Single-particle motion
• Plasma is a collection of charged particles under the influence of electromagnetic fields. The sources of the fields can be either external or
internal to the plasma.
• Before moving on to studying the collective phenomena in a plasma, it
is useful to get familiar with the motion of individual charged particles
in electromagnetic fields.
• A large part of phenomena we shall meet later can be understood (even
quantitatively) in terms of single-particle motion.
Guiding-center approximation
• Task: solve the equation of motion
d~
p
~ +~
~ +F
~non−EM
= q(E
v × B)
dt
~non−EM denotes all non-electromagnetic forces acting on the
where F
particle. In practice, this usually means gravitation,
~non−EM = m~
F
g.
• We will restrict to non-relativistic particles (γ = 1, p
~ = m~
v ).
• No general analytical solutions exit for arbitrary field configurations.
• For slow temporal changes and smooth spatial dependencies of the fields,
we can use perturbation theory, i.e., calculate the motion of particles
~, E
~ , and ~
starting from the solution with constant fields B
g.
• This way the particle trajectory, as averaged over many gyro-periods, is
obtained. This approach is called guiding-center approximation
~
Motion in a homogeneous and static B
~ =~
~ is constant. The eq. of motion reads
• Assume that E
g = 0 and B
d~
v
~
m
= q~v × B.
dt
• The magnetic field does no work on the particle ⇒ |~
v | = v is constant.
~ = B~
• Choose the coordinate system so that B
ez . Thus,
mv̇x
=
qvy B
mv̇y
=
−qvxB
mv̇z
=
0
and vz is constant.
• v and vz constants ⇒ v⊥ =
p
v 2 − vz2 =
q
vx2 + vy2 constant.
• Write vx = v⊥ cos φ and vy = v⊥ sin φ, to get
−mv⊥ sin φ φ̇
⇒
φ
=
qv⊥ sin φ B
=
φ0 − ωct.
⇒
φ̇ = −
qB
≡ −ωc
m
The motion of the particle is, therefore, gyromotion in the xy -plane!
• The gyromotion has an angular frequency of ωc and a radius of
rL =
v⊥
mv⊥
=
,
|ωc|
|q|B
which is commonly referred to as the Larmor radius or gyroradius.
• Positive [negative] charges gyrate in a left-handed [right-handed] sense
with respect to the magnetic field direction.
• The center of gyromotion is called the guiding center (GC).
• Particle motion in a homogeneous magnetic field can, thus, be divided in
two components: linear motion along the field lines (at speed vz = vk)
and gyromotion in the plane perpendicular to them.
• The sum of the two components is a helical path. The pitch angle α of
this helix is
vk
α = arccos
∈ [0, π]
v
• The coordinate system, where vk = 0 is called guiding center system
(GCS).
• Division of particle motion to the motion of the guiding center and the
gyromotion around it is called guiding center approximation.
• In GCS, the charge causes a current I = q|ωc|/2π . The associated
magnetic moment is
µ=
2
IπrL
=
2
1
mv
⊥
2
B
W⊥
=
.
B
• The magnetic moment is actually a vector:
µ
~ =
where
1
q~
rL × ~
v⊥,
2
~
m ~
v⊥ × B
~
rL = −
qB
B
is the gyro-radius vector (i.e., particle position in GCS).
• Direction of µ
~ always opposite to the magnetic field direction.
• Thus, gyrating charges tend to decrease the magnetic field, and plasma
is diamagnetic.
Drift motions due to non-magnetic forces
Electric drift or ExB drift
~ and E
~ are constant and non-zero.
• Assume that both B
• The parallel (to the B-field) equation of motion becomes
mv̇k = qEk.
⇒ particle experiences constant acceleration in the parallel direction.
• Usually plasma electrons very mobile in the parallel direction.
⇒ large-scale parallel E-fields typically very small.
~ ⊥E
~ k~
• Assume, therefore, that B
ex. This yields
v̇x
=
qE
ωcvy +
= ωc
m
v̇y
=
−ωcvx
E
vy +
B
• Making a transformation vy → vy0 = vy + E/B gives the original
equations of motion where the electric field does not appear. Thus the
guiding centers of the particles drift along the y -direction at the speed
E/B . In vector form, this drift speed is
~ ×B
~
E
.
~
vE =
B2
• The drift speed consistent with the Lorentz transformation of the electric
field to GCS:
~0 = E
~ +~
~
E
vE × B.
~ 0 = 0, we can solve for ~
By demanding that E
vE .
