EECE695: Computer Simulation (2005) Particle-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 III IIIIV V

I. Particle Equations of Motion  Newton-Lorentz equations of motion  In finite difference form, the leapfrog method Fig: Schematic leapfrog integration

I. Particle Equations of Motion Second order accurate Requires minimal storage Requires few operations Stable for

I. Particle Equations of Motion Boris algorithm

I. Particle Equations of Motion 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  1 st order error  Thermionic Emission  Fowler-Nordheim Field Emission  Child’s Law Field Emission  Gauss’s Law Field Emission

II. Particle Boundary  Secondary electron emission + –, – 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 Finite difference form in 1D planar geometry  Boundary condition : External circuit Fig: Schematic one-dimensional bounded plasma with external circuit

III. Electrostatic Field Model Short circuit Open circuit Voltage driven series RLC circuit From Kirchhoff’s voltage law, 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 Cubic spline Fig: Density distribution function of a particle at for various weightings in 1D Position (x)

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 For a pure Monte Carlo method, the timestep is chosen as the time interval between collisions. 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) Hence, traditional PIC-MCC codes are constrained by for accuracy.

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