The Influence of the Ionic Core on Structural and Thermodynamic Properties of Dense Plasmas

Abstract

In this paper, a new ion–ion screened potential was numerically calculated, which takes into account the ion core effect, i.e., the influence of strongly bound electrons. The pseudopotential model describing the shielding of ion cores and the screening using the density response function in the long wavelength approximation were used. To study the influence of this ion core effect on dense plasma’s structural and thermodynamic properties, the integral Ornstein–Zernike equation was solved in the hypernetted chain approximation. Our results show that the ion core has a significant impact on ionic radial distribution functions and thermodynamic properties when compared to the results obtained for the Yukawa potential, which does not take the ion core into account. Increasing the steepness of the core edge or decreasing the depth of the minimum leads to more pronounced screening due to bound electrons.

1. Introduction

The investigation of dense plasmas today is of particular interest because of its relevance in both astrophysical phenomena [1] and inertial confinement fusion applications [2,3], where the understanding of the interactions between charged particles is important in order to model properties of these plasmas [4]. The theoretical method of effective potentials, also known as pseudopotentials [5,6,7,8,9], is often used for modeling dense plasma, and, especially, effective potentials that describe the temperature differences between electrons and ions in dense plasmas in non-equilibrium states like the ones created by laser radiation [10,11] or inertial confinement fusion. The difficulties in this case are presented by differences in mass because heavier ions respond slower to perturbations than electrons, which leads to different energy and momentum transfer dynamics.

In recent studies, using a pseudopotential approach, researchers have investigated the influence of ionic cores on electron–ion scattering [12], ionic transport characteristics [13], and temperature relaxation [14] in dense plasmas. These results show that the impact of strongly bound electrons in ionic cores creates complex screening effects because of the resulting change in the effective charge of ions. Therefore, in this study, we want to add to the existing research by analyzing the impact of the ionic core effects on the structural and thermodynamic properties of dense plasmas.

In this study, we introduce a novel ion–ion screened potential that takes into account the ion core effect using a pseudopotential model with strongly bound electrons that have an impact on the shape of the pseudopotential near the ion core [15] in the form of the electron–ion potential. This approach allows us to consider both the shielding effects of ion cores and the screening derived from the exchange-correlation effect. By applying this new potential, we consider the impact of the ion core effect on the structural and thermodynamic properties of dense plasmas.

The paper is organized as follows: In Section 2, we describe the obtained screened ion–ion potential that takes into account the ion core effect due to bound electrons and screening. In Section 3, the method used to compute the radial distribution functions (RDFs) and non-ideality corrections to the thermodynamic properties (internal energy and equation of state) and the results of these calculations are presented. In Section 4, we discuss the results of the calculations.

2. Materials and Methods

In this work, we analyze the influence of strongly bound electrons, i.e., the ion core (a nucleus with bound electrons) effect, on the interaction between ions on the basis of the screened ion–ion interaction potential. The screened potential is related to the unscreened potential by the well-known formula $\widetilde{\Phi \left(k\right)}=\widetilde{\phi \left(k\right)}/\epsilon \left(k\right)$, where $\widetilde{\Phi \left(k\right)},\widetilde{\phi \left(k\right)}$ are the Fourier transforms of the screened and unscreened potentials, respectively. The Fourier transform is defined as $\widetilde{{\phi }_{ei}}\left(k\right)=4\pi /k\int r{\phi }_{ei}\left(r\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(kr\right)dr$, where $\epsilon \left(k\right)$ is the dielectric function of static screening. This relationship emphasizes how the initial Coulomb potential is modified by the screening effects in the plasma environment.

The interaction between free electrons and ions depends also on the occupied bound states, particularly core and valence electrons, which must create an antisymmetric state due to quantum mechanics. This antisymmetry impacts the overall electron cloud behavior around the ion. The core electrons create a region that effectively blocks nearby interactions, leading to increased Coulomb repulsion if another ion approaches. If core electrons are not explicitly considered, their effects can be captured by an effective electron–ion potential, which is strongly modified from the Coulomb potential near the core but resembles it at larger distances [16]. The simplest form of this pseudopotential is the empty core potential [17]. However, using a hard edge can introduce significant oscillations in Fourier space. To address this, we used a soft empty core potential, as proposed by Gericke et al. [15] (p. 2):

$${\phi }_{ei}\left(r\right)={\displaystyle \frac{Z{e}^2}{r}}\left[1-exp\left(-{\displaystyle \frac{r^{\alpha }}{r_{cut}^{\alpha }}}\right)\right].$$

(1)

