22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Analysis of recombination and relaxation of non-equilibrium air plasma
generated by short time energetic electron and photon beams
M. Maulois, M. Ribière, O. Eichwald and M. Yousfi
Université Paul Sabatier de Toulouse, CNRS, Laplace, 118 Route de Narbonne, FR-31062 Toulouse Cedex, France
CEA/DAM, FR-46500 Gramat, France
Abstract: The present contribution is devoted to the study of the evolution, recombination
and relaxation of non-equilibrium air plasma generated by short time energetic photon and
electron beams. We described the reaction scheme used and the numerical method
modeling the density evolution of the plasma species. The role of the main species and
reactions during the different stages of the air plasma evolution are identified and analyzed.
Keywords: Energetic electron and photon interactions, Non-equilibrium air plasma,
collision cross-sections
1. Introduction
The problem of energy deposition of electron and
photon beams in gases and the analysis of the associated
plasmas is of interest in the areas of for instance the
interactions of radiation with matters [1], the physics of
upper atmosphere [2], the electron-beam generated lasers
[3] or electrical discharges [4], the x-ray radiography [5].
Non equilibrium air plasmas generated by energetic
photon and electron beams can also be, as in the
framework of the present work, of great interest in the
electromagnetic perturbations of electronic devices due to
plasma-induced electromagnetic field (see e.g., [6]
showing the non-negligible effects of the plasma-inducedelectromagnetic field).
We have taken into account high energy photon and
electron beams simultaneously released in ambient air
during a short time (0.5 ns). Photon energy is chosen
equal to 0.35 MeV during this short time and the initial
electron energy to 1 MeV. During this short time of
particle beam application, strong ionization and
dissociation of the air molecules are expected thus leading
to air plasma generation with a certain ionization degree.
After this first stage of plasma formation lasting 0.5 ns,
there is a second stage dominated by recombination
processes
where
secondary
electrons
having
sub-ionization energy interact with the air plasma during
its slow relaxation towards its initial conditions. The
choice of such short time for particle beams is aimed to
validate the considered chemical kinetics reaction scheme
and the associated basic data.
Section 2, following this introduction, is devoted to the
description of the energy profile of the particle beams and
the considered reaction scheme for the study of the
formation of the evolution of the present air plasma
initially composed by synthetic air (i.e., 80% N 2 and
20% O 2 ). It is also devoted to the associated basic data
(reaction coefficients) taken either from literature or
calculated from collision cross sections. Formalism of
chemical kinetics model and the numerical scheme are
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described in section 3 while the obtained results on the
plasma formation and relaxation are discussed and
analyzed in section 4.
2. Energy profile of particles, reaction scheme and
basic data
2.1 Energy profile of particles
When short time pulsed beams of primary energetic
electrons and photons interact with air, energetic particles
of the beams lose energy mainly through inelastic
collision processes. The evolution of photon energy
during the beam application can be obtained from Monte
Carlo simulation taken into account the dominant inelastic
photons collisions corresponding to the present work
energy range (photo-ionization, Compton collision and in
a less degree pair production). However, in the case of a
very short time of the pulsed beam, the mean photon
energy can be assumed as quasi-invariant even though the
evolution of the mean photon energy E ph (t) from an initial
energy E ph (t=0) can be obtained from the following
relation considering the energy losses due to collisions
with the background gas:
E=
E=
0) e
ph ( t )
ph ( t
− v ph ,tot t
(1a)
where v ph,tot is the product of the background gas density
N, the light velocity c and the total collision cross-section
σ ph,tot (N x c xσ ph,tot ).
Furthermore, the time evolution of the electron mean
energy E e (t) from an initial energy E e (t=0) of primary
electrons can be approximated by the following relation
obtained from an analytic solution of the conservation
equation of electron mean energy assuming homogeneous
medium, constant total collision frequency v e,tot and
without considering the action of external forces [7]:
2m
3
3

 − ve ,tot t
0)
Ee (t ) =k BTgas +  Ee (t =−
k BTgas  e M
2
2


(1b)
1
m and M are respectively the electron and gas masses. It
is noteworthy that the electron energy relaxes in a long
time scale towards gas energy 3/2 k B T gas (Tgas and k B
being gas temperature and Boltzmann constant).
