EECE695: Computer Simulation

(2005)

Particle-in-Cell TechniquesParticle-in-Cell Techniques

HyunChul Kim and J.K. Lee

Plasma Application Modeling Group, 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)

PIC Overview

Applications of PIC model

• Basic plasma physics: waves and instabilities

• Magnetic fusion

• Gaseous discharges

• Electron and ion optics

• Microwave-beam devices

• Plasma-filled microwave-beam devices

PIC Overview

• PIC codes simulate plasma behavior of a large

number of charges particles using a few

representative “super particles”.

• These type of codes solve the Newton-Lorentz

equation of motion to move particles in

conjunction with Maxwell’s equations (or a

subset).

• Boundary conditions are applied to the

particles and the fields to solve the set of

equations.

• PIC codes are quite successful in simulating

kinetic and nonlinear plasma phenomenon like

ECR, stochastic heating, etc.

PIC Codes Overview

PIC-MCC Flow Chart

Fig: Flow chart for an explicit PIC-MCC scheme

• Particles in continuum space

• Fields at discrete mesh locations in space

• Coupling between particles and fields

I II

III IV

IV

V

ix, )( v

j, )( BE

I. Particle Equations of Motion

Newton-Lorentz equations of motion

)( BvEFu qmdt

d

vx dt

d

vu

In finite difference form, the leapfrog method

Fig: Schematic leapfrog integration

I. Particle Equations of Motion

)2

(2/2/2/2/

tt

ttttt

tttt

m

q

tB

uuE

uu

2/

2/

tt

ttttt

t

uxx

• Second order accurate

• Requires minimal storage

• Requires few operations

• Stable for 2twp

I. Particle Equations of Motion

m

tq ttt

22/ E

uu

• Boris algorithm

m

tq ttt

22/ E

uu

ttm

q

tBuu

uu

)(2

u

u uu

uu

m

tqt

tt

2)

2tan(ˆ

B

bt

I. Particle Equations of Motion

t' tuuu

tt

t

'tt

tuuu

1

2

u

u

ttu

'u

tt

t

'tt

tu

1

2

Finally,

II. Particle Boundary

• Conductor : absorb charge, add to the global σ

• Dielectric : deposit charge, weight q locally to mesh

Absorption

Reflection

• Physical reflection

• Specular reflection

1st order error

xx

bcbc

vv

xxx xreverse

)|v|

|x|2(vv

2/

t2/*2/ t

x

m

qEtt

bct

tttt

Thermionic Emission

Fowler-Nordheim Field Emission

Child’s Law Field Emission

Gauss’s Law Field Emission

II. Particle Boundary

Secondary electron emission

+

– se

ionphoton ex,electron–

• Ion impact secondary emission

• Electron impact secondary emission

Important in processes related to high-power

microwave sources

• Photoemission

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

Fig: Schematic one-dimensional bounded plasma

with external circuit

III. Electrostatic Field Model

• Short circuit

0)( specified, is )(0 tt J

• Open circuit

20010

2/1

x

xE

tttt

0

0 E J

JE

t

tt plasmattt dtJ00

• Voltage driven series RLC circuit

From Kirchhoff’s voltage law,

)()()(

)()()(

0

2

2

tttVC

tQ

dt

tdQR

dt

tQdL

J

A

QQdtJ

tttt

tt plasmattt

00

From Gauss’s law,

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

IV. Coupling Fields to Particles

Fig: Charge assignment for linear weighting in 2D

Areas are assigned to grid points; i.e., area a to grid point A, b to B, etc

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 x

j ijiT )()( where

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

iiTgi n

Rt

)()(

)1ln(

x

where 0< R< 1is a uniformly distributed random number.

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.

Thus, the total number of missed collisions (error in single-event codes)

i

i

k

ki P

PPr

1

2

Hence, traditional PIC-MCC codes are constrained by for accuracy.1max tv

))((max))x((max where max Tg

xnv

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