figure 2: Inelastic mean free path (a) and collision stopping power (b) of electrons in liquid
water. The continuous curves are the predictions of the default model in PENELOPE (Salvat
et al. 2008), whereas the dashed curves correspond to the fits by Emfietzoglou and Nikjoo
(2007) to the results of their optical-data model (Emfietzoglou et al. 2005). The circles are
values obtained by Dingfelder et al. (1998).