PHYSICAL REVIEW E 88, 063106 (2013)
Thomas-Fermi Z-scaling laws and coupling stabilization for plasmas Philippe Arnault,* Jean Cl´erouin, and Gregory Robert CEA, DAM, DIF, F-91297 Arpajon, France
Christopher Ticknor, Joel D. Kress, and Lee A. Collins Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 12 September 2013; published 17 December 2013) Extending the well-known Thomas-Fermi Z-scaling laws to the Coulomb coupling parameter, we investigate the stabilization of the ionic coupling in isochoric heating [Cl´erouin et al., Phys. Rev. E 87, 061101 (2013)]. This stabilization is restricted to a domain in atomic number Z, temperature, and density, including strong limitations on high couplings, that can only be obtained for high-Z elements. Contact is made with recent isochoric heating experiments. The consequences for corresponding states with respect to ionic coupling are also quantified via orbital free molecular dynamics simulations. This opens avenues for future isochoric heating experiments. DOI: 10.1103/PhysRevE.88.063106
PACS number(s): 52.27.Gr, 52.65.−y
I. INTRODUCTION
A realistic description of a hot dense plasma requires a model that includes all interactions explicitly (N -body problem) in contrast with weakly coupled plasmas that are well represented by only binary collisions and mean-field effects. For dense and fully degenerate plasmas, the one-component plasma (OCP) has served as the reference model [1,2], providing fits and simple expressions for the free energy and its derivatives, pressure, energy, and the transport properties such as diffusion and viscosity. In these expressions, the only variable is the Coulomb coupling parameter, =
Q2 e 2 , 4π 0 akB T
(1)
where Q is the ion charge state, e is the fundamental charge, a = (3/4π n)1/3 is the mean ion sphere radius, n is the plasma ionic density, and T is the temperature. Unfortunately, the OCP model supposes full electronic degeneracy, which is rarely the case, and a knowledge of the ionization Q. The need for a more precise representation of electron screening effects (finite degeneracy) has led to the development of molecular dynamics (MD) simulations using density functional theory (DFT) at finite temperature [3], which have, to considerable accuracy, determined hugoniots [4–6], electrical conductivities in the metal-nonmetal transition range [7,8], and optical properties of shocked materials [9–11]. However, DFTbased methods are strongly limited in temperature, mainly due to the introduction of orbitals to compute the electron kinetic energy in the Kohn-Sham (KS) ansatz. To bypass this limitation, orbital-free (OF) methods have been introduced that substitute a semiclassical formulation of the electron kinetic energy based on the local density approximation, the Thomas-Fermi (TF) functional, for its KS counterpart [12–15]. These OF methods are well adapted to describing dense systems at high temperature, where methods based on orbitals are prohibitively expensive. Several limited studies [16–19] indicate that the transition region between KS and OF calculations is sensitive to the system and its thermodynamic
*
[email protected]
1539-3755/2013/88(6)/063106(7)
conditions. The development of OF functionals remains an active field of research, particularly for finite temperature exchange-correlation functionals [20]. However, for compressions of up to a few times normal solid density, the ThomasFermi-Dirac (TFD) formulations, especially with the vonWeisz¨acker extension, agree within 10% or better with KS. Our main interest in the orbital-free molecular dynamics (OFMD) rests with extraction of the structure of the plasma and its comparison with the OCP. This introduces the concept of an effective OCP, in which the partial degeneracy and the corresponding electron screening is compensated by the introduction of an effective ionization. Actually, current threedimensional (3D) MD simulations of plasmas using DFT in either the KS or the OF formulation do not produce the ion charge state Q or the mean ionization as a standard output parameter since Q is not a quantum observable. Any definition of a charge state under these circumstances is ambiguous and may take different definitions [21]. On the other hand, the pair distribution function (PDF) g(r), which describes the statistics of nuclei interactions, is a standard output of a 3D simulation code. This function contains all the details of the screened interactions and can be easily compared with the OCP PDF, which is tabulated over a wide range of coupling parameters [22]. For pure repulsive systems, an adjustment with respect to can accurately reproduce the closest approach distance, the intensity of the main peak, and the location of the first minimum. This procedure defines an effective OCP model equivalent to the actual system [23]. From the temperature and the density, this determination of the effective coupling gives an estimation of the effective average charge Q. This effective mean ionization accounts for the electron screening of the nuclei interaction, at least for the PDF, and is in good agreement with spherically confined-atom methods when defined by the electron density ne (r) at the sphere radius Q = 4π a 3 ne (a)/3. In this approach, the effective ionization and corresponding effective OCP constitute a coarse-grained description of hot dense plasmas. Interestingly, the effective OCP approach leads also to reasonably accurate estimations of the transport coefficients of plasmas such as diffusion and viscosity [23,24]. In a recent paper [23], we have extracted the coupling parameters of tungsten from a series of OFMD simulations at
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increasing temperatures for a constant density of 40 g/cm3 and observed a quasiconstant coupling ( 20) over a wide range of temperatures between 60 and 800 eV (the plateau). This constant-coupling regime results from a balance between a rise in temperature and an increase in ionization, which is proportional to the square root of the temperature at a constant density. Well known for dilute plasmas [25,26], this behavior of the ionization is also confirmed by average-atom density functional models (see Fig. 2 of Ref. [21]). In this paper, we demonstrate that this effect is present for the simplest theoretical approach: the TF model of a single atom in a Wigner-Seitz cell [27]. As anticipated in Ref. [23], the existence of this plateau depends on the material through its atomic number Z, atomic mass A, and its mass density ρ. Taking advantage of the TF scaling laws, we can with simple formulas predict for any material and experimental conditions the occurrence of this plateau. This capability allows us to make contact with recent isochoric experiments [28–32]. In these experiments, temperatures ranging from 50 to 1000 eV have been reached by electron heating produced by picosecond laser pulses on a buried material [30,31]. The paper is organized as follows: we first review the TF scaling laws and extend them to the coupling parameter , defined using the TF ionization. We therefore introduce a Zscaled coupling parameter ∗ that allows us to build a universal diagram linking the coupling parameter to temperature for different densities. We discuss next the conditions for the occurrence of the plateau and propose simple fits to predict the plateau value of the coupling parameter, ∗ , and the corresponding range of temperature as a function of density. In the next section, we compare our predictions with OFMD simulations and analyze recent isochoric heating experiments. This comparison indicates that the TF ionization is accurate enough to define an effective OCP whose PDF is close to the actual one. The consequences for corresponding states with respect to ionic coupling are then explored. OFMD simulations are performed to corroborate these results.
