Helium-like Research Articles

Theory of X-Ray Anisotropy and Polarization Following the Dielectronic Recombination of Initially Hydrogen-Like Ions

The angular distribution and polarization of the x-ray photoemission of highly charged helium-like ions is studied following the K—LL dielectronic recombination of initially hydrogen-like ions. Calculation is carried out within the framework of the density matrix theory combined with the multiconfiguration Dirac—Fock approach. Attention is paid to magnetic sublevel alignment in the resonant intermediate state and to its nonuniform radiative decay processes. It is shown that the Breit interaction between the incident and target electrons plays a significant role for the alignment of the resonant state and thus causes a substantial change in the x-ray emission characteristic, when compared to the incorporation of only the (non-relativistic) Coulomb interaction. The most prominent difference in alignment parameter is found in the 2s2p1/2 J = 1 resonant state for a wide range of atomic numbers from 9 to 92. For this resonant state of helium-like ions, the Breit interaction becomes significant for ions with nuclear charge Z ∼ 30 already.

The effects of ionization potential depression on the spectra emitted by hot dense aluminium plasmas

Recent experiments at the Linac Coherent Light Source (LCLS) X-ray Free-Electron-Laser (FEL) have demonstrated that the standard model used for simulating ionization potential depression (IPD) in a plasma (the Stewart–Pyatt (SP) model, J.C. Stewart and K.D. Pyatt Jr., Astrophysical Journal 144 (1966) 1203) considerably underestimates the degree of IPD in a solid density aluminium plasma at temperatures up to 200 eV. In contrast, good agreement with the experimental data was found by use of a modified Ecker–Kröll (mEK) model (G. Ecker and W. Kröll, Physics of Fluids 6 (1963) 62–69). We present here detailed simulations, using the FLYCHK code, of the predicted spectra from hot dense, hydrogenic and helium-like aluminium plasmas ranging in densities from 0.1 to 4 times solid density, and at temperatures up to 1000 eV. Importantly, we find that the greater IPDs predicted by the mEK model result in the loss of the n = 3 states for the hydrogenic ions for all densities above ≈0.8 times solid density, and for the helium-like ions above ≈0.65 solid density. Therefore, we posit that if the mEK model holds at these higher temperatures, the temperature of solid density highly-charged aluminium plasmas cannot be determined by using spectral features associated with the n = 3 principal quantum number, and propose a re-evaluation of previous experimental data where high densities have been inferred from the spectra, and the SP model has been used.

The spectral decomposition of the helium atom two-electron configuration in terms of hydrogenic orbitals

The two-electron configuration in the helium atom is known to very high precision. Yet, we tend to refer to this configuration as a 1s↑1s↓ singlet, where the designations refer to hydrogen orbitals. The high precision calculations utilize basis sets that are suited for high accuracy and ease of calculation, but do not really aid in our understanding of the electron configuration in terms of product states of hydrogen orbitals. Since undergraduate students are generally taught to think of helium, and indeed the rest of the periodic table, in terms of hydrogenic orbitals, we present in this paper a detailed spectral decomposition of the two-electron ground state for helium in terms of these basis states. The 1s↑1s↓ singlet contributes less than 93% to the ground state configuration, with other contributions coming from both bound and continuum hydrogenic states.

Ground-state stability and criticality of two-electron atoms with screened Coulomb potentials using the B-splines basis set

We applied the finite-size scaling method using the B-splines basis set to construct the stability diagram for two-electron atoms with a screened Coulomb potential. The results of this method for two-electron atoms are very accurate in comparison with previous calculations based on Gaussian, Hylleraas and finite-element basis sets. The stability diagram for the screened two-electron atoms shows three distinct regions, i.e. a two-electron region, a one-electron region and a zero-electron region, which correspond to stable, ionized and double ionized atoms, respectively. In previous studies, it was difficult to extend the finite-size scaling calculations to large molecules and extended systems because of the computational cost and the lack of a simple way to increase the number of Gaussian basis elements in a systematic way. Motivated by recent studies showing how one can use B-splines to solve Hartree–Fock and Kohn–Sham equations, this combined finite-size scaling using the B-splines basis set might provide an effective systematic way to treat criticality of large molecules and extended systems. As benchmark calculations, the two-electron systems show the feasibility of this combined approach and provide an accurate reference for comparison.

Ground state for two-electron atoms in exponential-cosine-screened Coulomb potentials

We extend the Angular Correlated Configuration Interaction method to two-electron atoms placed in a plasma environment.

