The new model method of the electrostatic screening describing: three-component system of the 'closed' type

TL;DR

This paper introduces a new self-consistent model for electrostatic screening in fully ionized electron-ion plasmas of the 'closed' type, accounting for correlations and higher non-ideality effects, with broad parameter analysis.

Contribution

It develops the first comprehensive model for 'closed' type plasmas that includes electron-ion and ion-ion correlations, extending previous work on simpler systems.

Findings

Characteristics of plasmas across various densities and temperatures are calculated.

The model's results are compared with 'open' type systems.

Correlations significantly influence electrostatic screening in non-ideal plasmas.

Abstract

Within the problem of the finding of the mean potential energy of the charged particle in the plasma in this work a classification of physical systems (electrolytes, dusty plasmas, plasmas) is made based on consideration, or lack thereof, of a few special additional conditions. The system considered here, as well as other systems which are described with those additional conditions imposed are treated as the systems of the "closed" type, while the systems where those conditions are fully neglected - as being of the "open" type. In the our previous investigation one- and two component systems were examined. Here, as the object of investigation fully ionized electron-ion plasma with positive charged ions of two different kinds, including the plasmas of higher non-ideality, is chosen for the first time. For such plasma a new self-consistent model method of describing electrostatic screening is developed which includes all the necessary additional conditions. By this method such extremely significant phenomena as electron-ion and ion-ion correlations are included in the used model. The characteristics of the considered plasmas in a wide region of electron densities and temperatures are calculated. All the obtained results, including a comparison with systems of the "open" type are specifically discussed.

Equations

$$Z_{1}e\cdot N_1+Z_{2}e\cdot N_2-e\cdot N_e=0,$$

$$\frac{4\pi }{3}\cdot r_{1,2,e}^3=\frac{1}{N_{1,2,e}},\frac{4\pi }{3}\cdot r_{s;1,2}^3=\frac{Z_{1,2}}{N_e},r_{s;e}\equiv r_e.$$

$$p_1+p_2=1,p_{1,2}\equiv \frac{N_{1,2}\cdot Z_{1,2}}{N_e}=N_{1,2}\cdot \frac{4\pi }{3}{r}_{s;1,2}^3,$$

$$\underset{r\to \infty }{\lim }{n}_1^{(1,2,e)}(r)=N_1,\underset{r\to \infty }{\lim }{n}_2^{(1,2,e)}(r)=N_2,\underset{r\to \infty }{\lim }{n}_e^{(1,2,e)}(r)=N_e,$$

$${\rho }^{(1,2,e)}(r)=Z_{1}e\cdot n_1^{(1,2,e)}(r)+Z_{2}e\cdot n_2^{(1,2,e)}-e\cdot n_e^{(1,2,e)}(r),$$

$${\nabla }^2{\Phi }^{(1,2,e)}=-4\pi \left[Z_{1,2,e}e\cdot \delta (\vec{r})+{\rho }^{(1,2,e)}(r)\right],$$

$$\underset{r\to \infty }{\lim }{\Phi }^{(1,2,e)}(r)=0,\left|{\phi }^{(1,2,e)}\right|<\infty ;\text{vskip}3.0ptplus1.0ptminus1.0pt{\phi }^{(1,2,e)}\equiv \underset{r\to 0}{\lim }[{\Phi }^{(1,2,e)}(r)-\frac{Z_{1,2,e}e}{r}],$$

$$U^{(1,2,e)}=Z_{1,2,e}e\cdot {\phi }^{(1,2,e)}.$$

$${\Phi }^{(1,2,e)}(r)=-4\pi \underset{r}{\overset{\infty }{\int }}{\rho }^{(1,2,e)}(r)(r^{\prime })\left(\frac{1}{r}-\frac{1}{r^{\prime }}\right)r^{\prime 2}d{r}^{\prime },$$

$${\mu }_{\delta }(n_{\delta }^{(\delta )}(r),T_{\delta })+Z_{\delta }e\cdot {\Phi }^{(\delta )}(r)={\mu }_{\delta }(n_{\delta }^{(\delta )}(r_{st}),T_{\delta })+Z_{\delta }e\cdot {\Phi }^{(\delta )}(r_{st}),\delta =1,2,e,$$

$$n_{\delta }^{(\delta )}(r)-n_{\delta }^{(\delta )}(r_{st})=-\frac{Z_{\delta }e}{\partial {\mu }_{\delta }/\partial N_{\delta }}\cdot \left[{\Phi }^{(\delta )}(r)-{\Phi }^{(\delta )}(r_{st})\right],\frac{\partial {\mu }_{\delta }}{\partial N_{\delta }}\equiv {\left[\frac{\partial {\mu }_{\delta }(n,T_{\delta })}{\partial n}\right]}_{n=N_{\delta }},$$

$$\frac{|n_{\delta }^{(\delta )}(r)-N_{\delta }|}{N_{\delta }}\ll 1.$$

$$\underset{0}{\overset{\infty }{\int }}\left[N_{1,2,e}-n_{1,2,e}^{(1,2,e)}(r)\right]\cdot 4\pi r^{2}dr=\underset{0}{\overset{\infty }{\int }}\left[1-\frac{n_{1,2,e}^{(1,2,e)}(x\cdot r_{1,2,e})}{N_{1,2,e}}\right]\cdot 3x^{2}dx=1,$$

$$\underset{0}{\overset{\infty }{\int }}\left[N_{1,2}-n_{1,2}^{(e)}(r)\right]\cdot 4\pi r^{2}dr=\underset{0}{\overset{\infty }{\int }}\left[N_e-n_e^{(1,2)}(r)\right]\cdot 4\pi r^{2}dr=0.$$

$$\underset{0}{\overset{\infty }{\int }}\left[N_1-n_1^{(2)}(r)\right]\cdot 4\pi r^{2}dr=\underset{0}{\overset{\infty }{\int }}\left[N_2-n_2^{(1)}(r)\right]\cdot 4\pi r^{2}dr=0.$$

$$Z_{1,2,e}e+\underset{0}{\overset{\infty }{\int }}{\rho }^{(1,2,e)}(r)\cdot 4\pi r^{2}dr=0.$$

$$n_e^{(1,2)}(r)=N_e\cdot (1-{\alpha }_{e;1,2})+{\alpha }_{e;1,2}\cdot [Z_1{n}_1^{(1,2)}(r)+Z_2{n}_2^{(1,2)}],$$

$$N_1\cdot \frac{4\pi }{3}{l}_{s;1}^3+N_2\cdot \frac{4\pi }{3}{l}_{s;2}^3=1,$$

$$n_e^{(1,2)}(r)=n_{s;e}^{(1,2)}(r)+n_{ion;e}^{(1,2)}(r),$$

$${\int }_0^{l_{s;1,2}}[n_{s;e}^{(1,2)}(r)-N_e\cdot (1-{\alpha }_{e;1,2})]\cdot 4\pi r^{2}dr={\alpha }_{e;1,2}\cdot Z_{1,2},$$

$$n_{s;e}^{(1,2)}(r=l_{s;1,2})=N_e\cdot (1-{\alpha }_{e;1,2}),$$

$${\rho }^{(1,2)}(r)=e\cdot [Z_1\cdot (n_1^{(1,2)}(r)-N_1)+Z_2\cdot (n_2^{(1,2)}(r)-N_2)]\cdot (1-{\alpha }_{e;1,2}).$$

$${\rho }^{(1,2)}(r)=Z_{1,2}e\cdot (1-{\alpha }_{e;1,2})\cdot (1-{\alpha }_i)\cdot [n_{1,2}^{(1,2)}(r)-N_{1,2}],$$