Non-electromagnetic forces
• This coordinate transformation applies for all weak-enough, perpendicular
~⊥.
(to the B-field) non-magnetic forces, F
~ ←F
~⊥/q in the result, we get
• Substituting E
~
vD
~⊥ × B
~
F
=
qB 2
• Transformation possible only if F⊥/qB ¿ c. If F⊥/qB > c, guiding
center approximation fails!
• Example: gravitational drift,
~
m~
g×B
m
~
vg =
∝
.
2
q B
q
The gravitational field separates particles according to their m/q in the
~ ).
direction perpendicular to the fields (unless ~
gkB
• The gravitational drift leads to current flow!
Finite-Larmor radius correction
• If the E-field (or, analogously, g-field) changes substantially inside a
Larmor radius, the E×B drift has to be modified by
~
~
×B
1 2 2 E
.
~
vE = 1 + rL∇
4
B2
Such “finite Larmor radius effects” are important, for example, near
boundary layers separating different plasma regimes.
Polarization drift
• Same formalism can be used for a weakly time-dependent electric field,
~ =E
~0 + E
~˙ 0(t − t0),
E
~˙ 0 = const.
with E
~ is still constant. F
assuming that B
• First write the eqs. of motion in a coordinate system moving with speed
~
vE . There one sees that another transformation at speed
~˙ 0
~
E
1 dE
m
~
vP =
=
∝
ωc B
ωcB dt
q
restores the eqs. of motion without electric fields.
• Guiding center drifts ~
vd = ~
vE + ~
vP .
~
vP is the polarization drift velocity.
• Note that also the polarization drift separates ions from electrons leading
to a polarization current:
~
~
ne(mi + me) dE
nemi dE
~
JP = nee(~
vP i − ~
vP e) =
≈
,
B2
dt
B 2 dt
which is carried primarily by the more massive ions.
• Total time derivative of E-field causes the polarization drift
⇒ eq. for ~
vP holds also for weak spatial gradients. To see this, write
d/dt = ∂/∂t + ~
v·∇
~.
v≈~
vE , spatial term is of second order in E
• Since ~
⇒ Spatial term usually much smaller than the temporal term.
~
~
⇒ Polarization drift can be approximated by dE/dt
≈ ∂ E/∂t
.
Magnetic drift effects
• If the B-field changes slowly as a function of time, it induces an E-field
through Faraday’s law. ⇒ E-drifts (as described above).
• Consider a weakly inhomogeneous magnetic field, where changes are
small during one Larmor gyration, i.e.,
~ · (∇B)|
~ ¿B;
|r
L
~
~ ¿ B 2.
|(vk/ωc)B·(∇
B)|
~ · (∇C)
~ = (D
~ · ∇)C
~)
These criteria are energy-dependent! (Note D
• Then, expand the magnetic field at the position ~
r(t) of the particle as
~ r) = B
~ 0 + (~
~ 0 + ...
B(~
r−~
r0) · (∇B)
where the subscript 0 means that the quantity is evaluated at the
guiding-center position, ~
r=~
r0(t). Thus,
~ r) ≈ B
~0 + ~
~ 0
B(~
rL · (∇B)
• A straightforward calculation shows that the field inhomogeneity causes
a gyro-averaged force
~ = −µ∇B
F
on the particle.
• NOTE: this is an apparent force that is useful in the determination of
the GC motion. Bear in mind, that v remains constant under the action
of magnetic forces.
• The force component parallel to the magnetic field causes an acceleration
d~
vk
dt
=−
µ
∇kB
m
accompanied by a change in v⊥ to keep v constant.
Gradient and curvature drift
~⊥ = −µ(∇B)⊥ causes a drift:
• The perpendicular component F
~⊥ × B
~
F
µ ~
W⊥ ~
~
vG =
=
B
×
∇B
=
B × ∇B.
qB 2
qB 2
qB 3
This is called gradient drift. It is related to a current.
• If the field lines are curved, GCS is not an inertial frame. Then, there is
a centrifugal force
~C
R
2
~ = −mw
F
k 2 ,
RC
~ C is the radius of curvature (pointing inward) of the magnetic
where R
field and wk is the parallel component of the guiding-center motion.
(Note that wk ≈ vk but not exactly).
