Solution of time-dependent Boltzmann equation for electrons...

Solution of time-dependent Boltzmann equation forelectrons in non-thermal plasma

Z. Bonaventura, D. Trunec

Department of Physical ElectronicsFaculty of Science · Masaryk University

Kotlarska 2, Brno, CZ-61137, Czech Republic

Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondrejov, 18. 10. 2007 1 / 41

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 fluidand 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 gyroradiusZ. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondrejov, 18. 10. 2007 2 / 41

What will I talk about?

• weakly ionized plasma (small fraction of the atoms are ionized)

• moderate pressure (∼ 1 Torr)

• non-thermal or ’cold’ plasma (Te >> Tion = Tgas)

• non-magnetized plasma (B=0)

• solution of linear Boltzmann equation for electrons


What will I talk about?

• weakly ionized plasma (small fraction of the atoms are ionized)

• moderate pressure (∼ 1 Torr)

• non-thermal or ’cold’ plasma (Te >> Tion = Tgas)

• non-magnetized plasma (B=0)

• solution of linear Boltzmann equation for electrons

Are you still interested?


The Boltzmann equation

The Boltzmann equation is an equation for the time evolution of thedistribution function f (r , v , t) in phase space, where r and v areposition and velocity

∂f∂t

+ v · ∇r f + a · ∇v f =∂f∂t

∣

∣

∣

∣

c

We will discussed the Boltzmann equation with a collision term for

two-body collisions between particles of different types that areassumed to be uncorrelated prior to the collision: the linearBoltzmann equation .


The Boltzmann equation

The Boltzmann equation is an equation for the time evolution of thedistribution function f (r , v , t) in phase space, where r and v areposition and velocity

∂f∂t

+ v · ∇r f + a · ∇v f =∂f∂t

∣

∣

∣

∣

c

We will discussed the Boltzmann equation with a collision term for

two-body collisions between particles of different types that areassumed to be uncorrelated prior to the collision: the linearBoltzmann equation .

How we can solve it?


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


Monte Carlo methods

• a way how to obtain results with minimum of approximations

• MC:Solves a particle charge transport by the numerical simulation of largeparticle ensemble, collision processes are introduced by generatingappropriately distributed random numbers.


Monte Carlo methods

• a way how to obtain results with minimum of approximations

• MC:Solves a particle charge transport by the numerical simulation of largeparticle ensemble, collision processes are introduced by generatingappropriately distributed random numbers.

How does it work?


Consider n-dimensional integral

I =

∫ 1

0· · ·

∫ 1

0f (x1, . . . , xn) dnx ,

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 withstatistic estimate based on a numerical experiment.The mean value is estimated on an average over N trials

I ≃∑N

i=1 f (η1, . . . , ηn)

N, which is exact for N → ∞



I =

∫ 1

0· · ·

∫ 1

0f (x1, . . . , xn) dnx ,


I = 〈f 〉.


I ≃∑N

i=1 f (η1, . . . , ηn)


But why random numbers?



I =

∫ 1

0· · ·

∫ 1

0f (x1, . . . , xn) dnx ,


I = 〈f 〉.


I ≃∑N

i=1 f (η1, . . . , ηn)


Because a finite distribution of regular pointsgives a poor resolution whent the dimension n is high.


Propagation method

• Problem to be solved:⊲ determination of distribution function f of charged particles

based on the initial condition f0.⊲ Initial condition is a sum of mathematical particles (δ-functions)

which are propagated in time and space to represent a solutionat later times

• Consider N simulated superparticles, these representNreal real particles

w =Nreal

Nweight of superparticles


Propagation method

• Problem to be solved:⊲ determination of distribution function f of charged particles

based on the initial condition f0.⊲ Initial condition is a sum of mathematical particles (δ-functions)

which are propagated in time and space to represent a solutionat later times

• Consider N simulated superparticles, these representNreal real particles

w =Nreal

Nweight of superparticles

How to determine the density and the energy distribution function?


• Estimation of particle density in a given cellof the mathematical mesh

n =Ncellw

∆x∆y∆z

• Determination of energy distribution functionby introducing a mash on the kinetic energy axis


How to propagate particles?

