Advanced plasma physics

By Marino Dobrowolny

Il libro copre argomenti fondamentali nella fisica dei plasmi quali onde nei plasmi, instabilità e processi non lineari. Dopo una approfondita descrizione teorica vengono anche evidenziate osservazioni da satelliti di questi fenomeni in plasmi spaziali quali la ionosfera terrestre, la magnetosfera e il vento solare.

L’ultima parte del libro tratta le interazioni di corpi carichi con un plasma. Nel caso di misure da satellite abbiamo a che fare con un campo magnetico esterno e col moto del satellite nel plasma, Non essendoci una teoria esatta in queste circostante vengono usati metodi approssimati che tuttavia conducono a delle stime credibili della corrente raccolta. Anche per questi argomenti le teorie presentate sono raffrontate a misure di satellite e in particolare ai risultati della missione TSS (Tethered Satellite System).

• Language: English
• Publisher: Youcanprint
• Release date: May 1, 2019
• ISBN: 9788831614344

Book preview

theories.

Chapter 1

General properties of plasmas

1.1 Plasma as the fourth state of matter

We use the word plasma to describe a wide range of neutral substances which contain however free electrons and ionized atoms interacting through long range Coulomb forces.

Plasmas can be viewed as the fourth state of matter the other three being the solid, liquid and gaseous state. Solids, liquids and gases differ from one another because of the different strengths of the forces that keep together the constituent particles. These are very strong in solids, much weaker in liquids and almost absent in a gaseous state.

A substance is in one or another of these states according to its temperature. This determines the random kinetic energy of the constituents and the state is obtained by the equilibrium between the thermal energy and the binding forces between particles. If we heat a solid or a liquid, its atoms or molecules acquire more kinetic energy until that overcomes the potential energy which binds the particle constituents. At this point we obtain what is called a phase transition between solid and liquid state or liquid and gaseous state.

If we have a gas of molecules and continue to heat it, the molecules will dissociate into atoms, and, increasing the heating, the atoms will expel through collisions the outer electrons so that we end up with an ionized gas or a plasma. The transition from the gaseous to the plasma state is however obtained through a continuous process and is not a phase transition.

In the laboratory a plasma can be generated through the processes of photoionization or electrical discharge. In the photoionization one uses incident photons with energy greater than the ionization potential of the gas atoms. For example, the ionization potential energy for the external electrons of an Oxygen atom is 13.6 eV which can be obtained by ultraviolet radiation. The Earth’s ionosphere is a natural photo ionized plasma. On the other hand in a gas discharge we apply an electric field across the gas. If the gas is slightly ionized, the electric field will accelerate the free electrons to an energy sufficient to ionize neutral atoms.

We might think that the plasma state is a rare state of matter but just the opposite is true as in fact almost all the matter in the Universe is in the plasma state.

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between the electron density and the neutral density. This varies according to the altitude in the ionosphere.

1.2 Collective effects

In a plasma the basic interactions between the charged particles are electromagnetic. Due to the long range of the electromagnetic forces, each charged particle in a plasma interacts simultaneously with a large number of other charged particles resulting in important collective effects. It is this feature, namely the existence of collective effects, that makes a plasma different from an ordinary fluid.

A charged particle has its own electric field and it is this which acts on other charged particles following Coulomb law. Furthermore, if a charged particle is moving, it has also an associated magnetic field which in turn produces forces on other charges.

We can indeed distinguish between weakly and strongly ionized plasmas on the basis of the possible interactions taking place. A weakly ionized plasma, i. e. a plasma which contains many neutrals, is characterized mainly by charge neutral interactions which occur (contrary to the Coulomb interactions) over distances of the order of the atoms diameter. On the contrary in a strongly or fully ionized plasma the multiple Coulomb interactions are dominant and they have a long range. It is mainly to these plasmas we will refer to in this book.

