Solution of time-dependent Boltzmann equation for electrons in non-thermal plasma

Transcription

1 Solution of time-dependent Boltzmann equation for electrons in non-thermal plasma Z. Bonaventura, D. Trunec Department of Physical Electronics Faculty of Science Masaryk University Kotlářská 2, Brno, CZ-61137, Czech Republic Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

2 Description of plasma Kinetic description: most fundamental way providing f(r, v, t) achieved by solving the Boltzmann equation self-consistently determined the electromagnetic fields via Maxwell s equations Fluid description: description of the plasma based on macroscopic quantities fluid equations obtained by taking velocity moments of the BE Hybrid description: a combination of fluid and kinetic models treating some components of the system as a fluid and others kinetically Gyrokinetic Description: appropriate to systems with a strong background magnetic field the kinetic equations are averaged over the fast circular motion of the gyroradius Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

3 What will I talk about? weakly ionized plasma (small fraction of the atoms are ionized) moderate pressure ( 1 Torr) non-thermal or cold plasma (T e >> T ion = T gas ) non-magnetized plasma (B=0) solution of linear Boltzmann equation for electrons Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

4 What will I talk about? weakly ionized plasma (small fraction of the atoms are ionized) moderate pressure ( 1 Torr) non-thermal or cold plasma (T e >> T ion = T gas ) non-magnetized plasma (B=0) solution of linear Boltzmann equation for electrons Are you still interested? Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

5 The Boltzmann equation The Boltzmann equation is an equation for the time evolution of the distribution function f(r, v, t) in phase space, where r and v are position and velocity f t + v rf + a v f = f t We will discussed the Boltzmann equation with a collision term for two-body collisions between particles of different types that are assumed to be uncorrelated prior to the collision: the linear Boltzmann equation. c Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

6 The Boltzmann equation The Boltzmann equation is an equation for the time evolution of the distribution function f(r, v, t) in phase space, where r and v are position and velocity f t + v rf + a v f = f t We will discussed the Boltzmann equation with a collision term for two-body collisions between particles of different types that are assumed to be uncorrelated prior to the collision: the linear Boltzmann equation. How we can solve it? c Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

7 How to solve the Boltzmann equation? There are generally two ways: Monte Carlo methods (TPMC, PIC/MCC) A way how to solve it without solving it Actual solution of The Equation difficult but in simple cases and with approximations it is feasible Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

8 Monte Carlo methods a way how to obtain results with minimum of approximations MC: Solves a particle charge transport by the numerical simulation of large particle ensemble, collision processes are introduced by generating appropriately distributed random numbers. Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

9 Monte Carlo methods a way how to obtain results with minimum of approximations MC: Solves a particle charge transport by the numerical simulation of large particle ensemble, collision processes are introduced by generating appropriately distributed random numbers. How does it work? Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

10 Consider n-dimensional integral I = f(x 1,..., x n ) d n x, in n-cube of unit volume. It is the mean value I = f. MC gives us an approximation of exact value of the mean value with statistic estimate based on a numerical experiment. The mean value is estimated on an average over N trials I N i=1 f(η 1,...,η n ), which is exact for N N Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

11 Consider n-dimensional integral I = f(x 1,..., x n ) d n x, in n-cube of unit volume. It is the mean value I = f. MC gives us an approximation of exact value of the mean value with statistic estimate based on a numerical experiment. The mean value is estimated on an average over N trials I N i=1 f(η 1,...,η n ), which is exact for N N But why random numbers? Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

12 Consider n-dimensional integral I = f(x 1,..., x n ) d n x, in n-cube of unit volume. It is the mean value I = f. MC gives us an approximation of exact value of the mean value with statistic estimate based on a numerical experiment. The mean value is estimated on an average over N trials I N i=1 f(η 1,...,η n ), which is exact for N N Because a finite distribution of regular points gives a poor resolution whent the dimension n is high. Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

13 Propagation method Problem to be solved: determination of distribution function f of charged particles based on the initial condition f 0. Initial condition is a sum of mathematical particles (δ-functions) which are propagated in time and space to represent a solution at later times Consider N simulated superparticles, these represent N real real particles w = N real N weight of superparticles Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