10
3
10
T = T /Z
4/3
(2a) ,
1
Γ* 10
10
10
0
-1
-2
10
-4
10
-3
10
-2
10
-1
10
0
10
1
(2b)
10
2
10
3
T* (eV) FIG. 1. (Color online) Iso-ρ ∗ (= ρ/ZA) curves of ∗ = /Z as a function of T ∗ = T /Z 4/3 in the TF model.
with respect to which a corresponding state law gives the electron contribution to the thermodynamic properties of any element P (Z,ρ,T ) = Z 10/3 P1 (ρ ∗ ,T ∗ ), E(Z,ρ,T ) = Z
7/3
∗
∗
E1 (ρ ,T ),
(3a) (3b)
where quantities with 1 stand for the pressure (P) and the energy (E) of hydrogen (Z = 1) at reduced density ρ ∗ and temperature T ∗ . The quotidian equation of state (QEOS) model [34] employs such scaling laws. We propose to extend these Z-scaling laws to the coupling parameter , which is rescaled by the atomic number as follows: ∗ = /Z,
The TF model constitutes a simple framework widely used to compute properties of hot dense matter [27,33] although its drawbacks are well known. For instance, the ionization is overestimated at low density. However, the introduction of exchange (Thomas-Fermi-Dirac, TFD) and gradient (Thomas-Fermi-vonWeisz¨acker, TFW) corrections can lead to substantive improvements, leading to a very smooth transition between the TF and KS. As found in detailed studies by Mazevet et al. [16] and Danel et al. [17], the best agreement obtains for the extended Thomas-Fermi-Dirac-vonWeisz¨acker formulation. These improvements emerge at the expense of the loss of the Z-scaling laws. In this paper, we shall restrict ourselves to the simple TF model in order to exploit the power of such scaling laws. Following More et al. [34], we first define a reduced density ρ ∗ and a reduced temperature T ∗ by
∗
ρ* (mol/cm ) = 1 0.1 0.01 0.001 0.0001
2
10
II. Z-SCALING LAWS
ρ ∗ = ρ/ZA,
3
(3c)
using Eq. (2b), a = a1 /Z 1/3 , and the scaled ionization Q∗ , Q∗ = Q(Z,ρ,T )/Z = Q1 (ρ ∗ ,T ∗ ).
(3d)
The effective mean ionization at finite temperature Q1 (ρ ∗ ,T ∗ ) is given by More’s fit [35–37]. It is defined by the electron density at the ion sphere radius a. In Fig. 1, we plot ∗ versus T ∗ for different reduced densities ρ ∗ . For Z = 1, this figure corresponds to the hydrogen case. At density ρ ∗ = 1, the curve is monotonically decreasing. At ρ ∗ ∼ 0.01, we observe a pronounced inflection point, which turns into a plateau at lower density and eventually into a sigmoid curve for the lowest isodensity curve. To illustrate the -plateau feature, we show in Fig. 2 the coupling parameters of selected elements at densities chosen to satisfy ρ ∗ = 10−3 . In this diagram, aluminum at 0.351 g/cm3 is equivalent (same ∗ = /Z) to iron at 1.45 g/cm3 or to gold at 15.6 g/cm3 . For each element, temperatures are obtained by multiplying the abscissa of Fig. 1 by Z 4/3 . The actual coupling parameter in the plateau regime
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1 10 -3
5 10
0.332 0.309
-3
2 10
0.282 -3
1 10
Γ* 0.234 -4
2 10
0.2
0.146
0.01
0.1
1
10
T* (eV)
FIG. 2. (Color online) Coupling parameters of selected elements versus temperature for the ρ ∗ = 10−3 isochore.
ranges from 3 for aluminum to 18 for gold, and the range is shifted towards higher temperatures for high-Z materials.
FIG. 3. (Color online) Scaled coupling ∗ = /Z versus scaled temperature T ∗ = T /Z 4/3 for selected reduced densities ρ ∗ = ρ/AZ ranging from 2 10−4 to 10−2 mol/cm3 , from bottom to top. Horizontal lines mark the mean values defined as the plateaus. The gray area shows the domain where the plateau exists.
hence, for the time needed to heat the sample, allowing for diagnostics.
III. PLATEAU CONDITIONS
IV. PARAMETRIZATION
Figure 3 displays the part of Fig. 1 for values of the reduced density ρ ∗ in the vicinity of the plateau transition with ρ ∗ ranging from 2 × 10−4 (dashed line, bottom) to 1 × 10−3 (heavy solid line) to 0.01 (dashed line, top). This figure looks very similar to a liquid-vapor diagram with a critical point beyond which the plateau disappears. In our diagram, this point is located on the critical density curve ρc∗ = 0.0045 mol/cm3 at Tc∗ = 1.5 eV and c∗ = 0.335. Beyond this density, the coupling parameter decays monotonously with temperature. Below this point the reduced coupling exhibits a loop around a mean-value defined as the plateau. We recall that this curve is universal for TF. For a given atomic number Z, the range of accessible plateau is limited by ∗ < c∗ , namely < 0.335 Z, which means that certain strongly coupled configurations will miss the plateau regime. For example, a strongly coupled aluminum plasma with = 20 cannot exist in the plateau regime. This coupling parameter corresponds to a reduced coupling of ∗ = 1.54, well outside the plateau regime. During isochoric heating, such a coupling can only be obtained in a transient regime, not in a stabilized way as concerns the ionic coupling. Actually, coupling values higher than about 30 cannot be reached in the plateau regime. On the other hand, for gold at 16 g/cm3 , the reduced density ρ ∗ = 10−3 corresponds to a coupling parameter of 18.5 between 33 and 1500 eV ( between 16 and 21 if we consider the oscillation). The ionic coupling of a high-Z dense plasma can be stabilized in a wide range of temperature and,
Figure 3 shows that inside the plateau regime, a one-to-one correspondence exists between the plateau value ∗ and the reduced density ρ ∗ . In Fig. 4, we have plotted selected values of 0.4 0.714 + 0.0695 ln(ρ∗) 0.35
0.3
Γ* 0.25 0.2
0.15
0.1 0.0001
0.001
ρ*
0.01
FIG. 4. (Color online) Plateau value of the reduced coupling ∗ versus reduced density ρ ∗ and its fit. The error bars and the gray area represent the amplitude of the loop.
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the reduced density and the corresponding coupling parameter. The error bars and the gray area represent the amplitude of the loop. In a log-linear plot we find a linear relation which can be fitted by
1
g(r) 0.5
∗
∗ = b + a ln[ρ ],
(4)
where a = 0.0695 and b = 0.714. This fit is meaningless for ρ ∗ > ρc∗ and is not drawn in Fig. 4. The lowest density point (ρ ∗ = 2 10−4 ) with the strongest oscillation around the plateau is not taken into account for the fit. We have also parametrized the temperature extrema defining the plateau ∗ Tmin = 0.0445 exp(765 ρ ∗ ), ∗ Tmax
∗
= 5.66 exp(−248 ρ ).
0
200 eV
400 eV
OFMD ΓOCP = 9
OFMD ΓOCP = 9
800 eV
2000 eV
OFMD ΓOCP = 9
OFMD ΓOCP = 5
1
g(r) 0.5
(5a)
0
0
1
2
0
1
r/a
(5b)
2
3
r/a
FIG. 5. (Color online) PDF of silver at 2.5 g/cm3 and from left to right, top to bottom, 200, 400, 800, and 2000 eV. The red continuous lines are the OCP PDF best fits.
