Fundamentals of Plasmas, Nuclear Physics and Lasers
Homework #5
February 2016
1. The cross section for electron-neutral momentum transfer in Ar can be
approximated by the relation σ(u) = αu[eV ], with α = 1.37 × 10−20
m2 /eV (see figure, σ[10−16 cm2 ] vs. u [eV]). Calculate the mean collision
frequency for momentum transfer in an argon plasma at p = 5 Torr and
Tg = 20 o C, characterized by an electron temperature kTe = 1 eV, i.e,
assuming a Maxwellian distribution of velocities (pay attention to the
definition and normalization of the distribution!),
f (v) = 4πv
2
m
2πkTe
3/2
mv 2
exp −
.
2kTe
Compare with the value you would obtain if the cross section were
constant, with the value corresponding to the mean energy.
4.50E+01
4.00E+01
3.50E+01
3.00E+01
2.50E+01
2.00E+01
1.50E+01
1.00E+01
5.00E+00
0.00E+00
0
5
10
Real
15
20
25
Linear
Approxima8on
30
35
2. (F. F. Chen, 5.1) The electron-neutral collision cross section for 2 eV
electrons in He is about 6πa20 , where a0 = 0.53 × 10−8 cm is the radius
of the first Bohr orbit of the hydrogen atom. A positive column with
no magnetic field has p = 1 Torr of He (at room temperature), and
kTe = 2 eV.
(a) Compute the electron diffusion coefficient in m2 /s, assuming that
hσvi is equal to the product σv for 2 eV electrons.
(b) If the current density along the column is 2 kA/m2 and the plasma
density is 1016 m−3 , what is the electric field along the column?
~ =
3. Consider an axisymmetric cylindrical weakly-ionized plasma with E
~ = B~uz and ∇P
~ i,e = ∂Pi,e /∂r~ur . Neglect the convective term
E~ur , B
and consider the stationary case. Assume neutrality and the same
temperature for electrons ans ions.
(a) Obtain the expressions for the r and θ components of the electron
and ion velocities. Comment.
(b) Obtain the limit of the previous expressions for very intense B-field
(ωc νe , νi ).
(c) Find the expression of E that ensures ambipolarity along the radial direction.
4. Suppose that the electron distribution function in a homogeneous plasma
can be approximated by a superposition of two Maxwellians at different
temperature, i.e., f (~v ) = α1 f1 (~v ) + α2 f2 (~v ), where
fj (~v ) = n
m
2πkB Tj
3/2
mv 2
exp −
2kB Tj
,
with j = 1, 2, α1 + α2 = 1 e v = |~v |.
(a) Verify
RRR that3 the distribution function is correctly normalized, i.e.,
f (~v )d v = n.
(b) Show that the average value of the
qabsolute value of the velocity of
8k T
B j
each of the maxwellians is hvj i =
and calculate the average
πm
value of the absolute value of the velocity of the distribution.
(c) The cross section for electron-neutral momentum transfer can be
approximated by σm (u) = βm u, where βm is constant and u
is the electron energy. Show that the mean collision frequency
for momentum transfer associated to each Maxwellian is νm =
2N βm kB Tj hvj i and calculate the average value of the momentum
collision frequency of the distribution.
(d) The ionization cross section of the same gas can be approximated
by σi (u) = 0, if u < ui , and σi (u) = βi , if u ≥ ui , where ui is the
ionization threshold. Show that the ionization
associ
frequency
ui
ui
ated with each Maxwellian is νi = N βi hvj i kTj + 1 exp − kT
j
and calculate the mean ionization frequency of the distribution.
(e) Calculate the values in items c. and d. for kB T1 = 1 eV, kB T2 = 16
eV, α1 = 0.99, α2 = 0.01, βm = 10−20 m2 /eV, βi = 10−20 m2 ,
ui = 15 eV and N = 1023 m−3 . Comment the results.
5. (Lieberman and Lichtenberg 5.2) A steady-state argon plasma is created at high pressure between two parallel plates located at x = ±L/2
by illuminating the region between the plates with ultraviolet radiation
(which ionizes the neutrals). The radiation creates a uniform number
of electron-ion pairs per unit volume and pe unit time, G0 (m−3 s−1 ),
everywhere within the plates. The electrons and ions are lost by ambipolar diffusion to the walls.
(a) Show that in the limit Ti Te the ambipolar diffusion coefficient
is given by Da ' µi (kTe /e).
(b) Assuming Da uniform in space and constant in time, obtain the
stationary plasma profile, n(x), and the value of the density at
the center, n0 , assuming you can impose the boundary condition
n(x) ' 0 at the walls.