Problems for the Course F5170 Introduction to Plasma Physics
Jiří Šperka, Jan Voráč, Lenka Zajíčková
Department of Physical Electronics Faculty of Science Masaryk University
2014
Contents
1 Introduction 5
1.1 Theory............................... 5
1.2 Problems ............................. 6
1.2.1 Derivation of the plasma frequency........... 6
1.2.2 Plasma frequency and Debye length.......... 7
1.2.3 Debye-Hiickel potential ................. 8
2 Motion of particles in electromagnetic fields 9
2.1 Theory............................... 9
2.2 Problems ............................. 10
2.2.1 Magnetic mirror ..................... 10
2.2.2 Magnetic mirror of a different construction...... 10
2.2.3 Electron in vacuum - three parts............ 11
2.2.4 E x B drift........................ 11
2.2.5 Relativistic cyclotron frequency............. 12
2.2.6 Relativistic particle in an uniform magnetic field  ... 12
2.2.7 Law of conservation of electric charge......... 12
2.2.8 Magnetostatic field.................... 12
2.2.9 Cyclotron frequency of electron............. 12
2.2.10 Cyclotron frequency of ionized hydrogen atom   .... 13
2.2.11 Magnetic moment .................... 13
2.2.12 Magnetic moment 2................... 13
2.2.13 Lorentz force....................... 13
3 Elements of plasma kinetic theory 14
3.1 Theory............................... 14
3.2 Problems ............................. 15
3.2.1 Uniform distribution function.............. 15
3.2.2 Linear distribution function............... 15
3.2.3 Quadratic distribution function............. 15
3.2.4 Sinusoidal distribution function............. 15
3.2.5 Boltzmann kinetic equation............... 15
1
CONTENTS 2
4 Average values and macroscopic variables 16
4.1 Theory............................... 16
4.2 Problems ............................. 17
4.2.1 RMS speed........................ 17
4.2.2 Mean speed of sinusoidal distribution ......... 17
4.2.3 Mean speed of quadratic distribution.......... 17
4.2.4 The equilibrium temperature.............. 17
4.2.5 Particle density...................... 17
4.2.6 Most probable speed of linear distribution....... 17
4.2.7 Most probable speed of sinusoidal distribution   .... 17
5 The equilibrium state 20
5.1 Theory............................... 20
5.2 Problems ............................. 21
5.2.1 Gamma function..................... 21
5.2.2 ID Maxwell-Boltzmann distribution function..... 21
5.2.3 Two-dimensional Maxwell-Boltzmann distribution function ............................ 22
5.2.4 Three-dimensional Maxwell-Boltzmann distribution function ............................ 23
5.2.5 Exotic one-dimensional distribution function ..... 23
6 Particle interactions in plasmas 24
6.1 Theory............................... 24
6.2 Problems ............................. 25
6.2.1 Mean free path of Xe ions................ 25
6.2.2 Hard sphere model.................... 26
6.2.3 Total scattering cross section.............. 26
7 Macroscopic transport equations 27
7.1 Theory............................... 27
7.2 Problems ............................. 28
7.2.1 Afterglow......................... 28
7.2.2 Macroscopic collision term - momentum equation   . . 28
7.2.3 Macroscopic collision - momentum equation II  ... . 29
7.2.4 Simplified heat flow equation.............. 30
8 Macroscopic equations for a conducting fluid 31
8.1 Theory............................... 31
8.2 Problems ............................. 31
8.2.1 Electric current density................. 31
8.2.2 Fully ionised plasma................... 32
8.2.3 Diffusion across the magnetic field........... 32
CONTENTS 3
9 Plasma conductivity and diffusion 34
9.1 Theory............................... 34
9.2 Problems ............................. 35
9.2.1 DC plasma conductivity................. 35
9.2.2 Mobility tensor for magnetised plasma......... 36
9.2.3 Ohm's law with magnetic field ............. 36
9.2.4 Diffusion equation.................... 37
10 Some basic plasma phenomena 38
10.1 Theory............................... 38
10.2 Problems ............................. 39
10.2.1 Waves in non-magnetized plasma............ 39
10.2.2 Floating potential .................... 39
10.2.3 Böhm velocity ...................... 39
10.2.4 Plasma frequency..................... 39
11 Boltzmann and Fokker-Planck collision terms 41
11.1 Theory............................... 41
11.2 Problems ............................. 42
11.2.1 Collisions for Maxwell-Boltzmann distribution function 42
11.2.2 Collisions for different distributions........... 43
11.2.3 Collisions for Druyvesteyn distribution......... 43
List of Figures
1.1   Illustration of the problem no. 1.2.1............... 7
2.1   Sketch of the problem 2.2.3.................... 11
4.1 Diagram to the problem of the highest equilibrium temperature 4.2.4.............................. 18
4.2 Diagram to the problem of the highest particle density 4.2.5. 18
4
Preface
This document contains exercise problems for the course F5170 - Introduction to plasma physics. This work was supported by the Ministry of Research and Education of the Czech Republic, project no. FRVS 12/2013/G6. The most valuable source of information for this document was the book Fundamentals of Plasma Physics by J. A. Bittencourt [4]. The authors would be grateful for any notification about eventual errors.
The complete and up-to-date version of this document can be found at http://physics.muni.cz/~sperka/exercises.html.
Contacts
Jiří Sperka jewel@mail.muni.cz
Jan Voráč vorac@mail.muni.cz Lenka Zajíčková lenkaz@physics.muni.cz
Physical constants
Proton rest mass	mp	1,67-	1(T	27 kg
Electron rest mass	me	9.109	• 10	-31kg
Elementary charge	e	1.602	• 10	-19 C
Boltzmann's constant	k	1.38 •	io-	23 J K-1
Vacuum permittivity		8.854	• 10	-12A2 s^g-im-3
Used symbols
Vector quantities are typed in bold face (v), scalar quantities, including magnitudes of vectors are in italic (v). Tensors are usually in upper-case calligraphic typeface (V).
