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The limiting electric conductivity of plasma
Physics Letters A 319 (2003) 510–513 www.elsevier.com/locate/pla
The limiting electric conductivity of plasma E.I. Asinovskii, V.V. Markovets ∗ Institute for High Temperatures, Associated Institute for High Temperatures RAS, Izhorskaya 13/19, 125412 Moscow, Russia Received 20 October 2003; accepted 5 November 2003 Communicated by F. Porcelli
Abstract The results of all most known experiments on the electric conductivity of nonideal plasmas are analyzed. It is shown that the value of conductivity is limited by the plasma frequency. 2003 Elsevier B.V. All rights reserved. Keywords: Nonideal plasmas; Limiting electric conductivity
1. Introduction
Drude’s formula
Nonideal plasmas wherein the energy of Coulomb interaction between charged particles exceeds the thermal one fill a highly important place in current plasma studies. Generation of these plasmas under laboratory conditions and their theoretical and experimental investigations offer great difficulties. In this Letter are analyzed the results of all most known experiments on such plasmas spanning the wide range of magnitude in particle densities. On this basis it is shown that electric conductivity of strongly coupling Coulomb systems is governed by a general law.
σ=
e2 ne l, me ν¯e
where l is the electron free path. In Coulomb plasmas there exists the relation between the electron free path and the correlation length (Debye radius) l rD . On increasing the plasma particle density (ND → 1), the free path and Debye radius approach each other. In the limit of high densities l ≈ rD .
2. The limiting electric conductivity of plasma The notion of the limiting electric conductivity of plasma can naturally be developed starting from the * Corresponding author.
E-mail address: [email protected] (V.V. Markovets). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.11.004
In this case the formula for the electric conductivity takes a simple form: σ=
e2 ne rD ≈ ωp , me ν¯e
where ωp is the plasma frequency. This formula is valid both for classical and degenerate plasmas.
E.I. Asinovskii, V.V. Markovets / Physics Letters A 319 (2003) 510–513
Fig. 1. Experimental dependence of the electric conductivity of cesium plasma on temperature at pressure of 125 atm: 1—[14], 2—[2], 2a—the error of measurements [2], 3—calculation of the limiting electric conductivity by formula (1).
511
Fig. 2. The density dependence of the electric conductivity of hydrogen plasma [7]: white squares [15]; black squares [16]; solid line—calculation by formula (1).
3. Analysis of experiments 3.1. In paper [1], the analysis has been performed of experimental data [2] on the electric conductivity of a strongly ionized cesium plasma (see Fig. 1) and it was shown that at large values of the nonideality parameter Γ [3] electric conductivity σ reaches its lower limit given by formula ωp . (1) 4π Here plasma frequency ωp should be understood as the limiting reciprocal of the correlation time of current densities in plasma. σ=
3.2. In succeeding years some theoretical papers were published wherein the above-mentioned formula was obtained [3–6]. These theoretical papers can conceptually be divided into two groups. The first group includes papers [3,4] allowing for the percolation effect on calculating electric conductivity. In papers of the second group [5,6], the electron–plasma wave interaction is discussed. The theoretical models are not concerned with in this Letter. 3.3. In the last twenty years the results of four further experimental studies of relevance to the subject of this Letter were published.
Fig. 3. The current density dependence of the electric conductivity of a plasma of the electric discharge cooled to ∼ 5 K [8].
