Journal of the Korean Physical Society, Vol. 62, No. 3, February 2013, pp. 447∼452

Ionization and Electrical Conductivity of Dense Carbon Zhijian Fu∗ School of Electrical and Electronic Engineering, Chongqing University of Arts and Sciences, Chongqing 402160, China

Xiaowei Sun School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China

Weilong Quan National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, Chinese Academy of Engineering Physics, Sichuan 621900, China (Received 12 September 2012) The electrical conductivity of a dense carbon plasma has been calculated by using a linear mixture rule considering various interactions among electrons, atoms, and ions in the density and temperature ranges of 10−5 – 10 g cm−3 and 104 – 105 K, respectively. The non-ideal Saha equation is used to obtain the plasma composition and the degree of ionization successfully. The present calculation spans from the weakly coupled, nondegenerate region to the strong coupled, degenerate domain. The calculated conductivity is in reasonable agreement with the explosive wire measurement and the quantum kinetic calculation. A nonmetal-metal transition is predicted to occur at 0.56 g cm−3 at temperatures lower than 3 × 104 K based on the calculation of the electrical conductivity. PACS numbers: 52.25.Fi, 52.25.Jm, 52.27.Gr Keywords: Carbon plasma, Linear mixture rule, Ionization, Electrical conductivity DOI: 10.3938/jkps.62.447

I. INTRODUCTION

the electrical conductivity of copper plasmas [10], but the average electron-neutral momentum transport cross section in a copper plasma is given a fixed value which is not suitable for carbon plasmas. In order to calculate the cross section of the carbon plasma, we adopted an accurate fit, which has been employed to calculate the electrical conductivity of aluminum and copper plasmas [21]. The present calculation was verified with available experimental and theoretical results.

The electrical conductivity of warm dense matter is very important information for understanding the complex behaviors of matter at high temperature and high density [1–3]. Many experiments [4–9] and theoretical works [6, 10–18] have been concerned with metal plasmas. However, nonmetal plasmas, such as carbon plasmas, have been inadequately investigated. Haun et al. [19] employed an explosive wire experiment in a glass capillary to obtain the electrical conductivity of a carbon plasma in the range of 9 – 16 kK and 1021 – 1023 cm−3 . DeSilva et al. [20] measured the electrical conductivity of a carbon plasma by using an explosive wire experiment in a water bath in the range from 0.6 solid density down to 0.01 solid density and from 2 to 22 kJ gm−1 . Also, Haun et al. [19] calculated the electrical conductivity of a carbon plasma by solving the quantum kinetic equations. In this work, the non-ideal Saha equation is used to calculate the plasma composition. A linear mixture rule taking into account the interaction among electrons, atoms, and ions is applied to obtain the electrical conductivity, which has been successfully used to investigate ∗ E-mail:

II. THEORETICAL METHOD 1. Plasma Composition

The coupling and the degeneracy parameters are used for characterizing the plasma state. The ion-ion coupling parameter Γii is shown to be the ratio of the mean electrostatic potential energy to the mean kinetic energy of the ions, and the degeneracy parameter Θ is the ratio of the thermal energy to the Fermi energy, which determines the Fermi degenerate region, by estimating the importance of quantum statistical effects [2,3,22–24]:

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  2 2
e Zeff 4π i=1 ni , 4πε0 kB T 3 2me  2 − 23 3π ne , Θ = kB T
2

the particle number density of heavy particles, and αi = ni /nH is the molar fraction of i-fold ions. A dense carbon plasma satisfies conservation of charge and conservation of nuclei:

1 3

Γii =

(1)

where kB is the Boltzmann constant, T is the temperature, 0 is the permittivity of vacuum, ni is the particle number density of i-fold ions, ne is the particle number density of free electrons. me is the mass of an electron, and
is the reduced Planck’s constant. Zeff is the effective charge number of positive ions [1,25,26]. Usually, the regions where Γii
1 is the nearly ideal plasma region, where Γii ≤ 1 is the weakly coupled plasma region, and where Γii > 1 is the strongly coupled plasma region [2,3]. Θ > 1 indicates that the system is in the nondegenerate domain, and Θ < 1 indicates the degenerate system in which quantum effects play an important role [2]. Under local thermodynamic equilibrium, the Saha equation for a single elemental species plasma is express as   3/2 Ui 2πme kB T Iieff ni ne , = 2 exp − ni−1 Ui−1 h2 kB T (i = 1, . . . , Z).

