Exponential Cosine Screened Coulomb Potential Research Articles

Complete critical bound region of the generalized exponential cosine screened Coulomb potential

The complete critical bound region of the generalized exponential cosine screened Coulomb potential (ECSCP) V(r)=−r−1e−(βcos⁡θ)rcos⁡[(βsin⁡θ)r] is investigated by employing the generalized pseudospectral method. Highly accurate critical screening parameters are determined over the entire parameter space, spanning from the exponential screened Coulomb potential (θ = 0), via the oscillatory ECSCP ( θ=π/4 ), to the strongest oscillatory cosine Coulomb potential (CCP) ( θ=π/2 ). With an efficient extrapolation technique, the critical screening parameter for the CCP is estimated at about βc=2.34 . By analyzing the variation of the ground state energy in the critical bound region, we separate the parameter space into two regions, i.e. the exponential region at small and moderate values of θ where the exponential screening term dominates the critical behavior of system, and the cosine region for θ close to π/4 where the oscillatory cosine term dominates.

Hydrogen-like Plasmas under Endohedral Cavity

Over the past few decades, confined quantum systems have emerged to be a subject of considerable importance in physical, chemical and biological sciences. Under such stressed conditions, they display many fascinating and notable physical and chemical properties. Here we address this situation by using two plasma models, namely a weakly coupled plasma environment mimicked by a Debye-Hückel potential (DHP) and an exponential cosine screened Coulomb potential (ECSCP). On the other hand, the endohedral confinement is achieved via a Woods-Saxon (WS) potential. The critical screening constant, dipole oscillator strength (OS) and polarizability are investigated for an arbitrary state. A Shannon entropy-based strategy has been invoked to study the phase transition here. An increase in Z leads to larger critical screening. Moreover, a detailed investigation reveals that there exists at least one bound state in such plasmas. Pilot calculations are conducted for some low-lying states (ℓ=1−5) using a generalized pseudo spectral scheme, providing optimal, non-uniform radial discretization.

Hyperpolarizabilities of hydrogenlike atoms in Debye and dense quantum plasmas.

The hyperpolarizabilities of the hydrogenlike atoms in Debye and dense quantum plasmas are calculated using the sum-over-states formalism based on the generalized pseudospectral method. The Debye-Hückel and exponential-cosine screened Coulomb potentials are employed to model the screening effects in, respectively, Debye and dense quantum plasmas. Our numerical calculation demonstrates that the present method shows exponential convergence in calculating the hyperpolarizabilities of one-electron systems and the obtained results significantly improve previous predictions in the strong screening environment. The asymptotic behavior of hyperpolarizability near the system bound-continuum limit is investigated and the results for some low-lying excited states are reported. By comparing the fourth-order corrected energies in terms of hyperpolarizability with the resonance energies using the complex-scaling method, we empirically conclude that the applicability of hyperpolarizability in perturbatively estimating the system energy in Debye plasmas lies in the range of [0,F_{max}/2], where F_{max} refers to the maximum electric field strength at which the fourth-order energy correction is equal to the second-order term.

Double quantum ionization cross-sections for more general exponential cosine screened coulomb potential

Double-quantum ionization cross-sections for hydrogen atom have been calculated for the more general exponential cosine screened coulomb (MGECSC) potential. The ionization cross-sections are calculated as a function of the wavelength of the incident photons, as well as the potential parameters, using linearly and circularly polarized radiation. The transition matrix elements are calculated using the efficient pseudostate summation technique which has proved to be very accurate in these calculations. Numerical data is also presented for the double-quantum ionization cross-sections for hydrogen atom under the effect of MGECSC potential.

Critical screening parameters and critical behaviors of one-electron systems with screened Coulomb potentials

The critical screening parameters for one-electron systems screened by Hulthén, Debye–Hückel, and exponential cosine screened Coulomb potentials are calculated with an accuracy close to the precision of numerical arithmetic. The results for a H atom with an infinitely heavy nucleus are reported from the ground to high-lying excited states, and those for arbitrary two-body charged systems are derived from the Zm-scaling law. A thorough comparison of the critical screening parameters for the ground and the first p-wave excited states with previous predictions is made to demonstrate the accuracy of our calculations. The critical behaviors of system-bound and pseudo-continuum eigenenergies for s- and non-s-wave states are shown to follow the quadratic and linear laws, respectively. The variation of the corresponding wave functions is analyzed in detail. For systems with non-zero orbital angular momenta, the bound states convert into shape-type resonances when the screening parameter exceeds the critical value. The resonance energy shares the same linear law as the pseudo-continuum state, while the resonance width varies by an l-dependent power law. It is further shown that the different asymptotic behaviors of the resonance energy and width are consistent with the complex analog of the Hellmann–Feynman theorem.

