GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-11-4515-2018Evaluation of Monte Carlo tools for high-energy atmospheric physics
II: relativistic runaway electron avalanchesRelativistic runaway electron avalanche code evaluationSarriaDaviddavid.sarria@uib.nohttps://orcid.org/0000-0001-7892-587XRutjesCasperDinizGabrielLuqueAlejandrohttps://orcid.org/0000-0002-7922-8627IhaddadeneKevin M. A.DwyerJoseph R.ØstgaardNikolaiSkeltvedAlexander B.FerreiraIvan S.EbertUtehttps://orcid.org/0000-0003-3891-6869Birkeland Centre for Space Science, Department of Physics and Technology, University of Bergen, Bergen, NorwayCentrum Wiskunde & Informatica (CWI), Amsterdam, the NetherlandsInstituto de Física, Universidade de Brasília, Brasília, BrazilInstituto de Astrofísica de Andalucía (IAA-CSIC), P.O. Box 3004, Granada, SpainUniversity of New Hampshire Main Campus, Department of Physics, Durham, NH, USAEindhoven University of Technology, Eindhoven, the NetherlandsDavid Sarria (david.sarria@uib.no)13November201811114515453530April201814June201813September201812October2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://gmd.copernicus.org/articles/11/4515/2018/gmd-11-4515-2018.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/11/4515/2018/gmd-11-4515-2018.pdf
The emerging field of high-energy atmospheric physics studies how high-energy
particles are produced in thunderstorms, in the form of terrestrial γ-ray
flashes and γ-ray glows (also referred to as thunderstorm ground
enhancements). Understanding these phenomena requires appropriate models of
the interaction of electrons, positrons and photons with air molecules and
electric fields. We investigated the results of three codes used in the
community – Geant4, GRanada Relativistic Runaway simulator (GRRR) and Runaway
Electron Avalanche Model (REAM) – to simulate relativistic runaway electron
avalanches (RREAs). This work continues the study of
, now also including the effects of uniform
electric fields, up to the classical breakdown field, which is about
3.0 MV m-1 at standard temperature and pressure.
We first present our theoretical description of the RREA process, which is
based on and incremented over previous published works. This analysis confirmed
that the avalanche is mainly driven by electric fields and the ionisation and
scattering processes determining the minimum energy of electrons that can run away,
which was found to be above ≈10 keV for any fields up to the
classical breakdown field.
To investigate this point further, we then evaluated the probability to
produce a RREA as a function of the initial electron energy and of the
magnitude of the electric field. We found that the stepping methodology in
the particle simulation has to be set up very carefully in Geant4. For
example, a too-large step size can lead to an avalanche probability reduced
by a factor of 10 or to a 40 % overestimation of the average electron
energy. When properly set up, both Geant4 models show an overall good
agreement (within ≈10 %) with REAM and GRRR. Furthermore, the
probability that particles below 10 keV accelerate and participate in the
high-energy radiation is found to be negligible for electric fields below the
classical breakdown value. The added value of accurately tracking low-energy
particles (<10 keV) is minor and mainly visible for fields above
2 MV m-1.
In a second simulation set-up, we compared the physical characteristics of
the avalanches produced by the four models: avalanche (time and length)
scales, convergence time to a self-similar state and energy spectra of
photons and electrons. The two Geant4 models and REAM showed good agreement
on all parameters we tested. GRRR was also found to be consistent with the
other codes, except for the electron energy spectra. That is probably because
GRRR does not include straggling for the radiative and ionisation energy
losses; hence, implementing these two processes is of primary importance to
produce accurate RREA spectra. Including precise modelling of the
interactions of particles below 10 keV (e.g. by taking into account
molecular binding energy of secondary electrons for impact ionisation) also
produced only small differences in the recorded spectra.
IntroductionPhenomena and observations in high-energy atmospheric physics
In 1925, Charles T. R. Wilson proposed that thunderstorms could emit a
“measurable amount of extremely penetrating radiation of β or γ
type” , about 60 years before such radiation
was observed from the atmosphere and from space
. This and
subsequent observations and modelling are now being investigated within the
field of high-energy atmospheric physics (HEAP). A review is provided by
.
Observationally different types of high-energy emissions have been identified
coming from thunderclouds, naturally categorised by duration.
Microsecond-long bursts of photons, which were first observed from space
,
are known as terrestrial γ-ray flashes (TGFs). TGFs also produce bursts
of electron and positrons
that follow the
geomagnetic field lines into space and show longer durations. Two space
missions specifically designed to study TGFs and related phenomena will
provide new observations in the near future: ASIM (Atmosphere-Space
Interaction Monitor) , successfully launched in
April 2018, and TARANIS (Tool for the Analysis of Radiation from lightning
and Sprites) , which is to be launched at
the end of 2019.
Seconds to minutes or even hours long X and γ radiation has been
observed on the ground, from balloons and from aircraft
;
these are called γ-ray glows or thunderstorm ground enhancements. Some
modelling attempts of both γ-ray and electron observations are also
presented in .
TGFs were predicted to create a neutron emission on the millisecond duration,
with associated isotope production . Such
emission was observed from the ground
. A similar phenomenon was
modelled at higher altitudes by , who also proposed to
call it “TGF afterglow”.
Following the idea of , high-energy X and
γ radiation is created by runaway electrons, which may further grow by
the effect of Møller scattering in the form of so-called relativistic
runaway electron avalanches (RREAs) . For the
multiplication to occur, a threshold electric field of
Eth=0.28 MV m-1 (at standard temperature and pressure;
STP) is required
.
The difference in duration between TGFs and γ-ray glows can be explained
by two possible scenarios to create runaway electrons, which is traditionally
illustrated using the average energy loss or friction curve (see, e.g.
Fig. 1 of ). In this curve, there is a maximum at
around ε≈123 eV, illustrating the scenario that for
electric fields higher than a critical electric field, of
Ec≈26 MV m-1 at STP, thermal electrons can be accelerated into runaway regime, described in
the so-called cold runaway theory . The effective
value of Ec may be significantly lower, as electrons could overcome
the friction barrier due to their intrinsic random interactions
. Cold
runaway could happen in the streamer phase
or leader phase
of a transient discharge, explaining the high-energy electron seeding that
will evolve to RREAs and produce γ rays by bremsstrahlung emission from
the accelerated electrons. The cold runaway mechanism may be further
investigated with laboratory experiments, in high-voltage and pulsed plasma
technology, and may be linked to the not fully understood X-ray emissions
that have been observed during nanosecond pulsed discharge and the formation
of long sparks
(,
and references therein), with different possible production mechanisms that
were proposed and tested using analytical modelling
and computer simulations
. Alternatively, the
relativistic feedback discharge model is also proposed to explain TGF
production using large-scale and high-potential electric fields
, where the RREA initial seeding may be provided
by cosmic-ray secondaries, background radiation or cold runaway
.
For fields significantly below the thermal runaway critical electric field
Ec≈26 MV m-1 but above the RREA threshold electric
field of Eth=0.28 MV m-1 (at STP), runaway behaviour is
still observed in detailed Monte Carlo studies (see ,
and references therein). At thundercloud altitudes, cosmic particles create
energetic electrons that could run away in patches of the thundercloud where
the electric field satisfies this criterion. RREAs are then formed if space
permits and could be sustained with feedback of photons and positrons
creating new avalanches
.
The γ-ray glows could be explained by this mechanism, as they are observed
irrespectively of lightning or observed to be terminated by lightning
.
The fact that γ-ray glows are not (necessarily) accompanied by classical
discharge results in the conclusion that the electric fields causing them
are usually also below the conventional breakdown. The conventional (or
classical) breakdown field of Ek≈3.0 MV m-1 (at STP) is
where low-energy electrons (<123 eV) exponentially grow in number as
ionisation overcomes attachment. This exponential growth of charged particles
will affect the electric field, which requires a self-consistent simulation
to be properly taken into account. That is not something we want to test in
this study, since Geant4 is not capable of simulating it. Therefore, we will
focus on electric fields below the breakdown field
(Ek≈3.0 MV m-1) and above the RREA threshold
(Eth≈0.28 MV m-1).
As a note, one can find in the literature that Ek can be given between
2.36 and 3.2 MV m-1, the theoretical lowest
breakdown field being between 2.36 and 2.6 MV m-1seep. 338. The value of ≈3.2 MV m-1
is the measured breakdown field in centimetre gaps in laboratory spark
experiments seep. 135, which can be lower for longer
gaps.
