Maxwellian Collisions
Maxwell realized early on that the particular type of collision in which the cross-section
varies at Q∗rs ∼ 1/g offers drastic simplifications. Interestingly, this behavior is physically
correct for many charge-neutral collisions and moderate energies: The charge q polarizes the
neutral in proportion to the field (α ∼ q/r2 ) and the dipole α attracts the particle with a
force F ∼ α/r3 ∼ q/r5 . From our work on power-law potentials, this is the interaction type
that leads to Q∗ ∼ 1/g.
The simplification stems from the fact that the group gQ∗rs (g) appears in the integrals for
M rs and Ers , and can now be moved outside as a constant. We put,
M
g0 ∗
Q (g0 )
g rs
Q∗rs (g) =
then
M rs = µrs g0 Q∗ (g0 )
M
rs
´ ´
fr fs (w
Mr − w
M s )d3 wr d3 ws =
ws wr
ˆ
� ˆ
fs d3 ws
=
µrs g0 Q∗rs (g0 )
ws
"
w
M r fr d3 wr −
wr
-
ns
""
ˆ
ˆ
fr d3 wr
wr
-
nr ur
"
"
�
w
M s f s d3 w s
ws
-
nr
""
-
ns us
"
M rs = µrs g0 Q∗rs (g0 )nr ns (Mur − Mus )
M
Define νsr as the collision frequency of one s-particle will all r-particles,
νsr = nr gQ∗rs (g)
constant for Maxwellian collisions
(1)
Similarly, νrs = ns gQ∗rs (g) (Note: νsr /nr = νrs /ns )
M rs = µrs ns νsr (Mur − Mus ) = µrs nr νrs (Mur − Mus )
M
(2)
For other types of collisions the evaluation is much less straightforward, as it requires prior
M rs = µrs nr νrs (Mur − Mus ) can always be recovered,
solution for fr and fs . However, the form M
only the collision frequency νrs is generally not a constant, but a function of the electron
temperature, and is calculated from some of the existing models for fr and fs .
1
For energy transfer, we will deal directly with the internal energy transfer rate,
M rs
Ers = Ers − Mus · M
(3)
From the definitions,
ˆ
ˆ
∗
M − Mus ) · Mg d3 wd3 w1
fr1 fs gQrs
(g)(G
E
rs = µrs
(4)
w w1
and for Maxwellian collisions, the group gQ∗rs (g) is a constant and moves outside the in­
tegration. The velocity combination inside can be manipulated next. Define the random
velocities Mcs = w
M − Mus , Mcr = w
M 1 − Mur :
M − Mus ) · Mg =
(G
~ s +c
~s )+mr (u
~ r +c
~r )
ms (u
mr +ms
· (Mur + Mcr − Mus − Mcs ) − Mus · (Mur + Mcr − Mus − Mcs )
�
�
msMus + mr Mur
=
− Mus ·(Mur − Mus ) +
mr + ms
"
{z
"
mr
c2
mr +ms r
−
ms
c2
mr +ms s
+ (Terms linear in Mcr or Mcs )
mr
~ r −u
~ s)
(u
mr +ms
Calling for short mr + ms = m, and ignoring the linear terms, because they integrate to zero
(notice (Mcs )s = 0, (Mcr )r = 0),
2
2
M − Mus ) · Mg = mr (Mur − Mus )2 + mr cr − ms cr + (Linear terms)
(G
m
m
Substitute into (4):
E
rs
�
ˆ
ˆ
µrs
∗
2
fr1 fs d3 w1 d3 w + ...
=
(gQrs ) mr (Mur − Mus )
m
w w1
ˆ
ˆ
ˆ
ˆ
2
3
3
mr cr fr1 fs d w1 d w −
... +
ms c2s fr1 fs d3 w1 d3 w
w w1
w w1
2
(5)
The first of the integrals is simply nr ns . The second can be reorganized into
´
d3 wfs
w
0
´
fr1 mr c2r d3 w1 ,
w1
of which the inner integral yields 3kTr nr , while the outer one gives ns . With a similar argu­
ment for the third integral, we obtain
0
Ers =
µrs
0
0
∗
nr ns (gQrs
)[mr (Mur − Mus )2 + 3k(Tr − Ts )]
m
(6)
This has an interesting structure. The mr (Mur − Mus )2 term represents an irreversible internal
energy addition (heat) to species s from collisions with r, provided the two species drift at
0
0
different mean velocities. The second term, in (Tr − Ts ) is the transfer of heat from r to s
when the two species have different temperatures. It is reversible, depending on the sign of
0
0
Tr − Ts .
0
M rs ,
For completeness, we can now calculate the transfer of full kinetic energy, Ers = Ers +Mus · M
with the result
�
mr Mur + msMus
3k 0
0
∗
· (Mur − Mus ) + (Tr − Ts )
(7)
Ers = µrs nr ns (gQrs )
m
m
Some simple applications of the Momentum Equations
Electrons Ohm’s Law - Except for high-frequency effects (of the order of the Plasma Fre­
quency) or for very strong gradients (like in double layers), the inertia of electrons can be
neglected in their momentum balance. Assume collisions of electrons happen with one species
of ions and one of neutrals only:
0+
0
M + Mue × B
M ) + ne me [νei (Mui − Mue ) + νen (Mun − Mue )]
Pe = −ene (E
(8)
where we used µei ∼
= me , µen ∼
= me . In many cases, ui « ue , un « ue , and we can simplify
the equation by introducing the electron current density,
Mje = −eneMue
M+
ene E
Divide by
me
(νei
e
(9)
0
M + me (νei + νen )Mje
Pe = Mje × B
e
+ νen ) and define,
σ =
e2 ne
me (νei + νen )
βe =
(Electrical conductivity)
(10)
(Hall parameter)
(11)
eB
me (νei + νen )
3
_
0_
∇
P
e
M+
σ E
= Mje + Mje × βMe
ene
so that
(12)
Notice:
(a) Electron pressure gradients can drive electron current. This is sometimes called a “dia­
magnetic current”.
