Nonhydrogenic Atoms Research Articles

New definitions of the effective nuclear charge and its application to estimate the matrix element [formula omitted]

New definitions of the effective nuclear charge are proposed to evaluate the orbital wave function and the dipole matrix element ⟨n,l|r|n′,l′⟩ for the non-hydrogenic atoms or ions. It is found that the commonly used effective nuclear charge defined by the orbital energy is insufficiently precise to estimate the orbital wave functions and matrix elements ⟨n,l|r|n′,l′⟩. Instead, the effective nuclear charge defined by ⟨r⟩ or ⟨r2⟩ are more advantageous. It is shown that the effective nuclear charge method becomes increasingly precise to predict wave functions and matrix elements as the orbital angular momentum, the principle quantum number and the degree of ionization increase. Good accuracy is achieved when principal quantum numbers are identical n=n′ or when both orbital quantum numbers l and l′ are non-zero. When s-orbitals are involved (l or l′ equal to zero) the precision is decreasing.

Photoionization microscopy of the Rydberg Rb atom under a continuous infrared radiation laser field

The photoionization microscopy of the Rydberg Rb atom exposed to a continuous infrared radiation laser field is investigated based on the semiclassical open orbit theory. In contrast to the photoionization of the Rydberg hydrogen atom, the ionic core-scattering effect plays an important role in the photoionization of the Rb atom. Due to the core-scattering effect and the laser field, the electron trajectories become chaotic. A huge number of ionization trajectories from the ionic source to the detector plane appear, which makes the oscillatory pattern in the electron probability distribution become much more complicated. The ρ–θ curve on the detector plane exhibits a self-similar fractal structure for the ionization trajectories of the Rydberg Rb atom in the laser field. Due to constructive and destructive quantum interference of different electron trajectories, a series of concentric rings appear in the photoionization microscopy interference patterns on the detector plane. The electron probability density distributions on the detector are found to be changed sensitively with the scaled electron energy and the laser wavelength. Even as the detector plane is located at a macroscopic distance from the photoionization source, the photoionization microscopy interference patterns can be observed clearly. These calculations may provide a valuable contribution to the actual experimental study of the photoionization microscopy of non-hydrogenic Rydberg atom in the laser field.

One- and two-photon phase-sensitive coherent-control scheme applied to photoionization microscopy of the hydrogen atom

Photoionization microscopy (PM) is an electron imaging method designed for the measurement of the outgoing flux of slow electrons produced by photoionization of atoms in the presence of an external uniform static electric field. The high resolution of PM allows the observation of spatial quantum interference structures which are directly related to the squared modulus of the excited electron's wave function. The PM's range of interest lies above the saddle-point energy, where continuum Stark states are degenerate with quasibound ones (resonances). A principal aim of PM is to provide access to the wave functions of the latter, which in hydrogen atoms ionize exclusively via tunneling (in contrast to the continuum states where the electron escapes freely above the potential barrier). In nonhydrogenic atoms, however, quasibound states are coupled with the continuum ones. Among other consequences, this leads to comparable resonant and continuum excitation strengths and the weakening or disappearance of the resonant manifestations from the recorded PM images. Here we examine theoretically the possibility of bypassing these difficulties by applying two-excitation-pathway interference techniques. For this first case-study we employ the hydrogen atom but we, nevertheless, simulate a nonhydrogenic situation by selecting hydrogenic Stark resonances whose excitation strengths are comparable to or even smaller than the continuum ones. Specifically, we consider the interaction of ground-state atoms with an \ensuremath{\omega}/2\ensuremath{\omega} bichromatic laser field inducing one- and two-photon transitions to the final Stark states. It is shown that, under certain conditions and by appropriately adjusting the intensities of the two laser fields and their relative phase, the uncovering of resonant manifestations from PM images may indeed be achieved. Further work concerning theoretical extensions to complex atoms and possible experimental realizations are also discussed.

Near-saddle-point-energy photoionization microscopy images of Stark states of the magnesium atom

Photoionization microscopy (PM) is a photoelectron imaging method based on the measurement of the electron current probability density in the case of photoionization of an atom in an external uniform static electric field. The resolution of PM is high enough for observing the quantum oscillatory spatial structure of the outgoing electron flux. We examine the Stark states of the magnesium atom and focus on PM features which are specific to the energy range just above the saddle-point energy. Particularly, the existence of $m$-dependent saddle-point thresholds is clearly demonstrated by following the evolution of the recorded angular distributions with energy. Furthermore, the outer inflection point radii of the radial distributions of the images are generally found to increase with energy monotonically but occasionally abruptly, signaling parabolic channel openings (above which these channels are converted to continua). This observation is in accord with recently reported quantum mechanical PM calculations for hydrogenic systems. More importantly, there are a number of nonmonotonic variations (local maxima) of the outer inflection point. The latter are attributed to the local emergence of an intensity halo at the outer part of the images, a feature relevant to the excitation of quasibound Stark states (resonances). The emergence of the halo for a given resonance is found to depend on the static electric field strength. These observations along with the difficulties in observing resonant effects on the recorded images of nonhydrogenic atoms, such as the medium-size magnesium atom, are discussed in detail.