Theoretical understanding of RREAs
In the energy regime of a kilo-electronvolt (keV) to a hundred
mega-electronvolts (MeV), the evolution of electrons is mostly driven by
electron impact ionisation , as this energy
loss channel is much larger than the radiative (bremsstrahlung) energy loss.
However, the bremsstrahlung process does impact the shape of the electron
energy spectrum that can be understood by the straggling effect, which is
discussed in the next section. When the electric field is below the classical
breakdown Ek≈3.0 MV m-1 (at STP), the system can be
simplified, because the effect of the electrons below a certain energy can be
neglected, in particular the population that would otherwise (if E>Ek)
multiply exponentially and have an important effect on the electric field.
The part of the electron population that decelerates, and eventually
attaches, cannot contribute to the production of the high-energy radiation.
Let ϵ2min be the minimum energy for a secondary electron
to have a chance to run away and thus participate in the production of high-energy radiation. The subscript index i=2 indicates a secondary electron. A
precise value of ϵ2min will be evaluated in Sect. 3 with
the help of simulations, but, by looking at the friction curve, one can guess
it is located in the keV to tens of keV energy regime
seeFig. 1. As almost all energy loss of ionisation
is going into producing secondary electrons of lower energy (ϵ2≲ 200 eV),
it is reasonable to approximate that channel as
continuous energy loss or friction.
In the case of electric fields above the RREA threshold
(Eth=0.28 MV m-1 at STP), the electrons, when considered
as a population, will undergo avalanche multiplication. Some individual
electrons do not survive (because there can be hard bremsstrahlung or
ionisation collisions that will remove enough energy to get below
ϵ2min), but the ensemble grows exponentially as new
electrons keep being generated from the ionisation collisions on air
molecules, including a fraction with energy larger than
ϵ2min. The production of secondaries with energies much
larger than the ionisation threshold (a few kilo-electronvolts being a
reasonable value) can be described using the Møller cross-section, which
is the exact solution for a free–free electron–electron interaction (see,
e.g. , p. 321):
dσMdδ2=Z2πre2γ12-1(γ1-1)2γ12δ22(γ1-1-δ2)22γ12+2γ1-1δ2(γ1-1-δ2)+1,
where Z is the number of electrons in the molecule, the index i=1
indicates the primary electron, i=2 the secondary, γi is the
Lorentz factor, δi=γi-1=ϵi/(mec2) is the
kinetic energy divided by the electron rest energy (with rest mass
me) and re=14πϵ0e2mec2≈2.8×10-15 m is the classical electron radius. In the
case of δ2≪γ1-1 and δ2≪1, we observe that the term
∝1/δ22 is dominating. Thus, we can write
Eq. () as
dσMdδ2≈Z2πre2β121δ22,
with β1=v1/c the velocity of the primary particle. Integrating
Eq. () from δ2 to the maximum energy
(ϵ1/2) yields the production rate
σprod≈Z2πre2β121δ2∝1ϵ2,
using again ϵ2≪ϵ1. The remaining sensitivity of
σprod in units of area to the primary particle is given by
the factor β12, which converges strongly to 1 as the mean energy
of the primary electrons exceeds 1 MeV. In other words, as the mean energy
of the electrons grows towards even more relativistic energies, the
production rate σprod becomes independent of the energy
spectrum.
For illustrative purposes, we now consider the one-dimensional deterministic
case, which results in an analytical solution of the electron energy
spectrum. We make the system deterministic by assuming that the differential
cross-section is a delta function at ϵ2min (the minimum
energy at which a secondary electron can run away) and use
Λprod=1Nσprod as the constant
collision length, with N the air number density. In other words, every
length Λprod a secondary electron of energy
ϵ2min is produced. The derivation below is close to what
was presented by , ,
and references therein.
Consider a population of electrons in one dimension with space coordinate
z, a homogeneous and constant electric field E above the RREA threshold
and a friction force F(ϵ). The minimum energy
ϵ2min at which an electron can run away is given by the
requirement F(ϵ2min)≈qE (where q is the
elementary charge); that is to say, ϵ2min=function(F,E) is constant. Assuming that the mean energy of the
ensemble is relativistic results in a constant production rate
Λprod=Λprod(ϵmin). Thus,
in space, the distribution fe grows exponentially as
∂fe∂z=1Λprodfe.
While in energy, the differential equation is given by the net force:
dϵdz=qE-F(ϵ).
Solving for steady state means
dfedz=∂fe∂z+∂fe∂ϵdϵdz=0,
and using Eqs. () and () results in
∂fe∂ϵ=-1Λprod(qE-F(ϵ))fe.
For the largest part of the energy spectrum, specifically above 0.511 MeV
and below 100 MeV, F(ϵ) is not sensitive to ϵ (e.g. see
). Only at around ϵ≈100 MeV
electron energy F(ϵ) starts increasing again because of the
bremsstrahlung process. Thus, one may assume F(ϵ)≈F
constant, which yields that the RREA energy spectrum f(ϵ) at steady
state is given by
fe(ϵ)=1ϵ‾exp-ϵϵ‾,
with the exponential shape parameter and approximated average energy
ϵ‾(E) given by
ϵ‾(E)=Λprod(qE-F).
Equivalently, in terms of collision frequency νprod=βcΛprod, Eq. () can be written
as
ϵ‾(E)=βcνprod(qE-F),
with β the velocity v/c of the RREA avalanche front. For the 1-D
case, there is no momentum loss or diffusion, so β≈1. Note that
Λprod depends on ϵ2min (the minimum
energy at which a secondary electron can run away), which depends on the
electric field E as that determines the minimum electron energy that can go
into a runaway regime. In this analysis, we illustrate with
Eqs. () and () that
the full RREA characteristics, such as the mean energy ϵ‾
or the collision length Λprod (directly related to the
avalanche length scale λ discussed in
Sect. ), are driven by processes determining
ϵ2min.
In reality, there are important differences compared to the one-dimensional
deterministic case described previously, which we propose to discuss
qualitatively for understanding the Monte Carlo simulations evaluated in this
study. During collisions, electrons deviate from the path parallel to E.
Therefore, in general, electrons experience a reduced net electric field as
the cosine function of the opening angle θ, which reduces the net
force to qEcos(θ)-F and thereby the mean energy
ϵ‾ of Eq. (). In reality, the 3-D
scattering (with angle parameter θ) changes of the path of the
particle. Although the velocity remains still close to c (as the mean
energy is still larger than several MeV), the RREA front velocity parallel to
the electric field (E) is reduced again because of the opening angle as
function of its cosine:
β∥=βcos(θ),
which also reduces the mean energy ϵ‾. Note that θ
is not a constant and may change with each collision. Equivalently, the
avalanche scale length Λprod in 3-D changes to ≈Λprod×cos(θ). However, most importantly, the
momentum-loss of the lower energetic electrons results in a significant
increase of ϵ2min, as it is much harder for electrons to
run away. The increase of ϵ2min significantly increases
Λprod and thereby increases the characteristic mean energy
ϵ‾. On the other hand, the stochasticity creates an
interval of possible energies (ϵ2min) that can run away with
a certain probability and for thin targets a straggling effect
. A recent article discussed the influence of the
angular scattering of electrons on the runaway threshold in air
.
The effects discussed above prevent a straightforward analytical derivation
of the RREA characteristics in three dimensions, but what remains is the
important notion that the physics is completely driven by the intermediate
energy electron production. “Intermediate” means they are far above
ionisation threshold (≫123 eV) but much below relativistic energies
(≪1 MeV). The parameterisation of the electron energy spectrum, given by
Eq. (), turns out to be an accurate empirical fit, as
it was already shown in
, , and references
therein. Nevertheless, in these works, λmin(E), or
equivalently the velocity over collision frequency (βc/νprod), is fitted by numerical Monte Carlo studies, and the
final direct relation to ϵ2min is not executed.
calculated that νprod(E)∝E and
thus explains why ϵ‾(E) must saturate to constant value.
argue that ϵ2min(E) is given by
the deterministic friction curve F for which they use the Bethe formula
and an integration of a more sophisticated electron impact ionisation cross-section
(RBEB) including molecular effects, but that is only true in one
dimension without stochastic fluctuations. Other attempts to simulate RREA by
solving the kinetic equation instead of using Monte Carlo methods are
presented in
, , and
references therein. An analytical approach is provided by
.
