where the current density was varied within the range of 103–104 A cm−2, the current–voltage characteristics of the plasma was shown to be linear. Based on these measurements of the plasma–gap resistance, the plasma electrical conductivity was determined by employing numerical and electrostatic simulations of the corresponding electrostatic problem. The resulting accuracy of the measured plasma conductivity was estimated at 20–100%.

9.1.2Cylindrical geometry

Another experimental setup for shock wave compression (see Fig. 9.3) has been employed by Adamskaya et al. (1987) and Urlin et al. (1992, 1997). A cylindrical explosive 1 (40/60 trotyl/hexogen alloy) with outer diameter of 30 cm was initiated over the outer surface at 640 points. This generated a highly symmetric detonation wave at the inner surface of the explosive (variations in the arrival time were within 100 ns). The arrival of the wave at the inner surface caused the centripetal motion of the cylindrical steel impactor 2 with the initial velocity 5 km/s. The deceleration of the impactor against the metallic surface of the chamber 3 filled with the studied gas at initial pressures of up to 70 MPa generated a converging shock wave. The intensity of the wave increased as it approached the center, because of the geometric cumulation (Zababakhin 1979). Thereafter, successive reflections of the shock wave from the center of symmetry and from the moving inner surface of the chamber caused multiple shock compression. Similar to the case of planar geometry, the compression was close to isentropic.

The evolution of the profiles of the thermodynamic parameters caused by multiple compression was determined from 1D or 2D gas–dynamic calculations. Broad–range semiempirical equations of state for the explosive, materials used in the assembly, and target plasmas were employed in these calculations. In some experiments, the process of cylindrical compression was monitored by measuring the velocity of the impactor by electrocontact and fiber–light optical methods, as well as by using two sources of hard radiation (Pavlovskii et al. 1965) emitting beams crossed at 135◦ angle. This made it possible to track the dynamics of compression and to test the quality of gas–dynamic calculations and provided additional boundary conditions for the simulation codes. The parameters found for the shock–compressed plasma are: for deuterium, the pressure was 1.25– 1.44 TPa at temperatures of 12 500–14 000 K and densities of 2 g cm−3; for xenon, the pressure and density were 200 GPa and 13 g cm−3, respectively.

Brish et al. (1960) measured the electrical conductivity by implementing the classic two–point circuit diagram involving a reference resistance connected in parallel with the resistance of the studied sample. The resistance of hydrogen was determined by using two stainless steel electrodes 6 (of 2 mm diameter) placed on the axis of the setup with a gap of 6.5 mm in between. A high– capacitance capacitor was discharged through the resistance Rsh shunting the hydrogen sample. The decrease in the resistance of the compressed hydrogen led to a decrease in the total resistance, which in turn caused the voltage across the

63

2

1

7

+U0

Fig. 9.3. Experimental setup for multiple shock compression of condensed hydrogen and inert gases in cylindrical geometry (Fortov et al. 2003). 1, explosive, 2, casing, 3, chamber (steel), 4, hydrogen (gaseous), 5, air (p0 = 0.1 MPa), 6, measuring electrodes, 7, insulator.

measuring electrodes (recorded by an oscilloscope) to change. In these experiments, Rsh = 3 ohm was used. In order to determine the specific conductivity from the measured resistance values, the actual geometry of the current distribution between the electrodes was taken into account, with the evolving geometry of the electrodes and the profiles of the thermodynamic parameters of hydrogen obtained from hydrodynamic codes. The error in determining the specific conductivity was estimated at 50%.

The typical plasma parameters obtained in experiments with the planar and cylindrical geometry are shown in Table 9.1.

9.1.3Light–gas guns

Metallization of hydrogen and other dielectrics was studied with shock reverberation in light–gas guns (see Chapter 3 for details). A sketch of the experimental cell used by Nellis et al. (1999) is shown in Fig. 9.4. A two–stage light–gas gun accelerated impactor 1 made of Al or Cu to supersound velocities. Liquid hydrogen or another liquid dielectric in sapphire anvils 6 was placed in casing 5 made of Al and cooled down to 20 K. Upon collision of the impactor with the casing a strong shock was generated, which propagated through the studied substance and compressed it. The multiple shock reflection from the sapphire anvils was utilized to decrease the temperature of the sample and provide quasi–isentropic heating to temperatures above the melting value. The setup included also electrodes 4 to measure the conductivity of the sample. In addition to hydrogen, also deuterium was used for the investigations. This is because at temperature 20 K the initial

4

6 5

Fig. 9.4. Experimental setup for multiple shock compression of condensed hydrogen and inert gases in light–gas guns (Nellis et al. 1999). 1, metal impactor; 2, studied substance; 3, triggering contact; 4, measuring electrodes; 5, aluminum casing; 6, sapphire insulators.

densities of liquid H2 and D2 di er by a factor of two or even more, so that in the shock–compressed substance the values of the density, temperature and conductivity di er substantially as well. The maximum pressure in hydrogen achieved in light–gas guns with shock reverberation was 180 GPa. In accordance to the estimates made by the authors, the corresponding temperatures were relatively low, about 3 000 K.