V. SIMULATIONS AND EXPERIMENTS
The model predictions, based on the TF scaling laws, are utilized to analyze results of simulations and experiments for various chemical elements in Table I. The first columns give the properties of the element (atomic number Z, mass A, and normal density ρ0 ); the next columns present the corresponding reduced parameters ρ ∗ from Eq. (2a), and ∗ from Eq. (4), for the actual density ρ; and the following columns display the coupling parameter from Eq. (3c) (in bold) and the approximate temperature range for the plateau from Eqs. (5) and (2b). The last columns give the actual coupling parameters ref (in bold) and temperatures Tref obtained in simulations or experiments. In Ref. [23], a plateau was observed in OFMD simulations of tungsten at 40 g/cm3 . With the model, we find a coupling of 23, slightly higher than the one observed 20 ± 2 with a full OFMD-TFD formulation in Ref. [23]. This comes from the neglect of exchange in TF formulation leading to ionizations 2–4% higher than the TFD ones in this temperature range [21,23]. The temperature range of the plateau is also reproduced by the model. To check the existence of the plateau for lower-Z materials, we have performed OFMD simulations on silver at 2.5 g/cm3 . Within the model, this density corresponds to ρ ∗ = 5 10−4 and to a plateau at = 8.7 between 11 and 850 eV (cf. Table I). OFMD simulations of 432 silver nuclei
in a 3D box were performed for temperatures of 200, 400, 800, and 2000 eV. To be fully coherent with our bare TF approach, these simulations have been performed at TF level of DFT (no exchange, no gradients). The divergence of the electron-nucleus potential is regularized at each thermodynamic condition. The cutoff radius is chosen to be 30% of the Wigner-Seitz radius, sufficient to prevent overlap of the regularization spheres. The number of plane waves describing the local electronic density is then adjusted to converge the thermodynamic properties to within 1%. The MD time step is small enough to ensure good energy conservation [23]. The plateau is clearly observed in Fig. 5 for the considered range of temperatures. Between 200 and 800 eV, the PDFs are well represented by an effective OCP with = 9 ± 0.5 close to the predicted one. To get a significant decrease of correlations (OCP = 5) the system must be heated up to 2000 eV. In isochoric heating experiments, the temperature is usually inferred from the spectrally resolved line emission of the targets. To simplify the spectral analysis, low-Z targets are chosen: carbon [28], aluminum, silicon, germanium [31], titanium [31,32], and copper [29,30]. At normal density ρ0 , the reduced isochores ρ0∗ of these low-Z elements are generally above the plateau regime or close to the critical isochore at
TABLE I. Analysis of simulation or experimental results with the model based on TF scaling laws for different elements: tungsten in Ref. [23], silver of Fig. 5, carbon in Ref. [28], titanium in Refs. [31,32], copper in Refs. [29,41], and germanium in Ref. [31]. Z —
A amu
ρ0 g/cm3
ρ g/cm3
ρ∗ mol/cm3
∗ —
—
Tmin eV
Tmax eV
ref —
Tref eV
Ref.
W Ag
74 47
183.8 107.9
19.3 10.5
40 2.5
0.0029 0.0005
0.31 0.185
23 8.7
130 11
860 850
20 ± 2 9 ± 0.5
60–800 –1000
[23] This work
C Ti Ti Cu Ge Cu
6 22 22 29 32 29
12 47.9 47.9 63.5 72.6 63.5
2.27 4.51 4.51 8.96 5.32 8.96
0.03 4.51 4.51 8.96 5.32 1.
0.0004 0.0043 0.0043 0.0049 0.0023 0.00054
0.17 0.335 0.335 0.34 0.29 0.19
1 7.4 7.4 10 9.3 5.5
1 70 70 170 30 6
60 120 120 150 330 440
0.7 ± 0.2
45–50 780–850 –150 160 650–850 Future experiment
[28] [31] [32] [29] [31] [41]
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ρc∗ = 0.0045 mol/cm−3 : ρ0∗ = 0.032 for carbon, 0.0077 for aluminum, 0.0059 for silicon, 0.0043 for titanium, and 0.0049 for copper. Lower densities or higher Z elements should permit to reach a plateau, like germanium with ρ0∗ = 0.0023. We comment further on the following experimental situations: Carbon. In this experiment [28], a solid carbon sample is irradiated by a 100-fs laser pulse. After a rapid expansion, the ionic density stays constant for 200 ps at about 0.03 g/cm3 , which corresponds to a coupling of 0.7± 0.2 for a temperature of 45–50 eV [28]. With this ionic density, we estimate a coupling of 1 between 8 to 60 eV. Titanium. At normal density, titanium is close to the critical isochore and can reach a plateau around 7 in a small temperature range between 70 and 120 eV (cf. Table I). In the experiments of Ref. [31], the laser pulse intensities were so high that the plasma reached much higher temperatures, around 800 eV. In another experimental setup [32], thin wire targets were heated at the wire tip with lower laser pulse intensities leading to a gradient of temperature along the wire from 150 eV to less than 5 eV, including the whole range of temperatures of the plateau. Copper. In the experiments presented in Ref. [29], copper is heated isochorically at normal density up to few hundreds of eV. With a reduced density ρ ∗ = 0.0049 slightly above the critical one (ρc∗ = 0.0045), we extrapolate to a coupling parameter of about 10 close to the critical point, in good agreement with the values 8–9 in Ref. [29]. The temperature is also extrapolated to a value of about 160 eV. This extrapolated temperature is close to the reported values [29]. Germanium. Germanium is a good candidate to explore the plateau since it can reach a rather high coupling around 9 in a quite large temperature range between 30 and 330 eV (cf. Table I). Furthermore, its radiative spectrum has been extensively studied [38–40]. Unfortunately, with the same high intensity laser pulses as for titanium, the temperatures reached in the experiments of Ref. [31] are too high, around 750 eV, to explore the plateau. Copper foam.The last row in Table I corresponds to a proposal on copper foam at a density of 1 g/cm3 heated by protons on the Orion laser. Our model predicts a coupling of 5.5 between 6 and 440 eV [41].
TABLE II. Corresponding states with the same ionic coupling parameter 0 = 8.7. Z — Fe Zn Ag Au U
A amu
ρ0 g/cm3
∗ —
ρ∗ mol/cm3
ρ g/cm3
Tmin eV
Tmax eV
7.86 7.14 10.5 19.3 19.1
0.33 0.29 0.18 0.11 0.09
4.2 × 10−3 2.2 × 10−3 4.9 × 10−4 1.7 × 10−4 1.3 × 10−4
6.2 4.4 2.5 2.6 2.9
88 23 11 18 21
152 300 850 1800 2300
26 55.8 30 65.4 47 107.9 79 197. 92 238.
for a material with Z > 60. Next, let us consider that we want to determine all the elements that can reach a coupling 0 30 on a plateau. Among them, the lowest atomic number Z0 corresponds to the critical point c∗ ∼ 0 /Z0 . All the other elements reaching a 0 plateau have higher Z and corresponding lower ∗ ∼ 0 /Z but larger temperature range. Using the reciprocal of the rule (4), this corresponds to a reduced density ρ ∗ and an actual density ρ, using Eq. (2a). The temperature range of the plateau is also given using the value of ρ ∗ in Eq. (5) together with Eqs. (2b). Table II presents a few results of this analysis for 0 = 8.7. This coupling was chosen to allow for comparisons with iron. One can also see from Table II that the higher Z, the broader is the temperature interval for the plateau. Figure 6 illustrates such corresponding states for iron at 6.2 g/cm3 and 100 eV and silver at 2.5 g/cm3 and 200 eV. The PDFs have been obtained from 54, 128, and 432 atoms OFMD simulations. Despite the different thermodynamic conditions, the structure of the plasmas looks very similar [Fig. 6(a)].