Majorana Solutions to the Two-Electron Problem

The two-electron atom is the simplest nontrivial quantum system not amenable to exact solutions. Today, its relevance in the development of quantum mechanics and its pedagogical value within the realm of atomic physics are widely recognized. In this work, an historical review of the known different methods and results devised to study such a problem is presented, with an emphasis to the calculations of the ground state energy of helium. Then we discuss several, related, unpublished results obtained around the same years by Ettore Majorana, which remained unknown till recent times. Among them a general variant of the variational method appears to be particularly interesting, even for current research in atomic and nuclear physics: it takes directly into account, already in the trial wavefunction, the action of the full Hamiltonian operator of a given quantum system. Further relevant contributions, specialized to the two-electron problem, include the introduction of the remarkable concept of an effective nuclear charge different for the two electrons (thus generalizing previous known results) and an application of the perturbative method, where the atomic number Z was treated effectively as a continuous variable. Finally a survey of results, relevant mainly for pedagogical reasons, is given; in particular we focus on simple broad range estimates of the helium ionization potential, obtained by suitable choices for the wavefunction, as well as on a simple alternative to Hylleraas’ method, which led Majorana to first order calculations comparable in accuracy with well-known order 11 results derived, in turn, by Hylleraas.

Testing Three-Body Quantum Electrodynamics with TrappedTi20+Ions: Evidence for aZ-dependent Divergence Between Experiment and Calculation

We report a new test of quantum electrodynamics (QED) for the w (1s2p(1)P(1)→1s(2)(1)S(0)) x-ray resonance line transition energy in heliumlike titanium. This measurement is one of few sensitive to two-electron QED contributions. Systematic errors such as Doppler shifts are minimized in our experiment by trapping and stripping Ti atoms in an electron beam ion trap and by applying absolute wavelength standards to calibrate the dispersion function of a curved-crystal spectrometer. We also report a more general systematic discrepancy between QED theory and experiment for the w transition energy in heliumlike ions for Z>20. When all of the data available in the literature for Z=16-92 are taken into account, the divergence is seen to grow as approximately Z(3) with a statistical significance on the coefficient that rises to the level of 5 standard deviations. Our result for titanium alone, 4749.85(7) eV for the w line, deviates from the most recent ab initio prediction by 3 times our experimental uncertainty and by more than 10 times the currently estimated uncertainty in the theoretical prediction.

Angular distributions for two-photon double ionization of lithium

We present angular distributions for two-photon double ionization of lithium at photon energies of 50 eV ($\ensuremath{\lambda}$ $=$ 25 nm) and 59 eV ($\ensuremath{\lambda}$ $=$ 21 nm). The results are obtained from full-dimensional solution of the two-active-electron time-dependent Schr\odinger equation using the time-dependent close-coupling method. We investigate two different double ionization mechanisms. First, we consider direct double ionization of the Li ground state following the absorption of two photons. Secondly, we consider an initial photoexcitation of the $1s2s2p$ doubly excited state, followed by photoionization of the $2s$ and $2p$ electrons. We find significant differences between the angular distributions obtained for these two distinct processes. We also compare the characteristics of the angular distributions for Li with those of other two-electron atoms.

Atomic Properties of the Two-Electron System using Variational Monte Carlo Technique

Using variational Monte Carlo method we calculated the lowest order relativistic corrections for the ground state energies of the helium-like atoms, up to Z = 10, and also for some excited state energies of the helium atom. These relativistic corrections include: mass velocity e ect, orbit orbit interaction, spin magnetic and dipole moments of the two electrons and the Darwin e ect. Moreover, correction due to the nucleus motion has been also calculated. Our results were obtained by using two new types of compact and accurate trial wave functions for the helium ions. For excited states we used simple trial wave functions of good quality and accurate results. The obtained results are in good agreement with the most recent previous accurate values and also with the exact values.

