Relativistic Atom Research Articles

Faraday rotation method improves the upper limit of the electron electric-dipole-moment sensitivity.

The electron electric-dipole-moment (eEDM) is a powerful tool for exploring new particles. The candidates for eEDM search are heavy atoms and their molecules, which are well known for the obvious relativistic effect. Lead atom is considered to be the most ideal relativistic atom [Park et al., Nat. Commun. 11(1), 815 (2020)]. PbH molecule is an important representative of the Pb compound and is considered a cold candidate molecule due to the high diagonal Franck-Condon factors. We systematically investigated the (eEDM) searches of PbH using a two-component approach. The parity- and time-reversal symmetry violation constants of ground and excited states, including internal effective electric field Eeff, electron-nucleon scalar-pseudoscalar interaction constant WP,T, and nuclear magnetic quadrupole moment, were obtained and compared to other molecules. In addition, we designed two experimental methods to measure the sensitivity of the eEDM, indicating that the Faraday rotation method could greatly improve its sensitivity.

Relativistic artificial molecule of two coupled graphene quantum dots at tunable distances

In a molecule formed by two atoms, energy difference between bonding and antibonding orbitals depends on distance between the two atoms. However, exploring molecular orbitals of two natural atoms with tunable distance has remained an outstanding experimental challenge. Graphene quantum dots can be viewed as relativistic artificial atoms, thus offering a unique platform to study molecular physics. Here, through scanning tunneling microscope, we create and directly visualize the formation process of relativistic artificial molecules based on two coupled graphene quantum dots with tunable distance. Our study indicates that energy difference between the bonding and antibonding orbitals of the lowest quasibound state increases linearly with inverse distance between the two graphene quantum dots due to the relativistic nature of the artificial molecule. For quasibound states with higher orbital momenta, the coupling between these states leads to half-energy spacing of the confined states because the length of the molecular-like orbit is approximately twice that of the atomic-like orbit. Evolution from ring-like whispering-gallery modes in the artificial atoms to figure-eight orbitals in the artificial molecules is directly imaged. The ability to resolve the coupling and orbitals of the relativistic artificial molecule at the nanoscale level yields insights into the behavior of quantum-relativistic matter.

Coulomb impurities in graphene driven by ultrashort electromagnetic pulses: excitation, ionization, and pair creation

We provide a theory for electronic transitions induced by ultrashort electromagnetic pulses in two-dimensional artificial relativistic atoms which are created by a charged impurity in a gapped graphene monolayer. Using a non-perturbative sudden-perturbation approximation, we derive and discuss analytical expressions for the probabilities for excitation, ionization and electron-hole pair creation in this system.

Ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels

Motivated by the study of relativistic atoms, we consider the Hardy operator (−Δ)α/2−κ|x|−α acting on functions of the form u(|x|)|x|ℓYℓ,m(x/|x|) in L2(Rd), when κ≥0 and α∈(0,2]∩(0,d+2ℓ). We give a ground state representation of the corresponding form on the half-line (Theorem 1.5). For the proof we use subordinated Bessel heat kernels.

Ionisation in nanowire by ultra-short relativistic laser pulse

We show that the wakefield driven by fast electrons inside the nanowire when irradiated with an ultra-short relativistic laser pulse strips atoms to a higher charge state. Using particle-in-cell simulations, we demonstrate that the charge state agrees with the barrier suppression threshold of the wakefield and reaches a higher value via collision. The ionisation of gold nanowires occurs only via collisional-damped wakefield. We found that the collisional ionisation of high-Z nanowires depends on the onset of the z pinch. These results suggest a different ionisation mechanism of the structured target in the subfemtosecond regime.