(a)
(b)
1
g(r) OCP Γ=12 OCP Γ=10 OCP Γ=8 OCP Γ=6 Fe OFMD
0.5 Ag 2.5 g/cc Fe 6.17 g/cc
VI. CORRESPONDING STATES
0
The Z-scaling laws of Eqs. (2) have allowed us to investigate the -plateau features, but they can also be used to define corresponding states. Such corresponding states have equal reduced density ρ ∗ and temperature T ∗ . If a plasma is within the plateau regime, each of its corresponding states reaches also a plateau but with a different actual coupling over a temperature range. Examples of corresponding states, within the plateau regime or outside of it, are given in Fig. 2. We would like to address a different issue still using the Z-scaling laws, namely, What connection can be established between two different plasmas with equal coupling parameters reached on a plateau? This question is of special interest as regards to isochoric heating experiments. First, we recall that strongly coupled plasmas are difficult to produce in the plateau regime. The maximum coupling parameter that can currently be reached is around 30
(c)
(d)
1
g(r) 0.5 OFMD 54 OFMD 128 OFMD 432
OFMD 54 OFMD 128 OFMD 432 Ag 2.5 g/cc 200 eV
Fe 6.2 g/cc 100 eV 0
0
1
2
r/a
0
1
2
3
r/a
FIG. 6. (Color online) (a) Comparison between PDFs of iron and silver for corresponding states as given in Table II from 432 atoms simulations. (b) Comparisons between iron PDF and OCP results for decreasing from 12 to 6. (c) Comparisons among 54, 128, and 432 iron atoms simulations. (d) Same for silver.
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The best fit for iron is obtained by the effective OCP model at = 8 as shown in Fig. 6(b) where the sensitivity with respect to is tested. For silver, the best fit was = 9 (see Fig. 5). This slight difference in between iron and silver is noticeable in Fig. 6(a). In the next panels [Fig. 6(c) and Fig. 6(d)] finite-size effects are tested (54, 128, and 432 atoms). Although the structure of the iron plasma mimics the silver one, the thermodynamics differs and the iron pressure is higher (126 Mbar) than silver (84 Mbar). VII. EQUATION OF STATE
The concept of an effective OCP suggests the use of the OCP equation of state to describe the ions. Moreover, the effective coupling parameter gives an estimation of the effective ionization for given temperature and density. The pressure is obtained, as usual, by the ionic and electronic contributions PTot = POCP + Pele .
(6a)
The ionic OCP part accounts for the strong screened interactions between highly charged ions. We use the fit given by Slattery et al. [42] without the Madelung contribution, POCP /nkB T = 1 + 13 [b 1/4 + c −1/4 + d],
(6b)
with b = 0.94544, c = 0.17954, and d = −0.80049. For the electronic system, we use the ionization Q deduced from , predicted by Eq. (4) for the plateau, leading to an electronic density ne = Qn. For hot dense high-Z plasmas, the electronic component is usually weakly coupled and can be treated as an ideal gas. In order to extend the range of validity of our model, we use an interpolation formula between the Fermi gas and the perfect gas due to Nikiforov et al. [43] 9π 4 2 1/3 3 3/2 Pele /ne = (kB T ) + 3.36ne (kB T ) + n , (6c) 125 e where atomic units are used (1 a.u. of pressure =294 Mbar). We show in Fig. 7 the pressure predicted by this model for silver at 2.5 g/cm3 in the range 200–800 eV corresponding to a plateau of 8.7 constant between 200 and 800 eV (see Table II and Fig. 5). The agreement with the OFMD results and the SESAME equation 2720 [44] for silver is very good. We stress the fact that in this simple approach the equation-of-state is fully determined by the effective coupling parameter .
FIG. 7. (Color online) Pressure versus temperature for silver at 2.5 g/cm3 . The heavy dashed line is the SESAME EOS 2720; the full line is the result of Eqs. (6) and the circles are results of OFMD simulations.
coupling parameter for plasmas at constant density. This straightforward analysis confirms the importance of the atomic number Z in the existence of this plateau and in the range of plateau temperatures. For this constant coupling region, we have built a simple equation of state in close agreement with SESAME or with orbital-free simulations. Simple fit formulas are used to quantify the coupling parameter of recent isochoric experiments. The existence of corresponding states between different elements at different densities is also very promising. We are aware that the TF method can lead to large errors in the ionization for low-density, low-temperature cases; but this could be corrected a posteriori by a convenient rescaling of fit Eq. (4) to get more precise coupling parameters. ACKNOWLEDGMENTS
We have presented a simple model, based on the TF scaling laws, to predict the occurrence of a plateau regime in the
This work was done under the NNSA/DAM collaborative agreement P184. We especially thank Flavien Lambert for providing his OFMD code and L. Colombet for his assistance. The Los Alamos National Laboratory is operated by Los Alamos National Security, LLC for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396.
[1] J. P. Hansen, Phys. Rev. A 8, 3096 (1973). [2] J. P. Hansen, I. R. McDonald, and E. L. Pollock, Phys. Rev. A 11, 1025 (1975).
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VIII. CONCLUSION
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[5] J. D. Kress, S. Mazevet, L. A. Collins, and W. W. Wood, Phys. Rev. B 63, 024203 (2000). [6] L. A. Collins, S. R. Bickham, J. D. Kress, S. Mazevet, T. J. Lenosky, N. J. Troullier, and W. Windl, Phys. Rev. B 63, 184110 (2001). [7] V. Recoules, J. Cl´erouin, P. Renaudin, P. Noiret, and G. Z´erah, J. Phys. A: Math. Gen. 36, 6033 (2003). [8] J. Cl´erouin, P. Noiret, V. N. Korobenko, and A. D. Rakhel, Phys. Rev. B 78, 224203 (2008). [9] S. Mazevet, M. P. Desjarlais, L. A. Collins, J. D. Kress, and N. H. Magee, Phys. Rev. E 71, 016409 (2005). [10] Y. Laudernet, J. Cl´erouin, and S. Mazevet, Phys. Rev. B 70, 165108 (2004). [11] J. Cl´erouin, P. Renaudin, Y. Laudernet, P. Noiret, and M. P. Desjarlais, Phys. Rev. B. 71, 064203 (2005). [12] G. Z´erah, J. Cl´erouin, and E. L. Pollock, Phys. Rev. Lett. 69, 446 (1992). [13] F. Lambert, J. Cl´erouin, and S. Mazevet, Europhys. Lett. 75, 681 (2006). [14] J. Kress, J. S. Cohen, D. Kilcrease, D. Horner, and L. Collins, High Energy Density Phys. 7, 155 (2011). [15] F. Lambert, J. Cl´erouin, J.-F. Danel, L. Kazandjian, and S. Mazevet, Properties of Hot and Dense Matter by Orbital-Free Molecular Dynamics, Vol. 6, Recent Advances in Computational Chemistry (World Scientific, Singapore, 2013), pp. 165. [16] S. Mazevet, F. Lambert, F. Bottin, G. Z´erah, and J. Cl´erouin, Phys. Rev. E 75, 056404 (2007). [17] J. F. Danel, L. Kazandjian, and G. Zerah, Phys. Plasmas 19, 122712 (2012). [18] L. Burakovsky, C. Ticknor, J. D. Kress, L. A. Collins, and F. Lambert, Phys. Rev. E 87, 023104 (2013). [19] J. Cl´erouin and S. Bernard, Phys. Rev. E 56, 3534 (1997). [20] V. V. Karasiev and S. B. Trickey, Comput. Phys. Commun. 183, 2519 (2012). [21] M. S. Murillo, J. Weisheit, S. B. Hansen, and M. W. C. Dharma-wardana, Phys. Rev. E 87, 063113 (2013). [22] F. J. Rogers, D. A. Young, H. E. DeWitt, and M. Ross, Phys. Rev. A 28, 2990 (1983). [23] J. Cl´erouin, G. Robert, P. Arnault, J. D. Kress, and L. A. Collins, Phys. Rev. E 87, 061101 (2013). [24] P. Arnault, High Energy Density Phys. 9, 711 (2013). [25] Y. B. Zel’dovich and I. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. I (Academic Press, New York, 1966).