Symmetry-Breaking Assisted Landau-Zener Transitions in Rydberg Atoms.

We report the observation of a controlled Landau-Zener transition (LZT) in Rydberg atoms by breaking the symmetry of the underlying Hamiltonian. For a nonhydrogenic Rydberg atom inside a changing electric (F) field, a LZT occurs between the avoided crossing energy levels of neighboring Rydberg states only for a sufficiently high changing rate. If a transverse magnetic (B) field is applied as we implement, the atomic level symmetry is broken, which causes the Stark manifolds denoted by a different |m| (m is the magnetic quantum number) to interact with each other. The mixed state levels end up pushing the adiabatically repelled target states closer and additionally they serve as stepping stones for the sequential LZTs between the neighboring sublevels. Such a feature significantly decreases the changing rate required for an efficient LZT inside a pure electric field. We report experimental observations that support the above scenario. It opens a versatile approach for engineering a controlled LZT in more general systems.

Photoionization microscopy of the lithium atom: Wave-function imaging of quasibound and continuum Stark states

Photoionization of an atom in the presence of a uniform static electric field provides the unique opportunity to expand and visualize the atomic wave function at a macroscopic scale. In a number of seminal publications dating back to the 1980s, Fabrikant, Demkov, Kondratovich, and Ostrovsky showed that this goal could be achieved by projecting slow (meV) photoionized electrons onto a position-sensitive detector and underlined the distinction between continuum and resonant contributions. The uncovering of resonant signatures was achieved fairly recently in experiments on the nonhydrogenic lithium atoms [Cohen et al., Phys. Rev. Lett. 110, 183001 (2013)]. The purpose of the present article is the general description of these findings, with emphasis on the various manifestations of resonant character. From this point of view, lithium has been chosen as an illustrative example between the two limiting cases of hydrogen, where resonance effects are more easily identified, and heavy atoms like xenon, where resonant effects were not observed.

Wave-function visualization of core-induced interaction of a non-hydrogenic Rydberg atom in an electric field

In this paper we report a wave-function visualization of the core-induced interaction of a non-hydrogenic Rydberg atom in an electric field and its resultant redistribution of the spectral line intensities. The energy levels and oscillator strengths are calculated based on an effective potential where the radial part of the wavefunction is expanded in a B-spline basis. The calculation is in good agreement with the experimental measurements of the absorption spectrum of sodium atoms in an electric field of F = 840 V/cm, both below and above the saddle point Esp. The visualization of the wavefunction for the eigenstates can help us to see how they stem from the interaction between two or more red and blue hydrogenic states. An approximate localized symmetry close to the atomic core is also observed in the wavefunctions, which accounts for the alternation of oscillator strengths in the experiment.

Reply to “Comment on ‘Test of the Stark-effect theory using photoionization microscopy’ ”

The frame-transformation (FT) theory which describes the dynamics of nonhydrogenic atoms in an external uniform electric field was tested for the process of photoionization in a Stark field by Zhao et al. [Phys. Rev. A 86, 053413 (2012)]. Differential cross sections from the FT theory were found to be inconsistent with those from the fully quantum-mechanical coupled-channel theory. The discrepancy was attributed to the frame transformation of irregular wave functions. In a recent investigation, Giannakeas et al. [Phys. Rev. A 91, 067401 (2015)] draw a different conclusion. They show that the FT theory generates irregular wave functions in good agreement with exact solutions for low angular momenta, although an obvious disagreement is seen for high angular momenta. We performed test calculations for numerous Stark states and found that our original conclusion remains valid, namely, the Fano-Harmin frame transformation for the irregular wave function is inaccurate.

DYNAMICS OF RADIATIVE TRANSITIONS BETWEEN STARK SUBLEVELS FOR NON-HYDROGENIC ATOMS AND DIATOMICS IN DC ELECTRIC FIELD

The quantum approach for definition of the radiation transitions between Stark sublevels for non-hydrogenic atoms and diatomics in an external electric field is underlined and based on the operator perturbation theory and model potential method. The cited radiative phenomena can be basis for construction of new types of the quantum sensor devices.

