Effective Principal Quantum Number Research Articles

Lifetimes of fine structure levels of the Li atom for 20 < n < 31 by an extended Ritz formula

Lithium and lithium-like elements look like hydrogen atoms if their two electrons and the nucleus are considered as having a core around which a single electron is orbiting. The energy and radii expressions for hydrogen atoms can be used for lithium and lithium-like elements; an important modification is introducing an effective principal quantum number. The effective principal quantum number differs from the principal quantum number of hydrogen by the quantum defect. The quantum defect has respective values for various levels of lithium and lithium-like ions. In this study, we used extended Ritz formulas to calculate quantum defects required to calculate the energies of the ns, np, nd, and nf series up to n < 31. Using these energies, we calculated transition probabilities and then the lifetimes of the lithium levels. The lifetimes were compared with the available published data; an excellent agreement was noticed. The work also extended the available list of lifetimes to higher quantum numbers. Forty lifetimes are newly calculated and presented for the first time. A polynomial formula for each of the ns, np, nd, and nf series lifetimes has been developed that fits well the lifetime values.

Atomic Orbital Bases of Bunge and Koga Studied by Frobenius Products with Moscow–Aachen–Paris (MAP) Orbitals

The minimization of Frobenius angles between functional subspaces spanned by different sets of atomic functions is employed to determine the values of orbital exponents ξ characterizing minimum atomic parameters/Moscow–Aachen–Paris (MAP) basis sets providing the best representation of two Hartree–Fock based atomic basis sets: that of Bunge et al. available for elements H–Xe and that of Koga and Thakkar spanning H to Lr (Z = 103). So-extracted values of exponents follow piecewise linear laws as functions of the nuclear charge Z resembling the prescriptions set by Slater rules for the orbital exponents. In details, however, the rules proposed by Slater are not precisely followed, neither for effective principal quantum numbers n* nor screening increments σ. Nevertheless, the linear pieces of the ξ vs. Z follow the structure of the Periodic Table being specific for the segments corresponding to p-, d- (transition), and f- (Lanthanides and Actinides) elements, respectively.

Effective Principal Quantum Numbers of Hydrogen-Like States Belong to the Set of Common Fractions

Effective Principal Quantum Numbers of Hydrogen-Like States Belong to the Set of Common Fractions

Periodic Table of the Elements, History, Education and Evaluation

The periodic tables of transition metal thiophosphates MPS3, transition metal dichalcogenides MX2 and other materials, the origin of chemical elements and toxic trace elements in dried mushrooms are provided. The effective nucleus-electron attraction is proportional to the effective nuclear charge (Zeff) and inversely proportional to the effective principal quantum number (n*). The periodic arch is one of many modern visual displays that have been developed to augment the traditional periodic table of the chemical elements. The table is related with the multiparameter optimisation of N atom, nuclear magnetic resonance and everyday life. Educational activities are developed with evaluation.

Extended Bose-Hubbard models with Rydberg macrodimer dressing

Extended Hubbard models have proven to bear novel phases of matter, but their experimental realization remains challenging. In this work we propose to use bosonic quantum gases dressed with molecular bound states in Rydberg interaction potentials for the observation of these quantum states. We study the molecular Rabi coupling with respect to the effective principal quantum number and trapping frequency of the ground-state atoms for various molecular potentials of rubidium and potassium, and the hereby resulting dressed interaction strength. Additionally, we propose a two-color excitation scheme which significantly increases the dressed interaction and cancels otherwise-limiting ac Stark shifts. We study the various equilibrium phases of the corresponding extended Bose-Hubbard model by means of the cluster Gutzwiller approach and perform time evolution simulations via the Lindblad master equation. We find a supersolid phase by slowly ramping the molecular Rabi coupling of an initially prepared superfluid and discuss the role of dissipation.

Theoretical computation of normalised radii, density and global hardness as a function of orbital exponent

The recent work has an aim to establish a pivotal role of orbital exponent in the normalized atomic radii, atomic density and atomic hardness. These three periodic descriptors help to understand the real scenario of an element. Concerning the effective nuclear charge, screening constant and effective principal quantum number, we have developed a new relation between these periodic properties and invoked a new formula by which we can compute the normalized radii, density and global atomic hardness in terms of the orbital exponent. With comparison to the existing famous formulae originating from different concepts, we can conclude that our empirical computation has an inherent efficacy to predict periodicity.