5
Operators
scalar product a • b
vector product a x b
ith derivative with respect to x ^
partial derivative 7^-
1 ox
nabla operator V = (^, j|, )
Laplace operator A = V2
total time derivative ^ =     + u • V
Physical quantities
electron concentration ne electron temperature Te electron plasma frequency upe Debye length Ad Larmor radius rc Larmor frequency 0,c magnetic moment m force F electric field intensity E magnetic field induction B arb. quantity for one type of particles Xa distribution function f(Xa) mean velocity u charge denisty p mass density pm collision frequency v source term due to collisions Sa scalar pressure p tensor of kinetic pressure V mobility of particles Ma
6
Chapter 1
Introduction
1.1 Theory
Electron plasma frequency
= const y/n^.
(1.1)
describes the typical electrostatic collective electron oscillations due to little separation of electric charge. Plasma frequencies of other particles can be defined in a similar way. However, the electron plasma frequency is the most important because of high mobility of electrons (the proton/electron mass ratio mp/me is 1.8 x 103).
Note that plasma oscillations will only be observed if the plasma system is studied over time periods longer than the plasma period u~1 and if external actions change the system at a rate no faster than up. Observations over length-scales shorter than the distance traveled by a typical plasma particle during a plasma period will also not detect plasma behaviour. This distance, which is the spatial equivalent to the plasma period , is called the Debye length, and takes the form
The Debye length is independent of mass and is therefore comparable for different species.
7
CHAPTER 1.  INTRODUCTION 8
	ne [cm 3]	Tel	eV]	tlak [Pa]	Ref.
Plasma Displays	(2.5-3.7) xlO11 max 3 xlO12	0.8-	-1.8	(20-50) xlO3 (40-67) xlO3	[8] [22]
	(0.2-3) xlO13	1.6-	-3.4		[19]
Earth's ionosphere	max 106	max 0.26			[6]
				10~5	[2]
RF Magnetrons				0.5-10	[15]
	1-8 xlO9	2-9		0.3-2.6	[20]
DC Magnetrons	1018	1-5		0.5-2.5	[23]
RF Atmospheric	1013-1014			105	[10]
plasma					
		0.2-	-6	105	[12]
MW Atmospheric		1.2-	-1.9	105	[14]
plasma	3 x 1014				[11]
Welding arc	1.5 x 1017	1.5		105 105	[3] [21]
	1.6 x 1017	1.3		105	[18]
Low-pressure CCP	6 x 108			6-7	[24]
	(0.5-4.5) xlO10	1.4-	-1.6	4.7	[5]
Fluorescent lamps	10lu-10n	1		8 x 103	[7]
Table 1.1: An overview of typical values of the most important parameters for various plasmas.
1.2 Problems
1.2.1    Derivation of the plasma frequency
Consider a steady initial state with a uniform number density of electrons and an equal number of ions such that the total electrical charge is neutral. Neglect the thermal motion of the particles and assume that the ions are stationary. Show that a small displacement of a group of electrons leads to oscillations with the plasma frequency according to the equation (1.1).
Solution The situation is sketched in the figure 1.1. Assume that the electric field in the plane perpendicular to the x-axis is zero (just like in the case of an infinitely large charged plane or capacitor). Let us apply the Gauss's law to a closed cylindrical surface (only contour of which is sketched in the figure):
E.dS=Q = (S^e)X} (L3)
S e0      V eo
CHAPTER 1. INTRODUCTION
9
- elektron + kladný ion
			X
Ac +-+-+-+-+-+-+-+-+-+-1 +"+"+"+"+"+"+"+"+"+"	+■ +"	+- +- +- +■ +- +- +- +- +- +■ +- +■ +" +" +" +" +" +" +" +" +" +" +" +"	í
+++++++++++++++++++++++			
Figure 1.1: Illustration of the problem no. 1.2.1.
where S is the area of the base of the cylinder. The resulting electric field is
EX = (^) x. (1.4)
Inserting this electric field into the equation of motion of a single electron yields
i2 / 2
ax     I riQe
,.2+ — )x = 0, (1.5) dt2 \mee0'
Which is an equation of a harmonic oscillator with the frequency
2 \ 1/2
mee0
;i.6)
1.2.2   Plasma frequency and Debye length
Compute the plasma frequency and the Debye length for the following plasmas
(a) Earth's ionosphere with electron concentration ne = 106 cm~3 and electron temperature kTe = 0.2eV.
[ujp = 5, 6 x 107 rad • s"1 = 3, 5 x 108 Hz, AD = 3, 3 mm]
(b) A cell of a typical plasma display with electron concentration of 1013 cm~3 and electron temperature of 1 eV. The cell dimension is about 100 /tm. Is the condition that the system dimension should be much greater than the Debye length fulfilled?
pe
2, 3 x 1013 rad • s"1 =3,6 THz, AD = 21 nm]
(c) A welding arc with electron concentration ofl,6xl017 cm 3 and electron temperature of 1, 3eV
[tope = 2, 3 x 1013 rad • s"1 =3,6 THz, AD = 21 nm]
(d) A fluorescent lamp with electron concentration of 1010 cm~3 and electron temperature of 1 eV
[wpe = 5, 6 x 109 rad • s"1 = 0, 90 GHz, AD = 74 /mi]
CHAPTER 1. INTRODUCTION
10
1.2.3   Debye-Hiickel potential
Show that Debye-Hiickel potential
e    exP V Ar <p(r) = --(1.7)
4 7t£0 r
is solution of equation
where ro is Debye-Hiickel radius.
Remark: Debye-Hiickel potential which is called after Pieter Debye (1884-1966) and Erich Hiickel (1896-1980) who studied polarisation effects in electrolytes [9].
Solution Put simply the Debye-Hiickel potential into the equation (1.8) and calculate Laplace operator in spherical coordinates
1 d_ ( 2df\        1     d ( .    d£\        1 d2f
Chapter 2
Motion of particles in electromagnetic fields
2.1 Theory
The Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience the Lorentz force
F = g(E + vxB). (2.1)
The gyroradius (also known as Larmor radius or cyclotron radius) is the radius of the circular motion of a charged particle in the presence of a uniform magnetic field:
r» = W M
where rg is the gyroradius, m is the mass of the charged particle, v± is the velocity component perpendicular to the direction of the magnetic field, q is the charge of the particle, and B is magnitude of the constant magnetic field.