Visualizing the coupling between red and blue stark states using photoionization microscopy.

In nonhydrogenic atoms in a dc electric field, the finite size of the ionic core introduces a coupling between quasibound Stark states that leads to avoided crossings between states that would otherwise cross. Near an avoided crossing, the interacting states may have decay amplitudes that cancel each other, decoupling one of the states from the ionization continuum. This well-known interference narrowing effect, observed as a strongly electric field-dependent decrease in the ionization rate, was previously observed in several atoms. Here we use photoionization microscopy to visualize interference narrowing in helium atoms, thereby explicitly revealing the mechanism by which Stark states decay. The interference narrowing allows measurements of the nodal patterns of red Stark states, which are otherwise not observable due to their intrinsic short lifetime.

A classical analogue for adiabatic Stark splitting in non-hydrogenic atoms

For highly excited low-ℓ states of atoms, a rising electric field causes a mixing between angular momenta that gives rise to Stark states. A relatively simple situation occurs if the electric field is not strong enough to mix states with adjacent principle quantum numbers. If the initial state has slightly lower (higher) energy than the degenerate manifold, then the state adiabatically connects to a Stark state with the electron on the low (high) potential energy side of the atom. We show that purely classical calculations for non-hydrogenic atoms have an adiabatic connection to extreme dipole moments similar to quantum systems. We use a simple map to show that the classical dynamics arises from the direction of the precession of the Runge–Lenz vector when the electric field is off. As a demonstration of the importance of this effect, we perform classical calculations of charge exchange and show that the total cross section for charge transfer and for ionization strongly depend on whether or not a pure Coulomb potential is used.

Non-Hydrogenic Rydberg Atoms in Magnetic Fields

The classical and quantal dynamics of non-hydrogenic Rydberg atoms in magnetic fields are investigated. Previous attempts to infer classical behaviour from quantum properties produced conflicting results: at low scaled energies (ε=−0.5) the nearest-neighbour statistics (NNS) were found to be at the chaotic (Wigner) limit while quantum phase-space distributions suggested a high degree of regularity. Here the classical limit is investigated directly by solving the equations of motion of the Diamagnetic Kepler problem (DKP) with an additional non-Coulombic model potential. It is found that typically trajectories are, over a long time-scale, ergodic. However over a shorter time-scale—in between collisions with the core—classical trajectories remain confined on the tori of the DKP. The origin of a well-known resonance in the NNS of hydrogen at ε= −0.316 is clarified by the comparison with the non-hydrogenic behaviour. However, the classical model only partially explains the quantum behaviour. The difficulties of quantizing such a system are discussed.

Test of the Stark-effect theory using photoionization microscopy

We formulate the dynamics of electron-wave propagation in photoionization microscopy for nonhydrogenic atoms, based on a frame transformation between the spherical and parabolic coordinates used in Harmin's Stark-effect theory. An expression for the wave function for photoelectrons ejected from the nonhydrogenic atomic source has been derived. The spatial distributions of electron current densities or differential cross sections for Na are computed and compared to those from a recently developed coupled-channel theory. The difference between these two approaches is analyzed and attributed to the frame transformation for irregular wave functions. Since more detailed physical information can be extracted by comparing differential cross sections, rather than total cross sections, photoionization microscopy is proposed to test the Stark-effect theory.

Coupled-channel theory of photoionization microscopy

We develop a quantum-mechanical coupled-channel theory to simulate spatial distributions of electron current densities, produced in photoionization for nonhydrogenic atoms in the presence of a uniform external electric field. The coupled Schr\"odinger equations are numerically solved using the renormalized Numerov method. The expression for the outgoing wave function for photoelectrons ejected from the nonhydrogenic atomic source is derived. The theory is applied to investigations of photoionization for ground-state Li atoms. The distributions of electron current densities are computed and compared to the corresponding experimental images. Excellent agreement is obtained. It is, furthermore, found that the presence of the nonhydrogenic residual ion significantly changes the differential cross sections and/or electron current densities with respect to the hydrogenic case. Finally, the implications of the presence of the atomic core for quantum resonance tunneling are also analyzed.

Semiclassical calculation of ionisation rate for Rydberg helium atoms in an electric field

The ionisation of Rydberg helium atoms in an electric field above the classical ionisation threshold has been examined using the semiclassical method, with particular emphasis on discussing the influence of the core scattering on the escape dynamics of electrons. The results show that the Rydberg helium atoms ionise by emitting a train of electron pulses. Unlike the case of the ionisation of Rydberg hydrogen atom in parallel electric and magnetic fields, where the pulses of the electron are caused by the external magnetic field, the pulse trains for Rydberg helium atoms are created through core scattering. Each peak in the ionisation rate corresponds to the contribution of one core-scattered combination trajectory. This fact further illustrates that the ionic core scattering leads to the chaotic property of the Rydberg helium atom in external fields. Our studies provide a simple explanation for the escape dynamics in the ionisation of nonhydrogenic atoms in external fields.

