With the Bethe-Born Theory [10] , the GOS accounts for the probability of excitation from the initial state with the wavefunction $|{\psi }_o\rangle$ to a final excited state described by the wavefunction $|{\psi }_f\rangle$ obtained by means of the alternative length form, defined as [11] :

$\begin{array}{c}{F}_{of}\left(\omega ,q\right)=\frac{2\omega }{q^2}|{\displaystyle \underset{j=1}{\overset{N}{\sum }}{\displaystyle \int {\psi }_f\left({\stackrel{\to }{r}}_1{\stackrel{\to }{r}}_2\cdots {\stackrel{\to }{r}}_N\right)}}\times (\frac{1}{2\omega }[\exp \left(i\stackrel{\to }{q}\cdot {\stackrel{\to }{r}}_j\right)\left(q{\nabla }_j\right)\\ {\stackrel{}{}-\left(q{\stackrel{\leftarrow }{\nabla }}_j\right)\exp \left(i\stackrel{\to }{q}\cdot {\stackrel{\to }{r}}_j\right)])\times {\psi }_o\left({\stackrel{\to }{r}}_1{\stackrel{\to }{r}}_2\cdots {\stackrel{\to }{r}}_N\right)\text{d}{\stackrel{\to }{r}}_j|}^2\end{array}$ (5)

where ${\stackrel{\to }{r}}_j$ is the vector position of electron j, q is the transferred momentum, $\omega$ is the energy transferred. In this equation, the upper arrow in the Nabla operator indicates that it operates on the function standing to the left.

The atomic state wavefunctions ${\psi }_{of}\left({\stackrel{\to }{r}}_1{\stackrel{\to }{r}}_2\cdots {\stackrel{\to }{r}}_N\right)$ are Slater determinants built from spin orbitals. The spin orbitals are one-electron wave functions taking both their positions and spin angular momentums of atomic electrons which obey Pauli’s exclusion principle and Hund’s rule. The Slater determinant satisfies the anti-symmetry property because it can be expanded as a linear combination of one-electron functions. In the integration procedure of Equation (5), we have the normalization and orthogonality conditions that allow us to write the expression of $F_{of}\left(\omega ,q\right)$ as the sum of the GOS terms associated only with the transition of one electron from the state denoted by s to the state denoted by t.

$F_{of}\left(\omega ,q\right)={\displaystyle \underset{st}{\sum }{f}_{st}\left(\omega ,q\right)}$ (6)

where $f_{st}\left(\omega ,q\right)$ is the GOS term.

We consider that one electron of the initial state s is promoted to an excited state t under the assumption that the other electrons remain in the Debye plasma environment without leaving their initial subshells. In this case the GOS term is given by:

$f_{st}\left(\omega ,q\right)=\frac{2\omega }{q^2}{\left|{\displaystyle \underset{j=1}{\overset{N}{\sum }}{\displaystyle \int {\varphi }_t\left(\stackrel{\to }{r}\right)\left(\frac{1}{2\omega }\left[\exp \left(i\stackrel{\to }{q}\cdot \stackrel{\to }{r}\right)\left(q\nabla \right)-\left(q\stackrel{\leftarrow }{\nabla }\right)\exp \left(i\stackrel{\to }{q}\cdot \stackrel{\to }{r}\right)\right]\right){\varphi }_s\left(\stackrel{\to }{r}\right)\text{d}\stackrel{\to }{r}}}\right|}^2$ (7)

where ${\varphi }_s\left(\stackrel{\to }{r}\right)$ and ${\varphi }_t\left(\stackrel{\to }{r}\right)$ are the one-electron RHF wavefunctions of the initial and final states for each electron, respectively, when the energy conservation law $\omega ={\epsilon }_t-{\epsilon }_s$ is satisfied.