The parameter α regulates the steepness of the core edge (in our case, α = 2 and α = 6 were chosen for figures in the Section 3), $r_{cut}$ is the core’s cutoff radius, and $r$ is the distance between interacting particles (in this case, electron and ion). Their values were taken to for beryllium ion to correspond with Kohn–Sham density functional theory (KS-DFT), which has been shown to describe dense plasmas under conditions of partial degeneracy accurately ($r_{cut}=0.75Å\simeq 1.42a_B,a_B={\displaystyle \frac{{\hslash }^2}{\left(m_e{e}^2\right)}}$ is the first Bohr radius). We used the following dimensionless parameters to describe the state of the system: the degeneracy parameter $\theta =k_{B}T/E_F$, where $E_F$ is the Fermi energy, and the density parameter $r_s=a/a_B$ where $a_0={\left(3/(4\pi n\right)}^{1/3}$ is the mean inter-electronic distance (also known as Wigner–Seitz radius); $r$ has been made dimensionless by dividing it by $a_B$, and therefore, the potentials are measured in Hartree energy units [Ha]. The parameters were chosen for dense plasmas, and in particular, $r_s\precsim 2a_B$ as it is the distance at which potential (1) deviates from the Yukawa potential [15] (p. 2).

Potential (1) describes the screening of ion cores by strongly bound electrons but not from plasma electrons. Therefore, we obtain the screened potential of ion–ion interaction based on potential (1) using the following formula [18]:

$${\Phi }_{ii}\left(r\right)={\displaystyle \frac{Z^2{e}^2}{r}}+\int {\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^3}}{\left|\widetilde{{\phi }_{ei}}\left(k\right)\right|}^2{\chi }_e(k)e^{i\overrightarrow{k}\overrightarrow{r}},$$

(2)

where ${\chi }_e\left(k\right)$ is the static electron-density response function. Gericke et al. [15] (p. 2) define the screening function as:

$$q\left(k\right)={\chi }_e\left(k\right){\phi }_{ei}(k)=Z{\displaystyle \frac{k_e^2}{k^2+k_e^2}}cos\left(k{r}_{cut}\right),$$

(3)

where $k_e^2=\left(4e^2{m}_e\right)/\left(\pi {\hslash }^3\right)\int f_e\left(p\right)dp$ is the inverse screening length, with $f_e\left(p\right)$ being the Fermi distribution, the values of which are equal to the Debye and Thomas–Fermi expressions in the non-degenerate and zero temperature limits, respectively.

It should be noted that at α = 1, potential (1) reduces to the Deutsch potential, and obtaining on its basis the screened ion–ion potential from Equation (2) leads to the following effective potential:

$$\Phi \left(r\right)={\displaystyle \frac{Z_i^2{e}^2}{1-r_{cut}^2{k}_e^2}}\left({\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}(-k_{e}r)}{r}}-{\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}(-r/r_{cut})}{r}}\right),$$

(4)

which corresponds to the limiting formula for pseudopotentials of particle interactions as $r_{cut}\to {\lambda }_{\alpha \beta }$ (where $\alpha \mathrm{a}\mathrm{n}\mathrm{d}\beta$ are types of interacting particles). This formula accounts for both quantum-mechanical effects of diffraction at short distances and screening effects at large distances for a strongly coupled semiclassical plasma [19]:

$${\Phi }_{\alpha \beta }\left(r\right)={\displaystyle \frac{Z_{\alpha }{Z}_{\beta }{e}^2}{\sqrt{1-4{\lambda }_{\alpha \beta }^2{k}_e^2}}}\left({\displaystyle \frac{\exp \left(-Ar\right)}{r}}-{\displaystyle \frac{\exp \left(-Br\right)}{r}}\right),B^2={\displaystyle \frac{1+\sqrt{1-4{\lambda }_{\alpha \beta }^2{k}_e^2}}{2{\lambda }_{\alpha \beta }^2}},A^2={\displaystyle \frac{1-\sqrt{1-4{\lambda }_{\alpha \beta }^2{k}_e^2}}{2{\lambda }_{\alpha \beta }^2}}$$

(5)

In Reference [20], it was shown that in Equation (2), the incorporation of electronic correlation effects through the local field correction $G$, which modifies the density response function, can be expressed as:

$${\chi }_e^{-1}\left(k,\omega \right)={\chi }_0^{-1}(k,\omega )+{\displaystyle \frac{4\pi e^2}{k^2}}\left[G\left(k,\omega \right)-1\right].$$

(6)

In the case of the first-order approximation of the inverse ideal response function, the screened Coulomb potential, also known as Yukawa potential, can be derived from Equation (2):

$${\Phi }_Y={\displaystyle \frac{Ze}{r}}\mathrm{e}\mathrm{x}\mathrm{p}(-k_{e}r).$$

(7)

This potential is well-known for describing dense plasmas and corresponds to both the Thomas–Fermi and Debye–Hückel limits. Furthermore, Moldabekov et al. [20] (p. 4) showed that the second-order result of the long wavelength expansion of the inverse random phase approximation (RPA) response function recovers the Stanton and Murillo (SM) model, which includes corrections for nonlocal effects across different densities and temperatures and displays oscillating patterns. However, neither the Yukawa nor SM potentials account for non-ideality or correlation effects, while static local field correction is most often obtained from ab initio quantum Monte Carlo simulations, which are time- and computation-intensive.