Similarly, the frequency of this circular motion is known as the gyrofre-quency or cyclotron frequency, and is given by:
k\B , \
UJg = (2.3
m
Note: A cyclotron is a type of particle accelerator in which charged particles accelerate due to high-frequency electric field. The cyclotron was invented and patented by Ernest Lawrence of the University of California, Berkeley, where it was first operated in 1932.
11
CHAPTER 2. MOTION OF PARTICLES IN ELECTROMAGNETIC FIELDS12
2.2 Problems
2.2.1    Magnetic mirror
Magnetic mirrors are used to confine charged particles in a limited volume. The gradient of magnetic field induction can result in reversing the direction of drift of a charged particle.
Suppose we have an electron located at z = 0 with initial velocity vq and an initial pitch angle i/1. The magnetic field induction is given by
B(z) = B0 (l + (7^)2). (2.4) Calculate the turning point zt [13].
Solution We start with the conservation of kinetic energy and the magnetic moment. The kinetic energy conservation condition yields
vl = v2t. (2.5)
The z-component of the velocity at the turning point must be zero, which we immediately use in the equation describing the conservation of the magnetic moment
2 ™2„q _ „,2
me     sin i/1 me v{
2B0 2Bo(l + (7*t)
v20sm2^ (l + (7^t)2))   =   v2t {=vl)
2 2
7 h
1 — sin219 sin219
zt   =   —^—r (2.6) 7 tan 17
We see that the position of the point of reflection depends only on the gradient of the magnetic field and on the initial pitch angle.
2.2.2   Magnetic mirror of a different construction
Calculate the turning point for a charged particle in a magnetic mirror with induction given by
B(z) = B0 (l + (7^)4). (2.7) The initial pitch angle is it1.
CHAPTER 2. MOTION OF PARTICLES IN ELECTROMAGNETIC FIELDS13
1
V(t)
X X X X
X X X X
X X X X
X X X X
Figure 2.1: Sketch of the problem 2.2.3.
2.2.3   Electron in vacuum — three parts
(a) The time dependence of the position of the electron in the first part is expressed as x(t) = |t4 + ir. The electron remains in the first part for one second. Calculate the magnitude of the velocity vx that the electron acquires at the end of the first part.
[0.5 m/s]
(b) After that, the electron, having the velocity vx, enters the second part, where a transverse electric field E of the mag nitude 10~10V m 1 is applied. This field is generated by the plates of a capacitor with the length d = 1 m. What is the vertical displacement of the electron with respect to the starting position at the end of the second part? First derive the general solution.
[35.2 m]
(c) Finally, the electron enters a homogeneous magnetic field if of the magnitude 20.6 /xT (this is the magnitude of the horizontal component of the geomagnetic field induction in Brno). Calculate the Larmor radius, cyclotron frequency and the magnitude of the magnetic moment of the rotating electron.
[rc = 9.72 x 10~6 m, flc = 3.6 x 106 rad • s_1, |m| = 2.7 x 10~23 A • m2]
(d) What would be the result for proton, neutron and positron? For illustration see 2.1.
2.2.4   E x B drift
Suppose we have a vacuum chamber with electric field E = lkVm-1 perpendicular to magnetic field B = 1 mT. Calculate E x B drift speed for an electron inside the chamber.
CHAPTER 2. MOTION OF PARTICLES IN ELECTROMAGNETIC FIELDSU
2.2.5   Relativistic cyclotron frequency
What is the relativistic cyclotron frequency of an electron with velocity 0.8 c
(c denotes speed of light)? 6
To
[u = ^eB/m]
2.2.6   Relativistic particle in an uniform magnetic field
Derive the gyroradius, angular gyrofrequency $7£el, and energy of relativistic particle with speed v and charge q in an uniform magnetic field with magnitude of magnetic induction B.
Solution Gyroradius:
r = ~w (2-8)
Angular gyrofrequency:
nf = 1^ = ^ = ncVT^ = nji - (v-)2 (2.9)
Energy:
2
7—1 9 9 TYIC o , >
E~k = m^c — mc = — = — mc (2.10)
V 1 — w2/c2
2.2.7 Law of conservation of electric charge
Derive continuity equation from Maxwell's Equations.
"§? + v-j = o"
2.2.8 Magnetostatic field
Proof, that in presence of magnetostatic field total kinetic energy of charged particle      remains constant.
2.2.9   Cyclotron frequency of electron
What is a cyclotron frequency (in Hz) of electron in homogenous magnetostatic field:
a) \B\ = 0.01 T
b) \B\ = 0.1 T
c) \B\ = IT
d) \B\ = 5T
[a) 0.28 GHz ; b) 2.8 GHz; c) 28 GHz d) 140 GHz]
CHAPTER 2. MOTION OF PARTICLES IN ELECTROMAGNETIC FIELDS15
2.2.10 Cyclotron frequency of ionized hydrogen atom
What is a cyclotron frequency (in Hz) of ionized hydrogen atom in homogenous magnetostatic field:
a) \B\ = 0.01 T
b) \B\ = 0.1 T
c) \B\ = IT
d) \B\ = 5T
[a) 0.15 MHz ; b) 1.5 MHz; c) 15 MHz d) 76 MHz]
2.2.11 Magnetic moment
Suppose a planar closed circular current loop has area |5| = 10~3m2 and carries an electric current:
a) I = 1A
b) I = 2 A
c) J = 8A
Calculate the magnitude of its magnetic moment \m\.
[a) \m\ = 10~3Am2; b) \m\ = 2 x 10~3Am2; c)\m\ = 8 x 10~3 Am2 ]
2.2.12 Magnetic moment 2
How can be written the magnitude of the magnetic moment \rh\, which is associated with the circulating current of charged particle (charge q, angular frequency 0,c, mass m) in uniform magnetostatic field Bl
2.2.13 Lorentz force
Suppose a magnetostatic field B = (1, 2, 0) T. The velocity of an electron is v = (0, 2,1) ms_1. Calculate Lorentz force. [F = -e-(-2,l,-2)N]
Chapter 3
Elements of plasma kinetic theory
3.1 Theory
• Phase space is defined by six coordinations (x,y, z,vx,vy,vz).
• The dynamical state of each particle is appropriately represented by a single point in this phase space.
