Low Nuclear Charge Research Articles

Variational vs perturbative relativistic energies for small and light atomic and molecular systems.

Variational and perturbative relativistic energies are computed and compared for two-electron atoms and molecules with low nuclear charge numbers. In general, good agreement of the two approaches is observed. Remaining deviations can be attributed to higher-order relativistic, also called non-radiative quantum electrodynamics (QED), corrections of the perturbative approach that are automatically included in the variational solution of the no-pair Dirac-Coulomb-Breit (DCB) equation to all orders of the α fine-structure constant. The analysis of the polynomial α dependence of the DCB energy makes it possible to determine the leading-order relativistic correction to the non-relativistic energy to high precision without regularization. Contributions from the Breit-Pauli Hamiltonian, for which expectation values converge slowly due the singular terms, are implicitly included in the variational procedure. The α dependence of the no-pair DCB energy shows that the higher-order (α4Eh) non-radiative QED correction is 5% of the leading-order (α3Eh) non-radiative QED correction for Z = 2 (He), but it is 40% already for Z = 4 (Be2+), which indicates that resummation provided by the variational procedure is important already for intermediate nuclear charge numbers.

Electric Double‐Layer Capacitors Based on Non‐Aqueous Electrolytes: A Comparative Study of Potassium and Quaternary Ammonium Salts

AbstractImmense attention has been drawn towards electric double layer capacitors (EDLCs) as a viable intermittent energy storage solution, owing to their ultra‐fast charge/discharge rates and long cycle life. However, the high activation energy of ionic conductivity innate in conventional aprotic organic electrolytes has greatly impeded the feasibility of high‐performance EDLCs. Herein, we investigate and compare the physicochemical properties and performance of electrolytes based on potassium hexafluorophosphate (KPF6) and commercial triethylmethy‐lammonium tetra‐fluoroborate (TEMABF4) salts in EDLCs. Compared to commercial TEMABF4‐based electrolytes, KPF6 salt‐based electrolytes (in acetonitrile solvent) demonstrate outstanding rate performance. The KPF6 salt‐based electrolyte further manifests lower ionic resistance within activated carbon pores as well as lower interfacial resistance between electrode and electrolyte; an affirmation of not the high ionic conductivity but rather the eminence of low activation energy. The low activation energy can be attributed to the low effective nuclear charge of the K+ cations that allow anions to transverse easily in solvent state. This study not only underpins potassium‐ion (K+) as a fast charge carrier, but also a viable solution for the next‐generation non‐aqueous power devices relying on monovalent alkali cations.

Atomic-scale evidence for displacive disorder in bismuth zinc niobate pyrochlore

Pyrochlores characterized by the chemical formula A2B2O7 form an extended class of materials with interesting physical and chemical properties. The compound Bi1.5ZnNb1.5O7 is prototypical. Its excellent dielectric properties make it attractive, e.g. for capacitors, tunable microwave devices and electric-energy storage equipment. Bi1.5ZnNb1.5O7 shows an intriguing frequency-dispersive dielectric relaxation at 50 K ≤ T ≤ 250 K, which has been studied intensively but is still not fully understood. In this first study on a pyrochlore by atomic-resolution transmission electron microscopy we observe the Bi atoms on A sites since, due to their low nuclear charge, the contribution of Zn atoms to the contrast of these sites is negligible. We find in our [1¯00]and [112] oriented images that the position of the atomic intensity maxima do not coincide with the projected Wyckoff positions of the basic pyrochlore lattice. This supplies atomic-scale evidence for displacive disorder on split A-type sites. The Bi atoms are sessile, only occasionally we observe in time sequences of images jumps between individual split-site positions. The apertaining jump rate of the order of 0.1–1 Hz is by ten orders of magnitude lower than the values derived in the literature from Arrhenius plots of the low-temperature dielectric relaxation data. It is argued that these jumps are radiation induced. Therefore our observations are ruling out a contribution of Bi-atom jumps to low-temperature dielectric A sites-related relaxation. It is suggested that this relaxation is mediated by jumps of Zn atoms.

