In the present work, we focus on the fast electron inelastic scattering which we characterize by the generalized oscillator strength (GOS). This GOS [1] - [3] concept introduced by Bethe in atomic physics [1] provides information on the valence shell excitations of atoms and molecules. Interest in sodium atom in this study is related to its existence in the atmosphere and for its hydrogenlike electronic structure [4] . It widely exists in comets and stellar wind; it has attracted numerous interests of theoretical and experimental studies [5] [6] . To best of our knowledge, the study of free sodium atom has received main attention by many authors [7] - [9] . From the study in Debye plasma [10] - [20] , modifications of various atomic structure and spectrum parameters have been exhibited by using different types model potentials describing the screened interelectronic interactions when solving the Schrödinger equation. Janev et al. [10] provide, in their review a comprehensive overview of the fundamental theoretical studies of atomic physics in Debye plasmas modeled with screened interactions. As we know, Murillo et al. [11] have focused on the approach which can readily incorporate level shifts of the target as well as dynamic plasma effect before computing dipole oscillator strengths for atomic transitions in such potentials. Photoionization of hydrogen atom and helium He⁺ ion in Debye plasmas have been studied by Li et al. [12] by considering the exponential-cosine-screened Coulomb and screened Coulomb (Yukawa) potentials. Taking advantage of the complex coordinate rotation method, Sahoo et al. [13] have studied the influence of plasma on photoionization of atomic helium by means of the Debye Hückel model, that incorporates the effects of plasma. Mukherjee et al. [14] have shown the dependence of the energies of excited states of helium-like atoms embedded in plasma on both nuclear and Debye parameter. Investigation of oscillator strength, critical screening constant and polarizability of hydrogen-like plasma under Endohedral cavity has been made by Saptarshi et al. [15] .

By exanimating the competing effect of plasma charge density and temperature in weakly and strong coupled plasma, Mukherjee et al. [16] reported their calculated results of the oscillator strength and polarizabilities for considering 1s, 2s states of the H-like ions embedded in astrophysical plasmas. Using the relativistic treatment, Bahar [17] investigated energy spectrum, transition energies and oscillator strengths for atoms systems Li, Na and K in the presence of quantum plasma environment, under the influence of spherical encompassment and external field. Bahar also elucidated the effect of interaction parameters on the afore-mentioned observables with details analyses such as alternatives to each other, dominance, optimal ranges or critical values been carrying out. Recently, Bahar [18] has conducted the theoretical investigation on the persistent orbital charge-currents and induced magnetic field of Li and Na atom compressed by spherical confinement, embedded in a quantum plasma. Yue-Ying Qi et al. [19] reported their theoretical results of oscillator strengths and dipole polarizabilities of the 3s and 3p states of the sodium. The structure, dipole and generalized oscillator strengths of helium atoms in a plasma have been investigated by Martinez et al. [20] with the wavefunctions obtained by using the Debye-Hückel screened (DHS) potential and a more general exponential-cosine screened Coulomb (ECSC) potential when solving the Schrödinger equation in a restricted Hartree-Fock method. In the case of the GOS for ground state 2p⁶3s of sodium atom, theoretical work of Martínez-Flores [21] has been reported in the literature. This work has been carried out by using wavefunctions obtained with the pseudo-potential model modify the 3s valence in order to obtain the associated GOS. A report of this investigation stimulated full-electron studies of sodium atom in Debye-plasma where the inner electron exchange contributions are significant to account for generalized oscillator strengths of electronic excitations.

In order to supplement this previous work, two approaches have been proposed to take into account the screening interaction of many electron systems. It means here the inclusion of screening of the electron-electron interaction of sodium atoms in plasma studies. One of these methods is the Restricted Hartree-Fock (RHF) method which simplifies the exact Hartree-Fock one in screening environment investigations. This RHF approach has been fruitful in treating screening effect problems in two electron system structure and spectrum parameters with satisfactory results. An example is due to Martinez et al. [22] who investigated the effects of plasma on dipole and generalized oscillator strengths of helium. However, to the best of our knowledge, it appears that the influence of plasma environment on the energy levels and the dipole and generalized oscillator strengths of multi electron atomic sodium has not received little attention to date by means of the investigation within the framework of this RHF method.

