Abstract

The generalized oscillator strengths (GOSs) associated with the dipole 2s→2s⁰(2p, 3p, 4p) and quadrupole 2s→2s⁰(3d, 4d) excitations of a lithium atom confined within a spherical cavity and embedded in a quantum plasma are investigated within the Bethe-Born approximation. Radial wavefunctions are obtained from Bahar’s numerical solution of the radial Schrödinger equation using the More General Plasma Model Potential (MGPM), which incorporates length screening, oscillatory screening, and linear correction effects. The analysis, performed for a fixed confinement radius under different plasma conditions characterized by the screening parameters $a$ , $b$ , and $D$ , reveals the combined influence of these parameters on the GOSs. The quadrupole GOS amplitudes decrease with decreasing ($D$ ) and increase as $a$ and $b$ decrease. Moreover, the parameter $b$ generates additional maxima, while the peak positions shift toward higher $q^2$ values with increasing $a$ and toward lower $q^2$ values with increasing $D$ . Similar behavior is observed for the dipole excitations, where plasma screening has its strongest effect in the low squared momentum transfer region and becomes negligible at higher momentum transfers. Higher excited states are found to be more sensitive to the quantum plasma environment than lower excited states, although some transitions exhibit non-monotonic behavior due to the interplay between plasma screening and spherical confinement.

Keywords

Spherical Confinement, Quantum Plasma, Atomic Lithium, More Generalized Plasma Potential Model, Generalized Oscillator Strength

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1. Introduction

Studying atomic systems in plasma environments has attracted considerable interest in recent years because of its fundamental importance in atomic physics and its wide range of applications in astrophysical and laboratory plasmas. Plasma, often referred to as the fourth state of matter, constitutes a large fraction of the visible matter in the universe and occurs in diverse environments such as interstellar media, stellar atmospheres, fusion devices, laser-produced plasmas, and ionized gases. In such environments, interactions between charged particles and atoms significantly modify atomic properties, affecting energy levels [1], transition probabilities [1] [2], and inelastic collision dynamics [1]-[3]. Understanding these plasma-induced effects is therefore essential for accurate plasma diagnostics [4] and for developing reliable theoretical models of atomic systems.

Among the various topics in this field, the study of spherical compressed atoms confined in quantum plasma environments has received increasing attention. The combined effects of spatial confinement and plasma screening can substantially alter the electronic structure and dynamical properties of atomic systems, making them particularly relevant for diagnostic applications and for the modeling of astrophysical and laboratory plasmas. Investigations of these effects provide valuable understanding of the behavior of atoms under extreme conditions and contribute to a better interpretation of spectroscopic observations.

Lithium has emerged as an attractive model system for such studies because of its relatively simple electronic structure, which allows highly accurate theoretical treatments while preserving important many-body interactions. In addition, lithium is of considerable interest in plasma diagnostics, fusion research, and astrophysics. Its spectral lines are widely used to probe plasma characteristics in laboratory experiments, while observations of lithium in astrophysical environments provide important information about stellar atmospheres and cosmic matter. Consequently, investigating the influence of plasma environments on the atomic structure and transitions of lithium is crucial for understanding excitation processes and improving the description of plasma-related phenomena.

For example, Bahar [5] investigated the combined effects of spatial confinement and quantum plasma screening on the persistent orbital charge-current and the induced magnetic field of atomic sodium and lithium. Bahar’s results indicate that the above aforementioned observables can be controlled through quantum plasmas, allowing them to be increased or decreased as desired. While this study offered a deeper understanding of the electromagnetic properties of confined atoms in plasma environments, spectroscopic quantities such as generalized oscillator strengths (GOS) have received similar attention only in our recent study of the sodium atom [6].

From the preceding discussion, and to the best of our knowledge, the GOS as a function of the squared momentum transfer has not yet been investigated for a lithium atom under simultaneous spherical confinement and quantum plasma conditions. Motivated by this research gap, the present work examines the GOS of selected valence-shell transitions and explores the effects of confinement and plasma screening on atomic properties. The results contribute to the understanding of atomic processes in laboratory and astrophysical plasmas and may be useful for plasma diagnostics.

This paper is structured as follows. The theoretical model and procedure adopted in the present investigation is outlined in Section 2. Section 3 presents and discusses the obtained GOS results for multipole GOSs for the excitations to 2s⁰(2p, 3p, 4p, 3d, 4d, 4f, 6f), while the conclusions are given in the final section. Atomic units (a.u.) are used throughout the article unless explicitly stated otherwise.