Investigation on the DC Stark shifts of strontium autoionization states for isotope-selective resonance ionization

Three-step resonance excitation to autoionization states is a promising isotope-selective ionization technique for atomic strontium (Sr). We measured for the first time the DC Stark shifts of ten Rydberg levels (neff∼36−39,50−55) of 88Sr converging to the 4d 2D3/2 limit using the 5s21S0→ 5s5p 1P1∘→ 5p21D2→ (4d 2D3/2 Rydberg series) transition pathway. The polarizability α of each level was evaluated to be |α|∼10−30−10−28 [C2·m2·J−1] depending on the effective principal quantum number neff, which indicates that the external electric field E in the ionization region should be of the order of 0.1 V/cm or less to suppress its effect on the observed spectra. Our experimental data on the DC Stark shift and polarizability of the Rydberg levels are useful for future isotope-selective detection of the radioisotope 90Sr using three-step resonance excitation.

Observation of Cesium (nD 5/2+6S 1/2) Ultralong-Range Rydberg-Ground MoleculesSupported by the National Key R&D Program of China (Grant No. 2017YFA0304203), the National Natural Science Foundation of China (Grant Nos. 11434007, 61835007, 61675123, 61775124, and 11904215), Changjiang Scholars and Innovative Research Team in Universities of Ministry of Education of China (Grant No. IRT 17R70), and the 1331 Project.

Ultralong-range Cs2 Rydberg-ground molecules (nD 5/2 + 6S 1/2 F) (33 ≤ n ≤ 39, F = 3 or 4) are investigated by a two-photon photo-association spectroscopy of an ultracold Cs gas. Two vibrational ground molecular spectra of triplet 3 Σ and hyperfine mixed singlet-triplet 1,3 Σ molecular states and their corresponding binding energies are attained. The experimental observations are simulated by an effective Hamiltonian including low energy electron scattering pseudopotentials, the spin-orbit interaction of the Rydberg atom, and the hyperfine interaction of the ground-state atom. The zero-energy singlet and triplet s-wave scattering lengths are extracted by comparing the experimental observations and calculations. Dependences of the measured binding energies on the effective principal quantum number, n eff = n − δD (δD is the quantum defect of Rydberg D state), yield the scaling of , for deep triplet potential and , for shallow mixed singlet-triplet potential well. The simulations of low-energy Rydberg electron scattering show agreement well with the experimental measurements.

Уточнение связи между эффективными радиусами и энергиями ионизации атомов щелочных металлов

One of the key explanations for the peculiarities of the change in the first ionization energies (IE Ei1) of the elements of the periodic table is based on the postulation of an unambiguous relationship between Ei1 and atomic radii (rat). Due to the certain conventionality of the rat parameters, they do not belong to the category of attributive atomic characteristics. Usually they are found either by implementing various kinds of theoretical or semi-empirical calculation schemes, or by using experimental data on Ei1 for the corresponding approximations. As follows from the literature data, the known values of rat, even in the case of "quasi-hydrogen-like" alkali metals (AM), –relatively simple manyelectron systems – are characterized, as a rule, by a significant scatter. In addition, the canonical analytical formulas describing rаt within the framework of a hydrogen-like model with two effective interaction parameters (according to Slater) do not ultimately lead to adequate results in this set of elements. In this work, to correct the relationship rat = f (Еi1) related to the properties of the aggregate of alkaline elements, the publication of the effective Slater radii (reff) and Weber-Cromer orbital radii (rorb), and some results of later data are used. The effective principal quantum numbers n* for each of the AM (except for Li, for which it is assumed that n*=n=2) are estimated on the basis of the values of Ei1. A further step, by introducing n*, made it possible to use the criterion for optimizing the value of the effective radius r* and to correct the formula that determines its relationship with Ei1 in the form of a simple power function. At the same time, the radii Li, Na, and Fr have been refined towards some underestimation, compared with rorb. Arguments are presented in favor of the reliability of the proposed approach. It is shown that the obtained estimates of r* correlate better with the amplitudes of the solution of the wave equation covering all AMs than the known corresponding values of rorb and reff.