In the other hand, we also point out here the application of another technique of many electron problems. The specificity of the present method is the way that it takes into account virtual excitations of electrons from other subshells. This approach differs from the common calculations in the fact that we directly obtain the GOS without constructing first a pair of eigenfunctions from an integral equation which describes the collective multi-electron effects. To date no study of GOS has also been conducted on fast charged particle-sodium scattering for excitation process in plasma environment by using this second approach which is the random phase approximation with exchange (RPAE). The aim of the paper is to study in length form, the GOS of sodium atom planted in plasma environment both in the Restrict Hartree-Fock method and the Random Phase Approximation with Exchange approach. Furthermore, the present opportuneness of this study is that comparison may be made between our results and the earlier provided benchmark results in the reference [21] .

This work is organized as follows. Section 2 summarizes the procedure to find atomic orbitals with the Schrödinger equation in a restricted Hartree-Fock (RHF) approach to account for the sodium atom in a Debye plasma. The length formulation of GOS in the modified Hartree - Fock method and in the random phase approximation with exchange one, is the topic of Section 3. We provide, in the Section 4, the GOS results and make comparison with other work where it is possible. We draw the conclusion in the last section. Throughout the present paper, results are given in atomic units, unless explicitly indicated otherwise.

To study the GOS of the sodium atom in the inelastic scattering taking place in the presence of Debye-plasma, one needs to calculate the wavefunctions in the initial and final states. In order to investigate the effects of Debye-plasma on these wavefunctions, the radial Schrödinger equation for sodium atom in weak plasma can be written as [22]

$\left[-\frac{{\text{d}}^2}{2\text{d}{r}^2}+\frac{\ell \left(\ell +1\right)}{2r^2}+V_{eff}^D\left(r\right)\right]{{\rm P}}_{n\ell }\left(r\right)={\epsilon }_{n\ell }{{\rm P}}_{n\ell }\left(r\right)$ (1)

Here ${\epsilon }_{n\text{}\ell }$ is orbital energy; $V_{eff}^D\left(r\right)$ is the sum of the Debye-screened coulomb potential of a nuclei of charge Z that interacts with electron, the total interaction electron-electron Debye-screened coulomb potential and the Debye-screened exchange potential.

In this paper, we use the Debye-screened exchange potential which is approximated by a statistical free-electron expression with a Thomas-Fermi screening corrections [23] [24] . Then the value of the effective potential $V_{eff}^D\left(r\right)$, including the Debye screening tends towards zero as $r$ becomes larger [25] . In the Latter’s [23] [24] procedure, the asymptotic behaviour of the effective potential was corrected at large values of r from atomic nucleus.

The form of the effective potential $V_{eff}^D\left(r\right)$ for the field in which the electron moves can now be written as:

$\begin{array}{c}{V}_{eff}^D\left(r\right)=-\frac{Z}{r}{\text{e}}^{-\frac{r}{D}}+{\displaystyle \underset{n\ell }{\sum }\frac{N_{n\ell }}{D}[\frac{1}{\sqrt{r}}{K}_{1/2}\left(\frac{r}{D}\right){\displaystyle {\int }_0^r{\left|{{\rm P}}_{n\ell }\left(r^{\prime }\right)\right|}^2\frac{1}{\sqrt{r^{\prime }}}{I}_{1/2}\left(\frac{r^{\prime }}{D}\right)\text{d}{r}^{\prime }}}\\ +\frac{1}{\sqrt{r}}{I}_{1/2}\left(\frac{r}{D}\right){\displaystyle {\int }_r^{\infty }{\left|{{\rm P}}_{n\ell }\left(r^{\prime }\right)\right|}^2\frac{1}{\sqrt{r^{\prime }}}{K}_{1/2}\left(\frac{r^{\prime }}{D}\right)\text{d}{r}^{\prime }}]\\ -\frac{6}{2}{\left[\frac{3}{32{\pi }^2}{\displaystyle \underset{n\ell }{\sum }\frac{N_{n\ell }{\left[{{\rm P}}_{n\ell }\left(r\right)\right]}^2}{r^2}}\right]}^{1/3}\times F\left(\alpha \right)\end{array}$ (2)

if $r<r_0$ and

$V_{eff}^D\left(r\right)=-\frac{Z-N+1}{r}$ (3)

if $r\ge r_0$

Here $r_0$ is the value of $r$ when Equations (2) and (3) are equated. In the above equations and in the following, we use the notation that $Z$, $N_{n\ell }$ and $N$ are respectively the atomic number, the occupation number for orbital $n\ell$ and the number of atomic electrons more generally equal to ${\displaystyle \underset{n\ell }{\sum }{N}_{nl}}$.

