THE RANGE OF EXISTENCE OF NONIDEAL PLASMA

13

Pa	Pa
. . .	cm3 g-1

Fig. 1.5. Phase diagram of xenon (Iosilevskii et al. 2000). Lines I and II correspond to the limits of single and double ionization, respectively; dashed line show isotherms; C is the critical point; symbols correspond to the available measurements of thermodynamic parameters.

1.2.3Plasma of hydrogen and inert gases

As distinct from alkali metals and mercury, in the case of an inert gas plasma one manages to realize an extremely wide range of parameters and continuously follow the variation of the physical properties of such a plasma from gas (ne 1014 cm−3) to solid state (ne 1023 cm−3) densities. One deals with extremely high pressures, i.e., from hundreds to a million atmospheres at maximum temperatures up to 105 K. Under these strongly supercritical conditions, a plasma with developed ionization is realized.

Of particular interest is the investigation of properties of hydrogen (the most widely–distributed substance in the universe) plasma. At high pressures and densities the electrical conductivity of hydrogen sharply increases up to the values which are characteristic of metals, mostly due to pressure ionization. This a ects the properties of giant planets, for instance, the value of their magnetic field. The intriguing possibility of the existence of a metastable metallic and even superconductive phase of hydrogen at zero pressure is being investigated.

In the phase diagram of a xenon plasma, shown as an example in Fig. 1.5, solid curves indicate the regions of single and double ionization, while symbols correspond to thermodynamic measurements. In view of the high temperatures

14	NONIDEAL PLASMA. BASIC CONCEPTS

and peculiarities of electron shells, molecular and cluster formations are absent in such plasma, while the Coulomb interaction is the prevailing type of interparticle interaction because of the small number of neutrals. The nonideality parameter reaches, in this case, maximum values of about γ = 10. With the maximum plasma densities realized in experiment, ρ = 4.5–9.7 g cm−3 (which exceeds by several times the solid state density of xenon of ρ = 3 g cm−3), the electron component of such a plasma is partly degenerate, the parameter neλ3e reaching values close to 0.5.

A characteristic feature in the description of inert gas plasma when compressed to supercritical densities is the need to allow for atom–atom and ion–ion repulsion. Indeed, at ρ > ρc = 1.1 g cm−3, the characteristic interparticle distance in the plasma is comparable to the size of atoms and ions in the ground state (σAr = 3.4 · 10−8 cm, σXe = 4 · 10−8 cm) and, the more so, in excited states. The dimensionless parameter naσ3 characterizing this interaction reaches in xenon a value of 0.2–0.3.

Finally, in a high–temperature plasma (T > 3 · 104 K) the amplitude of the Coulomb scattering for the electron Ze2/T appears comparable to the characteristic ion size. This leads to non–Coulomb behavior of electron scattering. These conditions are attained during compression of a gas in strong shock waves and the conditions with more moderate parameters, during adiabatic compression. The principles of dynamical plasma generation are discussed in Chapter 3, while the properties of strongly compressed hydrogen and inert gases have discussed in Chapter 9.

1.2.4Plasma with multiply charged ions

Powerful pulsed energy contributions to condensed matter give rise to a nonideal superdense plasma with multiply charged ions. The density of such a plasma is close to that of the condensed state, the pressure reaches several terapascals, and the temperature is of the order of tens of electron volts. The plasma consists of electrons and highly charged ions, Z 10. In expanding, it passes through a whole gamut of peculiar states, in which the degree of degeneracy of the electron component also varies.

Relatively uniform volumes of such plasma are produced under laboratory conditions during compression of porous metals by powerful shock waves (neλ3e ≈ 0.5–2, γq = 10, Γ = 2).

During adiabatic expansion of shock–compressed metals, a wide range of plasma states is realized from a highly compressed metallic liquid to a weakly nonideal classical plasma.

