346	METALLIZATION OF NONIDEAL PLASMAS

	106
	5		Experiment					Minimummetalic
	10		-	1				condactivity
	4		-	2
	10		-	3		5
	3
1	10			T=30 kK	7	n	λ3	=1
1cm-	2					e	e
	10
	1					7
-	10
, ohm				T=20 kK
	0							6
	10					σSpitzer= ∞
σ	10-1
	10-2			T=10 kK 7
	10-3							4
	10-4	10-3		10-2		10-1		1
					ρ, g cm-3

Fig. 9.7. Electrical conductivity of helium versus density. Experimental data: 1, Ternovoi et al. (2002); 2, Ivanov et al. (1976); 3, Dudin et al. (1998). The dotted line corresponds to the notations in Fig. 9.6. Theoretical curves: 4, conductivity for a composition calculated from the model of ideal plasma; 5, composition calculated from the Debye–H¨uckel model; 6, composition calculated from the bounded atom model, with fixed atomic radius ra = 1.3a0, 7, Fortov et al. (2003).

standard Debye–H¨uckel model strongly overestimates the e ects of the Coulomb interaction, leading to the pressure–induced ionization at densities that are two orders of magnitude lower than the experimental values.

9.3Metallization of dielectrics

The “bounded atom model” by Gryaznov et al. (1980) can probably be considered as the most adequate theory of strongly nonideal plasmas. The model explicitly takes into account the finite size of the phase space for the realization of the bound states of atoms and ions. We used this theory in the previous chapters to describe the thermodynamics of the shock–compressed inert gases and cesium. Actually, this is a combination of the Wigner–Seitz solid state model and the plasma model of the ionization equilibrium. In the framework of this theory, atoms and ions are treated as rigid spheres, whose thermodynamic functions are constructed on the basis of MD and MC simulations, whereas the contribution of the bound electrons is described by the quantum–mechanical Hartree–Fock approximation. Figure 9.10 shows the calculated energy spectrum of a compressed hydrogen atom versus its radius rc. In the calculation, the radial component of the wavefunction satisfies the following boundary conditions:

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347

	105
	104	-	1		- 4
		-	2		- 5
	103	-	3
								n		3	=1
							5		e λe
	102
1					T=20 kK σSpitzer= ∞
-1 cm-	101
					T=10 kK		5
ohm	100
σ,	10-1
	10-2				T=5 kK		5
	10-3
											4
	10-4	-3	10	-2	10	-1		10	0
	10
					ρ, g cm-3

Fig. 9.8. Electrical conductivity of argon versus density. Experimental data: 1, Ivanov et al. (1976); 2, Gatilov et al. (1985); 3, Dolotenko et al. (1997). The dotted line corresponds to the notation in Fig. 9.6. Theoretical curves: 4, conductivity for a composition calculated from the model of ideal plasma; 5, Fortov et al. (2003).

∂fnl (r)

fnl (r)|r=rc = 0, ∂r r=rc = 0. (9.4)

In the framework of the solid–state model (Zaiman 1972), this corresponds to the upper and lower boundaries of the energy band to which the transition of the corresponding energy level of an isolated (rc → ∞) atom occurs as the result of compression. In this approach, the width of the forbidden energy gap, ∆E, can be taken equal to the di erence between the upper boundary of the ground state band (curve 1s in Fig. 9.10) and the lower boundary of the first excited state band (curve 2p). One can see that ∆E decreases with density (see Fig. 9.11), and its value is in good agreement with the data of Weir et al. (1996b) obtained from a direct analysis of experiments on the multiple compression of hydrogen and deuterium. A modification of this model was successfully used by Gryaznov et al. (1982, 1998) and Gryaznov and Fortov (1987) to describe the thermodynamics of metal plasmas in the region of high and ultrahigh (up to 400 TPa) pressures.