• This non-magnetic force, thus, causes a (current related) curvature drift:
~C × B
~
mvk2 R
~
vC = −
2 B2
q RC
• An exercise in differential geometry shows that
~ · ∇B)
~ ⊥
~C
R
(B
=
2
RC
B2
~ × (B
~ · ∇)B
~
mvk2 B
⇒ ~
vC =
q
B4
~ = 0, we can combine the
• In a region with negligible current, ∇ × B
~ · ∇)B
~ = B∇B and,
gradient and curvature drifts, since then (B
thus,
~
vGC =
W⊥ + 2Wk
qB 3
W (1 + cos2 α)
~
B × ∇B =
~
n ×~
t,
qBRC
~ and ~
~ C are unit vectors.
where ~
tkB
nkR
Adiabatic invariants
• Classical mechanics:
periodic motion → conserved quantity (exact invariant).
• Quasi-periodic motion → adiabatic invariant.
• Let p and q be a canonical momentum and its coordinate. If the motion
of the system is quasi-periodic with respect to q ,
I
I =
p dq
is an adiabatic invariant.
~.
~ = m~
v + qA
• In an electromagnetic field, the canonical momentum is p
A conjugate pair of (~
p⊥, ~
rL) may be used
I
I
Z
~ · dS
~
I =
p
~⊥ · d~
rL =
m~
v⊥ · d~
rL + q (∇ × A)
Z
2πrL
=
0
=
S
Z
~ · dS
~
B
mv⊥dl + q
S
2
2πmv⊥rL − |q|BπrL = 2π
m
µ
|q|
so that the magnetic moment of a charged particle is an adiabatic
invariant. Called the first adiabatic invariant.
• Let us next prove the invariance of the magnetic moment from the eq.
of motion of the GC and conservation of energy in a static and slowly
varying magnetic fields B.
~ = 0) ⇒ W = Wk + W⊥ is
• Assume a static magnetic field (and E
constant
⇒ 0
=
=
=
=
=
dWk
dvk
dW
dW⊥
d(µB)
=
+
= mvk
+
dt
dt
dt
dt
dt
dvk
dµ
dB vk m
+B
+µ
Fk = −µ∇kB
dt
dt
dt
dµ
dB
+µ
dt
dt
dµ
dB
−µ(vk∇kB) + B
+µ
dt
dt
vk(−µ∇kB) + B
−µ(vk∇kB) + B
dB
dt = vk∇kB
dµ
dµ
+ µ(vk∇kB) = B
dt
dt
• Thus, µ is constant in static magnetic fields (if GC approximation holds).
• Let us, next, consider a slowly time-varying magnetic field (with
∂/∂t ¿ ωc). According to Faraday’s law the electric field has to
be taken into account, and
dW⊥
~ · v⊥)
= q(E
dt
• The change in the perpendicular energy during one cyclotron period is
Z
∆W⊥
=
q
2π/|ωc |
I
0
C
Z
=
Z
~ · dS
~ = −q
(∇ × E)
q
S
=
|q|
~ · d~l
E
~ ·~
E
v⊥dt = q
S
~
∂B
~
· dS
∂t
|ωc|
2
∆B πrL = µ∆B
2π
On the other hand, ∆W⊥ = ∆(µB) so ∆µ = 0, and µ is again
constant. (Note that here the total energy is no longer a constant.)
• The invariance of µ therefore holds in a slowly varying magnetic field,
no matter how the field changes!
Magnetic mirror and magnetic bottle
• In a static B field, µ = W⊥/B and W = W⊥ + Wk constants.
2
⇒ (sin α)/B = const.
• Knowing the pitch angle α = α1 in a certain magnetic field B = B1
allows one to calculate α = α2 at any other field B = B2:
2
2
sin α2 = (B2/B1) sin α1
• Particle can move toward an increasing magnetic field until α = 90◦.
The force acting on the GC, Fk = −µ∇kB , always points toward the
decreasing magnetic field and turns the particle around.
– The field acts as a magnetic mirror
– The force Fk = −µ∇kB is called the mirror force.
• The value of the mirror field, Bm, depends on the pitch angle α0 at a
reference field B0 < Bm:
2
sin α0 = B0/Bm.
p
• Particles with sin α0 < B0/ max(Bm) are not reflected.
• A magnetic bottle is formed between two magnetic mirrors, which do
not have to be of equal strength.
• A particle remains trapped in the magn. bottle, if its pitch angle α0 in
the minimum field strength B0 inside the bottle is
s
s
B0
B0
◦
arcsin
≤ α0 ≤ 180 − arcsin
,
Bm
Bm
where Bm is weaker of the mirror fields.
• Otherwise, the particle leaks out from the bottle. It is in the loss cone.