Atomic physics constraints on the X boson

Recently, a peak in the light fermion pair spectrum at invariant q^2 approximately equal to 16.7 Me^2. has been observed in the bombardment of Li-7 by protons. This peak has been interpreted in terms of a protophobic interaction of fermions with a gauge boson (X boson) of invariant mass of approximately 16.7 MeV which couples mainly to neutrons. High-precision atomic physics experiments aimed at observing the protophobic interaction need to separate the X boson effect from the nuclear-size effect, which is a problem because of the short range of the interaction 11.8 fm, which is commensurate with a "nuclear halo". Here, we analyze the X boson in terms of its consequences for both electronic atoms as well as muonic hydrogen and deuterium. We find that the most promising atomic systems where the X boson has an appreciable effect, distinguishable from a finite-nuclear-size effect, are muonic atoms of low and intermediate nuclear charge numbers.

Hartree-Fock implementation using a Laguerre-based wave function for the ground state and correlation energies of two-electron atoms.

An implementation of the Hartree-Fock (HF) method using a Laguerre-based wave function is described and used to accurately study the ground state of two-electron atoms in the fixed nucleus approximation, and by comparison with fully correlated (FC) energies, used to determine accurate electron correlation energies. A variational parameter A is included in the wave function and is shown to rapidly increase the convergence of the energy. The one-electron integrals are solved by series solution and an analytical form is found for the two-electron integrals. This methodology is used to produce accurate wave functions, energies and expectation values for the helium isoelectronic sequence, including at low nuclear charge just prior to electron detachment. Additionally, the critical nuclear charge for binding two electrons within the HF approach is calculated and determined to be ZHFC=1.031 177 528.This article is part of the theme issue 'Modern theoretical chemistry'.

Interaction of octahedral Mg(II) and tetrahedral Al(III) substitutions in aluminium-rich dioctahedral smectites

Smectites are clay minerals of interest due to their natural availability, extraordinary surface properties, and high swelling capacity in the presence of water. The behaviour of smectites and other layered phyllosilicates is largely affected by the appearance of isomorphous substitutions with a lower nuclear charge in their layer structures. These substitutions are the origin of negative net layer charge, and their distribution in the mineral layer impacts on the electrostatic properties of the mineral surfaces. The present density functional theory study shows that octahedral Mg(II) and tetrahedral Al(III) substitutions in aluminium-rich dioctahedral smectites tend to avoid each other and disperse in the layer structure. This behaviour is explained and demonstrated to originate from geometry deformations around the substitutions which are larger than the unsubstituted species.

The splitting of atomic orbitals with a common principal quantum number revisited: np vs. ns

Atomic orbitals with a common principal quantum number are degenerate, as in the hydrogen atom, in the absence of interelectronic repulsion. Due to the virial theorem, electrons in such orbitals experience equal nuclear attractions. Comparing states of several-electron atoms that differ by the occupation of orbitals with a common principal quantum number, such as 1s(2) 2s vs. 1s(2) 2p, we find that although the difference in energies, ΔE, is due to the interelectronic repulsion term in the Hamiltonian, the difference between the interelectronic repulsions, ΔC, makes a smaller contribution to ΔE than the corresponding difference between the nuclear attractions, ΔL. Analysis of spectroscopic data for atomic isoelectronic sequences allows an extensive investigation of these issues. In the low nuclear charge range of pertinent isoelectronic sequences, i.e., for neutral atoms and mildly positively charged ions, it is found that ΔC actually reverses its sign. About 96% of the nuclear attraction difference between the 6p (2)P and the 6s (2)S states of the Cs atom is cancelled by the corresponding interelectronic repulsion difference. From the monotonic increase of ΔE with Z it follows (via the Hellmann-Feynman theorem) that ΔL > 0. Upon increasing the nuclear charge along an atomic isoelectronic sequence with a single electron outside a closed shell from Z(c), the critical charge below which the outmost electron is not bound, to infinity, the ratio ΔC/ΔL increases monotonically from lim(Z→Z(c)(+))ΔC/ΔL=-1 to lim(Z→∞)ΔC/ΔL=1. These results should allow for a more nuanced discussion than is usually encountered of the crude electronic structure of many-electron atoms and the structure of the periodic table.

