ECE586: Advanced E&M Simulation

(2004)

On PDX1 ProgramOn PDX1 Program

2004. 9. 16

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)

A Series of XPDX1*

r

~ LRC

Computation Space

* Developed by PTS group, UC BerkeleyAll are available at http://ptsg.eecs.berkeley.edu

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

PIC 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.

PIC Codes Overview

Computer Simulation of Plasma

Kinetic Description

Fluid Description

Vlasov, Fokker-Planck

Codes

ParticleCodes

Hybrid Codes

FluidCodes

• The particle-particle model

• The particle-mesh model

• The particle-particleparticle-mesh model

Particle codes

XPDx1 Flow Chart

Fig: Flow chart for an explicit PIC-MCC scheme

I II

III IV

IV

V

ix, )( v

j, )( BE

1 ,:Subcycling ktkt fastslowi

• Particles in continuum

space• Fields at discrete mesh

locations in space• Coupling between

particles and fields

I. Particle Equations of Motion

Newton-Lorentz equations of motion

)BvE(Fv qmdt

d

vx dt

d

In finite difference form, the leapfrog method

)B2

vvE(

vv 2/2/2/2/t

ttttt

tttt

m

q

t

2/vxx ttttt

t

• Second order accurate

• Stable for 2twp

I. Particle Equations of Motion

m

tq ttt

2

Evv 2/

• Boris algorithm

m

tq ttt

2

Evv 2/

t

m

q

tB)vv(

2

vv

v

v vv

vv

m

tq tt

2

B)

2tan(b̂t

I. Particle Equations of Motion

t' tvvv

tt

t

'tt1

t2vvv

v

v

ttv

'v

tt

t

'tt1

t2v

Finally,

XPDx1 Flow Chart

Fig: Flow chart for an explicit PIC-MCC scheme

I II

III IV

IV

V

II. Particle Boundary

Secondary electron emission

• Ion impact secondary emission

• Electron impact

secondary emission

+

– se

ionelectron–

• Conductor : absorb charge, add to the global σ

Absorption

XPDx1 Flow Chart

Fig: Flow chart for an explicit PIC-MCC scheme

I II

III IV

IV

V

III. Electrostatic Field Model

• In electrostatics,

Maxwell’s equation in vacuum

EDDt

BE 0,,

HBBt

DJH 0,0,

EE 0

0

2

(Poisson’s equation)

Or Gauss’ law

enclosedVSQdVsdD

III. Electrostatic Field Model

Possion’s equation

),,(),( tt xx

• Finite difference form in 1D planar geometry

,2

2

11

jjjj

x

Boundary condition : External circuit

20010

2/1

x

xE

tttt

0

0 E J

JE

A

QQdtJ

tttt

tt plasmattt

00

From Gauss’s law,

• Short circuit

0)( specified, is )(0 tt J

• Open circuit

t

tt plasmattt dtJ00

III. Electrostatic Field Model

• Voltage driven series RLC circuit

From Kirchhoff’s voltage law,

)()()(

)()()(

0

2

2

tttVC

tQ

dt

tdQR

dt

tQdL

J

― One second order difference equation

where

XPDx1 Flow Chart

Fig: Flow chart for an explicit PIC-MCC scheme

I II

III IV

IV

V

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

a point charge

grid point

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

NGP

Linear spline

Quadratic spline

1.0

0.5

0.0

Cubic spline

Fig: Density distribution function of a particle atfor various weightings in 1D

xxi xxi 2ixxxi xxi 2

Position (x)

ix

)( ixxSx

i

ijij xXSqX )()( j

ijjii xXSExqF )(

Weighting in 1D

• For linear weighting in cylindrical coordinates,

( 0 < j < N )

IV. Coupling Fields to Particles

XPDx1 Flow Chart

Fig: Flow chart for an explicit PIC-MCC scheme

I II

III IV

IV

V

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+)

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

])(exp[1 tnP iiTgi

j ijiT )()( where

• For a pure Monte Carlo method, the timestep is chosen as the time interval between collisions.

iiTgi n

Rt

)(

)1ln(

However, this method can only be applied when space charge and self-field effects can be neglected.