The [formula omitted]P[formula omitted] states of exponential cosine-screened heliumlike atoms

We investigate the singly-excited and doubly-excited 1,3Po states of screened-heliumlike atoms (Z=3-10), starting from Li+ to Ne8+ under the influence of exponential cosine-screened (ECS) Coulomb potentials. Pseudo-random-parameters dependent correlated exponential wave functions are used to represent the correlation effects between the charged particles. The singly-excited 1s2p 1,3P states are also estimated based on the Ritz variational method. The doubly excited resonance energies and widths are determined using the stabilization method. A total of 9 P-wave resonances including four members in the 2snp+ (2 ≤n≤ 5) series, three members in the 2snp−(3 ≤n≤ 5) series, and two members in the 2pnd (n=3, 4) series for each screened-systems below the 2s 2S thresholds of the respective one-electron subsystems are reported. The singly-excited state energies for each system and each screening parameter below the n=1 thresholds of the respective subsystems are also reported. The singly-excited states of ECS-heliumlike systems (Z= 6–10) and the doubly-excited 1,3Po states of the ECS-heliumlike systems are presented for the first time in the literature.

Confined hydrogenlike ions in plasma environments

The behavior of H-like ions embedded in astrophysical plasmas in the form of \emph{dense, strongly and weakly coupled} plasmas are investigated. In these, the increase and decrease in temperature is impacted with a change in confinement radius $(r_{c})$. Two independent and generalized scaling ideas have been applied to modulate the effect of plasma screening constant ($\lambda$) and charge of ion ($Z$) on such systems. Several new relations are derived to interconnect the original Hamiltonian and two scaled Hamiltonians. In exponential cosine screened Coulomb potential (ECSCP) (dense) and weakly coupled plasma (WCP) these scaling relations have provided a linear equation connecting the critical screening constant $(\lambda^{(c)})$ and $Z$. Their ratio offers a state-dependent constant, beyond which, a particular state vanishes. Shannon entropy has been employed to understand the plasma effect on the ion. With increase in $\lambda$, the accumulation of opposite charge surrounding the ion increases leading to a reduction in number of bound states. However, with rise in ionic charge $Z$, this effect can be delayed. The competing effect of plasma charge density ($n_e$) and temperature in WCP and ECSCP is investigated. A recently proposed simple virial-like theorem has been established for these systems. Multipole ($k=1-4$) oscillator strength (OS) and polarizabilities for these are studied considering $1s, 2s$ states. As a bonus, analytical closed-form expressions are derived for $f^{(k)}$ and $\alpha^{(k)} (k=1-4)$ involving $1s$ and $2s$ state, for \emph{free H-like ion}.

Ionization of many-electron atoms by the action of two plasma models

The Hartree-Fock equations for many-electron atoms embedded in a plasma medium are solved using two different plasma models: (a) Debye-Hückel screening (DHS) potential and (b) exponential cosine screened Coulomb (ECSC) potential. Roothaan's approach is implemented for these models after solving the inherent difficulties to evaluate integrals where screening appears explicitly. A corresponding computer code was developed using the method of global basis sets (GBS). The reliability of this approach was verified by solving the Hartree-Fock equations through implementation of the finite-differences and finite-element grid methods and applied to two-electron atoms, yielding excellent agreement with the Roothaan-GBS (RGBS) method. The RGBS method was used to study the energy evolution and ionization threshold of several closed- and open-shell many-electron atoms embedded either in weak or strong DHS or ECSC plasma conditions. In all cases, a critical value of the screening length is obtained for which ionization is achieved, being systematically larger for DHS conditions, indicating the effect of a more repulsive ECSC potential. For He-like atoms in the ground state, we report a comprehensive set of accurate total energy data as a function of the screening constant using the Lagrange mesh method, which includes the electron correlation effects. The electron correlation energy is estimated using this data with reference to the RGBS estimates of energy as the Hartree-Fock energy. The variation of correlation energy as a function of screening constant under the different plasma potentials is rationalized in terms of a conjectured comparison theorem. Finally, a discussion on the effect of plasma strength on localization or delocalization of the electronic density derived from the RGBS method is presented in terms of changes in the Shannon entropy, yielding consistent results for delocalization close to the ionization threshold.