On the role of atomic metastability in the production of Balmer line radiation from ‘cold’ atomic hydrogen, deuterium and hydrogenic ion impurities in fusion edge plasmas**This manuscript is dedicated to the memory of my esteemed colleague Dr. rer. nat. Manfred Korten (1940–2010).

Published arguments, which assign an important role to atomic metastability in the production of ‘narrow’ Zeeman component radiation from the boundary region of fusion plasmas, are examined critically in relation to l-redistribution by proton and electron collisions, and mixing of unperturbed atomic states by the ion microfield and microfield gradient. It is concluded that these important processes indeed severely constrain the contribution from ‘metastable’ states to the generation of the hydrogen Balmer spectra, for electron concentrations above 1012 cm−3, as pointed out before by the present author (Hey et al 1999 J. Phys. B: At. Mol. Opt. Phys. 32 3555). The analysis of collision-induced l-redistribution represents an extension of that used previously (Hey et al 1996 Contrib. Plasma Phys. 36 583), applicable up to higher electron densities. For comparison purposes, we also consider the question of metastability of ionized helium in a low-temperature plasma, and that of some common hydrogenic impurities (C5+ and Ne9+) in a hydrogen (deuterium) fusion plasma. While for low nuclear charge Z the metastability of 2s1/2 levels is quenched by the plasma environment, it is much reduced in high-Z ions owing to the rapid increase with Z of the two-photon electric dipole (2E1) and magnetic dipole (M1) spontaneous transition rates to the ground state, whereas the role of the plasma in these cases is less important. The main new principle elaborated in this work is the sensitivity of atomic line strengths, and hence collision strengths, to perturbation by the plasma environment for transitions between fine-structure sublevels of the same principal quantum number. As the plasma microfield strength grows, ‘allowed’ transitions diminish in strength, while ‘forbidden’ transitions grow. However, owing to violation of the parity selection rule, there is an overall loss of collision strength available to transitions, resulting from the appearance of significant ‘self-strength’ contributions, in accord with the sum rules for the line strengths, which remain valid over the range of fields considered. Thus, the relative effectiveness per perturber of both electron and ion collisions, for inducing population transfer between fine-structure sublevels, diminishes as the sublevels evolve from a fine-structure dominated to a Stark-effect-dominated regime. In the concluding discussion, we mention that this finding may have a bearing on discrepancies claimed between Stark broadening theory developed by Griem (1967 Astrophys. J. 148 547) and by Watson (2006 J. Phys.B: At. Mol. Opt. Phys. 39 1889), and the measurements of Bell and co-workers (2000 Publ. Astron. Soc. Pac. 112 1236; 2011 Astrophys. Space Sci. 335 451) for high-n radio recombination lines from galactic H II regions. In the absence of detailed modelling to test this suggestion, however, it would be premature to attempt to draw any firm conclusions along these lines.

ChemInform Abstract: Mean Amplitudes of Vibration of the Novel XeF- 5 Anion.

The first example of a chemical species adopting the unusual pentagonal planar geometry was the anion XeFj, reported in 1991 [1], Recently, Christe et al. [2] reported the existence of the IF|~ dianion, which is only the second known example of such a molecular geometry. In order to advance in the characterization of this very interesting species, we have now undertaken a calculation of its mean amplitudes of vibration and compared the results with those obtained for some related molecules and ions. The calculations were performed using the method of characteristic vibrations of Muller and co-workers [3-5], The vibrational frequencies were taken from [2] and an F I F angle of 72° was assumed. The mean amplitudes of vibration calculated for IF|~, in the temperature range between 0 and 1000 K, are presented in Table 1. For comparative purposes, Table 2 shows the corresponding values for the isoelectronic XeFJ anion [6] and other closely related species. The amplitude value for the I F bond in IFf~ is, as expected, appreciably higher than that of the X e F bond in XeFJ, in agreement with the additional negative charge present over the iodine species and its lower nuclear charge. These two factors evidently produce an enhancement of the I F bond polarity, generating a weakening of these bonds [2], which can also be visualized from the strong temperature dependence of its amplitude values (see Table 1), which in this case is yet more important than in XeFj [6]. On the other hand, the very strong temperature dependence also observed for the non-bonded in-plane