$${\rho }^{(1,2)}(r)={\kappa }_{as;1,2}^2\underset{r}{\overset{\infty }{\int }}{\rho }^{(1,2)}(r^{\prime })\left(\frac{1}{r}-\frac{1}{r^{\prime }}\right)r^{\prime 2}d{r}^{\prime },$$

$${\kappa }_{as;1,2}={\kappa }_{0;1,2}\cdot [(1-{\alpha }_{e;1,2})\cdot (1-{\alpha }_i){]}^{1/2},{\kappa }_{0;1,2}={\left[\frac{4\pi (Z_{1,2}e{)}^2}{\partial {\mu }_{1,2}/\partial N_{1,2}}\right]}^{1/2},$$

$${\rho }^{(1,2)}(r)={\rho }_{1,2}\cdot r_{b;1,2}\exp ({\kappa }_{as;1,2}{r}_{b;1,2})\cdot \frac{e^{-{\kappa }_{as;1,2}r}}{r},$$

$$n_{1,2}^{(1,2)}(r)=N_{1,2}+\frac{{\rho }_{1,2}\cdot r_{b;1,2}\exp ({\kappa }_{as;1,2}{r}_{b;1,2})}{Z_{1,2}e\cdot (1-{\alpha }_{e;1,2})(1-{\alpha }_i)}\cdot \frac{\exp (-{\kappa }_{as;1,2}r)}{r},$$

$$n_{2,1}^{(1,2)}(r)=N_{2,1}-\frac{{\rho }_{1,2}\cdot r_{b;1,2}\exp ({\kappa }_{as;1,2}{r}_{b;1,2})\cdot {\alpha }_i}{Z_{2,1}e\cdot (1-{\alpha }_{e;1,2})(1-{\alpha }_i)}\cdot \frac{\exp (-{\kappa }_{as;1,2}r)}{r},$$

$${\rho }^{(1,2)}(r)=\frac{{\kappa }_i^2}{r}\cdot \left[\underset{r}{\overset{r_{b;1,2}}{\int }}{\rho }^{(1,2)}(r^{\prime })(r^{\prime }-r)r^{\prime }d{r}^{\prime }+\frac{{\rho }_{1,2}\cdot (r_{b;1,2}-r)(1+{\kappa }_{as;1,2}{r}_{b;1,2})}{{\kappa }_{as;1,2}^2}\right]+{\rho }_{1,2},$$

$${\kappa }_i=[({\kappa }_{0;1}^2+{\kappa }_{0;2}^2)\cdot (1-{\alpha }_{e;1,2}){]}^{1/2},$$

Full text

††journal: Annals of Physics

1 Introduction

Thematically, this work is the natural extension of the research of the mechanism of plasma’s inner electrostatic screening whose results are presented in the papers [1, 2, 3], named here Part 1, Part 2 and Part 3, respectively. Namely, in these papers the single- and two- component systems were discussed, and here we will consider for the first time systems of the next level of complexity i.e. three-component systems which contain free electrons and positive charged ions of two different kinds. However, as for the method of investigation, we shall emphasize straight away that it will be completely different than in the previous papers. Because of that, we will remind that the conducted research had the following task: to investigate, within the problem of the finding of the mean potential energy of the charged particle in the plasma, whether the physical model of plasma’s inner electrostatic screening, introduced in [4], is already exhausted by Debye-Hückel’s (DH) method, described in the same paper, or it still allows for development of an alternative one. As in the previous papers here we keep in mind the electrostatic screening in fully ionized plasmas.

Although the paper [4] was devoted to electrolytes it left profound trace in plasma physics and in adjacent disciplines ([5, 6, 7, 8], see also [9]). Its influence is felt even today in various fields of plasma physics (laboratory, astrophysical, ionosphere). So, in numerous papers direct DH or DH-like methods are used, as well as such products of theirs as DH potential and DH radius (see Inv.Bre. refs. and [10, 11, 12, 13]). All this sparked the interest for the possibility of going beyond the sphere of influence of [4] and development of the mentioned alternative method. Also, another stimulus existed for development of alternative methods, connected with finding a characteristic length grater than Debye radius ([14, 15, 16, 17, 18, 19], see also Part 3).

Let us remind that the essential properties of the mentioned model are the following:

- the presence of an immobile probe particle, which represents one kind of charged particles in the real system (plasma, electrolyte);

- treatment of the considered components which contain free charged particles of different kinds as ideal gases in states of thermodynamical equilibrium, without the assumption about equality of all temperatures;

- treatment of the existing total electrostatic field in the considered system as external with respect to the considered ideal gas;

- finally, among the properties of this model is usage, as its relevant mathematical apparatus, of equations which describe the mean local electrostatic field and the conditions of conservation of thermodynamical equilibrium for the considered components. As in the previous papers, this model is treated here as the basic one.

Let us remind that in the source work [4] the authors did not assume the electro-neutrality of the considered system, it was obtained as an organic property of the DH solution. This fact deserves special attention, since the mentioned electro-neutrality was automatically provided regardless of the fact that DH solution is a highly approximated one and has a few nonconditional negative properties (see discussion in Part 2). It seems that in [4] the electro neutrality was obtained as a present, which was justified by the ingeniosity of the described basic model. In connection with this it is necessary to keep in mind that any improvement of DH method means absence of an automatical electro-neutrality of the considered system and requires completion of the basic model by an additional condition of electro-neutrality of the system. Consequently, this fact is taken into account in this investigation from the beginning.

The task formulated above itself enforced a special role of the mentioned DH method as a referent one whose predictions shall be compared with a possible new method that would arise as the result of the undertaken investigation. In accordance with this, the main aim of our previous research became creation of a ”self-consistent” method of describing the mentioned electrostatic screening mechanism which is completely free of DH method’s disadvantages. Let us note that the definition of a ”self-consistent” method implies that all the relevant characteristic are determined within this method itself and expressed only through its basic parameters i.e. the particle densities, temperature, etc.

However, it has been shown that, except for the case of single-component system (e.g. electron gas on a positively charged background), it was very difficult to finish the whole procedure of eliminating the disadvantages of DH method in a ”self-consistent” way. Analysis which was done later convinced us that this result is not accidental, since the outer differences between the DH and the presented method were not practically significant, and the principal differences between these methods are of conceptual nature.

It was connected with application of additional conditions, equivalent to the conditions of conservation of particle numbers in finite systems, which are applicable in the case of the basic model (see Section 3). Namely, it can be shown that the only important fact here is that in the presented method relevant additional conditions are taken into account, while within DH method they are completely neglected. Because of the above mentioned, the DH method lost its especial significance for this research. In accordance with this, further on we will treat it only as an example of a method where the mentioned additional condition are neglected and we will not deal with its disadvantages. Consequently, apart of the finding of the mean potential energy of the charged particles in the plasmas, the direct objective of this research became development of a self-consistent method of describing the electrostatic screening in the considered three-component system where the relevant additional conditions are included from the beginning. Let us note that in such form our task has practically shifted from a purely physical to a mosly mathematical domain, as physics is now due to appear only at the point of presentation of the final results. We have certainly taken good care in order that the approximations used within the developed procedures be physically sound, so that application of the obtained results to real plasmas could make sense.

In order to show the differences between the results of applying and neglecting the relevant additional conditions, we will remind of the behaviour of electron density in two cases of the electron-ion plasmas with the probe particle whose charge is equal to that of an ion, which was considered in Part 2, and is illustrated here by Figs. 1a and 1b. First of them illustrates application of the method developed in Part 2, which makes sure that the area of higher electron density is followed by an area of its lowering. This is in accordance with the role of the probe-particle approximation: the situation in the vicinity of the probe particle could be considered as reflection of the real situation in the vicinity of any ion in the considered plasma, and should enable usage of the results illustrated by the figure to the case of the real plasma, since they don’t influence the mean electron density. The systems with similar behaviour of the electron density are treated here as the ”closed” ones. In the figure 1b the behaviour of the DH electron density is shown in electron-proton plasma with the probe particle whose charge is equal to proton’s. From this figure monotonous increasing of the electron density can be seen with decrease of the distance from the probe particle from infinity to zero. In the considered case such behaviour causes creation of an excess of 1/2 electron and 1/2 proton in the vicinity of the probe particle. This phenomenon, which is unacceptable from the point of view of a method whith includes additional conditions, will be discussed especially in the Section 7.