348	METALLIZATION OF NONIDEAL PLASMAS

	104
	103	- 1		ne λ3e =1
		- 2
	102	- 3	T=20 kK
-1			σSpitzer = ∞
	101		T=10 kK
-1 cm
σ, ohm	1		T=7 kK
	10-1
	10-2		T=5 kK
	10-3		1	10
	0.1
			ρ, g cm-3

Fig. 9.9. Electrical conductivity of argon versus density. Experimental data: 1, Glukhodedov et al. (1999); 2, Veeser et al. (1998). The dotted line corresponds to the notation in Fig. 9.6. Theoretical curve 3, Fortov et al. (2003).

The parameter region investigated in the experiments is characterized by extremely complicated and diversified processes that require appropriate physical models. First of all, as the plasma is compressed by many orders of magnitude, the thermodynamic composition is changed and the interparticle interaction becomes strong, including the Coulomb interaction between electrons and ions, the polarization interaction between charged and neutral particles, and the short– range interaction between neutral particles. Since the typical interparticle distance in the plasma is comparable to the characteristic size of atoms and ions, the phase space occupied by them becomes inaccessible to other particles. Therefore, their kinetic energy grows, which provides the corresponding contribution to the free energy of strongly compressed disordered systems. Moreover, strong compression causes the energy spectrum of bound states in atomic and molecular systems to change. Also, one has to take into account the changes occurring in the continuous energy spectrum of electrons – the transition from the Boltzmann to Fermi statistics – because the degeneracy parameter (1.4) changes from ξ 5 · 10−2 to ξ 1.5 · 102 for the conditions of the experiments.

Fortov et al. (2003) calculated the plasma thermodynamics in the TPa pressure range as follows:

The free energy of a quasineutral mixture of electrons, ions, atoms and molecules was presented as a superposition of the ideal gas contribution and the term

METALLIZATION OF DIELECTRICS

349

						ρ, g cm −3
		0.6	0.07	0.01	0.005			0.002			0.0006
	14				3p	3d	3p	3s
			3d		2s
							3s
	12
			2s	2p
	10
eV	8		2p
),									- Rnl(rc ) = 0
c	6
(r
E					∆E				- R'	(r ) = 0
	4								nl	c
	2		1s
	0		1s
	0	2	4	6	8	10	12	14	16	18	20
						rc , a0

Fig. 9.10. Energy spectrum of the hydrogen atom.

responsible for the interparticle interaction. It was assumed that heavy particles (atoms, ions, molecules) obey Boltzmann statistics, whereas the electrons were treated as an ideal Fermi gas. A version of the pseudopotential model with multiple ionization was employed by Iosilevskii (1980) and Gryaznov et al. (2000) to include the Coulomb interaction. The principal point of this model is that the interaction of free charges at short distances deviates from the Coulomb form. This results in a noticeable positive shift not only in the potential energy, but also in the mean kinetic energy of free charges. The electron–ion pseudopotential in the Glauberman–Yukhnovskii form is given by Eq. (5.63). The contribution of the short–range repulsion of molecules, atoms, and ions is described phenomenologically within the soft–sphere approximation (Young 1977) generalized to the case of a multicomponent mixture.

The thermodynamic model of Fortov et al. (2003) provides the correct asymptotic behavior at low plasma densities, where it coincides with well–known theories of a rarefied plasma. In the region of extremely high densities, the applicability of the model was tested by comparing with the available experimental data.

9.3.1Hydrogen

In the phase diagram of hydrogen shown in Fig. 9.1, the transition to the metallic state at low temperatures is shown in accordance with the estimates given by

350	METALLIZATION OF NONIDEAL PLASMAS

	16	1atm
	14	Ry
			Single shock wave
			compression
	12
eV	10
			Egap
Energy,	8
	6		Eb	E0
			CA
		Ediss		CA
	4
	2				T~ 0.5 eV
		Multiple shock wave
	0	compression
		0.2	0.4	0.6	0.8
	0.0
			Density, g cm-3

Fig. 9.11. Energy gap of hydrogen.