If the electrons are nondegenerate, interaction of three di erent types is observed, namely, ion–ion, ion–electron, and electron–electron. The nonideality pa-

rameters for these interactions are
γZZ Z2e2ni1/3β,	γZe Ze2ne1/3β,	γee e2ne1/3β.	(1.19)

THE RANGE OF EXISTENCE OF NONIDEAL PLASMA

15

Z

Ry

Fig. 1.6. Diagram of the parameters of highly compressed matter (Kirzhnits et al. 1975). Curve 1 corresponds to electron degeneracy, curve 2 indicates the condition γee = 1, curve 3 indicates the condition γZe = 1, and curve 4 corresponds to the equality of the exchange and quantum–mechanical e ects to the correlation e ects.

If Z 1, then, in view of the quasineutrality condition, one can write the inequalities

γZZ γZe γee.

(1.20)

If the electrons are degenerate, the values of their interaction energy must be compared with that of the Fermi energy (1.5). The ions still remain very far from degeneracy. Therefore,

γZZ Z	2	2	1/3	β,	2	1/3	/εF,	2	1/3	/εF.
		e	ni		γZe Ze	ne		γee e	ne

and the hierarchy (1.20) is maintained. This implies the possibility of the existence of a number of interesting physical systems. These include a plasma formed by a quasicrystal system of multiply charged ions (γZZ 1) and an ideal electron gas (γee 1) with di erent intensity of interaction with ions.

In the diagram of the parameters of highly compressed matter (Fig. 1.6), a region is shaded in which the energy of interaction of a pair of electrons is low as compared with their kinetic energy. However, the electrons interact with nuclei, as well as with their numerous partners that are present in the sphere e ectively covered by the Coulomb forces. At Z 1, these interactions are far from being small. In the shaded region, the plasma is divided into Wigner cells and may be described by the Thomas–Fermi method with corrections or by the method of functional of the energy density (thermodynamic potential) (Kirzhnits et al. 1975).

16	NONIDEAL PLASMA. BASIC CONCEPTS

1.2.5Dusty plasmas

Over the last decade much interest has been attracted to investigations of properties of nonideal dusty plasmas (Yakubov and Khrapak 1989; Fortov et al. 1999; Bouchoule 1999; Shukla and Mamun 2002; Tsytovich et al. 2002; Piel and Melzer 2002; Merlino and Goree 2004; Vladimirov and Ostrikov 2004; Fortov et al. 2004; Ignatov 2005; Fortov et al. 2005), which consist of electrons, singly charged ions, and highly charged particles of the condensed disperse phase.

In a thermal plasma, the particles emitting electrons and acquiring a positive charge, can substantially increase the electron concentration in the plasma. In a nonequilibrium plasma of a gas discharge, the cold dust particles are charged negatively due to higher mobility of electrons in comparison with ions. The particle charge grows with size in both types of plasma and can achieve values of the order of Q (103–104)e for micron–size particles. The nonideality parameter describing dust–dust interactions γd is proportional to Q2n1d/3. For this reason, nonideality of the dust subsystem can be achieved much more easily than in the electron–ion subsystem even for relatively low dust densities. This allowed the realization of all possible states in dusty plasmas: ideal gas fully disordered state, liquid–like with short–range order in dust particle positions, and crystal–like with the clear long–range order.

At first glance dusty plasmas may seem to be a full analogue of multicomponent plasmas with multiply charged ion components. However, this is far from being true. One of the most important di erences is the variability of the particle charge. The point is that that the particle charge depends on the surrounding plasma parameters such as temperature, electron and ion concentration, dust particle concentration, etc. Moreover, charge fluctuations are always present due to the stochastic nature of the charging process. This leads to a variety of unexpected and interesting e ects, which can be observed in dusty plasmas. A detailed discussion of dusty plasma properties is given in Chapter 11.

1.2.6Nonneutral plasmas

Usually plasmas are assumed to be neutral or quasineutral systems with equal numbers of charges of opposite signs. However, plasmas consisting exclusively of particles with a single sign of charge can also exist. Examples of plasmas that have been realized in recent experiments include pure electron plasmas, positive–ion plasmas of one or more species, positron plasmas, and electron– antiproton plasmas. Obviously, because of the strong Coulomb repulsion, nonneutral plasmas must be confined. This can be achieved by using various types of electric and magnetic traps, e.g., Penning traps, r.f. or Paul traps, Kingdon traps, storage rings, and accelerators (Davidson 1990; Dubin and O’Neil 1999). Two–dimensional electron plasmas can be also realized above the surface of liquid helium (Cole 1974; Shikin and Monarkha 1989; Leiderer 1995) or on the inner surface of multielectron bubbles in the bulk of liquid helium (Volodin et al. 1977; Artem’ev et al. 1985). Pure electron and pure ion plasmas can be confined for hours and even days in a state of thermal equilibrium. They can be

