Solution of Time-dependent Boltzmann Equation

Transcription

1 WDS'5 Proceedings of Contributed Papers, Part III, , 5. ISBN MATFYZPRESS Solution of Time-dependent Boltzmann Equation Z. Bonaventura, D. Trunec, and D. Nečas Masaryk University, Faculty of Science, Department of Physical Electronics, Kotlářská, Brno, Czech Republic. Abstract. The time-dependent Boltzmann equation BE) is solved in a two-term and a multi-term approximation in order to study the time evolution of the electron energy distribution function EEDF) in argon plasma in a constant electric field. The initial distribution is taken to be isotropic Gaussian distribution, its center is set to the area of increasing elastic collision cross-section in order to study the effect of negative mobility. This effect is described and explained in detail. The results of the solution of the BE are compared with the results from the Monte Carlo simulation. Introduction In the last twenty years there have appeared several articles on the negative mobility of electrons in low-temperature plasma. This interesting phenomenon appears for example in a weakly ionized relaxing plasma in rare gases. At first it was theoretically predicted [McMahon et al., 1985] and this prediction enforced the publication of previous unpublished experiments [Warman et al., 1985]. In these experiments the phenomenon was observed on a nanosecond time scale in relaxing Xe plasma, ionized by a hard x-ray pulse. The possibility of steady state conditions leading to the negative mobility was theoretically studied in externally ionized gas mixtures [Rozenberg et al.,1988]. The theoretical studies mentioned above were made by solving an appropriate Boltzmann equation using so-called two-term approximation for the EEDF. The applicability of this approximation for a steady state was studied by [Dyatko et al., ] using two different methods: a Monte Carlo MC) simulation and the solution of Boltzmann equation in two-term approximation. The main aim of the presented paper is to demonstrate the applicability of a strict timedependent two-term approximation STTA) and higher order terms so called multi-term approximation MTA). The correctness of results of STTA is tested by the MC simulation. As a testing example the relaxing argon plasma was chosen. The reasons how can electrons move against the acting electric force which leads to a negative mobility) are discussed in more details. Model Suppose, we have a spatially homogeneous argon plasma. The neutral gas particles of mass M and density N are assumed to be at rest T gas = K). We are interested in the behavior of electrons mass and charge e ) under the action of external electric field E that is suddenly switched on at time t = s and remains constant. Electrons collide with neutral gas particles only, there is neither electron electron nor electron ion interaction. There is also no self-consistent space charge electric field. For simplicity only conservative inelastic collisions including ionization) were considered and the elastic and all inelastic collisions were assumed to be isotropic. 613

2 Boltzmann equation The time evolution of distribution function of electrons Fv,t) in our model can be described by the Boltzmann equation in a form t F e E v F = Cel + m C in m, 1) where C el and Cm in are elastic and inelastic collision integrals for the collisions of electrons with neutral gas particles. The electric field E is chosen to be parallel to the z-axis. In this case the distribution function Fv,t) has reduced velocity dependence Fv,v z /v,t) and it can be easily expanded in Legendre polynomials P n v z /v). This expansion is truncated after a sufficiently high number l of terms in order to approximate the distribution function properly Fv,v z /v,t) = l 1 n= F n v,t)p n v z /v). ) Substituting this expansion to the Boltzmann equation and replacing the velocity magnitude v by the kinetic energy U = v /, we obtain the hierarchy of partial differential equations [Loffhagen,1996] t f e 3 U U 1 U 3 U U + U in m U m Et) U Uf 1) M νel U)f ) + m ν in m U)f ν in mu + U in m)f U + U in m) = 3) t f n n n 1 n + 1 n + 3 m e + ν el U) + m [ e Et) U 1 [ e Et) ν in mu) ) U 1 U f n 1 n 1 ] U 1 f n 1 U f n+1 + n + ] U 1 fn+1 + f n =, 1 n l 4) t f l 1 l 1 l 3 + for the expansion coefficients ν el U) + m [ e Et) ) ν in mu) U 1 U f l l ] U 1 fl + f l 1 =, 5) )3 f n U,t) = π Fn vu),t). 6) Where U in m is the energy loss in the mth inelastic process. In deriving this hierarchy a further expansion of each collision integral with respect to the ratio /M of the electron-to-gas particle masses has been performed, only a leading term has been taken into account. The quantities ν el U) = U 1 NQ el U), 614

