Chapter 1

Introduction

1.1 Plasma: definitions

A plasma is a quasi-neutral gas of charged and neutral particles whichexhibits collective behaviour.

An equivalent, alternative definition:

A plasma is a globally neutral, (partially or totally) ionized gas, withcharacteristic dimension larger than the Debye length and with intrin-sic oscillation frequency (plasma frequency) greater than the collisionfrequency between individual plasma particles.

Ionization, Debye length (characterizing collective behaviour), and plasmafrequency are discussed in this Chapter. Collision frequency is dealt with inChapter 3.

We will also define ideal and non-ideal plasmas, classical and quantum plas-mas.

However, in this course, we shall almost exclusively treat fully ionized, ideal,classical plasmas.

Plasmas are very frequent in nature, since most stars consists of plasmas,and the interstellar medium is a plasma, too. Other natural plasmas are thesolar wind, and the ionosphere. Close to earth, natural plasmas are found rarely:a notable exception are lightning phenomena.

Plasmas are also produced by man, and have many applications. Theseinclude arc discharges, plasma screens, plasmas for material processing, forplasma-enhanced chemical vapour deposition, for medical treatment, and forspace propulsion. A large research e↵ort is devoted to plasmas for controllednuclear fusion, either using magnetic or inertial confinement.

1

2 CHAPTER 1. INTRODUCTION

1.2 Ionization. Saha equation for ionization equi-librium

Ionization, i.e. extraction of an electron from an atom, requires an energy atleast equal to the ionization energy of the electron. Such an energy can beprovided in an impact of the atom (with another atom or an ion or an electron),or by a photon, causing photoelectric e↵ect.

At thermodynamic equilibrium, ionization and recombination balance eachother, and the ionization degree of a given gas only depends on temperatureand density. We do not need to know the details (e.g. the cross-sections) of theindividual processes of ionization and recombination.

Here we only consider the case of hydrogen ionization. This is the simplestcase, and is also the most important both for fusion plasmas and astrophysicalplasmas. We consider a hydrogen plasma with atom number density n

0

, attemperature T . We call n

n

the density of non-ionized atoms (ions), ni

= n

0

�n

n

the density of ionized atoms. Of course, the electron density n

e

is, in this case,equal to the ion density. The densities of ionized and neutral atoms satisfy Sahaequation, which (for Hydrogen) reads

n

e

n

i

n

n

=

✓2⇡mk

B

T

h

2

◆3/2

exp(��/k

B

T ), (1.1)

with m the electron mass, kB

Boltzmann constant and � = 13.6 eV the ioniza-tion energy.

We are interested in computing the ionization degree, i.e. the ratio of theionized atoms to the total number of (ionized and non-ionized) atoms, i.e.

⌘ =n

i

n

0

=n

e

n

0

.

We can then rewrite Saha equation in the form

⌘

2

1� ⌘

= f(T, n0

) with f =1

n

0

✓2⇡mk

B

T

h

2

◆3/2

exp(��/k

B

T ) (1.2)

Let us see how the ionization degree changes with temperature and density.The dominating factor is the exponential. Ionization is vanishingly small ask

B

T ⌧ �, and approaches unity when k

B

T/� ⇠ 1. More quantitatively, thesolution of Eq. (1.2) is

⌘ =1

2

⇣�f +

pf

2 + 4f⌘.

For small and large f , we have, respectively,pf, f ⌧ 1;

⌘ =

1� 1

f

, f � 1.

1.2. SAHA EQUATION 3

Figure 1.1: Hydrogen degree of ionization vs temperature for di↵erent values ofthe density.

We find that the plasma can be considered nearly completely ionized (⌘ = 95%),when f = 18, or

T (eV) =13.6

66.2� ln(n0

) + (3/2) ln[T (eV)], (1.3)

with n

0

in units of m�3. 1 This equation (where the term with the tempera-ture is usually much smaller than the algebraic sum of the other terms in thedenominator) clearly shows that ionization is nearly complete at temperaturesof 0.5 –5 eV, depending on the density (see also Fig. 1.1). In our course we willonly consider plasmas at temperatures of hundred or thousands of eV. At suchtemperatures hydrogen plasmas are fully ionized.

1In plasma physics temperatures are often measured in units of eV. A temperature of 1 eVactually is the temperature corresponding to energy kBT = 1 eV. Therefore

T (eV) = T (K)1.38⇥ 1023

1.6⇥ 1019=

T (K)

11600.

4 CHAPTER 1. INTRODUCTION

1.3 Collective behaviour, Debye shielding, quasi-neutrality

A plasma is globally neutral. In this section we study how a globally neutralplasma reacts when a local charge unbalance is introduced, and whether localcharge unbalance is possible. It will turn out that plasmas act to shield exter-nally inserted charges, and - on small scales - allows some charge unbalance. Itcan then be considered quasi-neutral.