By inserting the operator $\exp \left(i\stackrel{\to }{q}\cdot \stackrel{\to }{r}\right)$ into the spherical wave expansion form and ${\varphi }_{s\left(t\right)}\left(\stackrel{\to }{r}\right)$ as a product of the radial, angular and spin functions, it allows the nabla operator in Equation (7) by using its spherical coordinate system to operate on the radial and spherical harmonic functions. Then, by integrating the angular and spin parts of the GOS term, we obtain that GOS $f_{st}\left(\omega ,q\right)$ can be expanded in terms of the total angular momentum $\ell$ in the form [12] [13] :

$f_{st}\left(\omega ,q\right)={\displaystyle \underset{\ell =\left|{\ell }_t-{\ell }_s\right|}{\overset{\left|{\ell }_t+{\ell }_s\right|}{\sum }}{f}_{st}^{\ell }\left(\omega ,q\right)}$ (8)

In Equation (8), $f_{st}^{\ell }$ is the $\ell$ multipole GOS, we have the total angular momentum $\ell$ of the electron-hole pair taking values: $\left|{\ell }_t-{\ell }_s\right|$; $\left|{\ell }_t-{\ell }_s\right|+2$; $\cdots$; $\left|{\ell }_t+{\ell }_s\right|$.

In the velocity formulation, the multipole GOS $f_{st}^{\ell }$ is defined as follow:

$\begin{array}{l}{f}_{st}^{\ell }\left(\omega ,q\right)=\frac{2\omega N_s\left(2\ell +1\right)}{q^2\left(2{\ell }_s+1\right)}|\frac{q\cdot \left({}_0^{{\ell }_t}{}_0^{\ell }{}_0^{{\ell }_s}\right)}{2\omega \left(2\ell +1\right)}\sqrt{\left(2{\ell }_t+1\right)\left(2{\ell }_s+1\right)}\\ \times \{{\displaystyle {\int }_0^{\infty }{P}_{n_s{\ell }_s}\left(r\right)\frac{j_{\ell -1}\left(q\cdot r\right)-j_{\ell +1}\left(q\cdot r\right)}{r}{P}_{n_t{\ell }_t}\left(r\right)[{\ell }_t\left({\ell }_t+1\right)-{\ell }_s\left({\ell }_s+1\right)]\text{d}r}\\ +{{\displaystyle {\int }_0^{\infty }\left[\left(\ell +1\right)j_{\ell +1}\left(q\cdot r\right)-\ell j_{\ell -1}\left(q\cdot r\right)\right]\left(P_{n_t{\ell }_t}\left(r\right)\frac{\text{d}{P}_{n_s{\ell }_s}\left(r\right)}{\text{d}r}-P_{n_s{\ell }_s}\left(r\right)\frac{\text{d}{P}_{n_t{\ell }_t}\left(r\right)}{\text{d}r}\right)\text{d}r}\}|}^2\end{array}$ (9)

In this Equation (9), $P_{n_s{\ell }_s}\left(r\right)$ and $P_{n_t{\ell }_t}\left(r\right)$ are also normalized radial functions of initial and final RHF wavefunctions, respectively and $j_{\ell }\left(q\cdot r\right)$ is the spherical Bessel function of the first kind. Equation (9) is used as an approximation to include only one electron of the screened atomic sodium that participes directly in the collision process. This approach, while describing the fast charged particle collisions with atomic sodium in the first-Born approximation, does not account for the exchange interaction of other atomic electrons. The impact of the incident charged particle on all the atomic electrons to which its energy is transferred, can be considered in the framework of the RPAE.

To introduce other means of electron excitation with the creation of an electron-hole pair in the calculation of the inelastic scattering GOS of a polyelectronic atom, one may use the RPAE description according to [13] . RPAE is based on a residual interaction because an electron excited from an atom by scattering can excite another atomic electron. The RPAE approximation enables us to treat the virtual transitions in this frame of a many-electron picture. The following considerations make it possible to write an equation for the matrix transition as [11] [13] [14] :