Let us note that D and $F\left(\alpha \right)$ in Equation (2) are respectively the screening length of a Debye plasma and the screening function. This screening function is a correction factor which is equal to [22] - [24] .

$F\left(\alpha \right)=1-\frac{{\alpha }^2}{6}-\frac{4}{3}\alpha {\tan }^{-1}\left(\frac{2}{\alpha }\right)+\frac{{\alpha }^2}{2}\left(1+\frac{{\alpha }^2}{12}\right)\ell n\left|1+\frac{2}{{\alpha }^2}\right|$ (4)

where $\alpha =\frac{k_s}{k_F}=\frac{1}{D{k}_F}$ is defined as the ratio of the Debye screening parameter $k_s\equiv \frac{1}{D}$ and the fermi momentum $k_F$. As can be expected from its formulae there is no screening for $k_s=0$ (i.e. $D\to \infty$). With the spherically average total electronic charge density, we set $\alpha$ equal to $0.646\times {\left[{\displaystyle \underset{n\ell }{\sum }\frac{N_{n\ell }{\left[{{\rm P}}_{n\ell }\left(r\right)\right]}^2}{4\pi r^2}}\right]}^{-1/6}$. It can be seen in the Equation (2), $I_{1/2}\left(r\right)$ and $K_{1/2}\left(r\right)$ which are obtained by using the modified Bessel functions of first kind $I_{k+1/2}\left(r\right)$ and second kind $K_{k+1/2}\left(r\right)$, respectively for the case of $k=0$.

In our previous development of unscreened modified Hartree-Fock method, we have used the method applied in a previous paper [26] based on finite difference approximations to numerically calculate the orbital energies and radial wavefunctions with good efficiency. Those orbital energies and radial wavefunctions and related physical quantities as generalized oscillator strengths (GOS) have been obtained for free neon atom [26] . The extension of unscreened numerical modified Hartree-Fock method to screened atomic sodium is a natural choice due to its simplicity and successfully previous usage. The radial wavefunctions for orbital eigen energy are numerically obtained by solving the modified non-relativistic HF Equation (1). In the present case, we have linearized this basic equation and calculated the radial integrals by the quadrature method. The Matlab Software has been used to compute the present data of sodium atomic structure. To test the quality of these radial wavefunctions of atomic sodium in Debye plasma, for example $D=100$, our present radial functions $P_{3s}$ ground state and $P_{3p}$ excited state are given in Figure 1 below and compared with the results provided by [21] where they used the pseudo-potential model approach. The inset graph, in the Figure 1 shows in detail the behaviour of 3s orbital ground state and 3p orbital excited state for each curve at small radius between 0 and 0.2 a.u.

We have found the other elaborate radial wavefunctions by using the same procedure and the obtained results are used to compute some dipole and GOS for excitation of an atomic sodium in Debye plasma environments.

The present results are those obtained by solving the Equation (1) with the screened modified Hartree-Fock method.

Since it has been introduced by Bethe [1] [27] and reviewed by Inokuti [3] , the generalized oscillator strength is previously an important property of a free atom. However, we learn from the physical nature of the plasma environment, that the properties of the atom in plasma can differ significantly from those of a free atom. Apart from the other properties, we have used here the generalized oscillator strength to describe the process of interaction between an incident particle and an atom embedded in plasma environment. In the case of the Debye plasma, this concept of generalized oscillator strength for excitation is defined in terms of the energy transfer $\omega$, momentum transfer $q$, initial state atomic wavefunction ${\Psi }_i$ and a particular atomic excited state wavefunction ${\Psi }_f$ as [1] [28] :

$G_{if}\left(\omega ,q\right)=\frac{2\omega }{q^2}{\left|{\displaystyle \underset{j=1}{\overset{N}{\sum }}{\displaystyle \int {\Psi }_f\left(r_1{r}_2\cdots r_N\right){\text{e}}^{iq\cdot r_j}{\Psi }_i\left(r_1{r}_2\cdots r_N\right)\text{d}{r}_j}}\right|}^2$ (5)

where $r_j$ is vector position of atomic electron $j$ and $\exp \left(iq\cdot r_j\right)$ the interaction operator of this electron with the incident charge particle, in the length formulae calculated in the early work of Bethe.

