Classical-Map Hypernetted Chain Calculations for Dense Plasmas

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1 Early View publication on (issue and page numbers not yet assigned; citable using Digital Object Identifier DOI) Contrib. Plasma Phys., No. X, 1 8 (2014) / DOI /ctpp Contribution of the Intern. Conf. on Strongly Coupled Coulomb Systems, Santa Fe, New Mexico, USA, from July 27th - August 1st, 2014 Classical-Map Hypernetted Chain Calculations for Dense Plasmas R. Bredow 1, Th. Bornath 1, W.-D. Kraeft 1, M.W.C. Dharma-wardana 2, and R. Redmer 1 1 Institut für Physik, Universität Rostock, Rostock, Germany 2 National Research Council of Canada, Ottawa, Canada Received 16 October 2014, revised 11 November 2014, accepted 11 November 2014 Published online XXX 2014 Key words CHNC, structure factor, pair distribution function, multi-component, non-equilibrium, plasmas. Warm dense matter is of interest for modeling the interiors of planets and Brown Dwarfs. Corresponding pumpprobe experiments are performed at free electron laser facilities such as FLASH, LCLS or the future XFEL in Hamburg. X-ray Thomson scattering is of special interest to extract the plasma parameters. In order to explain or predict the X-ray Thomson scattering spectra, simulations on the structural properties of plasmas are performed. While ab initio simulations are computationally expensive, semi-classical approaches can deliver results fast for pair distribution functions and static structure factors, even for dense systems. We solve the Ornstein-Zernike equation within the hypernetted chain approximation for dense multi-component plasmas using the classical-map method. This approach proposes to treat the quantum features of the electrons using an adapted temperature for the electron system while the ions are treated classically. Results for pair distribution functions and static structure factors are presented for dense hydrogen, beryllium, carbon and CH plasmas. 1 Introduction Matter under extreme conditions is a field of wide interest with application in astrophysics, material science and fusion research. On the experimental side X-ray Thomson scattering (XRTS) is a powerful tool to analyze matter under extreme conditions [1], and various materials, e.g. Be [2 4], B [5, 6], C [7], LiH [8], and Al [9], have succesfully been probed during the last decade. Warm dense matter as well as strongly-coupled plasmas can occur as matter in extreme states [10]. They fall into a parameter range which is defined by the coupling parameter Γ and the degeneracy factor Θ with Γ = ( ) 1/3 (Ze)2 3 1, Θ = k BT 1, E F = 2 (3π 2 n e ) 2/3, (1) 4πε 0 k B T 4πn i E F 2m e where E F is the Fermi energy of the electrons. This regime has recently been investigated with hypernetted chain (HNC) calculations, a simple but fast approach to derive the static structure factors which are needed for evaluating the Chihara dynamic structure factor [11, 12]. The latter is the key quantity for intepreting X- ray Thomson scattering spectra and allows to derive fundamental plasma properties like the equation of state (EOS) and the electrical conductivity. This makes HNC a powerful tool to be used with XRTS experiments for first evaluations of the spectra. However, the HNC approach fails to deliver reliable results for systems of high densities n cm -3 and comparatively low temperatures T < 2 3 ev. Under such conditions quantum effects become dominant and have to be accounted for by an improved ansatz. In this work we present results derived from the classical-map hypernetted chain (CHNC) technique [13] that has successfully been applied to a variety of systems [14 17], and compare with HNC [18 22], quantum Monte Carlo (QMC) [23] and quantum molecular dynamic simulations (QMD) [22, 24]. While QMC and QMD allow Corresponding author. richard.bredow@uni-rostock.de