Second adiabatic invariant
• The bounce motion in a magnetic bottle is also quasi-periodic, if the
field configuration does not change much per bounce period
Z
s0m
τb = 2
sm
ds
2
=
vk(s)
v
Z
s0m
sm
p
ds
1 − B(s)/Bm
,
where s is the arc length along the GC track and sm and s0m are the
coordinates of the mirror points.
• The corresponding adiabatic invariant is called longitudinal invariant
I
J =
pk ds,
where for an non-relativistic particle pk = mvk.
Since τb À τL = 2π/ωc, this invariant is weaker than µ.
• The conservation of J may lead to so-called Fermi acceleration, proposed
by E. Fermi (1949) as a model of cosmic ray acceleration.
Particle acceleration in a collapsing magnetic trap
• Let a magnetic trap consist of two identical mirrors (B = Bm) at the
ends of an otherwise constant field (B = B0 ¿ Bm).
• Thus, J = 2` |pk|, where ` = `0 − 2ut is the length of the trap.
J˙ = 0
⇒
d|pk|
dt
`˙
2u
= −|pk| = |pk|
`
`
• Since the field inside the trap remains constant and µ is conserved,
p⊥ = const. between reflections off of the mirrors.
• Thus, |pk| can only increase while
B0
p2⊥
2
=
sin
α
≥
,
2
2
p⊥ + pk
Bm
i.e., particle enters the loss cone, when
q
p ≈ pf = p⊥
Bm/B0.
• Final momentum fulfills
q
pf ≤ p0
Bm/B0,
where p0 is the initial momentum.
• Only interactions with many magnetic mirrors lead to a substantial
increase of the particle energy.
l
Third adiabatic invariant
• Also the cross-field motion can be periodic, if the field is axially symmetric, as in case of dipole magnetic field.
• The adiabatic invariant related to this periodicity is the magnetic flux
enclosed by the GC during its (periodic) drift motion across the field,
I
~ · d~l,
Φ=
A
where d~l is the arc element defined by the GC drift path. The drift
period has to be τd À τb À τL, so this invariant is weaker than µ
and J .
Invariant
velocity
time scale
conditions
µ
J
Φ
~
v⊥
w
~k
w
~⊥
τL
τb
τd
τ À τL
τ À τb À τL
τ À τd À τb À τL
• Each invariant is connected to an energy-change mechanism:
µ
W⊥ can be changed by changing B (temporally); betatron
acceleration
Wk can be changed by changing the length of the bottle;
J
Fermi acceleration
Φ
W can be changed by compressing/expanding the drift
surface
• In the inner magnetosphere of the Earth, µ is often an invariant, J is
an invariant for trapped particle populations, and Φ is an invariant only
for the high-energy particles in the radiation belts.
Particle motion in a dipole field
Dipole magnetic field (example: the Earth)
• Use “geomagnetically" defined spherical coordinate system:
~ E in the origin, pointing to the south.
– Earth’s dipole moment M
– Latitude λ = 0 at the equator, increases to the north
– Longitude (φ) increases to the east.
• ME measured in units of A m2. In the literature, ME often replaced
by k0 = µ0ME /4π , also called dipole moment.
ME
=
8 · 10
22
Am
k0
=
8 · 10
15
Wb m
=
8 · 10
25
G cm
=
0.3 GRE
3
2
3
(cgs units)
(RE = 6370 km)
• Dipole approximation: field-generating currents compressed to a singular
~ =0 ⇒ B
~ = −∇Ψ
point; elsewhere ∇ × B
1
sin λ
Ψ = −~
k0 · ∇ = −k0 2
r
r
scalar potential
• A straightforward calculation gives
~
~
3(
k
·
~
e
)~
e
−
k0
0
r
r
~ =
B
r3
2k
From which we get the components Br = − r30 sin λ ; Bλ =
k0
r3
cos λ ; Bφ = 0 and
k0
2
1/2
B = 3 (1 + 3 sin λ) .
r
• Magnetic field lines given by
dr
r dλ
=
.
Br
Bλ
⇒
2
r(λ) = r0 cos λ,
where r0 is the distance, where the field lines intersects the equator.
Some useful properties of dipole field
• Length of the field-line element
p
2
1/2
ds = dr 2 + r 2dλ2 = r0 cos λ(1 + 3 sin λ) dλ.
• Field line defined by its longitude φ0 and its
(equatorial) distance r0.
⇒ McIlwain’s L-parameter,
L ≡ r0/RE .
• The field line defined by L reaches the Earth’s
surface at
1
λe = arccos √ .