[26] R. P. Drake, High-Energy-Density Physics, Shock Waves and High Pressure Phenomena (Springer, Berlin, 2006). [27] P. Fromy, C. Deutsch, and G. Maynard, Phys. Plasmas 3, 714 (1996). [28] M. Nantel, G. Ma, S. Gu, C. Y. Cˆot´e, J. Itatani, and D. Umstadter, Phys. Rev. Lett. 80, 4442 (1998). [29] G. Gregori, S. B. Hansen, R. Clarke, R. Heathcote, M. H. Key, J. King, R. I. Klein, N. Izumi, A. J. Mackinnon, S. J. Moon et al., Contrib. Plasma Phys. 45, 284 (2005). [30] F. Perez, L. Gremillet, M. Koenig, S. D. Baton, P. Audebert, M. Chahid, C. Rousseaux, M. Drouin, E. Lefebvre, T. Vinci et al., Phys. Rev. Lett. 104, 085001 (2010). [31] C. R. D. Brown, D. J. Hoarty, S. F. James, D. Swatton, S. J. Hughes, J. W. Morton, T. M. Guymer, M. P. Hill, D. A. Chapman, J. E. Andrew et al., Phys. Rev. Lett. 106, 185003 (2011). [32] D. C. Hochhaus, B. Aurand, M. Basko, B. Ecker, T. Kuhl, T. Ma, F. Rosmej, B. Zielbauer, and P. Neumayer, Phys. Plasmas 20, 062703 (2013). [33] R. P. Feynman, N. Metropolis, and E. Teller, Phys. Rev. 75, 1561 (1949). [34] R. M. More, K. H. Warren, D. A. Young, and G. B. Zimmerman, Phys. Fluids 31, 3059 (1988). [35] R. M. More, Tech. Rep. UCRL-84991, Lawrence Livermore Laboratory (1983). [36] R. M. More, Adv. At. Mol. Phys. 21, 305 (1985). [37] D. Salzmann, Atomic Physics in Hot Plasmas (Oxford University Press, Oxford, 1998). [38] P. Arnault, T. Blenski, and G. Dejonghe, High Energy Density Phys. 3, 1 (2007). [39] J. W. O. Harris, L. M. Upcraft, D. J. Hoarty, B. J. B. Crowley, C. R. D. Brown, and S. F. James, High Energy Density Phys. 6, 95 (2010). [40] T. Blenski, G. Loisel, M. Poirier, F. Thais, P. Arnault, T. Caillaud, J. Fariaut, F. Gilleron, J.-C. Pain, Q. Porcherot et al., High Energy Density Phys. 7, 320 (2011). [41] E. T. Grumbell et al. (unpublished). [42] W. L. Slattery, G. D. Doolen, and H. E. DeWitt, Phys. Rev. A 21, 2087 (1980). [43] A. F. Nikiforov, V. G. Novikov, and V. B. Uvarov, QuantumStatistical Models of Hot Dense Matter (Birkauser, Basel, 2005). [44] S. P. Lyon and J. D. Johnson (eds.), SESAME : The Los Alamos National Laboratory Equation of State Database, Report No. LA-UR-92-3407 (Group T-1, 1992).
063106-7
Thomas-Fermi Z-scaling laws and coupling stabilization for plasmas Philippe Arnault,* Jean Cl´erouin, and Gregory Robert CEA, DAM, DIF, F-91297 Arpajon, France
Christopher Ticknor, Joel D. Kress, and Lee A. Collins Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 12 September 2013; published 17 December 2013) Extending the well-known Thomas-Fermi Z-scaling laws to the Coulomb coupling parameter, we investigate the stabilization of the ionic coupling in isochoric heating [Cl´erouin et al., Phys. Rev. E 87, 061101 (2013)]. This stabilization is restricted to a domain in atomic number Z, temperature, and density, including strong limitations on high couplings, that can only be obtained for high-Z elements. Contact is made with recent isochoric heating experiments. The consequences for corresponding states with respect to ionic coupling are also quantified via orbital free molecular dynamics simulations. This opens avenues for future isochoric heating experiments. DOI: 10.1103/PhysRevE.88.063106
PACS number(s): 52.27.Gr, 52.65.−y
I. INTRODUCTION
A realistic description of a hot dense plasma requires a model that includes all interactions explicitly (N -body problem) in contrast with weakly coupled plasmas that are well represented by only binary collisions and mean-field effects. For dense and fully degenerate plasmas, the one-component plasma (OCP) has served as the reference model [1,2], providing fits and simple expressions for the free energy and its derivatives, pressure, energy, and the transport properties such as diffusion and viscosity. In these expressions, the only variable is the Coulomb coupling parameter, =
Q2 e 2 , 4π 0 akB T
(1)
where Q is the ion charge state, e is the fundamental charge, a = (3/4π n)1/3 is the mean ion sphere radius, n is the plasma ionic density, and T is the temperature. Unfortunately, the OCP model supposes full electronic degeneracy, which is rarely the case, and a knowledge of the ionization Q. The need for a more precise representation of electron screening effects (finite degeneracy) has led to the development of molecular dynamics (MD) simulations using density functional theory (DFT) at finite temperature [3], which have, to considerable accuracy, determined hugoniots [4–6], electrical conductivities in the metal-nonmetal transition range [7,8], and optical properties of shocked materials [9–11]. However, DFTbased methods are strongly limited in temperature, mainly due to the introduction of orbitals to compute the electron kinetic energy in the Kohn-Sham (KS) ansatz. To bypass this limitation, orbital-free (OF) methods have been introduced that substitute a semiclassical formulation of the electron kinetic energy based on the local density approximation, the Thomas-Fermi (TF) functional, for its KS counterpart [12–15]. These OF methods are well adapted to describing dense systems at high temperature, where methods based on orbitals are prohibitively expensive. Several limited studies [16–19] indicate that the transition region between KS and OF calculations is sensitive to the system and its thermodynamic
*
[email protected]
1539-3755/2013/88(6)/063106(7)
conditions. The development of OF functionals remains an active field of research, particularly for finite temperature exchange-correlation functionals [20]. However, for compressions of up to a few times normal solid density, the ThomasFermi-Dirac (TFD) formulations, especially with the vonWeisz¨acker extension, agree within 10% or better with KS. Our main interest in the orbital-free molecular dynamics (OFMD) rests with extraction of the structure of the plasma and its comparison with the OCP. This introduces the concept of an effective OCP, in which the partial degeneracy and the corresponding electron screening is compensated by the introduction of an effective ionization. Actually, current threedimensional (3D) MD simulations of plasmas using DFT in either the KS or the OF formulation do not produce the ion charge state Q or the mean ionization as a standard output parameter since Q is not a quantum observable. Any definition of a charge state under these circumstances is ambiguous and may take different definitions [21]. On the other hand, the pair distribution function (PDF) g(r), which describes the statistics of nuclei interactions, is a standard output of a 3D simulation code. This function contains all the details of the screened interactions and can be easily compared with the OCP PDF, which is tabulated over a wide range of coupling parameters [22]. For pure repulsive systems, an adjustment with respect to can