2 2 R. Bredow et al.: Classical-map HNC calculations for a consistent quantum treatment but are computationally demanding, the semi-classical HNC approach relies on effective potentials and is computationally very inexpensive, and is also free of simulation-box finite-size effects. Effective potentials like the Kelbg potential [25] and the Deutsch potential [26] were derived from the evaluation of the two-particle Slater sum and account for short-range quantum effects. An improving ansatz was proposed by Dharma-wardana and Perrot [13] leading to an incorporation of quantum effects in the electron system in the HNC iteration scheme via a classical mapping. In this approach the input of the classical temperature T and the effective potential V (r) are adapted to be in agreement with Kohn-Sham reference calculations rather than those from Slater sums [27]. In the next chapter we present the theory of the HNC and CHNC approaches and afterwards, in section 3, the results are discussed for calculations on hydrogen, beryllium, carbon and hydrocarbon. In section 4a summary of our results will be given where we also envisage improving CHNC implementations in future work. 2 Theory 2.1 Ornstein-Zernike equations and hypernetted chain closure relation The structure factor S ab (k) and the pair distribution function g ab (r) are the key quantities that describe structural properties of plasmas with a, b donating the species (electrons, ions). Classically, solving the OZ equations for a many particle system, h ab (r 12 ) = c ab (r 12 ) + n c dr 3 c ac (r 13 )h cb (r 32 ), (2) c can deliver the pair distribution function and thus the static structure factor. Here h ab (r 12 ) is the total correlation function for two particles a and b containing the direct correlation c ab between these two particles and all indirect correlations provided by all other particles c at position r 3 in the system. A closure relation is needed to connect h ab (r 12 ) and c ab (r 12 ) in order to solve (2). It can be obtained by a cluster expansion. Here we use the HNC appoximation where bridge functions are neglected leading to an expression for the pair distribution function in the form g ab (r 12 ) = exp( βv ab (r 12 ) + N ab (r 12 )), (3) where N ab (r 12 ) is related to all unbridged ways of correlations represented by c n c dr3 c ac (r 13 )h cb (r 32 ) in equation (2) and V ab (r 12 ) describes the direct pair potential. Equations (2) and (3) lead to the HNC equations which can be solved iteratively: c ab (r 12 ) = g ab (r 12 ) 1 ln g ab (r 12 ) βv ab (r 12 ). (4) The HNC scheme includes classical screening automatically while quantum effects have to be introduced via effective potentials which can be extracted from a two-particle slater sum [28]. Typical effective potentials of choice are the Kelbg potential [25] and the Deutsch potential [26], the later being what we used in our former work and in this study for the HNC calculations: Vab Deutsch (r) = q ( ( aq b 1 exp r )) + δ ae δ be k B T ln 2 exp 4πϵ 0 r λ ab ( ln 2 π ( r λ ee ) 2 ). (5) The Deutsch potential includes quantum effects corresponding to the uncertainty principle and exchange effects. The temperature dependent de Broglie wavelength is given by λ ab = / 2m ab k B T with m ab = (m a m b )/(m a + m b ). A more detailed view on the theory of the Ornstein Zernike relation, the hypernetted chain approximation and effective potentials can be found in [10]. 2.2 Classical mapping Dharma-wardana and Perrot introduced an extended scheme for the hypernetted chain approximation based on Kohn-Sham reference calculations [13]. This classical-map hypernetted chain (CHNC) approximation uses a

3 Contrib. Plasma Phys., No. X (2014) / 3 quantum temperature T q to modify the physical temperature T so that the classical Coulomb fluid and the quantum fluid have the same Kohn-Sham correlation energy [13, 14]. It means that in multi-component cases like those considered in this work the electron temperature T e is substituted by a classical fluid temperature T cf in the form T cf = Te 2 + Tq 2, (6) while the temperature for heavy particles stays the same physical temperature T. In multi-component calculations one has to define an interaction temperature T ei since the electrons and ions differ in the temperatures used for the calculations even if the considered system is in physical equilibrium. We decided for the interaction temperature as defined for our non-equilibrium HNC calculations in [10]: T ei = T cf T i. (7) With suitably adapted OZ relations one can also use a mass weighted interaction temperature in the form T ij /m ij = T i /m i + T j /m j with m ij = m i m j /(m i + m j ) [14, 29] as well as other choices [27]. The CHNC scheme further proposes an effective potential containing a Pauli exclusion term. It is defined in the form ϕ ij (r) = V c ij(r) + P ij (r), (8) with Vij c(r) being the Coulomb potential and P ij(r) being the Pauli exclusion potential. The latter potential is identically zero for anti-parallel spin electrons. In effect, P ij (r) is defined such that the non-interacting electron pair distributions for parallel spins and anti-parallel spins are correctly recovered form the integral equations, thus extending the zero-temperature procedure of Lado [30] to arbitrary temperatures. This means one can directly derive P ij (r) by an inversion of the HNC iteration for the theoretically known non-interacting case, and incorporates exchange interactions essentially exactly. The key advantage of the CHNC scheme is the point that one can use the nearly unchanged HNC iteration concept, only extended with T cf and ϕ ij (r). 3 Results for multi-component equilibrium plasmas 3.1 Hydrogen Fig. 1 Pair distribution functions (left figure) and static structure factors (right figure) for a fully ionized two-component hydrogen plasma at T = 0.34 ev and an ion density of n i = cm -3. Results of CHNC (black) and QMC (blue) calculations are shown. For comparison reasons the ion-ion pair distribution functions and static structure factors are of main interest. It can be seen that both methods are in good agreement with only small shifts of the peak positions while the differences in the peak heights are more significant.

