solution of time-dependent boltzmann equation for electrons...
TRANSCRIPT
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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
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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
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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
Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondrejov, 18. 10. 2007 3 / 41
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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?
Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondrejov, 18. 10. 2007 4 / 41
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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 .
Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondrejov, 18. 10. 2007 5 / 41
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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?
Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondrejov, 18. 10. 2007 6 / 41
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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
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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.
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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?
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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 → ∞
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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 → ∞
But why random numbers?
Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondrejov, 18. 10. 2007 11 / 41
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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 → ∞
Because a finite distribution of regular pointsgives a poor resolution whent the dimension n is high.
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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
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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?
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• 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
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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”
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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”
How are collision times, tc, distributed?
Z. Bonaventura (UFE, F.Sci. MUNI) Simulation of Boltzmann equation Ondrejov, 18. 10. 2007 17 / 41
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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”
How are collision times, tc, distributed?The answer is simple for constant collision frequency!
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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
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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
And . . .
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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
• the collision is introduced by laws of physics,based on appropriate random distribution of impact parameters
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But . . .
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Collision frequency is not a constant!
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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
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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 velocityand its magnitude is determined
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Null collision technique
• Isotropic scattering can be assumed in many cases,momentum transfer cross section is used insted of DCS.
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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 solutionon the bases of physical analogy?
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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.
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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
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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
.
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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 . . .
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. . . 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)
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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
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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
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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
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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 –
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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]
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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.
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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
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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
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