Plasma diagnostics and simulations
Zero-dimensional plasma kinetics with
ZDPlasKin
Introduction
The dynamics of species density [Ni] is described by kinetic equation
d[Ni]
dt
=
j
Sij, (1)
where the source term for specie i consists of a sum of corresponding reaction sources Sij . For
example, the source term for reaction
aNi + bNl + [δ ] → a Ni + cNm + [δ ] (2)
can be expressed using reaction rate Rj and reaction constant rate kj as follows:
Rj = kj[Ni]a
[Nl]b
, (3)
Sij = (a − a)Rj. (4)
The evolution of the gas temperature can optionally be taken into account, through energy
loss/gain terms [δ ]. Solution of the heat transport equation under adiabatic isometric approximation
in the form
Ngas
γ − 1
dTgas
dt
=
j
±δ j · Rj + Ppelast · [Ne] + Qsrc (5)
gives for known specic gas heat ratio γ the evolution of gas temperature Tgas. The second term
in the right side Ppelast ·[Ne] is Joule heating due to the electron current and corresponds to elastic
election-neutral collisions. The third term Qsrc represents other heat sources.
ZDPlasKin and Bolsig+
ZDPlasKin is a Fortran 90 module designed to follow the time evolution of the species densities
and gas temperature in non-thermal plasmas with an arbitrarily complex chemistry described by
model equations above. A Boltzmann equation solver (BOLSIG+) incorporated in ZDPlasKin
provides values of electron transport and rate coecients when the electron energy distribution
function is non-Maxwellian. ZDPlasKin is now available as freeware and can be downloaded from
the following address: www.zdplaskin.laplace.univ-tlse.fr.
BOLSIG+ is a free and user-friendly computer program for the numerical solution of the
Boltzmann equation for electrons in weakly ionized gases in uniform electric elds, conditions
which occur in swarm experiments and in various types of gas discharges and collisional lowtemperature
plasmas. Under these conditions the electron distribution function is non-Maxwellian
and determined by an equilibrium between electric acceleration and momentum and energy losses
in collisions with neutral gas particles.
The main utility of BOLSIG+ is to obtain electron transport coecients and collision rate
coecients from more fundamental cross section data, which can then be used as input for uid
models.
The principles of BOLSIG+ can be summarized as follows:
Návody pro fyz. praktikum (verze June 7, 2018) 2
• Electron-neutral collision cross sections versus electron energy are read from input les.
• It is assumed that the electric eld and collision probabilities are constant in space and time
and that there are no boundaries.
• Angular dependence of the electron velocity distribution is approximated by two-term Legendre
expansion. Electron production/loss due to ionization/attachment is assumed to result
in exponential growth/decay of the electron number density.
Under the above assumptions, the Boltzmann equation reduces to a convection-diusion equation
with non-local source term in energy space, which is then discretized by an exponential scheme
and solved for the electron energy distribution function by a standard matrix inversion technique.
Various electron transport coecients and rate coecients are calculated and accessible in dierent
numerical/graphical forms and output le formats. Additional options are provided for AC electric
elds, crossed electric and magnetic elds, electron-electron and electron-ion collisions.
Linking of BOLSIG+ with ZDPlasKin is done automatically wia BOLSIG+ module distributed
within ZDPlasKin package.
BOLSIG+ as a separate executable program can be downloaded at https://www.bolsig.
laplace.univ-tlse.fr/
Cross section data
Cross section data can be found at www.lxcat.net (pronounced "elecscat"), an open-access website
for collecting, displaying, and downloading electron and ion scattering cross sections, swarm
parameters (mobility, diusion coecient, etc.), reaction rates, energy distribution functions, etc.
and other data required for modeling low temperature plasmas. The available data bases have
been contributed by members of the community and are indicated by the contributor's chosen title.
All download from this site data is compatible with ZDPlasKin package. Use of the data from
this site in publications should be accompanied by proper references. Original references should
be used where possible and reference should be made to the specic database(s) from which data
were retrieved, the LXCat site address, and the retrieved date.
Tasks
1. Download and nd out how to use ZDPlasKin and BOLSIG+.
2. Download cross section les for Ar from www.lxcat.net.
3. Run test case corresponding to an Ar plasma consisting of electrons, atomic ions, and
neutrals. The charged particles are supposed to be generated by direct electron impact
ionization and lost by 3-body recombination
e + Ar → e + e + Ar+
, k1 (6)
e + Ar+
+ Ar → Ar + Ar, k2 (7)
The recombination rate coecient assumed to be constant k2 = 10−25 cm6/s, and the
reduced eld strength, E/N = 50 Td, is also constant. This system has a simple asymptotic
solution.
4. Using BOLSIG+ separately, plot electron energy distribution function for Ar at E/N = 50
Td.
5. Using BOLSIG+ create tables for transport parameters and reactions rates as a function of
E/N from 10 to 1500 Td.