• The distribution function in phase space, fa(f,v,t), is defined as the density of representative points of the particles a in phase space:
• The number density, na(f, t), can be obtained by integrating /a(r, v, t) over all of velocity space:
• The differential kinetic equation that is satisfied by the distribution function, is generally known as the Boltzmann kinetic equation:
fa(r,v,t) =N^(r,v,t)/(d3rd3v).
(3.1)
(3.2)
dfg(f,v, t)
dt
+ v-Vfffa(f,v,t)+a- v^/a(r,v,t)
dfa(r,v, t) dt
collision
(3.3)
16
CHAPTER 3.  ELEMENTS OF PLASMA KINETIC THEORY
17
3.2 Problems
3.2.1 Uniform distribution function
Suppose we have system of particles uniformly distributed in space with constant particle number density n, which is characterised by one dimensional distribution function of speeds F(v):
F(v)   =   C   for v < vq F(v)   =   0 otherwise,
where C is positive non-zero constant. Express C using n and vq. [Solution: By integration n = C J dv we will get the solution C = ^.]
3.2.2 Linear distribution function
What is the normalizing constant C of the following distribution function of speeds?
F{v) = C v for v £ (0,1) and F(v) = 0 otherwise. [C = 2n (n denotes the particle density)]
3.2.3 Quadratic distribution function
What is normalizing constant C of following distribution function of speeds? F(v) = C v2 for v £ (0, 3) and F(v) = 0 otherwise. [C = n/9 (n denotes the particle density)]
3.2.4 Sinusoidal distribution function
What is the normalizing constant C of the following distribution function of speeds?
F{v) = C sin(u) for v 6 (0, tt) and F(v) = 0 otherwise. [C = n/2 {n denotes the particle density)]
3.2.5 Boltzmann kinetic equation
Consider the motion of charged particles, in one dimension only, in the presence of an electric potential ip(x). Show, by direct substitution, that a function of the form
is a solution of the Boltzmann equation under steady state conditions.
Chapter 4
Average values and macroscopic variables
4.1 Theory
• The macroscopic variables, such as number density, flow velocity, kinetic pressure or thermal energy flux can be considered as average values of physical quantities, involving the collective behaviour of a large number of particles. These macroscopic variables are related to the various moments of the distribution function.
• With each particle in the plasma, we can associate some molecular property Xair, v, t). This property may be, for example, the mass, the velocity, the momentum, or the energy of the particle.
• The average value of the property Xair, v, t) for the particles of type a is defined by
• For example, the average velocity (or flow velocity) ua(f,t) for the particles of type a is defined by
(4.1)
18
CHAPTER 4. AVERAGE VALUES AND MACROSCOPIC VARIABLES19
4.2 Problems 4.2.1   RMS speed
What is the rms speed of the following three electrons (\vi\ = 1, l^l = 2 and |«31 = 5)?
[VTo]
4.2.2   Mean speed of sinusoidal distribution
What is the mean speed of the following distribution function of speeds? f(v) =    s'm(v) for v 6 (0, tt) and f(v) = 0 otherwise, n denotes the particle density. [1]
4.2.3   Mean speed of quadratic distribution
What is the mean speed of the following distribution function of speeds? f{v) = 3nv2 for v £ (0,1) and f(v) = 0 otherwise n denotes the particle density. [3/4]
4.2.4   The equilibrium temperature
Consider Maxwell-Boltzmann distributions in Fig. 4.1. Which one has the
highest equilibrium temperature?
[c)]
4.2.5   Particle density
Consider Maxwell-Boltzmann distributions in Fig. 4.2. Which one has the
highest particle density?
[c)]
4.2.6   Most probable speed of linear distribution
Consider the following distribution function of speeds f{v) = nv for v 6 (0,1) and f(v) = 0 otherwise.
What is the most probable speed of this distribution? [1]
4.2.7   Most probable speed of sinusoidal distribution
Consider the following distribution function of speeds f(v) = ^ sin(u) for v £ (0,7r) and f(v) = 0 otherwise.
CHAPTER 4. AVERAGE VALUES AND MACROSCOPIC VARIABLES20
0.002
0.0015
= 0.001
0.0005
	-1	-1	I-1- a) -b)
			
	,--N		
			
0 1e+06       2e+06       3e+06       4e+06 5e+06
v [m/s]
Figure 4.1: Diagram to the problem of the highest equilibrium temperature 4.2.4.
0.002
0.0015
= 0.001
0.0005
-1	-1	-1	1-1- a) -b)
			
			
			
0 1e+06       2e+06       3e+06       4e+06 5e+06
v [m/s]
Figure 4.2: Diagram to the problem of the highest particle density 4.2.5.
CHAPTER 4. AVERAGE VALUES AND MACROSCOPIC VARIABLES21 What is the most probable speed of this distribution?
Chapter 5
The equilibrium state
5.1 Theory
• The equilibrium distribution function fa (r, v, t) is the time-independent solution of the Boltzmann equation in the absence of external forces.
• In the equilibrium state the particle interactions do not cause any changes in faq(r, v, t) with time and there are no spatial gradients in the particle number density.
• fa (r, v, t) is known as the Maxwell-Boltzmann distribution or Maxwell distribution (see problems 5.2.2-5.2.4).
Math useful for calculations
2
The "Gaussian integral" is the integral of the Gaussian function e~x over the entire real line. It is named after the German mathematician and physicist Carl Friedrich Gauss. The integral is (a, b denotes a constant):
/+oo poo r~T
e-x2dx = V^;     /    e-a<-x+V2dx= J-. (5.1) -oo J—oo V a
The gamma function T(n) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. That is, if n is a positive integer:
T(n) = (n-1)1 (5.2)
Other important formulas:
poo TV (n+l) \ /, \
/    xne-* dx = ^-f;       r(-   =V£. (5.3)
22
CHAPTER 5.  THE EQUILIBRIUM STATE
23
5.2 Problems
5.2.1 Gamma function
Starting from the definition of a Gamma function show that, if n is a positive integer, then
T(n + 1) = nl
Recipe: First, using integration by parts of r(n + l) = xne~x dx demonstrate that
r(a + 1) = aT{a). Next, it remains to show, that r(l) = 1.
