Introductory Transport Theory for
Charged Particles in Gases
R.E. Robson
Centre for Antimatter-Matter Studies
James Cook University, Townsville
Australian National University, Canberra
We also do physics!
Centre for AntimatterMatters Studies (CAMS)
Outline of this talk
Recent books
• Preliminaries
• Kinetic theory
• Ongoing issues
• Fluid theory
• Models
Acknowledgment
Dr. Lindsay Tassie, ANU (artwork)
REVIEWS
Robson, White and Petrović, Rev. Mod. Phys.
77, 1303 (2005)
Z Lj Petrović, M Šuvakov, Ž Nikitović, S
Dujko, O Šašić, J Jovanović, G Malović and V
Stojanović , PSST 16, S1-12 (2007)
1. Getting started
Gaseous Transport Phenomena
Gaseous electronics,
positronics
Atmospheric
physics, chemistry
Multi-wire
drift tubes
Ion chemistry
Swarm
experiments
Kinetic theory, fluid
model or Monte Carlo
Cross sections, V(r)
Studies go back ~ 100 years to the birth of modern atomic physics
(J.J. Thomson, Townsend, Franck & Hertz, …)
• For low density charged particles (e± , µ± , M±) in gases collisions are
predominantly with neutral molecules M
g’
e±
µ, g
k
j
• Collision cross sections averaged appropriately to give measureable
macroscopic properties
• May use a linear kinetic equation (Boltzmann 1872, Wang-Chang et al 1964)
Preliminaries, Definitions
a) Velocity space
cz
c = velocity of particle of
mass m, charge q
f (c) = density of points in
velocity space
cy
= “velocity distribution function”
cx
b) Differential cross section (Goldstein, “Classical mechanics” p. 81)
σ dΩ = number of particles scattered into dΩ per unit time
incident intensity
Beam experiments (energies > 0.5 eV)
(Brunger and Buckman, 2002)
Beam
cI
is ideally unidirectional and
monoenergetic
f(c) ~ δ (c - cI)
S
1
2
D
3
(R.W. Crompton, 1994)
Drift tube experiment (energies > 1/40 eV )
External electric force field F = q E
S
D
2
N
3
1
N-1
Gas T0 , n0
f(c) = ?
N >> 1
What is measured in experiment?
“Moments” of the charged particle velocity distribution
function, e.g.,
• Number density
n = ∫ f(c) dc
• Average velocity
v = ∫ c f(c) dc / ∫ f(c) dc
How is f(c) found?
From a kinetic equation (equation of continuity in phase space)
• Boltzmann equation 1872 (elastic collisions)
•Wang-Chang, Uhlenbeck, de Boer 1951 (semi-classical inelastics)
• Waldmann-Snider 1958 - 61 (quantum effects, density matrix)
2. Kinetic Theory of Gases
Ludwig Boltzmann (1844-1906)
H-theorem
S = k ln Ω
Second law of Thermodynamics
“I know much better how to integrate than to intrigue.”
