Continuum State Wave Functions Research Articles

A Study of the Atomic Processes of Highly Charged Ions Embedded in Dense Plasma

The study of atomic spectroscopy and collision processes in a dense plasma environment has gained a considerable interest in the past few years due to its several applications in various branches of physics. The multiconfiguration Dirac-Fock (MCDF) method and relativistic configuration interaction (RCI) technique incorporating the uniform electron gas model (UEGM) and analytical plasma screening (APS) potentials have been employed for characterizing the interactions among the charged particles in plasma. The bound and continuum state wavefunctions are determined using the aforementioned potentials within a relativistic Dirac-Coulomb atomic structure framework. The present approach is applied for the calculation of electronic structures, radiative properties, electron impact excitation cross sections and photoionization cross sections of many electron systems confined in a plasma environment. The present study not only extends our knowledge of the plasma-screening effect but also opens the door for the modelling and diagnostics of astrophysical and laboratory plasmas.

Elastic scattering of electrons by water: An ab initio study

In this work we devise a theoretical and computational method to compute the elastic scattering of electrons from a non-spherical potential, such as in the case of molecules and molecular aggregates. Its main feature is represented by the ability of calculating accurate wave functions for continuum states of polycentric systems via the solution of the Lippmann-Schwinger equation, including both the correlation effects and multi-scattering interference terms, typically neglected in widely used approaches, such as the Mott theory. Within this framework, we calculate the purely elastic scattering matrix elements. As a test case, we apply our scheme to the modelling of electron-water elastic scattering. The Dirac-Hartree-Fock self-consistent field method is used to determine the non-spherical molecular potential projected on a functional space spanned by Gaussian basis set. By adding a number of multi-centric radially-arranged s-type Gaussian functions, whose exponents are system-dependent and optimized to reproduce the properties of the continuum electron wave function in different energy regions, we are able to achieve unprecedented access to the description of the low energy range of the spectrum (0.001 < E < 10 eV) up to keV, finding a good agreement with experimental data and previous theoretical results. To show the potential of our approach, we also compute the total elastic scattering cross section of electrons impinging on clusters of water molecules and zundel cation. Our method can be extended to deal with inelastic scattering events and heavy-charged particles.

Study of electron impact excitation of H-like Si13+ ion in dense plasma environment

The relativistic configuration interaction technique has been employed to study the electron impact excitation collision strengths of H-like Si13+ ion in strongly coupled plasma environment. The analytical plasma screening b-potential has been adopted to determine the physical effects of the screened Coulomb interaction due to the influence of plasma on the atomic structure and dynamics affecting atomic processes. The modified Dirac equations are solved to obtain the bound and continuum state wave functions. We examine the plasma screening effect by considering the plasma conditions at the electron temperatures of 200, 600 and 1000 eV over electron density range of 2.0 × 1023 - 1.0 × 1024 cm−3. The present results highlight that the inclusion of plasma environment induces significant effect on the binding energies, transition energies, radiative transition probabilities, oscillator strengths and collision strengths. It has been observed that the collision strength decreases with increase in plasma electron density for a fixed temperature. Our results will be beneficial for laser produced plasmas, high energy density physics, astrophysical applications and will open the door to modelling and advanced diagnostics of solar and stellar plasmas.

Influence of strongly coupled plasma environment on photoionization of H-like O7+ion

The effect of strongly coupled plasma environment on the photoionization cross sections of the H-like O7+ ion is studied within the framework of relativistic configuration interaction technique implemented in the flexible atomic code. The analytical b-potential of Li and Rosmej [Phys. Lett. A 384, 126478 (2020)] and uniform electron gas model potential of Saha and Fritzsche [J. Phys. B 40, 259 (2007)] have been employed for incorporating the plasma shielding effects. The level energies for unscreened H-like O7+ ion are in excellent agreement with the National Institute of Standards and Technology values. The bound and continuum state wavefunctions are determined by solving the modified Dirac equations. The photoionization cross sections evaluated in the dipole approximation are obtained from the bound-free transition matrix element. Cross section calculations are performed over an electron density range of 1.0 × 1019–2.0 × 1023 cm−3 and a temperature range of 200–1000 eV. The plasma environment is found to considerably influence the photoionization cross sections. It has been observed that the ionization thresholds move to the low energy region with the increase in free electron densities. The present results will be beneficial for the interpretation of spectra observed in a variety of astrophysical and laboratory plasmas.