14 Propagation method Problem to be solved: determination of distribution function f of charged particles based on the initial condition f 0. Initial condition is a sum of mathematical particles (δ-functions) which are propagated in time and space to represent a solution at later times Consider N simulated superparticles, these represent N real real particles w = N real N weight of superparticles How to determine the density and the energy distribution function? Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

15 Estimation of particle density in a given cell of the mathematical mesh n = N cellw x y z Determination of energy distribution function by introducing a mash on the kinetic energy axis Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

16 How to propagate particles? performing a numerical solution of equations of motion under the effect of Lorentz force and random force which represents collision events between charged particles and neutral particles background dv t dr dt = v = q m E + F coll m collision events are instantaneous, successions of kicks Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

17 How to propagate particles? performing a numerical solution of equations of motion under the effect of Lorentz force and random force which represents collision events between charged particles and neutral particles background dv t dr dt = v = q m E + F coll m collision events are instantaneous, successions of kicks How are collision times, t c, distributed? Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

18 How to propagate particles? performing a numerical solution of equations of motion under the effect of Lorentz force and random force which represents collision events between charged particles and neutral particles background dv t dr dt = v = q m E + F coll m collision events are instantaneous, successions of kicks How are collision times, t c, distributed? The answer is simple for constant collision frequency! Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

19 Distribution of collision times for constant collision frequency α P(t c ) = α exp( αt c ) free flight time can be generated easily t c = lnη α. Implementation of propagation: particle collisionless equation of motion between (t, t + t c ) is solved dr = v dt dv = q t m E Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

20 Distribution of collision times for constant collision frequency α P(t c ) = α exp( αt c ) free flight time can be generated easily t c = lnη α. Implementation of propagation: particle collisionless equation of motion between (t, t + t c ) is solved dr = v dt dv = q t m E And... Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

21 Distribution of collision times for constant collision frequency α P(t c ) = α exp( αt c ) free flight time can be generated easily t c = lnη α. Implementation of propagation: particle collisionless equation of motion between (t, t + t c ) is solved dr = v dt dv = q t m E the collision is introduced by laws of physics, based on appropriate random distribution of impact parameters Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

22 But... Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

23 Collision frequency is not a constant! Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

24 Null collision technique A new collision frequency is introduced α max = max α(v, r). r,v w α(v, r) = p n(r)vσ p (v) α(v, r) is a sum of all contributions from different collision processes p A new null collision process is added in order to equate α(v, r) and α max Non-physical collisions are eliminated by considering null collision with probability 1 α(v, r)/α max Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

25 Null collision technique selection of an event with a probability p: Uniform η (0, 1] : (p > η? accepted : rejected) after the collision a new direction of velocity and its magnitude is determined Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

26 Null collision technique Isotropic scattering can be assumed in many cases, momentum transfer cross section is used insted of DCS. Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

27 Null collision technique Isotropic scattering can be assumed in many cases, momentum transfer cross section is used insted of DCS. Is the Monte Carlo technique merely a solution on the bases of physical analogy? Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

28 Formal view of Monte Carlo technique Consider linear Boltzmann equation in the form df dt = α maxf + α max Af A is a linear combination of unit operator and a Boltzmann collision operator The integral form: t f(t) = exp( α max t) + α max exp( α max τ)a(τ) f(τ) dτ Monte carlo method can be derived from numerical solution of this equation. MC proceure is applied to calculate this integral. 0 Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

29 Solving the BE by solving it Approximations: spatialy homogeneous electrons collide with neutral gas particles collisions are isotropic ionization is taken to be conservative gas particles at rest The Boltzmann equation solved by the expansion into the Legendre polynomials Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

30 Expansion of the BE into the LPs E z-axis BE has a form = f(v, t) f(v, θ, t) (rotational symmetry) f t + ee m f = f v z t. c Distribution function expanded into Legendre polynomials: f(v, θ, t) = f n (v, t) P n (cosθ). n=0 n=0 [ fn t + ee m + ( n f n 1 + n + 1 f n+1 2n 1 v 2n + 3 v (n + 2)(n + 1) f n+1 2n + 3 v n(n 1) 2n 1 f n 1 v + )] P n (cos θ) = f t. c Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