2. Main Points of Calculation

For fast nonrelativistic charged projectiles undergoing inelastic scattering by an atomic target, Bethe’s theory of inelastic scattering, based on the first-Born approximation (FBA), shows that the generalized oscillator strength (GOS) associated with a transition from an initial atomic state 0 to a final state f can be derived from the quantum-mechanical theory of inelastic scattering by combining Fermi’s Golden Rule with the first-Born approximation (FBA) [7] [8]:

${ℱ}_{0\text{f}}\left(\omega ,K\right)=\frac{2\left(E_{\text{f}}-E_0\right)}{K^2}{\left|{\displaystyle {\sum }_{j=1}^N{\displaystyle \int {\Psi }_{\text{f}}\left(r_i,\cdots ,r_N\right)\times {\text{e}}^{-\text{i}K\cdot r_j}{\Psi }_0\left(r_i,\cdots ,r_N\right)}}\right|}^2$ (1)

with ${\Psi }_0\left(r_i,\cdots ,r_N\right)$ and ${\Psi }_{\text{f}}\left(r_i,\cdots ,r_N\right)$ denote the initial and final wave functions of the atomic system, with corresponding eigenvalue energies $E_0$ and $E_{\text{f}}$ , respectively. Here, $r_j$ is the position vector of the $j$ -th electron in the $N$ -electron system and $K$ is the momentum transfer.

Under the single-electron transition approximation, Equation (1) can be simplified as:

$f_{\lambda \mu }\left(\omega ,q\right)=\frac{2\omega }{K^2}{\left|\langle {\varphi }_{\mu }|{\text{e}}^{-\text{i}K\cdot r}|{\varphi }_{\lambda }\rangle \right|}^2$ (2)

where ${\varphi }_{\lambda }$ and ${\varphi }_{\mu }$ are one-electron wave functions, and $\lambda$ and $\mu$ label the initial and final electron states, respectively, with energy eigenvalues ${\epsilon }_{\lambda }$ and ${\epsilon }_{\mu }$ . The transition energy is determined by the energy conservation relation $\omega ={\epsilon }_{\mu }-{\epsilon }_{\lambda }$ .

Since the wave functions are represented by normalized antisymmetrized products of spatial and spin coordinates, and by expanding the electron transition operator ${\text{e}}^{\text{i}q\cdot r}$ in terms of spherical waves, Equation (2) can be expressed as a sum of multipole contributions:

$f_{\lambda \mu }\left(\omega ,q\right)={\displaystyle {\sum }_{\ell }{f}_{\lambda \mu }^{\left(\ell \right)}}\left(\omega ,q\right),$ (3)

where $f_{\lambda \mu }^{\left(\ell \right)}\left(\omega ,q\right)$ denotes the $\ell$ -pole contribution to the generalized oscillator strength and is given by [8]:

$f_{\lambda \mu }^{\left(\ell \right)}\left(\omega ,q\right)=\frac{2\omega N_{\lambda }\left(2\ell +1\right)}{K^2}{\left|\sqrt{2{\ell }_{\mu }+1}\times \left(\begin{array}{ccc}{\ell }_{\mu }& \ell & {\ell }_{\lambda }\\ 0& 0& 0\end{array}\right){\displaystyle {\int }_0^{\infty }{P}_{n_{\mu }{\ell }_{\mu }}\left(r\right)j_{\ell }\left(qr\right)P_{n_{\lambda }{\ell }_{\lambda }}\left(r\right)\text{d}r}\right|}^2$ (4)

where $N_{\lambda }$ is the occupation number of the initial subshell, $\ell$ is the angular momentum transfer (multipole order), and $\left|{\ell }_{\mu }-{\ell }_{\lambda }\right|\le \ell \le \lfloor {\ell }_{\mu }+{\ell }_{\lambda }\rfloor$ which follows from the triangular selection rule. The function $j_{\ell }\left(qr\right)$ is the spherical Bessel function of the first kind of order $\ell$ , while $P_{n_{\lambda }{\ell }_{\lambda }}\left(r\right)$ and $P_{n_{\mu }{\ell }_{\mu }}\left(r\right)$ are the normalized radial wave functions of the initial and final states, respectively, calculated within the quantum-plasma model. The problem of determining that above radial functions was still a subject of considerable interest of the form of the model potential approach describing intensity of the interaction particles in the non-relativistic time independent Schrödinger equation which reduces to a one-dimension equation in terms of the radial variable given as [9]:

$\left[-\frac{1}{2}\frac{{\text{d}}^2}{\text{d}{r}^2}+\frac{\ell \left(\ell +1\right)}{2r^2}+V_{\text{effscr}}\left(r\right)\right]P_{n\ell }\left(r\right)={\epsilon }_{n\ell }{P}_{n\ell }\left(r\right)$ (5)

Here $V_{\text{effscr}}\left(r\right)$ is the effective screening model potential approach in quantum-plasma model.