Photoionization of H-like C[formula omitted] ion in the presence of a strongly coupled plasma environment

We have developed a model (namely EPPP) within the relativistic framework to calculate the photoionization process of highly charged ions in the presence of a strongly coupled plasma environment. The ion-sphere (IS) potential, including the screening on nuclear charge and the effect of confinement due to the neighbouring ions, is used to incorporate the plasma effects. Energies and wave functions of the (bound) continuum states are determined by solving the modified Dirac equations. As an application, we determine the bound and continuum state wave functions, energy levels, and transition parameters for plasma density ranges of Ne=1020–1024 cm−3, taking the H-like C5+ ion as an illustration example. The photoionization cross sections in the dipole approximation are calculated through the bound-free transition matrix elements. In addition, a theoretical attempt, based on the multiconfiguration Dirac-Fock approximation incorporating the same potential is presented for the atomic structure to serve as an independent check of the EPPP values and the results show fairly good agreement with the EPPP ones. Comparison of our calculated data with the results of other authors, when available, is made. Some interesting behaviors of the respective properties concerning the screening strength are noted. The present results are useful for line identification, plasma modeling and diagnostics in laser-produced plasmas and astrophysical plasmas.

(e, 2e) simple ionization of by fast electron impact: use of triangular three-center continuum and bound state wave functions

The variation of the multiply differential cross section of the (e, 2e) simple ionization of , with the incident and ejection energy values, as well as the directions of the ejected and scattered electrons, is studied. The calculations have been performed in the frame of the perturbative first Born procedure, which has required the development of equilateral triangular three center bound and continuum state wave functions. The results explore the optimal conditions and the particularities of the triangular targets, such as the appearance of interference patterns in the variation of the four fold differential cross section (FDCS) with the scattering angle for a fixed orientation of the molecule. The comparison between the results obtained by two H3+ ground wave functions, with and without a correlation term r12, shows that the effect of correlation on the magnitude of the triple differential cross section is not large, but it produces some modification in the structure of the FDCS.

Transition energies and polarizabilities of hydrogen like ions in plasma

Effect of plasma screening on various properties like transition energy, polarizability (dipole and quadrupole), etc. of hydrogen like ions is studied. The bound and free state wave functions and transition matrix elements are obtained by numerically integrating the radial Schrodinger equation for appropriate plasma potential. We have used adaptive step size controlled Runge-Kutta method to perform the numerical integration. Debye-Huckel potential is used to investigate the variation in transition lines and polarizabilities (dipole and quadrupole) with increasing plasma screening. For a strongly coupled plasma, ion sphere potential is used to show the variation in excitation energy with decreasing ion sphere radius. It is observed that plasma screening sets in phenomena like continuum lowering and pressure ionization, which are unique to ions in plasma. Of particular interest is the blue (red) shift in transitions conserving (non-conserving) principal quantum number. The plasma environment also affects the dipole and quadrupole polarizability of ions in a significant manner. The bound state contribution to polarizabilities decreases with increase in plasma density whereas the continuum contribution is significantly enhanced. This is a result of variation in the behavior of bound and continuum state wave functions in the presence of plasma. We have compared the results with existing theoretical and experimental data wherever present.

Precision calculation of above-threshold multiphoton ionization in intense short-wavelength laser fields: The momentum-space approach and time-dependent generalized pseudospectral method

We present an approach in momentum ($\mathcal{P}$) space for the accurate study of multiphoton and above-threshold ionization (ATI) dynamics of atomic systems driven by intense laser fields. In this approach, the electron wave function is calculated by solving the $\mathcal{P}$-space time-dependent Schr\odinger equation (TDSE) in a finite $\mathcal{P}$-space volume under a simple zero asymptotic boundary condition. The $\mathcal{P}$-space TDSE is propagated accurately and efficiently by means of the time-dependent generalized pseudospectral method with optimal momentum grid discretization and a split-operator time propagator in the energy representation. The differential ionization probabilities are calculated directly from the continuum-state wave function obtained by projecting the total electron wave function onto the continuum-state subspace using the projection operator constructed by the continuum eigenfunctions of the unperturbed Hamiltonian. As a case study, we apply this approach to the nonperturbative study of the multiphoton and ATI dynamics of a hydrogen atom exposed to intense short-wavelength laser fields. High-resolution photoelectron energy-angular distribution and ATI spectra have been obtained. We find that with the increase of the laser intensity, the photoelectron energy-angular distribution changes from circular to dumbbell shaped and is squeezed along the laser field direction. We also explore the change of the maximum photoelectron energy with laser intensity and strong-field atomic stabilization phenomenon in detail.