9.2Measurements of the electrical conductivity. Model of pressure–induced ionization.

The properties of a hydrogen plasma at high pressures – as well as of plasmas of the hydrogen isotopes – is of significant interest. Hydrogen is the most abundant chemical element in nature which has, in addition, the simples single-electron structure. Table 9.2 summarizes the chronology of hydrogen investigations, starting from its discovery in 1766.

In the experiments described below the multiple shock compression of hydrogen and inert gases is implemented, which makes it possible to obtain new physical information about an unexplored part of the phase diagram, which is depicted in Fig. 9.1. One can see that the region of pressures of up to 1.5 TPa and temperatures of 3000–7000 K was reached by means of dynamic compression. The achieved densities are one order of magnitude higher than those of solid hydrogen and solid inert gases at the triple point, where the mean spacing

˚

(about 0.74 A) and even isolated atoms in the ground state.

From the physical point of view, this region is interesting because it corresponds to a strong collective interparticle interaction and – when the ionization is high – to a strong Coulomb coupling (γ 10). The situation is additionally complicated by the fact that the type of statistics changes upon compression – electrons become degenerate and, instead of the temperature, the Fermi energy starts playing the role of the kinetic energy scale. All these features make a theoretical description of strongly nonideal states very complicated, preventing the employment of perturbation theory or parameter–free classical computer MC and MD methods (Zamalin et al. 1977) developed for Boltzmann statistics.

Experimental measurements of the electrical conductivity of shock–compress- ed hydrogen and inert gases are presented in Figs. 9.5–9.9, along with the results obtained on the basis of some theoretical models. Let us first point out some general features in the behavior of the electrical conductivity of strongly nonideal plasmas. The most prominent peculiarity is that, at the final stages of compression the electrical conductivity increases sharply (by three to five orders of magnitude) in a narrow range of “condensed” densities (ρ 0.3–1 g cm−3 for hydrogen and ρ 8–10 g cm−3 for xenon) at Mbar pressures, reaching values of about 102–103 ohm−1cm−1, which is typical for alkali metals. The measurements exhibit a pronounced threshold e ect in the density and are therefore in qualitative contradiction with models of weakly nonideal plasmas, which predict a monotonic decrease in the plasma electrical conductivity in response to the isothermal compression (see Chapter 6).

Indeed, at low degrees of the plasma ionization the electrical conductivity is determined by the scattering of electrons on neutrals and is qualitatively described by the Lorentz formula (6.5), according to which the electrical conductivity is proportional to the concentration of free electrons. In turn, the plasma composition is governed by the Saha ionization-equilibrium equation (6.5). Thus, based on the Lorentz and Saha formulas, one can conclude that in weakly ion-

Table 9.2

1766 Cavendish — discovery of “burning gas” – hydrogen

1898 Dewar — liquid and solid H2 – transparent (not a metal, as expected)

1927 Herzfeld — Clausius–Mossotti “dielectric catastrophe” at 0.6 g cm−3

1935 Wigner and Huntington — metallization at 25 GPa

1954 Abrikosov — metallization at 250 GPa

1968 Ashcroft — high–temperature superconductivity of atomic hydrogen

1971 Kagan — metallization of H2 at 300 GPa

1972 Kormer — explosive quasi–isentropic compression at 210–400 GPa

1978 Hawke — explosive magnetic compression at 200 GPa, 400 K, σ 1 ohm−1cm−1

1980 Mao, Bell, Hemley, and Silvera — static compression in diamond anvils at 10 GPa

1983 Ross — metallization at 300–400 GPa, based on shock–compression data

1987 Pavlovskii — explosive magnetic compression at 100 GPa, 300 K, σ 100 ohm−1cm−1

1990 Ashcroft — dissociation/metallization at 300 GPa

1991 Hemley — hcp nonmetallic phase

1993 Silvera, Mao, Hemley, and Ruo — solid H2, not a metal up to 250 GPa

1996 Nellis — light–gas gun – multiple shock compression of liquid H2 at 140 GPa, 2600 K – “metallic” conductivity

1997 Fortov and Ternovoi — explosive multiple shock compression of gaseous and liquid H2 at 150 GPa, 600 K, σ 103 ohm−1cm−1. Nonideal plasma