4 4 R. Bredow et al.: Classical-map HNC calculations As a first benchmark, calculations for hydrogen under extreme conditions were performed. Due to the high density of n i = cm -3 and the low temperature of T = 0.34 ev it can be valued as a serious test for the limits of CHNC concerning the lowest possible temperatures. HNC iterations which do not use the classical-map approach become computationally unstable for multi-component calculations in the range of low temperatures T < 1.0 ev and very high particle densities n i > cm -3. The chosen densities are larger than 450 times the solid density of hydrogen and allow comparison with quantum Monte Carlo calculation (QMC) by Liberatore et al. [23], see Fig. 1. It can be seen that for the ion-ion pair distribution functions g(r) and static structure factors S(k), both methods are in very good agreement concerning the primary structural information. These are mainly based on the peak positions of g(r) and S(k) which are in good agreement with only small shifts. However, the differences of the peak heights are more significant. Please note that in the original work [23] a factor of π/2 has to be included to the scaling of k for S(k) [31]. The structure itself shows typical features for ultra-dense matter with liquid-like peak oscillations. 3.2 Beryllium In [10] we presented HNC results for uncompressed beryllium at solid density (u-be) in the non-collective [2] and collective scattering regimes [3], as well as for three times compressed beryllium (c-be) [4]. These calculations were based on the corresponding experiments and were compared with QMD simulations [24]. First evaluations based on the Chihara formula [1,11,12] predicted plasma parameters of T u = 12 ev and n u i = cm -3 for u-be, and T c = 13 ev and n c i = cm -3 for c-be. At these temperatures the plasma composition can be described adequately by an effective degree of ionization (or effective ion charge) of Z u = and Z c = 2.21, as derived from the QMD simulations. It was found that the main trends of the QMD static structure factors could be reproduced by the HNC calculations but the latter underestimates the low-k range. Fig. 2 Pair distribution functions and static structure factors for a two-component beryllium plasma in uncompressed (u, black) and threefold compressed (c, blue) state. The effective charge was set to Z u = and Z c = 2.21, respectively. The plasma parameters for u-be are n u i = cm -3 and T u = 12 ev while for c-be T c = 13 ev and n c i = cm -3 apply. The HNC calculations (dashed-dotted lines) were performed using the Deutsch potential (5). In comparison with CHNC (solid line) it can be seen, that classical map leads to a good agreement with DFT-MD results of Plagemann et al. [24] (dashed lines) in the long wavelength limit, which is underestimated by HNC. In figure 2 results for CHNC, HNC and QMD simulations for these parameters are shown. With coupling parameters of Γ u = 4.7 and Γ c = 6.2 and degeneracy factors of Θ u = 0.78 and Θ c = 0.41 we consider the strongly coupled regime which is where CHNC should show its advantage. Considering the static structure factors, all three approaches are in good agreement for k > 1.0 but the long wavelength limit led to strong deviations between QMD and HNC. This deficit is not unexpected since the degeneracy factor Θ is smaller than unity for both, u-be and c-be, meaning that the limits of the semi-classical HNC approach are actually exceeded. However, the CHNC calculations can reproduce the QMD results for k = 0 with only slight shifts. This very

5 Contrib. Plasma Phys., No. X (2014) / 5 good agreement with QMD simulations for low values of k is a major advantage of CHNC since that region is of special interest for the evaluation of experimental XRTS spectra. 3.3 Carbon Carbon is one of the light elements like nitrogen and oxygen which are expected to occur under extreme conditions in ice giants like Neptune, Uranus and Neptune-like exoplanets. It is of interest as ablator material in inertial confinement fusion (ICF) capsules. It is a good candidate for investigating the influence of the electron-based improvements in the CHNC approach. This is because it occurs in various ionization states in most systems of dense plasmas considered, leading to multi-component calculations with four or more components. We considered a temperature of T = 10 ev and a total ion density of n i = cm -3. The composition was calculated using the COMPTRA package [32]. We show the results in figure 3. At a first view it immediately gets obvious that only small differences occur between HNC and CHNC. We performed more calculations on systems with four and more components and all showed the same behavior. While the electron-based mapping of the CHNC approach has a strong influence on the electron structure factors all heavy particle structure factors experience only slight shifts. In general the main trends of the structure factors stay unaffected by the classical mapping except for the electron-electron structure factor. The latter is shifted strongly from S ee (0) < 0.5 (HNC) to S ee (0) > 1.0 (CHNC). The heavy particle structure factors show the expected trends governed by the repulsive forces of equal charges and by the partial densities. Fig. 3 Results on static structure factors S(k) derived from CHNC (left) and HNC (right) calculations for a multi-component carbon plasma. The Deutsch potential (5) was applied in the HNC calculations. The systematical trends are governed by the charge states and the partial densities. Significant differences between CHNC and HNC can only be found for the electron subsystems. 3.4 Hydrocarbon We performed further multi-component calculations on hydrides of the light elements, which are of special interest in planetary science [33]. Here we also want to present results for hydrocarbon (CH) as a comparison between HNC and CHNC for a multi-element case. Under extreme conditions like those investigated in this work hydrocarbon is like carbon which is of interest as ablator material in ICF capsules [34]. The temperatures T and densities ϱ we have chosen are based on hydrodynamic simulations for pump-probe experiments at the National Ignition Facility (NIF) [35]. Assuming a 1 : 1 particle ratio of carbon to hydrogen the plasma compositions were calculated with the COMPTRA package [32] and can be found in table 1. An approach self-consistent with CHNC would be to obtain the plasma compositions themselves from the free energies evaluated from CHNC itself. But this is left for a future study. In figure 4 the most significant pair distribution functions and structure factors are shown for the three cases. With increasing density the typical behavior can be observed. The peaks are shifted to higher k-values. For this trend a difference between CHNC and HNC occurs. While the HNC calculation leads to a significant peak