Variability in X-ray line ratios in helium-like ions of massive stars: the radiation-driven case

Line ratios in "fir" triplets of helium-like ions have proven to be a powerful diagnostic of conditions in X-ray emitting gas surrounding massive stars. Recent observations indicate that these ratios can be variable with time. The possible causes of variation in line ratios are limited: changes in the radiation field or changes in density, and changes in mass-loss or geometry. In this paper, we investigate the ability of changes in the radiation field to induce variability in the ratio R=f/i. To isolate the radiative effect, we use a heuristic model of temperature and radius changes in variable stars in the B and O range with low-density, steady-state winds. We model the changes in emissivity of X-ray emitting gas close to the star due to differences in level-pumping from available UV photons at the location of the gas. We find that under these conditions, variability in R is dominated by the stellar temperature. Although the relative amplitude of variability is roughly comparable for most lines at most temperatures, detectable variations are limited to a few lines for each spectral type. We predict that variable values in R due to stellar variability must follow predictable trends found in our simulations. Our model uses radial pulsations as a mode of stellar variability that maximizes the amplitude of variation in R. This model is robust enough to show which ions will provide the best opportunity for observing variability in the f/i ratio at different stellar temperatures, and the correlation of that variability with other observable parameters. In real systems, the effects would be more complex than in our model, with differences in phase and suppressed amplitude in the presence of non-radial pulsations. This suggests that changes in R across many lines concurrently are not likely to be produced by a variable radiation field.

Correlated kinetic energy density functional of ground states of harmonically confined two-electron atoms for arbitrary interparticle interaction

Utilizing the earlier work of Holas et al (2003 Phys. Lett. A 310 451) and the more recent contribution of Akbari et al (2009 Phys. Rev. A 80 032509), we construct an integral equation for the relative motion (RM) contribution tRM(r) to the correlated kinetic energy density for modelling two-electron atoms with harmonic confinement but arbitrary interparticle interaction. It is stressed that tRM = tRM[f(G)], where f(G) is the atomic scattering factor: the Fourier transform of the density ρ(r). As a simple illustrative example of this functional relation for the correlated kinetic energy density, the harmonic Moshinsky case is investigated, the scattering factor then having a Gaussian form.

Analytic structure of ground-state energies and wave functions for the inhomogeneous electron liquid in non-relativistic He-like atomic ions with nuclear charge Ze

The ground state of the He atom for fixed nucleus remains intractable so far as regards exact analytic solutions. However, some important results already exist pertaining to its ground-state wave function Ψ and corresponding electron density n(r). Here, we extend the existing studies by focussing attention on the non-relativistic series of He-like atomic ions with nuclear charge Z. We then find it instructive to start from the energy E(Z) of such a two-electron spin-compensated problem. This is known to have non-analytic behaviour at a critical Z, say Z c , equal to 0.911028. A form of Darboux transformation going back at very least to Brändas and Goscinski [E. Brändas and O. Goscinski, Int. J. Quantum Chem. 6, 59 (1972)] is refined somewhat here, and compared with a more intuitive approach of Callan [E. Callan, Int. J. Quantum Chem. 6, 431 (1972)]. The important 1/Z expansion of E(Z) is also invoked. The electron density n(r) and the ground-state wave function Ψ are then treated in turn, in a related manner; especially their asymptotic behaviour far from the nucleus. Finally, two exact wave functions for analytically solvable two-Fermion models are shown to sum the infinite series proposed by Fock [V. Fock, Izv. Akad. Nauk SSSR, Ser. Fiz. 18, 161 (1954)].

The effects of the Pauli exclusion principle in determining the ionization energies of the helium atom and helium-like ions

For helium and helium-like ions, we have examined the differences between the values of the ionization energies as calculated from the Bohr theory and those measured in experiments. We find that these differences vary linearly with the atomic number of the system. Using this result, we show how the Bohr model for single-electron systems may be modified in a simple way so that it may be applied to two-electron systems as well.

Real-Time Observation of Interference between Atomic One-Electron and Two-Electron Excitations

We present results of real-time tracking of atomic two-electron dynamics in an autoionizing transient wave packet in krypton. A coherent superposition of two Fano resonances is excited with a femtosecond extreme-ultraviolet pulse. The evolution of the corresponding wave packet is subsequently probed with a delayed infrared pulse. In our specific case, we get access to the interference between one- and two-electron excitation channels in the launched wave packet, which is superimposed on its decay through autoionization. A simple model is able to account for the observed dynamical evolution of this wave packet.

Alternative route to Bose-Einstein condensation of two-electron atoms

We present a novel route to Bose-Einstein condensation devised for two-electron atoms, which do not admit practicable cooling techniques based upon narrow intercombination lines. A dipole trap for $^{40}$Ca atoms in the singlet ground state is loaded from a moderately cold source of metastable triplet atoms via spatially and energetically selective optical pumping permitting four orders of magnitude increase of the phase space density. Further cooling to quantum degeneracy is achieved by forced evaporation optimized to minimize three-body losses. In a combined loading and evaporation cycle of less than three seconds we are able to condense 3000 atoms.