Model reductions and previous study
Apart from analytical calculations, the physics behind TGFs, TGF afterglows
and γ-ray glows are also studied with the help of experimental data,
computer simulations and often a combination of both. Simulations
necessarily involve model reduction and assumptions. As we argued previously,
in scenarios where the electric field is below the classical
breakdown field (Ek≈3.0 MV m-1 at STP), electrons below a
certain energy can be neglected, because they will decelerate and eventually
attach, thus not contributing to the production of the hard radiation. In
Monte Carlo simulations, it is therefore common to apply a so-called
“low-energy cutoff” (or threshold), noted as εc, which is a
threshold where particles with lower energy can be discarded (or not
produced) to improve code performance. It is different from
ϵ2min (the minimum energy at which a secondary electron
can run away) as one is a simulation parameter and the other is a physical
value. Ideally, εc should be set as close as possible to ϵ2min.
A second simplification can be made for the
energetic enough particles that stay in the ensemble by treating collisions
that would produce particles below the low-energy cutoff as friction.
Both simplifications can be implemented in different ways, leading to
different efficiencies and accuracies.
benchmarked the performance of the Monte Carlo codes Geant4
, EGS5 and FLUKA
developed in other fields of physics, and of the
custom-made GRanada Relativistic Runaway simulator (GRRR) and MC-PEPTITA
codes within the parameter regime relevant for HEAP, in the
absence of electric and magnetic fields. In that study, they focused on basic
tests of electrons, positrons and photons with kinetic energies between
100 keV and 40 MeV through homogeneous air using a low-energy cutoff of
50 keV and found several differences between the codes and invited other
researchers to test their codes on the provided test configurations. We found
that the usage of average friction fails in the high-energy regime
(≳100 keV), as the energy loss is averaged too much, resulting in
an incorrect energy distribution .
As we indicated in Sect. , the
ionisation energy loss channel is much larger than the radiative
(bremsstrahlung) energy loss, by a few orders of magnitude. However, this is
only true for the average, and bremsstrahlung does have a significant effect
on the electron spectrum because of straggling .
This straggling effect was first studied by . If it
is not taken into account in the implementation of the low-energy cutoff,
the primary particle suffers a uniform (and deterministic) energy loss. This
means that only the energy of the primary particle is altered, but not its
direction. The accuracy of the assumed uniform energy loss is a matter of
length scale: on a small length scale, the real energy loss distribution (if
all interactions are considered explicitly) among the population would have a
large spread. One way to obtain an accurate energy distribution is by
implementing stochastic friction mimicking the straggling effect.
also indicated that including electric fields in
the simulations would potentially enhance the differences found by
introducing new errors, the simulation results being supposedly sensitive to
the low-energy cutoff. This effect is believed to be responsible of the
observed differences between the two Geant4 physics lists tested in
: for all fields between 0.4 and
2.5 MV m-1 (at STP), they found that the energy the spectrum and the
mean energy of runaway electrons depended on the low-energy cutoff, even when
it was chosen between 250 eV and 1 keV. In the following, this
interpretation is challenged.
Content and order of the present study
In the context of high-energy atmospheric physics, the computer codes that
were used are either general purpose codes developed by large collaborations
or custom-made codes programmed by smaller groups or individuals. Examples of
general purpose codes that were used are Geant4
e.g.
and FLUKA e.g.. Custom-made
codes were used in
, , , ,
, , , ,
and ,
among others. presented in their Sect. 1.3 the
reasons why different results between codes (or models) can be obtained and
why defining a comparison standard (based on the physical outputs produced by
the codes) is the easiest way (if not the only) to compare and verify the
codes. Here, we continue the work of , now with
electric fields up to the classical breakdown field (Ek≈3.0 MV m-1). As mentioned previously, we chose not to use larger
electric fields because that would produce an exponential growth of low-energy electrons (<123 eV) which would affect the electric field and
therefore require a self-consistent simulation that Geant4 is not capable
of. We aim to provide a comparison standard for the particle codes able to
simulate relativistic runaway electron avalanches, as simply and
informatively as possible, by only considering their physical outputs. These comparison
standards are described in the Supplement (Sects. S1 and S2).
In Sect. , we illustrated that the full
RREA characteristics, such as the mean energy ϵ‾ or the
collision length Λprod, are driven by processes determining
ϵ2min (the minimum energy at which a secondary electron
can run away). To prove this insight, and to benchmark codes capable of
computing RREA characteristics for further use, we calculated the probability
for an electron to accelerate into the runaway regime (see
Sect. ), which is closely related to the quantity
ϵ2min. From this probability study, it is directly clear
that it is safe to choose the low-energy cutoff εc higher
than previously expected by and
, given an electric field E<Ek. In
Sect. , we will demonstrate that the probability for
particles below 10 keV to accelerate and participate in the penetrating
radiation is actually negligible. Thus, in practice, an energy threshold value
of εc≈10 keV can be used for any electric field
below Ek. However, in Sect. , we will show that
step-length restrictions are of major importance (e.g. it can lead to an
underestimation of a factor of 10 of the probability to produce a RREA, in
some cases). The results of the comparison of several parameters of the RREAs
produced by the four tested codes is then presented in Sect. 4. We conclude
in Sect. .
The test set-ups of the two types of simulations (RREA probability and RREA
characteristics) are described in the Supplement, together with the data we
generated and figures in the Supplement comparing several characteristics of the
showers. The Geant4 source codes used in this study are also provided (see
Sects. 6 and 7).
Model descriptions
The data we discuss in the next sections were produced by the general purpose
code Geant4 (with several set-ups) and two custom-made codes – GRRR and Runaway
Electron Avalanche Model (REAM) – which we describe below. However, we do not describe comprehensively all the
processes, models or cross-sections used by the different codes, but provide,
in Sect. S13 in the Supplement, a table mentioning all implemented processes
and models, including all references.
Geant4
Geant4 is a software toolkit developed by the European Organization for
Nuclear Research (CERN) and a worldwide collaboration
. We use
version 10.2.3. The electromagnetic models can simulate the propagation of
photons, electrons and positrons including all the relevant processes and
the effect of electric and magnetic fields. Geant4 uses steps in distance,
whereas REAM and GRRR use time steps. In the context of this study, three main
different electromagnetic cross-section sets' implementations are included:
one based on analytical of semi-analytical models (e.g. using the Møller
cross-section for ionisation and Klein–Nishina cross-section for Compton
scattering), one based on the Livermore data set
and one based on the Penelope models . Each of them
can be implemented with a large number of different electromagnetic
parameters (binning of the cross-section tables, energy thresholds,
production cuts, maximum energies, multiple scattering factors and accuracy of
the electromagnetic field stepper, among others), and some processes have
multiple models in addition to the main three, e.g. the Monash University
model for Compton scattering .
used two different physics lists: low- and high-energy physics
(LHEP) and low background experiments (LBE).
The first one, based on parameterisation of measurement data and optimised for
speed, was deprecated since the 10.0 version of the toolkit. The LBE physics
list is based on the Livermore data, but it is not considered as the most
accurate electromagnetic physics list in the Geant4 documentation, which is
given by the Option 4 physics list (O4). This last uses a mix of different
models and in particular the Penelope model for the impact
ionisation of electrons. For this study, we will use two GEANT4 physics list
options: Option 4 (referred to as simply O4 hereafter), which is the most accurate
one according to the documentation, and Option 1 (referred to as simply O1
hereafter), which is less accurate but runs faster. In practice, O1 and O4
give very similar results for simulations without electric field and energies
above 50 keV, as produced in our previous code comparison study
.
By default, Geant4 follows all primary particles down to zero energy. A
primary particle is defined as a particle with more energy than a threshold
energy εcg (which is different from εc
described before). By default, εcg is set to 990 eV and was
not changed to obtain the results presented in the next sections. The LBE
physics list used by uses a threshold down to
250 eV (i.e. more accurate than using 990 eV, in principle) and this
parameter was thought to be responsible for a major change in the accuracy of
the obtained RREA energy spectra. In Sect. , we will argue
that the most important factor able to effect the spectra obtained from
Geant4 simulations is the accuracy of the stepping method for the tracking of
the electrons and not the low-energy threshold. Actually, we found that the
stepping accuracy of the simulation is indirectly improved by reducing
εcg, which explains why could
make this interpretation.
