(31)

where and are the left and right hand sides of Equation (28); is the number of evaluations (=90, 9 values of T and 10 values of n, in the range of QMD calculations) .

At the end of the fit, therefore the constrain is well obeyed. The

Table 1. Fitted EOS coefficients giving the pressure P in GPa. In italics, dependent parameters calculated with Equation (29) from the respective.

Table 2. Fitted EOS coefficients giving the internal energy U in Ha/atom.

average discrepancy in P is 5% (with 5 points and a maximum of 21%), and in U 2% with a maximum of 8%. Figure 12 shows the calculated QMD P points, as a function of T and ρ compared with the EOS fit. Small discrepancies are observed at low densities and temperatures. The biggest one, 21%, is at and.

The fitted parameters are presented in Table 1 and Table 2.

4.1.1. Isotherms

The EOS is studied experimentally in compression experiments such as modern diamond anvil cell techniques plus X-ray diffraction or older piston cylinder apparata. In these type of experiments the temperature T is constant, usually the room temperature, producing isotherms. The lowest isotherm in the present calculations is. Figure 13 shows experimental data (up to 21 GPa), a theoretical isotherm at, and our isotherm at.

Boettger and Trickey (BT) [35] calculated, by a LMTO technique, a theoretical cold EOS, without a T dependence. It made its way as the cold part of EOS 2293 [36] in the SESAME [37] database. Their cold EOS fits nicely the experimental points in Figure 13. It can be seen that the 10,000 K isotherm is less stiff (lower slope) at high densities (lower volumes). This trend continues in Figure 12 with lower and lower slopes at higher temperatures.

4.1.2. Hugoniot

Another technique to study EOS is by shock experiments. In these experiments, starting from an initial state, a final state is attained, in which all three thermodynamic quantities are changed. The Hugoniot curve is the locus of states of pressure P, volume V (or density ρ) and internal energy U,

Figure 12. Pressure in GPa as a function of T and ρ. QMD calculations-symbols. EOS fit-lines.

Figure 13. Experimental [33] [34] and theoretical [35] isotherm in Li compared with isotherm from the EOS above. The full range of V corre- sponds to the full range of, 0.1 - 2.5 cc/g, in calculations. In inset, only the range with existing experimental data.

which can be attained by applying different shock intensities, when starting from the same initial conditions. Noting by, and the quantities behind the shock front and by, and the initial conditions, the Hugoniot equation [38] is:

(32)

What is actually measured are two velocities: -shock velocity and - particles velocity, related to other quantities by [39] :

(33)

(34)

In Equation (32) apparently there is no explicit dependence on temperature but an EOS is an absolutely prerequisite for solving it (one needs 3 equations for 3 unknowns).

The chosen initial conditions where Ha/atom, based on a QMD run at 300 K, and, the STP density of Li. The left hand side of Equation (32) was minimized with MINUIT [40] at a fixed [the part should be multiplied by to get it in as the part] resulting in a value for. was sampled on a very fine mesh (1000 K steps) from 10,000 K to 50,000 K, the region where the EOS is valid. The solution was accepted if the minimum in the objective function was. The set of values, together with the derivated quantities define the Hugoniot curve, Figure 14.

Figure 14. Hugoniot curves. The pressure as a function of and the shock velocity versus the particle velocity. The experimental points of Bakanova et al. were obtained by digitizing Figure 9 of [41] .

Data from the LASL shock dat a library [42] are below the range of the calculated Hugoniot. Data of Bakanova et al. [43] have some overlap with the calculated range, but was criticized in [41] as being too soft (i.e. predicting too low pressures with increasing density) and probably in error. It is definitely below the calculated Hugoniot. The Hugoniot curve of Young and Ross [41] is based on only 4 points given in their paper, hence its fractured appearance. The large discrepancy between their curve and ours is, in part, due to lack of points between their last value at, and one before last at,.

Our values close to these V points are 194 GPa and 92 GPa, respectively, so while at the two curves agree rather well at Young and Ross’ Hugoniot climbs almost twice as fast becoming very stiff. Regarding vs, one can discern a change in the slope in respect with the experimental data [39] [42] . This is in agreement with a general feature of Hugoniot in metals, as was proposed by Johnson [44] , were a change in slope is always present near.

Numerical data for the calculated Hugoniot curve are presented in Table 3.