The systems with a similar behaviour of the electron density are treated here as ”open” ones. Let us note that only the systems of the ”closed” type are considered here in detail. The difference between ”open” and ”closed” systems will be discussed again at the end of this paper.

This work is dedicated to plasmas which are treated as fully ionized, including the plasmas of higher non-ideality. The region of electron densities from $10^{16}$cm*-3* to $10^{20}$cm*-3* is considered, and of temperatures from $1\cdot 10^{4}$ K to $3\cdot 10^{4}$ K. The material presented in this paper is distributed in seven Sections and five sections in Supplementary material. Section 2.1 contains the basic assumptions of the theory, Section 3, Section 4 and Section 5 - the description of the methods developed for the considered three-component systems in the case of positively and negatively charged probe particles respectively, Section 6 - the procedure used for determination of the theory parameters, Section 7 - the results and the necessary discussion, Supplementary material A, B, C contain a description of the procedures of obtaining some auxiliary expressions whose inclusion in the main part of the text could overload it, Supplementary material D - online materials with the tables and Supplementary material E - the future perspective of DH method.

2 Theory assumptions

2.1 The initial system and basic characteristic

A stationary homogeneous and isotropic system $S_{in}$ is taken here as the initial model of some real physical objects, suitable for applications of the results of this research. It is assumed that $S_{in}$ is constituted by a mix of a gas of free electrons and two gases of free ions of different kinds with positive charges $Z_{1}e$ and $Z_{2}e$, where $Z_{1,2}=1,2,3,...$, and $e$ is the modulus of the electron charge. The electron charge, $-e$, will be denoted also by $Z_{e}e$, where $Z_{e}=-1$. We will consider that these gases are in equilibrium states with: the mean densities $N_{1}$ and $N_{2}$ and the temperatures $T_{1}=T_{2}=T_{i}$ for the ions, and the mean density $N_{e}$ and the temperature $T_{e}\geq T_{i}$ for the electrons. All particles are treated as point-like, non-relativistic objects and their spins are taken into account only as factors which influence the chemical potentials of the considered gases. Satisfying the the condition

[TABLE]

is understood, which provides local quasi-neutrality of the system $S_{in}$. Let us emphasize that the case $Z_{1}=Z_{2}$ is also considered here (see subsection 7.1, Section 7), since it reflects the existence of some real systems with two physically different kinds of the ions with the same charge, e.g. H*+* and D*+, or H+* and He*+* etc.

In the subsequent considerations several characteristic lengths are used which are connected with the parameters $N_{1,2,e}$ and $Z_{1,2}$, namely the Wigner-Seit’s (WS) radii and the so-called ion self-spheres radii (see Part 2), denoted here by $r_{1,2,e}$ and $r_{s;1,2}$ respectively. They are defined by the relations

[TABLE]

From here it follows that Eq. (1) can be presented in the form

[TABLE]

where the parameters $p_{1}$ and $p_{2}$ describe the primary distribution of the space between the self-spheres of all ions of the first kind and all ions of the second kind.

2.2 The auxiliary systems

The properties and system conditions. In accordance with the basic model and the composition of the system $S_{in}$, the electrostatic screening of the charged particles is modeled here in three corresponding auxiliary systems. It is assumed that each of them contains: the electron component, two ion components with the same charges $Z_{1}e$ and $Z_{2}e$, and one immobile probe particle with the charge $Z_{p}e$, which is fixed in the origin of the used reference frame (the point $O$).

As in Part 1 and 2 only such cases are studied here where the probe particle can represent one of the charge particles of the system $S_{in}$, e.g. when $Z_{p}=Z_{1}$, $Z_{2}$ and $Z_{e}$. Two ion cases are denoted below with $(i1)$ and $(i2)$, the electron case - with $(e)$, while the corresponding auxiliary systems are denoted with $S_{a}^{(1)}$, $S_{a}^{(2)}$ and $S_{a}^{(e)}$ respectively.

All systems $S_{a}^{(1,2,e)}$ are treated below as isotropic and characterized by the corresponding mean local ion and electron densities: $n_{1}^{(1,2,e)}(r)$, $n_{2}^{(1,2,e)}(r)$ and $n_{e}^{(1,2,e)}(r)$, which retain the properties of the corresponding components in the system $S_{in}$ and satisfy the boundary conditions

[TABLE]

where $r=\vec{r}$, and $\vec{r}$ is the radius-vector of the observed point. Their other necessary characteristics are the mean local charge density $\rho^{(1,2,e)}(r)$ defined by relation

[TABLE]

and the mean local electrostatic potential $\Phi^{(1,2,e)}(r)$, which is treated as the potential of the external electrostatic field. Then, we will take into account the fact that $\Phi^{(1,2,e)}(r)$ and $\rho^{(1,2,e)}(r)$ have to satisfy the Poisson’s equation

[TABLE]

where $\delta(\vec{r})$ is the three-dimensional delta function [5], and $0\leq r\leq\infty$. Satisfying the boundary conditions

[TABLE]

is assumed, which guaranties physical sense of the mentioned electrostatic potential and connection with the system $S_{in}$, and is compatible with the electro-neutrality condition of the auxiliary systems. Since $\varphi^{(1,2,e)}$ is the mean electrostatic potential in the point $O$, the quantity

[TABLE]

is the mean potential energy of the probe particle is just the searched mean potential energy of the probe particle. In a usual way $U^{(1,2)}$ and $U^{(e)}$ are treated as approximations to the mean potential energies of the ion and electron in the initial system $S_{in}$.

In accordance with the basic model the electron and all ion components of all auxiliary systems are treated as ideal gases. Therefore we will encompass the characteristics of the auxiliary systems by chemical potentials $\mu_{1,2}(n_{1,2}(r),T_{i})$ and $\mu_{e}(n_{e}^{(e)}(r),T_{e})$ of the corresponding ideal ion and electron gases, which can depend on the corresponding particle spins; and by their boundary values, i.e. $\mu_{1,2}(N_{1,2}^{(1,2)},T_{i})=\lim\limits_{r\to\infty}\mu_{1,2}(n_{1,2}(r),T_{i})$ $=$ $\lim\limits_{r\to\infty}\mu_{1,2}(n_{1,2}^{(2,1)}(r),T_{i})$ and $\mu_{e}(N_{e},T_{e})=\lim\limits_{r\to\infty}\mu_{e}(n_{e}^{(e)}(r),T_{e})$. However, in accordance with Part 2, the ion components in the auxiliary system $S_{a}^{(e)}$ has to be considered in a special way which is described in the Section ”The case (e)”.

The system equations. It can be shown that on the basis of the procedure which was developed and described in detail in Part 1 and 2 for the case of the system which is electro-neutral as a whole, it is possible to switch from the Poisson’s equation to the equation for the potential $\Phi^{(1,2,e)}(r)$ which is more suitable for further consideration. This equation is given by

[TABLE]

and is taken here in such a form.