$\begin{array}{l}\langle {\varphi }_t\left(r\right)|V^{RPAE}\left(\omega ,q\right)|{\varphi }_s\left(r\right)\rangle \\ =\langle {\varphi }_t\left(r\right)|(\frac{1}{2\omega }\left[\exp \left(i\stackrel{\to }{q}\cdot {\stackrel{\to }{r}}_j\right)\left(q{\nabla }_j\right)-\left(q{\stackrel{\leftarrow }{\nabla }}_j\right)\exp \left(i\stackrel{\to }{q}\cdot {\stackrel{\to }{r}}_j\right)\right]|{\varphi }_s\left(r\right)\rangle \\ +\left({\displaystyle \underset{p\le F,r\ge F}{\sum }}-{\displaystyle \underset{r\le F,p\ge F}{\sum }}\right)\frac{\langle {\varphi }_r\left(r\right)|V^{RPAE}\left(\omega ,q\right)|{\varphi }_p\left(r\right)\rangle }{\omega -{\epsilon }_r+{\epsilon }_p+i\alpha \left(1-2{\eta }_r\right)}\\ \times \langle {\varphi }_p\left(r\right){\varphi }_t\left(r\right)|U|{\varphi }_s\left(r\right){\varphi }_t\left(r\right)\rangle \end{array}$ (10)

Here $V^{RPAE}\left(\omega ,q\right)$ is the multipole nonlocal operator in the velocity formulation that describes multi-electron correlations. $\langle {\varphi }_p\left(r\right){\varphi }_t\left(r\right)|U|{\varphi }_s\left(r\right){\varphi }_t\left(r\right)\rangle$ represents the combination of the direct and exchange matrix elements of the electron-electron interaction ${\varphi }_r$ ( ${\varphi }_p$) is the final (initial) virtual excitation state with their corresponding final (initial) orbital energy ${\epsilon }_r$ ( ${\epsilon }_p$). F is the Fermi level of the sodium atom and the Fermi step function ${\beta }_k$ appears as follows ${\eta }_r=1\left(0\right)$ for occupied (vacant) states. The complex number $i\alpha$ in the denominator of Equation (10), with the imaginary part $\alpha \to 0^{+}$ just gives us the direction of tracing the pole in integration.

Numerical study of the accuracy and efficiency of discrete excitations is very difficult because some terms of the denominators in Equation (10) become zero at $\omega ={\epsilon }_r-{\epsilon }_p$ in the calculations. The procedure described in detail in [15] to eliminate the divergent term from the sum in Equation (10) was used to calculate the GOS in the lowest order with respect to U using the following velocity expression:

$f_{st}^{RPAE}\left(\omega ,q\right)=\frac{{\left|{V^{\prime }}_{st}^{RPAE}\left(\omega ,q\right)\right|}^2}{1+{\displaystyle \underset{\left(st\right)\ne \left(rp\right)}{\sum }\frac{{\left|U_{strp}\right|}^2}{{\left(\omega -{\epsilon }_r+{\epsilon }_p\right)}^2}}}$ (11)

where, we consider ${V^{\prime }}_{st}^{RPAE}\left(\omega ,q\right)$ as a solution to Equation (10) with terms without the non vanishing denominator. Here $U_{strp}\equiv \langle {\varphi }_p\left(r\right){\varphi }_t\left(r\right)|U|{\varphi }_s\left(r\right){\varphi }_t\left(r\right)\rangle$

denotes the combination of direct and exchange matrix elements of the interaction electron-electron.

