1 1 Characteristics of Low-Teperature Plasas Under Nontheral Conditions A Short Suary Alfred Rutscher 1.1 Introduction The concept of a plasa dates back to Languir (1928) and originates fro the fundaental difference between regions of electrical gas discharges which are distant fro boundaries (bulk of the discharge) copared with regions which adjoin the boundaries (sheaths). 1.1.1 Definition Plasas are quasineutral particle systes in the for of gaseous or fluid-like ixtures of free electrons and ions, frequently also containing neutral particles (atos, olecules), with a large ean kinetic energy of the electrons and/or all of the plasa coponents (.2 ev to 2 MeV per particle) and a substantial influence of the charge carriers and their electroagnetic interaction on the syste properties. The interactions between the electric charges of the plasa coponents show two aspects: Coulob interaction aong the charge carriers. Owing to the long range of the Coulob force in the case of large charge-carrier densities (n e 1/λ D), each charge carrier interacts siultaneously with any others (collective interaction). Foration of acroscopic space charges (in the frae of quasineutrality) as a consequence of external influences and odification of charge-carrier oveent in the electrical field of these space charges. Related to the quasineutrality and the presence of free charge carriers, the ost intrinsic attribute of the plasa state is its tendency to iniize external electric and agnetic fields inside the bulk, in contrast to its behavior in the surrounding sheaths. Low Teperature Plasas. Fundaentals, Technologies, and Techniques (2nd Edn.) R. Hippler, H. Kersten, M. Schidt, K.H. Schoenbach (Eds.) Copyright c 28 WILEY-VCH Verlag GbH & Co. KGaA, Weinhei ISBN: 978--527-467-9
2 1 Characteristics of Low-Teperature Plasas Under Nontheral Conditions A Short Suary Table 1.1 Subdivision of plasas. Low-teperature plasa High-teperature plasa (LTP) (HTP) Theral LTP Nontheral LTP T e T i T 2 1 4 K T i T K T i T e 1 7 K T i T e 1 5 K e.g., arc plasa e.g., low-pressure e.g., fusion plasas at noral pressure glow discharge 1.1.2 Types of Plasas Plasas are frequently subdivided into low- (LTP) and high-teperature plasas (HTP). A further subdivision relates to theral and nontheral plasas (see Table 1.1). 1.2 Starting Point for Modeling the Plasa State There are three different basic approaches toward a theoretical description of the any-particle plasa state: single-particle trajectories, kinetic and statistical theory, and hydrodynaic approxiation. 1.2.1 Single-Particle Trajectories This odel is based on the otion of individual particles (e.g., under the influence of the Lorentz force). Proble: The electric and agnetic fields in the plasa ust be regarded as given and cannot be obtained in a self-consistent anner fro the cooperative oveent of the particles. Using Monte Carlo siulations the study of single-particle trajectories could be extended to kinetic ensebles taking into account the effect of collisions. This technique is an alternative to the kinetic theory. 1.2.2 Kinetic and Statistical Theory On the basis of kinetic criteria each particle enseble of the plasa is analyzed taking into consideration the specific conditions and generalizing the kinetic theory of neutral gases to plasas. The ultiate goal is to be able to calculate the space and tie dependence of all the interesting distribution functions by solving the kinetic equations. For nontheral low-teperature plasas the ost iportant of these is Boltzann s equation (1872) for the energy or velocity distribution of the electron coponent. This equation describes the balance of the particle density in phase space. The total tie derivative of the distribution function is the sole outcoe of particle collisions, contained in the so-called colli-
