Particle-in-Cell Modeling of Low-Temperature Plasma
Dmytro SydorenkoUniversity of Alberta, Edmonton, Canada
2
Motivation
Particle-in-cell simulations are important numerical tools in studying plasma properties.
They are invaluable in understanding kinetic effects, especially when experimental measurements are difficult to obtain or interpret.
3
Outline
Possible methods of kinetic numerical description of a plasma.
PIC basics: Charge and force weighting Leap-frog explicit algorithm Instability of the explicit algorithm
Direct implicit method Particle emission from walls Monte-Carlo model of electron collisions with
neutrals Coulomb collisions
4
Recommended literature
R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, Adam Hilger, Bristol and New York, IOP Publishing, 1988.
C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, Bristol and Philadelphia, IOP Publishing, 1991.
5
Kinetic description required for plasmas with low collision frequency Plasma density 1011-1012 cm-3 . Neutral gas pressure 1-100 mTorr . Electron temperature 1-50 eV,
corresponding thermal velocities 6x105-4.2x106 m/s .
Frequency of electron-neutral collisions 106-109 s-1.
Size of a device 10 cm. Electron mean free path 10-3-1 m.
3 kW Hall thruster in PPPL(from htx.pppl.gov)
ICP for material processing (University of Saskatchewan Plasma Physics Laboratory / PLASMIONIQUE Inc )
Fluid description
Kinetic description
6
The distribution function represents the number density of particles in 6-D coordinate-velocity space.
One can use EVDF to calculate density flux pressure
EVDF is described by the Vlasov-Boltzmann equation
Electron velocity distribution function),,( tvxf
),( vx
vdtvxftxn 3),,(),(
vdtvxfvtxj 3),,(),(
vdtvxfvmtxp 32 ),,(3
),(
COLLt
fvfBvE
mq
rfv
tf
v
dvvfdN )(
dv
f(v)
7
Direct solution of the Vlasov equation
Define EVDF on a grid in the coordinate-velocity space.
Advance EVDF using e.g. splitting scheme [Cheng and
Knorr, J.Comput.Phys., 1976] . flux-corrected transport
algorithm [Boris and Book, J.Comput.Phys., 1973].
semi-Lagrangian algorithm [Sonnendrucker et al., J. Comput.Phys., 1999].
The ideal method of advancing should preserve monotonicity, be conserving, do not disperse sharp gradients, adapt resolution to reproduce small scales.
Example of 1-D f(L,v)
Result of advancing with systematic interpolation error.
8
Lagrange (water-bag) scheme
A “water-bag” is a contour in the plane “coordinate –velocity” along which the EVDF is constant.
Such a contour is represented as a set of markers. Each marker is advanced according to the equations of motion.
From [Berk and Roberts, Phys. Fluids, 1967].
2/)()()( 2,
2, fvvLJ icici
9
The water-bag method: advantages and shortcomings
Good: No interpolation errors. Integration of EVDF is very
easy. With many contours, one can
approximate a smooth EVDF.
Bad: Contour processing
algorithms are complex. This method is very difficult to
extend to higher dimensions.
10
a water bag
11
Solving Vlasov equation for multiple dimensions is numerically very costly Consider a plasma with Te=1eV, ne=1011 cm-3, size L=1cm.
thermal velocity is 5.9x105 m/s, plasma frequency is 1.78x1010 s-1, Debye length is 3.3x10-3 cm, ratio L / Debye length is 300.
Assume that the desired resolution is 300 points (along each coordinate direction), 200 velocity points (along each velocity direction).
The number of variables is 1d1v :: 300 x 200 = 6x104, 1d3v :: 300 x 2003 = 2.4x109, 2d3v :: 3002 x 2003 = 7.2x1011, 3d3v :: 3003 x 2003 = 2.16x1014.
Compare: 3d3v PIC code with the same spatial resolution and 1000 particles per cell requires 6 x 3003 x 1000 = 1.62x1011
particle variables.
12
It’s easier to use particles…
Plasma is represented as a set of macroparticles.
Each macroparticle is a charged “cloud” representing many real charged particles (electrons or ions) “glued” together.
A macroparticle has the same charge-to-mass ratio (q/m) as the real charged particle.
Equations of motion are solved for each macroparticle.
The electric and magnetic fields are calculated self-consistently using charge densities and currents produced by the macroparticles.
ne=1012cm-3
L=10cm
Ne=1013
Q/e=106Nm=107
13
Particle-mesh method: Charges and currents are calculated
in a set of predefined points (nodes of computational grid).