The behavior of the RHF and RPAE GOS for the $3s\to 3p$ dipole transition of atomic sodium calculated in the present work for various Debye lengths ( ${\lambda }_D=\infty ,100,30,20$), is depicted in Figure 1 . Where they are compared with the results of other authors obtained in their experimental [16] [17] and theoretical studies [1] [7] . In Figure 1(a) , we quote the velocity results of the dipole $3s\to 3p$ transition for the Debye length ${\lambda }_D=\infty$. The curves of our results and those obtained by Martínez-Flores [7] for ${\lambda }_D=\infty$ have the same shape as those of the experimental [16] [17] and theoretical [1] results for the free case. In all cases in Figure 1 , we also note that the GOS converges to zero for all Debye lengths ${\lambda }_D$ when going from a larger momentum transfer q. This convergence is due to the fact that atomic electrons cannot receive momentum transfer greater than a limit determined by the uncertainty principle without recoiling out of the atom. In Figure 1(a) , there is agreement with the results above the momentum transfer $q=0.2\text{a}\text{.u}$. A discrepancy was noted between our calculated data and the results of Martinez-Flores [7] and Han et al. [1] for transferred momentum $q\le 0.2\text{a}\text{.u}$. This discrepancy does not exceed 6.20% and 9.40% for RHF-V and RPAE-V, respectively. This difference between may be attributed to the nature of the difference wavefunctions used in the work. The GOS from RHF-V shows a slight difference from that of RPAE-V at $q\le 0.2\text{a}\text{.u}$. The effects of electronic correlations on the GOS $3s\to 3p$ dipole transition, in this region are greatly appreciated. Their contribution to the present RPAE-V.

Framework is approximately 3.2%. In Figure 1(b) and Figure 1(c) , we find agreement between the RHF-V and RPAE-V calculations in the transferred momentum interval [0.2, 1] while a small disagreement between them is seen in the interval [0, 0.2] where we note that the RHF-V curves lie above the RPAE-V curves. The results of our two calculations agree well in the q region of 0 - 1 a.u. in Figure 1(d) . This situation can be explained by the fact that the correlation effects have less influence than the plasma Debye screening effects on the GOS when the Debye lengths decrease.

To observe the Debye screening effect on the GOS of the atomic sodium excitation from the ground state to 3p, the RHF-V and RPAE-V GOS results were obtained for Debye lengths ${\lambda }_D=\infty ,100,30$ and 20 a.u are shown in Figure 2 . Figure 2(a) shows the RHF-V results for these various Debye lengths while the RPAE-V ones are found in Figure 2(b) . As can be seen, the absolute values of this GOS for ${\lambda }_D=20$ are the smallest to the transferred momentum $q\le 0.5\text{a}\text{.u}$. A above this value of $q=0.5\text{a}\text{.u}$, all GOS curves are so close together. However, this is not the case when $q<0.5\text{a}\text{.u}$. Note that the amplitude of the GOS for the dipole transition diminishes as the Debye length ${\lambda }_D$ decreases. In this case where q tends towards zero, the RPAE-V GOS values are 1.066, 1.017 and 0.9600 for ${\lambda }_D=\infty$, ${\lambda }_D=100,30$ and ${\lambda }_D=20$ respectively. The present value 1.066 of the dipole oscillator strength is in the accordance with the theoretical value 1.05 listed in reference [18] because their agreement does not exceed 1.53%. From Figure 2(a) , a similar situation can be observed for the results obtained from the RHF-V calculations. This indicates that the Debye screening effects play a very important role in the modification of the GOS’s value at low q for the $3s\to 3p$ atomic sodium transition.

Once, the GOS of the atomic sodium dipole transition $3s\to 3p$ was calculated, we determined the GOS for dipole transition of the excitation 3s to 4p. Our computed data are shown for the $3s\to 4p$ atomic sodium in Figure 3 . Let us note that Figures 3(a)-(d) show respectively the curves of the GOS for ${\lambda }_D=\infty ,100,30$ and 20. Some of the data calculated in the present study are compared with those obtained in the length formulation by Martínez-Flores [7] in Figures 3(a)-(c) . We remark that the same shape of the GOS is observed in these figures and a decrease in their magnitude until reaching zero around the momentum transfer $q=1.5\text{a}\text{.u}$. The findings of GOS are in good agreement with the length results of Martínez-Flores [7] , except in the q region of 0 - 0.4206 a.u, as shown in Figure 3(c) . For $q\le 0.4206$, the velocity GOS values obtained in this case ${\lambda }_D=30$, were less important than those found theoretically by Martínez-Flores [7] . The difference between them does not exceed 6.92%. This discrepancy may be attributable to the variation in the wavefunctions for small atomic electron radius obtained in the two different calculations. Finally, Figures 3(a)-(d) also show that the RPAE-V curves lie slightly.