accurately reproduce the closest approach distance, the intensity of the main peak, and the location of the first minimum. This procedure defines an effective OCP model equivalent to the actual system [23]. From the temperature and the density, this determination of the effective coupling gives an estimation of the effective average charge Q. This effective mean ionization accounts for the electron screening of the nuclei interaction, at least for the PDF, and is in good agreement with spherically confined-atom methods when defined by the electron density ne (r) at the sphere radius Q = 4π a 3 ne (a)/3. In this approach, the effective ionization and corresponding effective OCP constitute a coarse-grained description of hot dense plasmas. Interestingly, the effective OCP approach leads also to reasonably accurate estimations of the transport coefficients of plasmas such as diffusion and viscosity [23,24]. In a recent paper [23], we have extracted the coupling parameters of tungsten from a series of OFMD simulations at
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increasing temperatures for a constant density of 40 g/cm3 and observed a quasiconstant coupling ( 20) over a wide range of temperatures between 60 and 800 eV (the plateau). This constant-coupling regime results from a balance between a rise in temperature and an increase in ionization, which is proportional to the square root of the temperature at a constant density. Well known for dilute plasmas [25,26], this behavior of the ionization is also confirmed by average-atom density functional models (see Fig. 2 of Ref. [21]). In this paper, we demonstrate that this effect is present for the simplest theoretical approach: the TF model of a single atom in a Wigner-Seitz cell [27]. As anticipated in Ref. [23], the existence of this plateau depends on the material through its atomic number Z, atomic mass A, and its mass density ρ. Taking advantage of the TF scaling laws, we can with simple formulas predict for any material and experimental conditions the occurrence of this plateau. This capability allows us to make contact with recent isochoric experiments [28–32]. In these experiments, temperatures ranging from 50 to 1000 eV have been reached by electron heating produced by picosecond laser pulses on a buried material [30,31]. The paper is organized as follows: we first review the TF scaling laws and extend them to the coupling parameter , defined using the TF ionization. We therefore introduce a Zscaled coupling parameter ∗ that allows us to build a universal diagram linking the coupling parameter to temperature for different densities. We discuss next the conditions for the occurrence of the plateau and propose simple fits to predict the plateau value of the coupling parameter, ∗ , and the corresponding range of temperature as a function of density. In the next section, we compare our predictions with OFMD simulations and analyze recent isochoric heating experiments. This comparison indicates that the TF ionization is accurate enough to define an effective OCP whose PDF is close to the actual one. The consequences for corresponding states with respect to ionic coupling are then explored. OFMD simulations are performed to corroborate these results.
10
3
10
T = T /Z
4/3
(2a) ,
1
Γ* 10
10
10
0
-1
-2
10
-4
10
-3
10
-2
10
-1
10
0
10
1
(2b)
10
2
10
3
T* (eV) FIG. 1. (Color online) Iso-ρ ∗ (= ρ/ZA) curves of ∗ = /Z as a function of T ∗ = T /Z 4/3 in the TF model.
with respect to which a corresponding state law gives the electron contribution to the thermodynamic properties of any element P (Z,ρ,T ) = Z 10/3 P1 (ρ ∗ ,T ∗ ), E(Z,ρ,T ) = Z
7/3
∗
∗
E1 (ρ ,T ),
(3a) (3b)
where quantities with 1 stand for the pressure (P) and the energy (E) of hydrogen (Z = 1) at reduced density ρ ∗ and temperature T ∗ . The quotidian equation of state (QEOS) model [34] employs such scaling laws. We propose to extend these Z-scaling laws to the coupling parameter , which is rescaled by the atomic number as follows: ∗ = /Z,
The TF model constitutes a simple framework widely used to compute properties of hot dense matter [27,33] although its drawbacks are well known. For instance, the ionization is overestimated at low density. However, the introduction of exchange (Thomas-Fermi-Dirac, TFD) and gradient (Thomas-Fermi-vonWeisz¨acker, TFW) corrections can lead to substantive improvements, leading to a very smooth transition between the TF and KS. As found in detailed studies by Mazevet et al. [16] and Danel et al. [17], the best agreement obtains for the extended Thomas-Fermi-Dirac-vonWeisz¨acker formulation. These improvements emerge at the expense of the loss of the Z-scaling laws. In this paper, we shall restrict ourselves to the simple TF model in order to exploit the power of such scaling laws. Following More et al. [34], we first define a reduced density ρ ∗ and a reduced temperature T ∗ by
∗
ρ* (mol/cm ) = 1 0.1 0.01 0.001 0.0001
2
10
II. Z-SCALING LAWS
ρ ∗ = ρ/ZA,
3
(3c)
using Eq. (2b), a = a1 /Z 1/3 , and the scaled ionization Q∗ , Q∗ = Q(Z,ρ,T )/Z = Q1 (ρ ∗ ,T ∗ ).
(3d)
The effective mean ionization at finite temperature Q1 (ρ ∗ ,T ∗ ) is given by More’s fit [35–37]. It is defined by the electron density at the ion sphere radius a. In Fig. 1, we plot ∗ versus T ∗ for different reduced densities ρ ∗ . For Z = 1, this figure corresponds to the hydrogen case. At density ρ ∗ = 1, the curve is monotonically decreasing. At ρ ∗ ∼ 0.01, we observe a pronounced inflection point, which turns into a plateau at lower density and eventually into a sigmoid curve for the lowest isodensity curve. To illustrate the -plateau feature, we show in Fig. 2 the coupling parameters of selected elements at densities chosen to satisfy ρ ∗ = 10−3 . In this diagram, aluminum at 0.351 g/cm3 is equivalent (same ∗ = /Z) to iron at 1.45 g/cm3 or to gold at 15.6 g/cm3 . For each element, temperatures are obtained by multiplying the abscissa of Fig. 1 by Z 4/3 . The actual coupling parameter in the plateau regime
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1 10 -3
5 10
0.332 0.309
-3
2 10
0.282 -3
1 10
Γ* 0.234 -4
2 10
0.2
0.146
0.01
0.1
1
10
T* (eV)
FIG. 2. (Color online) Coupling parameters of selected elements versus temperature for the ρ ∗ = 10−3 isochore.
ranges from 3 for aluminum to 18 for gold, and the range is shifted towards higher temperatures for high-Z materials.
FIG. 3. (Color online) Scaled coupling ∗ = /Z versus scaled temperature T ∗ = T /Z 4/3 for selected reduced densities ρ ∗ = ρ/AZ ranging from 2 10−4 to 10−2 mol/cm3 , from bottom to top. Horizontal lines mark the mean values defined as the plateaus. The gray area shows the domain where the plateau exists.
hence, for the time needed to heat the sample, allowing for diagnostics.