4.2. Sylvian Kahane

DC conductivity and Diffusion coefficient D

The DC conductivity is the static limit of. This quantity can be measured experimentally. In calculations it was obtained by a polynomial fit to

Table 3. Li Hugoniot data. in 10^{3} K, in g/cc and in GPa.

, in the range, as shown in Figure 10. Its dependence on density is shown in Figure 15 and compared with results from Kietzmann et al. [14] at lower temperatures.

Kietzmann’s decreases with increasing density in the region 0.53 - 3 g/cc, but increases with the density outside it, forming a region of inversion. This is considered typical of metals. No region of inversion is observed in Figure 15, except perhaps a hint of it at. For higher temperatures the conductivity increases all the way in the range of considered.

In the work of Desjarlais, Kress and Collins [9] on Al, an inversion region in the range 0.01 - 0.1 g/cc is seen at lower T, but it is wiped out at. Li behaves, thus, similarly. Moreover, Kietzmann et al. identified the inversion region being a fluid metal, where the ion-ion pair correlation function shows short range order typical of liquids. Not such order is seen in Figure 3 at higher temperatures.

Bastea and Bastea [45] and Fortov et al. [46] measured the conductivity in Li. Both works used the quasi-isoentropic technique in which a shock wave is traveling back and forth in the sample, reflected by the anvils (saphire or steel), increasing the pressure. In [45] the reported temperatures varied from 2000 K to 7000 K and P reached 180 GPa, while in [46] T was lower than 3000 K and P reached 210 GPa, thus both are below the present calculations range of T.

The conductivity and other quantities dependence on temperature is shown in Table 4.

The values of the diffusion coefficient D are very well reproduced by an Arhenius function only for. At 10,000 K the calculated D is substantially larger than Arhenius fit. The values of

Figure 15. DC conductivity as a function of density.

Table 4. Trends in conductivity, pressure and diffusion coefficients as a function of tem- perature.

are 4.33 and 2.82 eV for and 1.5 g/cc respectively, with and 0.177 cm^{2}/s.

The reduced diffusion coefficient is defined as with ―the ions plasma frequency,―Wigner-Seitz radius,―the ions number density,―atomic mass, and―the average ionization = 1 at the calculation temperatures. Its values are one order of magni- tude larger compared with the one component plasma fit given by Hansen et al. [47] .

4.3. Rosseland Mean Opacity

The absorption coefficient is sometimes called opacity, in particular in the Astronomy and Astrophysics (A&A) field, which is concerned with radiation transport through stellar envelopes. The Rosseland mean opacity is a harmonic i.e. of) weighted mean, depending on and, conveniently giving a single number figure of merit for the radiation transport. It was calculated with Equation (11).

Some of the present results are compared in Figure 16 with results based on opacities calculated with the atomic modeled plasma by collisional-radiative FLYCHK code [48] .

In the atomic model the attenuation of radiation involves electron transitions (bound-bound, bound-free, free-free) in an isolated atom. It is appropriate, hence, mainly for diluted plasmas or gases. When the density is larger the interaction between neighboring atoms begins to come into play. If this density effect is still weak, it can be treated as a perturbation in the framework of the atomic model, but when the density is large and the atoms close, the isolated model will fail and a more collective approach is needed. The QMD + FTDFT

Figure 16. Rosseland mean opacity. Blue lines―present QMD + FTDFT calculations, red lines―calculated with the atomic code FLYCHK [48] .

offers such alternative model, in which the interaction with the neighbors is built-in in the QMD step, while the electrons wavefunctions (needed for the transitions calculations) are obtained in the FTDFT step from a collective model, not resembling at all the isolated atom. The versatility of the QMD is illustrated in Figure 17 which shows the Li atoms at some position in time when 4 out of the 54 atoms clearly formed two Li_{2} dimmers in which they are very close and, hence, the electronic wavefunctions are severely distorted by the presence of the neighbor atom.

Figure 16 shows that the QMD + FTDFT Rosseland mean opacities vary much slower compared with the corresponding atomic ones. This is in qualitative agreement with the results for hydrogen from [50] . To understand more on the differences between the present and atomic approaches one has to look at the absorption coefficients in Figure 18.