Efficient Diffuse Basis Sets for Density Functional Theory

Eliminating all but the s and p diffuse functions on the non-hydrogenic atoms and all diffuse functions on the hydrogen atoms from the aug-cc-pV(x+d)Z basis sets of Dunning and co-workers, where x = D, T, Q, ..., yields the previously proposed "minimally augmented" basis sets, called maug-cc-pV(x+d)Z. Here, we present extensive and systematic tests of these basis sets for density functional calculations of chemical reaction barrier heights, hydrogen bond energies, electron affinities, ionization potentials, and atomization energies. The tests show that the maug-cc-pV(x+d)Z basis sets are as accurate as the aug-cc-pV(x+d)Z ones for density functional calculations, but the computational cost savings are a factor of about two to seven.

Scaled-energy spectroscopy of a |M| = 1 Rydberg barium atom in an electric field

We observe strong energy-dependent quantum defects in the scaled-energy Stark spectra for |M| = 1 Rydberg states of barium atoms at three scaled energies: ε = –2.000, ε = –2.500 and ε = –3.000. In an attempt to explain the observations, theoretical calculations of closed orbit theory based on a model potential including core effect are performed for non-hydrogenic atoms. While such a potential has been uniformly successful for alkali atoms with a single valence electron, it fails to match experimental results for barium atoms in the 6snp Rydberg states with two valence electrons. Our study points out that this discrepancy is due to the strong perturbation from the 5d8p state, which voids the simple approximation for constant quantum defects of principle quantum number n.

Arsenic toxicity: an atom counting and electrophilicity-based protocol

The atomic number (Z) and electrophilicity index (omega) have been utilized to explain the toxicity of various alkali and transition-metal ions as well as to predict that of the arsenic ions. The toxicity of two different training sets of arsenic derivatives is described using the global electrophilicity (omega) and number of nonhydrogenic atoms (N (NH)) along with the local philicity (omega(As)+) and the atomic charge (Q (As)) on the arsenic atom. Applying the regression models from the training sets, toxicity of some unknown arsenic derivatives is predicted.

METHOD FOR CALCULATING THE OPACITY OF THE ATOMIC LINES IN STELLAR ATMOSPHERES

A new approximate method for calculating the opacity of the atomic lines in the computation of stellar model atmospheres is presented. Transforming the sums in the equation of the opacity of the lines into integrals and considering the Kramers equation for the oscillator strengths for level transitions in the hydrogenic atoms as a continuous function of frequency, we can apply the mean value theorem for integrals in order to describe the variation of the global opacity of the lines of all the chemical elements. The high-lying levels in nonhydrogenic atoms as well as their hydrogen-like transitions can also benefit from this method. For the case of stellar atmospheres not in local thermodynamic equilibrium, we can apply our method directly. The compact analytic expressions obtained are easy to use. The application of the method to the hydrogen lines shows the classical line-blanketing effects.

An Atom Counting QSPR Protocol

A simple descriptor, viz the number of carbon (N C )/non-hydrogenic (N NH ) atoms present in a molecule, is proposed for the development of useful Quantitative Structure-Property-Relationship (QSPR) models. This work is prompted by that of Randic and Basak (J. Chem. Inf. Comput. Sci.2001, 41, 614-618) who have shown that this descriptor is superior to the simple connectivity index and is marginally worse than the variable connectivity index for analyzing the toxicity of aliphatic ethers. It is tested in the present work in models pertaining to the estimation of boiling point of alcohols, enthalpy of vaporization of Polychlorinated Biphenyls (PCBs), n-octanol/water partition coefficient of PCBs and chloroanisoles, pK a values of carboxylic acids, phenols, alcohols, etc. High values of various regression coefficients (R 2 , R 2 CV , R 2 AD ) suggest the significance of this descriptor which further improves in the resulting two-parameter QSPR models with electrophilicity or its local variant as an additional descriptor. Other statistical quantities are also analyzed for gaining additional insights into the proposed models. Although the atom number is not the best possible descriptor in a QSPR model in all cases, its simplicity provides it a special place in the development of those models. The present analysis extends the work of Randic and Basak by delineating the situations when N C /N NH performs well vis-a-vis the other descriptors.