NONIDEAL PLASMA IN NATURE. APPLICATIONS

17

cooled down to cryogenic temperatures, which allows one to observe liquid–like as well as crystal–like states. In Chapter 10 the properties of nonneutral plasmas, which can substantially di er from those of the usual quasineutral plasmas, are discussed.

1.3Nonideal plasma in nature. Scientific and technical applications

A high–density plasma with appreciable nonideality e ects is realized in numerous natural phenomena and technical devices. This includes the electron plasma in solid and liquid metals, semiconductors and electrolytes, superdense plasma of the matter of white dwarfs, the Sun, deep layers of the giant planets in the solar system, and astrophysical objects whose structure and evolution are defined by plasma properties. In studying the giant planets of the solar system, one will run, both in the literal and figurative sense, into a nonideal plasma formed as a result of hypersonic travel of space vehicles in the dense atmospheres of those planets.

Recently, growing purely pragmatic interest has been observed in high–pres- sure plasma studies in view of the realization of a number of major energy– related projects and devices which depend for their action on the pulsed local concentration of energy in dense media. A nonideal plasma appears to provide an advanced working medium in powerful continuous-operation and pulsed MHD generators (Biberman et al. 1982), power-generating facilities and rocket engines with gas–phase reactors (Gryaznov et al. 1980; Thom and Schneider 1971), in commercial–scale plasmochemical apparatus. A nonideal plasma is formed as a result of nuclear explosions (Ragan et al. 1977), explosive evaporation of liners of pinches and magnetocumulative generators, the e ect of powerful shock waves, laser radiation or electron and ion beams on condensed matter, and in a number of other cases.

A special need for information on the physical characteristics of nonideal plasma arises when realizing the idea of pulsed fusion accomplished by way of laser, electron, ion or explosion compression of spherical targets (Prokhorov et al. 1976), as well as when solving the most important problems related to high– velocity shocks.

Of decisive importance from the standpoint of physical analysis and calculation of the hydrodynamic consequences of such e ects are the data on the physical characteristics of plasmas in a wide range of phase diagram of matter from highly compressed condensed states to ideal degenerate and Boltzmann gases, including the high–temperature boiling curve and the neighborhood of the critical point.

In this section we shall mention very briefly the most representative practical situations in which a nonideal plasma is formed and utilized.

The most natural example of a nonideal plasma is provided by the plasma of conduction electrons in solid and liquid metals. We refer to a degenerate plasma (εF T ) with an electron concentration of ne = 1022–2.5 · 1023 cm−3. Because

18	NONIDEAL PLASMA. BASIC CONCEPTS

2 ≤ rs ≤ 5.6 (here rs is in atomic units), a strong Coulomb interaction between ions leading to crystallization is realized in such system. At the same time the interaction between electrons is weak (due to degeneracy), which allows us to consider them within the framework of an ideal degenerate gas. The e ect of powerful shock waves on metals (see Chapter 3) allows us to compress such a plasma by a factor of three–four to bring the maximum values of ne to about 6 · 1023 cm−3 and the maximum temperatures to 5 · 105 K, thereby approaching the limit of degeneracy.

Liquid electrolytes , in particular, ammoniacal solutions of alkali metals (Lepouter 1965) represent a strongly nonideal plasma in a very wide range of variation of the degeneracy and interaction parameters. This is attained by varying the fraction of metal dissolved in ammonia. Under these conditions, a powerful charge–neutral interaction is realized in the system along with the strong Coulomb interaction. These interactions have for their result unusual phase transitions and anomalously high electrical conductivities attained at moderate temperatures even with small fractions of metal in the solution.