Poisson’s and/or Maxwell equations are solved and give electromagnetic fields in the nodes.
The fields are then interpolated into the positions of the particles.
Particle-particle method: Electric and magnetic fields of a system of charged particles are (in a
static limit):
The numerical cost of calculating the individual fields grows with the particles number N as N2.
How to calculate the fields?
jjj BE
,,
)( BvEmq
dtd
i i
ii xx
xxqxE 30 ||4
1)(
i i
iii xx
xxvqxB 30
0
||4)(
jj J
,
14
Particle and force weighting
Distribute particle charge to neighbor nodes.
Get force, which is defined in the grid nodes, in the location of the particle.
15
Particle and force weighting Linear weighting to the grid:
This corresponds to a triangular shape function S.
Grid values are linearly interpolated into the particle’s location:
xxx
QQx
xxQQ ji
ijij
ij
11 ,
)(),()( 11
jijjijj xxSx
QxxSx
Qx
)()(
)(
11
11
ijjjij
jij
ijjii
xxSExxSExxx
Ex
xxEExE
0101
||0)(
hifxhhifxh
xhifhS
16
Bi-linear weighting on a 2D grid
17
Bi-linear weighting on a 2D grid
18
)()()()(
)()()()(
1,1
111,111,
kiyjixkjkiyjixjk
kiyjixkjkiyjixkj
yySxxSyx
QyySxxSyx
Q
yySxxSyx
QyySxxSyx
Q
Bi-linear weighting on a 2D grid
0101
||0)(
hifxhhifxh
xhifhSx
0101
||0)(
hifyhhifyh
yhifhSy
)()()()(
)()()()(
111,111,
1,1
kiyjixkjkiyjixkj
kiyjixkjkiyjixjki
yySxxSEyySxxSE
yySxxSEyySxxSEE
19
General sequence of a Particle-in-Cell (PIC) algorithm
Initial state
For all particles:accelerate+move
For all particles:collect grid values
j, Jj
For all grid nodes:calculate fields
Ej, Bj
t = t + t
20
Explicit leap-frog scheme
21
Explicit leap-frog scheme
Coordinates, charge densities, and fields are calculated at “integer” times .
Velocities are calculated at “half-integer” times
Equations of motion
Poisson’s equation
vdtxd
Fdtvdm
EqF
0
E
tvxx ni
ni
ni 2/11
txFvv ni
ni
ni )(2/12/1
2
011 2 x
njn
jnj
nj
xE jjn
j
211
tntn
tntn )2/1(2/1
22
Motion equation with magnetic field
To advance the velocity obtain v- from vn-1/2, rotate v- to obtain v+, obtain vn+1/2 from v+.
)( BvEqdtvdm
BvvEmq
tvv nn
nnn
2
2/12/12/12/1
22/1 t
mEqvv n
22/1 t
mEqvv n
Bvvmq
tvv
)(2
Centered-difference form
ttmqB
c
2
arctan2
Rotation by angle
This substitution cancels E completely [Boris, 1970].
22tan t
mqB
yxyyxx cvsvvsvcvv ,
212sin
s 2
2
11cos
c
For B directed along the z-axis, the rotation iswhere
23
Stability of the explicit leap-frog methodA harmonic oscillator can be described with the leap-frog method as follows
vdtdx
xdtdv 2
0
tvxx nnn 2/11
txvv nnn 20
2/12/1
220
11 2 txxxx nnnn
Assume that the solution is of the form then)exp( tnix 22
sin 0 tt
If then is complex and the solution grows exponentially.0
2
t
For the leap-frog explicit algorithm, the timestep must be small enough to resolve the electron plasma frequency: t<2/pe. Another important condition is the Courant criterion: t<x/v. Here x is the mesh size and v is the fastest speed of propagation of either a wave or a particle in the system. This criterion prevents a numerical instability and is applicable in many numerical schemes.
24
How to increase the time step?
Numerical schemes stable for large time steps, tpe>>1, are usually implicit.
In implicit schemes, calculation of the updated positions xn+1 requires knowledge of the fields En+1 at the same time.
Below we will consider a direct implicit algorithm for electrostatic simulations described in [Gibbons and Hewett, J.Comput.Phys., 1995].
