Effective Quantum Number Research Articles

Theoretical investigation of Rydberg states of He2 using the R-matrix method

Bound states of the He 2 molecule are determined from an electron-He 2 + collision calculation using the R-matrix method. The calculations are performed at a moderately dense grid of 35 internuclear separations are used to characterise He 2 Rydberg states with n ≤ 7 . Potential energy curves for singlet 1 Σ g + , 1 Σ u + , 1 Π g , 1 Π u and 1 Δ u and triplet 3 Σ g + , 3 Σ u + , 3 Π g , 3 Π u and 3 Δ u Rydberg states of He 2 as well as effective quantum numbers as a function of internuclear separation are calculated. Based on the potential energy curves of A 1 Σ g + , B 1 Π g , C 1 Σ u + , F 1 Π u and a 3 Σ u + , b 3 Π g , c 3 Σ g + , f 3 Π u , spectroscopic parameters ( T e , D e , R e , ω e , ω e x e , α e and B e ) of these states have been determined and compared with theoretical and experimental data available.

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.

Charge Transfer and Level Lifetime in Molecular Photon-Absorption upon the Quantum Impedance Lorentz Oscillator.

On the strength of the new quantum impedance Lorentz oscillator (QILO) model, a charge-transfer method in molecular photon-absorption is proposed and imaged via the numerical simulations of 1- and 2-photon-absorption (1PA and 2PA) behaviors of the organic compounds LB3 and M4 in this paper. According to the frequencies at the peaks and the full width at half-maximums (FWHMs) of the linear absorptive spectra of the two compounds, we first calculate the effective quantum numbers before and after the electronic transitions. Thus, we obtain the molecular average dipole moments, i.e., 1.8728 × 10-29 C·m (5.6145 D) for LB3 and 1.9626 × 10-29 C·m (5.8838 D) for M4 in the ground state in the tetrahydrofuran (THF) solvent. Then, the molecular 2PA cross sections corresponding to wavelength are theoretically inferred and figured out by QILO. As a result, the theoretical cross sections turn out to be in good agreement with the experimental ones. Our results reveal such a charge-transfer image in 1PA near wavelength 425 nm, where an atomic electron of LB3 jumps from the ground-state ellipse orbit with the semimajor axis ai = 1.2492 × 10-10m = 1.2492 Å and semiminor axis bi = 0.4363 Å to the excited-state circle (aj = bj = 2.5399 Å). In addition, during its 2PA process, the same transitional electron in the ground state is excited to the elliptic orbit with aj = 2.5399 Å and bj =1.3808 Å, in which the molecular dipole moment reaches as high as 3.4109 × 10-29 C·m (10.2256 D). In addition, we obtain a level-lifetime formula with the microparticle collision idea of thermal motion, which indicates that the level lifetime is proportional (not inverse) to the damping coefficient or FWHM of an absorptive spectrum. The lifetimes of the two compounds at some excited states are calculated and presented. This formula may be used as an experimental method to verify 1PA and 2PA transition selection rules. The QILO model exhibits the advantage of simplifying the calculation complexity and reducing the high cost associated with the first principle in dealing with quantum properties of optoelectronic materials.

Generalized effective-potential Landau theory for a tunable state-dependent hexagonal optical lattice

We analytically study the ground-state phase diagrams of ultracold bosons with various effective magnetic quantum number m in a state-dependent hexagonal optical lattice by using the generalized effective-potential Landau theory. While the first-order results correspond to mean field approximation, our third-order analytical results are in excellent agreement with the previous cluster Gutzwiller solutions. Furthermore, we reveal the reason why the regions of the Mott lobes (n,n) in phase diagrams for m=0.02 are unexpectedly expanded with increasing J/U in deep lattice. Besides, at the tips of the Mott lobes, the critical values Jc/U have also been obtained analytically as functions of the site-offset energy ϵ/U. Comparing the results obtained by our third-order analytical calculations and previous numerical solutions, we find that the relative deviations are less than 9% (5%) for Mott insulator with (half-)integer filling.