The dominance of collective effects leads us to a more quantitative definition of a plasma. This is obtained by recalling that a plasma is macroscopically neutral. This means that, in the absence of external fields, in a volume of the plasma sufficiently large as to contain many particles but sufficiently small compared to the lengths of variation of physical parameters such as density and temperature, the net electric charge is zero.

The neutrality is the result of a balance between the particle thermal energy, which tends to disturb charge neutrality, and the electrostatic energy which arises due to charge separation and tends to restore neutrality. The balance, as we will see, is obtained over a distance called the Debye length which is an important parameter of the plasma. This distance is given by

are electron temperature and density. The Debye length corresponds to the distance within which the field of an individual particle is felt by other charged particles inside the plasma. In other words, as it will be shown later, the Debye length is a characteristic distance over which the charges in a plasma collect around a given charge so as to screen its electric field.

A first criterion for the definition of a plasma, and therefore for the existence of collective effects, is given by

where L is a characteristic dimension for the variation of plasma parameters such as density and temperature.

Since the shielding effect is a consequence of the collective effects inside a Debye sphere, a second criterion is obtained by requiring the number of electrons in that sphere to be high, i. e.

1.3. Some fundamental plasma processes

Plasma dynamics can be investigated in a number of ways. We will go into that in Chapter 3 devoted to the possible levels of description. Here, on the other hand, we want to address a few fundamental plasma phenomena which can be described quite simply.

1.3a Electron plasma oscillations

The first phenomenon we consider is that of electron plasma oscillations. We have already said that, in equilibrium, a plasma tends to maintain a state of macroscopic neutrality. If, for some reason, we attempt to disturb this neutrality, the plasma reacts through electron oscillations which, on the average, maintain the electrical neutrality.

Suppose that a perturbation, in the form of a small negative charge is introduced in a small spherical region. The corresponding electric field will be radial and pointing towards the center of the region. Because of the electric field electrons will be moving radially outwards and, after some time, more electrons will leave the region than is necessary to maintain electrical neutrality. As a consequence an excess of positive charge is established within the region. Then the electrons will start moving radially inward.

This movement of electrons alternatively outwards and inwards is what constitutes an electron plasma oscillation. The total charge in the spherical region averaged over one period of these oscillations will be zero and it is in this way that electrical neutrality is maintained on the average. Notice that we have not talked about ions so far but this is legitimate as the frequency of the electron oscillations, as we will see, is high so that the motion of the ions which are are more massive than the electrons can in fact be neglected.

We will now describe mathematically these oscillations and, for the sake of simplicity, we will neglect electron thermal motions and electron pressure, We superimpose on the electron density a small perturbation by writing

with

are small perturbations so that we can use linearized equations. Regarding the electrons as a fluid, the linearized continuity and momentum equations for the electrons can be written as

(1.1)

(1.2)

Considering now singly charged ions, the overall charge density is

Therefore

(1.3)

we obtain

from eq.(1.3)

(1.4)

where

(1.5)

is called the electron plasma frequency. For the solution of eq. (1.4) we can write

varies harmonically in time with the plasma frequency. The same can be proved for all the other perturbed quantities. Furthermore for all the perturbations there is no change in phase from point to point which means that we do not have wave propagation or, in other words, the oscillations are stationary.

we obtain

The current density is given by

and from eq. (1.2) we obtain

equation we arrive at

having defined

so that the above equation reduces to

(1.6)

As the curl on any gradient function vanishes identically, we can then write that

(1.7)

where ψ is a magnetic scalar potential. Substituting (1.6) in (1.7) we obtain

However the only solution of this equation which is finite at infinity is

so that we have that

i.e. there is no magnetic field associated with the space charge oscillations.

In summary the electron plasma oscillations are stationary and electrostatic. They were first discovered by Tonsk and Langmuir [1] and are also called Langmuir oscillations. Notice that, if we include pressure gradient forces, the oscillations become propagating waves. We will see that in Chapter 4 where we will recover Langmuir waves from a kinetic treatment.