(2)

In Eq. (2), Ui is the internal partition functions of i-fold ions, and h is Planck’s constant. The effective ionization energy Iieff = Ii −∆Ii is relative to the ionization process i → (i + 1), where ∆Ii is the depression in the ionization potential. In this model, the depression in the ionization potential is written as ∆Ii =

(i + 1)e2 , 4πε0 Ri∗

(3)

where e is the electronic charge, i is the charge state of the ion, and Ri∗ is the characteristic radius with respect to the Debye length (λD ) and the ion-sphere radius ai [10],


2 2 ∗ 2 ai , (4) Ri = λD + 3 with the Debye length λD and the ion-sphere radius ai being defined as, respectively, ⎡ ⎤ 12 λD

⎢ ⎢ = ⎢ ⎢ ⎣ 

ai =

⎥ ⎥ kB T ε0  ⎥ ⎥ , Z  ⎦ 2 e2 nH Zav + αi × i i=1

3(i + 1) 4πnH (1 + Zav )

 13

.

(5)

In the above equations, Zav is the average ionization state, the degree of ionization of the plasma, nH , is

Z 

αi = 1,

i=0

Z 

iαi = Zav .

(6)

i=0

Combining the non-ideal Saha equation with conservation of charge and conservation of nuclei, we can obtain the average ionization state, Zav , and the molar fraction of various ions, αi , which are essential for the calculation of the electrical conductivity. In addition, the internal partition function Ui is expressed as 
∞  Ei Ui = , (7) gi exp − kB T i=1 where gi is the statistical weight and Ei is the ith excitation energy. If for the computation is to converge, the excitation energy needed is terminated by the relation Ei ≤ Iieff = Ii − ∆Ii . A large set of excitation energy levels for carbon has been applied in the computations of the internal partition function. These data come from an extensive database of electronic excited states compiled by the National Institute of Standards and Technology .

2. Electrical Conductivity

The electrical conductivity is calculated by using a linear mixture rule in which the electron-electron and electron-ion coulomb effect as well as the electron-atom interaction are considered. The linear mixture rule can be given by [26,29] 1 1 1 = + , σ σei een

(8)

where σei and σen are the electrical conductivities associated with the electron-ion and electron-neutral collisions, respectively. The electron-ion conductivity can be represented as σei =

ion ) T 1.5 γe (Zav . ion 38Zav ln Λ

(9)

ion Here, Zav is the average ionization state of ions, lnΛ is the Coulomb logarithm of the classical collision crosssection integral for interactions of electrons with ions [30, 31], π sin(1.5/Λ) ln Λ =
 2 Ci(1.5/Λ) 2 Si(1.5/Λ) + , × 1− π tan(1.5/Λ)    1/4 ¯0 4 , Λ ≈ (2.4287ΛB ) 1 + αi /2.4287ΛB b  ΛB = h/ 2πme kB T . (10)

Ionization and Electrical Conductivity of Dense Carbon – Zhijian Fu et al.

¯0 is an Where ΛB is the de Broglie wavelength and b ion ) is writaverage impact parameter from Ref. 32. γe (Zav ten as [33]
 153x2 + 509x 3π γe (x) = 1+ . (11) 32 64x2 + 345x + 288

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The electron-neutral conductivity can be written as  π zav e2 (12) σen = ¯ en , 8kB me T 0.5 α0 Q ¯ en is where α0 is molar fraction of neutral particles and Q the average electron-neutral momentum transport crosssection calculated in the Born approximation [21] and expressed as

 π 3 (αD /2r0 aB )2 4 , κ = 1/λ , r = αD aB /2Z 1/3 , D 0 A2κ + 3Bκ kr0 + 7.5Cκ (kr0 )2 − 3.4Dκ (kr0 )3 + 10.6668Eκ (kr0 )4 π 7 1 + 22κr0 − 11.3(κr0 )2 + 33(κr0 )4 , = 1 + 2κr0 + 2 (κr0 )2 + (κr0 )3 , Bκ = exp(−18κr0 ), Cκ = π 7 1 + 6κr0 + 4.7(κr0 )2 + 2(κr0 )4 1 + 28κr0 + 13.8(κr0 )2 + 3.2(κr0 )3 = , Eκ = 1 + 0.1κr0 + 0.3665(κr0 )2 . (13) 1 + 8 + 10(κr0 )2 + (κr0 )3

¯ en = Q Aκ Dκ

Fig. 1. (Color online) Coupling and degeneracy parameters of a carbon plasma as functions of density and temperature.