Multi-step laser excitation of the highly excited 5snf 1F3 states of cadmium

We report excitation energies, term energies and effective quantum numbers n of the 5snf 1F3 Rydberg series of cadmium from n = 14 to 45 using a three-step laser excitation scheme via the 5s5d 1D2 intermediate state. The experiment was performed using three dye lasers simultaneously pumped by the second (532 nm) and third (355 nm) harmonics of a Q-switch Nd: YAG laser in conjunction with an atomic beam setup. A two-parameter fit to the transition energies of the 5snf 1F3 series yields the binding energy of the 5s5d 1D2 level as 13,320.33 ± 0.03 cm−1 and consequently the first ionization potential of cadmium is determined as 72,540.06 ± 0.15 cm−1, which is in good agreement with the previously reported 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

On the population density of the argon excited levels in a high power impulse magnetron sputtering discharge

Population densities of excited states of argon atoms in a high power impulse magnetron sputtering (HiPIMS) discharge are examined using a global discharge model and a collisional-radiative model. Here, the ionization region model (IRM) and the Orsay Boltzmann equation for electrons coupled with ionization and excited states kinetics (OBELIX) model are combined to obtain the population densities of the excited levels of the argon atom in a HiPIMS discharge. The IRM is a global plasma chemistry model based on particle and energy conservation of HiPIMS discharges. OBELIX is a collisional-radiative model where the electron energy distribution is calculated self-consistently from an isotropic Boltzmann equation. The collisional model constitutes 65 individual and effective excited levels of the argon atom. We demonstrate that the reduced population density of high-lying excited argon states scales with (p*)−6, where p* is the effective quantum number, indicating the presence of a multistep ladder-like excitation scheme, also called an excitation saturation. The reason for this is the dominance of electron impact processes in the population and de-population of high-lying argon states in combination with a negligible electron–ion recombination.

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.

The corrections to quantum defect for high n Rydberg atoms

Quantum defect converges to constant value at high Rydberg series in quantum defect theory. In experiments, it is usually distorted by the shift of ionization limit, external magnetic field, or external electric filed. The shift of ionization limit has cubic power of effective quantum number correction to the quantum defect, while the magnetic and electric field has seventh and tenth power of effective quantum number correction, respectively. The simulation shows that the electric field correction to the quantum defect is very small at low Rydberg states, but it increases quickly at high Rydberg states and even exceeds the correction from the shift of ionization limit. The correction to quantum defect is up to 0.93 at n0*=80. The two scenarios can be distinguished by fitting the experimental data, and the energy spectrum can be obtained more accurately by fixing the corrections.

Quantum impedance Lorentz oscillator and its 1- and 2-photon-absorption applications

In this paper, a classical Lorentz oscillator is quantized via Bohr–Sommerfeld quantum theory and 1- and 2-photon absorption (1PA and 2PA) selection rules of quantum mechanics. Based on the Bohr–Sommerfeld model of a hydrogen-like atom in the adiabatic approximation, the computational formulas of the linear and nonlinear parameters and the damping coefficient of the quantized oscillator are derived and further expressed in terms of microphysical quantities, such as electronic charge and mass, Bohr radius, and effective quantum number. In accordance with Boltzmann thermal equilibrium distribution, here, the atom number density in general electric susceptibility is changed to the energy level transition one from the initial to the final state at equilibrium between atomic emission and absorption under light field. A new relationship is proposed to determine the transition eigenfrequency according to the peak frequency and full width at half maximum of an absorption spectrum. Our theoretical simulations of the 1PA spectra of atomic hydrogen and lithium and 1PA and 2PA spectra of two kinds of organic molecules turn out to be in good agreement with the experimental ones. These results suggest that our advancement in the quantization of the Lorentz oscillator is likely successful to make it available for use in the quantitative description of atomic or molecular 1PA and 2PA processes. Generally, the improved Lorentz oscillator may also be more suitable for approximating both linear and nonlinear properties of many dielectric or optoelectronic materials due to its relative simplicity.