In paper [7], are given the data on the electric conductivity of nonideal hydrogen plasma in a megabar range of dynamic pressures (see Fig. 2). In [8] the change in the electric conductivity mechanism in nonideal low-pressure helium plasma cooled to ∼ 5 K was established (see Fig. 3). In [9] the electric conductivity of aluminum foils at high level of the energy deposited (see Fig. 4) was measured. These findings allow verifying the con-
512
E.I. Asinovskii, V.V. Markovets / Physics Letters A 319 (2003) 510–513
Fig. 4. Results of [9] for the relative resistance of aluminum foils versus the specific energy released: solid curves are constructed by approximation formulae at different stages; points—experimental data; horizontal dash line—calculation of the relative resistance by formula (1) for the critical density.
clusions inferred from the experiments with cesium plasma through correlation with new data obtained in experiments on hydrogen plasma at megabar pressures, on low-pressure (∼ 10 torr) cryogenic helium plasma, and on aluminum plasma of the density approaching the critical point. In paper [10], dusty plasma under microgravitation conditions was investigated. The measured plasma parameters are given in Table 1 where the dusty particle parameters are marked off by subscript ‘d’, and Ze is their charge. They permit determining both ω and the dusty component electric conductivity σd =
Dd (Ze)2 nd kTd
using Einstein’s relationship [11]. 3.4. In Fig. 5 are collected the data obtained in all these five experiments. In addition, recent data [12] on the specific resistance of graphite (730 µ cm) and its density (0.55 g/cm3 ) extracted from electric explosion experiments are given in the graph.
Fig. 5. Experimental data on the limiting electric conductivity of strongly nonideal plasma for the maximum values of the nonideality parameter Γ [3] obtained in the experiments (the chosen points are marked off by circles in Figs. 1–4). He—cryogenic plasma of a glow discharge in helium: T = 5 K, ne = 8.9 × 108 cm−3 (Γ = 0, 4) [8]; Cs—cesium plasma: T = 10000 K, ne = 9 × 1019 cm−3 (Γ = 0, 5) [2]; H2 —hydrogen plasma: σ = 2 × 103 cm−1 , ρ = 0.8 g cm−3 [7]; Al—explosion of aluminum foil (the plasma density is assumed to approach the aluminum density at the critical point) [9]; C—explosion of graphite in a capillary [11]; dusty plasma under microgravitation condition [10]; solid line—linear approximation.
The formula for the limiting electric conductivity wholly satisfactorily describes the experimental results. Linear regression gives the value of the slope equal to 1.04 at a standard error of 0.001 which approaches index 1 for the limiting electric conductivity of plasma. Two special features should be emphasized: – The experimental schemes in all the studies mentioned are radically different (glow discharge at cryogenic temperatures [8]; electric explosion of aluminum foil [9], of graphite [12] and of cesium wire at high pressure [2]; dynamic compression of hydrogen [7]; dusty plasma [10]). – The range of changing the electron number densities spans 30 orders of magnitude and of ∼ 18 orders of magnitude in particle masses. 3.5. Taken together, the above-listed facts allow one to hypothesize that there exists some universal
E.I. Asinovskii, V.V. Markovets / Physics Letters A 319 (2003) 510–513
513
Table 1 Particle radius, µm
Specific weight, g cm−3
Diffusion coefficient Dd , cm2 /s
Z ∗ nd , cm−3
Z
Td , K
35
8.2
5 × 10−4
3 × 109
8 × 105
1 × 109
conduction mechanism in all the cases considered. Outside the limiting conductivity mode the ordinary collision channel l rD prevails.
4. Note Ioffe and Regel [13] proposed that de Broglie wavelength λdB may be used for the lower limit of the electron free path in semiconductors that is equivalent to the following relation: l λdB , and, correspondingly, the formula for the electric conductivity should take the form σ=
e2 ne e2 l . λ≈ me ν¯ e r¯ h
(2)
Comparison of formulae (1) and (2) results in different dependencies of σ on ne . In the first case 1/2
σ ∼ ne . In the second case 1/3
σ ∼ ne .
Acknowledgements The authors thank Prof. V.E. Fortov for fruitful discussions and Dr. O.A. Vaulina for kindly presenting the results of experiments on dusty plasma under microgravitation conditions.