The normalized radial wave functions employed in the present work were obtained from the theoretical calculations of Bahar, where the radial Schrödinger equation (5) for a compressed lithium atom embedded in a quantum-plasma environment was solved numerically using the More General Plasma Model Potential (MGPM) [9] [10]

$V_{\text{MGPM}}\left(r,Z,D,a,b\right)=V_{\text{model}}\left(r,Z\right)\left(1+br\right)\times {\text{e}}^{-r/D}\text{cos}\left(\frac{ar}{D}\right)$ (6)

as the effective screening model potential.

Where $V_{\text{model}}\left(r,Z\right)$ is the model potential defined in [5].

Within the one-electron transition approximation, lithium is represented as a closed-shell 1s² ionic core plus a single active valence electron. The two 1s electrons remain inactive during the transition and their average effect is incorporated into the effective model potential $V_{\text{model}}$ . The ground-state configuration 1s²2s is, with the 2s electron treated as the active electron undergoing the considered transitions. The 1s² core has zero net spin ($S=0$ ), while the total spin of the system is determined by the valence electron, $s=1/2$ . Explicit many-electron exchange and correlation effects beyond those represented by the effective model potential are neglected. The parameters $a$ , $b$ , and characterize the plasma screening effects. Specifically, $a$ is the oscillatory screening parameter that governs the frequency of the cosine modulation, $b$ is a linear correction parameter that improves the short-range behavior of the potential near the nucleus and for closely interacting particles, and $D$ is the Debye screening length that determines the rate at which the potential decreases with distance. The radial wavefunctions used in the present work were obtained from the numerical calculations of Bahar [5]. In his study, the radial Schrödinger equation was solved numerically using the tridiagonal matrix method, based on a second-order finite-difference discretization of the radial equation and a 1000-point numerical grid. The resulting radial wavefunctions, expanded in Slater-type functions were subsequently provided to us by the author for use in the present calculations. Prior to their use, their numerical normalization was verified according to ${\displaystyle {\int }_0^R{\left|P_{n \ell }\left(r\right)\right|}^2\text{d}r}=1$ , where $R$ denotes the spherical confinement radius imposed through the boundary condition [5]. The normalized reduced radial wavefunctions were then used directly in the numerical evaluation of the radial matrix elements entering the generalized oscillator strengths. The radial Schrödinger Equation (5) for a lithium atom confined within a spherical cavity of radius $R=25$ a.u. and embedded in a quantum plasma environment was solved numerically by Bahar using the procedure described in detail in his published work [5]. With the author’s permission, the resulting normalized radial wavefunctions provided to us in terms of Slater-type functions, were employed to calculate the generalized oscillator strengths (GOSs) considered in the present work.

The normalized radial wave functions obtained by Bahar [5] through the numerical solution of the radial Schrödinger Equation (5) for a lithium atom confined within a spherical cavity of radius $R=25$ a.u. and embedded in a quantum plasma environment were employed in the present study to calculate the GOSs. Specifically, dipole ($\ell =1$ ) oscillator strengths (OS) were evaluated for the excitations to 2s⁰ (2p, 3p, 4p), and quadrupole ($\ell =2$ ) GOSs for the 2s→2s⁰(3d, 4d) transitions. For each set of plasma parameters, the transition energy was consistently obtained from the energy difference between the corresponding initial and final-state eigenvalues associated with the supplied radial wavefunctions, $\omega ={\epsilon }_{\mu }-{\epsilon }_{\lambda }$ , and used in the corresponding GOS calculation.

3. Calculated Results for GOS

The generalized oscillator strengths (GOSs) corresponding to multipolarities $\ell =1$ and 2 from ground state of a spherically confined lithium atom in a quantum plasma were calculated using the above Equation (4) with the aid of MATLAB Online.

3.1. Dipole Excitations

The present section of work reports part of our results for the dipole transitions from the (2s) state to the (2p), (3p), and (4p) states of the lithium atom under spherical confinement and quantum plasma conditions. Equation (4) is the fundamental expression used to calculate the dipole ($\ell =1$ ) oscillator strength within the length-form Bethe first-Born theory, employing the atomic state wavefunctions generated by Bahar. To compute the dipole oscillator strength (DOS), we require only the radial wavefunctions of the ground state, $P_{2\text{s}}$ and the excited states $P_{2\text{p}}$ , $P_{3\text{p}}$ and $P_{4\text{p}}$ , of the embedded lithium atom, as obtained in the work of Bahar [5].