L
• The strength of the magnetic field on a given field line as a function of
latitude
q
B(λ) =
Br2(λ)
+
Bλ2 (λ)
k0 (1 + 3 sin2 λ)1/2
= 3
r0
cos6 λ
• In Earth’s field
3 · 10−5 T
k0
0.3 G
=
=
r03
L3
L3
⇒ Surface field: 0.3 gauss at equator, 0.6 gauss at poles.
• The radius of curvature of the dipole field lines is obtained from the
geometric formula 1/RC = |d2~
r/ds2| as
r0
(1 + 3 sin2 λ)3/2
RC (λ) =
cos λ
3
2 − cos2 λ
Guiding-center motion in a dipole field
• GC-approximation useful, if rL ¿ rC ∼ r0 ⇒ condition for the
magnetic rigidity, P⊥ = mv⊥/|q|, of the particle:
P⊥ ¿ r0B
Equatorial pitch-angle and loss cone
• Field strength:
k0 (1 + 3 sin2 λ)1/2
B(λ) = 3
r0
cos6 λ
• Conservation of µ ⇒
B0
cos6 λ
2
2
sin α(λ) =
sin α0 =
sin
α0
2
1/2
B(λ)
(1 + 3 sin λ)
2
Particle’s pitch-angle at the equator, α0 (magnetic field B0):
B0
cos6 λm
sin α0 =
=
B(λm)
(1 + 3 sin2 λm)1/2
2
λm mirroring latitude (independent on L!)
• If λm > λe = arccos L−1/2, particle hits the Earth’s surface before
mirroring and leaks out from the bottle. The boundary of the loss cone
is at
1
2
sin α0l =
(4L6 − 3L5)1/2
and the particle is in the loss cone if α0 < α0l or α0 > π − α0l .
Bounce motion
• The period of bounce motion in a dipole field is
Z
τb
=
λm
4
0
=
≡
4r0
v
Z
ds
=4
vk
λm
0
Z
λm
0
ds dλ
dλ vk
cos λ(1 + 3 sin2 λ)1/2
dλ
cos α(λ)
4r0
f (α0),
v
α0 = arcsin
cos3 λm
2
(1 + 3 sin λm
)1/4
,
where f (α0) ≈ 1.30 − 0.56 sin2 α0 for 30◦ ≤ α0 ≤ 90◦.
• Typical bounce periods,
τb ∼
L
4r0
≈ 0.085 ·
s,
v
β
– seconds for 1 keV electrons (β = v/c = 0.063)
– minutes for 1 keV protons (β = 1.5 × 10−3).
• Magnetosphere can change considerably in minutes
⇒ invariance of J for >keV protons questionable.
Drift motion around the Earth
• Gradient and curvature drifts drive particles across the magnetic field
around the Earth. The combined drift speed is
vGC (λ) =
W
2
[1 + cos α(λ)]
qB(λ)RC (λ)
• The angular drift speed around the Earth is
φ̇ =
vGC
vGC
=
r cos λ
r0 cos3 λ
which, when averaged over the bounce period, gives
hφ̇i =
4
τb
Z
τb /4
φ̇ dt =
0
1
=
r0f (α0)
=
2
3
mv
r0
2
qk0
Z
λm
0
4
vτb
Z
λm
φ̇
0
ds dλ
dλ cos α
4r0
f (α0)
τb =
v
vGC (λ)(1 + 3 sin2 λ)1/2
dλ
2
cos λ cos α(λ)
g(α0),
• The function g can be approximated as g(α0) = 0.7 + 0.3 sin α0
for 30◦ ≤ α0 ≤ 90◦ giving for the equatorial particles (α0 = 90◦)
3mv 2RE
hφ̇0i =
L.
2qk0
If the particles are relativistic, the equation generalizes to
3mc2RE
2
hφ̇0i =
Lγβ .
2qk0
• The drift period τd around the Earth is, thus,
τd
=
≈
2π
|hφ̇i|
=
4
1.0 · 10
2π
|hφ̇0i|g(α0)
=
4π |q|k0
1
3 mc2RE Lγβ 2g(α0)
1
me |q|
m e Lγβ 2g(α0)
s.
Particles in the keV/n-range have τd of a few hundred hours and particles
in the MeV/n range have τd of a few tens of minutes in the dipolar part
(L ≈ 2 − 7) of the field.
Particle motion in the field of a current sheet
• Interaction of the solar wind with the Earth’s magnetic field
⇒ the night side of the magnetosphere streched to long a magnetotail.