In order to find other necessary equations, we will consider the conditions of conservation of thermodynamical equilibrium (conservation of the electro-chemical potential) for those components which are represented by the corresponding probe particles, namely

[TABLE]

where, in accordance with the basic model, $\Phi^{(1,2,e)}(r)$ is treated as the potential of the external electrostatic field, and $r_{st}$ is the distance from the point $O$ of the chosen fixed (starting) point: $0<r_{st}\leq\infty$. From here, by means the usual linearization procedure, the necessary equations for the particle densities $n_{\delta}^{(\delta)}(r)$ are obtained in the form

[TABLE]

which is applicable under the condition

[TABLE]

Here we will use the fact that such equations can be applied not only to the classical cases, but also to the quantum-mechanical ones, including the case of ultra-degenerated electron gas [6, 8].

Let us note that the corresponding equations for the ion densities $n_{1,2}^{(e)}(r)$ are obtained and discussed in Section 5, and for $n_{2,1}^{(1,2)}(r)$ - in A.

The additional conditions. In Part 1 and 2 the conditions were already introduced for the component which is represent by the probe particle and in which the charge of the probe particle appears. In the considered three-component case these conditions are given by

[TABLE]

where $x\equiv r/r_{1,2,e}$. This means that the ratio in the expression inside the square brackets is of the order of magnitude of 1. This equation is especially important since it provides the continuity of the model. In order to show this fact it is enough to consider the situation when the charge density $n_{1,2}^{(2,1)}$ is negligible and the considered three systems for physical reasons can be treated as a two-component system. Then, as an approximation, we could replace the electron component by the negatively charged nonstructural background and come back to a one-component system. It is important that the single-component case is by now used for mathematical modeling of the plasma internal electrostatic screening (Iosilevskiy 2011, private communication). Let us note that in the single-component case (which was not considered in [4]) this first condition can be used instead of the electro-neutrality condition.

Then in Part 2 it the additional conditions for the electron density were introduced. Here the corresponding conditions are given by the equations

[TABLE]

The additional conditions for the ion components $n_{1}^{(2)}(r)$ and $n_{2}^{(1)}(r)$ are taken into consideration analogously to the electron components, since there are no principal differences between them. The corresponding relation are given by

[TABLE]

As it was mentioned (Introduction), the presented additional conditions are fully compatible with the basic model. Let us note that in this work the considered physical systems, as well as other physical systems which are described by means of additional conditions (13)-(15), are treated as systems of the ”closed” type, while the physical systems where they are fully neglected- as the systems of the ”open” type. The significance of these conditions and consequences of their application are discussed in Section 7.

As it is known, in our investigation we have to take care that our results be compared with the results of DH method, which provides electro-neutrality of the considered system as a whole. In principle this would justify introducing into consideration the corresponding electro-neutrality condition, namely

[TABLE]

However the fact is used that simultaneous satisfaction of the conditions (13) and (14) and (4) automatically provides satisfaction of this condition. Therefore this condition will not be used within this work.

3 (i1) and (i2) cases: auxiliary expressions

3.1 Needed remarks

Practically from the beginning of the research insufficiency (unsatisfactoriness) of equations and system equations from the previous section was a serious problem. Concerning the electron component, this problem was solved in Part 2 by elaboration of a complete acceptable procedure for its determination. We will take advantage of this fact and use the already obtained results in the next section with minimal necessary discussion.

As for the ion component, a similar problem is solved in this paper, and we need to present all the relevant considerations here. Because of the abundance of the necessary material a part of it is displayed in Supplementary material A and B.

For the sake of further consideration we will note that here, apart from $r_{s;1,2}$, $r_{1,2}$ and $r_{e}$, other characteristics lengths will be also used, namely: $l_{s;1,2}$ and $r_{b;1,2}>l_{s;1,2}$ in the cases $(i1)$ and $(i2)$, and $l_{e;1,2}$ in the case $(e)$. So, we will treat the regions $0<r<l_{s;1,2}$ and $r_{b;1,2}<r<\infty$ as those of small and large $r$ respectively and, consequently, the regions $l_{s;1,2}<r<r_{b;1,2}$ - as those of middle $r$.

3.2 The electron densities

The region of large and middle $r$. In Part 2 it the model of the electron component in the corresponding auxiliary system was introduced which can be applied in the case of the considered $S_{a}^{(1,2)}$ systems. It was obtained on the basis of the approximation where the heavy particles (ions) were treated as as an ideal gas with respect to the fixed probe particle, but as immobile with respect to the free light particles (electrons). Let us note that similar approximations were very often used in the physics of plasma (see e.g. [20]). Apart from that, in the region of small $r$ (the vicinity of the positively charged probe particle), equality of all positively charged ions of the same kind was used, as well as the symmetry of the considered system. Typical behaviour of the electron density in the three-component case is presented in Fig. 2.

Within this model the changes of the electron density in the regions of large and middle $r$ were expressed through changes of the total ion-charge density. For that purpose it was taken into account that in the considered regions of $r$ the changes of electron and ion densities are small and can be very well described in the linear approximations.

One can see that the mentioned model can be automatically applied to systems with more positively charged ion components. Namely, it is enough to use in the just mentioned relation the changes of the positive-charge densities of all the existing ion components. So, in the considered three-component case, the relevant relation should have the form: $[n_{e}^{(1,2)}(r)-N_{e}]\sim[(Z_{1}\cdot n_{1}^{(1,2)}+Z_{2}\cdot n_{2}^{(1,2)})-(Z_{1}\cdot N_{1}+Z_{2}\cdot N_{2})],$ whence it follows that the necessary expression for the electron densities in the regions of middle and large $r$ can be taken in the form

[TABLE]

where by $\alpha_{e;1}$ and $\alpha_{e;1}$ the corresponding proportionality coefficients are denoted, which are treated here as the corresponding electron-ion correlation coefficients ($0<\alpha_{e;1,2}<1$).

The region of small $r$. For such a region another adequate procedure for determination of electron density was developed in Part 2, which is also applicable to systems with more positively charged ion components. In the considered case the parameters $l_{s;1,2}$ are used for that procedure instead of $r_{s;1,2}$. It is understood that $l_{s;1,2}$ have to satisfy the condition

[TABLE]

which reflects the fact that transition from $r_{s;1,2}$ to $l_{s;1,2}$ has to represent only redistribution of the space between the ion components, which means the equality of the left sides of Eqs. (3) and (18). Let us note that in the case $Z_{1}=Z_{2}$ we have it that $l_{s;1,2}=r_{s;1,2}$.

Within the mentioned procedure the electron densities $n_{e}^{(1,2)}(r)$ in the regions $0<r<l_{s;1,2}$ are taken in the form

[TABLE]

where $n_{ion;e}^{(1,2)}(r)$ describes the distribution of electrons, whose presence is caused by the electron-ion correlation, and is obtained by the extrapolation of the second member in Eq. (17) into the considered regions. The member $n_{s;e}^{(1,2)}(r)$ has to satisfy the conditions

[TABLE]

[TABLE]

where Eq. (20) represents such a generalization of the corresponding condition from Part 2, which is adequate in both $Z_{1}\neq Z_{2}$ and $Z_{1}=Z_{2}$ cases (see Section 6).

The necessary expressions for $n_{s;e}^{(1,2)}(r)$ in the regions $0<r<l_{s;1,2}$ are obtained by means of the data from Part 2, by appropriate replacement of the origin designations. They are given below by Eqs. (43) and (45). As one can see, these expressions provide automatical satisfaction of the boundary conditions (21). This is important since in this case Eqs. (20) and (20) are enough for determination of the parameters $\alpha_{e;1,2}$ and $l_{s;1,2}$.

3.3 The ion densities

The region of large $r$. From Eqs. (1), (5) and (17) follows the general relation for the charge densities in the region of middle and large $r$, namely

[TABLE]

One can also see that this relation is insufficient for further investigation. Namely, it is not possible to satisfy the conditions (15) if $[n_{2,1}^{(1,2)}(r)-N_{2,1}]$ in the whole regions $l_{s;1,2}<r<\infty$ are described by means the equations similar to Eq. (11). Therefore a separate procedure for determination of $n_{2,1}^{(1,2)}(r)$ had to be developed here.