Dipole approximation in theL2,3electron excited spectra in3dtransition metals

A theoretical model based on the autoionization and characteristic decay processes following electron impact ionization of a core electron in solids that has previously been used in calculating electron-energy-loss spectra of transition metals near the $3p$-excitation edge has been extended to the $2p$-excitation edge for $^{21}\text{S}\text{c}$ through $^{27}\text{N}\text{i}$ as well. In the first set of calculations, magnetic effects were ignored and the relative scattering intensity was formulated in terms of the electrostatic interaction $U(p,d)$ between the $3p$ and $3d$ electrons of the intermediate resonant configuration state ${p}^{5}{d}^{n+1}$, using many-body perturbation theory that led to a generalized Fano-type formula for the intensity profiles. In the second set of calculations in which magnetic effects were included as well, an analysis based on the Bethe-Born formalism of inelastic scattering of electrons on atoms was used. The nature of the relative magnitudes of $U(p,d)$ and the spin-orbit parameters ${\ensuremath{\varsigma}}_{3p}$ and ${\ensuremath{\varsigma}}_{3d}$ and the localized nature of the $3p$ state necessitated the diagonalization of the intermediate configuration state ${p}^{5}{d}^{n+1}$ to determine the multiplet splitting and their corresponding intensities in the $LS$-coupling limit using fractional parentage scheme. The nonrelativistic multiconfiguration Hartree-Fock (MCHF) code was used in determining the ground and continuum state wave functions, and the itinerant $3d$ states in the solid were approximated with an atomic MCHF-wave function. The outline above is applied to the $2p$-excitation edge, except that because of the relative magnitudes of $U(p,d)$, ${\ensuremath{\varsigma}}_{2p}$, and ${\ensuremath{\varsigma}}_{3d}$, it is found that $LK$ coupling is suitable for Sc, Ti, and V, while $jK$ coupling is appropriate for Cr to Ni when it comes to the diagonalization of the configuration ${p}^{5}{d}^{n+1}$ to determine the multiplet splitting and their associated scattering intensities. In the dipole approximation, the scattering intensities separate into two distinct manifolds that arise from the ${p}_{3/2}$ and ${p}_{1/2}$ states. The branching ratios of the white lines are extracted from the spectra and compared with x-ray-absorption spectra.

E1-E2interference in the Coulomb dissociation of8B

We investigate the effects arising out of the $E1 - E2$ interference in the Coulomb dissociation of $^8$B at beam energies below and around 50 MeV/nucleon. The theory has been formulated within a first order semiclassical scheme of Coulomb excitation, in which both the ground state and the continuum state wave functions of $^8$B enter as inputs. We find that the magnitude of the interference could be large in some cases. However, there are some specific observables which are free from the effects of the $E1 - E2$ interference, which is independent of the models used to describe the structure of $^8$B. This will be useful for the analysis of the breakup data in relation to the extraction of the astrophysical factor $S_{17}(0)$.

Ionization Cross Sections of Hydrogen-like Atoms under Heavy Particle Impact

Cross sections for the ionization of hydrogen-like ions (He+, Li2+, Be3+, B4+, C5+) by impact of bare ions are determined in the energy range of 10 to 2000 keV/amu, using a continuum state wave function which incorporates the distortion due to the Coulomb fields of both the projectile and the residual target. Total and differential cross sections are calculated and compared with other existing theoretical findings as well as with experimental data.

Ionization of Atomic Oxygen in Collision with Proton and Alpha Particle

Ionization of atomic oxygen by impact of protons and alpha particles is studied in intermediate and high energy regions. The target oxygen atom is treated as having only one active electron and the interaction of the active electron with the target core and the passive electron(s) is described by a model potential. The continuum state wave function is represented by the product of two Coulomb wave functions, which considers the distortion due to the projectile and the residual target core nearly on equal footing

Impact parameters for ionization by high-energy electrons

Root-mean-square impact parameters for K- and L-shell ionization by electrons with energies between 60 and 400 keV, and pertinent to energy dispersive X-ray (EDX) analysis, are calculated for a wide range of atoms. Hartree–Fock wave functions with relativistic corrections are used for the bound states and Hartree–Slater wave functions for continuum states. These impact parameters differ substantially from those given by widely used approximate formulas. Impact parameters for K- and L-shell ionization fall on a single curve, suggesting that the impact parameters are a single-valued function of the ionization threshold energy for a given incident beam energy. For the range of incident beam energies considered, impact parameters for threshold energies up to 45 keV are given in a parameterized form.

Ionization of atoms by charged particles

Ionization of hydrogen atom by charged particle impact are studied at different collisional energies and the total and differential cross sections are calculated. In case of light particle impact the final-state wave function here considers all three two-body interactions on an equal footing and satisfies the exact Coulomb boundary conditions. The spin asymmetries are also found and the values are compared with other existing results. For heavy particle impact a final continuum state wave function which incorporates distortion due to the Coulomb fields of both the projectile and the target nuclei is employed. In this case the target hydrogen atom is considered in its ground as well as metastable 2s state. The results thus obtained are compared with the existing experimental findings as well as other theoretical predictions.