Precise measurements of polarizabilities of cesium nS Rydberg states in an ultra-cold atomic ensemble

We present precise measurements of polarizabilities of cesium nS1/2 (n = 65–75) Rydberg states by Stark spectroscopies. In experiment, Rydberg atoms are excited via a two-photon scheme of a standard magneto-optical trap and detected by the field ionization technique. The Stark shift is measured by analysing the spectroscopy under an external electric field. The polarizability, α, is acquired by fitting the experimental data of Stark shifts with . The theoretical model is applied to numerically simulate the Stark map and corresponding level shifts, related deviation between the experimental measurements and calculations are less than 2%. The scaling law of polarizabilities, (n* is effective principal quantum number), is attained, that shows a good agreement with the measurements.

Rydberg-positronium velocity and self-ionization studies in a 1T magnetic field and cryogenic environment

We characterized the pulsed Rydberg-positronium production inside the AEgIS (Antimatter Experiment: Gravity, Interferometry, Spectroscopy) apparatus in view of antihydrogen formation by means of a charge exchange reaction between cold antiprotons and slow Rydberg-positronium atoms. Velocity measurements on positronium along two axes in a cryogenic environment (10K) and in 1T magnetic field were performed. The velocimetry was done by MCP-imaging of photoionized positronium previously excited to the $n=3$ state. One direction of velocity was measured via Doppler-scan of this $n=3$-line, another direction perpendicular to the former by delaying the exciting laser pulses in a time-of-flight measurement. Self-ionization in the magnetic field due to motional Stark effect was also quantified by using the same MCP-imaging technique for Rydberg positronium with an effective principal quantum number $n_{eff}$ ranging between 14 and 22. We conclude with a discussion about the optimization of our experimental parameters for creating Rydberg-positronium in preparation for an efficient pulsed production of antihydrogen.

Theoretical transition probabilities, radiative lifetimes and Stark broadening parameters of singly ionized magnesium

ABSTRACTThe presence of spectral lines of singly ionized magnesium (Mg ii) in stellar atmospheres has been reported in different stars. Recently, the low-resolution spectrum obtained from Supernova 2014 J in M82, in which Mg ii absorption lines centred on 4400 Å as well as 7600 Å stand out, has been analysed. This is the motive for the atomic data calculations in this work, which are of much interest in the astrophysical area. In this article, ab initio relativistic Hartree–Fock calculations in an intermediate coupling formalism using Cowan’s code allowed us to obtain the required transition probabilities to calculate the theoretical radiative lifetimes for excited nS−, nP−, nD− and nF− states of singly ionized magnesium. An asymptotic dependence of lifetime (τnl) on the effective principal quantum number (n*) has been determined. Also, the Griem semi-empirical approach was used to obtain the theoretical Stark parameters (width and shift) of spectral lines; these data are displayed for an electron density of 1017 cm−3 and temperatures T = 10–100 (×103 K). We have compared the results of lifetimes for 16 levels and Stark parameters for seven spectral lines with previously reported experiments available in the literature. Finally, we discuss the behaviour of the Stark parameters versus temperature for three relevant spectral lines (2802.70, 2797.99 and 7868.04 Å).

Adiabatic potentials of cesium (nDJ)2 Rydberg–Rydberg macrodimers

Electrostatic multipole interactions generate long-range Rydberg–Rydberg macrodimers. We calculate the adiabatic potentials of cesium Rydberg macrodimers for principal quantum numbers n ranging from 56 to 62, for J = 3/2 and 5/2, and for the allowed values of the conserved sum of the atomic angular-momentum components along the internuclear axis, M. For most combinations (n, M, J) exactly one binding potential exists, which should give rise to Rydberg macrodimer states. We study the dependence of the adiabatic potentials on the size of the two-body basis sets used in the calculation, and on the maximal order, qmax, of the multipole terms included in the calculation. We determine the binding energies and lengths of the binding adiabatic potentials, investigate their scaling behaviors as a function of the effective principal quantum number, and discuss vibrational-state wave functions. We consider the applicability of the calculated potentials and excitation rates to an experimental scheme for preparing Rydberg-atom macrodimers using two-color double-resonant photoassociation.