Ludwig Boltzmann (1844-1906)
Edward A. Mason (1926-1994)
The Kinetic Equation
Kinetic equation valid for any m, m0 (ions, electrons, positrons, muons, …)
q[E(t ) + c × B] ∂f
∂f
+ c.∇f +
. = − J ( f , f0 )
m
∂t
∂c
(1)
J ( f , f 0 ) = ∑ ∫ [ f (r , c, t ) f 0 j (c0 ) − f (r , c ' , t ) f 0 k (c0' )]gσ ( jk ; gχ )dgˆ 'dc0
j,k
+
∑ f (r , c, t ) ∫ f
0j
(c0 ) gσ R ( j; g )dc0
j
σ ( jk ; gχ )
= differential cross section for the process j, c,c → k, c' , c '
0
f (c )
0j
0
= Maxwell-Boltzmann distribution function for the neutrals
f(c) ≠ Maxwellian
(far from it)
0
Beam experiment
f(c) ~ δ(c-cI ) is
prescribed
c’
c
k
Local operator
Cross sections
j
Most swarm experiments
Measured in swarm experiments
e.g., Franck-Hertz experiment
Plasma vs Swarms
Ion – neutral Boltzmann
collision operator
Kinetic equation for ions
S
H
Boundary +
initial conditions;
External fields
Maxwell’s equations;
self-consistent fields
Neutral gas
distribution
function
Coulomb collisions; FokkerPlanck collision operator
H
Kinetic equation for electrons
S
Electron-neutral
Boltzmann collision
operator
• Swarm (= “test particle”) limit - “switches” H are open, while the “switches” S are closed. Collective effects
are absent in this limit (λD > L)
• Hot, fully ionized plasmas - “switches” H are closed while the “switches” S are open. Coulomb interaction
and collective effects (waves, instabilities) dominate (λD << L)
• Low temperature, collision-dominated plasmas are intermediate between these two limits, and generally
all factors must be taken into consideration, i.e., all “switches” S+H are closed.
Plasma kinetic theory
Measurable quantities
Boltzmann equation for charged species s [1D]
 ∂  c z ∂  q s E/m s ∂  f s  −Jf s , F − ∑ Jf s , f s   (1)
∂t
∂z
∂c z
′
  i c  s ≡
′
s
1
n s
 d 3 c f s z, c, t  i c
 i c  1, mc, 12 mc 2 , mcc, . . .
• Input cross sections into right hand side
  2 c  s
 m s  c  s
≡ m s v s
  3 c  s
 1 m s c 2 
2
≡ s
• Prescribe boundary + initial conditions
• Solve for f(z,c,t) with Poisson’s equation:
∂E 
∂z
∑ n s q s /0
n s z, t ≡
 d 3 c f s z, c, t
s
Swarm (free diffusion) limit
• Low charge density, neglect non-linear terms ∑ ′ Jf s , f s ′ 
s
• Field is externally prescribed (Debye length large)
• Problem is linear
• Ions, m~M, modern era t > 1975 (Viehland, Mason, et al)
•Electrons m << M, modern era t > 1978 (many authors)
Solution Techniques
(a) Procedure for ALL types of charged particles
• Expand f(c) in terms of Burnett functions (wave functions of the 3-D harmonic oscillator)
φj (c), where j≡ (n,l,m), about a Maxwellian at an arbitrary basis temperature Tb (‘twotemperature theory’ - E.A. Mason et al, Ann. Phys. 1975,1978, J. Chem. Phys. 1979)
• Apply Talmi transformation from lab frame to centre of mass system in the collision
operator (K. Kumar, Aust. J. Phys. (1967))
• Take as many terms n, l, m as are needed to get required accuracy, and adjust Tb to speed
up convergence
• Physically meaningful quantities are low order velocity “moments”
<φj > = ∫ d3 c f(r,c,t) φi (c)
e.g., j=(0,1,m) Æ average velocity; j =(1,0,0) Æ average energy
(b) Special procedure for light charged particles (e±, µ±)
• Approximate J(f,f0) in differential form to order m/m0
• Solve resulting equations using standard numerical techniques for differential equations
• Often also assume near isotropy in c –space (!)
Ongoing basic issues
Issue # 2
Specify
accuracy
Two-term
approximation
Input
σ(g,χ)
Systematic
solution of B. Eqn.