1.3.1 Debye shielding and Debye length

We consider a homogeneous hydrogen plasma. A charge q is introduced inthe plasma at a point we take as the origin of a spherical coordinate system.Combining the first Maxwell equation

r ·E =⇢

"0

and the definition of electric potential �

E = �r�,

we write Poisson equation

r2�+⇢

"0

= 0, (1.4)

that relates potential and electric charge density ⇢. The latter is the sum of thecharge due to electrons, ions and the added charge q:

⇢ = ni

e+ ne

(�e) = (ni

� ne

)e+ q�(r),

where � is Dirac’s delta function. We then have to solve the equation

r2�+1

"0

(ni

� ne

)e, r > 0. (1.5)

In proximity of the origin (where the charge q is) the potential must approachthat of a single charge in vacuum, i.e.

limr!0

�(r) =q

4⇡"0

r. (1.6)

At thermodynamic equilibrium particle density should follow Boltzmann distri-bution

n↵

= n0

exp(�U↵

/kB

T ),

with ↵ = e,i, and U = q� the electrostatic energy and n0

the unperturbeddensity. Applying this general relation to plasma electrons and ions, we have,respectively,

ne

= n0

exp(e�/kB

T ), (1.7)

ni

= n0

exp(�e�/kB

T ). (1.8)

1.3. DEBYE SHIELDING AND QUASI NEUTRALITY 5

We shall show in the next section that in a plasma (actually, in an ideal plasma)U ⌧ k

B

T . This allows us to approximate the previous expressions as

ne

= n0

✓1 +

e�

kB

Te

◆, (1.9)

ni

= n0

✓1� e�

kB

Ti

◆. (1.10)

where we allow di↵erent temperatures for electrons and ions. Poisson equation(1.5) then becomes

r2�� n0

e2

"0

✓1

kB

Ti

+1

kB

Te

◆� = 0. (1.11)

Expressing the Laplacian in spherical coordinates and exploiting spherical sym-metry, we can finally write

1

r2@

@r

✓r2

@�

@r

◆� 1

�2

d

� = 0, (1.12)

where we have introduced the Debye length �d

, such that

1

�2

d

=n0

e2

"0

✓1

kB

Ti

+1

kB

Te

◆, (1.13)

and

�d

=

r"0

kB

Te

n0

e2Ti

Ti

+ Te

. (1.14)

The solution of Poisson equation (1.11), with the condition at the origin (1.6)is

� =q

4⇡"0

rexp(�r/�

d

). (1.15)

This is the potential of a charge in vacuum, attenuated (shielded) by the ex-ponential factor exp(�r/�

d

) (see Fig. 1.2). The perturbation caused by thecharge introduced in the plasma therefore vanishes at a distance of a few Debyelengths. This is the result of the collective behaviour of the plasma.

For order of magnitude estimates, the factor Ti

/(Ti

+Te

) can be simply takenequal to 1, and then

�d

'r

"0

kB

Te

n0

e2' 7430

sT (eV)

n(m�3)m. (1.16)

Values of the Debye length for a few characteristic plasmas are listed in Table 1.1.We see that in all cases the plasma dimension is much larger than the Debyelength.

6 CHAPTER 1. INTRODUCTION

Figure 1.2: Bare and shielded Coulomb potentials.

n⇥m�3

⇤T [eV] �

d

[m] size [m]Magnetic Con-finement fusionplasmas

1020 104 7.5⇥10�5 2

Inertial Con-finement FusionPlasma

1032 104 7⇥ 10�11 10�4

Ionosphere 1012 0.1 2⇥ 10�3 1000Solar Chromo-sphere

1018 2 5⇥ 10�6

Table 1.1: Debye length and characteristic linear dimensions of typical plasmas.

1.3. DEBYE SHIELDING AND QUASI NEUTRALITY 7

1.3.2 Plasma parameter. Ideal and non-ideal plasmas

In the above discussion on plasma screening we have implicitly assumed that ina volume of linear dimensions of the order of the Debye length there are manyparticles (otherwise there could not be any shielding), i.e.