We compute the energy eigenvalues of 2s, 2p, 3p and 4p states and the generalized oscillator strength with $K^2={10}^{-8}$ for excitations 2s⁰(2p, 3p, 4p). The GOS values of these transitions at $K^2={10}^{-8}$ are close to limiting behavior of the GOS as $K^2\to 0$ (oscillator-strength). A comparison of the energy eigenvalues obtained using the Exponential Screened Coulomb Potential (ECSC) without spherical confinement ($a=1$ , $b=0$ and $R\to \infty$ ) with the theoretical results reported in [11], is presented in Table 1. Very good agreement is observed for all the states considered. For $D=50$ , the differences between the present results and the theoretical values remain below approximately 0.5%, while for $D=25$ and $D=20$ , the deviations are also small, with maximum differences of about 0.51% and 0.54%, respectively. This close agreement is observed for the 2s, 2p, and 3p states over the range of Debye screening lengths considered. These results provide a validation of the present calculation of the energy eigenvalues for lithium under the considered plasma conditions.

For $a=0$ , $b=0$ , $D=\infty ,50,25$ and 15, and $R\to \infty$ , the energy of 2s, 2p, 3p and 4p orbitals and the oscillator strengths associated with the dipole transitions 2s→(2p, 3p, 4p) were also determined. The calculated results are summarized in Table 2, together with previously reported theoretical oscillator-strength values for these transitions [12]-[17].

Table 1. Energy of 2s, 2p, 3p and 4p orbitals for lithium under quantum plasma ($a=1$ , $b=0$ , $R\to \infty$ ).

^aTheoretical [11].

Table 2. Energy eigenvalues and oscillator strengths for dipole 2s→2s⁰(2p, 3p, 4p) transitions for lithium under plasma environment ($a=0$ , $b=0$ , $R\to \infty$ ).

^bPandey et al. [12]. ^cLi & Khar [13]. ^dShuai et al. [14]. ^eNIST data; see website www.physics.nist.gov, NIST data, 2008 [15]. ^fNing and Qi [16]. ^gOrdu & Bahar [17].

The theoretical data of the energy orbitals [12]-[14] are in good in good agreement (<1.94% ) with those found in this present work except in the case of 4p orbital for $D=15$ where the disagreement between the value listed in reference [13] and our finding one is about 22.6%. Table 2 compares the calculated oscillator strengths with the available theoretical values reported in previous studies and given also in those references [12]-[17].

In general, good agreement is obtained, particularly for the dominant 2s→2p transition. In the limit $D\to \infty$ , the present value of 0.744828 differs by less than approximately 1.1% from the available theoretical values. For the 2s→3p and 2s→4p transitions, the agreement depends more strongly on the reference considered. For 2s→3p, the present result at $D\to \infty$ , 0.00496633, agrees reasonably well with result in [13], 0.004741, whereas larger differences are observed with results found in [14] and [16]. Similarly, for the 2s→4p transition, the present value 0.00433032 is very close to the data in [13], 0.004278, and also remains relatively close to the values reported in the other calculations [14] [17]. At Debye screening length $D=50$ , the agreement with [13], is particularly good. For example, the present value of $f_{2\text{s}\to 2\text{p}}=0.747785$ is almost identical to the value 0.74777 reported by Ning et al. in [16]. The corresponding differences for the 2s→3p and 2s→4p transitions also remain small. However, somewhat larger deviations are observed with the results presented in [13] [14] particularly for the excitations to 2s⁰(3p, 4p). At Debye screening length $D=15$ , the present result for the strong 2s→2p transition remains in close agreement with the available theoretical values. However, for the weak 2s→3p and 2s→4p transitions, the discrepancies become more pronounced. This increased sensitivity may be associated with the stronger plasma screening at smaller Debye screening lengths, which affects high 2s→(23p, 4p) transitions more significantly.