for f(r,c,t)
Initial + boundary
conditions, identify
symmetries
r-space vs c- space
symmetries
Issue # 1
DL/DT
Wannier 1953
Wagner, Davis and Hurst, 1967
Skullerud, 1969; Lowke and Parker, 1969
Robson, 1972
Units
1 Td = 10-21V m3
Calculate relevant, physically meaningful
quantities to compare with experiment
“bulk” vs “flux”
transport
coefficients
Issue # 3
Isotropy in c-space: Two – term vs multiterm representation
f
(s)
lmax
∞
l
(r , c, t ) = ∑ ∑ f m( s ,l ) (r , c, t )Ym[ l ] (cˆ )
l = 0 m = -l
For light particles m/m0 <<1, truncation at lmax =1 (‘two-term approximation’) is generally
satisfactory (∼ 0.1%) if collisions are elastic. For ions m ~ m0 , this approximation is never made:
qE
Increasing
m/m0
D
• Both angles θ, φ are
generally needed
• Legendre polynomial
representation is generally
inadequate
‘Muliterm’ theory (l
max
≥ 2) required for m/m0 << 1 for inelastic collisions
(e,CH4) Ness and Robson, 1986
Model ions in parent gas (Wannier, 1953)
Difference may be more than quantitative!
Electrons in CH4 (r.f. electric field)
(lmax = 6)
White et al, J. Vac. Sci. Technol. A16, 316 (1998); ω/n0 =10-17, 10-15 and 10-14 rad m3 s-1
3. Transport coefficient definition: Tagashira-Sakai-Sakamoto effect
H. Tagashira et al, J Phys D 10, 1051 (1977)
Fick’s law:
Γ = n w– D ⋅∇n Æ ‘flux’ transport properties
Diffusion equation: ∂t n - nW⋅∇ + D:∇∇n = - nνR Æ ‘bulk’ transport coefficients
• “Flux” drift velocity w (not usually measured in swarm experiments)
• “Bulk” drift velocity W (measured)
• Fluid model for (Robson, J. Chem. Phys, 1986) Æ
W ≈ w−
2 < ε > d <ν R (ε ) >
2 < ε > d <ν R (ε ) > d < ε >
= w−
3e
3e
dE
d <ε >
dE
w may be < 0 (NAM), but W > 0 always (2nd Law)
Example: (e-, H2O)
W
w
Ness and Robson, Phys Rev A, 1988
3. Fluid equations
Fluid equation approach (both plasma and swarm)
“Short cut" method Æ   i c directly without directly solving Boltzmann’s
equation – computationally inexpensive
Cross
sections
Boltzmann Eqn
  i c 
Fluid model
• Form balance equations
 d3 c
B.E.

 i c
• BUT exact equations are not closed and not useful - there are always more unknowns than
equations to solve for – must truncate using an ansatz
•
Resulting equations are useful but approximate - how accurate are they?
•
Must benchmark (= test against known exact results)
Approximation of collision moments: Momentum Transfer Theory
Collision moments generally involve an infinite number of terms, e.g.,

 d 3 c  i c Jf, F  n ∑ J ij   j c 
j1
Momentum transfer theory (Wannier (1953), Mason and McDaniel (1988))
Æ Approximate r.h.s. by constant collision frequency expressions

∑ J ij   j c 
j1
e.g., momentum exchange term
i
≈
∑ J ij   j c 
j1
J11 = νm (ε)
Æ Many useful results in swarm limit, accuracy ~ 10%, e.g., Wannier
energy relation, generalised Einstein relations, corrections to Blanc’s
law, Tonks’ theorem, equivalent electric field, NDC
SOMETIMES EVEN USED FOR INVERTING SWARM DATA Æ
eE  m  m  v
  3 kT g  1 M v2
2
2
 m   N 2
m  m 
Momentum transfer cross sections for electrons in sodium vapour at T = 803° K, as calculated (i)
theoretically from the Fano profile formula (black line) and (ii) by inverting swarm data
(Nakamura and Lucas, J Phys D 11, 325 (1978)) using momentum transfer theory (red line ) (P.
Nicoletopoulos, http://arxiv.org/abs/physics/0307081, 2003). The black arrow indicates the
thermal energy 3kT/2 = 0.1 eV.