Above the curves of RHF-V particularly around the region of the maxima. As shown in Figure 3 , the correlation effects do not have much influence on the GOS for the $3s\to 4p$ dipole transition. Their contribution does not exceed 3%. Figure 4 presents the GOS of the atomic sodium $3s\to 4p$ dipole transition for Debye lengths values ${\lambda }_D=\infty ,100,30$ and 20. The results of our RHF-V calculations are compared in Figure 4(a) , while the comparison between the present RPAE-V GOS for different values of ${\lambda }_D$ are also shown in Figure 4(b) . As in RHF-V and RPAE-V, the calculations give almost the same amplitude of the GOS of the $3s\to 4p$ dipole transition for the values of ${\lambda }_D=\infty$ and 100.

Note also that the amplitude of the GOS for the dipole transition decreases with the decrease in Debye length ${\lambda }_D=100,30,20$ and the maxima shift for the last value of this selected set of the screening length. In the RPAE-V approximation, the maximum GOS obtained for the Debye length ${\lambda }_D=100$ is 1.1335, which is larger than the corresponding value for the Debye length ${\lambda }_D=30$ while this ratio becomes 1.1209 in the RHF-V method. We found that the amplitude of the GOS for Debye length value ${\lambda }_D=20$ was slightly less than that calculated for the value of ${\lambda }_D=30$. The ratio calculated in this case is approximately 0.9222 in the RPAE-V approach, whereas in the RHF-V method, we obtained 0.9229. The GOS of atomic sodium planted in plasma, to absorb energy from the transferred momentum to its electrons can be strongly affected by the increase in strong plasma interactions. We also note the accordance between our theoretical dipole oscillator value and the theoretical ones listed in references [6] [7] .

In Section 4.2, we present the GOS of our two velocity calculations for the atomic sodium dipole excitation to 5p in Figure 5 , along with the length form results of Martínez-Flores [7] . It can be observe from Figure 5(c) that the RHF-V and RPAE-V GOS values are smaller than those found in the theoretical work of Martínez-Flores [7] when $q<0.5055\text{a}\text{.u}$. For the dipole transition $3s\to 5p$ of atomic sodium in the Debye plasma environment, the difference between them in Figure 5(c) does not exceed 10.25% in this region of $q\le 0.5055\text{a}\text{.u}$. This difference can be explained by the different radial overlap of the wavefunctions in the calculated GOS. Above this value of $q=0.5055\text{a}\text{.u}$, it can be seen from Figure 5(c) that there is.

