1. Computational Plasma Physics Kinetic modelling: Part 2 W.J. Goedheer FOM-Instituut voor Plasmafysica Nieuwegein, www.rijnh.nl

2. Monte Carlo methods Principle: Follow particles by - solving Newton’s equation of motion - including the effect of collisions - collision: an event that instantaneously changes the velocity Note: The details of a collision are not modeled Only the differential cross section + effect on energy is used Example: Electrons in a homogeneous electric field Follow sufficient electrons for a sufficient time Obtain distribution over velocities etc.  f 0,f 1

3. Monte Carlo methods: Equation of motion Leap-frog scheme

4. Monte Carlo methods: B-field Problem with Lorentz force: contains velocity, needed at time t Solution: take average The new velocity at the right hand side can be eliminated by taking the cross product of the equation with the vector

5. Monte Carlo methods: Boris for B-field Equivalent scheme (J.P.Boris), (proof: substitution):

6. Monte Carlo methods: Collisions Number of collisions: N M  tot = 1/ per meter.   (x) =  (0)*exp(- N M  x) =  (0)*exp(-x/ ) dP(x)=fraction colliding in (x,x+dx)=exp(-x/ )(1-exp(-dx/ ))=(dx/ )exp(-x/ )  P(x)=(1-exp (-x/ )) Distance to next collision: L coll =- *ln(1-Rn) (Rn is random number,0<Rn<1) Number of collisions: N M  tot v= 1/  per second. Time to next collision: T coll =-  * ln(1-Rn)

7. Monte Carlo methods: Collisions Another approach is to work with the chance to have a collision on v  t: P c =v  t/ Ensure that v  t<< to have no more than one collision per timestep  Effect of collision just after advancing position or velocity  introduces only small error When there is a collision: Determine which one: new random number

8. Monte Carlo methods: Null Collision Problem: Mean free path is function of velocity Velocity changes over one mean free path Solution: Add so-called null-collision to make v*  tot independent of v Null-collision does nothing with velocity Mean free path thus based on Max (v*  tot ) Is rather time-consuming when v*  tot peaks strongly

9. Monte Carlo methods: Null Collision v  v*  v*  2 v*  1 v*  3 v*  tot Max v*  0 11 1+21+2 1+2+31+2+3  1 +  2 +  3 +..  N  1 +  2 +  3 +..  N +  0  ’s normalized to maximum: Draw random number

10. Monte Carlo methods: Effect of collision Determine effect on velocity vector Retain velocity of centre of gravity Select by random numbers two angles of rotation for relative velocity Subtract energy loss from relative energy Redistribute relative velocity over collision partners Add velocity centre of gravity

11. Monte Carlo methods: Effect of collision v 1,v 2 velocities in lab-frame prior to collision, w 1,w 2 in center of mass system

12. Monte Carlo methods: Effect of collision A collision changes the size of the relative velocity if it is inelastic A collision rotates the relative velocity Two angles of rotation:    and     usually has an isotropic distribution:  =Rn*   has a non-isotropic distribution Hard spheres:

13. Monte Carlo methods: Rotating the relative velocity Step 1: construct a base of three unit vectors: Step 2: draw the two angles Step 3: construct new relative velocity Step 4: construct new velocities in center of mass frame Step 5: add center of mass velocity

14. Monte Carlo methods: Applicability Examples where MC models can be used are: - motion of electrons in a given electric field in a gas (mixture) - motion of positive ions through a RF sheath (given E(r,t)) Main deficiency: not selfconsistent - electric field depends on generated net electric charge distribution - current density depends on average velocities - following all electrons/ions is impossible Way out: Particle-In-Cell plus Monte Carlo approach

15. Particle-In-Cell plus Monte Carlo: the basics -Interactions between particle and background gas are dealt with only in collisions -this means that PIC/MC is not! Molecular Dynamics -each particle followed in MC represents many others: superparticle -Note: each “superparticle” behaves as a single electron/ion -Electric fields/currents are computed from the superparticle densities/velocities -But: charge density is interpolated to a grid, so no “delta functions”