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1.3b Debye shielding

Debye shielding is another fundamental plasma phenomenon which can be explained quite simply. Let us focus on a given charge in the plasma. We will call it a test charge and suppose it is positive and amounts to +Q. Consider now a spherical coordinate system with its center at the position of the test charge. We want to determine the electrostatic potential which is established by this charge due to the combined effect of the test charge itself and of all the other charges of the plasma which are surrounding it.

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We can take maxwellian distributions for ions and electrons

having assumed that the electrons and the ions have the same temperature. The total charge density will then be given by

where δ is the Dirac delta function. Thus

From Maxwell equation

we arrive then at the differential equation

Suppose now that

Then the equation simplifies to

is the Debye length which we have already introduced. Since the problem has spherical symmetry, the electrostatic potential will depend from r only (and not the orientation of r). Therefore the equation for the potential can be rewritten as

(1.8)

To solve completely the problem let us first consider the case of an isolated charge +Q in free space. Then the electric field due to this charge will be directed radially outwards and will be given by

Consequently the electrostatic potential of this isolated charge will be given by

. Therefore it is appropriate to seek the solution of eq. (1.8) in the form

where the function F(r) must be such that F(r)→1 when r→0 . Furthermore Φ(r) will have to vanish at infinity. The differential equation that we obtain for F(r) is

and has the general solution

As Φ(r) must vanish for large r, we must have A = 0 and, because F(r)→1 when r→0, we must have B = 1. In all we obtain for the potential

(1.9)

. (see Figure 1.1). Hence we can say, at least approximately, that a charged test particle in a plasma interacts effectively only with particles situated less than one Debye distance away and, therefore, the same test charge has scarce influence on particles at distances larger than one Debye length. Furthermore it can be shown that the test charge is neutralized by the charge distribution which surrounds it.

in the solar wind.

Figure 1.1: Debye potential (solid line) versus Coulomb potential (dashed line)

1.4 Collisions in plasmas

Collisions in a plasma are quite different from those occurring in a neutral gas. This is due to the fact that the electrostatic forces between charged particles have a much longer range than the forces between neutral atoms. As a consequence, rather that close collisions modifying substantially the particle velocities, we have to consider distant encounters which, taken singly, produce only a very small effect. The cumulative effects of many of these collisions do lead however statistically to large deflection angles for the colliding particle and it is in relation to these that one introduces a collision frequency.

the collision frequency for particles of species s with particles of species s’, the total collision frequency for species s will be given by

for the ions by

Notice that these collision frequencies measure the frequency with which a particle trajectory undergoes a major angular change due to Coulomb interactions with the other particles. Therefore, as single collisions imply small angular deflections, the collision frequency is not the inverse of the typical time between collisions. Rather it is the inverse of the typical time needed for enough collisions to occur that the particle trajectory is deviated by 90o.

Spitzer [3] has derived for the electron collision frequency

where

for the solar wind.

It is clear that for high temperature diffuse plasmas the collision frequency will be small while the opposite is true for dense and low temperature plasmas. It turns out that most astrophysical plasmas fall in the first category. In particular this applies to the near Earth and solar wind plasmas which are the plasmas whose properties can be analyzed by means of measurements by satellites.

To decide if we can treat a plasma as collisionless or not (with respect to ordinary Coulomb collisions), we have to compare the above collision frequency with other typical frequencies which appear in the analysis of the various waves that a plasma can sustain (see Chapters 4, 5 and 6).

One of these frequencies that we have already encountered is the electron plasma frequency

Then we have the analogous frequency for the ions

In case the plasma has a magnetic field as it is the case for astrophysical plasmas, we further have the electron cyclotron frequency

which is the frequency of the circular motion of an electron around the magnetic field B, And obviously we have the analogous frequency for the ions

It turns out that for the space plasmas we mentioned before the electron collision frequency is much smaller than all the typical frequencies we have listed and, hence smaller than the frequencies of the oscillations that the plasma can sustain. Alternatively we can compare the mean free path for particle