1. Introduction Motivation Objective
DSMC (The Direct Simulation Monte Carlo method) is often used to analyse the plasma kinetics, since the behaviour of electrons and ions can be calculated correctly by this model even in the non-equilibrium region like electrode sheath.
In order to improve the accuracy of DSMC, it is necessary to describe the density and the velocity distributions in detail.
A number of slabs and the mesh points in the velocity distributions are needed, and this causes considerable computing time.
Objective
To develop DSMC coupled with LPWS in order to improve the accuracy of DSMC without increasing its computing time.
LPWS is adopted to describe the density distribution of the particles in discharge space.
The behaviour of electron swam in a parallel plate gap is traced and the swarm parameters calculated by this method, and the obtained parameters are compared with those calculated by conventional particle models.

2. DSMC (The Direct Simulation Monte Carlo method)
Discharge space is divided into slabs, and velocity distributions of the charged particles are defined in the slabs.
Each of the particles in the distributions is picked up, and its drift, acceleration and collision in unit time Dt are calculated by Monte Carlo technique, then the particle is stored in a new velocity distribution.
This procedure is repeated and the spatiotemporal evolution of the particles is given.
BEFORE
AFTER
n(z)
n(z) z z E
slab E
slab
f (vz,vr)
Cathode
Cathode
Anode
Anode
f (vz,vr)
Change in Dt is
calculated by MCS vr vz vz vr

3. EEDF sampling by conventional method and LPWS
Legendre Polynomial 　　　　　　　　　　　　　 Weighted Sampling
(simple counting)
-１≦χ≦１
Number of electrons in a bin is sampled.
To obtain EEDF in detail
fine mesh(bin) is needed
statistical fluctuation
a number of electron must be sampled
considerable computing time
Density gradient Gi(x) in a bin is described 　using Legendre polynomial.
To obtain EEDF in detail
accurate G(x) is needed
independent on bin size

4. Method of LPWS Density gradient in the i th bin ei ei+1 ei+2 Gi Gi+1
At boundary , Gi and Gi+1 are not necessarily continuous.
Solutions
To sample enough number of electron
To superpose Gi using B-spline
Superposition
Secondary B-spline
in general,
m : the order of B-spline Si : B-spline
0≦x≦１
( K.Satoh et al, T.IEE, Japan Vol.120-A, No.2, pp (2000) )

5. Direct Simulation of Monte Carlo with LPWS
LPWS gives detailed information and good statistics of density and velocity distributions with smaller number of particles and with wider size of bin than the conventional sampling. n
n(z)
1D LPWS
n(z)
1D LPWS can give accurate density distribution n(z) z E
slab
Cathode
Anode
2D LPWS
f (vz,vr) vr vz
2D LPWS can give accurate velocity distribution f(vz,vr)
f (vz,vr) vr vz

6. Calculation conditions
Gas : CF4
Electrodes : fully absorbing walls
Gap length : 3cm
Gas temperature : 0 ℃
Gas pressure p : 1Torr
Reduced electric field : 400Td
Initial electron number : 10,000
Sampling interval Dt : 0.1ns
Legendre polynomial : 8 terms
B-spline : primary
Number of slabs : 30
Description of f(vz,vr) :conventional method
Collision cross sections of CF4
H. Itoh et al, Proceedings of 21st ICPIG, Vol.1, p.245 (1993)
Initial density distribution z
n(z)
1cm
2cm
3cm
0cm
Gauss distribution with the center of mass of 0.15cm and s = is used.

7. Swarm parameters by conventional DSMC #1
Mean energy
Electron number
Velocity Wv
Ionization frequency

8. Swarm parameters by conventional DSMC #2
Center of mass (1st order moment)
2nd order moment Mz
The temporal variations of the mean energy, the drift velocities and the centre of mass (1st order moment) are not sensitive to the mesh number of the density and the velocity distributions.
The error in the ionisation frequency of the coarsest mesh (50x50x50) is not small, however, the error decreases when finer mesh is used.
The 2nd order moment is strongly affected by the number of the meshes.

9. Swarm parameters by DSMC with LPWS
2nd order moment Mz
Density distributions (T=40.0ns)
Although only 30 slabs are used for the density distribution in the DSMC coupled with LPWS, the temporal variation of the 2nd order moment approaches to that of the MCS.
Density distributions calculated by the conventional DSMC with coarse mesh are widened, so that they are strongly affected by numerical diffusion. However, density distribution calculated by DSMC coupled with LPWS is not affected by the numerical diffusion.

10. Electron swarm parameters and calculating time
The longitudinal diffusion coefficient DL is strongly affected by the number of the meshes and the value of error in DL is about 17% in the result of the conventional DSMC with the finest mesh (200x200x200) studied here. However, the value of error in DL obtained by DSMC coupled with LPWS descends to about 10%.
Computing time of DSMC coupled with LPWS is shorter than a quarter of that of the conventional DSMC with the finest mesh.
Computing time of DSMC coupled with LPWS enables improving the accuracy and reducing the computing time of DSMC.

11. DSMC with 1D-LPWS for n(z) and 2D-LPWS for f(vz,vr)
Ionisation frequency Ri
Longitudinal moment Mz
vz × vr × z CPU Time Ri (s-1) DL(cm2/s)
MCS(Thomas) min(DEC) , 59.2( ) , 0.94( )
30× 30× 30 (1D+2D) ,12.5min(DEC) , 59.2( 0% ) , 0.99( +5.3%)
200×200× 30 (1D) , 9.3min(DEC) , 59.7( +0.8% ) , 1.04( +10.3%)
By using 2D-LPWS for describing velocity distribution f(vz,vr) and 1D-LPWS for n(z), accuracy is also improved, however computing time increases. (about 1.5 times long)
1D-LPWS for n(z) + 2D-LPWS for f(vz,vr) is not suited practically for plasma simulation.

12. Conclusion DSMC coupled with LPWS is developed in this work.
This method reduces the error of swarm parameters and computing time.
Poisson’s equation and scaling technique for simulation of processing plasma are easily coupled with this method.
This method makes a contribution of improving the accuracy and accelerating calculation of particle model simulation for processing plasmas.

13. Thank you for your comments