In Eq. (13), αD is the dipole polarizability, aB is the Bohr radius, κ is the inverse screening length, k is the electron wave number, and r0 is the cutoff radius.

III. RESULTS AND DISCUSSION Figure 1 shows the coupling and the degeneracy parameters of a carbon plasma as a function of density and temperature in the density and the temperature ranges of 10−5 – 10 g cm−3 and 104 – 105 K. As can be seen from Fig. 1(a), in the low-temperature limit (around 104 K), the ion-ion coupling parameter Γii becomes larger than 1 at 0.5 g cm−3 , which means the carbon plasma is in the

Fig. 2. (Color online) Degree of Ionization αe of carbon plasma as functions of density at different temperatures.

strongly-coupled plasma region. The strongly-coupled region shifts towards lower density with increasing temperature. In particular, the ion-ion coupling parameter increases from 10 to 342 at densities higher than 1 g cm−3 over the whole temperature range, which indicates that the Coulomb interaction is very important and dominates in that region. At the same time, a carbon plasma is mostly a degenerate plasma at densities higher that 0.6 g cm−3 over the whole temperature range where the degeneracy parameter Θ is less than 1. In addition, the degeneracy effect can be neglected at densities up to 1024 cm−3 for the ion density in an aluminum plasma [34]. The present calculation for a carbon plasma is at densities considerably lower than the above ion density,

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Fig. 3. (Color online) Degree of ionization degree αe and plasma composition αi as functions of temperature for a particle number density 1.4 × 1022 cm−3

so the degeneracy effect is not considered in the work. As shown in the figure, the strongly coupled, degenerate region corresponds to the lower-temperature, higherdensity region. Figure 2 shows the degree of ionization of a carbon plasma as a function of density in the temperature range of 1.2 × 104 – 105 K. At constant temperature, the degree of ionization decreases as the density increases when the density is lower than 0.5 g cm−3 . Then, the ionization degree shows a clear increase until it reaches a peak. Meanwhile, at fixed density, the degree of ionization increases with increasing temperature at densities lower than 1 g cm−3 , but the trend is reversed; that is to say, the degree of ionization decreases as the temperature increases when the density is higher than 2 g cm−3 . In order to investigate the isochoric process in detail, we measured the plasma composition and the degree of ionization of a carbon plasma as functions of temperature in the range of 104 – 106 K at 1.4 × 1022 cm−3 , as shown in Fig. 3. The degree of ionization monotonically increases as the temperature is increased. The number of carbon atoms monotonically decreases with increasing temperature. Carbon atoms always exist at temperatures lower than 2 × 105 K, which means the carbon plasma is in the partially-ionized region, with the plasma becoming completely ionized at higher temperatures. C+1 , C+2 , C+3 , C+4 , and C+5 , respectively reach peak values as the temperature is increased. Figure 4 shows a comparison of the electrical conductivity of a carbon plasma with available experimental and theoretical results [19, 35]. The present calculation is in reasonable agreement with the explosive wire experiments at 12 and 16 kK and shows a relatively underestimated result compared with that at 9 kK [19]. This calculation agreed well with the simulation by Haun et al. [19]. The Spitzer model is appropriate in the low-density

Fig. 4. (Color online) Calculated conductivity compared with experimental and theoretical results. The present results at 9 kK, 12 kK, 16 kK, 20 kK, and 30 kK are marked as straight lines. The symbols show the measurements. The curves with solid symbols are Haun’s calculations. The lines with open symbols are SESAME results at 12.5 kK and 15.9 kK, and the Spitzer model at 30 kK, respectively.