GRRR
The GRRR is a time-oriented code for
the simulation of energetic electrons propagating in air and can handle
self-consistent electric fields. It is described in detail in the Supplement
of and its source code is fully available in a
public repository (see the “Code and data availability” section). In the scope of this
work, we want to point out three important features. (1) Electron ionisation
and scattering processes are simulated discretely, and the friction is
uniform and without a way to mimic the straggling effect. (2) Bremsstrahlung
collisions are not explicit and are simulated as continuous radiative losses,
without straggling. (3) GRRR uses a constant time step Δt both for
the integration of the continuous interactions using a fourth-order
Runge–Kutta scheme and for determining the collision probability of each
discrete process k as νkΔt, where νk is the collision rate
of process k. This expression assumes that νkΔt≪1 and
therefore that the probability of a particle experiencing two collisions
within Δt is negligibly small. The collisions are sampled at the
beginning of each time step, and therefore the rate νk is calculated
using the energy at that instant. In this work, we used Δt=0.25 ps
for the avalanche probability simulations and Δt=1 ps for the
simulations used to characterise the RREA. For both cases, the time steps are
small enough to guarantee a very accurate integration.
REAM
REAM is a three-dimensional Monte Carlo
simulation of the relativistic runaway electron avalanche (also referred to as
runaway breakdown), including electric and magnetic fields
. This
code is inspired by earlier work by and takes
accurately into account all the important interactions involving runaway
electrons, including energy losses through ionisation, atomic excitation and
Møller scattering. A shielded-Coulomb potential is implemented in order to
fully model elastic scattering, and it also includes the production of
X/γ rays from radiation energy loss (bremsstrahlung) and the
propagation of the photons, by including photoelectric absorption, Compton
scattering and electron–positron pair production. The positron propagation is
also simulated, including the generation of energetic seed electrons through
Bhabha scattering. The bremsstrahlung photon emissions from the newly
produced electrons and positron are also included.
In the scope of this study, it is important to point out that REAM limits the
time step size of the particles so that the energy change within one time
step cannot be more than 10 %. The effect of reducing this factor down to
1 % was tested and did not make any noticeable difference in the
resulting spectra. The comparative curves are presented in Sect. S10 in the Supplement.
Stepping methodologyGeneral method
In Monte Carlo simulations, particles propagate in steps, collide and
interact with surrounding media by means of cross-sections (and their
derivatives). A step is defined by the displacement of a particle between two
collisions. As it is presented in Sects.
and , the stepping methodology is responsible for most of the
differences we observed between the codes we tested. Simulations can be
either space-oriented or time-oriented, if the stepping is done
in space or in time. By construction, space-oriented simulations are thus not
synchronous in time. Usually, a single particle is simulated until it goes
below the low-energy threshold (εc) chosen by the user.
However, there are exceptions, like Geant4, that by default follow all primary
particles down to zero energy. The advantage of asynchronous simulations is
the ability to easily include boundaries to have particles step as far as
possible in the same material (minimising the overhead due to null
collisions) and smaller memory usage since there is no need to store all the
particles alive at a given time (which may be a million or more). However,
asynchronous simulations make it impossible to incorporate
particle-to-particle interactions, such as a space charge electric field, or
self-consistent electric fields.
During steps, charged particles can lose energy (and momentum) by collisions
and also change in energy (and momentum) when an electric fields is present.
To guarantee accuracy, energies should be updated frequently enough. An
accurate method would be to exponentially sample step lengths with
δℓ=minϵ(σt(ϵ)N)-1,
in space-oriented perspective or
δt=minϵ(v(ϵ)σt(ϵ)N)-1,
in time-oriented perspective, with v the velocity, σt the
total cross-section and N the number density of the medium. Then, at each
updated location (and energy), the type of collision must be sampled from
probability distributions. The probability of having a collision of the given
process (“pr”) can be calculated with
ppr=1-exp-N∫ifσprϵ(ℓ)dℓ,
where the index i refers to the beginning of the step and f to its end,
ℓ is the step length variable along the trajectory, and dℓ is an
infinitesimal step length. For time-oriented simulations, we have
equivalently
ppr=1-exp-N∫ifv(ϵ(t))σprϵ(t)dt.
Using these probabilities along a given step length or duration, there is a
chance that no interactions happen, but the energy of the particle is
guaranteed to be updated correctly.
The case of Geant4
In the Geant4 documentation, the stepping method presented in the previous
section is referred to as the “the integral approach to particle transport”.
This method is set up by default in Geant4 for impact ionisation and
bremsstrahlung. However, the way it is implemented does not exactly follow
what was described in the previous section. The description of the exact
implementation is out of the scope of this article but is presented in
detail in and
. The method relies on determining the maximum
of the cross-section over the step (σmax) using a parameter
αR (called “dR over range” in the Geant4 documentation)
that is also used to determine the step lengths. Another related parameter is
the maximum range parameter (ρmax), set to the default values of 1
and 0.1 mm for O1 and O4, respectively, and was never changed in the scope of
this study. The exact definition of these parameters is given in
and in the online Geant4 documentation (available
at https://geant4.web.cern.ch/support/user_documentation,
last access: 31 October 2018). The default value of αR is set
to 0.80 for O1 and to 0.20 for O4. We found that both values are not low
enough to be able to produce accurate results for the RREA probability
simulations presented in the next section. To make Geant4 able to produce
accurate RREA simulations using the multiple scattering algorithm, two
methods are possible.
The first method is to tweak the value of the αR parameter.
Its value is set to 0.80 by default for O1 and to 0.20 by default for O4. We
found that these default values are way too high to be able to produce
accurate RREA simulations, and values of αR<5.0×10-3 should be used, as presented in the next section.
The second method is to implement a step limiter process (or maximum
acceptable step). By default, this max step (δℓmax) is set to
1 km, and such a large value has no effect in practice, since the mean free
path of energetic electrons in STP air is orders of magnitude smaller.
Acceptable values of δℓmax depend on the electric field, and
we found that it should be set to 1 mm or less to produce accurate RREA
simulations, as presented in the next section. However, using this method
results in relatively long simulation time required to achieve an acceptable
accuracy, as the step is not adapted to the energy of the electrons. For
information, the relative impact on performance (in terms of requirements of
computation time) of tweaking the δℓmax and αR
parameters is presented in Appendix A.
Probability of generating RREA
As a first comparison test, we estimated the probability for an electron to
accelerate into the runaway regime and produce a RREA, given its initial
energy ϵ and some electric field magnitude E. Note that the
momentum of the initial electrons is aligned along the opposite direction of
the electric field, so that it gets accelerated. That gives maximum RREA
probabilities, as other alignments reduce the chance to produce a shower
(see, e.g. , Fig. 2.6). We defined this probability as
the fraction of initial (seed) electrons that created an avalanche of at
least 20 electrons above 1 MeV. Once this state is reached, there is no
doubt the RREA is triggered and can go on forever if no limits are set. The
number 20 is arbitrary to be well above 1 but small enough for computational
reasons. For some initial conditions, we also tested requirements of 30 and
50 electrons above 1 MeV, which resulted in very similar probabilities. This
study is somewhat similar to the works presented in
, , and , but
they all looked at the probability to have only a single runaway electron,
whereas we used the criterion of N=20 electrons above 1 MeV, which is a
stricter constraint. The difference between the two criteria is mainly
noticeable for low electric field (<0.4 MV m-1) and high seed
energies (>700 keV). A figure illustrating how the probability can change
with N is presented in Sect. S5.3 in the Supplement.
Relativistic avalanches probabilities calculated from Geant4
simulations for a specific point {ϵ=75keV,E=0.80MVm-1} (illustrated by a cross in
Fig. ) and for two stepping settings.
(a) Avalanche probability versus αR setting for
Geant4 O4 and Geant4 O1. δℓmax is set to the default value of
1 km. (b) Avalanche probability versus maximum step setting
(δℓmax) for Geant4 O4 and Geant4 O1. The parameter
αR is set to the default value of the models, which is 0.8 for
O1 and 0.2 for O4.