A direction of that development was chosen which is based on a physically justified assumption. Namely, we take into account that the rules found in the case of two kinds of positively charged ions should not change drastically if one kind of ions is replaced by a different one. In accordance with this, the next step should be choosing a system with two kinds of positively charged particles (which could be easily described at least qualitatively) and finding the expression for the corresponding charge densities for the region of large and middle $r$. After that it only remains for us to switch to the considered system, taking into account that in such procedures some parameters should appear which could be determined based on the corresponding physical conditions. This fact will lower the possible error in determination of the required charge density. Besides, we keep in mind that, similarly to the above case of the electron density, the changes of the ion densities in Eq. (22) are also very small in the region of large $r$ and could be well described in linear approximation.

As for the mentioned system which could be easily described, one can see that an obvious choice can be an imaginary system where the positively charged ions of one kind are replaced with positrons, which are treated here as a probe system. Certainly, instead of the electron component in that system the non-structured negatively charged background has to be taken, in order to eliminate the electron-positron interaction from the consideration.

Because of the great amount of the necessary material, the chosen probe system is described in details in Supplementary material A. The result of the procedure developed in this Supplementary material is the exceptionally significant relation, given by Eq. (78), where the parameter $\alpha_{i}$ exists which has the meaning of the ion-ion correlation coefficient. Namely, from this relation and Eq. (22) the expression

[TABLE]

is obtained, which provides the applicability of the procedure for determination of the charge and ion densities developed in Part 1 and 2. In accordance with that procedure this expression, by means Eq. (11) for $n_{1,2}^{(1,2)}(r)$ with $r_{st}=\infty$, and Eq. (9) for $\Phi^{(1,2)}(r)$, is transformed to an integral equation of Volterra’s type:

[TABLE]

[TABLE]

where the lower index $"as"$ points to the fact that it is just the ion screening constant $\kappa_{as;1,2}$ that determines the behaviour of $\rho^{(1,2)}(r)$ in the asymptotic region of $r$. According to Part 1 and 2, the solution of this equation can be taken in the form

[TABLE]

where $\rho_{1,2}\equiv\rho^{(1,2)}(r_{b;1,2})$ is determined from the condition of continuity of $\rho^{(1,2)}(r)$ in the points $r=r_{b;1,2}$. Finally, by means Eqs. (23), (78) and (26) we obtain the expressions for the ion densities

[TABLE]

[TABLE]

which are valid in the regions $r_{b;1,2}<r<\infty$.

The regions of middle and small $r$. In accordance with A the charge densities $\rho^{(1,2)}(r)$ in the regions $l_{s;1,2}<r<r_{b;1,2}$ can be determined directly from Eq. (22). Namely, by means Eq. (11) for $n_{1,2}^{(1,2)}(r)$ with $r_{st}=r_{b;1,2}$, Eq. (A.2) for $n_{2,1}^{(1,2)}(r)$ and Eq. (9) for $\Phi^{(1,2)}(r)$, Eq. (22) is transformed to the integral equation

[TABLE]

[TABLE]

where $\rho_{1,2}=\lim\limits_{r\to r_{b;1,2}}\rho^{(1,2)}(r)\equiv\rho^{(1,2)}(r_{b;1,2})$. In accordance with Eq. (81) from B the solutions of this equation are given by

[TABLE]

where $F_{1,2}(r)$ is given below by Eq.(41) with complete expressions for the ion densities, and $c_{1,2}>0$ is the unknown parameter. From here and Eqs. (11), (22) and (30) it follows that the ion densities in the regions $l_{s;1,2}<r<r_{b;1,2}$ are given by

[TABLE]

[TABLE]

where $f_{1,2}$ and $d_{1,2}$ are defined below by Eq. (41 together with $F_{1,2}(r)$. Then, we will introduce other unknown parameters $r_{0;1,2}^{(1,2)}$, taking $c_{1,2}$ in the form

[TABLE]

which provides that $n_{1,2}^{(1,2)}(r=r_{0;1,2}^{(1,2)})\equiv 0$.

Here we will take into account the fact that the behavior of the ions of both considered kinds in the regions of small $r$ has to be similar to that of positively charged ions in the vicinity of the positive charged probe particle in the single- and two-component systems. Therefore we will determine the ion densities in the regions $0<r<l_{s;1,2}$ using the procedure of extrapolation, which is described and discussed in detail in Part 1 and 2. This means the following: extrapolation of Eqs. (33) and (34) for $n_{1,2}^{(1,2)}(r)$ to the points $r_{0;1,2}^{(1,2)}$ and for $n_{2,1}^{(1,2)}(r)$ to the points $r_{0;2,1}^{(1,2)}$ such that $n_{2,1}^{(1,2)}(r_{0;2,1}^{(1,2)})=0$, and after that - extension to the point $r=0$ by zero.

4 (i1) and (i2) cases: complete expressions

4.1 The ion densities

The complete expressions for the ion densities follow just from the above mentioned and Eqs. (28), (31), (33), (31) and (34). Due to their importance they are presented separately for the case $(i1)$ in the form

[TABLE]

[TABLE]

and separately for the case $(i2)$ in the similar form

[TABLE]

[TABLE]

where the functions $F_{1,2}(r)$ and the coefficients $f_{1,2}$, $A_{1,2}$, $B_{1,2}$, $C_{1,2}$ and $d_{1,2}$ are given by the relations

[TABLE]

[TABLE]

[TABLE]

and the screening constants $\kappa_{as;1,2}$, $\kappa_{0;1,2}$ and $\kappa_{i}$ - by Eqs. (25) and (30).

4.2 The electron densities

The complete expressions for the electron densities are obtained from Eqs. (17), (19) and (21), and are presented here in the form

[TABLE]

where the ion densities $n_{1}^{(1,2)}(r)$ and $n_{2}^{(1,2)}(r)$ are given by Eqs. (35) - (37). The number $n_{s;e}^{(1,2)}(r)$, in accordance with the above, is given by the relations

[TABLE]

[TABLE]

[TABLE]

One can see that these expressions provide automatical satisfaction of the boundary conditions (21). This is important since in this case Eqs. (18) and (20) are enough for determination of the parameters $\alpha_{e;1,2}$ and $l_{s;1,2}$. Let us note that these parameters, as well as $r_{b;1,2}$, $r_{0;1}^{(1,2)}$, $r_{0;2}^{(1,2)}$ and $r_{0;1}^{(1,2)}$, are determined as it is described in Section 6.

5 The presented method: the case (e)

5.1 The partial expressions

The ion and electron densities: the region of large $r$. On the basis of discussion from Part 2 we will take into account that in the auxiliary system $S_{a}^{(e)}$ only the electron component can be considered an ideal gas, while the ion components can be treated as parts of some stationary, but non-homogenous positively charged background. This is a consequence of the fact that a fixed negatively charged probe particle, which certainly makes physical sense in the case of some electrolyte, cannot be the representative of a free electron (light particle), surrounded by positively charged ions (heavy particles), particulary for the fact that ions were already treated as immobile with respect to the electrons. In the previous Section it was mentioned that similar approximations have very often been used in the physics of plasma. As it is well known, the reason for this is very short time for which a free electron stays in the vicinity of any ion, which makes it possible to neglect the influence of the electron-ion interaction on the ion positions. Therefore we must not describe the ion components in the system $S_{a}^{(e)}$ in a way similar to that in the systems $S_{a}^{(1,2)}$. Namely, according to Part 2, the ion distributions in the system $S_{a}^{(e)}$ as a matter of fact have to reflect the properties of the electron distributions with respect to the positively charged ions in plasma, i.e. in the considered case - the properties of the already described electron distributions in the systems $S_{a}^{(1,2)}$, including the probe-particle self-spheres. This means that in the system $S_{a}^{(e)}$ the positive charge density has to be determined $\rho_{i}^{(e)}(r)=Z_{1}e\cdot n_{1}^{(e)}(r)+Z_{2}e\cdot n_{2}^{(e)}(r)$, where the functions $n_{1}^{(e)}(r)$ and $n_{2}^{(e)}(r)$ can be treated as ion densities, although they have only one task: to make possible treating the probe particle potential energy in the system $S_{a}^{(e)}$ as a good approximation of the mean electron potential energy in the system $S_{in}$.