1. The Role of Charge Carriers sion integral, which usually encloses a ultitude of ters for different collisions of electrons (elastic, inelastic, collective, etc.). After explicit replaceent of the external forces by the general Lorentz force, which then appears as self-consistently given by electric and agnetic fields of space charges and oving charge carriers, the collision-free approxiation of the Boltzann equation reduces to the Vlasov equation. The kinetic theory is the strongest instruent of plasa theory, e.g., for handling extree nonequilibriu conditions as well as deviations fro the Maxwell distribution function in any realistic plasas. 1.2. Hydrodynaic Approxiation This odel treats the plasa as a continuu and deterines the interesting acroscopic characteristics (density, flow, pressure, etc.) fro the balance equations of the nuber, energy, and oentu of each particle species. The balance equations are obtained as integrals of the appropriate kinetic equation. A special for of this approxiation is the so-called agnetohydrodynaics (MHD) odel, in which the plasa is considered as an electrically conducting liquid under the influence of agnetic fields. 1. The Role of Charge Carriers The existence of charge carriers as the doinating coponents of the plasa is connected with a series of characteristics which are also iportant in industrial applications. The ost active coponent of a nontheral low-teperature plasa (LTP) is the hot electron gas. The high ean kinetic energy of electrons results in the generation of electroagnetic radiation (lines and continua) and in the production of nuerous ionized, excited, and dissociated species of increased cheical activity. Applications are plasa light sources and plasa cheical reactors. Furtherore, the existence of charge carriers anifests itself in the following: occurrence of electrical conductivity, screening of electric fields, occurrence of a ultitude of oscillations and waves, typical for the plasa (Languir oscillations, ion acoustic oscillations, cyclotron oscillations, drift waves, surface waves, Alfven waves, etc.), as well as corresponding instabilities (plasa turbulence), interaction with agnetic fields. This is an iportant aspect of odern plasa physics. In the interaction with agnetic fields the whole spectru and variety of the typical plasa properties becoe effective, and foration of characteristic boundary sheaths due to the contact of plasas with solid surfaces. This is of particular iportance in the technology of plasa processing.
4 1 Characteristics of Low-Teperature Plasas Under Nontheral Conditions A Short Suary 1.4 Facts and Forulas 1.4.1 Electron Energy Distribution Functions (EEDF) Calculation of the distribution functions F for the velocity or energy of the electron coponent under existing conditions in each case is the central proble in nontheral low-teperature plasas. One approach is offered by solving the Boltzann equation adapted to plasas: df dt = F t + c F r + F L e F c where r is the position vector of the particle and c its velocity, and = C(F ) (1.1) F L = e o ( E + c B) (1.2) is the Lorentz force. The treatent of this equation has shown considerable progress during the last third of the 2th century, especially with the handling of the coplexity of the collision integral C(F ) and the tie and space variable ters (see, e.g., Chapter 2). However, the proble has not been astered at all and will continue into the 21st century. Soeties the approxiation of distribution functions by siple forulas is desirable. On this occasion the following standard ters for the electron energy distribution have proved to be valuable: and where ˆf (U) = a U /2 e exp [ 1 ( ) ] U U e with > (1.) a =(1/) ( 2)/2 /Γ(/2) (1.4) (see Table 1.2). A possible dependence on space and tie of the kinetic energy e U ay be incorporated in the distribution paraeter U e = U e( r, t). With =1, Eq. (1.) yields the Maxwell distribution, while = 2 represents the Druyvesteyn distribution. Of course the application of standard energy distributions in plasa physics is restricted to special cases, e.g., if only electrons with kinetic energies near the ean energy are of iportance. Table 1.2 Values of a for given. 1 2 a 1.128.97.798 1.4.2 Kinetic Teperature of Electrons The kinetic teperature T e of an electron gas is defined by eans of the ean energy U e, kte = e Ue with Ue = U /2 ˆf (U) du (1.5) 2
