Introduction. Chapter Plasma: definitions

Transcription

1 Chapter 1 Introduction 1.1 Plasma: definitions A plasma is a quasi-neutral gas of charged and neutral particles which exhibits collective behaviour. An equivalent, alternative definition: A plasma is a globally neutral, (partially or totally) ionized gas, with characteristic dimension larger than the Debye length and with intrinsic oscillation frequency (plasma frequency) greater than the collision frequency between individual plasma particles. Ionization, Debye length (characterizing collective behaviour), and plasma frequency are discussed in this Chapter. Collision frequency is dealt with in Chapter??. We will also define ideal and non-ideal plasmas, classical and quantum plasmas. 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 the solar 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. These include arc discharges, plasma screens, plasmas for material processing, for plasma-enhanced chemical vapour deposition, for medical treatment, and for space propulsion. A large research e ort is devoted to plasmas for controlled nuclear fusion, either using magnetic or inertial confinement. 1

2 2 CHAPTER 1. INTRODUCTION 1.2 Ionization. Saha equation for ionization equilibrium Ionization, i.e. extraction of an electron from an atom, requires an energy at least equal to the ionization energy of the electron. Such an energy can be provided 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 each other, and the ionization degree of a given gas only depends on temperature and density. We do not need to know the details (e.g. the cross-sections) of the individual processes of ionization and recombination. Here we only consider the case of hydrogen ionization. This is the simplest case, and is also the most important both for fusion plasmas and astrophysical plasmas. We consider a hydrogen plasma with atom number density n 0, at temperature T. We call n n the density of non-ionized atoms (ions), n i = 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 Saha equation, which (for Hydrogen) reads 3/2 n e n i 2 mkb T = n n h 2 exp( /k B T ), (1.1) with m the electron mass, k B Boltzmann constant and = 13.6 ev the ionization energy. We are interested in computing the ionization degree, i.e. the ratio of the ionized 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,n 0) with f = 1 3/2 2 mkb T n 0 h 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 as k B T, and approaches unity when k B T/ 1. More quantitatively, the solution of Eq. (1.2) is = 1 f + p f f. For small and large f, wehave,respectively, p f, f 1; = 1 1 f, f 1.

3 1.2. SAHA EQUATION 3 Figure 1.1: Hydrogen degree of ionization vs temperature for di erent values of the density. We find that the plasma can be considered nearly completely ionized ( = 95%), when y = 18.3, or T (ev) = ln(n 0 )+(3/2) ln[t (ev)], (1.3) with n 0 in units of m 3. 1 This equation (where the term with the temperature is usually much smaller than the algebraic sum of the other terms in the denominator) clearly shows that ionization is nearly complete at temperatures of ev, depending on the density (see also Fig. 1.1). In our course we will only consider plasmas at temperatures of hundred or thousands of ev. At such temperatures hydrogen plasmas are fully ionized. 1 In plasma physics temperatures are often measured in units of ev. A temperature of 1 ev actually is the temperature corresponding to energy k B T =1eV.Therefore T (ev) = T (K) T (K) =

4 4 CHAPTER 1. INTRODUCTION 1.3 Collective behaviour, Debye shielding, quasineutrality A plasma is globally neutral. In this section we study how a globally neutral plasma reacts when a local charge unbalance is introduced, and whether local charge unbalance is possible. It will turn out that plasmas act to shield externally inserted charges, and - on small scales - allows some charge unbalance. It can then be considered quasi-neutral Debye shielding and Debye length We consider a homogeneous hydrogen plasma. A charge q is introduced in the plasma at a point we take as the origin of a spherical coordinate system. Combining the first Maxwell equation and the definition of electric potential r E = " 0 E = r, we write Poisson equation r 2 + " 0 =0, (1.4) that relates potential and electric charge density. The latter is the sum of the charge densities due to electrons, ions and the added charge q: = n i e + n e ( e) =(n i n e )e + q (r), where is Dirac s delta function. We then have to solve the equation r " 0 (n i n e )e, r > 0. (1.5) In proximity of the origin (where the charge q is) the potential must approach that of a single charge in vacuum, i.e. lim r!0 (r) = q 4 " 0 r. (1.6) At thermodynamic equilibrium particle density should follow Boltzmann distribution n = n 0 exp( U /k B T ), with = e,i, and U = q the electrostatic energy and n 0 the unperturbed density. Applying this general relation to plasma electrons and ions, we have, respectively, n e = n 0 exp(e /k B T ), (1.7) n i = n 0 exp( e /k B T ). (1.8)