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

.1

2

i

i

k

ki P

PPr

Hence, XPDx1 is constrained byfor accuracy.

1max tv

))((max where max Tgnv

V. Monte-Carlo Collision Model

• Computing the collision probability for each particle each timestep is computationally expensive.

→ Null collision method

].exp[1 max tPT

1. The fraction of particles undergoing a collision each time step is given by

3. The type of collisions for each particle is determined by choosing a random number, .0 maxR

2. The particles undergoing collisions are chosen at random from the particle list.

Fig: Summed collision frequencies for the null collision method.

Null collision

Collision type 3

Collision type 1

Collision type 2

Tc PNN

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.

max/xt

pwt /2.0

/1t

Require # of superparticles per cell > 10. It should

be larger in simulations where particles remain

trapped for long times.

Example of XPDP1 Input FileRF 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]-- 2 400 1.8e6 8e-12 0.03 0.01 1.0 0 .0 0.0-rhoback[C/m^3]---backj[Amp/m^2]---dde--extR[Ohm]--extL[H]---extC[F]---q0[C]- 0.0 0.0 0.0 0.0 0.0 1.0 0.0-dcramped--source--dc[V|Amp]--ramp[(V|Amp)/s]---ac[V|Amp]---f0[Hz]--theta0[D]- 0 v 0.0 0.0 100 13.56e6 0--secondary--e_collisional---i_collisional---reflux---nfft--n_ave--nsmoothing--ntimestep-- 1 1 2 0 0 276549 6 0--seec(electrons)---seec(ions)---ion_species----Gpressure[Torr]---GTemp[eV]---imp-- 0.0 0.2 2 100e-3 0.026 0---GAS----psource--nstrt-- 1 0 0

SPECIES 1----q[C]-------m[Kg]---j0L[Amp/m^2]---j0R[Amp/m^2]----initn[m^-3]----k-- -1.602e-19 9.11e-31 0.0 0.0 5e14 1--vx0L[m/s]---vxtL[m/s]--vxcL[m/s]---vxLloader(0=RNDM,1=QS)-- 0.0 4.19e5 0.0 1 --vx0R[m/s]---vxtR[m/s]--vxcR[m/s]---vxRloader 0.0 4.19e5 0.0 1--v0y[m/s]---vty[m/s]---vyloader---v0z[m/s]---vtz[m/s]--vzloader-- 0.0 4.19e5 1 0.0 4.19e5 1--nbin----Emin[eV]----Emax[ev]---maxnp— 200 0.0 20.0 300000-For-Mid-Diagnostic---nbin----Emin[eV]---Emax[eV]----XStart--XFinish— 300 0.0 20.0 0.0 0.03

SPECIES 2----q[C] ------m[Kg]---j0L[Amp/m^2]---j0R[Amp/m^2]----initn[m^-3]----k- 1.602e-19 6.68e-26 0.0 0.0 5e14 1-vx0L[m/s]---vxtL[m/s]--vxcL[m/s]---vxLloader(0=RNDM,1=QS)-- 0.0 97.8 0.0 1 --vx0R[m/s]---vxtR[m/s]--vxcR[m/s]---vxRloader 0.0 97.8 0.0 1--v0y[m/s]---vty[m/s]---vyloader---v0z[m/s]---vtz[m/s]--vzloader-- 0.0 97.8 0 0.0 97.8 1--nbin----Emin[eV]----Emax[ev]---maxnp-- 100 0.0 100.0 300000-For-Mid-Diagnostic---nbin----Emin[eV]---Emax[eV]----XStart--XFinish-- 200 0.0 0.3 0.0 0.03

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.

Example of Result (driven by RF)

Vx vs. x for electrons

Density vs. x

Vx vs. x for ions

Potential vs. x

Ion flux vs. Ion Energy Electron Temperature vs. x