III. PLATEAU CONDITIONS
IV. PARAMETRIZATION
Figure 3 displays the part of Fig. 1 for values of the reduced density ρ ∗ in the vicinity of the plateau transition with ρ ∗ ranging from 2 × 10−4 (dashed line, bottom) to 1 × 10−3 (heavy solid line) to 0.01 (dashed line, top). This figure looks very similar to a liquid-vapor diagram with a critical point beyond which the plateau disappears. In our diagram, this point is located on the critical density curve ρc∗ = 0.0045 mol/cm3 at Tc∗ = 1.5 eV and c∗ = 0.335. Beyond this density, the coupling parameter decays monotonously with temperature. Below this point the reduced coupling exhibits a loop around a mean-value defined as the plateau. We recall that this curve is universal for TF. For a given atomic number Z, the range of accessible plateau is limited by ∗ < c∗ , namely < 0.335 Z, which means that certain strongly coupled configurations will miss the plateau regime. For example, a strongly coupled aluminum plasma with = 20 cannot exist in the plateau regime. This coupling parameter corresponds to a reduced coupling of ∗ = 1.54, well outside the plateau regime. During isochoric heating, such a coupling can only be obtained in a transient regime, not in a stabilized way as concerns the ionic coupling. Actually, coupling values higher than about 30 cannot be reached in the plateau regime. On the other hand, for gold at 16 g/cm3 , the reduced density ρ ∗ = 10−3 corresponds to a coupling parameter of 18.5 between 33 and 1500 eV ( between 16 and 21 if we consider the oscillation). The ionic coupling of a high-Z dense plasma can be stabilized in a wide range of temperature and,
Figure 3 shows that inside the plateau regime, a one-to-one correspondence exists between the plateau value ∗ and the reduced density ρ ∗ . In Fig. 4, we have plotted selected values of 0.4 0.714 + 0.0695 ln(ρ∗) 0.35
0.3
Γ* 0.25 0.2
0.15
0.1 0.0001
0.001
ρ*
0.01
FIG. 4. (Color online) Plateau value of the reduced coupling ∗ versus reduced density ρ ∗ and its fit. The error bars and the gray area represent the amplitude of the loop.
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the reduced density and the corresponding coupling parameter. The error bars and the gray area represent the amplitude of the loop. In a log-linear plot we find a linear relation which can be fitted by
1
g(r) 0.5
∗
∗ = b + a ln[ρ ],
(4)
where a = 0.0695 and b = 0.714. This fit is meaningless for ρ ∗ > ρc∗ and is not drawn in Fig. 4. The lowest density point (ρ ∗ = 2 10−4 ) with the strongest oscillation around the plateau is not taken into account for the fit. We have also parametrized the temperature extrema defining the plateau ∗ Tmin = 0.0445 exp(765 ρ ∗ ), ∗ Tmax
∗
= 5.66 exp(−248 ρ ).
0
200 eV
400 eV
OFMD ΓOCP = 9
OFMD ΓOCP = 9
800 eV
2000 eV
OFMD ΓOCP = 9
OFMD ΓOCP = 5
1
g(r) 0.5
(5a)
0
0
1
2
0
1
r/a
(5b)
2
3
r/a
FIG. 5. (Color online) PDF of silver at 2.5 g/cm3 and from left to right, top to bottom, 200, 400, 800, and 2000 eV. The red continuous lines are the OCP PDF best fits.
V. SIMULATIONS AND EXPERIMENTS
The model predictions, based on the TF scaling laws, are utilized to analyze results of simulations and experiments for various chemical elements in Table I. The first columns give the properties of the element (atomic number Z, mass A, and normal density ρ0 ); the next columns present the corresponding reduced parameters ρ ∗ from Eq. (2a), and ∗ from Eq. (4), for the actual density ρ; and the following columns display the coupling parameter from Eq. (3c) (in bold) and the approximate temperature range for the plateau from Eqs. (5) and (2b). The last columns give the actual coupling parameters ref (in bold) and temperatures Tref obtained in simulations or experiments. In Ref. [23], a plateau was observed in OFMD simulations of tungsten at 40 g/cm3 . With the model, we find a coupling of 23, slightly higher than the one observed 20 ± 2 with a full OFMD-TFD formulation in Ref. [23]. This comes from the neglect of exchange in TF formulation leading to ionizations 2–4% higher than the TFD ones in this temperature range [21,23]. The temperature range of the plateau is also reproduced by the model. To check the existence of the plateau for lower-Z materials, we have performed OFMD simulations on silver at 2.5 g/cm3 . Within the model, this density corresponds to ρ ∗ = 5 10−4 and to a plateau at = 8.7 between 11 and 850 eV (cf. Table I). OFMD simulations of 432 silver nuclei
in a 3D box were performed for temperatures of 200, 400, 800, and 2000 eV. To be fully coherent with our bare TF approach, these simulations have been performed at TF level of DFT (no exchange, no gradients). The divergence of the electron-nucleus potential is regularized at each thermodynamic condition. The cutoff radius is chosen to be 30% of the Wigner-Seitz radius, sufficient to prevent overlap of the regularization spheres. The number of plane waves describing the local electronic density is then adjusted to converge the thermodynamic properties to within 1%. The MD time step is small enough to ensure good energy conservation [23]. The plateau is clearly observed in Fig. 5 for the considered range of temperatures. Between 200 and 800 eV, the PDFs are well represented by an effective OCP with = 9 ± 0.5 close to the predicted one. To get a significant decrease of correlations (OCP = 5) the system must be heated up to 2000 eV. In isochoric heating experiments, the temperature is usually inferred from the spectrally resolved line emission of the targets. To simplify the spectral analysis, low-Z targets are chosen: carbon [28], aluminum, silicon, germanium [31], titanium [31,32], and copper [29,30]. At normal density ρ0 , the reduced isochores ρ0∗ of these low-Z elements are generally above the plateau regime or close to the critical isochore at
TABLE I. Analysis of simulation or experimental results with the model based on TF scaling laws for different elements: tungsten in Ref. [23], silver of Fig. 5, carbon in Ref. [28], titanium in Refs. [31,32], copper in Refs. [29,41], and germanium in Ref. [31]. Z —
A amu
ρ0 g/cm3
ρ g/cm3
ρ∗ mol/cm3
∗ —
—
Tmin eV
Tmax eV
ref —
Tref eV
Ref.
W Ag
74 47
183.8 107.9
19.3 10.5
40 2.5
0.0029 0.0005
0.31 0.185
23 8.7
130 11
860 850
20 ± 2 9 ± 0.5
60–800 –1000
[23] This work
C Ti Ti Cu Ge Cu
6 22 22 29 32 29
12 47.9 47.9 63.5 72.6 63.5
2.27 4.51 4.51 8.96 5.32 8.96
0.03 4.51 4.51 8.96 5.32 1.
0.0004 0.0043 0.0043 0.0049 0.0023 0.00054
0.17 0.335 0.335 0.34 0.29 0.19
1 7.4 7.4 10 9.3 5.5
1 70 70 170 30 6
60 120 120 150 330 440
0.7 ± 0.2
45–50 780–850 –150 160 650–850 Future experiment
[28] [31] [32] [29] [31] [41]
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ρc∗ = 0.0045 mol/cm−3 : ρ0∗ = 0.032 for carbon, 0.0077 for aluminum, 0.0059 for silicon, 0.0043 for titanium, and 0.0049 for copper. Lower densities or higher Z elements should permit to reach a plateau, like germanium with ρ0∗ = 0.0023. We comment further on the following experimental situations: Carbon. In this experiment [28], a solid carbon sample is irradiated by a 100-fs laser pulse. After a rapid expansion, the ionic density stays constant for 200 ps at about 0.03 g/cm3 , which corresponds to a coupling of 0.7± 0.2 for a temperature of 45–50 eV [28]. With this ionic density, we estimate a coupling of 1 between 8 to 60 eV. Titanium. At normal density, titanium is close to the critical isochore and can reach a plateau around 7 in a small temperature range between 70 and 120 eV (cf. Table I). In the experiments of Ref. [31], the laser pulse intensities were so high that the plasma reached much higher temperatures, around 800 eV. In another experimental setup [32], thin wire targets were heated at the wire tip with lower laser pulse intensities leading to a gradient of temperature along the wire from 150 eV to less than 5 eV, including the whole range of temperatures of the plateau. Copper. In the experiments presented in Ref. [29], copper is heated isochorically at normal density up to few hundreds of eV. With a reduced density ρ ∗ = 0.0049 slightly above the critical one (ρc∗ = 0.0045), we extrapolate to a coupling parameter of about 10 close to the critical point, in good agreement with the values 8–9 in Ref. [29]. The temperature is also extrapolated to a value of about 160 eV. This extrapolated temperature is close to the reported values [29]. Germanium. Germanium is a good candidate to explore the plateau since it can reach a rather high coupling around 9 in a quite large temperature range between 30 and 330 eV (cf. Table I). Furthermore, its radiative spectrum has been extensively studied [38–40]. Unfortunately, with the same high intensity laser pulses as for titanium, the temperatures reached in the experiments of Ref. [31] are too high, around 750 eV, to explore the plateau. Copper foam.The last row in Table I corresponds to a proposal on copper foam at a density of 1 g/cm3 heated by protons on the Orion laser. Our model predicts a coupling of 5.5 between 6 and 440 eV [41].