• performing a numerical solution of equations of motionunder the effect of Lorentz force and random force whichrepresents collision events between charged particlesand neutral particles background

drdt

= v

dvt

=qm

E +Fcoll

m

• collision events are instantaneous, successions of “kicks”




drdt

= v

dvt

=qm

E +Fcoll

m


How are collision times, tc, distributed?




drdt

= v

dvt

=qm

E +Fcoll

m


How are collision times, tc, distributed?The answer is simple for constant collision frequency!


Distribution of collision times• for constant collision frequency α

P(tc) = α exp(−αtc)

• free flight time can be generated easily

tc = −ln η

α.

• Implementation of propagation:⊲ ∀ particle collisionless equation of motion

between (t , t + tc) is solved

drdt

= v

dvt

=qm

E





tc = −ln η

α.



drdt

= v

dvt

=qm

E

And . . .





tc = −ln η

α.



drdt

= v

dvt

=qm

E

• the collision is introduced by laws of physics,based on appropriate random distribution of impact parameters


But . . .


Collision frequency is not a constant!


Null collision technique

• A new collision frequency is introduced

αmax = maxr ,v≤w

α(v , r ).

α(v , r ) =∑

p

n(r)vσp(v)

• α(v , r ) is a sum of all contributions from differentcollision processes “p”

• A new “null” collision process is added in orderto equate α(v , r ) and αmax

• Non-physical collisions are eliminated by considering“null” collision with probability

1 − α(v , r )/αmax



• selection of an event with a probability p:

Uniform η ∈ (0, 1] : (p > η ? accepted : rejected)

• after the collision a new direction of velocityand its magnitude is determined



• Isotropic scattering can be assumed in many cases,momentum transfer cross section is used insted of DCS.



• Isotropic scattering can be assumed in many cases,momentum transfer cross section is used insted of DCS.

Is the Monte Carlo technique merely a solutionon the bases of physical analogy?


Formal view of Monte Carlo technique

• Consider linear Boltzmann equation in the form

dfdt

= −αmaxf + αmaxAf

A is a linear combination of unit operatorand a Boltzmann collision operator

• The integral form:

f (t) = exp(−αmaxt) + αmax

∫ t

0exp(−αmaxτ)A(τ) f (τ) dτ

• Monte carlo method can be derived from numerical solution ofthis equation. MC proceure is applied to calculate this integral.


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 expansioninto the Legendre polynomials


Expansion of the BE into the LPs

• E ‖ z-axis =⇒ f (v , t) → f (v , θ, t) (rotational symmetry)

• BE has a form∂f∂t

+eEm

∂f∂vz

=∂f∂t

∣

∣

∣

∣

c

.

• Distribution function expanded into Legendre polynomials:

f (v , θ, t) =

∞∑

n=0

fn(v , t) Pn(cos θ).

∞∑

n=0

[

∂fn∂t

+eEm

(

n2n − 1

∂fn−1

∂v+

n + 12n + 3

∂fn+1

∂v+

+(n + 2)(n + 1)

2n + 3fn+1

v−

n(n − 1)

2n − 1fn−1

v

)]

Pn(cos θ) =∂f∂t

∣

∣

∣

∣

c

.


Expansion of BE into LPs

• Resulting hierarchy of PDEs

∂fn(v , t)∂t

=

−n

2n − 1

»

−eE(t)

mvn−1 ∂

∂v

„

fn−1(v , t)vn−1

«–

−

−n + 1

2n + 3

»

−eE(t)

m1

vn+2

∂

∂v

`

vn+2fn+1(v , t)´

–

+

+ vN(t)`

Qn(v) − Q0(v)´

fn(v , t) +

+mM

1v2

∂

∂v

h

v4N(t)`

Q1n(v) − Qn(v)´

fn(v , t)i

+

+X

k

»

(v kA)2

vN(t)Qk

n (v kA)fn(v k

A , t) −

− vN(t)Qk0 (v)fn(v , t)

–

,

n ≥ 0, f−1 ≡ 0,

where . . .