As a test case, we calculated the probability to produce RREAs as a function
of αR and δℓmax (these parameters are presented
in the previous section) for the configuration ϵ=75 keV,
E=0.80 MV m-1. This case was chosen because it showed a particularly
large sensitivity to the stepping methodology, as discussed later. The
results are presented in Fig. . Although
this configuration has a very low RREA probability for O1 and O4 by default
(where αR, respectively, is equal to 0.80 and 0.20, and δℓmax is 1 km for both), the probability increases as
αR decreases and converges to a value between 10 % and
12 % for both models when αR<5.0×10-3. The same
effect is observed when reducing δℓmax. In this case, the user
should not set δℓmax below the maximum range parameter, which is set to
1 mm for O1 and 0.1 mm for O4 by default (and never changed in the scope of
this article). When reducing the αR parameter to arbitrarily
small values, both Geant4 models converge to slightly different
probabilities: 10.7 % for O1 and 11.7 % for O4. We think this small
difference is not due to the stepping method, as reducing ρmax or
αR further does not produce a significant difference. It is
a probability due to other factors, in particular the difference in the
physical models and cross-section sets used. We encourage other researchers
to check if their simulations produce a RREA probability for this
{ϵ, E} setting that is consistent with our result.
(a) Relativistic avalanche probability comparison between
GRRR, REAM, O4 and O1. It shows three contour lines at 10 %, 50 % and
90 % as functions of seed (primary) energy ϵ and electric field
magnitude E. These contours are derived from the full probability scans
that are presented in the Supplement (Sect. S5). The cross at {ϵ=75keV,E=0.80MVm-1} highlights the point where we
studied the effect of the simulation stepping parameters (for O4 and O1)
on the probability; see Fig. .
(b) Five contour lines indicating the 0 %, 10 %, 50 %,
90 % and 100 % probabilities to generate a RREA as function of ϵ and E for the Geant4 O4 model
for which we could run a very large number of initial electrons
(> 50 000) to obtain curves with a very low noise
level.
As explained in Sect. , the final
electron spectrum is essentially driven by the minimum energy
ϵ2min of electrons that can create a RREA. Here, we can
clearly see this probability is strongly affected by the choice of the
αR and δℓmax simulation parameters, affecting
the accuracy of the stepping method, and that the values set by default for
these parameters are not precise enough to obtain correct RREA probabilities.
In order to help future researchers, we provide example Geant4 source codes
where the αR and δℓmax parameters can be
changed and their effect to be tested (see Sect. and the
“Code and data availability” section).
In Fig. a, we compare the contour lines of the
10 %, 50 % and 90 % probability of triggering a RREA as function
ϵ and E for the four models: Geant4 O4 (αR=1.0×10-3), Geant4 O1 (αR=1.0×10-3), GRRR and
REAM. The full RREA probability results in the ϵ, E domain for
each model are presented in Sect. S5 in the Supplement.
The most important difference between Geant4 and GRRR is present for energies
>200 keV and E fields <0.5 MV m-1. At 1 MeV, the level curves
are significantly different between the Geant4 models and GRRR: the 50 %
probability to trigger RREA for GRRR is approximately located at the 10 %
probability for O4 and at the 90 % probability for GRRR is located at
the 50 % probability for O1. The reason is probably similar to a point we
raised in our previous study : GRRR does not
include a way to simulate the straggling effect for the ionisation process.
By looking at Fig. 2 of , we can see that
200 keV is roughly the energy from where the difference in the spectrum of
GRRR, compared to codes that simulate straggling, starts to become
significant.
For low electron energy (<40 keV) and high electric field
(>2 MV m-1), GRRR and O4 present good agreement; however, O1
deviates significantly from O4. We investigated the effects of the stepping
parameters (αR, δℓmax and ρmax) and it
is clear that they were not involved in this case. We think the Møller
differential cross-section (with respect to the energy of the secondary
electron) used by O1 and extrapolated down to low energies leads to the
production of secondary electrons with average energies lower than the
Penelope model (used by O4), which includes the effects of the atomic electron
shells, and hence is probably more accurate. This hypothesis is confirmed by
looking at the shape of the differential cross-sections of impact ionisation,
whose plots are presented in Sect. S11.4 in the Supplement.
The RREA probability data for REAM are also displayed in Fig. 2a as the red
curves. The three REAM level curves show significantly higher noise than
the Geant4 data, mainly because the latter used 1000 electron seeds, whereas
the former used only 100. The algorithms used to calculate the level curves
were also found to impact the noise level. Nevertheless, the noise level is
low enough to be able to evaluate the consistency between the codes. REAM
shows a consistency with Geant4 (O1 and O4) within less than 12 % in the
full parameter range and less than 5 % in some part of it. The most
apparent deviations between REAM and Geant4 O1/O4 can be noticed for a seed
electron energy range between 50 and 300 keV, for the 50 % and 90 %
level curves, where there is a systematic, statistically significant
difference in the probability for REAM compared to Geant4 (REAM requiring
about a 10 % larger electric field or primary electron energy to reach the
90 % or 50 % contour levels). However, we do not expect such a
small difference to significantly affect the characteristics of the RREA
showers, such as the multiplication factors or the mean energies of the RREA
electrons. To test this quantitatively, a detailed comparison of the most
important characteristics of the RREA showers obtained with the four models
is presented in the following section.
In Fig. b, we show the 0 %, 10 %,
50 %, 90 % and 100 % probability contour lines for the Geant4 O4
model where we could run a very large number of initial electrons (“seeds”)
to obtain curves with a very low noise level. These are the most accurate
probabilities we could obtain. From this figure, it is clear that the RREA
probability for an electron of less than ≈10 keV is null for any
electric field below Ek≈3.0 MV m-1. Therefore, 10 keV is a
reasonable a lower boundary of ϵ2min (the minimum energy
at which a secondary electron can run away), and any simulation with an
electric field below Ek≈3.0 MV m-1 could use an energy
threshold (εc) of this value while keeping accurate
results. If electric fields with lower magnitude are used, it is also
reasonable to increase this energy threshold by following the 0 % level
curve showed in Fig. 2b.
Characterisation of RREA showers
We compared the output of the four models over 12 different electric field
magnitudes from E=0.60 to E=3.0 MV m-1. Two types of simulations
were set: record in time and record in distance (or space). This last choice
was made because the resulting spectra can change significantly depending on
the record method, as presented in Fig. 10 of .
All the curves presenting the simulation results are presented in the
Supplement, as well as the complete details on how the simulation should be
set up. In the following section, we discuss only the most important
differences we found between the four codes. We show the comparison of
avalanche scales in space and time in Sect. and in
Sect. the evolution to self-similar state. Finally, in
Sect. , we show the comparison of the self-similar
energy spectra of electrons and photons of the RREA.
Avalanche timescales and length scales
Figures and show
the avalanche length scales and timescales as a function of electric fields, for the
four models, together with their relative difference with respect to REAM.
Note that we could not compute any values for electric fields below
0.60 MV m-1, as we only used 200 initial electron seeds of 100 keV,
which could not produced enough showers. The choice of 200 initial electrons
is purely due to computational limitations. The avalanches' length and times
of the different models agree within ±10 %. There is also a
systematic shift of about 7 % between the two Geant4 models for both
timescales and length scales. The Geant4 O4 model is in principle more accurate than the
O1 model, since it includes more advanced models. For most of the electric
fields, O1 tends to be closer to REAM and O4 tends to be closer to GRRR.
Following , the avalanche length and time can
be fitted by the empirical models
λ(E)=c1E-c2,τ(E)=c3E-c4,
where c1 is in V, c2 and c4 in V m-1 and c3 in
s V m-1. The c2 and c4 parameters can be seen as two estimates
of the magnitude of the electric field of the minimum of ionisation for
electrons along the avalanche direction and also of the electric field
magnitude of the RREA threshold; both values are close. However, we note
that these fits neglect the sensitivity of the mean energy and velocity to
the electric field. These empirical fits are motivated from the relations
presented in Eqs. ()
and (), derived for the one-dimensional case.
First results of such fits were presented in
and , and they obtained consistent results.
Here, we will compare our results against Coleman and Dwyer.
The best fit values of the two models to the simulation data are given in
Table . The c1 parameter is directly linked to
the average energy of the RREA spectrum, though the definition of this
average energy can be ambiguous as energy spectra change significantly if
recorded in time or in space. The values given by all the codes are located
between 6.8 and 7.61 MV, and are all consistent with each other within a
95 % confidence interval, with the exception of O4, which slightly deviates
from O1. Combining the four values gives
c1‾=7.28±0.10MV.
By “combining”, we mean that the four values are averaged and the rule
σcomb=σ12+σ22+σ32+σ42/4
is used to “combine” the four uncertainty ranges. The value
c1‾ is consistent with the value of 7.3±0.06 MV given
in . Also, all the estimated values of the c2
and c4 are consistent with each other within a 95 % confidence
interval. Combining all the values of c2 and c4 gives
c2‾=279±5.6kVm-1c4‾=288±4.8kVm-1.