5.2.2 ID Maxwell-Boltzmann distribution function
Gas composing of particles of one kind moving in only one dimension x is characterised by the following homogeneous isotropic one-dimensional Maxwell-Boltzmann distribution function:
f{vx) = C ■ exp
m v% ' 2kT
(5.4)
(a) Calculate the constant C.
(b) Derive the ID Maxwell-Boltzmann distribution function of speeds.
(c) Calculate the most probable speed.
(d) Calculate the mean speed.
(e) Derive the relation for the number of particles passing through a unit of length in a unit of time from one side (the flux of particles from one side).
Solution
(a) Integrate the distribution function over the whole velocity space. The condition that the integral equals the concentration of particles n yields
n = C   / exp
m v% ' 2kT
Avx = C\l2—. (5.5)
m
(5.6)
CHAPTER 5.  THE EQUILIBRIUM STATE
24
(b) Distribution of particle speeds F(v) from the summation over the both possible directions is
F(v) = 2n
m
2kTir
exp
m v ' 2kT
(5.7)
(c) From the condition that the derivation of the distribution F(v) must equal zero
0 = v exp
mv
2kT
we will get that the most probable speed is zero.
(d)
(v) = J v F(v) dv o
2kT
Tim
(5.8)
(5.9)
vx f(vx)dvx = n
kT
2-Km
o
(5.10)
5.2.3   Two-dimensional Maxwell-Boltzmann distribution function
Solve the tasks of the preceding problem with two-dimensional Maxwell-Boltzmann distribution function
f{vx,vy) = C ■ exp
m(vl +vy)
2kT
(5.11)
Results:
(a) C
2irkT
(b) F(v) =2irv f(v) = ffv exp
m v ' 2kT
(c) Most probable speed v = \j — ■
(d) Mean speed (v) = J^.
CHAPTER 5.  THE EQUILIBRIUM STATE
25
5.2.4   Three-dimensional Maxwell-Boltzmann distribution function
Solve the tasks of the preceding problem with three-dimensional Maxwell-Boltzmann distribution function
Results: (a) C = n
f(vx,vy,vz) = C -exp
\3/2
m {vl + v% + vp 2kT
(5.12)
2irkT)
m v ' 2kT
2kT
8kT
(b) F(«)=47rn(^)3/Vexp
(c) Most probable speed v
(d) Mean speed (v) =
(e) r = nJ^g
5.2.5   Exotic one-dimensional distribution function
Solve the tasks of the preceding problem with the following function (Cauchy/Lorentz distribution):
f(v) = -T^Pr • (5-13)
V2 + ML-
Results:
(a) C = n
kT
(b) F{v)=2nJ-
kT 1
m-k2 v2 + —
m
(c) Most probable v = 0 speed.
(d) Mean speed v is not defined, [1] see Cauchy distribution.
(e) Not defined.
Chapter 6
Particle interactions in plasmas
6.1 Theory
Collisional phenomena can be divided into two categories:
• elastic - conservation of mass, momentum and energy is valid in such a way that there are no changes in the internal states of the particles involved and there is neither creation nor annihilation of particles.
• inelastic - the internal states of some or all of the particles involved are changed and particles may be created as well as destroyed. A charged particle may recombine with another to form a neutral particle or it can attach itself to a neutral particle to form a heavier charged particle. The energy state of an electron in an atom may be raised and electrons can be removed from their atoms resulting in ionization.
The total scattering cross section can be obtained by integrating a{x,z)d£l over the entire solid angle:
In the special case, when the interaction potential is isotropic (e.g. Coulomb potential), we can get the total scattering cross section using the formula
0
For the same case, when the interaction potential is isotropic, we can get the momentum transfer cross section using the formula:
7T
26
CHAPTER 6.  PARTICLE INTERACTIONS IN PLASMAS
27
crm = 2?r j (1 - cosx)cr(x)sinxdx. (6-3) 0
6.2 Problems
6.2.1    Mean free path of Xe ions
Scattering cross section a for elastic collisions of Xe+ ions with Xe atoms is approximately independent on their energy with cross section value of a = l(T14cm2.
A) Calculate mean free path I of Xe+ ions for elastic collisions in a weakly ionized plasma in xenon atmosphere at room temperature (20 °C) at the pressure:
a) 1000 Pa
b) 10 Pa
c) 0.1 Pa
B) How long is the time period between two subsequent collisions, if the mean temperature of Xe ions is T = 1000 K?
Solution:
A) The mean free path is defined as
x = ±.
n a
Density of particles can be calculated from the equation of state p = nkT, so
p a
So the final results for given pressures are: a) 4 • 10~6m b) 4 • 10~4m c) 4 • 10~2m.
B) The thermal velocity of ions is v = \/^jr-- Mass of Xe ion is approx-
M
1-27-'
imately 131 amu (1 amu = 1.66 • 10 kg). The time period between two subsequent collisions equals to the fraction of mean free path and thermal
CHAPTER 6.  PARTICLE INTERACTIONS IN PLASMAS
28
velocity:
m
t = X
3kT
So the results are: a) 17 • 1(T9 s b) 17 • l(T7s c) 17 -l(r5s.
6.2.2 Hard sphere model
What is the total scattering cross section for the hard sphere model (two
elastic spheres, radius R\ and -R2)?
[tv(R1+R2)2}
6.2.3 Total scattering cross section
Differential cross section is given by
<r(x) = ^o(3 cos2 x + 1) (6.4)
Calculate the total cross section and the momentum transfer cross section. [4tto-0, 4tto-0]
Chapter 7
Macroscopic transport equations
7.1 Theory
From different moments of Boltzmann equation, the following macroscopic transport equations can be derived:
• From the condition of conservation of mass the continuity equation
at
+ V ■ (pmaua) = Sa, (7.1)
where pma is the mass density of type-a particles and Sa describes the creation or destruction of particles due to collisions (ionization, recombination, etc.).