25
Electrostatic direct implicit algorithm (1)
The finite difference equations are
Bvvmtqatvv nnnnn
2/12/12/12/1
2
2/11 nnn vtxx
11
21 nnn E
mqaa
Note that we must know En+1 to get xn+1.
The velocity equation can be transformed as follows:
1112/12/1
22nnnn E
mtqatAvKv
Here matrices K and A-1 contain coefficients depending on the magnetic field.
vdtxd
Fdtvdm
Note, the unknown updated electric field is separated.
26
Electrostatic direct implicit algorithm (2)
Acceleration and displacement split in two steps. Step 1 (pre-push, all values in the RHS are known):
Step 2 (final push):
The advanced electric field must be found between the two steps.
1112/12/1
22nnnn E
mtqatAvKv
112/1
2~
nn atAvKv
vtxx n ~~
vtxxn ~1
11
2
nEAmtqv
vvv n ~2/1
27
Electrostatic direct implicit algorithm (3)
To define En+1 it is necessary to predict the charge density n+1.
.)()(~2
)(~~~
~~
~)~()(
112
,,,,
,,
,,,
,,,
1
sj
nsjs
s
sj
isisj
jisisj
s
isisj
isisisj
s
isisisj
sj
n
xEAxmqtxxxS
xvtxxS
xq
xxSx
xxxSx
qxxxSx
qx
Here S is the shape function which defines charge distribution into the grid nodes, subscript s denotes particle species, subscript i denotes particle number.
Introduce implicit susceptibility: .
The field equation becomes .
ssjs
s
sj Ax
mqtX 1
2
)(~2
~10 X
28
29
Advantages and limitations of implicit algorithms Implicit simulation are usually less prone to numerical
instability than the explicit ones. Implicit algorithms are more complex and more costly
numerically. Time step may be increased but:
Debye shielding will not be reproduced correctly if kvtht>1. If pet>>1, waves must be longer than the Debye length, kD<<1
[Langdon, Cohen, and Friedman, J.Comput.Phys., 1983]. There is always the Courant criterion t<x/vmax.
With improper resolution in space and time, an implicit simulation may look stable but the results will be unphysical.
30
Main code features: Active material surfaces with secondary
electron emission induced by incident electrons and ions.
Monte-Carlo collisions between electrons and neutrals.
Electron-electron and electron-ion collisions. 107 particles, 1000s spatial cells, 1000s
particles per cell, abundant diagnostics. MPI parallelization.
The code was used to study EVDF anisotropy and plasma-wall interaction in Hall thrusters [Sydorenko, Kaganovich, Raitses, and Smolyakov, Phys.Rev.Lett, 2009].
Recently, the code was used to study two-stream instability and multi-peak EVDF formation in dc-rf discharges.
Schematic of PIC simulations,plane geometry approximation
Hall thruster, cylindrical geometry
EDIPIC – electrostatic 1d3v code based on the direct implicit algorithm
31
Processing of probabilistic events
Let an event has probability P<1. To define whether the event occurs or not: take a random number 0<R<1, the event occurs if R<P.
Let some variable is described by a probability density function f(v). To select the value of this variable: take a random number 0<R<1,
solve for v.
1)(
dvvf
v
dvvf ')'(
v
dvvfR ')'(
To reduce numerical cost, one can precalculate sets of vi(Ri) and interpolate: for random number R, Ri<R<Ri+1, ii
ii
ii
ii RR
RRvRRRRvv
11
1
1
32
Secondary electron emission
Emission coefficient , where 1 is the primary electron flux
and 2 is the secondary electron flux.1
2
The primary electron can be absorbed, be reflected elastically, be reflected inelastically, cause the emission of “true”
secondary electrons.
The secondary electron emission is an important process which modifies the structure of near-wall sheath and affects intensity of plasma cooling due to wall losses.
1
,,;2,,
iet
iet
iet
The partial emission coefficients are
The total emission coefficient is
iet ,2,2,22
33
Partial emission coefficients For primary electron energies above 10s of eV,
the total emission coefficient is [Vaughan, IEEE Trans.Electron. Dev., 1989]:
Total emission coefficient
Incident electron energy
Total (1) and partial emission coefficients: elastic (1), inelastic (2), true (3). Markers are BN experimental data [Dunaevsky, Raitses, Fisch, Phys.Plasmas, 2003]
20max,max
0max
0
2max
1)(,)(
),(
),(1exp),(2
1),(
s
ksV
kwwww
wwwv
wvwvkw
max
max
25.062.0
wwifwwif
kw and are the primary electron’s energy and angle of incidence.