In addition, both values are also consistent with each other, leading to the final
value of c2,4‾=283.5±3.69kVm-1. These
values slightly deviate from the value of 276.5±2.24 obtained from
, if the values they obtained for the fits of
λ and τ are combined. The work of
used the REAM model too, in a version that should not have significantly
changed compared to the one used here. Thus, we think this difference is
purely attributed to differences in the methodology that was used to make
these estimates from the output data of the code. Concerning the c3
parameter, combining all the estimates gives
c3‾=26.8±0.32 ns MV m-1, which is slightly lower than
the value of 27.3±0.1 ns MV m-1 of
, but none of the values are consistent within
the 95 % confidence interval. For this case, we also think the slight
difference can be attributed to differences in methodology. Furthermore, the
ratio c1‾/c3‾ can also be used to determine an
average speed of the avalanche ≈β∥c along the
direction of the electric field (which also corresponds to the z direction),
and we can estimate β∥≈0.90, which is very close to what
was found in previous studies.
(a) Avalanche multiplication length as function of ambient
electric field for each of the codes included in this study.
(b) The relative difference of all other models with respect to
REAM. Table indicates the values of the fit
parameters.
(a) Avalanche multiplication time as function of ambient
electric field for each of the codes included in this study.
(b) The relative difference of all other models with respect to
REAM. Table indicates the values of the fit
parameters.
Values of the parameters of the fits (with 95 % confidence
intervals) for the simulations' data for avalanche scale in space and time,
using the models described by Eqs. ()
and (). See Figs.
and for the corresponding
curves.
CodeAvalanche length Avalanche time c1c2c3c4(MV)(kV m-1)(ns MV m-1)(kV m-1)REAM7.43±0.18290±9.527.6±0.91293±13G4 O17.50±0.10276±5.627.6±0.44290±6.3G4 O46.93±0.13285±7.525.9±0.28288±4.2GRRR7.25±0.30266±1826.2±0.76282±12Evolution to self-similar state
The photon and electron energy spectra of a RREA is known to converge in time to a self-similar solution,
where its shape is not evolving anymore, even if the number of particles
continues growing exponentially. It may also be referred to as the
“self-sustained state” or the “steady state” in the literature. At least
five avalanche lengths (or avalanche times) are required to be able to assert
that this state is reached. We propose to estimate this time by looking at
the mean electron energy evolution as a function of time. Notice that, as
already mentioned in the beginning of Sect. , this mean energy
recorded in time is different from the one recorded in distance, which is used in the
next section. We arbitrarily choose to evaluate this mean by averaging all
the energies of each individually recorded electron from 10 keV and above.
This choice of a 10 keV energy threshold (instead of a higher value, like
511 keV or 1 MeV) does not affect significantly the final estimate of this
time to self-similar state.We started with a mono-energetic beam of 100 keV
electrons, which is considered low enough compared to the self-similar state
mean energy of 6 to 9 MeV. To define the time to self-similar state
(Ts), we fitted the time evolution of the mean electron energy
ϵ‾ with the model
ϵ‾(t)=b1-b2×exp(-t/b3),
where b1 and b2 have a dimension of energy and b3 has a dimension of time, and
we define Ts=5b3, which is five e-folding times, i.e.
converged to 99.3 %. The evolution of electron spectra to self-similar
state is illustrated for the Geant4 O4 model in the Supplement
(Sect. S12.4). The values of Ts we estimated for the different
models are presented in Fig. , together
with relative differences of the models with respect to REAM. The relatively
high uncertainty (within 95 % confidence intervals) that can be seen in
the estimate of Ts is due to a combination of the confidence
interval from the exponential fit from the statistics of the number of seed
electrons that could produce a RREA and from the statistics of the particle
counts. For most cases, 200 initial seed were used, but for REAM, only
16 seeds were simulated for E≥2.2 MV m-1, and for GRRR, only
20 seeds were simulated above E≥2.0 MV m-1 because of
computation time limitations.
In Fig. , Geant4 O1, O4, GRRR and REAM
show consistent times to reach the self-similar state for all the
E fields. Notice that, for them, T(=Ts/5) is close to the
avalanche time value τ given in the top panel of
Fig. . For the low electric field of
0.60 MV m-1, it seems to take about 5 times longer to reach
self-similar state. For this field, there were only three electron seeds
that could produce a RREA, giving a large uncertainty on the estimate of
Ts, making it impossible to conclude on an inconsistency. From
0.60 to 1.8 MV m-1, where all data from codes have good
statistics, the times to self-similar state are consistent. From 2.0 to
2.4 MV m-1, the two Geant4 models and REAM are consistent, but GRRR
presents lower times by about -20 % to -50 %, but it is impossible
to conclude an inconsistency given the large confidence intervals. For
E-field magnitudes of 2.6 to 2.8 MV m-1, O1 and O4 present times
to self-similar state lower than REAM by about 50 %, which is significant
given the uncertainty intervals, whereas GRRR and REAM are consistent. We
could not find a clear explanation for it.
(a) Time to self-similar state as function of ambient
electric field for each of the codes included in this study.
(b) Relative difference with respect to
REAM.
RREA spectra
The Supplement (Sect. S6) presents all the comparison spectra we obtained for
photons, electrons and positrons for the electric field between 0.60 and
3.0 MV m-1. In this section, we discuss the most important differences
we could find between the four models.
Electrons
After the RREA electron spectra have reached self-similar state (which requires
at least five avalanche lengths or times), we recorded the energy spectrum in a
plane at a given distance (which is different for each electric field). Then,
we fitted it with an exponential spectrum model
∝exp(-ϵ/ϵ‾) (see also
Eq. ). Note that, for an exponential
distribution, the mean of the energy distribution is an estimator of its
parameter ϵ‾, justifying the bar notation. We chose to
evaluate the mean energy ϵ‾ for record at distances
because, contrary to time records, it produces spectra that can be perfectly
fit with an exponential distribution over the whole energy range (0 to
100 MeV). Therefore, in this case, only the mean RREA electron energy is
uniquely defined and does not depend on an arbitrarily chosen energy
threshold or fitting method. The mean energy ϵ‾ of the
exponential spectrum is calculated for the several codes as a function of
electric field E, as presented in Fig. . For
Geant4 O1, the entire simulation and analysis were done twice for maximum
allowed step length settings of δℓmax=1 cm and δℓmax=1 mm, to show that the first case generates totally
incorrect spectra, which is consistent with having incorrect RREA
probabilities (presented in Sect. 3). In addition, values of the mean energy
ϵ‾ for O1 with αR=1.0×10-3 and
δℓmax=1 cm are presented in Sect. 7 in the Supplement.
The data of Fig. 6 were fit following the model,
ϵ‾fit(E)=λ(E)(qE-F),λ(E)=βca1qEFa2+a3-1,
motivated by the facts that ϵ2min is roughly a power law
of E (see Fig. ) and λ is a
power law of ϵ2min (see Eq. ). It
has three adjustable parameters: a1, a2 and a3. We set F=0.28 MeV m-1, which is approximately the RREA threshold. The speed
β is set constant, equal to 0.90, because the RREA velocity does not
change more than 5 % over the range of electric fields we tested. This
model is in general agreement with the calculations of
, where λ(E) presents an approximately
linear relation with the electric field. Table
gives the parameters' best fits (with confidence intervals) for the different
models, and Fig. shows the corresponding
curves.
Mean electron energies at self-similar state (for distance record)
for different electric field magnitudes. The data points are fitted with the
model presented in Sect.
(Eq. ). The values of the fitted parameters are
presented in Table . To highlight the importance of
including step limitations, Geant4 O1 values are presented for two different
max step (δℓmax) settings: one that is not acceptable
(1 cm) and one that is acceptable (1 mm). The parameter αR
is set to its default value of 0.8 for O1 and 0.2 for
O4.
Values of the parameters of the fits (with 95 % confidence intervals)
for the electron mean energies using
Eq. (). F is set to
0.28 Me Vm-1. The corresponding curves are shown in
Fig. .