Agreement between the calculated data in our two velocity calculations and the length form existing theoretical values of Martínez-Flores [7] are good. We find agreement between the present RPAE-V calculations and the length form calculations performed earlier in the work of of Martínez-Flores [7] in Figure 5(a) and Figure 5(b) which correspond to the cases with screening lengths ${\lambda }_D=\infty$ and ${\lambda }_D=100$. There is also an obvious difference in absolute values between the RPAE-V GOS and our theoretical RHF-V values around their maxima, as seen in Figure 5(a) and Figure 5(b) , which depict the results of the transition $3s\to 5p$ for the Debye lengths ${\lambda }_D=\infty$ and 100. The electronic-correlations contribute respectively 5.55% and 5.99% to the ∞ and 100 values, respectively. They are unimportant when the Debye length takes values of 30 and 20 because the situation is reversed, as observed in Figure 5(c) and Figure 5(d) . The effect of the correlations on the GOS for dipole transition is not pronounced for high plasma screening. The reason reported above can explain that why the present RPAE-V slightly underestimates the results of our RHF-V. This phenomenon may also be explained by the fact that the terms chosen to be neglected in the velocity matrix element transition become important as ${\lambda }_D$ decreases. In Figure 6 , we plot the dipole transition GOS as a function of the momentum transfers for Debye lengths ${\lambda }_D=\infty ,100,30$ and 20. The curves in Figure 6(a) represent the RHF-V GOS, whereas those obtained using the RPAE-V approach are shown in Figure 6(b) . For both Figure 6(a) and Figure 6(b) , we observe a decrease in the $3s\to 5p$transition GOS amplitude with decreasing Debye length ${\lambda }_D$ and a shift of the maxima positions towards greater momentum transfer in the case of ${\lambda }_D=20$. For both the RHF-V and RPAE-V calculations, the ratio of the GOS magnitude obtained with ${\lambda }_D=\infty$ and the maximum GOS for ${\lambda }_D=100$ is less than 1.08. In addition, for the considered Debye lengths ${\lambda }_D=100$ and ${\lambda }_D=30$ this ratio does not exceed 1.41 while it increases to almost 3 for values of 30 and 20 of Debye length ${\lambda }_D$. Therefore, we conclude that the screening effect strongly affects the GOS of the dipole transition when the plasma Debye length ${\lambda }_D$ approaches the critical plasma screening length value, which is given in reference [6] for each sub-shell of sodium atoms confined in the plasma environment. The free value ( ${\lambda }_D\to \infty$) of the GOS transition $3s\to 5p$ reaches 0.002115 a.u. and 0.002158 a.u. in the RHF-V and RPAE-V calculations respectively. We find agreement between these values of the present theoretical calculations and the dipole oscillator strength with values of 0.00216 listed in reference [6] [7] and 0.00221 measured by Wiese et al. [19] .

In Figure 7 , we present the fast charge particle impact excitation of the atomic sodium transition GOS as a function of the momentum transfer and Debye length. The GOS results are shown in Figures 7(a)-(c) for values of ${\lambda }_D=\infty ,100$ and 30, respectively. Here, we cannot now turn to the results of the $3s\to 6p$ transition GOS of the sodium atom confined by plasma for ${\lambda }_D=20$ because this value is smaller than the critical ${\lambda }_D$ value 23.380 listed in Table 1 of [6] . It can be seen that all the GOS curves have a similar profiles. They also tend towards zero as q increases which is due to the variation of the radial wave functions for a high atomic electron radius.

We note that the correlation effects also have a poor influence on the GOS of the excitation to 6p sub-shell of sodium atoms confined by the Debye plasma environment, except around their maxima. In this region, where a difference in the absolute values of the RPAE and RHF data is observed, the gap between them does not exceed 4.6%, 6.08% and 3.17% for values ∞, 100 and30 of ${\lambda }_D$, respectively. Results of the $3s\to 6p$ transition GOS for different Debye plasma screening lengths ${\lambda }_D$ considered in this study are shown in Figure 7(c) . The results in Figure 7(c) provide an interesting comparison. In the RPAE-V approach, the peak of the GOS curve for ${\lambda }_D=100$ is 1.0747 times lower than that of GOS curve for ${\lambda }_D=\infty$ and the curve’s peak of GOS for ${\lambda }_D=30$ is 2.0188 times smaller than the peak of the GOS curve for ${\lambda }_D=100$.

In the RHF-V method, the maximum GOS of dipole transition for ${\lambda }_D=\infty$ is 1.0928 larger than that of the GOS for ${\lambda }_D=100$. The maximum GOS for the value 100 was 1.7540 times that of GOS for ${\lambda }_D=30$. We note that the magnitude of GOS consistently decreases with decreasing Debye length ${\lambda }_D$.

Regarding the present investigation of the GOS, our results have a considerable effect on the sodium atoms planted in the Debye plasma environment. For ${\lambda }_D=\infty$, the limiting behavior of the $3s\to 6p$ transition GOS as $q\to 0$ was also 0.0006412 a.u and 0.0006549 a.u in the present RHF and RPAE velocity calculations, respectively. Both of these values are between the theoretical data of [6] (0.00067) and the theoretical data of [20] (0.000593).