Quantum Defect and Common Fractions

The common fractions N1/N2, where N1 and N2 – the small integers, quite often are used at the quantum-mechanical consideration of microcosm objects (for example, fractional charges of quarks and some quantum characteristics, such as particles spin). Recently the fractional quantum Hall effect was discovered, and common fractions have considerably expanded their presence in microcosm physics. The theory of the fractional quantum Hall effect has appeared nontrivial, so the Nobel Prize on physics in 1998 was awarded not only for discovery of the effect in 1982 (Daniel Tsui and Horst Stӧrmer) but also for the theory creation in 1983 (Robert Laughlin). And now one more sensational discovery: common fractions were «detected» at the analysis of experimental characteristics of hydrogen-like atoms and ions (with only one electron on an outer shell). It has appeared, that the effective main quantum number of outer shell electron, that is, subject to quantum defect (Rydberg correction), can be expressed in common fractions.

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.

Competition between heating and cooling effects in an optomechanical oscillator using a squeezed field

Squeezed light is a useful phenomenon that can be exploited to improve the sensitivity of specific classes of detectors based on optomechanical effects. In this study, the effect of squeezed light was evaluated by explicitly examining the role of the squeezing parameters on the final effective temperature of the oscillator. The results show that the observed cooling and heating effects are strongly dependent on the squeezing parameters and the phase. Using an oscillator of 10.1 MHz driven by a 1064-nm laser, the lowest effective temperature and quantum number are three orders of magnitude smaller compared to the case of no squeezing; especially, these minimum values are obtained at the squeezing phase of about 0.8π. This study highlighted important insights for the optimization of cooling efficiency using squeezed light.

Квантовый дефект и простые дроби

The common fractions N1/N2, where N1 and N2 - the small integers, quite often are used at the quantum-mechanical description of microcosm objects (for example, fractional charges of quarks and some quantum characteristics, such as particles spin). Recently the fractional quantum Hall effect was discovered, and common fractions have considerably expanded their presence in microcosm physics. The theory of the fractional quantum Hall effect has appeared nontrivial, so the Nobel Prize on physics in 1998 was awarded not only for discovery of the effect in 1982 (Daniel Tsui and Horst Störmer) but also for the theory creation in 1983 (Robert Laughlin). And now one more sensational discovery: common fractions were «detected» at the analysis of experimental characteristics of «hydrogen-like» atoms and ions (with only one electron on an outer shell). It has appeared, that the effective main quantum number of outer shell electron, that is, subject to quantum defect (Rydberg correction), can be expressed in common fractions.

Scattering of charged fermion to two-dimensional wormhole with constant axial magnetic flux

Scattering of charged fermion with (1+2)-dimensional wormhole in the presence of constant axial magnetic flux is explored. By extending the class of fermionic solutions of the Dirac equation in the curved space of wormhole surface to include normal modes with real energy and momentum, we found a quantum selection rule for the scattering of fermion waves to the wormhole. The newly found momentum–angular momentum relation implies that only fermion with the quantized momentum k=m'/asqrt{q} can be transmitted through the hole. The allowed momentum is proportional to an effective angular momentum quantum number m' and inversely proportional to the radius of the throat of the wormhole a. Flux dependence of the effective angular momentum quantum number permits us to select fermions that can pass through according to their momenta. A conservation law is also naturally enforced in terms of the unitarity condition among the incident, reflected, and transmitted waves. The scattering involving quasinormal modes (QNMs) of fermionic states in the wormhole is subsequently explored. It is found that the transmitted waves through the wormhole for all scenarios involving QNMs are mostly suppressed and decaying in time. In the case of QNMs scattering, the unitarity condition is violated but a more generic relation of the scattering coefficients is established. When the magnetic flux phi =mhc/e, i.e., quantized in units of the magnetic flux quantum hc/e, the fermion will tunnel through the wormhole with zero reflection.

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.