At high energies, elastically / inelastically reflected electrons comprise abour 3% / 7% of the emitted current [Gopinath, Veboncoeur, Birdsall, Phys.Plasmas, 1998].
VtViLEeVe w 9.0,07.0),(03.0 ,
Additional term which increases elastic reflection at low energies.
34
Angular and energy distributions of secondary electrons Elastically reflected electrons:
energy w2=w1 specular reflection random reflection
emission angle 2= arcsin(R) azimuthal angle 2= R2
Inelastically backscattered electrons: energy w2=Rw1 emission angle 2= arcsin(R) azimuthal angle 2= R2
True secondary electrons: energy correspond to a half-Maxwellian
distribution of temperature Tt, emission angle 2= arcsin(R) azimuthal angle 2= R2
From [Seiler, J.Appl.Phys, 1983]
Energy spectrum of secondary electrons in simulation
Everywhere, R is a random number, 0<R<1.
35
Probabilistic model of secondary electron emission Assume that no true secondary electrons are emitted if the primary
electron reflects either elastically or inelastically. A particle collide with a wall. Then
calculate particle energy w and angle of incidence calculate the total and the partial e, i, t emission coefficients, return if =0 take a random number R, 0<R<1 if R<e inject elastically reflected electron, return if R< e+i inject inelastically backscattered electron, return if <1 then
if R< e+i +e inject true secondary electron return
calculate *=t / (1-e-i), must be *>1 inject INT(*) true secondary electrons take random number R, 0<R<1 if R< *-INT(*) inject a true secondary electron return
36
Electron-neutral collisions
37
Electron-neutral collisions
Collision processes: elastic scattering, excitation, ionization.
Assumptions: only two-particle collisions occur, only a small fraction of particles collides at each
timestep, a particle cannot collide more than once during
the timestep.
38
Probability of an electron-neutral collision is where is the collision frequency, t is the time interval, u is the particle velocity, T is the total scattering cross-section, nT is the density of neutrals (targets).
A straightforward and numerically ineffective way is to check whether each particle at each timestep makes a collision: calculate probability P, take a random number R, collision occurs if R<P.
Alternatively, if we know how many particles collide at each timestep, we can randomly choose these particles and perform collision procedures only for them.
Monte-Carlo model of electron-neutral collisions
)exp(1 tP TT nmuu )2/( 2
39
Calculate maximal frequency of collisions max and corresponding probability Pmax.
Calculate the number of colliding particles
Prepare the “accumulated probabilities” Pk.
At each timestep: select randomly Ncoll particles take random number R collision of type k occurs if
if the collision to occur is the null collision, do nothing.
The null collision methodTotal cross sections of elastic (1) excitation (2) and ionization (3) collisions in Xenon.
Accumulated “probabilities” of collisions [normalized]. The zero corresponds to the null collisions.
1;
;;;0
4max
3213
max
212
max
110
PP
PPP
[Vahedi and Surendra, Comput. Phys. Comm., 1995]
totcoll NPN max
kk PRP 1
40
Differential cross-section and the scattering angle are: [Surendra, Graves, and Jellum, Phys.Rev.A,
1990], w is the energy in eV
[Okhrimovskyy, Bogaerts, and Gijbels, Phys.Rev.E, 2002], is w/27.21
Scattering angle is .
Selection of scattering angles
vinc is the electron velocity before scattering, vsc is the velocity of scattered electron, and are the scattering angles.
)1ln()]2/(sin1[4)(),(
2 www
ww
www R)1(22cos
2cos441481
)(),(
)1(8121cos
RR
2R
Everywhere, R is a random number, 0<R<1.
41
Rotation and energy transfer
The vector velocity scattered by angles and is
To account for the energy transfer, the scattered velocity must be multiplied by factor vinc is the electron velocity
before scattering, vsc is the velocity of scattered electron, and are the scattering angles.
cossinsin||
sinsinsin
cosinc
incincincincsc v
kvvkvvv
cos121 Mm
42
Processing an inelastic (excitation) e-n collision The energy of the scattering electron decreases by the
excitation threshold. For example, for xenon it is wexc=8.32 eV.
The processing sequence is as follows: calculate the modified energy wsc = winc – wexc , calculate scattering angles and with wsc , rotate the initial velocity vector (use same formula as
in an elastic collision), multiply the rotated vector by a factor .
inc
exc
ww
1
43
Energies of primary and secondary electrons in ionization collisions Only electrons with energies above the ionization
threshold w>wion can do the ionization. For xenon wion=15eV.