References [1] E.I. Asinovskii, A.A. Valuev, High Temp. (USSR) 18 (6) (1980) 1318. [2] I.Ya. Dikhter. Equation of state and electric conductivity of dense strongly ionized plasma. PhD Thesis, Moscow, 1977. [3] V.E. Fortov, I.T. Iakubov, Nonideal Plasma, Energoatomizdat, Moscow, 1994. [4] V.S. Vorob’ev, High Temp. (USSR) 25 (3) (1987) 430. [5] Yu.K. Kurilenkov, A.A. Valuev, Beitr. Plasmaphys. 24 (3) (1984) 161. [6] A.A. Valuev, A.S. Kaklyugin, G.E. Norman, Zh. Eksp. Teor. Fiz. 113 (3) (1998) 880. [7] V.E. Fortov, V.Ya. Ternovoi, S.V. Kvitov, et al., Pis’ma Zh. Eksp. Teor. Fiz. 69 (12) (1999) 874. [8] E.I. Asinovskii, A.V. Kirillin, V.V. Markovets, V.E. Fortov, Dokl. Akad. Nauk 378 (3) (2001) 326. [9] G.A. Mesyats, Ectons in Vacuum Discharge: Breakdown, Sparks, Arc, Nauka, Moscow, 2000. [10] A.P. Nefedov, O.S. Vaulina, O.F. Petrov, et al., JETP 95 (4) (2002) 673. [11] C. Kittel, Elementary Statistical Physics, Wiley, New York, 1958. [12] V.N. Korobenko. Experimental investigation of properties of liquid metals and carbon. PhD Thesis, Associated Institute for High Temperatures, Moscow, 2001. [13] A.F. Ioffe, A.R. Regel, Radiation. Electrons. Semiconductors, in: Selected Scientific Works by A.F. Ioffe, Vol. II, Nauka, Leningrad, 1975. [14] P.P. Kulik, E.K. Rozanov, V.A. Ryabay, High Temp. (USSR) 15 (2) (1977) 415. [15] V.Ja. Ternovoi, A.S. Filimonov, V.E. Fortov, et al., Physica B 265 (1999) 6. [16] S.T. Weir, A.C. Mitchell, W.J. Nellis, Phys. Rev. Lett. 76 (1996) 1860.
The limiting electric conductivity of plasma E.I. Asinovskii, V.V. Markovets ∗ Institute for High Temperatures, Associated Institute for High Temperatures RAS, Izhorskaya 13/19, 125412 Moscow, Russia Received 20 October 2003; accepted 5 November 2003 Communicated by F. Porcelli
Abstract The results of all most known experiments on the electric conductivity of nonideal plasmas are analyzed. It is shown that the value of conductivity is limited by the plasma frequency. 2003 Elsevier B.V. All rights reserved. Keywords: Nonideal plasmas; Limiting electric conductivity
1. Introduction
Drude’s formula
Nonideal plasmas wherein the energy of Coulomb interaction between charged particles exceeds the thermal one fill a highly important place in current plasma studies. Generation of these plasmas under laboratory conditions and their theoretical and experimental investigations offer great difficulties. In this Letter are analyzed the results of all most known experiments on such plasmas spanning the wide range of magnitude in particle densities. On this basis it is shown that electric conductivity of strongly coupling Coulomb systems is governed by a general law.
σ=
e2 ne l, me ν¯e
where l is the electron free path. In Coulomb plasmas there exists the relation between the electron free path and the correlation length (Debye radius) l rD . On increasing the plasma particle density (ND → 1), the free path and Debye radius approach each other. In the limit of high densities l ≈ rD .
2. The limiting electric conductivity of plasma The notion of the limiting electric conductivity of plasma can naturally be developed starting from the * Corresponding author.
E-mail address: [email protected] (V.V. Markovets). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.11.004
In this case the formula for the electric conductivity takes a simple form: σ=
e2 ne rD ≈ ωp , me ν¯e
where ωp is the plasma frequency. This formula is valid both for classical and degenerate plasmas.