Therefore, in this work, to account for the local field correction (exchange-correlation effect) in the long wavelength limit, we used the parameter $\gamma$ in the following form [21], where the inverse screening length is:

$$k_e^2={\displaystyle \frac{k_{id}^2}{1-k_{id}^2\gamma }},k_{id}^2=k_{TF}^2{\theta }^{{\displaystyle \frac{1}{2}}}{I}_{-{\displaystyle \frac{1}{2}}}(\eta )/2,\gamma =-{\displaystyle \frac{k_F^2}{4\pi e^2}}{\displaystyle \frac{{\partial }^{2}n{f}_{xc}\left(n,T_e\right)}{\partial n^2}},$$

(8)

where $I_{-1/2}$ is the Fermi integral of the order $-1/2$, $\eta =\mu /k_B{T}_e$ is the reduced chemical potential, $k_{TF}^2=3{\omega }_p^2/{\upsilon }_F$ is the Thomas–Fermi wavenumber (${\omega }_p$ is the plasma frequency, ${\upsilon }_F$ is the Fermi velocity), and $f_{xc}(n,T_e)$ is the electron exchange-correlation free energy density of the uniform electron gas (UEG) [22].

3. Results

3.1. Ion–Ion Interaction Potential

Screened ion–ion interaction potential was calculated numerically from Equation (2) using potential (1) to account for the ionic core effect and the density response function of the UEG in the long wavelength approximation to account for screening due to plasma electrons.

Figure 1 shows the comparison of the calculated screened ion–ion interaction potential from Equation (2) at fixed values of $\theta =1$ and $r_s=2$ for different values of parameters $r_{cut}$ and $\alpha$: the left panel (a) is for fixed $\alpha$ = 6 with $r_{cut}=0.25$ (black solid line) and $r_{cut}=0.75$(red dashed line); the right panel (b) is for fixed $r_{cut}=0.75$ with $\alpha$ = 6 (black solid line) and $\alpha$ = 2 (red dashed line). These potentials were compared with the Yukawa potential (blue dash-dotted line).

3.2. The Radial Distribution Functions of Ions

The ionic core effect on a system’s structural properties was analyzed by solving the Ornstein–Zernike equation [23]:

$$h\left(r\right)=c\left(r\right)+n\int c\left(r^{\prime }\right)h\left(\right|\overrightarrow{r}-\overrightarrow{r}\prime \left|\right)d\overrightarrow{r}\prime ,$$

(9)

in the hypernetted chain (HNC) approximation:

$$g_{HNC}\left(r\right)=exp\left[-\beta \phi \left(r\right)+h\left(r\right)-c\left(r\right)\right],$$

(10)

where $\beta ={\left(k_{B}T\right)}^{-1}$ and $\phi \left(r\right)$ is the interaction potential, $h\left(r\right)=g\left(r\right)-1$ is the total correlation function, and $c\left(r\right)$ is the direct correlation function. Equations (9) and (10) were solved by numerical methods in combination with Fourier and iteration methods [24,25] ([20], p. 14).

It is important to note that while the radial distribution functions (RDFs), $g\left(r\right)$, calculated via the HNC approximation utilize an effective screened ion–ion potential as a micropotential—accounting for the medium through the polarization function of electrons—the ion screening effect is not double counted. Potential (2) only considers the screening from the electron background, not from the ions themselves. This is confirmed by the comparison of radial distribution functions carried out in Reference [26], where the results on the basis of MD and HNC (using a different ionic core potential) coincide.

Figure 2 shows the RDFs in the case (a) α = 6 for different values of $r_{cut}$ ($r_{cut}=0.25$—black solid line and $r_{cut}=0.75$—red dashed line) and in the case (b) $r_{cut}=0.75$ for different values of α (α = 2—black solid line and α = 6—red dashed line). The RDFs on the basis of the Yukawa potential (blue dash-dotted lines) were also calculated using the Ornstein–Zernike Equation (9) in HNC approximation (10).