The spectrum of secondary electrons is described as
Constant B is known from measurements (for example, B=8.7eV for xenon).
The energy of a secondary electron can be found as follows [from a mini-course “Particle-in-Cell Technique by J.P.Verboncoeur, 2002”]:
The energy of the scattered primary electron is then
Differential cross-section as a function of the secondary (ejected) electron energy. From [Opal, Peterson, and Beauty, J.Chem.Phys, 1971].
22
21
121
2arctan
)(),(wB
Bww
Bwwwion
i
BwwRBw ion
2arctantan 1
2
R is the random number, 0<R<1.
21 wwww ionscat
44
Processing an ionization collision Once the energies of scattered and
secondary electrons are found, the processing sequence is as follows: get the secondary electron velocity v2 :
calculate 2 and 2 , rotate v1 by angles 2 and 2 , multiply by factor (w2/w1)1/2;
get the scattered electron velocity vsc : calculate 1 and 1 , rotate v1 by angles 1 and 1 , multiply by factor (wsc/w1)1/2;
take ion velocity from the velocity distribution of the neutral gas.
Angular distribution of secondary electrons as function of their energy. From Opal, Peterson, and Beauty, J.Chem.Phys, 1971.
45
Collisions with neutrals are treated largely like scattering of spheres
46
Coulomb collisions need different approach The differential cross section for Coulomb scattering is [see e.g.,
Lieberman and Lichtenberg, Principles of plasma Discharges and Materials Processing]
Straightforward integration gives an infinite total cross section.
Compared to electron-neutral collisions, electron-electron [Coulomb] collisions are much more frequent and are characterized by small scattering angles.
Application of the traditional Monte-Carlo approach is both ineffective numerically and physically inappropriate.
)2/(sin)8(),( 44
022
0
42
0
vmeZv
R
0
0 sin),(2 dvtot
47
Electron-electron Coulomb collisions The effect of Coulomb collisions can be represented as a result of
dynamical friction and stochastic diffusion – scattering on many particles can be substituted by scattering off the grid [Jones, Lemons, Mason, Thomas, and Winske, J.Comput.Phys., 1996; Manheimer, Lampe, and Joyce, J.Comput.Phys.,1997].
)()(21)()(
2
vfvDvv
vfvFvt
fd
ee
)(4
)( 220
4
vHvm
nevFd
)(4
)(2
220
4
vGvvm
nevD
|~|)~(~2)( 3
vvvfvdvH
|~|)~(~)( 3 vvvfvdvG
Fokker-Planck equation for e-e scattering:
Dynamic friction
Velocity diffusion coefficient:
describes change of electron mean directed velocity.
describes electron spreading in the velocity space.
H and G are the first and the second Rosenbluth potentials [Rosenbluth, MacDonald, and Judd, Phys.Rev., 1957]
48
Drag force and diffusion coefficients
Calculations simplify if the EVDF is isotropic in the electron flow frame ue.
||
0
2222
0
4
)~(~~||
12||
)(euv
ee
ed wfwwd
uvmen
uvuvvF
euvw
~~
w
w
wfwwdwfwwwwdwm
enwnDwDwD0
222322
0
4
12211 )~(~~2)~()~3(~~13
)()()(
w
w
wfwwdwfwwdwm
enwnDwD0
4322
0
4
333 )~(~~)~(~~13
2)()(
The drag force
Velocity of a scatterer electron in the flow frame The diffusion coefficient tensor becomes diagonal in the [primed] frame where the 3rd axis is directed along : euv
49
Velocity correction due to diffusion
The Fokker-Planck equation is equivalent, to the first order of accuracy, to the Langevin equation:
QtFv d
tDQQ
tDQ
DDtQ
11
22
21
33
23
2/13311
2/3 2''
2'exp
)2(1)'(
)2/'()( 3332
'
3
3
tnDQgydydydyRQ
x
dxg )exp()( 22/1
)(2
2
tdtxd
dtxdm
The original Langevin equation describes Brownian motion:
drag force noise term
In the “primed” frame coordinates Q’1,2,3 of a vector Q’ correspond to the distribution
To find, for example, component Q’3, one has to solve equation
where R is a random number, 0<R<1, and .
To reduce numerical load, solution of R=g(x) can be tabulated.