In intrinsic and impurity semiconductors the number of electrons and holes is varied over a wide range by varying the temperature and concentration of impurities. Under conditions of intense light irradiation, electrons optically excited to the conduction band form a plasma. In a number of cases, the interparticle interaction in this plasma is so strong as to lead to a phase transition, i.e., formation of an exciton liquid (Je ries and Keldysh 1983).

According to Shatzman (1977), the parameters of the iron plasma in the center of the Sun are extremely high: ρ ≈ 120 g cm−3, T ≈ 13 · 106 K. The nonideality parameter γ ≈ 40 realized under these conditions is apparently insu cient for plasma crystallization. Calculations using the method of molecular dynamics give the crystallization limit of γ ≈ 170. However, even at lower values of γ, partial crystallization of the plasma is possible, thus causing, in particular, a variation of its optical properties determining the internal structure of the Sun.

A plasma with extreme parameters is realized at late stages of the evolution of stars in the so–called white dwarfs (Zel’dovich and Novikov 1971), decaying stars with a mass of less than 1–1.2 times the mass of the Sun. In this case, the matter is in the state of equilibrium when the pressure of a quasiuniform degenerate gas balances out the gravity forces compressing the star. At high densities, pyknonuclear reactions (quantum tunnelling of nuclei through the Coulomb barrier upon “cold” compression above the density of 1034 cm−3) occur in such a plasma and, at su ciently high temperatures, thermonuclear reactions occur. The e ects of nonideality increase the rates of nuclear reactions and define the structure, stability, and evolution of these exotic objects (Ichimaru 1982; Slattery et al. 1980).

Experimental investigations of the giant planets of the solar system using unmanned spacecraft provide rich information about their physical properties, which stimulates construction of modern models, involving the theory of nonideal plasmas. The point is that the strong gravitational field of those planets

NONIDEAL PLASMA IN NATURE. APPLICATIONS

19

forms a very dense atmosphere. Under conditions of hypersonic travel of space craft in such a dense atmosphere, a shock wave is formed in front of the spacecraft. Within the shock wave, there occur compression and irreversible heating of the plasma to tens of thousands of degrees at pressures of hundreds and thousands of atmospheres. In order to provide e ective protection of the spacecraft against the e ects of such a plasma, as well as to ensure reliable radio communication, one needs reliable data on the thermodynamic, transport, and radiation characteristics of strongly nonideal shock–compressed plasmas.

Of most importance among the numerous technical applications of nonideal plasmas are energy–related applications because the development and realization of a whole series of advanced energy-related projects are associated with ionized high–density plasmas. Along with thermonuclear systems of magnetic confinement of hot plasma, inertial controlled thermonuclear fusion is being developed as an alternative approach (Kadomtsev 1973; Brakner and Djorna 1973). With this approach, the thermonuclear reaction is accomplished as a “microexplosion” over a short time (several nanoseconds) defined by the time of inertial expansion of the hot plasma. The energy threshold of initiation in systems of inertial fusion is achieved by compressing the thermonuclear fuel of the target to a density approximately 1000 times higher than that of a solid. Diverse possibilities are considered as regards the compression and heating of a deuterium/tritium mixture in spherical microtargets, namely, powerful laser or “soft” X–ray radiation, streams of relativistic electrons and light and heavy ions, and macroscopic liner shock. In so doing, an energy of the order of 106 J must be delivered to a composite layered target of about 0.1 cm over a time of about 10−9 s. This results in the emergence of a complicated unsteady state flow of dense nonideal plasma with a pressure of up to 10 TPa. In order to calculate the hydrodynamics of such flow and develop the optimum structures of thermonuclear microtargets, one needs extensive data on the thermodynamic, optical, and transport properties of nonideal plasmas of various composition in an extremely wide range of pressures and temperatures.

A strongly nonideal plasma characterized by a wide range of parameters emerges also in the case of interaction between powerful pulsed sources of radiation and matter. When the surface of a solid material is subjected to a high–power laser pulse, a region of inhomogeneous nonideal plasma occurs, in which a wide range of states is realized. A relatively rarefied heated plasma (ne < 1021 cm−3, kT 1 keV) moves toward the laser beam and absorbs its energy. This energy is transferred to the region of strongly nonideal plasma (ne ≈ 1021–1023 cm−3), where at “critical” density (at which the frequency of laser radiation becomes equal to the characteristic plasma frequency) unsteady–state hydrodynamic motion emerges. The strong recoil reaction causes the target material to compress, and a region of superdense matter forms in the target (ne > 1023 cm−3), see Fig. 1.7 (More 1983).