6 6 R. Bredow et al.: Classical-map HNC calculations shift between state 2 and state 3, this result can not be found with the CHNC approach. The latter only leads to a small loss in the peak heights. However, the main difference between both methods can mainly be found for the electron static structure factors. This is what we already found for a carbon multi-component plasma. The electron-electron structure factors derived from HNC start at low values for k = 0 and rise to small peaks for all three densities. On the other hand, CHNC calculated higher values in the long wavelength limit and the curves rise to unity with no significant peaks. The reason is to be found in the electron based mapping ansatz. The main quantum information that is needed in the original HNC approximation is numerically implemented via the quantum temperature T q, and the Pauli potentials and diffraction corrections that are only of relevance for the electron sub system. In multi-component calculations the results for the ion-ion static structure factors are dominated by the heavy particle interactions, with only a small influence of the electron system through the Ornstein-Zernike relation. A further effect that can be observed is based on the connection of the long wavelength limit with the compressibility. With increasing density the compressibility should decrease. This is found to be true for the main component C 4+, but not for C 3+ due to the heavy decrease of its relative partial density. Ion T = 4 ev T = 6 ev T = 8 ev ϱ = 3 g/cm 3 ϱ = 6 g/cm 3 ϱ = 9 g/cm 3 H C C C C Table 1 Composition (in % for H and C separately) of hydrocarbon plasmas at different temperatures and densities calculated with the COMPTRA program package [32]. Neutral particles are not listed. Fig. 4 Static structure factors for a multi-element hydrocarbon plasma at three density and temperature states (see table 1) calculated with CHNC and HNC (using the Deutsch potential). The trend of right shifted peaks with increasing density can be found for HNC between all the states, but not for state two and three derived from CHNC. Only a small loss in the peak height occurs. Beyond, only the electron static structure factors show significant differences with higher long wavelength limits and no peaks calculated with CHNC. 4 Summary We have performed calculations based on the approach of classical-map to improve the hypernetted chain approximation used in our former work for numerical investigations on the structure of warm dense plasmas [10]. As a first test for the region of applicability of classical-map HNC our results were compared to calculations performed by Liberatore et al. [23]. Good agreement was found for the ion-ion pair distribution functions and the static structure factors with differences only in the peak heights. A good improvement to the results derived from HNC calculations was found for beryllium on using CHNC. In our previous work [10] we showed that the HNC calculations lead to good agreement with QMD simulations

7 Contrib. Plasma Phys., No. X (2014) / 7 by Plagemann et al. [24] concerning the main trends of the curves but fail to recover the long wavelength limit. For both, solid density and threefold compression, the low-k values of S(k) were below the results of QMD. However, the CHNC approach led to very good agreement with QMD for the long wavelength limit. Further comparisons between HNC and CHNC were presented for a multi-component carbon plasma consisting of four different ion species. The major systematics for the structure factors, which are governed by the charge states and partial densities, can be found for both approaches with only small differences for the heavy particles but strong deviations for the electron structure factors. Finally, results for a multi-element plasma, hydrocarbon, were shown. The same behavior like for carbon occurs. Only the electron subsystem exhibits a significant influence from the quantum effects brought into the HNC-calculation by the classical mapping. This can be explained by the CHNC ansatz itself. While the heavy particle subsystems are treated classically, the quantum effects that are introduced to the system are numerically coupled to the electron subsystem. In plasmas with few or only two components this leads to a strong influence on all substructures, electrons and ions, through the Ornstein-Zernike relation. However, an increasing number of components results in heavy particle substructures that are more dominated by the ion subsystems themselves. This means, for the pair distribution functions and the static structure factors the differences between CHNC and HNC remain more or less the same for electrons, while the heavy-particle results converge towards each other with increasing number of ionic components. 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