Multipole polarizabilities for hydrogen-like atoms in Hulthén potential with and without spherical confinement

The multipole polarizabilities for hydrogen-like atoms with Hulthen potential are calculated with the sum-over-states formalism where the system bound and pseudocontinuum states are produced by employing the generalized pseudospectral method. Energies for nonzero angular momentum states, transition oscillator strengths, and dipole polarizabilities for the ground state of H atom with HP are calculated with high accuracy and compared with previous calculations. Quadrupole, octupole, and hexadecapole polarizabilities for both the ground and excited states are reported for the first time. The combined effect of Hulthen potential and impenetrable sphere on the polarizabilities is investigated, and the relation of such system with the confined H atom and particle in a box is discussed. Further comparison among Hulthen, screened Coulomb, and exponential cosine screened Coulomb potentials reveals that although the ground state energies follow the comparison theorem very well, their first-order energy corrections show complicated variations.

Dipole and generalized oscillator strengths-dependent electronic properties of helium atoms immersed in a plasma

Atoms subjected to extreme environmental conditions are of fundamental importance due to the modification of their electronic properties. In this work, we study the helium atom when immersed in a plasma environment. In order to describe the plasma medium, we use two models when solving the Schrodinger equation in a restricted Hartree–Fock approach: the Debye–Huckel screened (DHS) potential and a more general exponential-cosine screened Coulomb (ECSC) potential. The plasma length parameter, $$\lambda $$ , in both model potentials characterizes the plasma screening effects which cause an increase or decrease in the electronic properties of the helium atom. We report results for the total electronic ground state energy, orbital energy, dipole oscillator strengths, generalized oscillator strengths (GOS), mean excitation energy, electrostatic dipole polarizability, and electronic stopping cross section. We find that the ECSC plasma model produces a less bound system than the DHS plasma model at the same value of the screening length, $$\lambda $$ . However, the ECSC model potential has a stronger dipole transition from the 1s to the 2p and 3p states than the DHS model potential. Also, the ECSC potential predicts a higher static dipole polarizability than the DHS model potential, with a consequent lower mean excitation energy. We also find a larger GOS for the ECSC than for the DHS for the same momentum transfer at the same value of the screening length, $$\lambda $$ . Consequently, the ECSC mode potential produces a larger electronic stopping cross section than the DHS for the same $$\lambda $$ and projectile velocity. In the limit of $$\lambda \rightarrow \infty $$ , we have excellent agreement with the free helium properties. These quantitative values for the electronic properties would be useful for the investigations of the atomic structure and collisions of helium atoms immersed in plasmas.

Structural properties of Li atom under quantum and classical plasmas: A composite variational framework

AbstractStructural properties of 1s2nl (2L) [n = 2–5, l = 0–4; where, n and l are the principal quantum number and orbital angular momentum quantum number, respectively] states of Li atom embedded in classical weakly coupled plasma (WCP) and dense quantum plasma (DQP) have been discussed. The Debye‐Hckel potential or the screened‐Coulomb potential (SCP) and exponential‐cosine‐screened Coulomb potential (ECSCP) have been used to mimic the WCP and DQP, respectively. Li atom has been treated as a composite system with a frozen core Li+ ion and a chemically active valence electron. The Rayleigh‐Ritz variational method with Hylleraas‐type basis set has been used to estimate the energy eigenvalue of 1s2 (1S) state of Li+ ion core and a pure exponential basis has been considered to compute the energy of nl (2L) states of the valence electron of Li atom. The influence of ECSCP and SCP on the radial probability distribution of the valence electron of the Li atom has also been studied.

