COMPARISON WITH EXPERIMENTS

217

5.8Comparison with experiments

By now, a fairly large body of experimental data has been accumulated on the thermodynamics of a nonideal plasma of various elements under conditions of developed ionization. The great majority of these data were obtained using dynamic methods (see Chapter 3), by shock and adiabatic compression of cesium, as well as compression of argon, xenon, and copper in powerful shock waves.

The data obtained for cesium plasma (Kunavin et al. 1974; Bushman et al. 1975; Iosilevskii and Gryaznov 1981; Alekseev et al. 1981; Iakubov 2000) relate to various regions of the phase diagram (see Fig. 3.2) that partly overlap at the boundaries and agree with each other within the claimed experimental accuracy. Experiments on adiabatic compression by Kunavin et al. (1974) enabled one to advance, as compared with static measurements (Kalitkin and Kuzmina 1975, 1976; Alekseev et al. 1981), to higher temperatures, T 4000 K. This, however, turned out to be insu cient for noticeable thermal ionization of the plasma. Under these conditions the charge–neutral interaction prevails, but its contribution to the equation of state remains within the measurement errors. The principal conclusion by Kunavin et al. (1974) drawn on the basis of these experiments consists in the absence of phase separation (see Section 5.9) caused by the metal–dielectric transition. Experiments on cesium compression by direct and reflected shock waves provided further extension of the temperature range, T (2.6–20)·103 K (see Fig. 3.18), where the Coulomb interaction is strong, Γ 0.2–2.2, and defines the physical properties of the plasma with developed ionization.

Considerably higher plasma parameters were attained as a result of explosive compression of heavy rare gases (Bespalov et al. 1975; Fortov et al. 1976; Gryaznov et al. 1980; Fortov 1982), namely, pressures of up to 6 GPa and temperatures of up to 6 ·104 K. The obtained plasma densities of ρ 0.4 g cm−3 and ne 3 · 1021 cm−3 approach the density of condensed xenon, and even exceed it (ρ 4.5 g cm−3) in experiment by Mintsev et al. (1980). The characteristic interparticle distances in plasma (Gryaznov et al. 1980) were about (6–7)·10−8 cm, which is comparable with the ion and atom size of (3–4)·10−8 cm, whereas the maximum nonideality parameters Γ 5 were close to the maximum possible for nondegenerate plasmas value (see Fig. 1.1). The experimental data for argon and xenon (Bespalov et al. 1975; Fortov et al. 1976; Gryaznov et al. 1980; Fortov 1982) relate to the region of developed single (xAr 0.7) and double (xXe 1.8) ionization. These data allow us to advance into the region of condensed densities (see Fig. 1.5) and approach extremely high pressures (p 0.13 TPa) and compressions (ρ 9.6 g cm−3) of a plasma obtained by shock compression of liquid xenon and heated to T 30 · 103 K (Lundqvist and March 1983).

All thermodynamic measurements (Figs. 5.22–5.26, Tables 5.2–5.5) exhibit a clear tendency: The measured enthalpy or internal energy (curves I in the figures) is lower than that obtained with the traditional plasma calculation represented by the ring Debye approximation in a grand canonical ensemble (5.50)–(5.52) (curves II). In Figures 5.22–5.26 and Tables 5.2–5.5, the experimental data are

218 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

Pa
		Pa
		v cm		g
		.
		v
Pa	v cm	g
		v	cm	g

Fig. 5.22. Results of the shock compression of cesium plasma. T – isotherms; S – isentropes; 1 and 2 – incident and reflected shock waves, respectively. Dashed regions show the measurements errors.

compared with a number of other plasma approximations: Ideal plasma model with Fb = 0 and Σk = 2 (curves III); Fb = 0 in Eq. (5.40) and Σk from Eq. (5.73) (curves IV); Debye theory in a small canonical ensemble, Eq. (5.43) (curves V); pseudopotential Monte Carlo model, Eqs.(5.58)–(5.61) (curves VI); and “bound” atom model, Eqs.(5.77)–(5.83) (curves VII). The relative contribution by other models to the equation of state was analyzed by Rajagopal (1980).

The performed analysis shows that, although the majority of the used theoretical models do not contradict the experimental isotherms of cesium plasma within the measurement accuracy, these models cannot provide a consistent description of the thermal and caloric data. Inert gas plasmas correspond to substantially higher parameters of nonideality and, in this case, the traditional plasma models, Eqs. (5.41)–(5.47), (5.50)–(5.52), (5.58)–(5.61), are in contradiction not only with the caloric, but also with the thermal measurements.