Nd

= n�3

d

� 1. (1.17)

The quantity Nd

is called plasma parameter.2 Using the expression of Debyelength we can write

Nd

=

✓"0

kB

T

e2

◆3/2

n�1/2 = 4.1⇥ 1011[T (eV)]3/2

[n(m�3]1/2. (1.18)

Hence the condition Nd

� 1, i.e the condition of ideal plasma becomes

n(m�3) ⌧ 1.7⇥ 1023[T (eV)]3. (1.19)

We now compare average electrostatic energy and average electron kineticenergy in an ideal plasma:

Ues

Uth

' e2

4⇡"0

d

1

(3/2)kB

T, (1.20)

where d is the average interparticle distance, which we estimate as d ' n�1/3.Equation (1.20) can then be written as

Ues

Uth

' e2n1/3

6⇡"0

kB

T=

1

6⇡

1

N3/2

d

. (1.21)

This proofs that the average electrostatic energy is much smaller than the ther-mal energy in an ideal plasma.

1.3.3 Quasi-neutrality

A plasma is globally neutral. On smaller spatial scales (and on short temporalscales) however charges of opposite signs do not necessarily balance. To deter-mine the scale over which charge unbalance can occur we consider the simplecase of a small spherical region, of radius L with a net electric charge in anotherwise neutral and homogeneous plasma (see Fig. 1.3). In this small spherethe electron density is lower than the ion density, n

e

= ni

� n = n0

� n. UsingGauss theorem we find the electric field 3 and, in particular its maximum value

Emax

= E(L) =en

3"0

L. (1.24)

2Some authors define a plasma parameter � = 1/Nd

3Gauss theorem: Z

S

E · ndS =

1

"0

Z

V

⇢dV,

with charge density ⇢ = e(ni � ne) = en inside a sphere of radius L. We obtain

8 CHAPTER 1. INTRODUCTION

Figure 1.3: Charged spherical volume inside a neutral plasma.

We then obtain the maximum electrostatic energy density,

Umax

= "0

E2

max

2= U(L) =

n2e2

18"0

L2. (1.25)

It follows that

✓n

n0

◆2

=18"

0

e2L2

Umax

="0

kB

T

n0

e218U

max

L2n0

kB

T=

✓�d

L

◆2

27Umax

Uth

. (1.26)

Since (see previous subsection) the electrostatic energy is always much smallerthan thermal energy in an ideal plasma, we finally obtain that the relativecharge unbalance is always very small for L � �

d

, while it can be quite largefor distances L smaller than the Debye length.

1.4 Classical vs quantum plasma

As a simple rule of thumb, we are allowed to use classical physics (insteadof quantum physics) when interparticle distances d ' n�1/3 are substantiallylarger than the reduced de Broglie wavelength �� = h/p, where p is particlemomentum. For (nonrelativistic) plasma electrons

�� ' h

me

vte

' hp2m

e

kB

Te

, (1.27)

E(r) =

en

3"0r L L, (1.22)

E(r) =

en

3"0

L2

r3L � L. (1.23)

1.5. PLASMA OSCILLATIONS AND PLASMA FREQUENCY 9

where vte

=pkB

Te

/2me

is a characteristic electron thermal velocity. It followsthat a plasma behaves classically if

kB

Te

� h2n2/3

e

2me

. (1.28)

or, inserting the values of the constants,

Te

(eV) � 3.8⇥ 10�20n2/3

e

, (1.29)

with the electron density in units of m�3. It turns out that most plasmas areclassical. Some quantum e↵ects occur in strongly compressed inertial confine-ment fusion plasma, and in the plasmas of the stellar cores.

1.5 Plasma oscillations and plasma frequency

We now study the temporal response of a plasma to small local charge perturba-tions. We consider a simple one-dimensional plasma. We can assume that, dueto their larger mass, ion are always at rest, while an electron layer is displacedfrom its initial position (see Fig. 1.4). Let us call d the displacement, and n

0

the unperturbed charge density.

Figure 1.4: Model for the study of plasma oscillations. Charge density distri-bution and electric field distribution

10 CHAPTER 1. INTRODUCTION

Following this charge displacement, a double layer is generated, with surfacecharge density � = n

0

ed and areal mass n0

me

d; the electric field distributionshown in the Figure. Such an electric fields acts as a spring on the electronlayer. According to Newton’s second law we then have

n0

me

dd = �n0

edn0

ed

"0

, (1.30)

or

dd+ edn0

e2

me

"0

d = 0. (1.31)

Equation 1.31 is the equation of a harmonic oscillator. The electrons oscillatewith angular frequency4

!p

=

sn0

e2

me

"0

(1.32)

and frequency (oscillations per unit time)

fp

=!p

2⇡=

1

2⇡

sn0

e2

me

"0

. (1.33)

Note that the plasma frequency only depends on plasma density. Numerically

fp

=1

2⇡

1.6⇥ 10�19

p8.9⇥ 10�12 · 9.1⇥ 10�31

pn[m�3] = 8.98

pn[m�3] Hz. (1.34)

Plasma frequency of a few important plasmas are listed in the table below.

n⇥m�3

⇤fp

[Hz]Magnetic fusionplasma

1020 9 · 1010

Inertial fusionplasma

1028 1015

Ionosphere 1012 107

We shall see later in this course that electromagnetic waves can only propagatein a plasma if their frequency is higher than the plasma frequency.