Tsallis entropy: A comparative study for the 1s2-state of helium atom

The Tsallis entropies are reported, as a function of the Tsallis index q and the screening parameter μ, in position space, for the ground state of the helium atom. Three different potentials, namely Coulomb, Yukawa, and exponential cosine screened Coulomb potentials and their combinations are used in the calculations. The simple Hylleraas correlated wave function, with a three-parameter, is applied to focus on getting the most characteristic physical features of the electron-electron repulsion results. The values are calculated in one- and two-electron charge densities. Moreover, for a comparative study, the Hartree-Fock approximation, which implements a numerical grid method, is employed. Additionally, the modified Tsallis index, Qq, is proposed and calculated to predict the best value of q in the combined potentials, and it was found that 0.5≤q<1.0.

The information theory of the helium atom in screened Coulomb potentials

AbstractIn this work, we show Hartree–Fock (HF) results for the Shannon entropy, Rényi entropy, Tsallis entropy, Onicescu information energy, and ground‐state properties of the helium atom in screened Coulomb potentials. We solve the HF equations through a numerical grid method where the screened functions model two plasma environments: the Debye–Hückel screening (DHS) and the exponential‐cosine‐screened‐Coulomb (ECSC) model potentials. For plasma conditions near to the energy ionization, for the Shannon entropy, we report an increase of the ∼80% and ∼60% for the DHS and ECSC potentials concerning the unscreened case. A similar situation is found for the Rényi and Tsallis entropies where an increase of ∼86% and ∼57% is reported in the presence of the DHS and ECSC potential functions. Our results allow us to obtain knowledge of the information theory parameters in the description of the electron localization for the helium atom screened by the DHS and ECSC potentials. Finally, our information theory findings in the absence of the screening functions are in excellent agreement with the available results found in the literature.

Critical stability and quantum phase transition of two-electron system under exponential-cosine-screened-Coulomb interaction

The stability of two electron system under the influence of the exponential-cosine-screened-Coulomb (ECSC) potential has been studied as a function of continuously varying nuclear charge (Z) and screening constant (μ). Inter-electronic correlation has explicitly been considered by adopting Hylleraas type basis set under Ritz variational framework. The correlated basis integrals originating from the elements of Hamiltonian and overlap matrices are evaluated analytically. Critical nuclear charges (Zc), the minimum amount of nuclear charge required to form at least one bound state, corresponding to different μ values have been predicted. Different structural properties along with two-particle densities (TPD) for all possible orientations of the electrons have been estimated using the optimized wave function. Prominent signature of quantum phase transition (QPT) of the system around the Zc values has been reported. A comparative study has been done on the critical stability of the system under the influence of ECSC potential, screened Coulomb (SC) potential and pure Coulomb (PC) potential.

Rényi, Fisher, Shannon, and their electron correlation tools for two-electron series

Information entropies, such as Rényi, Fisher, and Shannon, are calculated for the 1s2-state of the two-electron series, in position space. It is considered that the exponential cosine screened Coulomb potential (ECSCP) describes the medium. Explicit correlated Hylleraas functions with 3-parameter are used to focus on getting the most distinctive physical features of the inter-electronic repulsion results. In one- and two-electron charge densities, the numerical values of the information entropies and their ratios are calculated. Fisher-Shannon, and Fisher-Rényi information products that measure the electron-electron correlation strength, are calculated and discussed. Also, we used the contour maps and 3D-graphs to visualize how the interplay between inter-electronic repulsion and electrons-nuclear charge interaction determines the variation of information entropies as a function of the nuclear charge and the screening parameter. In this work, an expression for the divergence in Hylleraas coordinates has been derived and the results of the information quantities for two-electron systems with ECSCP have been calculated, in one-and two-electron densities, for the first time.

Generalized oscillator strengths for the ground state [formula omitted] of sodium atom embedded in a plasma medium

The dipole and generalized oscillator strengths (DOS and GOS) of the free and confined atomic system are related to the description of several interesting chemical physical properties, e.g. static polarizability, photoionization, and stopping cross section (energy loss), among others. In this work, we show results for the 3s→3p,4p DOS electronic transition, the 3s→4s,5s,6s, and the 3s→3p,4p,5p GOS electronic transition of the sodium atom in the presence of a weak and strong plasma environment describe by a Debye-Hückel (DHS) and an exponential-cosine-screened-Coulomb (ECSC) potentials as a function of the plasma screening length, λ. We solve the time-independent Schrödinger equation implementing a numerical approach using a pseudo-potential model to describe the sodium core electrons. We find that the 3s energy level becomes delocalized due to the screening plasma interaction for lower values of λ~2.43 and ~4.10 a.u., for the DHS and ECSC potentials, respectively. As a consequence, the f3s→3p and f3s→4p DOS transitions for both DHS and ECSC plasma environments are reduced as the screening length is decreased observing clear differences between plasma mediums. Similar behavior is observed for the GOS electronic transitions as a function of the transferred momentum where the ECSC plasma medium decreases the amplitude transition as the screening length reaches the energy delocalization. In the absence of a plasma environment, we compare our DOS and GOS results observing a good agreement in comparison with other theoretical and experimental findings.