Since the adequate procedure of determination of $\rho_{i}$ in the system corresponding to $S_{a}^{(e)}$, which was developed in Part 2, allows generalization to the systems with more positively charged ion components, it is just that procedure that is used here, certainly with the necessary modifications. So, the probabilities of changes of the partial charge densities $Z_{1,2}e\cdot n_{1,2}^{(e)}(r)$ with the corresponding changes of $n_{e}^{(e)}(r)$ are not equal to $\alpha_{e;1,2}$, but to $\alpha_{e;1,2}\cdot p_{1,2}$. Consequently, Eqs. (78) and (35) from Part 2 are transformed to the form

[TABLE]

From here and Eq. (5) the relation for $\rho^{(e)}(r)$ follows, namely

[TABLE]

Then, from this relation and Eq. (11) for $n_{e}^{(e)}(r)$, with $r_{st}=\infty$ and $\Phi^{(e)}(r)$ given by Eq. (51) from Part 1, the necessary integral equation is obtained:

[TABLE]

where $\kappa_{e}$ is the electron screening constant. Since it is similar to Eq. (25), the charge and electron densities in the region $l_{e;max}<r<\infty$ can be taken in the known form

[TABLE]

which provides that $n_{e}^{(e)}(r=r_{0;e})\equiv 0$, where $r_{0;e}$ is the unknown parameter. The electron density in the region $0<r<l_{e;max}$ will be determined by means of the procedure similar to the one which is used in the previous Section for the ion components in the region of small $r$.

The ion densities: the region of small $r$. In accordance with Part 2 the ion densities $n_{1,2}^{(e)}$ inside the probe particle self-sphere are taken here in the form

[TABLE]

where $n_{el;1,2}^{(e)}(r)$ describes the distribution of the ions, whose presence is caused by the electron-ion correlation, and is obtained by extrapolation of the second member in Eq. (46) into the regions $0<r<l_{e;1,2}$. The member $n_{s;1,2}^{(e)}(r)$ has to satisfy the conditions

[TABLE]

[TABLE]

where the second one represents a generalization of the condition (41) from Part 2 which is applicable in both $Z_{1}\neq Z_{2}$ and $Z_{1}=Z_{2}$ cases (see Section 6). The introduction of two characteristic lengths $l_{e;1,2}$ instead of one $r_{e}$ is caused by the presence of two ion components. The expressions for $n_{s;1,2}^{(e)}(r)$ are obtained here by corresponding replacement of the origin designations of Eq. (43) from Part 2, and are given below by Eq. (56).

5.2 The electron and ion densities: complete expressions

Here we can repeat the procedures from Part 2 word-for-word. So, starting from Eq. (48), we obtain the expression for the electron density

[TABLE]

which determines $n_{e}^{(e)}(r)$ in the whole region $0<r<\infty$. Then, in accordance with the previous subsection, we obtain the expressions for the ion densities in the same regions by means of Eqs. (46) and (50), the conditions (51) and (52), and Eq. (43) from Part 2. The obtained expressions are given here by the relations

[TABLE]

[TABLE]

[TABLE]

where $a_{1,2}$ and $b_{1,2}$ are given by Eq. (45). One can see that in such a form $n_{s;1,2}^{(e)}(r)$ automatically satisfy the boundary condition (51), as well as the condition (52). Let us note that the the ratios $l_{s;1,2}/r_{s;1,2}$ and the characteristic length $r_{0;e}$ are determined as it is described in the next Section.

6 Determination of the parameters

The parameters $\alpha_{e;1,2}$ and $l_{s;1,2}$ are determined differently in the cases $Z_{1}=Z_{2}=Z_{i}$ and $Z_{1}\neq Z_{2}$. Namely, within the used procedure the first case, where $r_{s;1}=r_{s;2}\equiv r_{s;i}$, is equivalent (from the point of view of determination of $\alpha_{e;1,2}$) to the case of the two-component plasma with the same $Z_{i}$, $N_{e}$ and $T_{e}$. Consequently, in the case $Z_{1,2}=Z_{i}$ the following relations

[TABLE]

are valid, where in accordance with Part 2, $\alpha(x)$ is defined by

[TABLE]

However, in other case we have it that $r_{s;1}\neq r_{s;2}$ and, consequently, $l_{s;1}\neq l_{s;2}$. Therefore, in the case $Z_{1}\neq Z_{2}$ the parameters $\alpha_{e;1,2}$ and $l_{s;1,2}$ have to be determined together from Eqs. (20) and (21). For that purpose these equations have to be transformed first, by means Eqs. (43) and (45), to the form

[TABLE]

[TABLE]

where $x_{l;1,2}$ is given by Eq. (45). Then, the variables $x_{l;1,2}$ in Eq. (59) have to be replaced with $x_{s;1,2}\cdot l(s;1,2)/r_{s;1,2}$, where $x_{s;1,2}=k_{0;e}\cdot r_{s;1,2}$. After that the ratios $l_{s:1,2}/r_{s;1,2}$ can be used as variables in both equations. Obtaining their solution is aided by the relations

[TABLE]

which are established by direct calculations. Also, it is useful to keep in mind the fact that the left side of Eq. (59) approaches to $\alpha(x_{s;i})$ if $l(s;1,2)$ approaches to $r_{s;i}$, and that it is always $l_{s;1}>r_{s;1}$ and $l_{s;2}<r_{s;2}$ if $Z_{1}<Z_{2}$.

It is important that the electron-ion correlation coefficient $\alpha_{e;1,2}$ and the characteristic lengths $l_{s;1,2}$ are determined, as it is described below, independently of all other parameters. Namely, that is why the existing conditions are sufficient for determination of the characteristic lengths $r_{0;1,2,e}^{(1,2,e)}$, $r_{0;2,1}^{(1,2)}$ and $r_{b;1,2}$, and the ion-ion correlation coefficient $\alpha_{i}$.

So, $r_{0;1,2}^{(1,2)}$ are determined as functions of $\alpha_{i}$ and $r_{b;1,2}$ from Eq. (13) for $n_{1,2}^{(1,2)}$, and $r_{0;2,1}^{(1,2)}$ - as the roots of the equations $n_{2,1}^{(1,2)}(r)=0$ which can be find in the interval $[0,r_{b;1,2}]$. The very important parameters $r_{b;1,2}$, i.e. the distances from the point $O$ at which the manner of describing the ion densities changes, are determined from Eq. (15) for $n_{2,1}^{(1,2)}$ through a procedure where it is taken that

[TABLE]

where $\eta_{1,2}$ are new parameters which are used in the calculations in such a way that they vary with the small steps $\Delta\eta_{1,2}=1/K_{1,2}$ where $K_{1,2}\gg 1$. As the results of this procedure we get the values of the parameters $r_{b;1,2}$, i.e. the main one of the considered characteristic lengths, which corresponds to the current value of the ion-ion correlation coefficient $\alpha_{i}$. The final value of this coefficient itself is determined through a procedure which implies scanning $\alpha_{i}$ with a very small step in the interval from 0 to 1, and examining at each step whether the equation

[TABLE]

is satisfied, which provides the physical meaning of the obtained solutions. The whole procedure ends when the equation is satisfied.