Figures 1-3 show the variation of the dipole oscillator strengths for the 2s→2p, 2s→3p and 2s→ 4p transitions, respectively as the squared momentum transfer $K^2$ varies from 0 to 1.2 a.u. In all three panels (α), (β) and (γ) of Figure 1, the DOS decreases rapidly with increasing $K^2$ , indicating that the dipole transition is mainly concentrated in the low-squared momentum-transfer region and becomes negligible at large $K^2$ . The first panel (α) illustrates the effect of the oscillatory screening parameter $a$ . For fixed values of $b=1$ and $D=80$ , increasing from parameter $a$ 0 to 5 and 10 progressively reduces the GOS in the low-$K^2$ region. The largest value is obtained for an atom embedded in a plasma without oscillatory screening ($a=0$ ), whereas the introduction and subsequent increase of the oscillatory screening $a=5$ and $a=10$ suppress the transition probability. As $q^2$ increases, the three curves gradually converge and become nearly indistinguishable, showing that the influence of parameter $a$ is significant only at small squared momentum transfer. The results described in the. second panel (β) of Figure 1 demonstrate the effect of the linear correction parameter $b$ as the oscillatory screening parameter $a$ and the screening length $D$ were fixed to 1 and 80, respectively. For that above fixed two parameters without the linear correction parameter ($b=0$ ) the embedded lithium atom result exhibits the largest GOS throughout the entire momentum-transfer range. In contrast, when $b$ agreement increases to 1 and 1.5, the GOS is reduced by several orders of magnitude, as highlighted by the insets. The enlarged views reveal that the screened curves exhibit a small maximum at very low $q^2$ , followed by a gradual decrease. The close between the $b=1$ and $b=1.5$ curves indicate that further increasing $b$ beyond unity produces only a minor additional effect. As seen in the panel (γ) of this Figure 1, DOS curve depicts the results showing the effect of the Debye length $D$ for fixed $a=1$ and $b=0.5$ . Increasing the Debye length from 10 to 250 enhances the dipole oscillator strength, particularly in the low-$K^2$ region. Since a larger $D$ corresponds to weaker plasma screening, the transition probability increases as $K^2<0.3171$ au. This situation is reversed above $K^2=0.3171$ au. This behavior can be understood in terms of the delicate interference between the oscillatory radial components of the initial and final state wavefunctions. For $D=80$ and 250, the wavefunction overlap progressively decreases with increasing $K^2$ . This reduction arises because a stronger attractive potential increases the curvature of the wavefunctions, leading to stronger spatial localization and, consequently, a smaller overlap between the initial and final states. However, as in the previous cases, the differences between the curves diminish with increasing momentum transfer, and all curves converge toward zero at large $K^2$ . For the dipole 2s→3p transition of lithium embedded in a quantum plasma, the dipole oscillator strength results shown in Figure 2 as a function of the squared momentum transfer $K^2$ , similarly to the 2s→2p transition, are concentrated at low values of $K^2$ and decrease rapidly with increasing $K^2$ . Clearly, panel (α) of Figure 2 presents the calculated DOS for the 2s→3p excitation of a lithium atom confined within a spherical cavity of radius $R=25$ and embedded in a quantum plasma described by the MGPM potential. For fixed values of $b=1$ and $D=80$ , the oscillatory screening parameter $a$ significantly affects the dipole oscillator strength in the low-$K^2$ region. At very small momentum transfers ($K^2<0.0266219$ a.u.), the DOS for $a=5$ is slightly larger than that for $a=10$ . However, at $K^2\approx 0.0266219$ a.u., the two curves intersect, and for $K^2>0.0266219$ a.u., the $a=10$ curve becomes the largest, followed by the $a=5$ and $a=0$ curves. Thus, increasing $a$ generally enhances the dipole oscillator strength over most of the low-$K^2$ region, except in the very small squared momentum-transfer interval before the crossover. As $K^2$ increases further, the differences between the three curves gradually diminish, and all of them approach zero. For panel (β) given in Figure 2 where $a$ and $D$ are been fixed to 1 and 80, respectively, the embedded lithium atom case result ($b=0$ ) exhibits a strong peak at $K^2=0$ , whereas the screened cases ($b=1$ and $b=1.5$ ) are reduced by several orders of magnitude, as illustrated by the enlarged insets. The insets further reveal a non-monotonic behavior for the screened cases, characterized by a weak maximum at very small $K^2$ , followed by a broad secondary maximum around $K^2\approx 1.5$ , before decreasing again at larger squared momentum transfers. Increasing $b$ from 1 to 1.5 produces only small quantitative changes, indicating that the influence of $b$ becomes saturated. Figure 2 (γ) illustrates the influence of the length screening $D$ on the DOS for the 2s→3p transition while the oscillatory screening parameter $a$ and the linear correction parameter $b$ were fixed at values 1 and 0.5, respectively. The DOS reaches its maximum value in the optical limit ($K^2\to 0$ ) and decreases rapidly with increasing squared momentum transfer, approaching nearly zero for $K^2\gtrsim 0.3$ a.u. for three values of $D=10,80$ , and 250. The length screening has a pronounced influence on the DOS in the low-$K^2$ region. As the Debye length increases from $D=10$ to $D=80$ and $250$ a.u, the DOS in the low-$K^2$ region is generally enhanced. The highest peak is obtained for $D=80$ , whereas the lowest corresponds to $D=10$ . Interestingly, the peak for $D=250$ is slightly smaller than that for $D=80$ , although it remains higher than that for $D=10$ , indicating that the dependence on the Debye length is not strictly monotonic. This behavior reflects the stronger screening associated with shorter Debye lengths, which weakens the transition probability by reducing the overlap between the initial- and final-state wavefunctions. The slight reduction in the DOS peak for $D=250$ relative to $D=80$ may result from the competing effects of quantum plasma screening and spherical confinement, whose relative contributions change with the screening length. Furthermore, in panel (γ) of Figure 2, a different trend is observed. In the $K^2$ range from 0.0591279 to 0.317152 a.u., the DOS for $D=10$ becomes the largest, whereas the smallest values are obtained for $D=250$ . The results for $D=80$ lie between these two cases throughout this interval. This reversal in the relative magnitudes of the DOS suggests that the dominant physical mechanism changes with increasing momentum transfer, possibly because of variations in the overlap between the initial and final-state radial wavefunctions and the competing influences of plasma screening and spherical confinement. At larger squared momentum transfers, the three curves nearly coincide, indicating that the influence of plasma screening becomes negligible in this region.