Electron-impact excitation of ions within a quantum plasma

We develop a new relativistic model which can describe the dynamics on electron-impact excitation (EIE) of ions within a quantum plasma environment. An exemplary excitation 1s1/2 → 2p3/2 of Fe XXVI by electron impact with the exponential cosine-screened Coulomb potential (ECSCP) is presented, i.e., 25 times ionized atoms of Fe, exist in laser-produced plasmas, astrophysical objects, and electron beam ion trap experiments. For the atomic structure of target ion, using the same potential, a Ritz variational method with a trial wave function that contains unknown variational parameters is also proposed for comparison purposes. In this method, an analytical formula of bound energy is derived from the energy equation, and the variational parameters are determined by the minimization of the energy. The energies obtained from our two methods are compared with each other and other theoretical predictions based on the non-relativistic model and found to be in good agreement. For the collision dynamics, the continuum wave function is determined by solving the modified Dirac equations, in which the ECSCP is included for the solution of the continuum orbital. The total cross sections, magnetic sublevel cross sections, and the degrees of linear polarization of emitted x-rays are calculated as a function of the incident energy and screening parameter by using the distorted-wave method in the relativistic frame. Since there are no experiments or other calculations for EIE cross sections on this ion, the obtained results have been compared with those for an isolated ion. The effect of quantum plasma leads to a reduction in the total and magnetic sublevel cross-sections and thus reduces the degrees of linear polarization of emitted x-rays. Comparative analysis allows to estimate the quality of the proposed approximations. Our dynamic model is practicable and easy to apply, which may provide an alternative route towards information with fluorescence polarization in a quantum plasma environment.

S-wave resonances below the Ps($$n=2$$) threshold in positronic sodium interacting with screened Coulomb potentials

S-wave resonance states of the positronic sodium atom ( $$e^+\hbox {Na}$$ ) interacting with screened Coulomb potentials (SCP) are studied. Two different SCPs are tested, namely static screened Coulomb potential (SSCP) and exponential cosine screened Coulomb potential (ECSCP). Resonance states are calculated by the stabilization method. Four S-wave resonance states are found to exist lying below the Ps( $$n=2$$ ) excitation threshold ( $$E_\mathrm{ex}^{\mathrm{Ps}(2)}$$ ). For free atomic system, our reported results of resonance energy compare nicely with the reliable results available in the literature. To the best of our knowledge, our present results will be the first reported results of resonance of positronic–sodium atom under a screened environment.

Stark resonances of a hydrogen-like atom under exponential cosine screened Coulomb potential

We have developed a computational method to solve the complex scaled Schrödinger equation by basis expansion with a set of Laguerre polynomials, which results in the Hamiltonian of a hydrogen-like atom under the exponential cosine screened Coulomb (ECSC) potential imposed by an external electric field becoming a tridiagonal form. We have checked the convergence of computations with respect to the size of basis and stability against the complex scaling parameter in satisfactory results. Furthermore, we have explored the Stark resonances for a hydrogen-like atom under the ECSC potential, the available energies and widths are in agreement with those in the other methods.

Scaling behaviour of Fisher and Shannon entropies for the exponential-cosine screened coulomb potential

ABSTRACTThe scaling laws are given for the entropies in the information theory, including the Shannon's entropy, its power, the Fisher's information and the Fisher–Shannon product, using the exponential-cosine screened Coulomb potential. The scaling laws are specified, in the r-space, as a function of |μ − μc, nℓ|, where μ is the screening parameter and μc, nℓ its critical value for the specific quantum numbers n and ℓ. Scaling laws for other physical quantities, such as energy eigenvalues, the moments, static polarisability, transition probabilities, etc. are also given. Some of these are reported for the first time. The outcome is compared with the available literatures’ results.