The effective relativistic coupling by asymptotic representation approach for molecules with multiple relativistic atoms

The Effective Relativistic Coupling by Asymptotic Representation (ERCAR) approach is a method to generate fully coupled diabatic potential energy surfaces (PESs) including relativistic effects, especially spin–orbit coupling. The spin–orbit coupling of a full molecule is determined only by the atomic states of selected relativistically treated atoms. The full molecular coupling effect is obtained by a diabatization with respect to asymptotic states, resulting in the correct geometry dependence of the spin–orbit effect. The ERCAR approach has been developed over the last decade and initially only for molecules with a single relativistic atom. This work presents its extension to molecules with more than a single relativistic atom using the iodine molecule as a proof-of-principle example. The theory for the general multiple atomic ERCAR approach is given. In this case, the diabatic basis is defined at the asymptote where all relativistic atoms are separated from the remaining molecular fragment. The effective spin–orbit operator is then a sum of spin–orbit operators acting on isolated relativistic atoms. PESs for the iodine molecule are developed within the new approach and it is shown that the resulting fine structure states are in good agreement with spin–orbit ab initio calculations.

Giant orbital magnetic moments and paramagnetic shift in artificial relativistic atoms and molecules.

Materials such as graphene and topological insulators host massless Dirac fermions that enable the study of relativistic quantum phenomena. Single quantum dots and coupled quantum dots formed with massless Dirac fermions can be viewed as artificial relativistic atoms and molecules, respectively. Such structures offer a unique testbed to study atomic and molecular physics in the ultrarelativistic regime (particle speed close to the speed of light). Here we use a scanning tunnelling microscope to create and probe single and coupled electrostatically defined graphene quantum dots to unravel the magnetic-field responses of artificial relativistic nanostructures. We observe a giant orbital Zeeman splitting and orbital magnetic moment up to ~70 meV T-1 and ~600μB (μB, Bohr magneton) in single graphene quantum dots. For coupled graphene quantum dots, Aharonov-Bohm oscillations and a strong Van Vleck paramagnetic shift of ~20 meV T-2 are observed. Our findings provide fundamental insights into relativistic quantum dot states, which can be potentially leveraged for use in quantum information science.

Measurement of the scalar curvature of high-power lasers

High-power lasers develop high energy per unit time, and as energy curves space, we expect atomic energy levels to change. The fluorescence spectrum is a good measurement of the matrix elements involved in the Rabi oscillation and consequently allows us to determine the scalar curvature. At high Z, electrons oppose ionization even for strong intensities. Because high-power lasers address relativistic atoms, the wave functions involved must be solutions of the Dirac equation in a curved space-time. The paper can be seen as a way to check whether the Einstein’s gravitational theory is valid in the dimension of laboratory.

Study of the Klein–Gordon equation for a hydrogenic model of dyons

This paper presents the generalization of a zero spin hydrogen atom to a relativistic atomic model of hydrogen with dyons using the Klein–Gordon equation. The derivation of the Klein–Gordon equation for the particle of relative motion is shown. In addition, the analytical solutions of the equation are calculated in terms of Whittaker functions and Jacobi weighted polynomials. The discrete spectrum of energy and the charge density of the orbiting dyon are presented. For a system of positive magnetic and electric charges in the nucleus and negative charges for the orbiting particle, and considering the first allowed values of [Formula: see text] and [Formula: see text], it was found that the dyon atom acts with a greater force of interaction between the charges of the nucleus and the secondary particle compared to the standard atom. It was obtained by comparing the distance between the nucleus and charge density concentrations from the dyon atom with the relativistic pionic atom.

Front Cover: The Cesium Oxide Mercuride Cs18Hg8O6 (Z. Anorg. Allg. Chem. 10/2022)

The cover picture shows crystal structure and electronic structure of the compound Cs18Hg8O6. It contains the first isolated mercuride anions, cubic [Hg8]6− entities which are coordinated by Cesium cations. As further anions oxide anions O2− are present in the structure and form a 3D scaffolding network of [Cs6O] octahedra. The ′mercubanide′ anions are arranged in a cubic body-centered topology. The calculated electronic structure is shown as a spaghetti representation. The energy bands below the Fermi level can be attributed to either mercuride or oxide anions, emphasizing the picture of an ionic double salt. The mercuride Cs18Hg8O6 complements a series of compounds with anions from relativistic atoms, showing peculiar features of the chemical bonding. Below, a photograph shows a typical rod-shaped crystal with dark metallic luster.(DOI: 10.1002/zaac.202100389).