5 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 n e = n 0 1+ e, (1.9) k B T e e n i = n 0 1, (1.10) k B T i where we allow di erent temperatures for electrons and ions. Poisson equation (1.5) then becomes r 2 n 0 e 2 " k B T i k B T e =0. (1.11) Expressing the Laplacian in spherical coordinates and exploiting spherical symmetry, we can finally write r 2 @r 1 2 d =0, (1.12) where we have introduced the Debye length d, such that 1 2 = n 0e , (1.13) d " 0 k B T i k B T e and d = r "0 k B T e n 0 e 2 T i T i + T e. (1.14) The solution of Poisson equation (1.11), with the condition at the origin (1.6) is = q 4 " 0 r exp( r/ d). (1.15) This is the potential of a charge in vacuum, attenuated (shielded) by the exponential factor exp( r/ d ) (see Fig. 1.2). The perturbation caused by the charge introduced in the plasma therefore vanishes at a distance of a few Debye lengths. This is the result of the collective behaviour of the plasma. For order of magnitude estimates, the factor T i /(T i +T e ) can be simply taken equal to 1, and then d ' r "0 k B T e n 0 e 2 ' 7430 s T (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 Debye length.

6 6 CHAPTER 1. INTRODUCTION Figure 1.2: Bare and shielded Coulomb potentials. n m 3 T [ev] d [m] size [m] Magnetic Confinement fusion plasmas Inertial Confinement Fusion Plasma Ionosphere Solar Chromosphere Table 1.1: Debye length and characteristic linear dimensions of typical plasmas.

7 1.3. DEBYE SHIELDING AND QUASI NEUTRALITY Plasma parameter. Ideal and non-ideal plasmas In the above discussion on plasma screening we have implicitly assumed that in a volume of linear dimensions of the order of the Debye length there are many particles (otherwise there could not be any shielding), i.e. N d = n 3 d 1. (1.17) The quantity N d is called plasma parameter. 2 Using the expression of Debye length we can write 3/2 "0 k B T N d = e 2 n 1/2 11 [T (ev)]3/2 = (1.18) [n(m 3 ] 1/2 Hence the condition N d 1, i.e the condition of ideal plasma, becomes n(m 3 ) [T (ev)] 3. (1.19) We now compare average electrostatic energy and average electron kinetic energy in an ideal plasma: U es U th ' e2 1 4 " 0 d (3/2)k B 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 U es ' e2 n 1/3 U th 6 " 0 k B T = N 3/2 d. (1.21) This proofs that the average electrostatic energy is much smaller than the thermal energy in an ideal plasma Quasi-neutrality A plasma is globally neutral. On smaller spatial scales (and on short temporal scales), however, charges of opposite signs do not necessarily balance. To determine the scale over which charge unbalance can occur we consider the simple case of a small spherical region, of radius L, with a net electric charge in an otherwise neutral and homogeneous plasma (see Fig. 1.3). In this small sphere the electron density is lower than the ion density, n e = n i ñ = n 0 ñ. Using Gauss theorem we find the electric field 3 and, in particular its maximum value E max = E(L) = eñ 3" 0 L. (1.24) 2 Some authors define a plasma parameter = 1/N d 3 Gauss theorem: Z E nds = 1 Z dv, S " 0 V with charge density = e(n i n e)=eñ inside a sphere of radius L. Weobtain

8 8 CHAPTER 1. INTRODUCTION Figure 1.3: Charged spherical volume inside a neutral plasma. We then obtain the maximum electrostatic energy density, U max = " 0 E 2 max 2 = U(L) =ñ2 e 2 18" 0 L 2. (1.25) It follows that ñ n 0 2 = 18" 0 n 2 0 e2 L 2 U max = " 0k B T n 0 e 2 18U max L 2 n 0 k B T = 2 d 27 U max. (1.26) L U th Since (see previous subsection) the electrostatic energy is always much smaller than thermal energy in an ideal plasma, we finally obtain that the relative charge unbalance is always very small for L d, while it can be quite large for 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 (instead of quantum physics) when interparticle distances d ' n 1/3 are substantially larger than the reduced de Broglie wavelength = h/p, wherep is particle momentum. For (nonrelativistic) plasma electrons ' h h ' p, (1.27) m e v te 2me k B T e E(r) = eñ 3" 0 r L apple L, (1.22) E(r) = eñ 3" 0 L 2 r 3 L L. (1.23)

9 1.5. PLASMA OSCILLATIONS AND PLASMA FREQUENCY 9 where v te = p k B T e /2m e is a characteristic electron thermal velocity. It follows that a plasma behaves classically if or, inserting the values of the constants, h 2 n 2/3 e k B T e. (1.28) 2m e T e (ev) n 2/3 e, (1.29) with the electron density in units of m 3. It turns out that most plasmas are classical. Some quantum e ects occur in strongly compressed inertial confinement 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 perturbations. We consider a simple one-dimensional plasma. We can assume that, due to their larger mass, ion are always at rest, while an electron layer is displaced from 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 distribution and electric field distribution