Distorted Coulomb field of the scattered electron

Experimental results for the radiation emission from ultrarelativistic electrons in targets of 0.03%--5% radiation length is presented. For the thinnest targets, the radiation emission is in accordance with the Bethe-Heitler formulation of bremsstrahlung, the target acting as a single scatterer. In this regime, the radiation intensity is proportional to the thickness. As the thickness increases, the distorted Coulomb field of the electron that is the result of the first scattering events, leads to a suppressed radiation emission per interaction, upon subsequent scattering events. In that case, the radiation intensity becomes proportional to a logarithmic function of the thickness, due to the suppression. Eventually, once the target becomes sufficiently thick, the entire radiation process becomes influenced by multiple scattering and the radiation intensity is again proportional to the thickness, but with a different constant of proportionality. The observed logarithmic thickness dependence of radiation intensity at intermediate values of the thickness can be directly interpreted as a manifestation of the distortion of the electron Coulomb field resulting from a scattering event. The Landau-Pomeranchuk-Migdal effect is explored with high primary energy using materials with low nuclear charge ($Z$). Also, targets that should give rise to the claimed interference effect in high-energy radiation emission from a structured target of thin foils are investigated.

Self-energy correction to the bound-electrongfactor ofPstates

The radiative self-energy correction to the bound-electron g factor of 2P_1/2 and 2P_3/2 states in one-electron ions is evaluated to order alpha (Z alpha)^2. The contribution of high-energy virtual photons is treated by means of an effective Dirac equation, and the result is verified by an approach based on long-wavelength quantum electrodynamics. The contribution of low-energy virtual photons is calculated both in the velocity and in the length gauge and gauge invariance is verified explicitly. The results compare favorably to recently available numerical data for hydrogenlike systems with low nuclear charge numbers.

Self-energy correction to the hyperfine splitting and the electrongfactor in hydrogenlike ions

The hyperfine structure (hfs) and the $g$ factor of a bound electron are caused by external magnetic fields. For the hfs, the magnetic field is due to the nuclear spin. A uniform-in-space and constant-in-time magnetic field is used to probe the bound-electron $g$ factor. The self-energy corrections to these effects are more difficult to evaluate than those to the Lamb shift. Here, we describe a numerical approach for both effects in the notoriously problematic regime of hydrogenlike bound systems with low nuclear charge numbers. The calculation is nonperturbative in the binding Coulomb field. Accurate numerical values for the remainder functions are provided for $2P$ states and for $\mathit{nS}$ states with $n=1,2,3$.

Electron Self-Energy in the Presence of a Magnetic Field: Hyperfine Splitting andgFactor

A high-precision numerical calculation is reported for the self-energy correction to the hyperfine splitting and to the bound-electron g factor in hydrogenlike ions with low nuclear charge numbers. The binding nuclear Coulomb field is treated to all orders, and the nonperturbative remainder beyond the known Zalpha-expansion coefficients is determined. For the 3He+ ion, the nonperturbative remainder yields a contribution of -450 Hz to the normalized difference of the 1S and 2S hyperfine-structure intervals, to be compared with the experimental uncertainty of 71 Hz and with the theoretical error of 50 Hz due to other contributions. In the case of the g factor, the calculation provides the most stringent test of equivalence of the perturbative and nonperturbative approaches reported so far in the bound-state QED calculations.

Two-photon decays reexamined: cascade contributions and gauge invariance

The purpose of this paper is to calculate the two-photon decay rate corresponding to the two-photon transitions nS → 1S and nD → 1S in hydrogen-like ions with a low nuclear charge number Z (for principal quantum numbers n = 2, …, 8). Numerical results are obtained within a nonrelativistic framework, and the results are found to scale approximately as (Zα)6/n3, where α is the fine-structure constant. We also attempt to clarify a number of subtle issues regarding the treatment of the coherent, quasi-simultaneous emission of the two photons as opposed to one-photon cascades. In particular, the gauge invariance of the decay rate is shown explicitly.