1.4 Facts and Forulas 5 This results in T e (kelvin) ˆ= 774 U e (volt) Using Eq. (1.), we obtain U e = εu e with ε = 1/ Γ(5/2)/Γ(/2) (1.6) (see Table 1.). For the Maxwell distribution we thus obtain T e (kelvin) ˆ= 11, 6 U e (volt) In ost cases the approxiation 1 V 1 4 K is sufficient. Table 1. Values of ε for given. 1 2 ε 1.5 1.46.856 1.4. Coefficients for Particle and Energy Transport The electron flow density depends on the electron density n e and the drift velocity v e and is given by j e = n e v e = n e μ e E grad (ned e) (1.7) The first ter on the right-hand side represents the electrical field drift, the second one cobines the action of diffusion and therodiffusion. Using the usual approxiations the obility μ e and the diffusion coefficient D e are given by μ e = 1 ( ) 1/2 2e e λ eu ˆf U du (1.8) D e = 1 ( ) 1/2 2e λ e U ˆf du (1.9) e The particle flow is connected with a flow of energy, j e = n eu e v e = n e μ eu ee grad (ne U ede) (1.1) U e μ e = 1 ( ) 1/2 2e e λ eu 2 ˆf U du (1.11) U ede = 1 ( ) 1/2 2e λ eu 2 ˆf du (1.12) e
6 1 Characteristics of Low-Teperature Plasas Under Nontheral Conditions A Short Suary Using a siple exponential approxiation for the energy dependence of the free path λ e for oentu transfer of electrons: 1 λ en = σ(u) = au n (1.1) where N is the particle density of the neutral gas and σ (U) is the cross section for oentu transfer; the standard distribution equation (1.) results in the following transport coefficients: μ e = ( ) 1/2 μ 2e Ue (2n+1)/2 a N e D e = αu eμ e Nernst Townsend Einstein relation De = α U eμ e (1.14) μ e = γμ e with μ = 2 2n 1 2 Γ ( ) 1+ n Γ ( ) α = 1 2 Γ ( ) 2 n Γ ( ) (1.15) 1+ n α = 1 Γ ( ) 2 Γ ( ) 5 2 Γ ( ) n Γ ( 1+ n ) γ = Γ ( ) 2 Γ ( 5 2 ) Γ ( 2+ n Γ ( 1+ n ) ) (1.16) 1.4.4 Generalized Boltzann Equilibriu In front of insulating walls or floating etallic surfaces a plasa shows, as a atter of principle, inhoogeneities which are siilar to the Boltzann equilibriu in a neutral gas under the action of external forces. These inhoogeneities should not be confused with sheath regions because there is no violation of the plasa conditions. In particular, the quasineutrality reains in force. Of course iportant differences fro the Boltzann equilibriu with neutral gases exist in the plasa: The origin of the forces is space charges in the plasa. Deviations fro the Maxwell distribution function ust be taken into consideration. The distribution function ay vary in space. The first condition for the Boltzann equilibriu is a vanishing particle flow in the flow direction. Then Eq. (1.7) yields E = 1 grad (n ed e) (1.17) μ e n e In the case of a Maxwell distribution which is constant in space, we have E = U e grad n e n e = grad V (1.18)
1.4 Facts and Forulas 7 and the concentration n e( r) shows the well-known exponential behavior under the action of the potential V ( r) according to the baroetric forula: n e ( r) n e() ( ) V ( r) = exp U e (1.19) In this case the properties of the Maxwell distribution also provide for the energetic equilibriu of the electrons ( j e = ). Every generalization (e.g., U e = U e( r)) has to ensure additionally that j e =, resulting in E 1 = U e μ e grad (n e U e D e) n e (1.2) Equations (1.17) and (1.2) deterine the necessary conditions of the spatial variables U e( r), V ( r), n e( r) for generalized equilibriu. Hence for the standard distributions (Eq. (1.)) and with Eq. (1.1) it follows that grad U e U e = δ grad ne α αγ δ = (1.21) n e (n 1/2)(α αγ) α ( ( ) ) 1 E = α 1+ 2 n Ue grad n e δ n e U e( r) U e() = ( ) δ ne( r) (1.22) n e() For the Maxwell distribution, which holds for U e = const., we have δ =. For non-maxwell distributions (δ )we obtain n e( r) n e() = ( ) 1/δ δ V ( r) α(1+(1/2 n)δ) U +1 (1.2) e() Consideration of the spatial variations of the electron distribution function is of utost iportance in the case of deviations fro the Maxwell distribution. Such deviations are coon in plasas, ostly as a consequence of collisions with heavy particles. Consequently a detailed analysis of the energetic relations of the electron gas is necessary. Very recently the nonlocal coplex nature of the power and oentu balance in space-dependent plasas was studied for the first tie, starting fro the Boltzann equation (Winkler 1996, see also Chapter 2). Neglecting collisions, the predoinance and stability of the Maxwell distribution (e.g., in low-pressure discharges, Languir paradox) should be a consequence of an energetic quasiequilibriu j e. Table 1.4 shows nuerical values for the particle and energy transport, using standard distributions, while Figs. 1.1 and 1.2 contain soe illustrations of the Boltzann equilibriu and its generalization. 1.4.5 Abipolar Diffusion Within the plasa the oveents of ions and electrons are interconnected via electric space charges. In the absence of external forces these space charges provide for equal electron and
