Physics 138
Plasma Physics 2003
Seminar 1 presentation (mb)
ωpe, νc, and Λ (mb): Show the connection among Debye length λD,
plasma parameter Λ, plasma frequency ωpe, and collision frequency νc.
Saha equation: Before we get started, it’s useful to consider the Saha
equation which tells us the ionization fraction for a gas in thermal equilibrium
at temperature T:
ne
nneutral
∼= 2.4 × 1021 T
3/2
K
ne
e−Ui/kT
where T is in Kelvin, the densities are in particles per m3
, and Ui is the
ionization potential. For air in this room, T = 300K, nneutral = 3×1025
m−3
,
and Ui = 14.5 eV (nitrogen). We find that ne/nneutral = 10−122
. Since there
are only a few 1000 Avogadro numbers worth of particles in the room (maybe
1027
), there are effectively no electrons from thermal ionization. The derivation
is here: http://scienceworld.wolfram.com/physics/SahaEquation.html
Debye length λD: First let’s write down the Poisson equation for the
electrostatic potential (MKS):
2
φ = −ρ/ 0 = e(ne − ni)/ 0.
Now assume that the ions and electrons are separately in thermal equilibrium...
ie they each have a temperature and a Boltzmann factor like
n = n0 exp(eφ/kT). If eφ/kT 1 then we can expand the exponential and
find:
2
φ =
1
r2
d
dr
r2 dφ
dr
=
n0e2
k 0
1
Te
+
1
Ti
φ.
We define the Debye length for each species
λD =
0kT
n0e2
1/2
(MKS) =
kT
4πn0e2
1/2
(CGS)
and the total Debye length as the sum of the inverse squares.
From here on out, we’ll incorporate Boltzmann’s constant into the temperature
T and talk about temperature like an average particle energy measured
in electron volts. Also, we’ll combine constants like k, 0, e and write
λD = 740(T/n)1/2
cm where T is measured in eV and n is in units of particles
per cm3
. So a plasma at 1 eV and 1 cm−3
(say the solar wind) has a Debye
length of 740 cm.
Now we can solve for the potential:
1
r2
d
dr
r2 dφ
dr
=
φ
λ2
D
.
The solution is
φ(r) =
q
r
e−r/λD
.
The interpretation is that if the Debye length is large (say very low plasma
density like the air in this room) then the exponential is about unity and the
potential falls off like 1/r. If the Debye length is short (low temperature and
high density like in a metal), then point charges are immediately shielded and
the potential drops to zero in a few Debye lengths (λD 1˚A in a metal).
When we say that free charges reside on the surface of a metal, we should
really say that free charges reside a few Debye lengths from the surface of a
metal.
plasma parameter Λ: A useful dimensionless number is the number of
particles in a Debye cube (or sphere). This is called the plasma parameter:
Λ = nλ3
D 1
and the requirement that there are lots of particles in a Debye cube is one of
the definitions of a plasma (for statistical reasons if nothing else). It turns out
that Λ has several other interpretations (see below and other presentations).
Numerically we get Λ = 4 × 108
T3/2
n−1/2
.
plasma frequency ωpe: Consider a slab ’o plasma (ions and electrons
in equal numbers) with area A and thickness L (ie volume = AL). If we
displace the electrons in the slab a small distance δ from their equilibrium
positions (say to the right), then we’ll expose a layer of ions of thickness δ
on the left. An electric field of magnitude E = σ/ 0 now points to the right
where the surface charge σ = neδ. The force acting on the electron fluid is
F = QtotE = (neAL)
neδ
0
=
n2
e2
ALδ
0
= Mtot
¨δ = nmLA¨δ.
From which we can write:
¨δ =
ne2
m 0
δ.
We can immediately identify the plasma frequency
ωp =
ne2
m 0
1/2
(MKS) =
4πne2
m
1/2
(CGS).
Numerically, νpe = 9000n1/2
e Hz so a typical laboratory plasma at 1010
cm−3
wiggles at 1 GHz. The ionosphere has an electron density of 105
cm−3
so
its plasma frequency is about 3 MHz which is intermediate between AM and
FM radio.
collision frequency νc: Imagine a charged particle (mass m, charge q,
velocity v0) approaching another charged particle at rest (mass M m,
charge q0). If v0 is small, then the incoming particle won’t be able to get
too close to the target particle without getting deflected say 90o
. The more
kinetic energy the incoming particle has, the closer it can approach the target
particle so we see there’s a scale we can associate with the energy:
mv2
0
2
=
qq0
4π 0δ
.
The parameter δ = 2qq0
4π 0mv2
0
is sometimes called the Landau length.
If we have a population of particles of density n0, velocity v0, adn charge
e all heading for our target particle (also charge e), the rate at which they get
scattered (say 90o
or more) is roughly the flux of particles that pass within
a radius δ of the target:
νc = (πδ2
)nv0 = π
4e4
(4π 0)2m2v4
0
nv0 =
e4
n
m2v3
0(4π 2
0)
.
If you do this more carefully (considering small angle collisions) you get an
extra factor of 2 ln(Λ) ∼= 20, also in CGS the factor of (4π 0) downstairs
is replaced by a factor of (4π) upstairs. The important feature to notice is
that the collision frequency goes like v−3
0 or, if the particles are a thermal
distribution, like T−3/2
.
The connection: Finally, the point of all this is to show the relation
among these parameters. Look at the ratio of the plasma frequency to the
collision frequency:
ωp
νc
=
ne2
m 0
1/2
m2
v3
0(4π 2
0)
e4n
=
4π
3/2
0 m3/2
v3
n1/2e3
.
Notice now that the plasma parameter can be written:
Λ = nλ3
D = n
0T
ne2
3/2
=
3/2
0 m3/2
v3
n1/2e3
where we note that
m < v2
>
2
=
3T
2
=
3mv2
2
.
We find that
ωp
νc
= 4πΛ.
The interpretation here is that a plasma will oscillate many times at ωp before
it is damped by collisions. This really emphasizes the point that plasmas are
dominated by collective effects (ωp) rather than single particle effects (νc).