Relativistic Hirshfeld atom refinement of organo-gold(I) and organo-mercury(II) compounds

Relativistic Hirshfeld atom refinement of organo-gold(I) and organo-mercury(II) compounds

Relativistic Hirshfeld atom refinement of an organo-gold(I) compound.

The main goal of this study is the validation of relativistic Hirshfeld atom refinement (HAR) as implemented in Tonto for high-resolution X-ray diffraction datasets of an organo-gold(I) compound. The influence of the relativistic effects on statistical parameters, geometries and electron density properties was analyzed and compared with the influence of electron correlation and anharmonic atomic motions. Recent work in this field has indicated the importance of relativistic effects in the static electron density distribution of organo-mercury compounds. This study confirms that differences in electron density due to relativistic effects are also of significant magnitude for organo-gold compounds. Relativistic effects dominate not only the core region of the gold atom, but also influence the electron density in the valence and bonding region, which has measurable consequences for the HAR refinement model parameters. To study the effects of anharmonic motion on the electron density distribution, dynamic electron density difference maps were constructed. Unlike relativistic and electron correlation effects, the effects of anharmonic nuclear motion are mostly observed in the core area of the gold atom.

Relativistic quantum crystallography of diphenyl- and dicyanomercury. Theoretical structure factors and Hirshfeld atom refinement

Quantum crystallographic refinement of heavy-element-containing compounds is a challenge, because many physical effects have to be accounted for adequately. Here, the impact and magnitude of relativistic effects are compared with those of electron correlation, polarization through the environment, choice of basis set and treatment of thermal motion effects on the structure factors of diphenylmercury(II) [Hg(Ph)2] and dicyanomercury(II) [Hg(CN)2]. Furthermore, the individual atomic contributions to the structure factors are explored in detail (using Mulliken population analysis and the exponential decay of atomic displacement parameters) to compare the contributions of lighter atoms, especially hydrogen atoms, against mercury. Subsequently, relativistic Hirshfeld atom refinement (HAR) is validated against theoretical structure factors of Hg(Ph)2 and Hg(CN)2, starting from perturbed geometries, to test if the relativistic variant of HAR leads to multiple solutions. Generally, relativistic HAR is successful, leading to a perfect match with the reference geometries, but some limitations are pointed out.

Equivalence of Sobolev Norms Involving Generalized Hardy Operators

Abstract We consider the fractional Schrödinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces, which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schrödinger operators and have an important application in the treatment of large relativistic atoms.

Relativistic two-dimensional hydrogen-like atom in a weak magnetic field

A two-dimensional (2D) hydrogen-like atom with a relativistic Dirac electron, placed in a weak, static, uniform magnetic field perpendicular to the atomic plane, is considered. Closed forms of the first- and second-order Zeeman corrections to energy levels are calculated analytically, within the framework of the Rayleigh–Schrödinger perturbation theory, for an arbitrary electronic bound state. The second-order calculations are carried out with the use of the Sturmian expansion of the two-dimensional generalized radial Dirac–Coulomb Green function derived in the paper. It is found that, in contrast to the case of the three-dimensional atom (Stefańska, 2015), in two spatial dimensions atomic magnetizabilities (magnetic susceptibilities) are expressible in terms of elementary algebraic functions of a nuclear charge and electron quantum numbers. The problem considered here is related to the Coulomb impurity problem for graphene in a weak magnetic field.