4. A favoured model
The benchmark model
• Electrons emitted at a steady rate from source into infinite gas
• Swarm limit, external electric field, B=0, no ionisation/attachment
• Diffusion equation Æ unphysical results (both ions and electrons)
• Exact analytical solution of Boltzmann equation known for νm =constant
Example: (e-, Hg) Franck-Hertz oscillations
Cross sections
Hanne, Amer. J. Phys. (1988)
England and Elford, Aust. J. Phys. (1991)
Kinetic theory
Robson, Li and White, J Phys B (2000)
Normalised distance from the cathode Æ
Why Fick’s Law is of no use for this Problem
Fluid equations with momentum transfer theory approximation
∂Γ  0
∂z
This Æ Fick’s law only if
it can be assumed that
2 ∂n  neE − nm m v
3 ∂z
∂J q
− 1 v ∂  2 ∂v  1
e
3 ∂z n ∂z
∂z
Inelastic collision term
∂ε/∂z =0
  − 1 M v2  
2

∑ I
 I −  I /  e 
I
To close the equations, need to express heat flux J q in terms of n, v, ε
Ansatz
Jq  − 2 ∂
3m ∂z
n
 m 

5 − 2p na
− 5 Γ
3
 m  3
d ln m
(p 
d ln
Expression exact only for  m    and then   2
• Fourier ansatz Jq = - λ ∂ε /∂ z
• Jq = 0
)
Fluid model: Mean energy in the “window”
Step function model
σm= 6 A2
σI = 0.1 A2
εI = 2 eV
T0=0 K, m0= 4 amu
Mean electron energy at E/N = 6.5 Td as a function of normalized
distance downstream from the source. These Franck-Hertz
oscillations are characteristic of all physical properties in the
“window” region of E/N - mean velocity, number density, heat
flux, although there are phase differences
Nicoletopoulos and
Robson, PRL 2008
We could go on to deal with condensed matter…
a) Crystalline semiconductors
Collisions of electrons with phonons ≡ collisions with atoms, therefore transport in 1 to
correspondence with gases
b) Amorphous materials -fractional kinetics
Memory effects, fractional diffusion equation, anomalous diffusion
c) Liquids, soft condensed matter
•
Charged particles interact coherently with many other constituent molecules
•
Double differential cross section
Single DCS
Dynamic structure factor
The “signature” of anomalous diffusion
n(x,t)
Long-lived
sharp
gradient!
Density distribution for particles originally
located at a well defined point for
crystalline ( ) and amorphous( )
materials
• Analysis proceeds according to the
classical and fractional diffusion
equations respectively
After
Sokolov, Klafter and Blumen,
Phys Today, Nov, 2002
• In the classical case gradients become
weak asymptotically, therefore diffusion
equation OK
• In the amorphous case the distribution
retains a cusp characteristic of “strange
kinetics”
γ
2
0 ∂ t n + v d,γ · ∇ n − D γ ∇ n = 0
What is the regime of validity of this fractional diffusion equation?
… but if you’d rather be in the laboratory
Thank you for your attention!
FINALLY
Do we need a new kinetic equation?
The great (e,H2) vibrational cross section controversy
What could be wrong?
EITHER
Beam experiments?
Q.M. calculations?
Swarm experiments?
Solution of kinetic equation?
Investigations
over ~ 30 years
stalemate!
OR
Kinetic equation itself?
Only
alternative?
Re-examination of Boltzmann’s equation
a) Tests of solution techniques
White, Morrison, Mason, J. Phys. B (2002)
•“Multi-term” spherical harmonic decomposition of f(c)
•Angular dependence of DCS
Discrepancy
unresolved!
•Monte-Carlo benchmarks, etc.
b) Implications of Waldmann-Snider kinetic equation:
Robson, White and Morrison, J. Phys. B (2003); White et al, IOP Conf. (2007)
Rotational angular momentum of H2 treated as
a vector property
•
• Electron swarm generates a pronounced
polarisation ‘echo’ in neutral gas
Non-local collision
theory required?