– pressure–induced ionization

1997 Da Silva, Cauble et al. — laser–induced shocks at 200 GPa, 4500 K – nonideal plasma

2001 Trunin, Fortov et al. — explosion–induced spherical shocks at 100 GPa

2001 Asay, Knudson et al. — shocks in Z–pinch at 100 GPa

ized ideal plasmas the electrical conductivity under the conditions of isothermal compression scales as σ 1/ρ. Nonideality, which must be taken into account under the discussed conditions, is introduced to the Saha equation via a density– dependent shift of the ionization potential, ∆I. This results in a nonthermal growth of the ionization fraction and, hence, causes the plasma electrical conductivity to increase upon isothermal compression. Then the curve representing the electrical conductivity versus density at T = const. exhibits a minimum, and its depth decreases with temperature. At kT I the minimum levels out, because the thermal ionization e ects become more pronounced than e ects associated with pressure–induced ionization (the latter are significant at kT I).

As the density and/or temperature increases further, ionization processes described by the Saha formula are completed. Then, instead of the Lorentz formula, one should employ the Spitzer formula (6.8) – when the Boltzmann statistics is applicable, or relation σ ne/Λ – in the case of Fermi statistics (where Λ is the

Fig. 9.5. Electrical conductivity of hydrogen versus density. Experimental data: 1, planar geometry (Fortov et al. 2003); 2, cylindrical geometry (Fortov et al. 2003); 3 and 4, magnetic confinement (Hawke et al. 1978), and (Pavlovskii et al. 1987), respectively; 5, light–gas guns (Nellis et al. 1998, 1999; Weir et al. 1996a).

Coulomb logarithm). Thus, at high temperatures the exponential dependence on the electron density is changed to a weaker, logarithmic or linear, scaling. To estimate the conductivity in this case, one can use the so–called “minimal metal” Regel–Io e conductivity (7.14), which is widely used in the theory of simple metals and semiconductors (Seeger 1977),

where rs is the Wigner–Seitz radius and vT is the mean thermal velocity of electrons. The Regel–Io e limit is applicable when the electron mean free path becomes comparable to the distance between atoms.

One can see that the exponential growth of the electron density caused by the reduction of the ionization potential due to strong interparticle interaction is the main reason behind the sharp increase in the measured electrical conductivity. At the same time, the increase of the frequency of electron collisions with atoms and ions is not important and can be estimated from the standard models discussed in Chapters 6 and 7. It should be emphasized that the pressure–induced ionization model discussed here leads to an exponential variation of the electrical conductivity with temperature,

Fig. 9.6. Electrical conductivity of xenon versus density. Experimental data: 1, Adamskaya et al. (1987); 2, Ivanov et al. (1976); 3, Mintsev and Fortov (1979); 4, Mintsev et al.

(1980); 5, Mintsev et al. (2000); 6, Eremets et al. (2000); 7, Dudin et al. (1998). Also shown are lines of electron degeneration (neλ3e = 1) and the constant value of the coupling parameter, Γ = 1, where the conductivity calculated from the Spitzer formula diverges. Line 8 corresponds to the conductivity calculated with the model proposed by Fortov et al. (2003).

similar to what the semiconductor thermal-excitation model yields (see, for example, Seeger 1977), where the energy gap ∆(ρ) decreases with density. This model was used by Ternovoi et al. (1999) and Nellis et al. (1999) to analyze experiments with light–gas guns. In the pressure range studied, the conductivity can be well approximated with the following dependence of the energy gap on density:

where ∆ is measured in eV, ρ is in mol cm−3, and the conductivity σ0 is taken to be constant and equal to 90 ohm−1cm−1. For a density of 0.32 mol cm−3, which corresponds to a pressure of 120 GPa, the energy gap becomes equal to2600 K. The authors suggest that the metallization of hydrogen (∆ = 0) occurs at p = 140 GPa and T = 2600 K. The pressure dependence practically vanishes at higher p,

Thus, the experimental data on the electrical conductivity obtained by multiple shock compression at kT I provide a unique opportunity to make an adequate choice of the thermodynamic model for the reduction of the ionization potential. For example, analysis of the data in Figs. 9.5–9.9 reveals that the