0
0
M + vPe = 0, E
M = − vPe ,
(b) As a limit, if boundary conditions forbid currents, je = 0, then E
ene
ene
0
which means density gradients can set up a field − the Ambipolar field. If Te ∼
= const.
0
0
kT ∇ne
kT
ne
− φ
= − e
→ φ = φ0 + e ln
e
ne0
e ne
(13)
Which strongly resembles the kinetic Boltzmann relationship (except this time we only look
at averages).
(c) The Hall parameter is the ratio β = ωνcee of election gyro frequency to electron collision
frequency. It can be large in low-density plasmas, even with moderate B fields.
(d) The current is not aligned with the driving fields. Additional deviations from the electric
0
M∗ = E
M + vPe
field result from E
ene
M∗ = E
M +
(e) Eq. (12) can be solved for Mje in terms of E
products) times βMe :
0
vPe
.
ene
Start by multiplying (cross
M ∗ = βMe × Mje + βMe × (Mje × βMe )
σβMe × E
"
-{z
"
~e (βe ·j
~e )
βe2~je −β
M , so that B
M e · Mje = 0:
consider only the current perpendicular to B
M∗
Mje × βMe = βMe2Mje⊥ − σβMe × E
M ∗ = Mje⊥ + β 2Mje⊥ − σβMe × E
M ∗ , or
and substitute this into (12): σE
e
M
je⊥ =
_
_
σ
∗
∗
M
M
M
E + βe × E⊥
1 + βe2 ⊥
4
M ∗
plus Mje = σE
(14)
M :
This is sometimes organized as a tensor equation. With z taken along B
⎧ ⎫
⎧ ∗⎫
Ex ⎪
⎪
⎪
⎪
j
ex ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎛
⎞ ⎪
⎪
⎪
⎪
⎪
β
1
⎪
⎪
⎪
⎪
e
⎪
⎪
⎪
⎪
0
2
2
⎨
⎬
⎨
⎬
1+βe
1+βe
⎜
⎟
∗
βe
1
Ey
jey
= σ ⎝
−
1+β 2 1+β 2 0
⎠
e
e
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
0
0
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
j ⎭
⎩ ∗⎪
⎭
Ez
ez
(15)
which makes the anisotropy of the situation more clear. In Ionospheric Physics, this is also
put as a conductivity tensor
⎛
⎞
σP σH 0
MMσ
=
⎝
−σH σP 0 ⎠
M ∗ ·
Mje = MMσE
(16)
0
0 σk
σP = “Pedersen conductivity” (very small in the ionosphere, βe » 1)
σH = “Hall conductivity” (intermediate)
σk = σ “Parallel conductivity” (very large in the ionosphere)
Ambipolar Diffusion
Consider a simple case with B = 0, negligible inertia. Write both, electron and ion momen­
tum equations:
mi ne
DMui
0
M e + ne [me νie (Mue − Mui ) + µin νin (Mun − Mui )]
+ ∇Pi = eEn
Dt
0
M
e + me ne [νei (Mui − Mue ) + νen (Mun − Mue )]
∇Pe = −eEn
Add together, note ne νei = ni νie (and also ne = ni ),
m i ne
DMui
0
0
+ ∇(Pi + Pe ) = nei µin νin (Mun − Mui ) + me ne νen (Mun − Mue )
-
{z
"
"
Dt
me
me
usually smaller ∼
or
mi
mi
Also, normally ∇Te�0 /Te�0 « ∇ne /ne . In addition let us assume that ion inertia can be also
neglected in comparison with the other terms in the momentum balance (although keeping
the term would be more general),
0
0
k(Te + Ti )∇ne = − ne µin νin (Mui − Mun )
or,
0
ne (Mui − Mun ) = −
5
0
k(Te + Ti )
∇ne
µin νin
Sometimes neutrals return from recombination of ions, so,
_
_
ne
ne
neMui = −nnMun then
ne (Mui − Mun ) = Mui + Mui ne = (nn + ni ) Mui
nn
nn
0
0
nn k(Te + Ti )
∇ne
neMui = −
ni + nn µin νin
νin = nn gin Qin
0
neMui =
ni + nn =
ρ
mi
; µin =
mi
2
0
nn k(T + Ti )
�
∇ ne
− mi ρ e
� �
n Q g
2 �
mi n in in
neMui = −Da ∇ne
0
2k(Tei + Ti )
Da =
ρQin gin
Ambipolar diffusivity
Back to the electron equation, if we neglect both collision forces,
0
M e
∇ Pe ∼
= −eEn
0
kTe
∇ne ∼
= e∇φ
ne
0
∇(eφ − kTe ln ne ) = 0
0
kTe
ne
φ − φ0 =
ln
e
ne0
Equivalent Boltzmann equilibrium
6
MIT OpenCourseWare
http://ocw.mit.edu
16.55 Ionized Gases
Fall 2014
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