Ebeling et al. (1991) at a pressure of about 300 GPa. The triple point, at which the metal phase coexists with solid molecular hydrogen and a molecular liquid, is predicted by Ebeling et al. 1991 to occur at p = 100 GPa and T = 1500 K. Both critical points lie in the molecular liquid phase. The first critical point as well as the curve of gas–liquid coexistence are well known and lie in the low– temperature region. The locations of the second critical point and the coexistence curve – which is associated with an abrupt change in the dissociation and degree of ionization and is of major interest – are not known precisely. According to the estimates of Ebeling et al. (1991), Tc2 = 16 500 K, pc2 = 22.8 GPa, and ρc2 = 0.13 g cm−3. Also shown in Fig. 9.1 are the estimates made by other authors (Robnik and Kundt 1983; Haronska et al. 1987; Saumon and Chabrier 1989, 1992; Beule et al. 1999; Mulenko et al. 2001) for the coexistence curve and the critical point for this plasma phase transition (curves 12–16).

One can see that the parameter regions accessible in experiments with multiple shock compression in planar systems (Fortov et al. 1999a, 2003) (region 9) as well as in cylindrical systems (Mostovych et al. 2000; Knudsonet al. 2001; Fortov et al. 2003) (curves 4, 5) and in experiments with light–gas guns (Saumon and Chabrier 1989, 1992) (curves 7, 8) partly overlap a rather large region of the possible plasma phase transition. The shock adiabats of singly compressed liquid hydrogen (deuterium) that were obtained in experiments with high–power lasers

METALLIZATION OF DIELECTRICS

351

(Da Silva et al. 1997; Mostovych et al. 2000) (curves 10, 11), in a high–current Z– pinch (Knudson et al. 2001) (curve 3), and in explosive spherical systems (Belov et al. 2002) (stars 6), also locate in this region, but at higher temperatures. In experiments on the isentropic compression by strong magnetic fields in explosive magnetic compression systems (Hawke et al. 1978; Pavlovskii et al. 1987), temperatures of about 700 K were realized (boxes 1, 2). Pressures of up to 300 GPa were obtained via the isothermal compression of hydrogen (T 300 K) in diamond anvils (Maksimov and Shilov 1999) (curve 17). The regions where strong Coulomb interaction and the degeneracy of the electron component are important lie above the curves γ = 1 and neλ3e = 1, respectively. Curve 19 represents the shock adiabat for liquid hydrogen (Da Silva et al. 1997), while curve 18 corresponds to parameters of the atmosphere of Jupiter (Nellis 2000). Figure 9.1 also displays the regions of typical parameters achievable with ordinary (Gaydon and Hurle 1963) and explosive (Mintsev and Fortov 1982) shock tubes in discharges and usual low–current pinches.

For hydrogen, there exists a large “monomolecular” region (ρ ≤ 0.3 g cm−3), where the thermodynamics is almost entirely determined by H2–H2 interaction. In the framework of the soft–sphere model (Young 1977), the parameters of H2– H2 interaction were chosen by Fortov et al. (2003) to be close to those of the rigorous “nonempirical” atom–atom approximation (Yakub 1990, 1999), and the anisotropy of the interaction was neglected. The calculations where the “soft– core” repulsion V (r) r−6 was used revealed satisfactory agreement with the molecular part of the T = 0 isotherm (“cold curve”), as well as with a considerable part of the shock wave experiments and results of precise MC simulations of the H2+H thermodynamics (Yakub 1990, 1999).