E.I. Asinovskii, V.V. Markovets / Physics Letters A 319 (2003) 510–513
Fig. 1. Experimental dependence of the electric conductivity of cesium plasma on temperature at pressure of 125 atm: 1—[14], 2—[2], 2a—the error of measurements [2], 3—calculation of the limiting electric conductivity by formula (1).
511
Fig. 2. The density dependence of the electric conductivity of hydrogen plasma [7]: white squares [15]; black squares [16]; solid line—calculation by formula (1).
3. Analysis of experiments 3.1. In paper [1], the analysis has been performed of experimental data [2] on the electric conductivity of a strongly ionized cesium plasma (see Fig. 1) and it was shown that at large values of the nonideality parameter Γ [3] electric conductivity σ reaches its lower limit given by formula ωp . (1) 4π Here plasma frequency ωp should be understood as the limiting reciprocal of the correlation time of current densities in plasma. σ=
3.2. In succeeding years some theoretical papers were published wherein the above-mentioned formula was obtained [3–6]. These theoretical papers can conceptually be divided into two groups. The first group includes papers [3,4] allowing for the percolation effect on calculating electric conductivity. In papers of the second group [5,6], the electron–plasma wave interaction is discussed. The theoretical models are not concerned with in this Letter. 3.3. In the last twenty years the results of four further experimental studies of relevance to the subject of this Letter were published.
Fig. 3. The current density dependence of the electric conductivity of a plasma of the electric discharge cooled to ∼ 5 K [8].
In paper [7], are given the data on the electric conductivity of nonideal hydrogen plasma in a megabar range of dynamic pressures (see Fig. 2). In [8] the change in the electric conductivity mechanism in nonideal low-pressure helium plasma cooled to ∼ 5 K was established (see Fig. 3). In [9] the electric conductivity of aluminum foils at high level of the energy deposited (see Fig. 4) was measured. These findings allow verifying the con-
512
E.I. Asinovskii, V.V. Markovets / Physics Letters A 319 (2003) 510–513
Fig. 4. Results of [9] for the relative resistance of aluminum foils versus the specific energy released: solid curves are constructed by approximation formulae at different stages; points—experimental data; horizontal dash line—calculation of the relative resistance by formula (1) for the critical density.
clusions inferred from the experiments with cesium plasma through correlation with new data obtained in experiments on hydrogen plasma at megabar pressures, on low-pressure (∼ 10 torr) cryogenic helium plasma, and on aluminum plasma of the density approaching the critical point. In paper [10], dusty plasma under microgravitation conditions was investigated. The measured plasma parameters are given in Table 1 where the dusty particle parameters are marked off by subscript ‘d’, and Ze is their charge. They permit determining both ω and the dusty component electric conductivity σd =
Dd (Ze)2 nd kTd
using Einstein’s relationship [11]. 3.4. In Fig. 5 are collected the data obtained in all these five experiments. In addition, recent data [12] on the specific resistance of graphite (730 µ cm) and its density (0.55 g/cm3 ) extracted from electric explosion experiments are given in the graph.
Fig. 5. Experimental data on the limiting electric conductivity of strongly nonideal plasma for the maximum values of the nonideality parameter Γ [3] obtained in the experiments (the chosen points are marked off by circles in Figs. 1–4). He—cryogenic plasma of a glow discharge in helium: T = 5 K, ne = 8.9 × 108 cm−3 (Γ = 0, 4) [8]; Cs—cesium plasma: T = 10000 K, ne = 9 × 1019 cm−3 (Γ = 0, 5) [2]; H2 —hydrogen plasma: σ = 2 × 103 cm−1 , ρ = 0.8 g cm−3 [7]; Al—explosion of aluminum foil (the plasma density is assumed to approach the aluminum density at the critical point) [9]; C—explosion of graphite in a capillary [11]; dusty plasma under microgravitation condition [10]; solid line—linear approximation.
