- •Preface
- •Contents
- •1 Nonideal plasma. Basic concepts
- •1.1 Interparticle interactions. Criteria of nonideality
- •1.1.1 Interparticle interactions
- •1.1.2 Coulomb interaction. Nonideality parameter
- •1.1.4 Compound particles in plasma
- •1.2.2 Metal plasma
- •1.2.3 Plasma of hydrogen and inert gases
- •1.2.4 Plasma with multiply charged ions
- •1.2.5 Dusty plasmas
- •1.2.6 Nonneutral plasmas
- •References
- •2.1 Plasma heating in furnaces
- •2.1.1 Measurement of electrical conductivity and thermoelectromotive force
- •2.1.2 Optical absorption measurements.
- •2.1.3 Density measurements.
- •2.1.4 Sound velocity measurements
- •2.2 Isobaric Joule heating
- •2.2.1 Isobaric heating in a capillary
- •2.2.2 Exploding wire method
- •2.3 High–pressure electric discharges
- •References
- •3.1 The principles of dynamic generation and diagnostics of plasma
- •3.2 Dynamic compression of the cesium plasma
- •3.3 Compression of inert gases by powerful shock waves
- •3.4 Isentropic expansion of shock–compressed metals
- •3.5 Generation of superdense plasma in shock waves
- •References
- •4 Ionization equilibrium and thermodynamic properties of weakly ionized plasmas
- •4.1 Partly ionized plasma
- •4.2 Anomalous properties of a metal plasma
- •4.2.1 Physical properties of metal plasma
- •4.2.2 Lowering of the ionization potential
- •4.2.3 Charged clusters
- •4.2.4 Thermodynamics of multiparticle clusters
- •4.3 Lowering of ionization potential and cluster ions in weakly nonideal plasmas
- •4.3.1 Interaction between charged particles and neutrals
- •4.3.2 Molecular and cluster ions
- •4.3.3 Ionization equilibrium in alkali metal plasma
- •4.4 Droplet model of nonideal plasma of metal vapors. Anomalously high electrical conductivity
- •4.4.1 Droplet model of nonideal plasma
- •4.4.2 Ionization equilibrium
- •4.4.3 Calculation of the plasma composition
- •4.5 Metallization of plasma
- •4.5.3 Phase transition in metals
- •References
- •5.1.1 Monte Carlo method
- •5.1.2 Results of calculation
- •5.1.4 Wigner crystallization
- •5.1.5 Integral equations
- •5.1.6 Polarization of compensating background
- •5.1.7 Charge density waves
- •5.1.8 Sum rules
- •5.1.9 Asymptotic expressions
- •5.1.10 OCP ion mixture
- •5.2 Multicomponent plasma. Results of the perturbation theory
- •5.3 Pseudopotential models. Monte Carlo calculations
- •5.3.1 Choice of pseudopotential
- •5.5 Quasiclassical approximation
- •5.6 Density functional method
- •5.7 Quantum Monte Carlo method
- •5.8 Comparison with experiments
- •5.9 On phase transitions in nonideal plasmas
- •References
- •6.1 Electrical conductivity of ideal partially ionized plasma
- •6.1.1 Electrical conductivity of weakly ionized plasma
- •6.2 Electrical conductivity of weakly nonideal plasma
- •6.3 Electrical conductivity of nonideal weakly ionized plasma
- •6.3.1 The density of electron states
- •6.3.2 Electron mobility and electrical conductivity
- •References
- •7 Electrical conductivity of fully ionized plasma
- •7.1 Kinetic equations and the results of asymptotic theories
- •7.2 Electrical conductivity measurement results
- •References
- •8 The optical properties of dense plasma
- •8.1 Optical properties
- •8.2 Basic radiation processes in rarefied atomic plasma
- •8.5 The principle of spectroscopic stability
- •8.6 Continuous spectra of strongly nonideal plasma
- •References
- •9 Metallization of nonideal plasmas
- •9.1 Multiple shock wave compression of condensed dielectrics
- •9.1.1 Planar geometry
- •9.1.2 Cylindrical geometry
- •9.3 Metallization of dielectrics
- •9.3.1 Hydrogen
- •9.3.2 Inert gases
- •9.3.3 Oxygen
- •9.3.4 Sulfur
- •9.3.5 Fullerene
- •9.3.6 Water
- •9.3.7 Dielectrization of metals
- •9.4 Ionization by pressure
- •References
- •10 Nonneutral plasmas
- •10.1.1 Electrons on a surface of liquid He
- •10.1.2 Penning trap
- •10.1.3 Linear Paul trap
- •10.1.4 Storage ring
- •10.2 Strong coupling and Wigner crystallization
- •10.3 Melting of mesoscopic crystals
- •10.4 Coulomb clusters
- •References
- •11 Dusty plasmas
- •11.1 Introduction
- •11.2 Elementary processes in dusty plasmas
- •11.2.1 Charging of dust particles in plasmas (theory)
- •11.2.2 Electrostatic potential around a dust particle
- •11.2.3 Main forces acting on dust particles in plasmas
- •11.2.4 Interaction between dust particles in plasmas
- •11.2.5 Experimental determination of the interaction potential
- •11.2.6 Formation and growth of dust particles
- •11.3 Strongly coupled dusty plasmas and phase transitions
- •11.3.1 Theoretical approaches
- •11.3.2 Experimental investigation of phase transitions in dusty plasmas
- •11.3.3 Dust clusters in plasmas
- •11.4 Oscillations, waves, and instabilities in dusty plasmas
- •11.4.1 Oscillations of individual particles in a sheath region of gas discharges
- •11.4.2 Linear waves and instabilities in weakly coupled dusty plasmas
- •11.4.3 Waves in strongly coupled dusty plasmas
- •11.4.4 Experimental investigation of wave phenomena in dusty plasmas
- •11.5 New directions in experimental research
- •11.5.1 Investigations of dusty plasmas under microgravity conditions
- •11.5.2 External perturbations
- •11.5.3 Dusty plasma of strongly asymmetric particles
- •11.5.4 Dusty plasma at cryogenic temperatures
- •11.5.5 Possible applications of dusty plasmas
- •11.6 Conclusions
- •References
- •Index
4
IONIZATION EQUILIBRIUM AND THERMODYNAMIC PROPERTIES OF WEAKLY IONIZED PLASMAS
4.1Partly ionized plasma
4.1.1Classical low–temperature plasma
Let us consider a degenerate low–temperature plasma, for which the following inequality is valid:
βRy 1, Ry = me4/2 2, β = 1/kT, |
(4.1) |
where Ry is the ionization energy of the hydrogen atom. Such a plasma is classical, because inequality (4.1) implies that e2β λ. The quantity e2β is the mean scattering amplitude (Landau’s length), which is equal to within an order of magnitude to the square root of the mean e ective scattering cross–section. Therefore, not only does the system follow classical statistics, neλ3 1, but the motion of individual particles in the system may be described within the framework of classical mechanics.