. . . where

Qn(v) =

Z 2π

0dφ

Z

π

0σ

el(v , θ)Pn(cos θ) sin θ dθ,

Q1n(v) =

Z 2π

0dφ

Z

π

0σ

el(v , θ)P1(cos θ)Pn(cos θ) sin θ dθ,

Qkn (v) =

Z 2π

0dφ

Z

π

0σ

k(v , θ)Pn(cos θ) sin θ dθ,

where σel(v) σk (v) are differential collision cross sections for elastic and k-th inelasticcollision process and

v kA =

„

v2 +2Ak

m

«

, Ak> 0

Ak is excitation energy of k-th state.

• For numerical solution kinetic energy U used instead ofvelocity v :

U = (v/γ)2, γ = (2/m)1/2.

• Isotropic scattering (introducing mt)


Different approximations of the expansion of BE

• Two-term approximations:⊲ conventional two-term approx.

∂f1/∂t ≡ 0, f1(U, t) ∼ E(t)∂f0(U, t)/∂Uyou cannot choose f1(U, 0)!

⊲ strict time-dependent two-term approx.f2(U, t) ≡ 0

• Multi-term approximation:Expansion truncated after arbitrary (> 2) number of terms

• Effective field approximation (2-term):E(t) = E0 exp jωtTime variation f0 a f1 “slow enough” compared to ω.Generally: ω/2π ∼ GHz or higher


Solution procedure

• Hierarchy of PDEs solved as a initial–boundary value problem∂f0∂U

∣

∣

∣

∣

U=0= 0, fn(U = 0, t) = 0, n ≥ 1

fn(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


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

σel = 6.0 × 10−20 m2

⊲ cross section for inelastic collisions

σin =

{

0 ε ≤ 0.2 eV10(ε − 0.2) × 10−20 m2 ε > 0.2 eV

• All collisions assumed to be isotropic

• Thermal motion of gas particles neglected


Reid ramp model: results

• Time independent electric field

• Initial distribution of electrons: Maxwellian 300 K.

• Steady state for E/N = 24 Td (1 Td = 10−21 V m−1)

BE MC Reid Ness

Drift velocity (104 m s−1) 8.892 8.88 8.89 8.886

Mean energy (eV) 0.4079 0.408 0.413 –


Reid ramp model : E/N = 24 Td, 1 Torr

0.0

5.0

10.0

15.0

20.0v d

[104 m

/s]

BEMC

0.00

0.10

0.20

0.30

0.40

0.50

10-9 10-8

ε [e

V]

time [s]


Rare gases and the effect of ‘negative mobility’

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 0

5

10

15

20

f n

Q(ε

) [1

0-20 m

2 ]

energy [eV]

f0initial

f0final

Qel

Argon

-0.5

0.0

0.5

1.0

10-11 10-10 10-9 10-8 10-7 10-6 10-5 5.1

5.2

5.3

5.4

5.5

5.6

v d (

104 m

/s)

ε (e

V)

time (s)

vd

ε

Argon:

• initial distribution: Gaussian, center 5.5 eV, width 1 eV

• gas density 3.45 × 1022 m−3 (pressure 1 Torr, temperature 273 K)E = 2 kV m−1, E/N = 58 Td.


Rare gases and the effect of ‘negative mobility’

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

0 5 10 15 20 25

ε1/2

f0/n

e

[eV

-1],

|ε

f1|/n

e

[eV

-1/2

]

co

ll. c

ross s

ectio

n [

a.u

.]

ε [eV]

7.07×10-10

s f0f1 > 0

f1 < 0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

10-11 10-10 10-9 10-8 10-7 0

2

4

6

8

10

12

14

16

v d (

104 m

/s)

ε (e

V)

time (s)

vdε

Xenon:

• initial distribution: Gaussian, center 15.0 eV, width 1 eV

• gas density 3.45 × 1022 m−3 (pressure 1 Torr, temperature 273 K)E = 2 kV m−1, E/N = 58 Td


Conclusions

For a solution of time-dependent Boltzmann equation

� the Monte Carlo Method : is easy to implement, ratherstraightforward method, even for complex boundary conditions,may be rather time consuming

� Solution of the Equation : is a challenge, it is a fun, it gives goodresults (multi-term), for many problems it is impossible


Solution of time-dependent Boltzmann equation for electrons...

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