Figure 1. Dipole oscillator strength for the excitation to 2s⁰2p of lithium surrounded by a sphere of radius R = 25 and embedded in quantum plasma modeled by MGPM potential with: b = 1, D = 80 and three different a-values for panels (α); a = 1, D = 80 and three different b-values for panels (β) and a = 1, b = 0.5 and three different D-values for panels (γ).

Figure 2. Dipole oscillator strength for the excitation to 2s⁰3p of lithium surrounded by a sphere of radius R = 25 and embedded in quantum plasma modeled by MGPM potential with: b = 1, D = 80 and three different a-values for panels (α); a = 1, D = 80 and three different b-values for panels (β) and a = 1, b = 0.5 and three different D-values for panels (γ).

Similar to Figure 2, Figure 3 presents the variation of the dipole oscillator strength as a function of the squared momentum transfer, $K^2$ , for the 2s→4p transition of a lithium atom confined within a spherical cavity of radius $R=25$ a.u and embedded in a quantum plasma. In contrast to the 2s→2p and 2s→3p transitions, the present transition exhibits a pronounced maximum at finite momentum transfer, whose position and magnitude depend strongly on the plasma parameters. In all cases, the DOS decreases toward zero as $K^2$ increases. As shown in panel (α) of Figure 3, for fixed values of $b=1$ and $D=80$ a.u, increasing the oscillatory screening parameter $a$ from 0 to 5 and 10 shifts the principal maximum toward higher values of $K^2$ . The lithium atom embedded in a quantum plasma without oscillatory screening ($a=0$ ) yields the highest and narrowest peak at very low squared momentum transfer ($K^2\approx 0.05$ ). As $a$ increases to 5 and 10, the peak amplitude decreases while its position shifts toward larger momentum transfers, indicating that the plasma environment redistributes the oscillator strength over a broader $K^2$ range. Beyond the principal peak, all curves decay rapidly, leaving only a weak tail at larger squared momentum transfers. The shift of the principal maximum toward higher $K^2$ with increasing oscillatory screening $a$ reflects the redistribution of the transition probability over a wider momentum-transfer range due to the modification of the effective plasma potential and the resulting changes in the overlap between the initial and final state wavefunctions. To understand the effect of the linear correction parameter $b$ on the dipole oscillator strength for atomic lithium confined within a spherical cavity of radius $R=25$ a.u and embedded in a quantum plasma, panel (β) in Figure 3 depicts our findings for $f_{\lambda \mu }^{\left(\ell =1\right)}\left(\omega ,q\right)$ of the 2s→4p transition for fixed oscillatory screening $a=1$ and length screening $D=80$ and three different values of linear correction parameter $b$ which are $b=0,1$ and 1.5. The embedded atomic lithium case result without the linear correction ($b=0$ ) shows a large dipole oscillator strength concentrated at very low squared momentum transfer. In contrast, the screened cases ($b=1$ and $b=1.5$ ) are reduced by several orders of magnitude, as shown by the enlarged insets. The insets reveal an oscillatory structure consisting of several small maxima separated by minima, with the first pronounced peak occurring around $K^2\approx 1.2$ , followed by weaker secondary maxima at larger squared momentum transfers. The curves corresponding to $b=1$ and $b=1.5$ are very similar, indicating that further increasing $b$ produces only minor quantitative changes. From results drawn in the panel (γ) of Figure 3, we can see that at fixed oscillatory screening $a=1$ and linear correction parameter $b=0.5$ , the dipole oscillator strength is progressively enhanced in the low-$q^2$ region with increasing length screening from $D=10$ to $D=80$ and 250 a.u. The largest value is obtained for $D=250$ , followed by $D=80$ , whereas the strongest screening ($D=10$ ) yields the smallest DOS. The peak also shifts toward lower squared momentum transfer as $D$ increases, reflecting the weakening of plasma screening. At larger momentum transfers, the three curves rapidly converge toward zero.