Comparison of classical and second quantized description of the dynamic Stark shift

We compare the derivation of the dynamic Stark shift of hydrogenic energy levels in a classical framework with an adiabatically damped laser-atom interaction, which is equivalent to the Gell-Mann-Low-Sucher formula, and a treatment based on time-independent perturbation theory, with a second-quantized laser-atom dipole interaction Hamiltonian. Our analysis applies to a laser that excites a two-photon transition in atomic hydrogen or in a hydrogenlike ion with low nuclear charge number. Our comparisons serve to demonstrate why the dynamic Stark shift may be interpreted as a stimulated radiative correction and illustrates connections between the two derivations. The simplest of the derivations is the fully quantized approach. The classical and the second-quantized treatment are shown to be equivalent in the limit of large photon numbers.

TECHNIQUES IN ANALYTIC LAMB SHIFT CALCULATIONS

Quantum electrodynamics has been the first theory to emerge from the ideas of regularization and renormalization, and the coupling of the fermions to the virtual excitations of the electromagnetic field. Today, bound-state quantum electrodynamics provides us with accurate theoretical predictions for the transition energies relevant to simple atomic systems, and steady theoretical progress relies on advances in calculational techniques, as well as numerical algorithms. In this brief review, we discuss one particular aspect connected with the recent progress: the evaluation of relativistic corrections to the one-loop bound-state self-energy in a hydrogenlike ion of low nuclear charge number, for excited non-S states, up to the order of α(Zα)6 in units of the electron mass. A few details of calculations formerly reported in the literature are discussed, and results for 6F, 7F, 6G and 7G states are given.

Second-order electron self-energy loop-after-loop correction for low-Z hydrogen-like ions

The second-order electron self-energy loop-after-loop correction is investigated for hydrogen-like ions in the region of low nuclear charge numbers Z. Both irreducible and reducible parts of this correction are evaluated for the 1s1/2-state within the Fried–Yennie gauge. We confirm the result obtained first by Mallampalli and Sapirstein. The reducible part of this correction is evaluated numerically for the first time and it is consistent with the corresponding analytical αZ-expansion.

Electron self-energy for theKandLshells at low nuclear charge

A nonperturbative numerical evaluation of the one-photon electron self-energy for the K- and L-shell states of hydrogenlike ions with nuclear charge numbers $Z=1$ to 5 is described. Our calculation for the ${1S}_{1/2}$ state has a numerical uncertainty of 0.8 Hz in atomic hydrogen, and for the L-shell states ${(2S}_{1/2},$ ${2P}_{1/2},$ and ${2P}_{3/2})$ the numerical uncertainty is 1.0 Hz. The method of evaluation for the ground state and for the excited states is described in detail. The numerical results are compared to results based on known terms in the expansion of the self-energy in powers of $Z\ensuremath{\alpha}.$

Radiative Corrections in Highly Charged Ions and Tests of QED in Strong Electric and Magnetic Fields

To provide predictions of the Lamb shift of highly charged ions on the level of accuracy of about 10-6 has been achieved after exact results for the contributions of all two-photon self-energy diagrams have been performed. We report on the present status of our Lamb-shift calculations including all QED-corrections of first- and second-order in the finestructure constant α and all relevant nuclear effects. An excellent agreement between the most recent experimental data for Lamb shift of the 1s-ground state in hydrogenlike uranium can be stated. This can serve as a sensitive test of QED in the strongest electric fields accessible in nature. In a second part of this article we report about an all-order numerical evaluation of the one-photon selfenergy at low nuclear charge (Z = 1, ..., 5). A sensitive comparison of our numerical approach with analytical approach to the one-photon selfenergy confirms the consistency of these two different approaches (numerical and analytic) to very high precision.

Calculation of QED Effects in Hydrogen

Atomic mass differences are influenced by QED corrections, and a reliable understanding of these corrections is therefore of importance for the current and next generation of high-precision mass determinations based on Penning traps. We present a numerical evaluation of the self-energy correction, which is the dominant contribution to the Lamb shift, in the region of low nuclear charge. Our calculation is nonperturbative in the binding field and has a numerical uncertainty of 0.8 Hz in atomic hydrogen for the ground state and of 1.0 Hz for L-shell states (2S1/2, 2P1/2, and 2P3/2).