3.3. Thermodynamic Properties

The internal energy and the equation of state [27] were calculated on the basis of potential (2) and RDFs (9) and (10) using the following expressions:

$$E=E_{id}-\pi \sum _{\alpha =i,e}{n}_{\alpha }\sum _{\beta =i,e}{n}_{\beta }{\int }_0^{\infty }{g}^{\alpha \beta }(r){\phi }^{\alpha \beta }(r)r^{2}dr=E_{id}-∆E$$

(11)

$$P=P_{id}-{\displaystyle \frac{2}{3}}\pi \sum _{\alpha =i,e}{n}_{\alpha }\sum _{\beta =i,e}{n}_{\beta }{\int }_0^{\infty }{\displaystyle \frac{\partial {\phi }_{\alpha \beta }(r)}{\partial r}}{g}_{\alpha \beta }(r)r^{3}dr=P_{id}-∆P$$

(12)

where $N$ is the number of particles in the system, $E_{id}=3/2N{k}_{B}T$ is the internal energy of an ideal plasma, and $P_{id}=n{k}_{B}T$ is the equation of state of an ideal plasma.

Figure 3 shows the ionic correlation energies dependent on density, while Figure 4 shows the pressures dependent on temperature. The results were calculated on the left panels (a) for fixed values of $\alpha$ = 6 with $r_{cut}=0.25$ (black solid lines) and $r_{cut}=0.75$(red dashed lines); the right panel (b) is for fixed values of $r_{cut}=0.75$ with $\alpha$ = 6 (black solid line) and $\alpha$ = 2 (red dashed line). On both panels, the results for the Yukawa potential are presented in blue dash-dotted lines.

4. Discussion

Figure 1 shows that potential (2) tends to the Yukawa potential (i.e., screened Coulomb potential) at large distances. At smaller distances, the ion core effect leads to weaker screening compared to the Yukawa potential due to weakening of the interaction between particles due to shielding by strongly bound electrons near the ion core.

In Figure 2, for fixed values of $\alpha$, decreasing the $r_{cut}$ radius leads to strengthened screening, which is shown as the RDF curve for $r_{cut}$ = 0.25 lying higher than the RDF for $r_{cut}$ = 0.75. This is explained by the $r_{cut}$ radius controlling the depth of the minimum and causing deeper negative minimum values of initial non-screened electron–ion potential, which are due to stronger attractive interactions between electrons and ions, i.e., more pronounced screening.

On the other hand, for a fixed $r_{cut}$, increasing $\alpha$ shifts the position of the potential minimum closer to the ion, i.e., the potential becomes steeper, because of stronger screening effects. This leads to the RDF for $\alpha$ = 6 being slightly higher than for $\alpha$ = 2. However, the difference between the RDFs for values of $\alpha$ = 6 and $\alpha$ = 2 is slight, while the deviation between the results for potential (2) and Yukawa is more pronounced because the Yukawa potential does not take into account the ion core effect. That is also the reason why the Yukawa potential’s RDFs (blue solid lines) reach unity faster than the RDFs for all other potentials.

Figure 3 and Figure 4 show the influence of the ion core on the ionic non-ideality corrections to the thermodynamic properties (correction to the internal energy, i.e., correlation energy and correction to the equation of state, which is incorporated in values of pressure measured in Mbars, respectively). Reducing $r_{cut}$ values for fixed $\alpha$ or increasing $\alpha$ for fixed $r_{cut}$ leads to more pronounced screening due to a stronger attraction between bound electrons to the ions. The thermodynamic properties deviate for the Yukawa potential because it has no core effects present. This leads to the small values of the corrections for non-ideality to the correlation energy and to the equation of state.

Our study, while grounded in the experimental results evaluated by Moldabekov et al., Table 1 of [20], such as cryogenic DT implosion on OMEGA [28,29], direct-drive ignition at the NIF [30], solid Be heated by 4–5 keV pump photons [31], laser-driven shock compressed aluminum [32], and Be sample [33,34], focuses on a simpler case to maintain generality, particularly as many relevant experiments involve non-isothermal plasmas, which we aim to explore in future work. Although our ion–ion screened potential model effectively describes interactions in dense plasmas with heavy ions, it has limitations in extreme conditions, such as very high densities or low temperatures, where strong quantum effects and inter-particle correlations can lead to significant deviations from expected behavior. While it is possible to adapt this model to more complex systems with multiple ion species, this remains beyond the current scope of our work. Such an adaptation would entail a more nuanced approach to the unique interactions and contributions of each species to the overall screening effects, along with addressing the increased numerical complexity to ensure stability and convergence in simulations.

5. Conclusions

In this study, we introduced a novel ion–ion screened potential that takes into account the ion core effect through a pseudopotential model, which modifies the electron–ion potential near the ion core under the influence of strongly bound electrons. This approach allows us to account for both the shielding effects of ion cores and the screening from the exchange-correlation effect in the long wavelength limit. Our results show that the ion core significantly influences ionic RDFs and non-ideality corrections to the thermodynamic properties compared to the results calculated for the Yukawa potential, which has no ion core effect, with reductions in $r_{cut}$ or increases in $\alpha$ enhancing screening through stronger electron–ion attraction. These results show the necessity of taking into account the role of ionic core effects in understanding the structural and thermodynamic properties of dense plasmas.