• The field points Earthward in the northern part and anti-Earthward in
the southern part of the magnetotail.
ex points Sunward, ~
ez to the north ⇒ field can be approximated by
• ~
~ = Bx(z)~
B
ex + Bn~
ez
~ = 0 (here Bn is constant).
where Bx and Bn must be such that ∇ · B
• Bx(z) goes from a negative value at z → −∞ to a positive value at
z → ∞. In the magnetotail |Bx(z)| À |Bn| when z → ±∞.
• Ampère’s law:
~
∇×B
1 ∂Bx
~
J =
=
~
ey
µ0
µ0 ∂z
flowing along the positive y -axis and concentrated in the region where
Bx changes sign. This is called a current sheet.
• Harris model:
~ = B0 tanh z ~
ex + Bn~
ez ,
B
L
x
x
with constants Bn ¿ B0. Constant
L gives the thickness of the current
sheet:
J~ =
B0
µ0L cosh2(z/L)
.
z
1D
z
2D
• Current sheets found all over the universe:
– planetary magnetotails
– boundaries of magnetic flux tubes in solar and stellar atmospheres
– heliospheric current sheet near the ecliptic, etc.
• Particle motion in Harris field more complicated than in a dipole field:
GC approximation breaks down near the current sheet!
y
B
B
B
∆
∆
• Basic forms of the motion in the 1D case
(Bn = 0)
– simple cyclotron motion far away
(|z| À L) from the current sheet
– gradient-drifting of particles near the
sheet but not crossing it; note the
direction of the drift!
– particles crossing the current sheet
moving back and forth along y .
– particles in a monotonic motion along
the y axis: positive [negative] charges
moving in the +[−] y direction
(Speiser motion). These particles carry
the current in the Harris field!
Jy
Jy
B
x
z
• 2D-case: a dipole-type field stretched to a Harris field; what happens?
– particles adiabatic, as long as RC /rL & 10
Bn
L;
RC ≈
B0
vm
rL ≈
eBn
Lωc0
⇒v.
10(B0/Bn)2
⇒ bouncing motion across the current sheet.
– system stretched more (i.e., BnL is made smaller)
⇒ particles of smaller velocities become non-adiabatic.
⇒ chaotic trajectories!
Collisions and plasma conductivity
• Theory of single particle motion does not describe any interactions
between the plasma particles
• The next step toward the inclusion of collective effects is to deal with
particle interactions as collisions.
• Collisions allow us to get a handle of an important property of a plasma:
its electrical conductivity.
• Most plasmas we are going to study can be characterized as being
collisionless meaning that the collision frequencies are much smaller
than any other relevant frequencies of the problem.
• Important exceptions
– stellar cores (where the physics is primarily nuclear physics)
– weakly ionized upper parts of planetary atmospheres, i.e., the
ionospheres.
• Collisional plasmas are divided in weakly and fully ionized plasmas.
– Weakly ionized plasma (e.g., ionosphere): collisions predominantly
on neutral particles (i.e., atoms and/or molecules)
– Fully ionized plasma (i.e., a plasma with more than 10 % degree of
ionization): Coulomb collisions between charged particles themselves
dominate.
Collision frequencies
Weakly ionized plasmas
• A charged particle interacts with a neutral particle only through direct
collisions.
• Let nn be the density of neutral particles, σn = πd20 the cross section
of the neutral with an effective radius of d0, and hvi the mean speed of
the charged particles relative to the neutrals.
• The mean free path of the charged particles (between collisions):
lmfp =
1
.
n n σn
• The collision frequency of the charged particles on the neutral particles:
νn = nnσnhvi.
• The collision frequency does not depend on the density of the charged
particles. Its inverse, the collision time τn = 1/νn, gives the time
between collisions for an individual charged particle in the plasma.
• Atoms and molecules have cross sections of about
10
−20
2
−19
m . σn . 10
2
m .
• Collisions can be elastic, but often they also lead to ionizations or
charge exchange between the collision partners (inelastic collisions). For
example, collisions between neutrals and energetic ions can yield so-called
energetic neutral atoms (ENAs), like in
pfast + Hslow
→
HENA + pslow
pfast + Oslow
→
HENA + Oslow
+
• Inelastic collisions also lead to excitation of the neutrals → aurora
Fully ionized plasmas
Rutherford scattering formula
• In fully ionized plasmas, the interactions between the particles occur via
the Coulomb force.
• Any charged particle in a plasma constantly interacts with a large number
(∼ Λ) of other particles.
• On the other hand, most of the collisions occur with far-away particles.