31 Expansion of BE into LPs Resulting hierarchy of PDEs f n(v, t) = t n» ee(t) 2n 1 m n + 1» ee(t) 2n + 3 m v n 1 v 1 v n+2 v fn 1 (v, t) v n 1 «`v n+2 f n+1 (v, t) + + vn(t)`q n(v) Q 0 (v) f n(v, t) + + m 1 h i v 4 N(t)`Q M v 2 1n (v) Q n(v) f n(v, t) + v + X» (v k A ) 2 N(t)Q v n(v k A)f k n(va, k t) k vn(t)q0(v)f k n(v, t), n 0, f 1 0, where... Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

32 ... where Q n(v) = Q 1n (v) = Q k n(v) = Z 2π 0 Z 2π 0 Z 2π 0 dφ dφ dφ Z π 0 Z π 0 Z π 0 σ el (v, θ)p n(cos θ) sin θ dθ, σ el (v, θ)p 1 (cos θ)p n(cos θ) sin θ dθ, σ k (v, θ)p n(cos θ) sin θ dθ, where σ el (v) σ k (v) are differential collision cross sections for elastic and k-th inelastic collision process and «va k = v 2 + 2Ak, A k > 0 m A k is excitation energy of k-th state. For numerical solution kinetic energy U used instead of velocity v: U = (v/γ) 2, γ = (2/m) 1/2. Isotropic scattering (introducing mt) Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

33 Different approximations of the expansion of BE Two-term approximations: conventional two-term approx. f 1 / t 0, f 1 (U, t) E(t) f 0 (U, t)/ U you cannot choose f 1 (U, 0)! strict time-dependent two-term approx. f 2 (U, t) 0 Multi-term approximation: Expansion truncated after arbitrary (> 2) number of terms Effective field approximation (2-term): E(t) = E 0 exp jωt Time variation f 0 a f 1 slow enough compared to ω. Generally: ω/2π GHz or higher Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

34 Solution procedure Hierarchy of PDEs solved as a initial boundary value problem f 0 U = 0, f n (U = 0, t) = 0, n 1 U=0 f n (U U, t) = 0, n 0, U sufficiently large energy, by the method of lines BE discretized in energy space on uniform mesh U (0, U ) system of ODEs received ODEs integrated by Runge Kutta method of order 4(5) with step-size control Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

35 Case study: Reid ramp model Reid ramp model: Classical benchmark problem for solvers of BE, (Reid I. D., 1979, Aust. J. Phys. 32 p 231) Hypothetical gas mass: 4 amu cross section for elastic collisions σ el = m 2 cross section for inelastic collisions { 0 ε 0.2 ev σ in = 10(ε 0.2) m 2 ε > 0.2 ev All collisions assumed to be isotropic Thermal motion of gas particles neglected Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

36 Reid ramp model: results Time independent electric field Initial distribution of electrons: Maxwellian 300 K. Steady state for E/N = 24 Td (1 Td = V m 1 ) BE MC Reid Ness Drift velocity (10 4 m s 1 ) Mean energy (ev) Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

37 v d [10 4 m/s] Reid ramp model: E/N = 24 Td, 1 Torr BE MC ε [ev] time [s] Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

38 Rare gases and the effect of negative mobility f n f 0 final f 0 initial Argon Q el Q(ε) [10-20 m 2 ] v d (10 4 m/s) v d ε ε (ev) Argon: energy [ev] time (s) initial distribution: Gaussian, center 5.5 ev, width 1 ev gas density m 3 (pressure 1 Torr, temperature 273 K) E = 2 kv m 1, E/N = 58 Td. Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

39 Rare gases and the effect of negative mobility ε 1/2 f 0 /n e [ev -1 ], ε f 1 /n e [ev -1/2 ] Xenon: s f 0 f 1 > 0 f 1 < ε [ev] coll. cross section [a.u.] v d (10 4 m/s) ε 2.0 v d time (s) ε (ev) initial distribution: Gaussian, center 15.0 ev, width 1 ev gas density m 3 (pressure 1 Torr, temperature 273 K) E = 2 kv m 1, E/N = 58 Td Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41

40 Conclusions For a solution of time-dependent Boltzmann equation the Monte Carlo Method: is easy to implement, rather straightforward method, even for complex boundary conditions, may be rather time consuming Solution of the Equation: is a challenge, it is a fun, it gives good results (multi-term), for many problems it is impossible Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondřejov, / 41