It is also possible to consider a more rigorous time
evolution of the electron mean energy from the solution
of a more complete energy conservation equation. Such
equation can be written by considering individually each
process (elastic, attachment, recombination, excitation,
ionization, Bremsstrahlung radiative loss and also the
possible source term S ph due to photon impacts. This
formalism is detailed in reference [8].
These relations (1a) and (1b) of the time evolution of
photon E ph (t) and electron E e (t) energies have been
considered in the case of short time pulsed particle beam.
2.2 Reaction scheme of air plasma
As soon as the particle beams impact the background
gas, this leads to the generation of different plasma
species that in turn can disappear or generate new species
following various collision processes. The electron
creations by photon impacts are due to photo-ionization,
Compton collision and pair production while energetic
electron impacts lead to air ionization, dissociation and
also creation of long living excited states (or metastables)
that can in turn be ionized or dissociated thus generating
new or further species. Therefore, the species thus
formed through interactions with air molecules or atoms
by high energetic particles are secondary electrons and
photons, single and multi-ionized atoms and molecules,
dissociated molecules and metastable states. All these
species constitute a kind of non-equilibrium plasma with
ionization degree depending on the energy and the time
duration of the initial pulsed particle beams.
Obviously the different plasma species can interact with
background air and also with themselves.
These
interactions, occurring mainly in a longer time scale than
the short time beam duration, can be for instance
recombination, charge transfer, electron detachment, ion
conversion, stepwise and Penning ionizations. During
this longer time scale, there are also collisions occurring
between secondary electrons and background gas. The
reactions corresponding to all these primary and
secondary processes that have been considered in the
present work are detailed in [8]. In fact, we have
considered 25 species (photons: hν, electrons: e-, neutrals
in background states: N 2 , N, O 2 , O, NO, positive ions:
N 2 +, N 2 ++, N+, N++, O 2 +, O 2 ++, O+, O++, N 2 O 2 +, N 4 +, N 3 +,
NO+, O 4 +, negative ions: NO-, O 2 -, O-, and nitrogen
metastables: N 2 (A3Σ+ u ) and N 2 (a’Π g )) interacting
following 175 reactions. The main creation and loss
processes between the 25 considered species can be
summarized as following:
- Reactions involving photons as photo-ionization
processes leading to single and double ions and
Compton interactions
- Reactions involving electrons as ionizations by
electron impacts leading to single and double ions,
2
-
electronic dissociation, nitrogen metastable formation,
stepwise ionizations, electron attachments, radiative
electron recombination, two- and three-body electron
recombination
Electron detachments
Recombination involving positive and negative ions
Two and three body charge transfers
Recombination involving metastable and neutral
Penning ionizations
Two and three body neutral recombination.
2.3 Basic data
The determination of the density evolution during the
considered time scale, lasting between the plasma
generation towards the plasma recombination and its
relaxation, for the different plasma species needs the a
priori knowledge of the reaction coefficients of each
considered interaction. In the case of the photon and
electron impacts taken into account in the present reaction
scheme, the collision cross sections are generally known
and come from the literature. Therefore, using the known
data of collision cross sections, the reaction coefficients
can be calculated from relation (2a) for a given photon
reaction number i having a photon mean energy E ph (t)
already determined from previous relations (1) or (2):
k ph ,i ( E ph (t )) = σ ph ,i ( E ph (t ))ϕ ( E ph (t ))
(2a)
k ph,i is the reaction coefficient of first order ( s-1) of the
photon reaction number i while σ ph,i (in cm2) is the
corresponding collision cross section depending on
photon energy. As photon velocity is constant, there is no
need to consider a photon distribution as for instance in
the case of electrons since we considered mono-energetic
photons (i.e., corresponding to a Dirac photon distribution
at energy E ph (t)) and the reaction coefficient is therefore
directly obtained from the photon flux ϕ (in cm-2s-1). The
photon flux is representative of the studied case of energy
deposition of the considered ionizing radiation beams;
ϕ is obtained from Monte Carlo simulation method using
similar procedures and processes as PENELOPE code [9].