Then, $r_{0;e}$ is determined from Eq. (13) for $n_{e}^{(e)}(r)$, as in Part 1 and 2. Therefore it can be presented in two equivalent forms, namely

[TABLE]

where $x=\kappa_{e}r_{e}$, $r_{\kappa;e}\equiv 1/\kappa_{e}$, and the coefficients $\gamma_{s;e}(x)$ and $\gamma_{\kappa;e}(x)$ are connected with the electron non-ideality parameters $\Gamma_{e}=e^{2}/(kT_{e}r_{e})$ and $\gamma_{e}=e^{2}/(kT_{e}r_{\kappa;e})$ as it is described in Part 3.

Finally, let us note that the partial electron and ion densities $n_{s;e}^{(1,2)}$ and $n_{s;1,2}^{(e)}$, due to the structure of the equation for coefficients $a_{1,2}$ and $b_{1,2}$, 45, can be determined by Eqs. (43), (45) and (56) in both $Z_{1,2}=Z_{i}$ and $Z_{1}\neq Z_{2}$ cases. Therefore it is necessary to take the corresponding values of $\alpha_{e;1,2}$ and $l_{s;1,2}$ only in these expressions, e.g. $l_{s;1,2}=r_{s;i}$ if $Z_{1,2}=Z_{i}$.

The behaviour of the characteristic length $l_{s;1,2}$ and electron-ion correlation coefficients $\alpha_{e,1,2}$ is illustrated by Tabs. 1 and 2 in the online material, while the behaviour of the parameters $r_{b;1,2}$ and ion-ion correlation coefficients $\alpha_{i}$ is shown there in Tabs. 3 and 4.

These tables cover the regions of $N_{e}$ from $10^{16}$ cm*-3* to $10^{19}$ cm*-3* for $T=3\cdot 10^{4}$ K. They show that the values of all parameters are within the expected boundaries. So from these tables one can see that mostly $l_{s;1,2}\approx r_{s;1,2}$, electron-ion correlation coefficient $l_{e;1,2}\approx 1$, $r_{b;1,2}\sim r_{s;1,2}$ and ion-ion correlation coefficient $\alpha_{i}\approx 1$; a significant increase in the correlation coefficients values $\alpha_{e,1,2}$ and $\alpha_{i}$ is registered in the region of extremely high electron density ($N_{e}\approx 10^{19}$ cm*-3*). These tables cover the region of electron densities $10^{16}$ cm*-3* $\leq N_{e}\leq 10^{20}$ cm*-3* for $T=3\cdot 10^{4}$ K.

Remark: at this point all the calculations are performed for the case $T_{i}=T_{e}=T$.

7 Results and discussions

7.1 The properties of the obtained solutions

One of the most important results of this research is the performed classification of the considered systems from the point of view of application of the above described additional conditions:

• –

the systems of the ”closed” type, which are investigated here, and which are described by means of the mentioned additional conditions;

• –

the systems of the ”open” type, where those conditions are fully neglected.

It is just for the systems of the ”closed” type that the method of describing the inner electrostatic screening is developed in this work. As a continuation of our previous research (Part 1 and Part 2), as the systems of the ”closed” type in this work fully ionized electron ion plasmas are chosen with the positive ion charges of two different kinds. Such choice if especially important since increasing the number of ion components further would not cause appearance of any new phenomena. The obtained results are discussed in this section, including the results of comparison with the systems of the ”open” type,

One can see that the procedures of obtaining the Eqs. (35)

• (45) and Eqs. (53) - (56), as well as the values of the existing parameters, provide that these expressions: are self-consistent; satisfy all the conditions from Section 2.1, including Eq. (15) and Eq. (63); can be applied not only to the classical, but also to the quantum-mechanical systems (see Part 2.), including here the plasmas of higher non-ideality. Since the presented expressions do not contain the particle masses they can be used also for describing some other systems (the corresponding electrolytes and dusty plasmas). The behaviour of the ion and electron densities is illustrated by Fig’s. 2 and 3 on the examples of the cases $(i1)$ and $(i2)$ for $Z_{2}\neq Z_{1}$ and $Z_{2}=Z_{1}$ respectively.

Since Eq. (43) and Eq.(55) show that the solutions $n_{e}^{(1,2)}(r)$ and $n_{i;1,2}^{(i)}(r)$ are singular in the point $r=0$, it is useful to note that the existence of singularities in model solutions is fully acceptable, if it does not have other non-physical consequences. Such solutions are well known in physics: it is enough to mention, for example, Thomas-Fermi’s models of electron shells of heavy atoms ([21, 22]; see also [23]), which are used in plasma research until now (see e.g. [24]).

One more comment is due with respect to the figures 2 and 3. Namely they show that the ion densities $n_{1,2}^{(2,1)}(r)$ in the points $r_{b;1,2}$ do not keep the smutting. It makes it actually impossible that within any similar ion-ion correlation model there be any other equations which would be better then Eq. (A.2).

All potential applications of the obtained solutions are connected with the charge density $\rho^{(1,2,e)}(r)$, which in general is given by Eq. (5) and described in detail in Supplementary material C. Namely, if $\rho^{(1,2,e)}(r)$ is known then by means of Eq. (9) the electrostatic potential $\Phi^{(1,2,e)}(r)$ can be determined, and by means of Eq. (7) - the potential $\varphi^{(1,2,e)}$ at the point $O$. Finally, by means of the potential $\varphi^{(1,2,e)}$ the probe particle potential energies given by Eq.(8) are determined.

Except for the potential $\Phi^{(1,2,e)}(r)$ and $\varphi^{(1,2,e)}$ the systems $S_{a}^{(1,2,e)}$ are certainly characterized by radial charge densities $P^{(1,2,e)}(r)\equiv 4\pi r^{2}\cdot\rho^{(1,2,e)}(r).$ According to Part 3, each of the functions $|P^{(1,2,e)}(r)|$ has at least one strongly expressed maximum, whose position is an important characteristic of the distribution of charge in the neighborhood of the probe particle.

In order to demonstrate the very large differences between the alternative and DH-like characteristics, we will compare the asymptotic behaviour of the potential $\Phi^{(1,2,e)}(r)$ and DH-like potential $\Phi^{(1,2,e)}_{DH}(r)$. For this purpose we will take into account that from Eqs. (26), (49), and the Supplementary material E, it follows that

[TABLE]

where the ion screening constants $\kappa_{as;1,2}$ are given by Eqs. (25), (45) and (48) and the DH screening constant $\kappa_{DH}=(\kappa_{0;1}^{2}+\kappa_{0;2}^{2}+\kappa_{0;e}^{2})^{1/2}$, where $\kappa_{0;1,2}$ and $\kappa_{0;e}$ are determined by Eqs. (25) and (45).

Here, we will consider the case of classical plasma with $T_{i}=T_{e}=T$, where $\partial\mu_{1,2,e}/\partial N_{1,2,e}=kT/N_{1,2,e}$ and, consequently, the relations

[TABLE]

are valid. These relations show that the ion asymptotic screening constants $\kappa_{as;1,2}$ always have to be significantly smaller than $\kappa_{DH}$, and at the same time the corresponding screening radii $r_{as;1,2}$ always have to be significantly larger then $r_{DH}$. It is important that a similar result obtained in Part 2 was noted there as an evident shortcoming of the DH solution.

From Eq. (66) it follows that $\Phi^{(1,2,e)}(r)$ has a completely different asymptotic behavior compared to $\Phi_{DH}^{(1,2,e)}(r)$. Let us note that we will reach the same conclusion by comparing the behavior of the radial charge density $P^{(1,2,e)}(r)$ to its DH-like analog.