An analysis of thermodynamic data suggests the presence of interparticle repulsion in a strongly compressed plasma, which is not described by the plasma theories, Eqs. (5.41)–(5.47), (5.50)–(5.52), (5.58)–(5.61). Although one can attain good agreement with the thermal equation of state by introducing modifications to the correction F c, such a procedure leads to even greater discrepancy with the caloric data. It was found by Bushman et al. (1975), Fortov et al. (1976),

COMPARISON WITH EXPERIMENTS

219

kJ g

a

v

.

.

.

v

			kJ g
.	.
		.	.
.

v

kJ g

.	.	.	.	.	pv	kJ g
	J g

b

.

.

.

MPa

Fig. 5.23. Isochore of cesium plasma, v = 103 cm3 g−1 (a) and v = 200 cm3 g−1 (b), the error band is dashed. I – experiment; II – ring Debye approximation; III – bound atom model; IV – pseudopotential model (5.63); V – superposition of III and IV.

220 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

.

.

. . MPa

Fig. 5.24. Thermal equation of state of cesium plasma (Iosilevskii 1980). The error band is dashed. The solid line is the ring Debye approximation, the dashed line is the pseudopotential model (5.63), the dash–dotted line is the asymptotic model (Krasnikov 1977).

and Gryaznov et al. (1980) that the consistent description of thermal and caloric data could be achieved by modifying the contribution of the bound states to the thermodynamic functions of dense plasma – an e ect ignored by most of the traditional plasma approximations.

Irrespective of this conclusion, the experimental results indicate that the Debye and similar theories, (5.43)–(5.47), overestimate considerably the corrections to thermodynamic functions for the interaction in a continuous spectrum. The best properties of extrapolation to region Γ 1 are exhibited by the theories whose nonideality corrections do not exceed the corrections of the ring approximation in the grand canonical ensemble (5.50)–(5.52). This conclusion is in qualitative agreement with the results obtained by the Monte Carlo method for the one–component model (see Section 5.1), which indicates the validity of asymptotic approximations up to Γ 1.

In the considered region of high plasma densities, the mean interparticle distances are comparable with the characteristic sizes of atoms and ions. This fact, along with a strong Coulomb interaction of free charges, can cause significant deformation of energy levels.

Pa	Pa		Pa	Pa
	a	.	Pa	b
		.
		.		Pa
		.
		.
			Pa
		.		Pa
.			.	.
			Pa	Pa

.	.	.
v cm g	. . .
	v cm	g
v cm g	v cm g

Fig. 5.25. Phase diagram of argon (a) and xenon (b). Dashes mark the two–phase region boundaries, C indicates the critical point. Experiments: ◦ — (Gryaznov et al. 1980), — (Fortov et al. 1976), ♦ — (Bespalov et al. 1975), dashed lines show the isotherms of the initial states, solid lines show the boundaries of single (I) and double (II) ionization. Given on the right is a comparison between the experimental shock adiabats (dots) and theoretical data: Solid lines represent calculations with the ring Debye approximation in the grand canonical ensemble (5.50)–(5.52); dash–dotted lines represent the additional inclusion of the atomic interaction with the second virial coe cient; dashed lines in (a) represents the “bound” atom model (5.77)–(5.83), and, in (b), the pseudopotential model (5.63)–(5.71).

EXPERIMENTS WITH COMPARISON

221

222 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

.

.

.

.

.

.

.

.

.

.

. . . .

.

MPa

Fig. 5.26. Thermal equation of state of argon plasma. Vertical lines represent experimental data (Bespalov et al. 1975; Fortov et al. 1976); ◦ – calculation with the ring approximation in a grand canonical ensemble (5.50)–(5.52) for given pexp and Texp; 1–3 stand for calculated isotherms (T = 16 000 K and T = 20 000 K) in the approximations: 1 – pseudopotential model (5.63)–(5.71), 2 – with additional inclusion of the second and third virial coe cients, 3 – from the “bound” atom model (5.77)–(5.83).

The description of this e ect required implementation of the quantum–mecha- nical model of a “bound” atom (5.78)–(5.83), which is unconventional for plasma physics. The model takes into account the e ect of the plasma environment on the discrete spectrum of atoms and ions in a strongly compressed plasma. It is seen from a comparison of this model with experiments that the model correctly recovers the experimentally revealed tendency of the somewhat overestimated repulsion e ect at near–critical plasma densities. Under these conditions one should apparently use the more adequate “soft” sphere model (Bushman and Fortov 1983). Later, model (5.77)–(5.83) will be employed for the description of the optical characteristics of nonideal plasmas.