Also directed toward the utilization of nonideal plasma is another futuristic energy project, that of the gas–phase nuclear reactor (Ievlev 1977). This is a

Fig. 1.7.

20

NONIDEAL PLASMA. BASIC CONCEPTS

Results of interaction between radiation and matter (More 1983). 1, laser pulse; 2, corona flame (ρ ≤ 10−2 g cm−3, kT ≈ 1 keV); 3, nonideal plasma of ablation products (ρ ≈ 0.01–1 g cm−3, kT ≈ 100 eV); 4, superdense matter (ρ = 10–50 g cm−3, kT ≈ 10–100 eV); 5, cold matter of normal density.

cavity-type reactor (see Fig. 1.8) with a high–pressure uranium plasma in the center. Flowing between the uranium and the walls is a working medium heated by the thermal radiation of the uranium plasma. The mixing of the working medium and uranium is suppressed by a stabilizing magnetic field, profiling of the velocity field and by other means. Such an apparatus can provide a basis for the development of nuclear power plants, compact space-borne power-generating facilities, rocket engines, etc. (Ievlev 1977). The high temperatures and requirements of criticality result in the need to have a pressure of hundreds of atmospheres in the gas–phase reactor cavity. Under these conditions, at temperatures of tens of thousands degrees, uranium and the working media (mixtures of alkali metals and hydrogen) are in the state of nonideal plasma.

A high level of electrical conductivity with appreciable compressibility, characteristic of a nonideal plasma, renders such a plasma a suitable working medium for magnetohydrodynamic generators (Iakubov and Vorob’ev 1974; Nedospasov 1977). The operating principle of the magnetohydrodynamic (MHD) generator provides for the travel of a conducting medium in a transverse magnetic field. The power which can be extracted per unit active volume is, other things being equal, proportional to the electrical conductivity of the working medium. The electrical conductivity of a nonideal plasma may exceed the ideal plasma conductivity by several orders of magnitude at relatively moderate temperatures. A closed-cycle MHD scheme utilizing a nonideal cesium and sodium plasma is treated by Iakubov and Vorob’ev (1974).

Biberman et al. (1982) proposed a MHD power plant scheme wherein the MHD generator operates in a closed cycle utilizing a nonideal plasma of saturated cesium or potassium vapors. For the case of cesium, the thermodynamic parameters of the working medium are at the level of T ≤ 1800 K, p ≤ 7 MPa. The facility operates on the Rankine cycle. The cycle performed by the alkali metal (see Fig. 1.9) includes heating and evaporation in the steam generator, ex-

NONIDEAL PLASMA IN NATURE. APPLICATIONS										21

Fig. 1.8. Schematic of a gas-phase nuclear reactor (Ievlev 1977). 1, neutron reflector; 2, fissile uranium plasma; 3, flow of the working medium.

Fig. 1.9. Schematic of a MHD facility utilizing alkali metal vapors (Biberman et al. 1982). 1, steam generator; 2, MHD unit (nozzle, MHD channel, di user); 3, condensing heat exchanger; 4, liquid metal pump; 5, water pump; 6, to the turbogenerator.

pansion in the MHD unit, condensation in the condensing heat exchanger with heat transfer to the steam–water loop, and an increase of pressure in the pump.

A nonideal plasma of cesium can also be regarded as the working medium for plasma thermoelements. This is favored by the high values of the thermoelectric coe cient of cesium (S = 103 V K−1) at moderate pressures and temperatures (p ≤ 10 MPa, T ≤ 2000 K) (Alekseev et al. 1970). The thermal conversion element is a sleeve of niobium in which there is inserted a beryllium oxide tube inside which cesium is poured. On top, the tube is closed with a metal plug serving as one of the electrodes. The temperature di erence is developed by a resistance furnace placed in a protective casing under the desired pressure of an inert gas. The thermoelement may have a power of up to 10 W cm−2 and an