This means that the particles change their propagation direction only by
a small amount per collision. We talk about small-angle collisions.
• Consider a collision of an electron (q = −e) off an ion (q = +e).
Let b be the impact parameter of the collision, and ve be the speed of
the electron (wrt. ion) prior to (and after) the collision. The scattering
angle φ as a function of these two is given by the Rutherford-scattering
formula
b0(ve)
φ
e2
≡
tan =
,
2
4π²0mebve2
b
where b < b0(ve) defines the impact parameter range for large-angle
collisions.
b
r
θ0
θ
z
φ
Large-angle collisions
• The cross section for large-angle scatterings is, thus,
σc =
2
πb0
e4
e4
=
≈
2
2
4
16π²0me ve
16π²20m2e hvei4
and the corresponding scattering frequency is
νei = niσchvei
ni =ne
=
nee4
16π²20m2e hvei3
• Replace the electron speed by the average energy, kB Te = 12 mehvei2,
and use the definition of the plasma frequency to get
√
4 2 ωpe kB Te −3/2
νei =
64π ne
me
Small-angle collisions and Coulomb scattering frequency
x, x’
v
y
z
y’ ϕ
θ0
θ
θ0
φ
z’
v0
• The derived formula for νei is not complete, since it does not take
account of the small-angle collisions.
• Let us fix the coordinate system so that electron velocity prior to the
collision is given by
~
v0 = ve(0, sin Θ0, cos Θ0).
• Let the electron scatter into an angle φ relative to its original direction.
• An exercise in spherical geometry shows that the new velocity vector ~
v
fulfills
~
v ·~
ez = ve(cos Θ0 cos φ + sin Θ0 sin φ cos ϕ) ≡ ve cos Θ,
where ϕ is the scattering angle measured around the original direction
of propagation.
• Since ϕ is uniformly distributed, we have on average
∆ cos Θ = − cos Θ0(1 − cos φ).
• Scatterings to the angle φ occur at rate nive dσ , where dσ = 2π b db
is the differential cross section. The average rate of change of cos Θ is,
thus,
h∆ cos Θi
∆t
Z
=
− cos Θ0
(1 − cos φ) nive dσ
Z
=
bmax
− cos Θ0 nive 2π
(1 − cos φ) b db
bmin
2
=
− cos Θ0 nive πb0
Z bmax/b0
φ
φ
2 φ
4
sin
cot d cot
2
2
2
bmin /b0
=
cos Θ0 nive πb0
φ 2 arctan(b0/bmax)
4 ln sin
.
2 φ=2 arctan(b0/bmin)
2
• The values for bmax and bmin have to be determined.
– The field of the ion is screened at r > λD , so we take bmax = λD .
Since λD À b0, we have sin(φ/2) ≈ b0/λD at the upper limit.
– The value of bmin is assumed to be much smaller than b0. This gives
sin(φ/2) ≈ sin(π/2) = 1 at the lower limit.
• Therefore,
h∆ cos Θi
=−
∆t
λD
2
nive πb0 4 ln
b0
cos Θ0 ≡ −νei cos Θ0
where we have identified the rate of change of cos Θ as the scattering
frequency, i.e., d cos Θ/dt = −νei cos Θ. The correction due to
small-angle scatterings is, thus, an extra factor of 4 ln(λD /b0) =
4 ln(16πΛ) relative to the large-angle result.
• Thus, the scattering frequency is
νei ≈
ωpe ln Λ
32π Λ
and the mean free path is
lmfp
hvei
2ωpeλD
Λ
=
=
≈ 64π λD
.
νei
νei
ln Λ
• The Coulomb logarithm ln Λ is between 10 and 30 for almost all plasmas
found in nature. (Thus, replacing ln(16πΛ) by ln Λ results in an error
of less than 40%.)
• The assumption we made on the smallness of bmin needs to be justified.
For protons, the ion radius (beyond which electrons see a Coulomb
potential) is about
−15
bmin ∼ 10
m.
The potential energy of an electron at this distance, Umin =
e2/4π²0bmin ∼ 1 MeV. If the electron temperature well is below
this value (∼ 1010 K), the assumption of bmin ¿ b0 is justified, since
b0 =
λD
Umin
= bmin
.
16πΛ
2kB T
• For other ions, if they are not fully stripped, the situation can be different,
and corrections to the value of Coulomb logarithm result. Note, however,
that since always bmin ¿ λD , our result should give the correct order
of magnitude for νei.