10 10 CHAPTER 1. INTRODUCTION Following this charge displacement, a double layer is generated, with surface charge density = n 0 ed and areal mass n 0 m e d; the electric field distribution shown in the Figure. Such an electric fields acts as a spring on the electron layer. According to Newton s second law, in the limit of very small displacement, we then have n 0 m e d d = n 0 ed n 0ed, (1.30) " 0 or d d + ed n 0e 2 m e " 0 d =0. (1.31) Equation 1.31 is the equation of a harmonic oscillator. The electrons oscillate with angular frequency 4 s n 0 e! p = 2 (1.32) m e " 0 and frequency (oscillations per unit time) s f p =! p 2 = 1 n 0 e 2. (1.33) 2 m e " 0 Note that the plasma frequency only depends on plasma density. Numerically f p = p p n[m 3 ]=8.98 p n[m 2 3 ] Hz. (1.34) Plasma frequency of a few important plasmas are listed in the table below. n m 3 f p [Hz] Magnetic fusion plasma Inertial fusion plasma Ionosphere We shall see later in this course that electromagnetic waves can only propagate in 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 linear perturbation theory. Again, we consider the ions at rest, and also assume that electrons only move due electric fields, i.e. we neglect electron thermal motion (hence the name cold fluid model). The motion of this electron fluid is 4 often simply called plasma frequency, omitting the adjective angular.

11 1.6. PLASMA OSCILLATIONS FROM A FLUID MODEL 11 then described by momentum and mass conservation equations, and by the first Maxwell equation mn dv +(v r) v = dt ene + r (n v) = (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, E the electric field. For simplicity, we consider a simple one-dimensional system, so that Eqs. (1.35) (1.37) become mn dv dt + v dv = ene + d (nv) = 0 dx (1.39) de dx = e (n i n) " 0 (1.40) We now use linear perturbation theory. We write each quantity f as the sum of its equilibrium value f 0 and a perturbation: f 1,i.e. f = f 0 + f 1. We assume that this perturbation is so small that the product of two perturbations can be neglected, being a second order term. In addition, the assumption of small perturbations allows us to use the superposition principle, i.e., to describe the perturbed fluid motion as the sum of linearly independent Fourier modes of the form f 1 = f 1 e i(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 the exponential notation allows us to formally replace the derivatives In the present case the equilibrium values are = i!, (1.42) = ik. (1.43) v 0 = (...) 0 = 0 = 0 (1.46) E 0 = 0 (1.47)

12 12 CHAPTER 1. INTRODUCTION In addition, the ion density is constant and equal to the equilibrium electron density, n i = n 0. We start by considering momentum equation (1.38), which becomes m (n 0 + n (v 0 + v 1 )+(v 0 + v (v 0 + v 1 )= e (n 0 + n 1 )(E 0 + E 1 ), and then 0 1 mn v 1 = 0 E 1 + n 1 E 1 A. } {z } } The terms indicated by braces can be neglected, being of second order. Next, we consider modes of the form (1.41), and we obtain mn 0 ṽ 1 ( i!) = en 0 Ẽ 1. (1.48) Analogously, continuity equation and first Maxwell equation become, respectively, i!ñ 1 + ikn 0 ṽ 1 = 0 (1.49) and ikẽ1 = e " 0 ñ 1. (1.50) We can now easily solve the system of three linear equations. From Eq. (1.49) we obtain the perturbed density which we substitute into Eq. (1.50) ñ 1 = kn 0ṽ 1!, ikẽ1 = e " 0 kn 0 ṽ 1!, to obtain the perturbed electric field as a function of the perturbed velocity: e n 0 Ẽ 1 = i" 0! ṽ1. Finally we substitute this last expression into Eq. (1.48): i!mn 0 ṽ 1 en 0 i e " 0 n 0! ṽ1 =0 By dividing both members by the perturbed velocity ṽ 1, and rearranging terms, we finally have! 2 = n 0e 2 m" 0, which is just the result obtained with the simple double layer model. As in any linear perturbation theory, the amplitude of the perturbations do not appear in the final relation. Also, notice, that we assumed modes in the form of (propagating) waves [see Eq. (1.41)]; we instead do not find propagating waves, but oscillations (the wavenumber k does not appear in the final dispersion relation).

13 1.7. PLASMAS IN DENSITY-TEMPERATURE SPACE Natural and man-made plasmas in densitytemperature space To conclude this introductory chapter in Fig. 1.5 we show the position a few important plasmas in the density-temperature plane. We observe that most of these plasmas are fully ionized, classical, ideal plasmas. Figure 1.5: Natural and artificial hydrogen plasmas in the temperature-density plane.

14 14 CHAPTER 1. INTRODUCTION