Measurement of quantum defect of cesium nP<sub>3/2</sub> (n = 70—94) Rydberg states by using ultraviolet single-photon Rydberg excitation

A narrow-linewidth continuous-wave single-frequency tunable 318.6-nm ultraviolet laser system with watt-level output power is developed in our experiment based on well-developed fiber lasers, fiber amplifiers, and efficient laser frequency conversion technique. Cesium 6S<sub>1/2</sub>—<i>n</i>P<sub>3/2</sub> (<i>n</i> = 70—94) single-photon Rydberg excitation in a room-temperature cesium atomic vapor cell is realized by using our ultraviolet laser system. The single-photon Rydberg excitation signal is obtained via the V-type three-level atomic system which contains 6S<sub>1/2</sub> (<i>F</i> = 4) ground state, 6P<sub>3/2</sub> (<i>F</i> = 5) excited state and one of <i>n</i>P<sub>3/2</sub> (<i>n</i> = 70—94) Rydberg states. When cesium atoms populated on the ground state are partially excited to Rydberg state by the ultraviolet laser, absorption of 852.3-nm probe beam which is locked to 6S<sub>1/2</sub> (<i>F</i> = 4)—6P<sub>3/2</sub> (<i>F</i> ′ = 5) hyperfine transition will decrease. In this way, the cesium Rydberg states are detected. The quantum defects for cesium <i>n</i>P<sub>3/2</sub> (<i>n</i> = 70—94) Rydberg states are experimentally measured with a high-precision wavemeter. The variation trend of experimentally measured data deviates from that of calculated values. Due to the fact that the cesium vapor cell is positioned in a magnetic shielding tank, the Zeeman effect can be ignored. Considering that the polarizability of Rydberg atoms is proportional to (<i>n</i>*)<sup>7</sup>, in which <i>n</i>* is the effective principal quantum number, the Rydberg screen effect of cesium atomic vapor cell cannot completely protect cesium atoms from being perturbed by an external DC electric field. Therefore the residual DC electric field existing inside the cesium vapor cell will have a significant influence on quantum defect measurement of Rydberg atoms. Using the theoretical model of Stark effect and the relationship between polarizability of Rydberg atoms and the effective principal quantum number <i>n</i>*, the corrected experimental value of quantum defect for cesium <i>n</i>P<sub>3/2</sub> (<i>n</i> = 70—94) Rydberg states is found to be ~(3.5591 ± 0.0007). The corrected experimental value of quantum defect is consistent with the calculation.

Isoelectronic Theory for Cationic Radii.

Ionic radii play a central role in all branches of chemistry, in geochemistry, solid-state physics, and biophysics. While authoritative compilations of experimental radii are available, their theoretical basis is unclear, and no quantitative derivation exists. Here we show how a quantitative calculation of ionic radii for cations with spherically symmetric charge distribution is obtained by charge-weighted averaging of outer and inner radii. The outer radius is the atomic (covalent) radius, and the inner is that of the underlying closed-shell orbital. The first is available from recent experimental compilations, whereas the second is calculated from a "modified Slater theory", in which the screening (S) and effective principal quantum number (n*) were previously obtained by fitting experimental ionization energies in isoelectronic series. This reproduces the experimental Shannon-Prewitt "effective ionic radii" (for coordination number 6) with mean absolute deviation of 0.025 Å, approximately the accuracy of the experimental data itself. The remarkable agreement suggests that the calculation of other cationic attributes might be based on similar principles.

Line-Profile Analysis of Excitation Spectroscopy in the Even 3p5(2P1/2)nl′[K′]J(l′=1,3) Autoionizing Resonances of Ar

The even-parity autoionizing resonance series 3p5np′[3/2]1,2, 3p5np′[1/2]1, and 3p5nf′[5/2]3 of Ar have been investigated exciting from the two metastable states 3p54s[3/2]2 and 3p54s′[1/2]0 in the photon energy range of 32500–35600 cm−1 with an experimental bandwidth of ∼0.1 cm−1. The excitation spectra of the even-parity autoionizing resonance series show typical asymmetric line shapes. New level energies, quantum defects, line profile index and resonance widths, resonance lifetime and reduced widths of the autoionizing resonances are derived by a Fano-type line-shape analysis. The line profile index q and the resonance widths Γ are shown to be approximately proportional to the effective principal quantum number n*. The line separation of the 3p5np′ autoionizing resonances is discussed.