limit, but produces an overestimated result at moderate densities. The SESAME data [35] show a considerable deviations from experiment and other models. In addition, large discrepancies from Haun’s results still exist at 9 kK when the particle number density is higher than 1021 cm−3 . This can be attributed to neglecting the delocalization of electrons in the present calculation. In the lower temperature, higher density region, a carbon plasma is mostly partially degenerate, and an increase in the number of conducting electrons leads to a relatively strong quantum effect. Also, neglecting the quantum effect results in an underestimated conductivity. When the present result is compared with Haun’s calculations point by point at 1.2 × 1021 cm−3 in Table 1, good agreement is observed. In order to obtain a detailed understanding of the trend in the electrical conductivity with changing density and temperature, the isotherms and isochores for the electrical conductivity of a carbon plasma are shown in Fig. 5. The electrical conductivity monotonically increases with increasing temperature at densities lower than 1 g cm−3 while it decreases with increasing temperature at densities higher than 1 g cm−3 . At the same time, a minimum exists at temperatures lower than 3 × 104 K. The position of the minimum is shifted towards lower density as the temperature is increased. When the temperature is higher than 3 × 104 K, the conductivity monotonically increases with increasing density. It should be noted that the reversal of the electrical conductivity with temperature is connected to the nonmetalto-metal transition, which has been a concern in other plasmas such as aluminum [36], copper [37], and hydrogen [38]. As shown in Fig. 5(b), the nonmetal-to-metal

Ionization and Electrical Conductivity of Dense Carbon – Zhijian Fu et al.

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Table 1. Calculated electrical conductivity (σ) compared with σHaun of Haun et al. at 1.2 × 1021 cm−3 . The degree of Ionization (αe ), ion-ion coupling parameter (Γii ), and degeneracy parameter (Θ) are shown. T (K) 9.0 × 103 1.2 × 104 1.6 × 104 2.0 × 104 2.4 × 104 3.0 × 104 3.9 × 104 5.0 × 104

αe 1.29 × 10−3 1.20 × 10−2 6.62 × 10−2 0.168 0.296 0.483 0.715 0.978

σ (S m−1 ) 1.12 × 102 1.10 × 103 7.02 × 103 2.04 × 104 3.84 × 104 6.39 × 104 9.15 × 104 1.13 × 105

σHaun (S 2.33 × 1.75 × 8.51 × 1.94 × 3.31 × 5.25 × 7.73 × 1.02 ×

m−1 ) 102 103 103 104 104 104 104 105

Γii 0.346 0.546 0.724 0.791 0.797 0.760 0.725 0.798

Θ 159 48 20.5 13.7 11.3 10.2 10.2 10.6

Figure 6 shows the isotherm and the isochore of the average electron-neutral momentum transport cross section. On the one hand, the cross section decreases with increasing temperature at densities lower than 0.5 g cm−3 while it increases with increasing temperature at densities higher than 0.5 g cm−3 . On the other hand, for a fixed temperature, the cross section is almost constant at densities lower than 0.56 g cm−3 and monotonically decreases with increasing density at densities higher than 0.56 g cm−3 . The trend of the cross section with density and temperature also indirectly indicates that a nonmetal-to-metal transition occurs in the warm dense matter regime.

Fig. 5. (Color online) Electrical conductivity as a function of (a) temperature or (b) density.

IV. CONCLUSION A simple and effective method, including the non-ideal Saha equation and a linear mixture rule, has been used to investigate the ionization and the electrical conductivity of a dense carbon plasma. The coupling and the degeneracy parameters are used to characterize the plasma from the weakly coupled, nondegenerate region to the strongly coupled, degenerate region. The maximum degree of ionization reaches four in a warm dense carbon plasma, and the change in the degree of ionization with density and temperature corresponds to the trend in the electrical conductivity over the whole range. The comparison with experimental and theoretical results demonstrates that the present model is valid for simulating the transport property of a warm dense carbon plasma in the considered region. A nonmetal-to-metal transition is shown at 0.56 g cm−3 in the lower temperature range.

Fig. 6. (Color online) Isothermal and isochore average electron-neutral momentum-transport cross sections.

transition for a carbon plasma takes place at 0.56 g cm−3 at temperatures lower than 3 × 104 K, and the corresponding conductivity is 2.7 × 105 S m−1 .

ACKNOWLEDGMENTS We would like to thank Professor Mofreh R. Zaghloul for helpful comments. This work was supported by the National Natural Science Foundation of China under

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Grant No. 11074226, the Science and Technology Development Foundation of China’s Academy of Engineering Physics under Grant No. 2012B0101001, the National Basic Research Program of China under Grant No. 2011CB808201, and the Foundation of National key Laboratory of Shock Wave and Detonation Physics Research, China’s Academy of Engineering Physics under Grant No. 9140C670104120C6703.

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