Electronic correlation in the ionization of molecular hydrogen.

Cross sections for double ionization and ionization plus excitation of ${\mathrm{H}}_{2}$ by the high-energy charged particles are calculated as a function of the orientation of the ${\mathrm{H}}_{2}$ internuclear axis. The contributions of the double collision, the shake-off or shake-up, and ground-state correlation mechanisms have been considered. The calculated cross sections have been compared with the data of Edwards et al. [Phys. Rev. A 42, 1367 (1990); 44, 797 (1991); 46, 6970 (1992)]. Our results for double ionizations are acceptable, while the ionization-excitation cross sections have a strong dependence on the bound-state and continuum-state wave functions.

Photoionization of two-electron atoms via the hyperspherical artificial-channel method: Application to H- and He.

An application of the artificial-channel method with hyperspherical coordinates to the calculation of photoionization cross sections of two-electron systems is presented. The method yields ``bound-free'' transition amplitudes and bound-state energies of any two-electron system. It obviates the separate generation of bound- and continuum-state wave functions by calculating directly the desired amplitudes. As applications, the photodetachment cross section of the ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ ion and the photoionization cross section of the He atom have been calculated. The computations are compared with other theoretical calculations and with experiments. The results obtained are in excellent agreement with experimental results for all the energies considered. Of equal significance is the close agreement, attesting to the high accuracy of the method, between results obtained by using the length form for the field-matter interaction and those obtained with the acceleration form.

Effects of an electric field on the continuum energy levels in InxGa1-xAs/GaAs quantum wells terminated with thin cap layers.

Electroreflectance was used to study the influence of an electric field on the bound and the continuum states of In x Ga 1-x As/GaAs quantum-well structures of various well widths, terminated by thin cap layers. Optical transitions involving only bound states were not significantly affected for the cap-layer thicknesses studied, and are Stark shifted as expected by the electric field. However, strong oscillations are observed in the spectra at energies larger than the barrier band gap, and are due to quantum interference in the continuum state wave functions which is related to the finite size of the cap layer. These oscillations, which do not follow a Franz-Keldysh relation, shift linearly with the electric field

Oscillatory behavior of the continuum states in InxGa1-xAs/GaAs quantum wells due to capping-barrier layers of finite size.

Photoreflectance spectra of ${\mathrm{In}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As/GaAs single quantum wells have shown above-barrier oscillations, of amplitude comparable to those of transitions involving only bound states. The spacing between the oscillation extrema increases with energy according to the relation ${\mathit{E}}_{\mathit{n}}$=${\mathit{E}}_{0}$+\ensuremath{\alpha} ${\mathit{n}}^{2}$ for the nth extremum. These results demonstrate experimentally that the continuum states are influenced by the finite extent of the barrier on the side of the well which terminates the device (the cap layer). The high potential at the surface of the cap and the finite size of the cap layer produce quantum interference in the continuum state wave functions which leads to oscillations in the probabilities of finding the carriers in the various regions of the structure. The modulation is expected to be determined mainly by ${\mathrm{sin}}^{2}$(${\mathit{k}}_{\mathit{b}}$t), where t is the thickness of the cap layer and ${\mathit{k}}_{\mathit{b}}$ is the wave vector in the barrier, and includes the carrier effective mass. Depending on the value of t, these cap-related oscillations are observed for the various types of carriers in photoreflectance experiments; their visibility depends on the well width.

Triple-differential cross sections for the electron- and positron-impact ionization of helium in an improved second Born approximation.

The triple-differential cross sections for the electron-impact ionization of helium in coplanar asymmetric geometry have been calculated by use of a second Born approximation which employs an improved choice for the target continuum-state wave function. The results are compared with the absolute experimental data of K. Jung et al. [J. Phys. B 18, 2955 (1985)] at an incident energy of 600 eV and are found to be in very good agreement with them. The corresponding results for the positron-impact case are also presented.

Triple-differential cross sections for the ionization of helium by fast electrons.

The triple-differential cross section for the single ionization of helium in the coplanar asymmetric geometry has been calculated by using a second Born approximation which employs an improved choice of the target continuum-state wave function. The contribution of still higher-order terms of the scattering amplitude is also considered via the usual Glauber approximation. The results are compared with the recent data of Jung et al. at an incident energy of 600 eV and are found to be in very good agreement with them.