In Fig. , it is clear that the Geant4 O1 model
with δℓmax=1 cm presents a significantly higher
ϵ‾(E) than the other codes, with values ranging from 9.5
to 12.5 MeV. From the previous RREA probability simulations (see
Sect. ), we know that this δℓmax
parameter is not low enough, and so the results of this model can be
disqualified. However, when δℓmax is reduced to 1 mm,
the results of both Geant4 models are close. There seems to be a consensus
between Geant4 (O1 and O4) and REAM, which gives mean energy that is between
8 and 9 MeV and can vary up to 10 % depending on the electric field. For
all electric field magnitudes, GRRR shows a smaller average energy from
about 10 % less at 1 MV m-1 to about 20 % less at
2.8 MV m-1. The reason is certainly because GRRR only includes
radiative energy losses as continuous friction. This is actually a similar
difference to what has been observed and discussed in
concerning the high-energy electron beams, and
one can read the discussion therein for more details.
Figure compares the electron spectra
recorded at z=128 m (the electric field has a non-null component only in
the z direction, so that electrons are accelerated towards positive z)
for an electric field magnitude E=0.80 MV m-1, for a RREA generated
from 200 initial (“seed”) electrons with ϵ=100 keV. This record
distance was chosen because it corresponds to about 8.5 avalanche lengths,
giving a maximum multiplication factor of about 5000 for which there is no
doubt the RREA is fully developed and has reached self-similar state. This
electric field of E=0.80 MV m-1 was chosen because it is where we
could observe the most interesting differences between the models, and it
also happens to be the lowest for which we could build spectra with enough
statistics on all the models to be able to present a precise comparison. The
choice of 200 initial electrons is purely due to computational limitations.
In Fig. b, the error bars
represent the uncertainty due to the Poisson statistics inherent when
counting particles. The four models are consistent within 10 % between
20 keV and 7 MeV. Below 20 keV, we think the discrepancy is not physical
and can be attributed to the recording methods set up for the different
codes, which are not perfect and have a more or less important uncertainty
range (which is not included in the display errors bars and only based on Poisson
statistics). Above 7 MeV, O1 remains consistent with REAM overall, but O4
and O1 deviate significantly: up to 50 % for O4 and up to 90 % for
GRRR. For the last bin between 58 and 74 MeV, O4 and GRRR are inconsistent,
which is explained by the fact that GRRR does not include straggling for
bremsstrahlung (i.e. either explicit bremsstrahlung collision or some
stochastic fluctuations mimicking straggling). The deviations for the
high-energy part (>7 MeV) in the electron spectrum are significant for this
particular field (E=0.80 MV m-1); however, this is not true for all
electric fields, where the codes are overall roughly consistent, as seen in
the Supplement (Sect. S6). In principle, O4 should be more precise than O1
, as it includes more advanced models, yet we cannot
argue that O4 is more accurate than REAM. One way of deciding which model is
the most accurate might be to compare these results with experimental
measurements. However, in the context of TGFs and γ-ray glows, it is
complicated to get a proper measurement of electron spectra produced by RREA.
However, photons have much longer attenuation lengths than electrons and can
be more easily detected, e.g. from mountains, planes, balloons or satellites.
In the next section, we present and discuss the corresponding photon spectra.
(a) Electron (kinetic) energy spectra of Geant4 (O4 and
O1), REAM and GRRR for E=0.80 MV m-1, recorded at z=128 m. The
RREA is generated from 200 seed electrons of ϵ=100 keV.
(b) Relative difference between REAM and the three other models. The
error bars are calculated from the Poisson
statistics.
Photons
In Fig. , the photon spectra recorded at
z=128 m (the electric field has a non-null component only in the z
direction) for a magnitude E=0.80 MV m-1 are given for Geant4 O1/O4
and REAM, together with the relative difference with respect to REAM. The
reasons why these z and E values were chosen are given in the previous
section.
The error bars in the relative differences represent the uncertainty due to
the inherent Poisson statistics when evaluating particle counts. The Geant4
O1 and O4 models are consistent for the full energy range, except a small
discrepancy below 20 keV, which can be attributed to different physical
models, with O4 being more accurate in principle. In this case, it cannot be
attributed to recording methods, since they are exactly the same for both
Geant4 models. At 10 keV, the two Geant4 spectra are about 80 % larger
than REAM. With increasing energy, the discrepancy reduces and reaches
0 % at 100 keV. Above 100 keV, the three models show consistent
spectra. There may be some discrepancy above 30 MeV, but it is hard to
conclude since the uncertainty interval is relatively large.
As just presented, the main noticeable discrepancy between O1/O4 and REAM is
present below 100 keV. As far as we know, there is no reason to argue that
Geant4 gives a better result than REAM in this range, or vice versa. One way
to find out which model is the most accurate could be to compare these
results with real measurements. Are such measurement possible to obtain? Any
photon that an instrument could detect has to travel in a significant amount
of air before reaching detectors. The average path travelled in the atmosphere
by a 100 keV photon in 12 km altitude air is 1540±806 m. It
decreases for lower energies and is 671±484 m at 50 keV and 63.0±61.5 m at 20 keV. Note that these lengths have been evaluated from
precise Geant4 simulations and are smaller than the attenuation lengths at
the same energies, because photons gradually lose energy due to stochastic
collisions. These average travelled paths are too small for the photons to
have a reasonable chance to escape the atmosphere and to be detected by a
satellite. However, we cannot exclude that they may reach an airborne detector
located inside or close to a thunderstorm. As a side note, we want to
indicate that the vast majority (if not all) of the photons observed from
space with energies below a few hundred kilo-electronvolts (e.g. by the
Fermi space telescope; see ) had very likely more than
1 MeV when they were emitted. They lost some part of their energy by
collisions (with air molecules in the atmosphere and/or with some part of the
satellite) before being detected by the satellite. For information, a figure
presenting the probability of a photon to escape the atmosphere as function
of its primary energy for a typical TGF is presented in Sect. S14 in the Supplement.
(a) Photon energy spectra of Geant4 (O4 and O1) and REAM
for E=0.80 MV m-1, recorded at z=128 m. (b) Relative
difference between Geant4 (O1 and O4) and REAM. The error bars are calculated
from the Poisson statistics.
Other differences
In addition to what is presented so far in this article, the following points
should also be mentioned when comparing the results of the codes. The
corresponding plots are available in the Supplement.
The mean parallel (to the E-field direction) velocity β∥
of the avalanche is shown in Sect. S4.2 of the Supplement (labelled “mean Z
velocity”). We observe that GRRR is giving β∥ faster than all
the other codes, and O4 is systematically slower than REAM and O1, though the
differences are less than 2 %. The variation of β∥ towards
the electric field E is small, about 10 % for all codes. For increasing
E fields, electrons are less scattered and more focused in the field
direction, hence slightly increasing β∥.
The electron to (bremsstrahlung) photon ratio re/p was also calculated and compared
for different distance record in the RREA shower, and the corresponding plots
are presented in Sect. S3 in the Supplement. GRRR is excluded because it does
not include photons. For any electric field, the same discrepancy is
observed. At the beginning of the shower (<4 avalanche lengths),
re/p appears to be about 20 % larger for REAM compared to O1
and O4; then, the three models are consistent at a given distance, and finally
for more than about four avalanche lengths, the tendency is inverted and REAM
presents a re/p about 20 % smaller than Geant4. The magnitude
of this discrepancy is largely reduced for increasing electric fields. We did
not fully understand the reasons of these differences, and it may be due to
the bremsstrahlung models used. More investigations are
required.
The positron spectra have relatively low statistics (on the order of a few
hundred particles recorded) and are all quite consistent within the
relatively large uncertainties.
In the photon spectra obtained from particle records at fixed times,
REAM seems to show significantly less (at least a factor of 10) photon counts
than the two Geant4 models for most of the electric field magnitudes. For
some fields, it even shows a lack of high-energy photons, with a sharp cut at
about 30 MeV. It seems to point to a problem in the record method,
explaining why we chose not to discuss these spectra in the main article. The
spectra produced by the Geant4 O1 and O4 models for this case are consistent
with one another for all the E fields.
Conclusions
We have investigated the results of three Monte Carlo codes able to simulate
RREAs, including the effects of
electric fields up to the classical breakdown field, which is
Ek≈3 MV m-1 at STP. The Monte Carlo codes REAM, GRRR and
Geant4 (two models: O1 and O4) were compared. The main difference between the
Geant4 O4 and O1 models is the inclusion of more precise cross-sections for
low-energy interactions (<10 keV) for O4.