Multi-solitary wave solutions to the general time fractional Sharma–Tasso–Olver equation and the time fractional Cahn–Allen equation

The space time fractional Sharma–Tasso–Olver (STO) equation and the time fractional Cahn–Allen (CA) equation are well-designated to fission and fusion phenomena including those for solitons, electromagnetic interactions, quantum relativistic atom theory, phase isolation in several components bass system, the relativistic energy-momentum relation. In this article, we have exerted the exact solution to STO and CA equation in the light of fractional derivative (e.g. modified Riemann–Liouville derivative). By using wave transformation the ODE of integer order is generated by the fractional partial differential equation (FPDE). We investigated the travelling wave solution to these equations through the recently established double-expansion method. The results obtained in view of hyperbolic, trigonometric and rational functions containing parameters. We have demonstrated that this method is convenient, effective and powerful tools for solving nonlinear fractional differential equations (NLFDEs).

Calculation of Silicon Plasmas with a Relativistic Collisional Radiative Average Atom Code

Calculation of Silicon Plasmas with a Relativistic Collisional Radiative Average Atom Code

"Through-Space" Relativistic Effects on NMR Chemical Shifts of Pyridinium Halide Ionic Liquids.

We have investigated, using two-component relativistic density functional theory (DFT) at ZORA-SO-BP86 and ZORA-SO-PBE0 level, the occurrence of relativistic effects on the 1 H, 13 C, and 15 N NMR chemical shifts of 1-methylpyridinium halides [MP][X] and 1-butyl-3-methylpyridinium trihalides [BMP][X3 ] ionic liquids (ILs) (X=Cl, Br, I) as a result of a non-covalent interaction with the heavy anions. Our results indicate a sizeable deshielding effect in ion pairs when the anion is I- and I3- . A smaller, though nonzero, effect is observed also with bromine while chlorine based anions do not produce an appreciable relativistic shift. The chemical shift of the carbon atoms of the aromatic ring shows an inverse halogen dependence that has been rationalized based on the little C-2s orbital contribution to the σ-type interaction between the cation and anion. This is the first detailed account and systematic theoretical investigation of a relativistic heavy atom effect on the NMR chemical shifts of light atoms in the absence of covalent bonds. Our work paves the way and suggests the direction for an experimental investigation of such elusive signatures of ion pairing in ILs.

Second-order Stark effect and polarizability of a relativistic two-dimensional hydrogenlike atom in the ground state

The second-order Stark effect for a planar Dirac one-electron atom in the ground state is analyzed within the framework of the Rayleigh-Schr\"odinger perturbation theory, with the use of the Sturmian series expansion of the generalized Dirac-Coulomb Green function. A closed-form analytical expression for the static dipole polarizability of that system is found. The formula involves a generalized hypergeometric function ${}_{3}F_{2}$ with the unit argument. Numerical values of the polarizabilities for relativistic planar hydrogenic atoms with atomic numbers $1\leqslant Z\leqslant68$ are provided in a tabular form. A simple formula for the polarizability of a nonrelativistic two-dimensional hydrogenic atom, reported previously by several other authors, is recovered from our result in the nonrelativistic limit.

Long‐range relativistic heavy atom effect on 1H NMR chemical shifts of selenium‐ and tellurium‐containing compounds

AbstractPreviously unknown manifestation of heavy atom effect on the NMR chemical shifts of β‐ and γ‐protons initiated by the relativistic effects of the tellurium and selenium atoms has been investigated in the representative series of selenium‐ and tellurium‐containing compounds. To approve the four‐component density functional approach to be the appropriate tool for the investigation of the heavy atom on light atom effect (HALA), the benchmark calculations of the proton chemical shifts have been performed at the CCSD level using comprehensively chosen locally dense basis set with taking into account solvent, vibrational, and relativistic corrections. A good agreement with the experimental data was achieved. The magnitudes of the relativistic HALA corrections to β‐ and γ‐proton chemical shifts were found to vary in a wide range, namely from −3.08 ppm for the γ‐proton of methyltelluraldehyde to 14.51 ppm for β‐proton in benzotelluraldehyde.