3 ν in mu) = me U 1 NQ in m U) 7) are the collision frequencies for elastic and inelastic collisions, where Q el and Q in m denote the cross-sections for elastic and mth inelastic collision. The system of equations was solved as an initial-boundary value problem. The boundary conditions are given by relations f U U,t) = f n U =,t) = n 1 f n U U,t) = n 1, 8) where U is a sufficiently large energy. The system of equations was discretized in energy space on a uniform grid using finite differences. The resulting system of ordinary differential equations for the values of f n in the grid points was obtained and it was solved by an explicit Runge-Kutta method of order 4)5 DOPRI5 procedure [Hairer, 1993]). This method of solution is known as the method of lines. The electron number density has to be constant during the time evolution since ionization is taken to be conservative and this fact was used to check the accuracy of the numerical calculations. Monte Carlo Motion of electrons in an external electric field E is simulated using Monte Carlo method also. Appropriate initial velocity distribution of electrons is generated. The trajectories are treated one by one. Electrons collide with neutral gas particles of zero temperature this is for the consistency with kinetic simulation described above. Time of a free flight between two successive collisions is generated using a null-collision method [Longo, ]. Collisions are assumed to be binary and instantaneous. The result of Monte Carlo simulation is a distribution function Fv,t) at a given time t. Results and discussion All the calculations were done for E/N = 6Td. Neutral gas number density N corresponds to pressure 1Torr, and temperature T = 73.15K. This temperature is used just to calculate the value of N, all the gas particles are actually at rest as supposed in the model. The electrons starts from the isotropic Gaussian distribution in the form ) ) n e exp U Uc U w f U,) = Ū 1 Ū Uc ) ), exp U w dū where U c = 5.5eV, U w = 1.eV. The initial distribution can be seen in fig. 3. The values of U c and U w were chosen in order to fit the initial distribution in the area of rapidly growing elastic collision cross-section. When the electric field is switched on the electrons move in the direction of the acting force DAF) and their drift velocity grows in absolute value) and is negative due to a negative sign of the electron charge, see Fig. 1. Then at time 1 1 s that is approximately equal to a time of a free flight, the drift velocity turns to zero again and what more it changes its sign in the time interval from.6 to 4ns. Note that the electrons move against the DAF at this time and a majority of electrons that are situated in the area close to the Ramsauer minimum of the elastic collision cross-section can be seen in figure 3. The reason is as follows: We suppose to have an isotropic scattering so that after each collision the scattered electron has an equal probability to go in the DAF and to go in the 615

4 1 TA 4 TA MC v d 1 4 m/s) time s) Figure 1. Drift velocity of electrons in Ar as a function of time. Electrons starts form an isotropic distribution v d = ) at time t = s. Drift velocity is negative due to a negative sign of the electron charge. It reaches its maximal negative value, then it decreases and v d changes the direction at time interval from.6 to 4ns. Then it is negative again and it reaches its steady state value. Circles represent the results of Monte Carlo MC) simulation, the solid line represents the solution of BE 1) in four term approximation 4 TA), and the broken line represents the solution of BE in two term approximation TA). opposite direction. In the first case it is accelerated and its energy grows, it climbs up the collision cross-section and since nearly all electrons are situated in the area where dqu)/du > their time of a free flight swiftly decreases. If the electron is scattered in the direction opposite to DAF it decelerates, it goes down the cross-section and its time of a free flight grows. As a result of this, there is a large group of electrons going in the direction opposite to the DAF. In figure there is depicted the distribution of electrons Fv,v z /v) as a function of a magnitude of velocity v and of a projection of v to the direction of electric field drawn as a gray map: the darker the color, the higher the value. The left part of the plot with v z /v [ 1,] corresponds to the electrons, that are moving in the direction of electric force are accelerated. The right part of the plot with v z /v [,1] corresponds to the electrons that are decelerated by the electric force. It can be seen that the left part of this distribution corresponding to the electrons running in DAF goes to higher energies than the electrons running in the direction opposite to DAF. The negative mobility is due to the group of electrons situated in the lower right part of this distribution function. The function Uf 1 U) at the same time can be seen in fig. 4 also. The reason for the different signs of the peaks is the following: the electrons at a given energy move mostly in the DAF where the sign of this function is negative, and in opposite to DAF where the sign is positive. The area under this function is directly connected to the value of the drift velocity by the relation v d t) = 1 U f 1 U,t)dU. 9) 3n e t) The values of drift velocity in Fig. 1 were computed in two different approximations of BE. It can be seen that the strict time dependent two term approximation exhibits some artificial oscillations at time near 1 8 s and due to this reason it is not a sufficient approximation. 616

