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The main ion-atomic collision treatment methods based on Monte-Carlo simulation are considered and discussed. We have proposed an efficient scheme for simulation of time between collisions taking into account cross-section dependence on ion velocity and random generation of ion velocities and scattering angles after collisions. The developed algorithm of simulation of interval between collisions takes into account the change of relative velocity of ion-atom pair as well as the change of cross-section of collision and atomic concentration. At the same time, unlike the widely used “null-collision” method, both the probability of collision and change of particles’ state which determines this probability are taken into consideration for each particle independently in time. The simulation results according to the techniques proposed are found to be close to the theoretical values of ion drift velocities. It is revealed that the “null-collision” method results in exceeding of drift velocity in strong and intermediate fields. At the same time the proposed method of accumulation of probability under the same conditions gives values close to theoretical ones. In weak fields calculated values of drift velocity in both methods exceed theoretical values to some small extent.

The simulation of plasma by Particle-in-Cell Monte-Carlo (PIC-MC) method has received widespread application in calculations of ion current on probe and dust particle and plasma simulation of glow and high-frequency discharges [

The drift time of an ion up to the next collision has probabilistic nature: probability of collisions for the time

Let us assume by this

where

Equation (2) is probability of collision since the time

The following methods of simulation of the interval between collisions are applied.

Equation (3) is approximation of constant time between collisions

This approximation is correct at inverse dependence of cross-section on velocity.

In the supposition based on Equation (3)

with the help of uniformly distributed at [0,1] random value “rand”. Equation (4) shows the expression for random value “rand”

In Equation (4)

Equation (5) is the expression for path length in this case

Let us replace a variable in Equation (1)

For each ion path length is simulated after each collision. Equations (7) are the expressions for ion’s path length

The achievement of individual path length by each ion is checked by summing up its ways on each time step

The approximation of constant path length takes into account the change of ion velocity, however the cross- section of collisions is considered to be constant here that can be accepted with some approximation at resonant charge exchange. In [

For the time interval

While each ion is simulated, if

respond to the current time point

This algorithm is considered in details in [

product

tion (9) is constant conditional collision probability for the time

Then, in case of total number of ions N on each time step, the number of ion is simulated P_{0}N times. For this ion rand is simulated again and the type of collision or the absence of it is defined. Equation (10) is the condition of event that k-type of collision occurs:

At

Treatment methods of collision probability are combined in the “null-collision” technique. At first, the number of the ion is determined in the ensemble, and then the type of collision or its absence is defined through the change of ion state in time. The number of arithmetic operations

In [

There is a variety of revisions of the “null-collision” algorithm used for simulation of electron-atom collisions when cross-sections are strongly dependent on electrons’ energy. Thus, in the method of constant time between

collisions the time interval is determined by maximum possible value of

It is possible to offer the following method taking into account the change of value

then we will have

Equations (13) are the expressions if individual ion is simulated:

Summing up by steps in time lasts up to the reaching of equality in Equation (13). In such a way, this algorithm takes into account the change of

Direction of ion motion after the collision is characterized by taking into account deviation from original direction

At elastic scattering of ion on atom in these works scattering goes only forward. Equation (15) is the formula for

Equation (16) is energy after scattering:

In works [

Let us consider Monte-Carlo algorithm of simulation of elastic ion-atom collisions on the model of elastic spheres. In this model in

Further we consider ion motion in their own gas

of ion and atom velocities to normal. For the angle

rameter,

For the density of probability we have

For strong fields if we ignore atom velocity

However, in weak fields an ion can get significant additional velocity from an atom in collisions.