8 1 Characteristics of Low-Teperature Plasas Under Nontheral Conditions A Short Suary Table 1.4 Nuerical values for particle and energy transport, calculated for three different standard distributions. =1(Maxwell distribution) n = 1 1/2 1/6 1 /2 α 1 1 1 1 1 1 α 2 1.667 1. 1.222.667. γ 2 1.667 1. 1.222.667. δ μ.752.5.76.54.76.667 =2(Druyvesteyn distribution) n = 1 1/2 1/6 1 /2 α.627.697.798.842 1.25 2.9 α.956.956.956.956.956.95 γ 1.797 1.667 1.524 1.476 1.198 1 δ.24.275.15..444.54 μ.647.5.46.8.24. =4 n = 1 1/2 1/6 1 /2 α.489.571.692.746 1.282 2.44 α.659.716.789.82 1.6 1.269 γ 1.71 1.667 1.616 1.579 1.498 1.427 δ.45.488.522.54.598.66 μ.59.5.429.49.4.7 ion drifts. For instance in the direction toward isolating walls the steady-state drift velocities of electrons and ions converge to the coon velocity v a of abipolar diffusion (n e n i = n) where D a = v a = D a grad n n μe Di + μi De μ e + μ i (1.24) is defined as the abipolar diffusion coefficient. Generally we have μ e μ i, which results in D a αμ i U e (1.25) The regie of abipolar diffusion (j e = j i) shows soe analogy to the Boltzann equilibriu (j e = j i =). Therefore the forulas of Section 1.4.4 are also valid approxiately for abipolar diffusion. Instead of forula (1.17) we obtain for the internal electric abipolar field: E a = 1 ) (1 μi grad (ned e) (1.26) μ e μ e n e
1.4 Facts and Forulas 9 Fig. 1.1 Equilibriu values of the electron concentration and teperature for given electrical potentials (full curves: Boltzann equilibriu, i.e., Maxwell distribution with U e = const.; dashed curves: Generalization for a Druyvesteyn distribution with U e = U e(r) and n =in Eq. (1.1)). 1.4.6 Condition of Quasineutrality In order to guarantee the status of free particles for electrons and ions in the plasa the field energy of space charges is liited to values uch less than the kinetic energy of the charge carriers. This results in tolerable deviations Δn e = n eo n e fro exact neutrality n eo = n io: Δn e /n eo λ D/L (1.27) where λ D is the Debye screening length (see Section 1.4.7), and L is a characteristic plasa length (e.g., the radius of a plasa colun). Within the Debye length considerable deviations fro neutrality ay occur in plasas. In this case the dynaics of such deviations is governed by the Languir plasa frequency. 1.4.7 Debye Screening Length The electrical potential distribution of a charge carrier inside a plasa is different fro the corresponding distribution in a vacuu. In a plasa each charge-carrier polarizes its surroundings and thereby reduces the interaction length of the Coulob potential V c which is copensated in part by the space charge potential V R (see Fig. 1.). In the case of ions of charge Ze and with e V D kt i the screened potential is
1 1 Characteristics of Low-Teperature Plasas Under Nontheral Conditions A Short Suary Fig. 1.2 Equilibriu values of the electron teperature and the electrical potential for a preset concentration profile n e(r)/n e()=1 (r/r ) 2 ; curves according to the conditions of Fig. 1.1. V D(r) = 1 e 4πε r ( exp r ) λ D λ 2 D = ε kt e e 2 ne (1 + ZTe/Ti) (1.28) Outside the Debye length λ D the potential ay be neglected (V D ). This cutoff is typical for plasa conditions and of great iportance for the interaction of charge carriers. Equation (1.28) is based on the assuption that in a sphere of radius λ D any charge carriers exist, i.e., 4π ne λ D 1 (1.29) Then the plasa state is tered ideal. In this case the Coulob interaction energy between two charge carriers at the ean distance is uch saller than the theral energy. For nonideal plasas the electrostatic energies exceed the theral energy. Figure 1.4 displays calculated values of the Debye length.