The major issue in applying the chemical model to describe nonideality (including the case of dense hydrogen) is the correct choice of the entire set of e ective potentials for the interaction between species. This concerns the interactions involving both charged and neutral particles – first of all, interactions of H2–H and H–H pairs. It is important is that the e ective interaction of free atoms di ers drastically from the singlet (attractive) and triplet (repulsive) branches given by the rigorous theory for the total potential of H–H interaction. This is because the contribution of H–H pairs interacting via the singlet branch has been already partially taken into account in the intramolecular motion. The same is true for the e ective interaction involving (free) charged particles, since the contributions of free and bound states must be consistent in the chemical model. At present, there exist serious contradictions in the suggestions given by di erent approaches for the form and parameters of the e ective potentials. O the monomolecular region, the major issue is the choice of parameters of the short– range repulsion in H–H and H2–H pairs. We note that, according to Fortov et al. (2003), the appropriate choice of parameters for the H2–A± interaction (where A± stands for all charged components) is equally important. One possible choice follows from the nonempirical atom–atom approximation (Yakub 1990, 1999), which leads to relatively large “eigenvolumes” of the hydrogen atom. In terms

352 METALLIZATION OF NONIDEAL PLASMAS

of the modified soft-sphere model (Young 1977), the obtained results correspond almost exactly to the “additive volume” approximation, with (DH2 )3 2(DH)3,

			4					≤	ρ		0.3 g cm−3, the results
where D is the atom or molecule diameter. For ρ
at T	≤	10		K are in very good	agreement with the precise MC calculations
						4	K they coincide with the nonanomalous
(Yakub 1990, 1999), whereas at T ≥ 10

part of the results obtained from quantum MC simulations (PIMC, Pierleoni et al. 1974). At such temperatures, the data are also in satisfactory agreement with the results of other ab initio approaches, including the methods of quantum MD simulations (TBMD, Collins et al. 1995, Lenosky et al. 1997) and “wave packets” (WPMD, Knaup et al. 1999).

Figure 9.12 presents the entire set of currently available experimental data on single shock compression of liquid deuterium. Pressures of up to 25 GPa (circles 1) were achieved by Nellis et al. (1983, 1998) and Weir et al. (1996a) with direct shock waves generated in light–gas guns. In the experiments of Da Silva et al. (1997) and Mostovych et al. (2000) on the shock generation with high–power lasers (2, 3), pressures of up to 300 GPa were achieved and an anomalously high compressibility of deuterium was discovered at pressures p > 40 GPa. However, more recent experiments performed by Knudson et al. (2001) with Z–pinches (4) and by Belov et al. (2002) with explosive spherical systems (5) did not confirm this anomaly up to p 70 GPa.

The shock adiabats calculated with the SESAME code (Kerley 1972) (curve 6) do not predict this anomaly in the shock compressibility, nor does it emerge in calculations with semiempirical equations of state (Grigor’ev et al. 1978). Moreover, the anomaly is not expected from ab initio approaches, like the quantum MC method (Pierleoni et al. 1974) (curve 7) and the MD method (Collins et al. 1995, Lenosky et al. 1997). Ross (1998) proposed an interpolation equation of state for deuterium (curve 10), which provides a qualitative description of the experimental results obtained with lasers.

The approach of Fortov et al. (2003) also does not reproduce this abrupt change of the compressibility (σmax ≡ ρmax/ρ0 6.5 versus expected σmax 4) in the behavior of the deuterium shock adiabat at p 50–200 GPa (curve 11), nor does it lead at ρ ≥ 1 g cm−3 to anomalies typical of phase transitions.

The thermodynamics of compressed hydrogen (deuterium) exhibits quite different behavior if the H–H (D–D) interaction is calculated with the H–H potential (which is widely used for approximate calculations, Ross et al. 1983), and the H2–H interaction is derived from standard composition rules. In terms of the modified soft sphere model of Fortov et al. (2003), this corresponds to a much smaller ratio of H and H2 “eigen–volumes” (DH/DH2 0.4 → 2VH /VH2 0.13). Such a choice of the atom “eigen–size” immediately leads to “pressure–induced dissociation” at ρ ≥ 0.3 g cm−3, which is accompanied by a dip in the deuterium shock adiabat (curve 12).