As the main characteristics of the considered plasmas we take here the probe particle mean potential energies $U^{(1,2,e)}$ which later get identified with the mean potential energies of ions in the real plasmas. In order to determine these ion energies $U^{(1,2,e)}$ it is necessary to know the values of the potential $\varphi^{(1,2,e)}$. For the calculation of $\varphi^{(1,2,e)}$ the equations (7) are taken in the form

[TABLE]

where the charge density $\rho^{(1,2,e)}(r)$ is described in detail in Supplementary material C. After the determination of $\varphi^{(1,2,e)}$, the mentioned energies are obtained by means of Eq. (8). Let us remind that the case $Z_{1}=Z_{2}=1$ can correspond to the case of plasma with the ion H*+, He+*(1s), etc. Within this work the energies $U^{(1)}$ and $U^{(2)}$ are determined for two cases: $Z_{1}=1$ and $Z_{2}=2$ and $Z_{1}=Z_{2}=1$. In both cases the calculations are performed for plasmas with the electron densities $10^{16}\rm{cm^{-3}}\leq N_{e}\leq 10^{20}\rm{cm^{-3}}$, for $T=3\cdot 10^{4}$ K. The obtained results are presented in Tabs 5 and 6 in the online material with quite small ion-density steps: from $0.5\cdot 10^{16}\rm{cm^{-3}}$ in the region of electron density $10^{17}\rm{cm^{-3}}$ to $0.5\cdot 10^{19}\rm{cm^{-3}}$ in the region of electron density $10^{20}\rm{cm^{-3}}$. In order to investigate the dependance of the energies of the systems on the temperature the calculations of $U^{(1)}$ and $U^{(2)}$ were performed here in the same regions of electron densities and with the same ion-density steps as it Tabs 5 and 6, but for the temperatures $T=1\cdot 10^{4}$ K, $1.5\cdot 10^{4}$ K, $2\cdot 10^{4}$ K and $2.5\cdot 10^{4}$ K. The corresponding results are presented in the tables 7-14 in the online material. From these results one can see that the potential energies $U^{(1)}$ and $U^{(2)}$ are sensitive to considerable lowering of the temperature ($T=1\cdot 10^{4}$K). Let as note that, for the readers’ convenience, the potential energies $U^{(1)}$ and $U^{(2)}$ in all the tables are given in [eV].

Remark: at this point all calculations are also performed in the case $T_{i}=T_{e}=T$.

7.2 Interpretation of the obtained results

Results such as those presented in the previous subsection (see 66) might leave an impression that they mean an absolute advantage over the DH-like methods (or other similar ones). Such an impression will be wrong since, as a matter of fact, from the presented results follows only an absolute advantage of the presented method in the case of a system of the ”closed” type. In the same context it is necessary to interpret the phenomenon which were described in the Introduction concerning the results presented in the Figs. 1a and 1b. Namely, that phenomenon can be interpreted as physically unacceptable when DH or DH-like method is used on the system of the ”closed” type. However the system gets treatment as ”closed”-type only in the case when it is described by means of the above mentioned (the point 2.2) additional conditions. Consequently, in the opposite case (when additional conditions are absent) the system has to be treated as an ”open” one and can be successfully described by means of DH-like or similar methods. Hence the conclusion follows that the way of describing the physical system depends on its treatment i.e. as an ”open” type or a ”closed” type. Concerning this, see Fig. 4a,b which shows the results of the application of DH method to the considered plasma (a- $Z_{1}=1$ and $Z_{2}=2$, b- $Z_{1}=Z_{2}=1$). It is useful to compare this figure with the Figs. 2 and 3. In this context let us draw attention to the fact that such a treatment itself has to bee determined based on the properties of the considered system and the physical problem which can be solved using that system.

At the end of this point it would be useful to linger on such influence of the additional conditions on the properties of the obtained solutions which could be treated as manifestation of their deviation from thermodynamic equilibrium. Here we mean the necessity of substituting the equation obtained from the condition of thermodynamic equilibrium by the equations obtained in different ways - the electron charge density $n_{e}^{(1,2)}$ in the region of large $r$, and the ion charge density $n_{1,2}^{(2,1)}(r)$ - also in the region of large $r$. However, we draw attention to the fact that all changes have purely phenomenological character and do not influence the thermodynamic properties of the considered gases: gas with temperature $T$ remains gas with the temperature $T$. Apart from that, we will remind of the fact that in all regions of $r$, where the considered components can be treated independently from one another, their state was described by means of equations obtained from the condition of thermodynamic equilibrium.

From above presented material it follows that source basic model can generate only DH or some DH-like metod. In this sense this model has already exhausted its potential but as one can see it enables, with a minimal deviation from the basic model (in the area of mathematical apparatus) to leave DH-like sphere and to develop a new model method of describing the plasma’s inner electrostatic screening.

7.3 The possible ion-ion probe systems

From the presented matter it follows that the main properties of the considered three-component system come about from the analogy with the properties of positron-ion probe system. This fact certainly deserves one special additional consideration. Concerning this it is useful to note the fact which was established by means the molecular dynamic (MD) simulation of a dense electron-proton plasma in [25]. In this work some characteristics of the considered plasma were determined as the results of averaging over all ion configurations possible under the considered conditions, for different values of the ratio $m_{e}/m_{p}$, where $m_{e}$ and $m_{p}$ are the electron and proton masses. These values were changed from $1/1836$ to $1/100$, but the changes of the results of MD simulations could be neglected. This can be very interesting even just for the similarity of the procedures which were used here and in [25]. However, it can be particularly important under the assumption that the similar conclusion is valid in the case of plasma which contains electrons and ions of some heavy atoms, particulary if the non-negligible probability is taken into account that the value of the mentioned ratio can be even greater than $1/100$. Namely, under this assumption Eq. (A.2) could be confirmed directly for such more realistic systems as plasmas whose ion components contain H*+* and ions of atoms heavier than Tc, for example Ag2+, or D*+* and ions of atoms heavier than Au, for example Hg2+.

8 Conclusions

As one of the most important results of this research a classification of physical systems (electrolytes, dusty plasmas) is performed in this work based on consideration, or lack thereof, of a few special additional conditions. The system considered here, as well as other systems which are described by means of the mentioned additional conditions are treated here as the systems of the ”closed” type, while the systems where that conditions are fully neglected - as being of the ”open” type. As the object of investigation here fully ionized electron-ion plasma is chosen with positively charged ions of two different kinds, including here the plasmas of higher non-ideality. The direct aim of this work is to develop, within the problem of the finding of the mean potential energy of the charged particle for such plasma, a new model self-consistent method of describing of the electrostatic screening which includes all the necessary additional conditions. With minimal derivation from the source basic model (in the area of mathematical apparatus) this aim has been realized here. Within the method presented such extremely significant phenomena as the electron-ion and ion-ion correlations are included in the used model, and all types of the necessary conditions are clearly defined. The characteristics of the considered plasmas in a wide region of the electron densities and temperatures are calculated. All obtained results, including the comparing with the systems of the ”open” type are specifically discussed at the end of this work. Here the case of the three-component systems was considered which is especially important since further increase of the number of the ion components did not cause appearance of any new phenomena.

Acknowledgments

The authors would like to express their gratitude to Professor Y. Vitel for many years of cooperation and very helpful initial discussions, Professor V.E. Fortov for truly priceless help in this research, and also to Professor V.M. Adamyan, for his assistance in the calculations in the region of very dense plasmas. The authors are thankful to the Ministry of Education, Science and Technological Development of the Republic of Serbia for the support of this work within the projects 176002 and III44002.