1.4 Facts and Forulas 11 1.4.8 Degree of Ionization Calculation of the degree of ionization in a closed for is only possible for plasas in the exact equilibriu state. The Saha Eggert equation then holds. For single ionization the following equation is valid: x 2 1 x 2 =2 ( 2πe h ) /2 g 1E 5/2 g p ( kt E ) 5/2 ( ) exp E kt (1.) where x = n e/(n + n e) is the degree of ionization, p =(n +2n e)kt is the kinetic pressure, g and g 1 are statistical weights, and E is the ionization energy. Figure 1.5 shows according to Eq. (1.) soe curves of constant degree of ionization. Under the nonequilibriu conditions of LTP the calculation of n e or x requires a detailed analysis of the corresponding balances. The energy balance of the electrons then results in a very siple and useful expression for the estiation of the degree of ionization. In the steady state and for x 1-9 4 V 2 Coulob Potential Debye Potential V λ = C D V λ = D D e 1 πε r λd e 1 πε r λ e 4 4 Space Charge Potential VRλD = e 1 πε r λ 4 V = V + V D C R D D r λd r λd ( e 1) V C 1 V λ D V D 1 2 r / λ D -1 V R Fig. 1. Variation of the electrical potential around an ion ibedded in a plasa.
12 1 Characteristics of Low-Teperature Plasas Under Nontheral Conditions A Short Suary 1-2 λ = D ε kt 2 2 e ne 1-4 1-6 r 1 8 K 1 6 K 1 4 K K r = 4π n e 1-8 λ D 1-1 2 a n 1-12 e 1 14 1 16 1 18 1 2 1 22 1 24 1 26 1 28 - Fig. 1.4 Debye screening length λ D versus electron concentration ( r, ean distance of electrons; a, radius of the first Bohr orbit). n e n it reads n e n 2 1 δ loss τ e n e U e P/V n 2 (1.1) where τ e is the ean free tie between electron collisions, P/V is the power density supplied to the plasa and δ loss is the ean fraction of energy that the electrons lose in a single collision. For elastic collisions, δ loss = δ el =2 e/m is of the order of δ el 1 4 1 5. The inelastic energy loss δ inel is typically larger by one to two orders of agnitude. The quantity τ en/δ loss is a function of U e, but often its variations are rather sall. Copared to the range of P/V n 2 it ay be regarded as constant. This is a rather good approxiation at higher gas pressures (p 1 Pa). 1.4.9 Electrical Conductivity Under the action of an electrical field E the free electrons and ions of the plasa gain drift velocities and generate an electric current density of j = e (n e ν e n i ν i) = e (n e μ e + n i μ i) E (1.2) Generally, since of μ e μ i and n e n i, only the contribution of electrons deterines the current density. Then the electrical conductivity σ of a plasa is given by σ = e n e μ e (1.) The concentration n e and obility μ e of the electrons are given by, e.g., Eqs. (1.8) and (1.1). The eleentary kinetic approxiation for the conductivity reduces to σ = e 2 n eτ e/ e (1.4)
1.4 Facts and Forulas 1 Fig. 1.5 Curves for constant degree of ionization in confority with the Saha Eggert equation. With regard to the degree of ionization two cases ay be distinguished: Weakly ionized plasas. The ean free tie of flight τ e is defined by electron ato collisions and is independent of n e. Consequently, σ n e. Fully ionized plasas. τ e is deterined by Coulob collisions with τ e 1/n e and the conductivity is constant. For a fully ionized theral plasa the corresponding equation is naed the Spitzer forula σ = 64 2πε 2 (kt e) /2 e 2 e ln Λ (1.5) where ln Λ is the Coulob logarith; its nuerical value for the ajority of plasas lies in the range fro 15 to 2.
14 1 Characteristics of Low-Teperature Plasas Under Nontheral Conditions A Short Suary 1.4.1 Plasa Frequency Representative of the ultitude of dynaic processes in plasas is the longitudinal electrical oscillations (Languir 1928). The occurrence of space charges generates in general a quasielastic coupling of the electrons to the ionic background and results in oscillations with a frequency ω P given by ωp 2 = e2 n e (1.6) ε e The electron plasa frequency (Eq. (1.6)) is critical for the propagation of electroagnetic waves in plasas. In the range ω<ω P the daping of the waves is strong. For ω = ω P electroagnetic waves show strong reflection at the plasa interface. This is related to the refractive index n 2 =1 (ω P /ω) 2 (Eccle relation).