50
Transformation to the laboratory frame
Vector Q’ must be transformed to the laboratory frame as follows:
z
y
x
z
y
x
QQQ
QQQ
'''
cossin0cossincoscossin
sinsinsincoscos
where
2222
22
cos,sin,cos,sinyx
y
yx
xzyx
ww
w
www
ww
www
and wx,y,z are the components of the scattering electron velocity in the flow frame.
euvw
51
Electron-electron Coulomb collisions, algorithm summary Calculate the electron flow velocity ue. Calculate the EVDF f(w) in the flow frame, w is the absolute
value of the velocity. Tabulate drag force coefficient Fd and the velocity diffusion
coefficients D1 and D3 as functions of w. For each electron
calculate velocity correction due to the drag force Fdt calculate velocity correction due to the velocity diffusion Q, apply the corrections, accumulate the kinetic energy of all electrons before (Wbefore) and
after (Wafter) the corrections are applied. For each electron
multiply the velocity by the factor (Wbefore/Wafter)1/2 to ensure energy conservation.
52
Electron-electron Coulomb collisions, examples
In this test, the Coulomb collisions transform the initially rectangular anisotropic EVDF into a Maxwellian isotropic EVDF.
Average energy of electron motion in x (1), y (2), and z (3) directions vs time.
Initial 1-D EVDFs.
Final1-D EVDFs.
Initial (left) and final (right) electron velocity phase space.
53
Electron-ion coulomb collisions The procedure is similar to the one for the electron-electron
collisions. The integrals in coefficients H and G are simplified because the ion velocity is much slower than the electron one.
Then
To avoid very large corrections for particles with small velocities, electrons with speed v<vthr are scattered off ions with constant coefficient Fd,thr=Fd(vthr), where the threshold is obtained from condition :
||,/1)( vvvvH vvG )(
222
0
4 14
)()(vm
envvvnF
vvvF dd
vmenwnDwDwD 1
4)()()( 22
0
4
12211 0)()( 333 wnDwD
)( thrdthr vtFnv 3/1
220
4
4
metnvthr
54
55
ReferencesC. Z. Cheng and G. Knorr, J. Comput. Phys., 22, 330-351 (1976).J. R. Boris and D. L. Book, J. Comput. Phys., 11, 38-69 (1973).E. Sonnendrucker, J. Roche, P. Bertrand, and A. Ghizzo, J. Comput. Phys., 149, 201-220 (1999).H. L. Berk and K. V. Roberts, Phys. Fluids, 10, 1595-1597 (1967).J. P. Boris, Princeton University, PPL Report MATT-769, March 1970.M. R. Gibbons and D. W. Hewett, J. Comput. Phys., 120, 231-247 (1995).A. B. Langdon, B. I. Cohen, and A. Friedman, J. Comput. Phys., 51, 107-138 (1983).D. Sydorenko, I. Kaganovich, Y. Raitses, and A. Smolyakov, Phys. Rev. Lett, 103, 145004 (2009).V. P. Gopinath, J. P. Veboncoeur, C. K. Birdsall, Phys. Plasmas, 5, 1535-1540 (1998).A. Dunaevsky, Y. Raitses, and N. J. Fisch, Phys. Plasmas, 10, 2574-2577 (2003).H. Seiler, J. Appl. Phys., 54, R1-R18 (1983).V. Vahedi and M. Surendra, Comput. Phys. Comm., 87, 179-198 (1995).M. Surendra, D. B. Graves, and G. M. Jellum, Phys.Rev.A, 41, 1112-1125 (1990).A. Okhrimovskyy, A. Bogaerts, and R. Gijbels, Phys. Rev. E, 65, 037402 (2002).J. P. Verboncoeur, Particle-in-Cell Technique, 2002.C. B. Opal, W. K. Peterson, and E. C. Beauty, J. Chem. Phys., 55, 4100-4106 (1971).M. A. Lieberman and A. J. Lichtenberg, Principles of plasma Discharges and Materials Processing,
published by John Wiley & Sons, Inc., Hoboken, NJ, 2005.M. E. Jones, D. S. Lemons, R. J. Mason, V. A. Thomas, and D. Winske, J. Comput. Phys., 123, 169-
181 (1996).W. M. Manheimer, M. Lampe, and G. Joyce, J. Comput. Phys., 138, 563-584 (1997).M. N. Rosenbluth, W. M. MacDonald, and D. L. Judd, Phys. Rev., 107, 1-6 (1957).
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