1. Plasma Application Modeling, POSTECH
ECE586: Advanced E&M Simulation (2004)
On PDX1 Program
HyunChul Kim and J.K. Lee
Plasma Application Modeling, POSTECH
References:
Minicourse by Dr. J. P. Verboncoeur (PTS Group of UC Berkeley) in IEEE International Conference on Plasma Science (2002)
“Plasma Physics via Computer Simulation” by C.K. Birdsall and A.B. Langdon (Adam Hilger, 1991)

2. A Series of XPDX1*
XPDx1: X window (using xgrafix library), Plasma Device, 1 Dimensional (1d3v), Bounded (with external circuit drive), Electrostatic Code
XPDP1 (x=P) : Planar Configuration
XPDC1 (x=C) : Cylindrical Configuration
XPDS1 (x=S) : Spherical Configuration r ~
LRC
Computation Space
* Developed by PTS group, UC Berkeley
All are available at

3. PIC Overview PIC Codes Overview
Plasma behavior of a large number of charges particles are simulated by using a few representative “super particles”.
PIC codes solve fundamental equations, the Newton-Lorentz equation of motion to move particles in conjunction with Maxwell’s equations (or a subset) with few approximations.
PIC codes are quite successful in simulating kinetic and nonlinear plasma phenomenon like ECR, stochastic heating, etc.

4. Computer Simulation of Plasma
Kinetic
Description
Fluid
Description
Vlasov,
Fokker-Planck
Codes
Particle
Codes
Hybrid
Codes
Fluid
Codes
Particle codes
The particle-particle model
The particle-mesh model
The particle-particleparticle-mesh model

5. XPDx1 Flow Chart I II V IV III IV Particles in continuum space
Fields at discrete mesh locations in space
Coupling between particles and fields I II V IV
III IV
Fig: Flow chart for an explicit PIC-MCC scheme

6. I. Particle Equations of Motion
Newton-Lorentz equations of motion
In finite difference form, the leapfrog method
Second order accurate
Stable for

7. I. Particle Equations of Motion
Boris algorithm

8. I. Particle Equations of Motion
Finally,

9. XPDx1 Flow Chart I II V IV III IV
Fig: Flow chart for an explicit PIC-MCC scheme

10. II. Particle Boundary + – – Absorption
Conductor : absorb charge, add to the global σ
Secondary electron emission
Ion impact secondary emission + –
Electron impact secondary emission –

11. XPDx1 Flow Chart I II V IV III IV
Fig: Flow chart for an explicit PIC-MCC scheme

12. III. Electrostatic Field Model
Maxwell’s equation in vacuum
In electrostatics,
(Poisson’s equation)
Or Gauss’ law

13. III. Electrostatic Field Model
Possion’s equation
Finite difference form in 1D planar geometry
Boundary condition : External circuit
From Gauss’s law,
Short circuit
Open circuit

14. III. Electrostatic Field Model
Voltage driven series RLC circuit
From Kirchhoff’s voltage law,
― One second order difference equation
where

15. XPDx1 Flow Chart I II V IV III IV
Fig: Flow chart for an explicit PIC-MCC scheme

16. IV. Coupling Fields to Particles
Particle and force weighting
: connection between grid and particle quantities
Weighting of charge to grid
Weighting of fields to particles
grid point
a point charge

17. IV. Coupling Fields to Particles
Nearest grid point (NGP) weighting
 fast, simple bc, noisy
Linear weighting
: particle-in-cell (PIC) or cloud-in-cell (CIC)
relatively fast, simple bc, less noisy
Higher order weighting schemes
slow, complicated bc, low noisy
Quadratic spline
NGP
1.0
Linear spline
Cubic spline
0.5
0.0
Position (x)
Fig: Density distribution function of a particle at
for various weightings in 1D

18. IV. Coupling Fields to Particles
Weighting in 1D
For linear weighting in cylindrical coordinates,
( 0 < j < N )

19. XPDx1 Flow Chart I II V IV III IV
Fig: Flow chart for an explicit PIC-MCC scheme

20. Collisions Electron-neutral collisions
Elastic scattering (e + A → e + A)
Excitation (e + A → e + A*)
Ionization (e + A → e + A+ + e)
Ion-neutral collisions
Elastic scattering (A+ + A → A+ + A)
Charge exchange (A+ + A → A + A+)

21. V. Monte-Carlo Collision Model
The MCC model statistically describes the collision processes, using cross sections for each reaction of interest.
Probability of a collision event
For a pure Monte Carlo method, the timestep is chosen as the time interval between collisions.
However, this method can only be applied when space charge and self-field effects can be neglected.