5 Conclusion The time-dependent Boltzmann equation was solved in a two-term and a four-term approximation in order to study the time evolution of the electron energy distribution function in argon plasma in a constant electric field. Argon was chosen as a model gas because of the shape of elastic collision cross-section with Ramsauer minimum and a strong dqu)/du > behavior. This shape of cross-section is common in rare gases. Due to this dqu)/du > shape the isotropic scattering can exhibit apparent anisotropic behavior and for some short time it can create a large number of electrons running in the direction that is opposite to the direction of acting force. This leads to the phenomenon of negative mobility as discussed. This effect exists only within a very short time since a relatively strong group of high energy electrons running along the DAF is created and the effect of apparent anisotropic behavior in the area where dqu)/du > is no longer strong enough. It was shown that the ordinary two-term approximation of the expansion of the Boltzmann equation into the Legendre polynomials is not sufficient since it exhibits some artificial nonphysical oscillations. v [,v max ] v z /v [-1,1] Figure. A plot of Fv,v z /v). The left part of the plot with v z /v [ 1,] corresponds to the electrons, that are moving in the direction of electric force are accelerated. The right part of the plot with v z /v [,1] corresponds to the electrons that are decelerated by the electric force. Acknowledgments. We are grateful to the Czech Science Foundation for financial support under contract No /3/87 and No /3/H

6 1 1 1 QU) U 1/ f /n e ev -1 ) QU) 1 - m ) t = s s s U ev) Figure 3. Energy dependence of a function Uf U) of the zeroth expansion coefficient from eqn. 6) at three different times various thin lines) together with a collision cross-section for electrons in argon thick full lines). The area under this function corresponds to the electron concentration. The dashed-and-dotted line represents the Uf U) at the time of a maximal negative mobility. It can be seen that a great deal of the electrons is in the area where the slope of a collision cross-section has a large positive dqu)/du > value. The thin full line represents an initial and the dotted line represents a steady state of Uf U) QU) U f 1 /n e ev -1/ ) ) -) s s QU) 1 - m ) U ev) Figure 4. Energy dependence of the magnitude of the function Uf 1 U) of the first expansion coefficient froqn. 6) at two different times various thin lines) together with a collision cross-section for electrons in argon thick full lines). The dashed-and-dotted line represents the Uf 1 U) at the time of a maximal negative mobility and +) and ) shows its sign. The final Uf 1 U) is negative in all the energy range. The function Uf 1 U) du is proportional to a contribution of electrons in the vicinity du near the energy U to a drift velocity v d, see eqn. 9). 618

7 References N. A. Dyatko, A. P. Napartovich, S. Sakadzic, Z. Petrovic and Z. Raspopovic, On the possibility of negative electron mobility in a decaying plasma, J. Phys. D: Appl. Phys., 33, , R. Hairer, S.P. Norsett and G. Wanner, Solving ordinary differential equations, I. Nonstiff problems. nd edition, Springer series in computational mathematics, Springer-Verlag, D. Loffhagen and R. Winkler, Time-dependent multi-term approximation of the velocity distribution in the temporal relaxation of plasma electrons, J. Phys. D: Appl. Phys., 9, , D. Loffhagen and R. Winkler, Multi-term treatment of the temporal electron relaxation in He, Xe and N plasmas, Plasma Sources Sci. Technol., 5, , S. Longo, Monte Carlo models of electron and ion transport in non-equilibrium plasmas, Plasma Sources Sci. Technol.,9, , Z. Rozenberg, M. Lando, and M. Rokni, On the possibility of steady state negative mobility in externally ionised gas mixtures, J. Phys. D: Appl. Phys., 1, , 1988 D. R. A. McMahon and B. Shizgal, Hot-electron zero-field mobility and diffusion in rare-gas moderators, Phys. Rev. A, 31, , 1985 J. M. Warman, U. Sowada, and M. P. De Haas, Transient negative mobility of hot electrons in gaseous xenon, Phys. Rev. A, 31, ,