There is a sharp contrast below ~3 - 4 eV (the plasma frequency for the re- spective and is), where the QMD + FTDFT stops growing as, reverses course and declines slightly. This behavior is dictated by the conductivity (see Equation (9)) which approaches, quite flatly Figure 10), the zero-frequency limit, while the index of refraction varies only by one order of magnitude (Figure 11). On the other hand the atomic climbs higher and higher as approaches zero (this behavior is due to the free-free transitions), i.e. when the photon has very little energy and is not able to induce any electronic transition, the medium is totally opaque. The QMD + FTDFT prediction that for a photon of vanishing energy () the plasma is slightly more transparent than for a photon of 3 eV (Figure 18) is not so clear. This kind of behavior was received consistently in other works also [10] [50] . The plasma is permanently ionized at these tem- peratures (average 1^{+}) so a low energy photon does not have to actually excite a

Figure 17. Formation of Li_{2} dimmers in the course of the QMD simulations (a pseudo electron density iso-surface, created by the XCrysDen [49] program, is displayed). The distances between the Li ions in the dimmers are 0.65 Å and 0.75 Å.

Figure 18. Absorption coefficient compared with FLYCHK calculation.

bound electron to induce a transition, so one can expect some flat opacity at low energies below.

The K-edge of the FLYCHK presents some strong oscillations due to the bound-bound transitions, but afterwards declines exponentially as expected. The QMD + FTDFT K-edge is shifted toward lower energies, reaches the same values as the atomic one and begins to decay exponentially. At 80 - 90 eV drops sharply due to the finite number of states calculated in the FTDFT step.

Neither the order of magnitudes differences in the absorption coefficient at low energies, or the differences at the K-edge, are really influencing the QMD + FTDFT vs. FLYCHK Rosseland means. As can be seen in Figure 18 the weighting function samples mainly in a small range around.

4.4. Experimental Optical Data

Experimental optical data on Li metal was taken from an Internet source [51] , without proper credits, from Callcot and Arakawa (C&A) [52] and from Mathewson and Myers [53] . The data was measured, most probably, at room temperature. It is hard to estimate if the density is the nominal density (), for example the work of C&A uses thin films of unspecified density.

In Figure 19 these experimental data are compared with the QMD + FTDFT calculation at and. The goal of the calculations is obviously not to reproduce experimental data at room temperature, nevertheless it is instructive to compare. In spite of the very different conditions, the calculation does not depart wildly from the experiment. The C&A [52] data is

Figure 19. Experimental optical properties of Li metal, the index of refraction: real and imaginary. Blue + Ref. [51] , red # Ref. [52] , only the reliable part. Full line QMD + FTDFT calculation at and.

devided by its authors in two regions. In the range, The data is considered by the authors as very reliable. In the range below 6 eV, C&A working with two different substrates for their thin films, obtained two different branches of data, one numerically larger than the other. It seems that [51] includes the reliable part of [52] .

5. Summary

This work presents Quantum Molecular Dynamics and Finite Temperature DFT calculations, from which optical and electrical properties of warm Lithium plasma are obtained. It covers a range of temperatures and densities not in- vestigated previously bringing, therefore, fresh new information on dense plasma

Figure 20. Experimental dielectric function. Symbols: red [53] ; magenta # [52] , from the unsafe region, the upper branch; purple [52] , reliable data. Lines: maron-experimental data from [51] ; blue―QMD + FTDFT calculation at and. becomes negative at. The inset shows this region on a linear scale.

characteristics.

Detailed theoretical backgrounds were discussed:

・ Specifically the connections between the calculations and the dielectric func- tion.

・ Extraction of the optical properties from the dielectric function.

・ Use of pseudopotentials both in the QMD and the DFT calculations.

・ Strength and the problems in using the PAW pseudopotential for the DFT and the dielectric function calculations.

Moreover, also other computational techniques, of more heuristic approach, were employed resulting in a formula for Lithium Equation of State at high temperature and densities.

Whenever possible comparison with experimental data was shown, even when the temperature range was different.

Conclusion: New theoretical data for Rosseland absorption mean, indexes of refraction, , dielectric function and an equation of state are offered for Lithium in an unexplored range of temperatures and pressures.

Acknowledgements

I am grateful to Dr. Yuri Ralchenko from NIST, for his help with the FLYCHK program.

Conflicts of Interest

The authors declare no conflicts of interest.

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