The need to include in the thermodynamics of dense plasma the variation of the discrete electron spectrum is clearly demonstrated in experiments with shock compression of liquid argon and xenon at pressures up to 0.13 TPa (Lundqvist and March 1983; Sin’ko 1983). The interpretation of these measurements has shown (Fig. 5.27) that neglect of the discrete spectrum (excitation energy) leads to a drastic overestimation of the calculated pressures (curve C). Curves A and B take into account the thermodynamic e ects of thermal excitation of electrons from the valence band 5s to the conduction band 5d, and the variation of the forbidden band width from 7 to 4 eV with the density increase (Liberman 1979; Lundqvist and March 1983).

COMPARISON WITH EXPERIMENTS

223

GPa

v cm g

Fig. 5.27. Comparison of theoretical models with experimental data on the shock compressibility of xenon. • — (Lundqvist and March 1983); ◦ — (Klyuchnikov and Lyubimova 1987); C — calculations by Liberman 1979 without taking into account the electronic excitation ; A and B – two examples where the discrete spectrum deformation is included.

A peculiarity of the quasi–chemical description of a plasma under conditions of strong nonideality is a “conditional” division of particles into free and bound ones. Therefore the e ect regarded as a distortion of the excited states contribution may, in the case of a di erent separation into species, be interpreted as a manifestation of the quantum nature of the electron–ion interaction at small distances. The pseudopotential (5.63) was implemented by Iosilevskii (1980) and Gryaznov et al. (1989) in order to describe this interaction, and on this basis, using the conditions of local electroneutrality (5.65) and(5.66), the semiempirical model of a nonideal plasma was constructed. By selecting an appropriate value for the single parameter of the model – the pseudopotential depth Φie(0), which is equal to the energy separating the particles into free and bound ones – a consistent description of currently available thermodynamic experimental data has been performed for a “gaseous” plasma (Bushman et al. 1975; Bespalov et al. 1975; Fortov et al. 1976; Mintsev et al. 1980; Iosilevskii and Gryaznov 1981).

In conclusion of this paragraph, we shall dwell on the comparison between the quasiclassical model (Section 5.5) and plasma experiments, as shown in Fig. 5.28 for the cesium gaseous plasma from the shock–wave (Bushman et al. 1975) and electric explosion (Dikhter and Zeigarnik 1977) experiments.

In accordance with the conclusions of Section 5.5, the worst accuracy of the Thomas–Fermi approximation is attained in the region of temperatures corresponding to the ionization from the filled electron shell (beginning of the secondary cesium ionization). Note that the plasma investigated by Bushman et al. (1975) and Iosilevskii and Gryaznov (1981) has a noticeable degree of nonideality, Γ 1.

Table 5.2 Thermodynamic parameters of cesium behind an incident shock wave

			Experiment I							Theory
							II			III
			v1,
p0,	T0, K	u,		p1,	H1,	T1, K	H,	T , K	H,	ne,	α	Γ	T , K
105 Pa		km s−1 cm3 g−1 105 Pa			102 J g−1		102 J g−1		102 J g−1 1013 cm−3
0.130	780	0.83	1200	1.40	4.20	2600	4.20	2600	4.10	0.006	0.002	0.02	2600
0.147	780	0.935	1000	2.00	5.10	3100	5.40	3100	5.10	3.03	0.007	0.34	3100
0.107	760	1.23	1030	2.80	8.3	4100	10.0	4200	8.6	0.250	0.056	0.62	4200
0.160	780	1.05	800	2.90	6.3	3500	6.6	3500	6.0	0.09	0.015	0.48	3500
0.074	750	1.53	1160	3.10	12.4	5100	15.7	5000	13.0	0.56	0.140	0.73	5000
0.240	830	0.95	630	3.30	5.30	3200	5.50	3200	5.20	0.046	0.006	0.41	3200
0.063	740	1.87	1060	4.20	18.0	5600	22.0	5700	18.0	1.05	0.250	0.83	5700
0.051	700	2.23	1130	5.1	25.0	6400	30.0	6600	25.6	1.60	0.41	0.85	6400
0.180	790	1.33	580	5.4	9.6	4600	9.5	4600	9.5	0.53	0.070	0.8	4600
0.380	860	1.03	390	6.0	6.2	3600	6.5	3600	6.0	0.13	0.01	0.59	3600
0.078	750	2.10	800	6.5	22.4	6100	26.4	6300	22.0	1.80	0.320	0.93	6200
0.080	740	2.05	770	6.6	21.6	6100	25.4	6200	21.0	1.70	0.290	0.94	6100
0.044	690	2.77	1270	6.9	39.0	7900	44.0	8500	41.0	2.50	0.70	0.76	8000
0.180	790	1.49	530	7.0	11.8	5200	15.6	5200	12.3	1.00	0.12	0.93	5200
0.056	710	2.50	950	7.1	32.0	7000	35.0	7300	31.0	2.40	0.50	0.89	7000
0.044	690	2.83	1260	7.2	40.0	8100	45.0	8700	43.0	2.60	0.73	0.75	8200
0.240	820	1.55	410	9.4	12.7	5500	17.0	5400	13.2	1.45	0.130	1.04	5400
0.290	840	1.49	350	10.4	12.0	5300	15.0	5200	11.7	1.30	0.10	1.05	5200
0.360	860	1.92	240	22.0	19.0	6700	25.0	6800	20.0	4.60	0.24	1.35	6700
0.440	870	1.87	200	25.0	18.0	6700	24.0	6600	18.3	4.80	0.21	1.40	6600