In the case of reaction coefficient k e,j (in cm3/s)
corresponding to the electron impact number j for an
electron mean energy E e (t) already determined from
previous relations (1) or (2), we can write:
ke, j ( Ee (t )) =
1
2
σ e, j (e e )e e 2 f ( Ee (t ), e e )d e e
∫
m
(2b)
σ e,i is the associated collision cross-section depending on
electron kinetic energy e e and f(E e (t), e e ) is the electron
distribution function assumed Maxwellian with a mean
energy E e (t).
The reactions coefficients versus mean energy are
displayed in figure 1 in the case of photons and in figure 2
in the case of electrons.
The collision cross-sections of photon impacts used to
calculate the photon collision frequencies for N 2 and O 2
displayed in figure 1 are taken from reference [10]. The
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references for the other species are detailed in [8]. It is
noteworthy that when it was necessary such collision
cross sections are extended towards high energy range (up
to several MeV) in coherence with appropriate high
energy formalisms.
The collision cross-sections of electron impacts used for
the calculation of electron reaction coefficients displayed
in figure 2 are detailed in reference [8]. Here also
collision cross-sections have been extended when that
was necessary towards high energy range from
appropriate formalisms.
Furthermore, in the case of reactions involving heavy
particles as charge transfer, recombination between
positive and negative ions, electron detachment, reaction
between excited species and neutral recombination, the
reaction coefficients are taken from the literature using
Arrhenius formulae for the reaction number h between
heavy species at temperature T s .
Due to the short time duration of the energetic particle
beams, the temperature T s of each considered heavy
species is assumed close to the initial air temperature. In
fact, a real increase of the temperature of heavy species
requires a time scale much more higher, of several microseconds in air at atmospheric pressure in order to
accumulate a huge number of interactions between more
particularly free electrons and ions. Indeed, it is known
[11] that such Coulomb interactions, particularly efficient
when ionization degree increases, highly contribute to the
ion heating thus leading to a global temperature rise of the
neutral species due to their interactions with the heated
ions.
3. Chemical kinetics model: formalism and numerical
method
Assuming that the space gradients can be neglected
(mean volume approximation), the evolution of the
density of the plasma species generated by the ionizing
radiation (energetic photons and electrons) is governed by
a strongly coupled ordinary system of first order time
dependent differential equations. This system can be
written in a general form as:
 

dn (t ) / dt = f (n (t ), t )
Fig. 1. Calculated single photo-ionization collision
frequencies of O 2, NO, N 2, O and N versus photon energy
with double photo-ionization collision frequencies of O
and N versus photon energy.
Fig. 2. Calculated ingle ionization coefficients by
electron impacts of NO, N 2 , O 2, O and N with double
ionization coefficients by electron impacts of N, O, NO,
N 2 and O 2 versus electron mean energy.
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
n (t ) is the vector of densities ni (t ) , i varying from 1 to
the total number of considered particles (26 in our case).
 

f (n (t ), t ) is the vector of source terms f i (n (t ), t ) which
takes into account all the reactions involved in the
creation and loss of the corresponding particles i. For a

given species i, f i ( n (t ), t ) can be written as follow in for
instance the case of two body kinetics reactions:

f i (n (t ), t ) =
∑ k gain, j (t )n1 j (t )n2 j (t ) −∑ kloss,m (t )n1m (t )n2m (t )
j
m
k gain,j (t) is the reaction coefficient of the reaction j
involved in the creation of the species i, while n 1j (t) and
n 2j (t) are the densities of the two species interacting in the
two body reaction j. k loss,m (t) is the reaction coefficient of
the reaction m involved in the disappearance of the
species i, while n 1m (t) and n 2m (t) are the densities of the
two species interacting in the two body reaction j. In this
case, the reaction coefficients are expressed in cm3/s. In
the case of a three body reaction, the reaction coefficient
is expressed in cm6/s and the corresponding source term is
the product of a reaction coefficient with the densities of
the 3 interacting species.
The numerical solution of such strongly coupled system
of equations requires some precautions in order to avoid
the numerical instabilities because of the stiffness of those
equations.
Stiffness is due to the very different
magnitudes of the reaction coefficients leading to very
different time scales for the evolution of the various
considered. Similar numerical method, already used
elsewhere [12] and [13] in non-equilibrium plasma
chemistry, is detailed in reference [14].