The figures clearly demonstrate that the plasma environment, characterized by the parameters $a$ , $b$ , and $D$ , has a significant influence on the dipole generalized oscillator strengths. The effect is strongest in the low-$K^2$ region, where the electron collision is more sensitive to the screened atomic potential. As the momentum transfer increases, the influence of plasma screening gradually weakens, and the GOS becomes nearly independent of the plasma parameters. Furthermore, the sensitivity to the plasma environment increases with the principal quantum number of the excited state, the 2s→4p transition showing much larger modifications than the 2s→2p transition. This behavior reflects the greater spatial extension of highly excited states, making them more susceptible to plasma screening effects.

Figure 3. Dipole oscillator strength for the excitation to 2s⁰4p of lithium surrounded by a sphere of radius R = 25 and embedded in quantum plasma modeled by MGPM potential with: b = 1, D = 80 and three different a-values for panels (α); a = 1, D = 80 and three different b-values for panels (β) and a = 1, b = 0.5 and three different D-values for panels (γ).

3.2. Quadrupole Excitations

Figure 4 illustrates the variation of the quadrupole generalized oscillator strengths (GOS) for the 2s→3d and 2s→4d transitions as a function of the squared momentum transfer, $K^2$ , under different plasma conditions. For $b=1$ , $D=80$ and $R=25$ a.u, panel (α)shows the behaviour of the generalized oscillator strength of the quadrupole 2s→3d transition for $a=0,5$ and 10 while panel (β) depicts the quadrupole GOS of the embedded atomic lithium excitation to 2s⁰4d for same values of the oscillatory screening parameter $a$ . The GOS curves of panel (α) in for the quadrupole transition 2s→3d have one a single pronounced maximum at low squared momentum transfer, followed by a monotonic decrease toward zero as $K^2$ increases. The effect of the plasma the oscillatory screening parameter $a$ mainly reflected in the peak height and width. Increasing $a$ reduces the maximum value of the GOS while shifting the peak slightly toward larger values of $K^2$ , resulting in a broader distribution. Consequently, the quadrupole 2s→3d GOS decreases more slowly at intermediate squared momentum transfers as $a$ increases from 0 to 5 and 10. The panel (β) presents the corresponding results for the 2s→2s⁰4d transition. Compared with the 2s⁰3d excitation of the embedded atomic lithium, this higher excited state exhibits a much stronger sensitivity to the plasma oscillatory screening parameter $a$ . Figure 4(β) shows that increasing the oscillatory screening parameter $a$ from 0 to 5 and 10 drastically suppresses the large peak observed for $a=0$ , resulting in a much smaller and broader distribution. This behavior indicates that the quadrupole excitation to the 4d state is highly sensitive to the quantum plasma environment. This can be explained by the stronger oscillatory screening, which modifies the radial wavefunctions of the embedded lithium atom and reduces the spatial overlap between the initial and final states, leading to a lower quadrupole transition probability. After discussing the influence of the oscillatory screening parameter $a$ on the octupole GOS, we now examine the effect of the linear correction parameter $b$ on the quadrupole transitions. At fixed values of $a=1$ , $D=80$ , and $R=25$ a.u., the results for the 2s→3d and 2s→4d quadrupole GOS, shown in panels (γ) and (δ) of Figure 2, exhibit a single pronounced maximum when the calculations are performed using the radial wavefunctions obtained without the linear correction parameter ($b=0$ ). It is also observed from the plot that the difference in the peak amplitudes remains negligible as the parameter $b$ varies from 1 to 1.5. However, as illustrated in the insets of Figure 4(γ) and Figure 4(δ), the inclusion of the linear correction parameter ($b=1$ and $b=1.5$ ) strongly modifies the GOS profiles. Instead of a single dominant peak, the quadrupole excitations to the 3d and 4d states develop several additional maxima at higher values of $K^2$ , revealing that the oscillatory component of the MGPM potential redistributes the transition strength over a broader squared momentum-transfer region. This peculiar behavior quadrupole GOS in the inset graph might be attributed to the fluctuations in the wavefunctions generated by the potential which includes also the spherical confinement effect. Figure 4(η) and Figure 4(κ) show the calculated quadrupole GOS of the 2s→2s⁰3d and 2s→2s⁰4d transitions, respectively, obtained for $a=1$ , $b=0.5$ , $R=25$ a.u and three different values of $D=10,80$ and 250. All GOS curves of the panel (η) obtained with $D=10,80$ and 250 for the transition 2s→2s⁰3d have the same shape contrary to the case of the embedded atomic lithium quadrupole excitation to 2s⁰4d, the profile of GOS for $D=10$ is different to those calculated with $D=80$ and 250, presenting the similar profile in panel (κ). The quadrupole GOS for the 2s→2s⁰3d transition in the panels (η) of Figure 4 are characterized by a single dominant maximum for all lengths screening considered here. The highest peak is obtained for $D=80$ , whereas the amplitudes decrease for both $D=10$ and $D=250$ . In addition, as seen in panel (η) of this Figure 4, the peak shifts progressively toward lower $K^2$ values as the length screening increases from 10 to 80 and 250, indicating that weaker plasma screening enhances the transition probability at smaller squared momentum transfers.