In carrying out the present restricted Hartree-Fock approach, the atomic wavefunctions ${\Psi }_{if}$ are the Slater determinant of spin orbitals which satisfies the antisymmetry property. Taking into account the normalization and orthogonality conditions in integration procedure, Equation (5) is reduced to the sum of the GOS term describing the transition of one electron to a particular excited state after interaction with the incident particle while other atomic electrons remain in their states. Then, the inelastic collision GOS term of an atomic sodium in Debye plasma for this one electron transition from an initial state denoted by $m$ to the final state denoted by $t$ is given by [29]

$g_{mt}\left(\omega ,q\right)=\frac{2\omega }{q^2}{\left|{\displaystyle \int {\psi }_t\left(r\right){\text{e}}^{iq\cdot r}{\psi }_m\left(r\right)\text{d}r}\right|}^2$ (6)

The exponent ${\text{e}}^{iq\cdot r}$ in Equation (6) can be expanded into the following series [30]

${\text{e}}^{iq\cdot r}={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\left(2k+1\right)j_k\left(qr\right)P_k\left(\cos \theta \right)}$ (7)

Here $j_k\left(qr\right)$ is the spherical Bessel function of the first kind of order k Taking into account the spherical symmetry of the system, Equation (6) with the help of (7) simplifies considerably after integrating it over the angles, summing over the spin projection and orbital angular momentum in the final state, and averaging it in the initial state. Then the expression of Equation (6) is reduced to the following relation [28] :

$g_{mt}\left(\omega ,q\right)={\displaystyle \underset{k}{\sum }{g}_{mt}^k}\left(\omega ,q\right)$ (8)

where the length form multipole GOS is given by

$\begin{array}{c}{g}_{mt}^k\left(\omega ,q\right)=\frac{2\omega N_m\left(2k+1\right)}{q^2\left(2{\ell }_m+1\right)}|\sqrt{\left(2{\ell }_t+1\right)\left(2{\ell }_m+1\right)}\left({}_0^{{\ell }_t}{}_0^k{}_0^{{\ell }_m}\right)\\ \times {{\displaystyle {\int }_0^{\infty }{P}_{n_m{\ell }_m}\left(r\right)j_k\left(qr\right)P_{n_t{l}_t}\left(r\right)\text{d}r}|}^2\end{array}$ (9)

with $N_m$ the number of electrons in the excited state, $P_{n_m{\ell }_m}\left(r\right)$ and $P_{n_t{l}_t}\left(r\right)$ being respectively, the final and initial normalized radial functions of RHF wavefunctions. Note that the total angular momentum $k$ of the electron-hole pair, satisfies the triangular rule $\left|{\ell }_m+{\ell }_t\right|\ge k\ge \left|{\ell }_m-{\ell }_t\right|$.

According to Equation (9), the GOS can be studied in estimating only the influence of the Debye plasma on sodium atomic structure and the direct excitation of the atomic electron under consideration. In reality, the interaction in a sodium atomic electron involves many electrons simultaneously. In order to obtain a better mathematical treatment of the effects of the dynamical perturbation of the self-consistent field by the fast incoming charged particle, we have adopted an approach based on a collective description of the interaction between atomic electrons. The main idea of this approach is to represent the amplitude transition as a combination of the direct and exchange matrix elements of the inter-electron interaction. The theoretical calculations including the mechanism due to the electron-electron interaction is taken into consideration here within the random phase approximation with exchange (RPAE) [28] [30] . We proceed to calculate the GOS by using the following matrix element defined in RPAE length formulation [28] [30]

$\begin{array}{l}\langle {\psi }_t\left(r\right)|L^{RPAE}\left(\omega ,q\right)|{\psi }_m\left(r\right)\rangle \\ =\langle {\psi }_t\left(r\right)|{\text{e}}^{iq\cdot r}|{\psi }_m\left(r\right)\rangle +\left({\displaystyle \underset{n\le F,k\ge F}{\sum }}-{\displaystyle \underset{k\le F,n\ge F}{\sum }}\right)\frac{\langle {\psi }_k\left(r^{\prime }\right)|L^{RPAE}\left(\omega ,q\right)|{\psi }_n\left(r^{\prime }\right)\rangle }{\omega -{\epsilon }_k+{\epsilon }_n+i\alpha \left(1-2{\beta }_k\right)}\\ \times \langle {\psi }_n\left(r^{\prime }\right){\psi }_t\left(r\right)|V|{\psi }_k\left(r^{\prime }\right){\psi }_m\left(r\right)\rangle \end{array}$ (10)

where $\langle {\psi }_n\left(r^{\prime }\right){\psi }_t\left(r\right)|V|{\psi }_k\left(r^{\prime }\right){\psi }_m\left(r\right)\rangle$ stands for the combination of the direct and exchange matrix elements of the electron - electron interaction ${\psi }_k\left({\psi }_n\right)$ is the final (initial) virtual excitation state with their corresponding final (initial) orbital energy ${\epsilon }_k\left({\epsilon }_n\right)$. F is the Fermi level of the sodium atom and the Fermi step function ${\beta }_k$ looks as follows: for unoccupied states ${\beta }_k=0$ and for occupied states ${\beta }_k=1$. The complex number $i\alpha$ in denominator of Equation (10), with imaginary part $\alpha \to 0^{+}$ just gives us the direction of tracing the pole in integrating.