From conservation of momentum the momentum transfer equation
PmJj^- = naqa (E + ua x B) + pmag - V • Va + Aa - ua Sa (7.2)
ua is the mean velocity, ^ = + ua ■ V is the total time derivative operator, na is the particle density, qa is the charge of a single particle, E and B are the electric and magnetic fields, g is the gravitational acceleration, Va is the kinetic pressure dyad,
Aa = -pma ^ *v(ua - Ug) (7.3)
is the collision term with uap being the collision frequency for the momentum transfer between the particles of type a and particles of type (3. From conservation of momentum during a collision follows
Pma Va/3 = Pm/3 vf3a- (7.4)
29
CHAPTER 7. MACROSCOPIC TRANSPORT EQUATIONS
30
• From the energy conservation the energy transport equation
Dl(^)+^V-Ua + (P-V)-Ua + V-qa =
= Ma - ua ■ Aa + ^u^Sa, (7.5)
where pa is the scalar pressure, qa is the heat flux vector and Ma represents the rate of energy density change due to collisions.
7.2 Problems
7.2.1 Afterglow
Consider a homogeneous plasma afterglow consisting of electrons and one type of singly charged positive ions. In this case, the continuity equation is
—£ = -krneni, (7.6) at
where kr is the rate coefficient for recombination. The spatial derivatives vanish because of the spatial uniformity. The concentration of electrons at t = 0 is uq. Calculate ne{t > 0). Remember the quasineutrality condition.
ne(t)
ng kr t+1
7.2.2   Macroscopic collision term — momentum equation
Consider a uniform mixture of different fluids (all spatial derivatives vanish), with no external forces, so that the equation of motion for the a species reduces to
= -vap (ua - up). (7.7)
Assume that the mass density of (3 species is much greater and thus neglect the temporal change of up. Notice that at equilibrium (dua/d£ = 0) the velocities of all species must be the same.
solution The situation is identical in all spatial coordinates, thus, only the solution in the x direction will be presented.
—--h vapuax(t) = ua/3 upx (7.8)
This simple differential equation can be solved by the method of variation of parameter. First, look for the particular solution of the homogeneous equation
duaXjp(t)
dt
+ VotpUax,p{t) = 0 (7.9)
CHAPTER 7. MACROSCOPIC TRANSPORT EQUATIONS 31
This is obviously
uax,p{t)=CeTv^t (7.10)
We now take the parameter C to be time-dependent C = C(t) and calculate the deriative
duUtl = dC(t)    Vapt _ c{t) v -uaPt (7-n)
At           dt                  y '   1 y ' inserting this into the original equation (7.8) yields
dC(t)
■ e
-vapt
At
from which we obtain by integrating
C(t)=uPxeu^t + K
where K is an arbitrary integration constant. The solution is then
uax(t)=uPx + Ke-v^t (7.12)
And similarly for all three spatial components. The velocity ua will exponentially approach to the velocity ug with the rate given by the collision frequency for momentum transfer vap.
7.2.3   Macroscopic collision — momentum equation II
Recalculate the task of the previous problem without the assumption ug = const. In this case, the velocities ua, up are described by a pair of coupled differential equations
Aua(t) ^ _ (7
At dug(t)
Uap(up(t) -Ua(t)), (7.14)
dt Pm/3
where pma, pmp are the mass densities of particles a, /3. Suppose that ua and ug are parallel and ua(t = 0) = 2up(t = 0).
(a) Calculate the time dependence of the difference u = ua — up.
(b) Calculate ua(t) and up(t).
Results:
(a) u(t) = ua(0) ■ exp
1 + t
Pm/3 '
(b) «*(*) = i^h    •exp [~v<* i1 + S) *] + p™)
Up(t) = u(t) +ua(t)
CHAPTER 7. MACROSCOPIC TRANSPORT EQUATIONS
32
7.2.4   Simplified heat flow equation
Suppose the simplified equation for heat flow in a stationary electron gas
Tt)*h^=(iL (7-i5)
Assume the collision term given by the relaxation model
^       =~Hfe-fe0) (7-16) St / coll
and the ideal gas law pe = nekTe. Show that the heat flow equation can be written as
^ (qe x B) = -K0VTe + (fe - fe0), (7.17) v
where
Ko = ^ (7-18)
2 me v
is the thermal conductivity.
Chapter 8
Macroscopic equations for a conducting fluid
8.1 Theory
The equations governing the important physical properties of the plasma as a whole can be obtained by summing the terms for the particular species. If also several simplifying assumptions are made, the following set of so called magnetohydrodynamic equations can be derived:
The continuity equation
dt
The momentum equation
+ V-(pmu) = 0 (8.1)
/)m^=JxB-Vp (8.2)
• Generalised Ohm's law
J = oo(E + u x B) - — J x B. (8.3)
n e
The electric and magnetic fields are further bound by the Maxwell equations. In these equations, viscosity and thermal conductivity are neglected.
8.2 Problems
8.2.1    Electric current density
The mean velocity of plasma u is defined as a weighted average of the mean velocities of the particular species
EPma (a A\ -ua (8.4)
33
CHAPTER 8. MACROSCOPIC EQUATIONS FOR A CONDUCTING FLUID34
where pm is the total mass density of the plasma. Each species has concentration na, charge qa and the so called diffusion velocity wa = ua — u. Calculate the total electric current density J in terms of the total electric charge density p and the particular densities, charges and diffusion velocities. Note, that due to the definition of the mean velocity of plasma, the result is not simply J = pu.
J = pu + ^2 na qaw0
8.2.2 Fully ionised plasma
From the equation for electric current density in fully ionised plasma containing electrons and one type of ions with charge e
J = ^ na qa ua = e(rii u« - ne ue) (8.5)
a
and form the equation for the mean velocity of the plasma as a whole
u = — (pmeue +pmiUi) (8.6)
Pm
derive the drift velocities Uj and ue.
1      Pmi   \ me        e J '        e      pme   \ nii        e J '     r me+rrn
8.2.3 Diffusion across the magnetic field
From the momentum conservation equation with the magnetohydrody-namic approximation
pm^=JxB-Vp (8.7)
and the generalised Ohm's law in the simplified form and without considering the Hall effect term
J = o-0(E + uxB) (8.8)
derive the equation for the fluid velocity u.
Assume E = 0 and p = const, and calculate the fluid velocity perpendicular to the magnetic field B.