16. Particle-In-Cell plus Monte Carlo: Bi-linear interpolation x i =i  xx i+1 =(i+1)  x x s, q s =eN s  i :=  i +(x i+1 -x s )q s /  x  i+1 :=  i+1 +(x s -x i )q s /  x z i =i  z z i+1 =(i+1)  z x j =j  xx j+1 =(j+1)  x  ij :=  ij +(z i+1 -z s ) (x j+1 -x s ) q s /(  x  z) zszs xsxs

17. Particle-In-Cell plus Monte Carlo: Solution of Poisson equation Boundary conditions on electrodes, symmetry, etc. Electric field needed for acceleration of particle: (bi)linear interpolation, field known in between grid points

18. Particle-In-Cell plus Monte Carlo: Full cycle, one time step Collision new v Interpolate charge to grid Solve Poisson equation Interpolate field to particle Check loss at the walls Move particles F  v  x

19. Particle-In-Cell plus Monte Carlo: Problems Main source of problems: Statistical fluctuations Fluctuations in charge distribution: fluctuations in E average is zero but average E 2 is not  numerical heating Sheath regions contains only few electrons Tail of energy distribution contains only few electrons large fluctuations in ionization rate can occur

20. Particle-In-Cell plus Monte Carlo: Problems Solutions: -Take more particles (NB error as N -1/2 ), parallel processing! -Average over a long time -Split superparticles in smaller particles when needed requires a lot of bookkeeping, different weights!

21. Particle-In-Cell plus Monte Carlo: Stability Plasmas have a natural frequency for charge fluctuations: The (angular) Plasma Frequency: And a natural length for shielding of charges: The Debye Length: Stability of PIC/MC requires:

22. Power modulated discharges Modulate RF voltage (50MHz) with square wave (1 - 400 kHz) Observation in experiments UU) optimum in deposition rate

23. Modulated discharges Results from a PIC/MC calculation: Cooling and high energy tail

24. 1-D Particle-In-Cell plus Monte Carlo Simulation of a dusty argon plasma Dust particles with a homogeneous density distribution are present in two layers This resembles certain experiments done under micro-gravity Dust particles do not move, they only collect and scatter plasma ions and electrons The charge of the dust results from the collection process The charge of the dust is defined on the grid needed for the Poisson equation

25. RF Void Crystal (2  10 10 m -3 ) 7.5  m radius 1-D Particle-In-Cell plus Monte Carlo Simulation of a dusty argon plasma Capture cross section Scattering: Coulomb, truncated at d L/4 L/8 w is energy electron/ion

26. Charging of the dust upon capture of ion/electron The total charge is monitored on the gridpoints Charge of captured superparticle is added to nearest gridpoints Division according to linear interpolation Superparticle is removed Local dustparticle charge is total charge divided by nr. of dust particles This number is: density*dz*  a 2, with a the electrode radius For Monte Carlo the maximum  v is computed for all available dust particle charges Null-collision is used

27. Simulation for Argon, 50MHz, 100mTorr, 70V, L=3cm dustfreewith dust V d  6V

28. Simulation for Argon, 50MHz, 100mTorr, 70V, L=3cm dustfreewith dust

29. Simulation for Argon, 50MHz, 100mTorr, 70V, L=3cm Generation of internal space charge layers An internal sheath is formed inside the crystal Ions are accelerated before they enter the crystal This has consequences for the charging + shielding

30. Particle-In-Cell plus Monte Carlo: What if superparticles collide? Example: recombination between positive and negative ions Procedure: number of recombinations in  t: N + N - K rec  t corresponds to removal of corresponding superparticles randomly remove negative ion and nearest positive ion but: be careful if distribution is not homogeneous A more sophisticated approach: Direct Simulation Monte Carlo

31. DSMC: Basics Divide the geometry in cells Each cell should contain enough testparticles (typically 25) Newton’s equation: as before, but keep track of cell number Collisions: choose pairs (in same cell!) and make them collide Essential: the velocity distribution function is sum of  -functions Only small fraction of pairs collides in one time step

32. DSMC: Choosing the pairs Add null collision Chance of collision of particle i with j is P c =(N pp /V cell )*Max(v  )  t Number of colliding pairs: n(n-1)* P c /2 Select randomly particle pairs (make sure no double selection) See if there is no null collision (again with random number) Perform the collision

33. DSMC: An example Relaxation of a mono-energetic distribution to equilibrium 20000 particles, hard sphere collisions. All particles are in the same cell. Distribution at various time steps