Finite-size scaling for quantum criticality using the finite-element method

Finite size scaling for the Schrödinger equation is a systematic approach to calculate the quantum critical parameters for a given Hamiltonian. This approach has been shown to give very accurate results for critical parameters by using a systematic expansion with global basis-type functions. Recently, the finite-element method was shown to be a powerful numerical method for ab initio electronic-structure calculations with a variable real-space resolution. In this work, we demonstrate how to obtain quantum critical parameters by combining the finite-element method (FEM) with finite size scaling (FSS) using different ab initio approximations and exact formulations. The critical parameters could be atomic nuclear charges, internuclear distances, electron density, disorder, lattice structure, and external fields for stability of atomic, molecular systems and quantum phase transitions of extended systems. To illustrate the effectiveness of this approach we provide detailed calculations of applying FEM to approximate solutions for the two-electron atom with varying nuclear charge; these include Hartree-Fock, local density approximation, and an "exact" formulation using FEM. We then use the FSS approach to determine its critical nuclear charge for stability; here, the size of the system is related to the number of elements used in the calculations. Results prove to be in good agreement with previous Slater-basis set calculations and demonstrate that it is possible to combine finite size scaling with the finite-element method by using ab initio calculations to obtain quantum critical parameters. The combined approach provides a promising first-principles approach to describe quantum phase transitions for materials and extended systems.

ChemInform Abstract: The Structure and Reactivity of Iron Nitride Complexes

AbstractReview: 67 refs.

Series expansions for an exact two-electron wave function in terms of Löwdin's renormalized natural orbitals

In recent developments on the pair density needed to treat the non-Hartree-Fock-like part of interparticle repulsion, the natural orbitals and sign-correct expansion coefficients play a central role. Since, in principle, an infinite number of natural orbitals must be included, the convergence of expectation values due to finite-term approximations is an important issue. Here we discuss quantitatively this convergence problem based on an exactly solvable two-electron model atom, where the Schr\"odinger wave function for the ground state is expressible in terms of L\"owdin's natural orbitals and sign-correct expansion coefficients. Using properly renormalized truncated series expansions for such an exact decomposition, the corresponding expectation values of the Schr\"odinger Hamiltonian are calculated analytically. A rapid and uniform convergence is found in these expectation values at given values of the coupling in the interparticle repulsion.

Effects of photon momentum in nonrelativistic(γ,2e)processes

We study the effects of nonzero photon momentum on the triply differential cross section (TDCS) for ($\ensuremath{\gamma},2e$) processes. Due to the low value of the photon momentum, these effects are weak and manifest only in special kinematical conditions such as the back-to-back emission of the electrons with equal energy sharing. Helium and a few light heliumlike ions are treated in detail. Quite unexpectedly, the magnitude of these effects is maximal for relatively small photon energies. However, although this effect on the TDCS remains rather small, of the order of a few mb eV${}^{\ensuremath{-}1}$ sr${}^{\ensuremath{-}2}$, it is sufficient to be observed experimentally.

Effect of strongly coupled plasma on the doubly excited states of heliumlike ions

The effect of strongly coupled plasma occurring in astrophysical context has been studied for the first time to estimate the energy levels of the autoionizing states of highly stripped astrophysically important ions Al11+, Si12+, P13+, S14+ and Cl15+ and also C4+ isoelectronic to helium. The transition energies corresponding to 1s 2:1Se → 2s 2:1Se, 2p 2:1De, 2s2p:1Po, 2s3d:1De and 2p3d:1Fo are analyzed with respect to different plasma densities using the ion sphere (IS) model of strongly coupled plasma. Transition energies are obtained from the position of the poles of a variational functional based on frequency dependent perturbation calculation probing the collective oscillation modes of the plasma embedded two electron ions. For the free ions corresponding to zero plasma coupling our calculated data agree well with those obtained from spectroscopic data while for the plasma embedded ions the data are new but follow systematic trend. The work has been performed in the domain of linear response theory. The analytical wave function of the doubly excited states have been obtained and may be useful for calculating various transition properties of the plasma embedded ions and also for estimating the rate coefficients for dielectronic recombination processes which play a major role in maintaining equilibrium in high temperature astrophysical or laser produced plasmas.