Computational Plasma Physics

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Computational Plasma Physics Kinetic modelling
Part 2 W.J. Goedheer FOM-Instituut voor
Plasmafysica Nieuwegein, www.rijnh.nl
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Monte Carlo methods
Principle Follow particles by - solving
Newtons 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. ? f0,f1
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Monte Carlo methods Equation of motion
Leap-frog scheme
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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
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Monte Carlo methods Boris for B-field
Equivalent scheme (J.P.Boris), (proof
substitution)
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Monte Carlo methods Collisions
Number of collisions NM?tot 1/? per meter. ?
?(x) ?(0)exp(- NM?x) ?(0)exp(-x/?) dP(x)fr
action colliding in (x,xdx)exp(-x/?)(1-exp(-dx/?
))(dx/?)exp(-x/?) ? P(x)(1-exp
(-x/?)) Distance to next collision
Lcoll-?ln(1-Rn) (Rn is random
number,0ltRnlt1) Number of collisions NM?tot v
1/? per second. Time to next collision
Tcoll-? ln(1-Rn)
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Monte Carlo methods Collisions

• Another approach is to work with the chance
• to have a collision on v?t Pcv?t/?
• Ensure that v?tltlt? 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

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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
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Monte Carlo methods Null Collision
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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
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Monte Carlo methods Effect of collision
v1,v2 velocities in lab-frame prior to collision,
w1,w2 in center of mass system
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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

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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
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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

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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

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Particle-In-Cell plus Monte Carlo Bi-linear
interpolation
xs
zi1(i1)?z
xs, qseNs
zs
zii?z
xii?x
xi1(i1)?x
?i?i(xi1-xs)qs/?x ?i1?i1(xs-xi)qs/?x
xjj?x
xj1(j1)?x
?ij?ij(zi1-zs) (xj1-xs) qs/(?x ?z)
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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
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Particle-In-Cell plus Monte Carlo Full cycle,
one time step
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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 E2 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
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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!

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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
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Power modulated discharges
Modulate RF voltage (50MHz) with square wave (1 -
400 kHz)
Observation in experiments UU) optimum in
deposition rate
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Modulated discharges
Results from a PIC/MC calculation Cooling and
high energy tail
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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
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1-D Particle-In-Cell plus Monte Carlo
Simulation of a dusty argon plasma
Capture cross section
Crystal (2?1010 m-3) 7.5 ?m radius
Scattering Coulomb, truncated at ?d
Void
L/8
L/4
RF
w is energy electron/ion
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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 densitydz?a2, with a
the electrode radius For Monte Carlo the maximum
?v is computed for all available dust
particle charges Null-collision is used
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Simulation for Argon, 50MHz, 100mTorr, 70V, L3cm
dustfree
with dust
Vd?6V
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Simulation for Argon, 50MHz, 100mTorr, 70V, L3cm
dustfree
with dust
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Simulation for Argon, 50MHz, 100mTorr, 70V, L3cm
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
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Particle-In-Cell plus Monte Carlo What if
superparticles collide?
Example recombination between positive and
negative ions Procedure number of
recombinations in ?t NN-Krec ?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
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DSMC Basics
Divide the geometry in cells Each cell should
contain enough testparticles (typically
25) Newtons 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
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DSMC Choosing the pairs
Add null collision Chance of collision of
particle i with j is Pc(Npp/Vcell)Max(v?)?t
Number of colliding pairs n(n-1) Pc/2 Select
randomly particle pairs (make sure no double
selection) See if there is no null collision
(again with random number) Perform the collision
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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