Equation (19) is ions’ velocity after collision in this case

Let us consider atom

Equation (20) is the expression for simulation after introducing the new parameter

The integral can be replaced by an analytic expression, having approximation with accuracy up to 3%. Equation (23) is the approximation for

The results of calculations presented in

Equation (24) is density of probability of

It determines

Angle

Let us have angle

It is here that we need angle

At resonant charge exchange angle

To compare “null-collision” method and method of accumulation of probability the simulation experiment of ion motion in constant electric field

rand_{3} | 5.88 ´ 10^{−3} | 4.38 ´ 10^{−2} | 0.131 | 0.266 | 0.427 | 0.590 | 0.730 | 0.837 | 0.910 | 0.954 | 0.991 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.4 | |

0.185 | 0.400 | 0.616 | 0.832 | 1.041 | 1.243 | 1.440 | 1.630 | 1.815 | 1.993 | 2.343 |

The number of ions was taken as equal to 10^{5}. Original ion velocities had Maxwell distribution with the temperature of atoms by Equations (23) and (24). The interval of time

Equations (27) are non-dimensional variables accepted:

With equations of motion

Obtained average drift velocities were compared with theoretical ones [

Equations (28) are drift velocities for resonant charge exchange and elastic collision in weak fields when drift velocity is less than thermal velocity:

at resonant charge exchange and elastic interaction correspondingly, where

Equation (30) is the approximation used in intermediate fields:

The results of simulation are given in

10^{-6} | 0 | 0.02 | 7.98 ´ 10^{−4} | 8.0 ´ 10^{−4} | 8.0 ´ 10^{−4} |

10^{−4} | 0 | 0.2 | 7.98 ´ 10^{−3} | 8.0 ´ 10^{−3} | 8.3 ´ 10^{−3} |

10^{−3} | 0 | 0.47 | 2.52 ´ 10^{−2} | 2.5 ´ 10^{−2} | 2.9 ´ 10^{−2} |

10^{−2} | 0 | 0.86 | 7.98 ´ 10^{−2} | 8.0 ´ 10^{−2} | 0.12 |

10^{−6} | 5.8 ´ 10^{−3} | 0.02 | 8.1 ´ 10^{−5} | (8.4 - 9) ´ 10^{−5} | (8.6 - 9.2) ´ 10^{−4} |

10^{−4} | 3.66 ´ 10^{−2} | 0.2 | 1.2 ´ 10^{−3} | (1.3 - 1.4) ´ 10^{−3} | (1.6 - 1.8) ´ 10^{−3} |

10^{−5} | 0 | 0.02 | 3.63 ´ 10^{−3} | 3.6 ´ 10^{−3} | 3.6 ´ 10^{−3} |

10^{−4} | 0 | 0.05 | 1.15 ´ 10^{−2} | 1.16 ´ 10^{−2} | 1.2 ´ 10^{−2} |

10^{−3} | 0 | 0.15 | 3.63 ´ 10^{−2} | 3.6 ´ 10^{−2} | 3.9 ´ 10^{−2} |

10^{−4} | 0.016 | 0.08 | 5.22 ´ 10^{−3} | (5.3 - 5.4) ´ 10^{−3} | (5.3 - 5.5) ´ 10^{−3} |

10^{−3} | 0.016 | 0.18 | 3.09 ´ 10^{−2} | (3.1 - 3.3) ´ 10^{−2} | (3.2 - 3.5) ´ 10^{−2} |

It is necessary to mention that computing time in “null-collision” method (in the same conditions) turned out to be 2 - 3 times less than in method of accumulation of probability. However, in case of large

Method of accumulation of probability in strong and intermediate fields gives values close to theoretical ones. Accuracy is limited only by fluctuations of average velocity increasing when the total number of ions decreases and their temperature goes up. In weak fields calculated values of drift velocity in both methods exceed theoretical values to some extent (up to 5%). It can be caused by the fact that when Equations (28) are obtained inte-

grated square velocity is accepted as an average relative velocity:

The given effective models of calculating time between collisions by the method of accumulation of probability and ion angle velocities’ simulation after collision on the basis of solid spheres result in ion drift velocities close to theoretical ones and can be applied in the process of simulation of ion motion in heterogeneous plasma with non-constant concentration of atoms and ions and dependence of cross-section on velocity.

The work is done by the support of Program of strategic development of Petrozavodsk State University from 2012 to 2016, Ministry of Education of Russian Federation, contract No. 3.757.2014/K.