3
4. Results and discussions
The initial conditions, before the impacts of energetic
photon and electron beams, required for the chemical
kinetics model concerns:
- initial background gas composed by synthetic air at
atmospheric pressure (i.e., 80% N 2 and 20%O 2 ) and
ambient temperature (300 K)
- initial density assumed equal to 103 cm-3 for the most
considered plasma species.
Energy profiles of photon and electron beams are
already given in section 2 and the reaction coefficients of
each considered process are evoked in section 2.
Figure 3 display time evolution of electron density with
some neutral densities when ambient air is irradiated by
the short time (0.5 ns) photon and electron beams while
figure 4 corresponds to the case with only the electron
beam (1 MeV for initial energy). As expected, the
creation of new electron by photon processes starts early,
at about femtosecond for Compton processes while
electron ionization occurs 2 decades later at tenth of
picosecond (about 0.1 ps). This is simply due to the
higher magnitude of photon collision frequency (around
1015 s-1) in comparison to the electron one (7x1012 s-1).
Furthermore, these results lead to several remarks.
N 2 ionization by electron impacts at the same energy is
much higher (around 10-18 cm2).
The magnitude of electron density is also correlated to
the magnitude of the neutral target species. Indeed, in the
case of Fig. 3 with photon and electron beams, molecule
dissociation start early due to photon interactions and the
densities of neutral species (N 2 , O 2 , N and O) are
necessary lower. This obviously favors a lower electron
source term. For instance, at the end of irradiation
(0.5 ns), N 2 density in Fig. 3 is around 106 cm-3 and O 2
density around 1.9 x105 cm-3 while without photon, these
densities are higher, around 3 decades for both molecules.
Results on ion densities (not shown here) confirm this
trend since gas irradiation with both photon and electron
beams generally leads to stronger ionization processes
and therefore higher ion densities.
At the end of the irradiation (0.5ns), due to the strong
dissociation processes, the magnitude of the neutral
densities “n” displayed in figs 3 and 4 follow this order:
n N > n O > n N2 > n O2 > n NO . Then after the beam
irradiation we assumed that the air plasma is governed by
secondary electrons having sub-ionization energy without
any effect of photon interactions which are neglected.
In these conditions, the air plasma enters mainly in a
recombination and relaxation stage. For instance, the
time evolution of electron density that obviously depends
on the magnitude of secondary electron energy (in both
tested cases: 1 eV in fig. 3 and 10 eV in fig. 5) relaxes
during this transient stage towards to its initial value
under mainly the effect of two and three body electron
recombination processes.
Fig. 3. Electron density with neutral species densities in
the case of both photon beam with initial energy of
0.35 MeV and electron beam with initial energy of 1 MeV
applied during 0.5ns (secondary electron energy = 1 eV).
During the beam irradiation corresponding to the air
plasma formation, electron density at the end of the beam
(0.5 ns) is of about 4.9 x 1019 cm-3 when photon and
electron beams are both applied (Fig. 3) while it becomes
about 8% higher (about 5.3 x 1019 cm-3) in the case of the
alone electron beam (Fig. 4). As the source term of
electron density depends both on collision frequency and
the densities of colliding species (particularly N 2 , O 2 , N
and O), the magnitude of electron density can be first
correlated to the magnitude of the associated collision
cross sections. For instance, single N 2 photo-ionization is
about 5x10-27cm2 at 0.35 MeV and N 2 Compton
cross-section of about 5x1024 cm-27cm2 at 0.35 MeV while
4
Fig. 4. Electron density with neutral species densities in
the case of only electron beam with initial energy of
1 MeV applied during 0.5 ns (Secondary electron energy
= 1 eV).
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[4]
[5]
[6]
[7]
[8]
[9]
[10]
Fig. 5. Electron density with neutral species densities in
the case of both photon beam with initial energy of
0.35 MeV and electron beam with initial energy of 1 MeV
applied during 0.5 ns (Secondary electron energy = 10 eV).
Last, this coherent relaxation behavior shows to a kind
of validation of the chosen kinetics reaction scheme and
the associated basic data.
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