IONIZATION DEVELOPED WITH PLASMAS OF THERMODYNAMICS 224

Table 5.3 Thermodynamic parameters of cesium behind a reflected shock wave

					Experiment I																Theory
																II									III
p0,				u,	v1,	v2,	p2,			H2,				H,					H,				ne,
			T0, K										1 T2, K					T , K								3 α	Γ	T , K
10	5	Pa		km s−	1 cm3 g−1cm3 g−1			5	Pa	10	2	J g−		10	2	J g−	1		10	2	J g−	1	10	13	cm−
							10
0.230			820	0.90	680	250	12.0			9.0			4 400	10.4				4 500	8.5				0.75			0.04	1.0	4 500
0.155			790	1.10	820	260	15.0			13			5 600	17				5 600	13				2.1			0.12	1.2	5 600
0.094			760	1.94	700	170	60			42			9 600	45				10 700	42				16			0.60	1.4	9 900
0.047			690	2.52	1150	300	64			73			15 600	73				17 300	74				14			0.95	0.66	15 700
0.166			790	1.70	450	110	70			32			8 500	35				9 000	30				16			0.38	1.7	8 600
0.320			840	1.58	280	75	95			27			8 700	33				8 800	26				19			0.32	1.9	8 500
0.180			800	1.96	420	100	100			42			10 000	44				11 000	41				25			0.56	1.6	10 300
0.064			710	2.75	860	240	104			90			19 700	85				21 400	90				19			0.97	0.54	19 500
0.130			790	2.30	490	144	130			64			15 200	70				17 400	71				28			0.90	0.95	15 500
0.440			870	1.87	190	50	200			38			11 000	41				11 200	37				43			0.45	2.1	10 500

EXPERIMENTS WITH COMPARISON

225

Table 5.4 Thermodynamic properties of shock–compressed argon

			Experiment														Theory
											Ring Debye approximation												Bounded atom
																							(5.77)–(5.80)
					v,
	p0,	D,	u,	p,		H,			v,		T , 10		3		K	ne,		Γ	nσ		3		v,		T , 10			3	K
	105 Pa	km s−1 km s−1 108 Pa cm3 g−1 10−3					J g−1cm3 g−1									1020 cm−3							cm3 g−1
	0.78	7.78	6.94	0.692	86.0	30.1	88.2				22.0					0.642		0.843	0.00422				90.6		21.6
	1.0	6.5	5.59	0.597	87	20.9	81.7				19.5					0.401		0.798	0.0057				85.5		19.1
	3.0	7.93	6.97	2.73	25.0	31.1	24.0				24.3					2.23		1.35	0.016				25.6		23.0
	5.0	6.51	5.52	2.96	18.8	20.9	17.45				21.3					1.63		1.4	0.0277				19.9		19.9
	10.0	7.6	6.52	8.19	8.78	28.4	7.8				25.1					5.405		2.01	0.055				9.06		22.3
	20.0	7.35	6.28	15.4	4.47	26.6	4.05				25.3					8.68		2.51	0.113				5.19		20.5
	20.0	7.65	6.4	16.3	5.01	28.6	4.00				26.0					9.77		2.54	0.11				5.0		21.3
				Table 5.5 Thermodynamic properties of shock–compressed xenon
			Experiment														Theory
										Ring Debye approximation													Pseudopotential
																						model (5.63)–(5.71)
					v,
	p0,	D,	u,	p,		H,			v,		T , 10	3		K		ne,		Γ	nσ	3		v,		T , 10		3	K
	105 Pa	km s−1 km s−1 108 Pa			cm3 g−1 10−3		J g−1cm3 g−1									1020 cm−3						cm3 g−1
	1.0	4.58	4.11	1.02	19.3	10.4	18.8				21.0					1.33		1.31	0.0163			21.0		21.6
	3.0	7.81	7.00	8.99	6.43	30.2	6.00				42.0					9.71		1.38	0.051			7.07		41.7
	5.0	6.35	5.39	9.49	5.56	19.7	3.77				33.1					11.4		1.99	0.0812			4.62		33.7
	10.0	6.23	5.4	19.2	2.38	19.1	1.83				34.0					22.0		2.64	0.168			2.34		35.1
	10.0	8.93	7.69	39.2	2.48	39.1	1.94				55.8					34.9		1.78	0.158			2.40		55.4
	10.0	4.11	3.52	8.27	2.56	8.32	2.04				21.6					7.2		2.91	0.150			2.4		23.4