Figure 4. Generalized oscillator strength for the quadrupole excitations to 2s⁰(3d, 4d) of lithium surrounded by a sphere of radius R = 25 and embedded in quantum plasma modeled by MGPM potential with: b = 1, D = 80 and three different a-values for panels (α) and (β); a = 1, D = 80 and three different b-values for panels (γ) and (δ); a = 1, b = 0.5 and three different D-values for panels (η) and (κ).

The non-monotonic variation of the peak amplitude with ${\lambda }_D$ reflects changes in the overlap between the initial- and final-state wavefunctions induced by plasma screening. For panel (κ) of the same Figure 4, the appearance of two maxima for $D=10$ may be attributed to the stronger screening, which substantially modifies the radial wavefunctions of the initial and final states. Consequently, the overlap integral entering the transition matrix element changes sign in different radial regions because of the nodal structure of the 4d wavefunction, giving rise to constructive and destructive interference and producing the observed double-peak structure. In contrast, for the weaker screening cases ($D=80$ and 250), the wavefunctions are less distorted, resulting in a single dominant maximum at low squared momentum transfer. It is observed that, for all values of the plasma shielding parameters, the octupole generalized oscillator strengths decrease and eventually approach zero as the squared momentum transfer $q^2$ increases. As been noted in [18], this behavior may be attributed to the existence of a limiting momentum transfer imposed by the uncertainty principle, beyond which the atomic electrons cannot absorb additional momentum without being ejected from the atom. This decrease of the octupole GOS toward zero with increasing $q^2$ can be attributed to the increasing oscillatory behavior of the momentum-transfer operator in the transition matrix element, which leads to enhanced cancellation in the radial integral at large squared momentum transfer.

4. Conclusions

In this work, the generalized oscillator strengths (GOSs) for the dipole 2s→2s⁰(2p, 3p, 4p) and quadrupole 2s→2s⁰(3d, 4d) transitions of a lithium atom confined within a spherical cavity and embedded in a quantum plasma were calculated using the radial wavefunctions obtained by Bahar through the numerical solution of the radial Schrödinger equation with the More General Plasma Model Potential (MGPM), which incorporates the combined effects of length screening, oscillatory screening, and linear correction. The results demonstrate that the plasma shielding parameters ($a,b,D$ ), strongly influence the generalized oscillator strengths, particularly in the low-squared momentum transfer region where the electron–atom interaction is most sensitive to the effective screened potential. The oscillatory screening parameter $a$ modifies both the magnitude and the position of the GOS maxima, while the linear correction parameter $b$ significantly redistributes the transition strength and gives rise to additional maxima, especially for the quadrupole excitations. As reported in [2] [18] [19], a decrease in the screening length $D$ generally leads to a reduction in the generalized oscillator strengths (GOSs) and shifts their principal maxima toward higher squared momentum transfer values, although some transitions exhibit non-monotonic behavior due to the interplay between plasma screening and spherical confinement. The investigation lets us see now the influence of the plasma environment becomes more pronounced for higher excited states, reflecting their larger spatial extension and greater sensitivity to modifications of the effective potential. For all investigated transitions, the generalized oscillator strengths decrease rapidly with increasing squared momentum transfer and progressively approach zero at large ($K^2$ ), indicating that high-momentum collisions contribute only weakly to the excitation process.

The present study provides new theoretical data for generalized oscillator strengths of confined lithium atoms in quantum plasmas and demonstrates the importance of simultaneously accounting for spherical confinement and quantum plasma screening. These results should be useful for modeling electron-atom collision processes and may contribute to plasma diagnostics and the interpretation of spectroscopic measurements in laboratory and astrophysical plasmas.

Author Contributions

Conceptualization, L. GOMIS, M. COULIBALY and C. DIATTA.; methodology, L. GOMIS, C. DIATTA and R. GOMIS; software, M. COULIBALY, I. KLADOUM and A. KOSSI.; validation, C. DIATTA, M. S. DIOMBATY, and I. G. FAYE; formal analysis, L. GOMIS and M. COULIBALY; investigation, Y. DIOUF; resources, L. GOMIS and M. COULIBALY; writing—original draft preparation, L. GOMIS, C. DIATTA, Y. DIOUF and I. G. FAYE; writing—review and editing, L. GOMIS, M. COULIBALY and C. DIATTA; visualization, C. DIATTA, A. C. WADE; supervision, L. GOMIS and M. S. TALL.

All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References