Solution   The equation for u is
/)m^ = ffoExB + ff0(uxB)xB-Vp. (8.9)
Assuming E = 0 and p = const., it reduces to
Pm ^ = o-o (u x B) x B (8.10)
CHAPTER 8. MACROSCOPIC EQUATIONS FOR A CONDUCTING FLUID35
To calculate the vector (u x B) x B we define the coordinates such that the z-axis is parallel with the magnetic field. In these coordinates, the cross product is
y -
The equations for x and y component of the velocity are thus of the same form. Writing only the equation in x
(u x B) x B = (-uxB2, -uvB2, 0). (8.11)
Dux —<jqB
2
ux (8.12)
Dt pr
This has a form of a simple decay problem. The solution is
ux(t) = ux(0) exp f-^-t] . (8.13)
V      Pm J
The same holds for uy, so the time dependence of the component of the velocity perpendicular to the magnetic field u± = ^Ju2+u2 is
u±(t) = u±(0) exp (-t/r), (8.14)
where
r = ^ (8.15) is the characteristic time for diffusion across the magnetic field lines.
Chapter 9
Plasma conductivity and diffusion
9.1 Theory
In weakly ionised cold plasma, the equation of motion for electrons takes the simple form of the so called Langevin equation
me~f^~ = -e(E + ue x B) -ucmeue, (9.1)
where vc is the collision frequency for momentum transfer between the electrons and the heavy particles.
In the absence of a magnetic field, the current produced by moving electrons is
J = — ene ue (9.2)
and the DC conductivity is
and the electron mobility
o-o = (9.3)
me vc
Me = = -«L. (9.4)
me uc       ne e
When an external magnetic field is present, the plasma becomes anisotropic, and the DC conductivity and electron mobility are described by tensors (see problem 9.2.2).
In weakly ionised plasma with relatively high density of neutrals, the diffusion equation for charged species a is
^ = BVV (9.5)
36
CHAPTER 9.  PLASMA CONDUCTIVITY AND DIFFUSION
37
The diffusion coefficient De for electrons in an isotropic plasma with no internal electric fields is
De = -^k (9.6)
me vc
In the magnetised plasma, the De becomes a tensor similar to the DC conductivity or electron mobility.
In plasma, electrons usually diffuse faster than ions due to their lower mass and thus higher mobility. As a result, internal electric field is produced, slowing down the diffusion of electrons and speeding up the diffusion of ions. This effect is called ambipolar diffusion. If the relation between the ion concentration rij and ne is
nt = Cne (9.7) where C is a constant, the ambipolar diffusion coefficient Da is
k(Te + CTi) , ,
Da =-1 '        l)   , 9.8
me vce +6 mi vci
where uci, uce are the collision frequencies for momentum transfer between neutrals and ions or electrons, respectively.
9.2 Problems
9.2.1    DC plasma conductivity
From the Langevin equation for electrons in the absence of magnetic field and in the steady state
— eE — mei/cue = 0 (9.9)
derive the expression for the DC conductivity of the plasma.
Solution   The electric current density is defined as
J = —ene ue (9.10)
Inserting this into the Langevin equation (9.9), we obtain the expression for the current density J
2
nP e
The Ohm's law states
the DC conductivity is thus
■E (9.11)
mevc
J = o-0E (9.12)
CHAPTER 9.  PLASMA CONDUCTIVITY AND DIFFUSION
38
9.2.2   Mobility tensor for magnetised plasma
In magnetised plasma, the Ohm's law obtains a matrix form
		J	= S ■ E		
Jx 1	\ j	( *±	-<jh	° \	/ Ex
Jy				°	
J Z )	f \	\ o	0		\ Ez
(9.14)
where the components of the DC conductivity tensor S are
,,2
ao (9.15)
aH =   o , o9 -o (9.16)
2
nP e
^||=^o = ^—, (9.17)
where i/c is the collision frequency for momentum transfer between electrons and heavy particles and 0,ce is the electron cyclotron frequency due to the external magnetic field. Find the components of the mobility tensor Me defined as
ue = AVE. (9.18)
Results:
j	/ M±	-MH	0
Me =	MH	M±	0
1	0	0	M
M» = -^JfnS) (9-20)
M\\=--— (9.21)
9.2.3   Ohm's law with magnetic field
Consider the equation J = S ■ E as in the preceding problem. Suppose
that E = (E±, 0, E\\) and B = (0, 0, Bq). Calculate J. Note what direction of the electric current is governed by what component of S. [3 = {a±E±, aHE±, a\\ E\\)]
CHAPTER 9. PLASMA CONDUCTIVITY AND DIFFUSION 39
9.2.4   Diffusion equation
Solve the diffusion equation for one spatial dimension
9n{x, t)     n d2n{x,t)
—d^ = D—dx^- (9'22) by separation of variables, assuming
n(x,t) = S(x)T(t). (9.23)
Results:
• Tk{t)=T0 exp(-DkH)
• S(x) = c(k) exp(i kx), k is a separation constant
n
(x,t) = J c(k) exp(—i k x - D k2 t) dk
Chapter 10
Some basic plasma phenomena
10.1 Theory
In a paper from 1923, an American chemist and physicist wrote, that electrons are repelaed from negative electrode, whereas positive ions are attracted towards it. Langmuir concluded, that around every negative electrode, a sheath of defined thickness containing only positive ions and neutral atoms exists. Morever, Langmuir observed, that also the glass wall of the discharge chamber is negatively charged and repels (or reflects) almost all electrons [16].
The fact, that insulated objects inside plasma are negatively charged (in respect to plasma) to floating potential, is caused by higher mobility of electrons than ions. The thermal velocity of electrons {k-QTe/me)1/2 is at least 100 times higher than the thermal velocity of ions (k-QTi/Mi)1/2 [17]. The first reason for different mobility is higher mass of ions. If we consider only proton (the lightest ion that can appear in a plasma), than the ratio between the mass of proton and electron mp/me is 1836. This ratio corresponds approximately to the ratio of masses of heavy bowling ball (5 kg) and ping pong ball (2,7 g). Another reason for higher thermal velocity of electrons in low-temperature plasma is their higher temperature in respect to the ions.