IONIZATION DEVELOPED WITH PLASMAS OF THERMODYNAMICS 226

Fig. 5.28.

COMPARISON WITH EXPERIMENTS

227

kJ g

.	MPa
MPa
	.	eV
pv		kJ g

Caloric equation of state for a cesium plasma (Iosilevskii and Gryaznov 1981), p = 12.5 − 50 MPa: 1 – experiment (Bushman et al. 1975); 2 – experiment (Dikhter and Zeigarnik 1977); 3 – calculation with the ring approximation in a grand canonical ensemble (5.50)–(5.52); 4 – Thomas–Fermi approximation with quantum and exchange corrections.

The ultrahigh–pressure region was investigated by Vladimirov et al. (1984) and Avrorin et al. (1987) using the technique of powerful shock waves. Comparison of these results and the results obtained with the plasma methods is given in Fig. 5.29 (Nikiforov et al. 1979). The plasma parameters are specified for some points. Curves 1, 2, 4, and 5 in Fig. 5.29 indicate the data by Nikiforov et al. (1979), 1 is the modified Hartree–Fock–Slater method (Nikiforov et al. 1979 and Novikov 1985), 3 is the self–consistent field method (Sin’ko 1983), and 4 is the Thomas–Fermi theory with quantum and exchange corrections (Kalitkin and Kuzmina 1975, 1976). Curve 2 indicates the plasma model of bound atom (see Section 5.4) which takes into account the electron degeneracy, Coulomb nonideality, excitation of bound states, and short–range repulsion (Nikiforov et al. 1979), whereas 5 is the same, but with the contribution made by the equilibrium radiation.

One can see that the plasma model of the bound atom reproduces reasonably well the states of ultrahigh energy density. Unfortunately, the precision of measurements performed by Vladimirov et al. (1984) was insu cient for a thorough analysis of advantages and peculiarities of models describing the superdense compression and, in particular, the role of the shell e ects in thermodynamics. However, these experiments enable one to follow the asymptotic properties of the theories. During the compression in experiments by Vladimirov et al. (1984), the parameter of plasma nonideality decreased from 4.3 to 0.05, although the ther-

228 THERMODYNAMICS OF PLASMAS WITH DEVELOPED IONIZATION

. TPa

	.	.
		cm
	.
	.	J cm
		.
		.
		cm
	.	.
		J cm
	cm
	J cm
		Ideal gas

Fig. 5.29. Shock adiabats for aluminum at ultrahigh pressures (Vladimirov et al. 1984;

Avrorin et al. 1987) (see text for notations): and – light–gas guns and explosives, respectively; ◦ — (Volkov et al. 1980; Akkerman et al. 1986); • — (Vladimirov et al.

1984); dashed region is from (Model’ et al. 1985); ◦ — (Vladimirov et al. 1984); —

(Simonenko et al. 1985).

modynamic parameters turned out to be exotically high, namely, p 400 TPa, T 7 · 106 K, and ne 3.6 · 1024 cm−3, with the specific energy density close to 1 GJ cm−3. One can see from Fig. 5.29 that the pressure range of thousands of TPa is the most interesting one for revealing the role of the shell e ects, since the estimates of the bound states contribution based on di erent models have the worst accuracy in this range. Figure 5.30 gives the results of measurements by Avrorin et al. (1987) of the relative compressibility of aluminum and copper (taking iron as the standard), as well as the comparison of these results with the quasiclassical model of Kalitkin and Kuzmina (1975, 1976) (curve 2) and with the cell model of the self–consistent field of Sin’ko 1983 (curve 1). The experimental results clearly demonstrate the substantial contribution by the bound states that cause nonmonotonic behavior of the thermodynamic functions of dense plasma. Naturally, this cannot be described by the Thomas–Fermi model with the quantum and exchange corrections (Kalitkin and Kuzmina 1975, 1976).