TABLE II. Corresponding states with the same ionic coupling parameter 0 = 8.7. Z — Fe Zn Ag Au U
A amu
ρ0 g/cm3
∗ —
ρ∗ mol/cm3
ρ g/cm3
Tmin eV
Tmax eV
7.86 7.14 10.5 19.3 19.1
0.33 0.29 0.18 0.11 0.09
4.2 × 10−3 2.2 × 10−3 4.9 × 10−4 1.7 × 10−4 1.3 × 10−4
6.2 4.4 2.5 2.6 2.9
88 23 11 18 21
152 300 850 1800 2300
26 55.8 30 65.4 47 107.9 79 197. 92 238.
for a material with Z > 60. Next, let us consider that we want to determine all the elements that can reach a coupling 0 30 on a plateau. Among them, the lowest atomic number Z0 corresponds to the critical point c∗ ∼ 0 /Z0 . All the other elements reaching a 0 plateau have higher Z and corresponding lower ∗ ∼ 0 /Z but larger temperature range. Using the reciprocal of the rule (4), this corresponds to a reduced density ρ ∗ and an actual density ρ, using Eq. (2a). The temperature range of the plateau is also given using the value of ρ ∗ in Eq. (5) together with Eqs. (2b). Table II presents a few results of this analysis for 0 = 8.7. This coupling was chosen to allow for comparisons with iron. One can also see from Table II that the higher Z, the broader is the temperature interval for the plateau. Figure 6 illustrates such corresponding states for iron at 6.2 g/cm3 and 100 eV and silver at 2.5 g/cm3 and 200 eV. The PDFs have been obtained from 54, 128, and 432 atoms OFMD simulations. Despite the different thermodynamic conditions, the structure of the plasmas looks very similar [Fig. 6(a)].
(a)
(b)
1
g(r) OCP Γ=12 OCP Γ=10 OCP Γ=8 OCP Γ=6 Fe OFMD
0.5 Ag 2.5 g/cc Fe 6.17 g/cc
VI. CORRESPONDING STATES
0
The Z-scaling laws of Eqs. (2) have allowed us to investigate the -plateau features, but they can also be used to define corresponding states. Such corresponding states have equal reduced density ρ ∗ and temperature T ∗ . If a plasma is within the plateau regime, each of its corresponding states reaches also a plateau but with a different actual coupling over a temperature range. Examples of corresponding states, within the plateau regime or outside of it, are given in Fig. 2. We would like to address a different issue still using the Z-scaling laws, namely, What connection can be established between two different plasmas with equal coupling parameters reached on a plateau? This question is of special interest as regards to isochoric heating experiments. First, we recall that strongly coupled plasmas are difficult to produce in the plateau regime. The maximum coupling parameter that can currently be reached is around 30
(c)
(d)
1
g(r) 0.5 OFMD 54 OFMD 128 OFMD 432
OFMD 54 OFMD 128 OFMD 432 Ag 2.5 g/cc 200 eV
Fe 6.2 g/cc 100 eV 0
0
1
2
r/a
0
1
2
3
r/a
FIG. 6. (Color online) (a) Comparison between PDFs of iron and silver for corresponding states as given in Table II from 432 atoms simulations. (b) Comparisons between iron PDF and OCP results for decreasing from 12 to 6. (c) Comparisons among 54, 128, and 432 iron atoms simulations. (d) Same for silver.
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The best fit for iron is obtained by the effective OCP model at = 8 as shown in Fig. 6(b) where the sensitivity with respect to is tested. For silver, the best fit was = 9 (see Fig. 5). This slight difference in between iron and silver is noticeable in Fig. 6(a). In the next panels [Fig. 6(c) and Fig. 6(d)] finite-size effects are tested (54, 128, and 432 atoms). Although the structure of the iron plasma mimics the silver one, the thermodynamics differs and the iron pressure is higher (126 Mbar) than silver (84 Mbar). VII. EQUATION OF STATE
The concept of an effective OCP suggests the use of the OCP equation of state to describe the ions. Moreover, the effective coupling parameter gives an estimation of the effective ionization for given temperature and density. The pressure is obtained, as usual, by the ionic and electronic contributions PTot = POCP + Pele .
(6a)
The ionic OCP part accounts for the strong screened interactions between highly charged ions. We use the fit given by Slattery et al. [42] without the Madelung contribution, POCP /nkB T = 1 + 13 [b 1/4 + c −1/4 + d],
(6b)
with b = 0.94544, c = 0.17954, and d = −0.80049. For the electronic system, we use the ionization Q deduced from , predicted by Eq. (4) for the plateau, leading to an electronic density ne = Qn. For hot dense high-Z plasmas, the electronic component is usually weakly coupled and can be treated as an ideal gas. In order to extend the range of validity of our model, we use an interpolation formula between the Fermi gas and the perfect gas due to Nikiforov et al. [43] 9π 4 2 1/3 3 3/2 Pele /ne = (kB T ) + 3.36ne (kB T ) + n , (6c) 125 e where atomic units are used (1 a.u. of pressure =294 Mbar). We show in Fig. 7 the pressure predicted by this model for silver at 2.5 g/cm3 in the range 200–800 eV corresponding to a plateau of 8.7 constant between 200 and 800 eV (see Table II and Fig. 5). The agreement with the OFMD results and the SESAME equation 2720 [44] for silver is very good. We stress the fact that in this simple approach the equation-of-state is fully determined by the effective coupling parameter .
FIG. 7. (Color online) Pressure versus temperature for silver at 2.5 g/cm3 . The heavy dashed line is the SESAME EOS 2720; the full line is the result of Eqs. (6) and the circles are results of OFMD simulations.
coupling parameter for plasmas at constant density. This straightforward analysis confirms the importance of the atomic number Z in the existence of this plateau and in the range of plateau temperatures. For this constant coupling region, we have built a simple equation of state in close agreement with SESAME or with orbital-free simulations. Simple fit formulas are used to quantify the coupling parameter of recent isochoric experiments. The existence of corresponding states between different elements at different densities is also very promising. We are aware that the TF method can lead to large errors in the ionization for low-density, low-temperature cases; but this could be corrected a posteriori by a convenient rescaling of fit Eq. (4) to get more precise coupling parameters. ACKNOWLEDGMENTS
We have presented a simple model, based on the TF scaling laws, to predict the occurrence of a plateau regime in the
This work was done under the NNSA/DAM collaborative agreement P184. We especially thank Flavien Lambert for providing his OFMD code and L. Colombet for his assistance. The Los Alamos National Laboratory is operated by Los Alamos National Security, LLC for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396.
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VIII. CONCLUSION
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