To incorporate the effects of correlations into the calculation of the GOS, we just use a relation similar to (6). In this new GOS’s expression, the factor form is the transition matrix element including the mechanism describing the response of the interacting multielectron atomic sodium given in (10). Therefore, the length form of GOS of the RPAE contribution due to different electron interactions is introduced:

$g_{mt}^{RPAE}\left(\omega ,q\right)=\frac{2\omega }{q^2}{\left|{\displaystyle \int {\psi }_t\left(r\right)L^{RPAE}\left(\omega ,q\right){\psi }_m\left(r\right)\text{d}r}\right|}^2$ (11)

Due to the fact that some terms of the denominators in Equation (10) become zero at $\omega ={\epsilon }_k-{\epsilon }_n$ in the calculations, it is difficult to investigate the discrete excitations numerically in accuracy and efficiency. To avoid this singularity at $\omega ={\epsilon }_k-{\epsilon }_n$, it can be derived that the expression of GOS under the procedure described in detail in [31] [32] is in the first non-vanishing order:

$g_{mt}^{RPAE}\left(\omega ,q\right)=\frac{{\left|L_{mt}^{\text{'}RPAE}\left(\omega ,q\right)\right|}^2}{1+{\displaystyle \underset{\left(mt\right)\ne \left(kn\right)}{\sum }\frac{{\left|V_{mtkn}\right|}^2}{{\left(\omega -{\epsilon }_k+{\epsilon }_n\right)}^2}}}$ (12)

where $L_{mt}^{\text{'}RPAE}\left(\omega ,q\right)$ is the regular solution of the Equation (10) at $\omega ={\epsilon }_k-{\epsilon }_n$ by dropping the singular terms and $V_{mtkn}$ represents, symbolically, the last factor in Equation (10) which denotes a combination of the direct and exchange matrix elements of the interaction electron-electron $V_{mtkn}=\langle {\psi }_n\left(r^{\prime }\right){\psi }_t\left(r\right)|V|{\psi }_k\left(r^{\prime }\right){\psi }_m\left(r\right)\rangle$.

The formulas (9) and (12) are reduced to a form suitable for numerical computation in this paper. The results for GOS of valence-shell excitations of atomic sodium in Debye plasma, computed here from these formulas by using Matlab software are presented and compared with those obtained by other authors.

The present calculations have been performed for the transitions 3s to 3p, 4p, 5p and 6p of the sodium atom in Debye plasma. In our calculations, presenting the GOS of these transitions for screening length $D=\infty$, 100, 30 and 20, appears to work quite well in investigating the competition between shell electron interactions and plasma screening effects and also offers a significant advantage to compare them with available theoretical and experimental data [19] [21] [33] .

We have investigated both the plasma Debye screening effects and the correlation effects on the GOS of the dipole transition of the sodium atom in plasma environment. The results calculated in both restricted Hartree-Fock and random phase approximation with exchange for the screened sodium discrete excitations from the ground state 2p⁶3s¹ to 2p⁶3s⁰ (3p, 4p, 5p, 6p) have proved that the dipole GOS is decreased in going from higher to lower Debye length D. At $D=\infty ,100$ and 30; the many-electron effects are mildly quite important only around the region of the dipole GOS maxima of the excitation to 2p⁶3s⁰ (4p, 5p, 6p) while they are not significant in the excitation to 2p⁶3s⁰3p. The investigation lets us see now the correlation effects estimations are negligible than the plasma Debye screening effects for D = 20 when we apply the RPAE treatment. The new data of the GOS for dipole transition of three excitations considered in the present work have been added for the screening length which may be close to its critical values where transition is no longer bound because the plasma screening interactions become usually stronger. Also, with the present study, the RHF and RPAE calculations give, to the best of our knowledge for the different values of Debye lengths D, theoretical data of GOS for the transition 3s → 6p of Na atom in Debye plasma. So, further theoretical and experiments data are needed to test the accuracy of our RHF and RPAE calculations.