Plasma conductivity
• Collisions change the momentum of the particles ⇒ a friction-like term
in the particles’ ensemble averaged equation of motion,
m
dh~
vi
~ + h~
~ − mνc(h~
= q(E
v i × B)
vi − ~
u),
dt
where h~
v i is the ensemble-averaged particle velocity and ~
u is the
(constant) velocity of the collision targets.
Unmagnetized plasma
~ = 0 in steady state: choose a coordinate system with ~
• Case B
u=0
⇒
~ − meνc~
0 = −eE
ve.
=
−
=
nee2 ~
−ene~
ve =
E. Ohm’s law
meνc
• Solution:
~
ve
⇒
J~
e ~
E
meνc
• The conductivity of an unmagnetized plasma is
2
²0
ωpe
nee2
σ=
=
.
meνc
νc
• The reciprocal of conductivity, η = 1/σ , is called resistivity. If νc is
not due to particle collisions, we talk about anomalous resistivity. An
example of mechanisms leading to anomalous resistivity is scattering of
particles by wave fields (wave-particle interactions)
Magnetized plasma
~ 6= 0.
• Let us turn to the case B
• In the simplest case, the conductivity can still be regarded as a scalar in
the frame co-moving with the plasma,
~ 0,
J~ = σ0E
where
nee2
σ0 =
me ν c
is the conductivity of an unmagnetized plasma and
~0 = E
~ +V
~ ×B
~
E
~ in
is the electric field in the frame of the plasma, flowing at velocity V
the laboratory frame.
• Thus, the Ohm’s law becomes
~ +V
~ × B),
~
J~ = σ0(E
which is often called generalized Ohm’s law in plasma physics.
• In the limit of σ0 → ∞ (νc = 0), we can write this in the form
~ +V
~ ×B
~ = 0,
E
which is the form of Ohm’s law used in the so-called
ideal magnetohydrodynamics.
Conductivity tensor
• Electron motion parallel to the magnetic field is very different from the
motion perpendicular to it, if the magnetic field is strong enough, i.e., if
ωce > νc. ⇒ Conductivity not necessarily a scalar.
• Electron eq. of motion: stationary case, ion rest frame:
ve
~ +~
~ = − mνc~
E
ve × B
.
e
• Thus, the Ohm’s law becomes (E)
~−
J~ = σ0E
e ~
~
J × B,
mνc
nee2
σ0 =
meνc
~ = B~
• Choose the coordinate system so that B
ez . Use the electron–
cyclotron frequency, ωce = eB/me, to give the Ohm’s law as
Jx
=
Jy
=
Jz
=
ωce
σ0 E x −
Jy
νc
ωce
Jx
σ0 E y +
νc
σ0 E z
• This is easy to solve for the components of J~ as (E)
=
ωceνc
νc2
σ
E
−
σ0 E y
0 x
2
2
2
2
νc + ωce
νc + ωce
Jy
=
νc2
ωceνc
σ0 E y + 2
σ0Ex
2
2
νc2 + ωce
νc + ωce
Jz
=
σ0 E z
Jx
which can be given in matrix form as
~
J~ = σ · E.
• The conductivity tensor is

σP
σ =  −σH
0
σH
σP
0

0
0 .
σk
The components of the conductivity tensor are
σP
=
σH
=
σk
=
νc2
σ0 ,
2
νc2 + ωce
ωceνc
σ0 ,
2
νc2 + ωce
nee2
σ0 =
,
meνc
Pedersen conductivity
Hall conductivity
parallel conductivity
~ and parallel to E
~ ⊥.
• σP is the conductivity perpendicular to B
~ and E
~ ⊥.
• σH is the conductivity perpendicular to both B
~
• σk is the conductivity parallel to B
~ ⊥ and J~H =
• The corresponding perpendicular currents, J~p = σP E
~ ⊥, are the Pedersen and Hall currents.
σH ~
eB × E
– Pedersen current results, when collisions are so frequent that they
prevent the gyromotion around the magnetic field lines. Thus, the
particles move along the perpendicular electric field.
~ ×B
~ drift motion of electrons
– Hall current is related to the E
across the magnetic field (perpendicular to the electric field). In a
collisionless situation, all species would drift at same speed and no
current could fly. Hall current is, thus, important only in situations
where the drift of ions is impeded, e.g.,
∗ by collisions or
~ ⊥ fluctuates at a frequency ωci ¿ ω ¿ ωce.
∗ if E
~ k is the field-aligned current. In plasma physics, the field• J~k = σkE
~ ·∇×B
~ 6= 0, are very important.
aligned currents, for which B