1.6 Plasma oscillations from a fluid model

Plasma oscillations can also be studied using a cold fluid model and linearperturbation theory. Again, we consider the ions at rest, and also assume thatelectrons only move due electric fields, i.e. we neglect electron thermal mo-tion (hence the name cold fluid model). The motion of this electron fluid is

4often simply called plasma frequency, omitting the adjective angular.

1.6. PLASMA OSCILLATIONS FROM A FLUID MODEL 11

then described by momentum and mass conservation equations, and by the firstMaxwell equation

mndv

dt+ (v ·r)v = �enE (1.35)

@n

@t+r (n · v) = 0 (1.36)

r ·E =e (n

i

� n)

"0

(1.37)

where m is the electron mass, n the electron density, v the electron velocity, Ethe electric field. For simplicity, we consider a simple one-dimensional system,so that Eqs. (1.35)–(1.37) become

mndv

dt+ v

dv

dx= �enE (1.38)

@n

@t+

d

dx(nv) = 0 (1.39)

dE

dx=

e (ni

� n)

"0

(1.40)

We now use linear perturbation theory. We write each quantity f as thesum of its equilibrium value f

0

and a perturbation: f1

, i.e. f = f0

+ f1

. Weassume that this perturbation is so small that the product of two perturbationscan be neglected, being a second order term. In addition, the assumption ofsmall perturbations allows us to use the superposition principle, i.e., to describethe perturbed fluid motion as the sum of linearly independent Fourier modes ofthe form

f1

= f1

· ei(kx�!t), (1.41)

where i is the imaginary unit, k the wave number and ! the angular frequency.We can then consider individual modes separately. In addition, the use of theexponential notation allows us to formally replace the derivatives with multipli-cations:

@

@t= �i!, (1.42)

@

@x= ik. (1.43)

In the present case the equilibrium values are

v0

= 0 (1.44)

@

@t(. . .)

0

= 0 (1.45)

@n0

@x= 0 (1.46)

E0

= 0 (1.47)

12 CHAPTER 1. INTRODUCTION

In addition, the ion density is constant and equal to the equilibrium electrondensity, n

i

= n0

. We start by considering momentum equation (1.38), whichbecomes

m (n0

+ n1

)@

@t(v

0

+ v1

) + (v0

+ v1

)@

@x(v

0

+ v1

) = �e (n0

+ n1

) (E0

+ E1

) ,

and then

mn0

@v1

@t+mn

1

@v1

@t| {z }+ v

1

@v1

@x| {z }= �e

0

@n0

E1

+ n1

E1| {z }

1

A .

The terms indicated by braces can be neglected, being of second order. Next,we consider modes of the form (1.41), and we obtain

mn0

v1

(�i!) = �en0

E1

. (1.48)

Analogously, continuity equation and first Maxwell equation become, respec-tively,

�i!n1

+ ikn0

v1

= 0 (1.49)

andikE

1

= � e

"0

n1

. (1.50)

We can now easily solve the system of three linear equations. From Eq. (1.49)we obtain the perturbed density

n1

=kn

0

v1

!,

which we substitute into Eq. (1.50)

ikE1

= � e

"0

kn0

v1

!,

to obtain the perturbed electric field as a function of the perturbed velocity:

E1

= � e

i"0

n0

!v1

.

Finally we substitute this last expression into Eq. (1.48):

�i!mn0

v1

� en0

i

e

"0

n0

!v1

= 0

By dividing both members by the perturbed velocity v1

, and rearranging terms,we finally have

!2 =n0

e2

m"0

,

which is just the result obtained with the simple double layer model. As in anylinear perturbation theory, the amplitude of the perturbations do not appear inthe final relation. Also, notice, that we assumed modes in the form of (prop-agating) waves [see Eq. (1.41)]; we instead do not find propagating waves, butoscillations (the wavenumber k does not appear in the final dispersion relation).

1.7. PLASMAS IN DENSITY-TEMPERATURE SPACE 13

1.7 Natural and man-made plasmas in density-temperature space

To conclude this introductory chapter in Fig. 1.5 we show the position a fewimportant plasmas in the density-temperature plane. We observe that most ofthese plasmas are fully ionized, classical, ideal plasmas.

Figure 1.5: Natural and artificial hydrogen plasmas in the temperature-densityplane.