General model of depolarization and transfer of polarization of singly ionized atoms by collisions with hydrogen atoms

Simulations of the generation of the atomic polarization is necessary for interpreting the second solar spectrum. For this purpose, it is important to rigorously determine the effects of the isotropic collisions with neutral hydrogen on the atomic polarization of the neutral atoms, ionized atoms and molecules. Our aim is to treat in generality the problem of depolarizing isotropic collisions between singly ionized atoms and neutral hydrogen in its ground state. Using our numerical code, we computed the collisional depolarization rates of the p-levels of ions for large number of values of the effective principal quantum number n* and the Unsöld energy Ep. Then, genetic programming has been utilized to fit the available depolarization rates. As a result, strongly non-linear relationships between the collisional depolarization rates, n* and Ep are obtained, and are shown to reproduce the original data with accuracy clearly better than 10%. These relationships allow quick calculations of the depolarizing collisional rates of any simple ion which is very useful for the solar physics community. In addition, the depolarization rates associated to the complex ions and to the hyperfine levels can be easily derived from our results. In this work we have shown that by using powerful numerical approach and our collisional method, general model giving the depolarization of the ions can be obtained to be exploited for solar applications.

Rydberg states of the HeH+ ion calculated by ab initio methods: potential energies and effective principal quantum numbers

More than 80 excited electronic states of the hydrohelium ion HeH+ of 1, 3Σ+, 1, 3Π, 1, 3Δ, 1, 3Φ and 1, 3Γ symmetry have been calculated ab initio up to n = 6 for internuclear distances ranging from 0.5 to 100 bohr. The computations involve a configuration interaction (CI) treatment based on a home-made suite of programs that uses special basis sets designed for the representation of molecular Rydberg states. The results are compared with previous computations where these are available (up to n = 4), and it is found that except for the very lowest excited states, the present energies are consistently lower than those obtained previously, with an average lowering corresponding to several hundred cm−1. It is shown that with the exception of its ground state, HeH+ is an effective one-electron system having an overall electronic structure similar to H+2. The interaction of the excited electron with the He+ 1s core electron causes small singlet–triplet splittings to appear and ℓ-mixing interactions to occur, that are not present in H+2.

Lifetimes of Rydberg states of Eu atoms**Project supported by the National Natural Science Foundation of China (Grant No. 11174218).

The radiative lifetimes of the Eu 4f76snp (8PJ or 10PJ) Rydberg states with J = 5/2 and 11/2 are investigated with a combination of multi-step laser excitation and pulsed electric field ionization, from which their dependence on the effective principal quantum number is observed. The lifetimes of 21 states are reported along with an evaluation of their experimental uncertainty. The influence of blackbody radiation, due to the oven temperature, on the lifetime of the higher-n states is detected. The non-hydrogen behavior of the investigated states is also observed.

Potentials of long-range cesium Rydberg molecule

Rydberg atom, with a large principal quantum number n, has big size, long lifetime, strong long-range interactions, and so on. These properties make Rydberg atoms potential candidate of quantum gate and single-photon source. Rydberg electron can interact with nearby ground-state atom, which is polarized by the Rydberg electron and is bound to the orbit of Rydberg electrons forming Rydberg molecule. As the kinetic energy of the Rydberg electron is very low, only the lowest partial waves will contribute to the molecular potential.#br#In this paper, the low electron-atom scattering with the semi-classical approximation is introduced, and the pseudopotential of interaction between Rydberg electron and ground-state atom is used to describe the long-range Rydberg molecular potential. Molecular potential curves for cesium (nS, n=30-60) are plotted according to the results of numerical computation, from which the outermost potential depth De and the equilibrium distance r0 of long-range cesium Rydberg molecule are deduced. Potential curves of cesium Rydberg molecules are consistent with the distribution curves in radial probability densities of cesium Rydberg electrons. Dependences of De and r0 on the principal quantum number n are investigated, this has an important role for the experimental measurements. The size of a Rydberg molecule depends on the equilibrium distance r0 and is proportional to the square of effective principal quantum number (n-δ )2. The calculated outermost potential depth De of Rydberg molecule becomes smaller with the increase of principal quantum number n. Rydberg molecule is very sensitive to the external field and can be used to measure and monitor weak signals.