The formula for the limiting electric conductivity wholly satisfactorily describes the experimental results. Linear regression gives the value of the slope equal to 1.04 at a standard error of 0.001 which approaches index 1 for the limiting electric conductivity of plasma. Two special features should be emphasized: – The experimental schemes in all the studies mentioned are radically different (glow discharge at cryogenic temperatures [8]; electric explosion of aluminum foil [9], of graphite [12] and of cesium wire at high pressure [2]; dynamic compression of hydrogen [7]; dusty plasma [10]). – The range of changing the electron number densities spans 30 orders of magnitude and of ∼ 18 orders of magnitude in particle masses. 3.5. Taken together, the above-listed facts allow one to hypothesize that there exists some universal
E.I. Asinovskii, V.V. Markovets / Physics Letters A 319 (2003) 510–513
513
Table 1 Particle radius, µm
Specific weight, g cm−3
Diffusion coefficient Dd , cm2 /s
Z ∗ nd , cm−3
Z
Td , K
35
8.2
5 × 10−4
3 × 109
8 × 105
1 × 109
conduction mechanism in all the cases considered. Outside the limiting conductivity mode the ordinary collision channel l rD prevails.
4. Note Ioffe and Regel [13] proposed that de Broglie wavelength λdB may be used for the lower limit of the electron free path in semiconductors that is equivalent to the following relation: l λdB , and, correspondingly, the formula for the electric conductivity should take the form σ=
e2 ne e2 l . λ≈ me ν¯ e r¯ h
(2)
Comparison of formulae (1) and (2) results in different dependencies of σ on ne . In the first case 1/2
σ ∼ ne . In the second case 1/3
σ ∼ ne .
Acknowledgements The authors thank Prof. V.E. Fortov for fruitful discussions and Dr. O.A. Vaulina for kindly presenting the results of experiments on dusty plasma under microgravitation conditions.
References [1] E.I. Asinovskii, A.A. Valuev, High Temp. (USSR) 18 (6) (1980) 1318. [2] I.Ya. Dikhter. Equation of state and electric conductivity of dense strongly ionized plasma. PhD Thesis, Moscow, 1977. [3] V.E. Fortov, I.T. Iakubov, Nonideal Plasma, Energoatomizdat, Moscow, 1994. [4] V.S. Vorob’ev, High Temp. (USSR) 25 (3) (1987) 430. [5] Yu.K. Kurilenkov, A.A. Valuev, Beitr. Plasmaphys. 24 (3) (1984) 161. [6] A.A. Valuev, A.S. Kaklyugin, G.E. Norman, Zh. Eksp. Teor. Fiz. 113 (3) (1998) 880. [7] V.E. Fortov, V.Ya. Ternovoi, S.V. Kvitov, et al., Pis’ma Zh. Eksp. Teor. Fiz. 69 (12) (1999) 874. [8] E.I. Asinovskii, A.V. Kirillin, V.V. Markovets, V.E. Fortov, Dokl. Akad. Nauk 378 (3) (2001) 326. [9] G.A. Mesyats, Ectons in Vacuum Discharge: Breakdown, Sparks, Arc, Nauka, Moscow, 2000. [10] A.P. Nefedov, O.S. Vaulina, O.F. Petrov, et al., JETP 95 (4) (2002) 673. [11] C. Kittel, Elementary Statistical Physics, Wiley, New York, 1958. [12] V.N. Korobenko. Experimental investigation of properties of liquid metals and carbon. PhD Thesis, Associated Institute for High Temperatures, Moscow, 2001. [13] A.F. Ioffe, A.R. Regel, Radiation. Electrons. Semiconductors, in: Selected Scientific Works by A.F. Ioffe, Vol. II, Nauka, Leningrad, 1975. [14] P.P. Kulik, E.K. Rozanov, V.A. Ryabay, High Temp. (USSR) 15 (2) (1977) 415. [15] V.Ja. Ternovoi, A.S. Filimonov, V.E. Fortov, et al., Physica B 265 (1999) 6. [16] S.T. Weir, A.C. Mitchell, W.J. Nellis, Phys. Rev. Lett. 76 (1996) 1860.