22. V. Monte-Carlo Collision Model
There is a finite probability that the i-th particle will undergo more than one collision in the timestep.
Since XPDx1 deals with only one collision in the timestep, the total number of missed collisions
Hence, XPDx1 is constrained by
for accuracy.

23. V. Monte-Carlo Collision Model
Computing the collision probability for each particle each timestep is computationally expensive.
→ Null collision method
1. The fraction of particles undergoing a collision each time step is given by
2. The particles undergoing collisions are chosen at random from the particle list.
3. The type of collisions for each particle is determined by choosing a random number,
Null collision
Collision type 3
Collision type 2
Collision type 1
Fig: Summed collision frequencies for the null collision method.

24. Numerical Parameters
Choose Δx and Δt to resolve the smallest important physical feature
 Require Δx < Debye length, sheath length, wave length, Larmor radius, boundary feature, etc.
 Require for all species (“particle Courant”) for accurate sampling of fields.
 Require for accuracy of explicit leap frog mover or for accuracy when space charge forces are important.
 Require when collisions are important.
Require # of superparticles per cell > 10. It should be larger in simulations where particles remain trapped for long times.

25. Example of XPDP1 Input File
RF DISCHARGE(IN MKS UNITS) Voltage-driven with electron-neutral collisions (Argon atom) -nsp---nc---nc2p---dt[s]---length[m]--area[m^2]--epsilonr---B[Tesla]---PSI[D] e6 8e rhoback[C/m^3]---backj[Amp/m^2]---dde--extR[Ohm]--extL[H]---extC[F]---q0[C] dcramped--source--dc[V|Amp]--ramp[(V|Amp)/s]---ac[V|Amp]---f0[Hz]--theta0[D] v e secondary--e_collisional---i_collisional---reflux---nfft--n_ave--nsmoothing--ntimestep seec(electrons)---seec(ions)---ion_species----Gpressure[Torr]---GTemp[eV]---imp e GAS----psource--nstrt
SPECIES q[C] m[Kg]---j0L[Amp/m^2]---j0R[Amp/m^2]----initn[m^-3]----k e e e vx0L[m/s]---vxtL[m/s]--vxcL[m/s]---vxLloader(0=RNDM,1=QS) e vx0R[m/s]---vxtR[m/s]--vxcR[m/s]---vxRloader e v0y[m/s]---vty[m/s]---vyloader---v0z[m/s]---vtz[m/s]--vzloader e e nbin----Emin[eV]----Emax[ev]---maxnp—
For-Mid-Diagnostic---nbin----Emin[eV]---Emax[eV]----XStart--XFinish—
SPECIES q[C] m[Kg]---j0L[Amp/m^2]---j0R[Amp/m^2]----initn[m^-3]----k e e e vx0L[m/s]---vxtL[m/s]--vxcL[m/s]---vxLloader(0=RNDM,1=QS) vx0R[m/s]---vxtR[m/s]--vxcR[m/s]---vxRloader v0y[m/s]---vty[m/s]---vyloader---v0z[m/s]---vtz[m/s]--vzloader nbin----Emin[eV]----Emax[ev]---maxnp For-Mid-Diagnostic---nbin----Emin[eV]---Emax[eV]----XStart--XFinish

26. Some Input Parameters nsp : Number of species.
nc: The number of spatial cells. Δx=length/nc
nc2p: Superparticle to actual particle weight. The initial number of superparticles is N=initn·area·length/nc2p.
dt: Timestep for simulation in seconds.
length: The length of the system (distance between electrodes) in meters.
B: Applied homogeneous magnetic field in Tesla
PSI: Angle of the B-field in degrees
extC: The external circuit capacitance in Farads. Used in conjuction with extL, extR and the driving source.
source: Either a voltage (v) or current (i) source
f0: AC frequency of the source.
GAS: The type of gas, important when collisions are turned on. Helium = 1, Argon = 2, Neon = 3, Oxygen = 4.
Gpressure : Background gas pressure in Torr.
q: Charge of the particle in Coulombs.
m: Mass of the particle in Kgs.
initn: Initial particle number density
For details, refer the source code itself or the manual inside the package of source file.

27. Example of Result (driven by RF)
Vx vs. x for electrons
Vx vs. x for ions
Density vs. x
Potential vs. x
Ion flux vs. Ion Energy
Electron Temperature vs. x