We first proposed a theoretical description of the RREA process that is
based on and incremented over previous published works. Our analysis confirmed
that the relativistic avalanche is mainly driven by electric fields and the
ionisation and scattering processes determining ϵ2min,
the minimum energy of electrons that can run away. This is different from some
of the previous works that speculated that the low-energy threshold
(εc), when changed from 1 keV to 250 eV, was the most
important factor affecting the electron energy spectra
.
Then, we estimated the probability to produce a RREA from a given electron
energy (ϵ) and a given electric field magnitude (E). We found that
the stepping methodology is of major importance, and the stepping parameters
are not set up satisfactorily in Geant4 by default. We pointed out which
settings should be adjusted and provided example codes to the community (see
Sect. and the “Code and data availability” section). When properly
set up, the two Geant4 models showed good overall agreement (within
≈10 %) with REAM and GRRR. From the Geant4, GRRR and REAM
simulations, we found that the probability for the particles below
≈10 keV to accelerate and participate in the penetrating radiation
is actually negligible for the full range of electric field we tested (E<3 MV m-1). It results that a reasonable lower boundary of the low-energy threshold (εc) can be set to ≈10 keV for
any electric field below Ek≈3 MV m-1 (at STP), making it
possible to have relatively fast simulations. For lower electric fields, it
is possible to use larger εc, following a curve we
provided (Fig. 2b).
The advantage of using more sophisticated cross-sections able to accurately
take into account low-energy particles could be probed by comparing directly
the O1 and O4 models. They showed minor differences that are mainly visible
only for high E fields (E>2 MV m-1), where low-energy particles
have more chances to run away.
In a second part, we produced RREA simulations from the four models and
compared the physical characteristics of the produced showers. The two Geant4
models and REAM showed good agreement on all the parameters we tested. GRRR
also showed overall good agreement with the other codes, except for the
electron energy spectra. That is probably because GRRR does not include
straggling for the radiative and ionisation energy losses, hence implementing
these two processes is of primary importance to produce accurate RREA
spectra. By comparing O1 and O4, we also pointed out that including precise
modelling of the interactions of particles below ≈10 keV provided
only small differences; the most important being a 5 % change in the
avalanche multiplication times and lengths. We also pointed out a discrepancy
from Geant4 (O1 and O4) compared to REAM, which is a 10 % to 100 %
relative difference in the low-energy part (<100 keV) of the photon energy
spectrum for an electric field of E=0.80 MV m-1. However, we argued that
it is unlikely to have an impact on spectra detected from satellites.
Recommendations
From the experience of this study, we give the following general
recommendations concerning RREA simulations:
Codes should be checked/tested/benchmarked using standard test set-ups. In the
Supplement, we provide a precise description of such tests. In Sect. 7 of
this article, we provide links to download the full data set we obtained for
the codes we tested (Geant4 with two set-ups, REAM and GRRR), as well as
processing scripts. We also provide the source code of the Geant4 codes.
Custom-made codes should be make available to other researchers or at least the results
they give for standard tests.
In order to make it possible to compare results from different studies, the methodology
used to derive a given quantity should be rigorously chosen and presented clearly somewhere.
Extending the recommendations of , we concluded that to get
an accurate RREA electron spectra above 10 MeV, radiative loss
(bremsstrahlung) should not be implemented with uniform friction only:
straggling should be included. Straggling should also be included for
ionisation energy loses below the energy threshold.
Concerning the usage of Geant4 for simulating RREA:
Default settings are not able to simulate RREA accurately. To get accurate RREA results,
one of the following tweaks is possible:
Changing the αR (“dR over range”) parameter of the electron/positron
ionisation process to 5.0×10-3 or less gives the
best ratio between accuracy and computation time. Leave the “final range”
parameter at 1 mm (default value) or less.
Setting up a step limitation process (or a maximum acceptable step) to 1 mm or
less will significantly increase the required computation time.
Using the single (Coulomb) scattering model instead of multiple scattering
(the two previous tweaks relying on the multiple scattering algorithm)
will substantially increase the necessary computation time. This is because
multiple scattering algorithms were invented to make the simulation run
faster by permitting to use substantially larger (usually >10 times) step
lengths compared to a pure single scattering strategy, while keeping a
similar accuracy.
In the “Code and data availability” section, we provide a link to Geant4 example source codes implementing
these three methods.
Compared to using the default Møller/Bhabha scattering models for ionisation,
the usage of more accurate cross-sections, e.g. taking into account the electrons' molecular
binding energies (as done for the Livermore or Penelope models), only leads to minor differences.
The full simulation output data of the four
models are available through
the following link: https://filesender.uninett.no/?s=download&token=738a8663-a457-403a-991e-ae8d3fca3dc3 (Sarria, 2018a).
The scripts used to process these data to make the figures of the Supplement
are available in the following repository:
https://gitlab.com/dsarria/HEAP2_matlab_codes.git (Sarria, 2018b).
The full GRRR source code is available in the following repository:
https://github.com/aluque/grrr/tree/avalanches (last access: 30 October 2018; Luque, 2014).
The Geant4 source code for the RREA probability simulations is available in
the following repository: https://gitlab.com/dsarria/av_prob.git (Sarria, 2018c).
The Geant4 source code for the RREA characterisation simulations is available
in the following repository:
https://gitlab.com/dsarria/RREA_characteristics.git (Sarria, 2018d).
Geant4 relative performance
Table presents the relative computation times it takes to
complete the simulation with an electric field magnitude of
1.2 MV m-1, 100 initial (“seed”) electrons with initial energy
ϵ=100 keV and a stop time (physical) of 233 ns. The fastest
simulation uses Geant4 with the O1 physics list and δℓmax=10 cm and took 4.53 s to complete on one thread with the microprocessor
we used. The simulations with the O4 physics list with δℓmax=1 mm require about 400 times more computation time.
Setting up δℓmax=1 mm or lower is necessary to
achieve correct simulation of the RREA process, as argued in
Sect. . To achieve it for the full range of electric fields
we tested (in a reasonable amount of time), it required the use of the
Norwegian Fram computer cluster. The simulations with δℓmax=0.1 mm for all electric fields could not be achieved in
a reasonable amount of time, even by using the computer cluster.
On the other hand, if δℓmax is left at its default value
(1 km) and the αR parameter is tweaked instead, accurate
simulations can be achieved with a value of αR=5.0×10-3 or lower. It requires almost an order of magnitude less computation
time compared to using δℓmax=1 mm.
Computation time needed by different Geant4 configurations for the
simulation of the same physical problem, relative to the Geant4 O1
δℓmax=10 cm case. Two parameters are tested: the maximum
allowed step (δℓmax) and the “dR over range”
(αR).
Model Option 1 (O1)Option 4 (O4)δℓmax10 cm16.491 cm11.527.21 mm2223930.1 mm21003740αR (default)0.800.20αR0.80≈12.440.202.617.660.0507.1236.50.005021.01260.001041.7224δℓmax (default)1 km1 km
The supplement related to this article is available online at: https://doi.org/10.5194/gmd-11-4515-2018-supplement.
DS, CR and GD designed the tests. DS, CR and GD wrote most of the manuscript.
DS, GD and CR conducted the data analysis and made the figures and tables. DS carried out
the Geant4 simulations and provided the data. AL carried out the GRRR simulations and provided
the data. JRD and KMAI carried out the REAM simulations and provided the data. NO, KMAI, JRD,
UE, ABS and ISF provided important feedback on and review of the text.
The authors declare that they have no conflict of interest.
Acknowledgements
This work was supported by the European Research Council under the European
Union's Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 320839
and the Research Council of Norway under contracts 208028/F50 and
223252/F50 (CoE). For part of the results of this work, it was necessary to
use the Fram computer cluster of the UNINETT Sigma2 AS, under project no.
NN9526K.
Gabriel Diniz is supported by the Brazilian agency CAPES. Casper Rutjes acknowledges funding
by FOM project no. 12PR3041, which also supported Gabriel Diniz's 12-month stay in
the Netherlands. Ivan S. Ferreira thanks CNPqs grant PDE(234529/2014-08) and also
FAPDF grant no. 0193.000868/2015, 03/2015.
This material is based in part upon work supported by the Air Force Office of
Scientific Research under award no. FA9550-16-1-0396. The authors
would like to thank the two referees, Ashot Chilingarian and an anonymous referee, for their valuable
comments and suggestions that helped to improve the quality of this work.
Edited by: Josef Koller
Reviewed by: Ashot Chilingarian and one anonymous referee
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