From the standpoint of problems treated in this book, partly ionized plasma is of most interest. It is also referred to as three–component plasma, bearing in mind the atoms and free electrons and ions that remain unbound. The emergence of atoms in the initial electron–ion system is due to the electron/ion interaction which, if βRy 1, becomes strong even at relatively low values of the density. This interaction results in the formation of bound electron–ion pairs, i.e., atoms.
It is important that the electron and ion bound in an atom are spaced from each other to a distance of the order of Bohr’s radius a0, which is the least characteristic dimension in the system, a0 n−1/3, a0 e2β, a0 λ. Therefore, the electron and ion, which make up an atom, shield one another such that, to a good approximation, the interaction of such pairs with surrounding particles may be fully ignored.
4.1.2Three–component electron–ion–atomic plasma
The concentrations of components are related by the condition of ionization equilibrium. Let us write this condition – the Saha’s equation and first corrections to it. This may be done within the second virial approximation developed in the initial electron–ion system.
The second virial correction to Gibbs’ thermodynamic potential of an electron–ion system β∆Ω has the form (Vedenov and Larkin 1959; Kopyshev 1968)
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where ξi and ξj denote the fugacity of the plasma particles, |
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Here µj , µi, gj , gi, m, M denote their chemical potentials, statistical weights, and masses, respectively, mij is the reduced mass, ν is the quantum number of relative motion of the i– and j–particles, and Eijν stands for the corresponding energy of continuous and discrete spectra.
We retain in (4.2) the major term under condition βRy 1, and write β∆Ω in the form
−β∆Ω = ξi + ξe + ξiξe(2π 2β/mei)3/2(ga/gige) exp(βE1). |
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Here mei = mM (M + m)−1 and ga is the statistical weight of the state with energy E1 = Ry.
Now we proceed to the three–component description of the plasma, by taking atoms into account. For this purpose, we define the activity of atoms by the following expression:
ξa = exp(βµa)[(M + m)/2π 2β]3/2ga exp(βE1),
and then use the condition of chemical equality µa = µe + µi, where µa is the chemical potential of the new particle. Then, we derive, instead of (4.3) the expression
−β∆Ω = ξj , (4.4)
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describing a three–component ideal plasma (j = e, i, a). By using the formula nj = −ξj ∂(β∆Ω)/∂ξj , this yields the particle density, nj = ξj , and the equation of ionization equilibrium – Saha’s formula,
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Because an atom and an ion are multilevel systems, it is natural that gi and ga should be replaced by the corresponding internal partition functions, Σ+ and
PARTLY IONIZED PLASMA |
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Σ. The ionization potential of a complex atom I is di erent from Ry, and the reduced mass is mei m. Saha’s equation has the final form
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taken over all bound states ofthe atom (ion), where Eka and Eki are the energies and gka and gki are the statistical weights of the kth state of the atom and ion, respectively.
However, the three–component model in this form is inadequate. First, the contribution to the second virial coe cient due to the interaction between free electrons and ions – which was not taken into account – diverges. Second, the partition functions over bound states diverge as well. This is because of the long–range nature of the Coulomb potential.
4.1.3Second virial coe cient and partition function of atom
Let us discuss the origin of the divergence of the second virial coe cient corresponding to interactions between free electrons and ions at large distances,
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where k, j assume the values of e, i. This divergence is eliminated if one takes into account the many–particle aspect of the interaction. For this purpose, integration must be restricted to a distance less than the Debye length rD. As a result, we have the Debye–H¨uckel correction ∆(βΩ)D,
β∆ΩD = (12π)−1(4πβe2 ξj )3/2, j = e, i. (4.6)
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If (4.6) is included in constructing the ionization equilibrium, the following correction (Likal’ter 1969) appears to the ionization potential in (4.5):
∆I = −2kT ln[1 + Γϕ(Γ)/2],
where Γ = e2/rDkT is the Debye parameter of the interaction and the function ϕ(Γ) is derived from the expression Γ = 2(1 − ϕ2)/ϕ3. Thus, the ionization potential decreases.
The sum over bound states
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=gk exp(−βEk )
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diverges at small energies, which are hydrogen–like, Ek = −Ryk−2 and gk = k2. This divergence, occurring in a discrete spectrum, is compensated, in the case