The slowest possible speed of ions at the sheath edge is called the Bohm speed «b- lonts are accelerated to this speed in a quasi-neutral pre-sheath, where small electric field exists. The Bohm criterion of plasma sheath is described by following equation
40
CHAPTER 10. SOME BASIC PLASMA PHENOMENA
41
10.2 Problems
10.2.1   Waves in non-magnetized plasma
Plasma of so called E layer of Earth's ionosphere has electron density approximately 105 cm~3 and is at altitude of approximatelyo 100 km.
a) Which electromagnetic waves can be reflected from this layer?
b) Calculate the dielectric constant of plasma for the waves with frequencies of 100 Mhz and 1000 Hz.
b) Calculate the skin depth of the wave with frequency of 1000 Hz. Solution:
a) All electromagnetic waves with frequency lower than the plasma frequency (2 839 725 Hz) will be reflected.
b) The dielectric constant of plasma is defined as
For 100MHz e = 0.9991 (positive value, elmag. waves propagate), for 1000 Hz e = —8064037 (negative value, imaginary refraction index, reflection).
c) Skin depth 5 appropriately equals to c/^Jujp —uj2, where c is speed of light. The skin depth for 1000 Hz is 16.8m.
10.2.2 Floating potential
Explain why insulated object inserted to plasma will aquire a negative potential with respect to the plasma itself.
10.2.3 Bohm velocity
Calculate the Bohm velocity for hydrogen ion in plasma with electron temperature of Te = 1 eV. [9 787.2 m/s]
10.2.4 Plasma frequency
When the macroscopic neutrality of plasma is instantaneously perturbed by external means, the electrons react in such a way as to give rise to oscillations at the electron plasma frequency. Consider these oscillations, but include also the motion of ions. Derive the natural frequency of oscillation of the net charge density in this case. Use the linearized equations of continuity and of momentum for each species, and Poisson equation, considering only
CHAPTER 10. SOME BASIC PLASMA PHENOMENA
the electric force due to the internal charge separation. [u> = (u,*+u>*)1'2, where ,1 = /g]
Chapter 11
Boltzmann and Fokker-Planck collision terms
11.1 Theory
Under several simplifying assumptions (mainly homogeneous and isotropic distribution function of electronic velocities, molecular chaos, considering only binary collisions and ignoring external forces), so called Boltzmann collision integral can be derived
g = |v — vi| is the relative speed of the electron and its collision partner, a is the differential cross section for this type of collisions, depending on the solid angle $7. Two types of distribution functions are considered here - the electronic distribution function /e(v) and that of the particular kind of collision partners /i(vi). If more kinds of collision partners should be considered, the collision term is expressed as a sum of terms similar to eq.
The first term expresses the amount of electrons with initial velocity v' that undergo collisions with the collision partner with velocity vj. After this collision, the electrons have velocity v and their collision partners have velocity vi, i.e. they add to the electronic distribution function at the velocity v. The second term expresses an inverse collision, which leads to loss of particles of the velocity v and is thus negative.
If only collisions leading to small-angle deflections are considered, as expected for long-range Coulomb interactions, the Fokker-Planck collision term can be derived
(11.1)
(11.1).
(^)coii = -E^(^(A^) + i E^(/«(AÄ), (11.2)
43
CHAPTER 11. BOLTZMANN AND FOKKER-PLANCK COLLISION TERMSU
where
Avi g a(£l)d£lfßl d3Vl
n Jvi
(AviAvj)
j/av
AviAvj g a(ü)düfßl d3vx
(11.3)
(11.4)
are the coefficients of dynamical friction and diffusion in velocity space, respectively.
11.2 Problems
11.2.1    Collisions for Maxwell-Boltzmann distribution function
Consider a plasma in which the electrons and the ions are characterised, respectively, by the following distribution functions
fe = n0
fi = n0
mP
2-KkTe
2-KkT
3/2
exp
3/2
exp
me[y - ue 2 kTe
rrn v - u,;
2kT,}
(11.5)
(11.6)
(a) Calculate the difference (/e(v') fi(vi) - /e(v) /j(vi)).
(b) Show that this plasma of electrons and ions will be in the equilibrium state, that is, the difference (/e(v')/«(vi) — /e(v)/«(vi)) will vanish if and only if ue = Uj and Te = Tj.
Solution
(/e(v/)/,(v/1)-/e(v)/,(v1))=n2
2-wk
3 / \ 3/2
TeTr
mP v
x I exp
exp
Ue)2      Vfli {v[
u,
2kTP
2kT;
me (v - ue)2     mi (vi - Uj)2
2kTP
2kT
(11.7)
(b) For the difference to vanish, the term in the parentheses must equal zero. This will happen if the arguments of the exponentials will be equal. Let us rewrite the arguments, omitting the factor —(2 A;)-1:
m,
'-(v'2 -2V ■ue + u2p) +
m. Ti
1 („.12
2 v[ ■ Uj + uj
Hi
CHAPTER 11. BOLTZMANN AND FOKKER-PLANCK COLLISION TERMS45
^ (v2 - 2 v • ue + ul) + ^(vj - 2 vx • u, + u2) (11.9)
From the derivation of the Boltzmann collision term follows, that the pairs of velocities v, vi and v', v'x can be considered as pairs of velocities before and after an elastic two-body collision. Thus, they are bound by the conservation laws:
me v2 + rrij v?     me v'2 + rrij v'?
_e-t-i_i = _e-t-(11.10)
me v + rrii vi = me v' + rrii v'x (11.11)
It is now obvious from the last four equations, that the collision term will vanish if and only if Te = T{ and ue = Uj. In other words, the distribution function fe will be changed by the collisions only if the plasma is out of equilibrium - the collisions tend to bring the plasma to the state of equilibrium.
11.2.2    Collisions for different distributions
Recalculate the task (a) of the preceding problem with Druyvesteyn-like distribution function for electrons and Maxwell-Boltzmann-like distribution for ions (Ce, ae and C% are constants)
fe = Ce exp[-ae m2e (v - ue)4] (11.12)
^2"
fi = Ci exp
rrii v-i - u,:
2kTi
Will the difference (/e(v') fi(v[) - /e(v) /j(vi)) be zero for ue = Uj?
11.13)
11.2.3   Collisions for Druyvesteyn distribution
Recalculate task (a) of the first problem for Druyvesteyn-like distribution for both electron and ion velocities (see eq. (11.12)). Can the collision term be equal zero for ue = Uj? Is it possible to find equilibrium state of plasma described by the Boltzmann kinetic equation with Boltzmann collision term in form of Druyvesteyn-like distribution?
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