Principal Quantum Number Research Articles

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.

Non-dissociative ionization cross section of the electronically excited H2 and D2 with atomic-limit principal quantum number n[formula omitted] 3 and 4

Electron-impact cross sections for non-dissociative ionization via the ground vibrational level (v=0) of the electronically excited H2 and D2 with atomic-limit principal quantum number n= 3–4, including HH¯1Σg+, B′1Σu+, D1Πu, GK1Σg+, I1Πg, J1Δg, O1Σg+, B′′B¯1Σu+, D′1Πu, P1Σg+, R1Πg, S1Δg, h3Σg+, e3Σu+, d3Πu, g3Σg+, i3Πg, j3Δg, f3Σu+, k3Πu, r3Πg, and s3Δg, are calculated by Gryzinski method in the present study. The cross sections for these states show a significant dependence on the threshold energy. The isotope effect of the cross sections for H2 and D2 is discussed. The computed cross sections of the electronic states with n= 1–4 from threshold up to 10 keV are fitted. The corresponding Maxwellian rate coefficients are derived from those cross sections with electron temperature from 1 to 10 eV.

Perpendicular magnetic anisotropy induced by 6p atomic layers: Crucial role of interface structural order

Two-dimensional atomic layers composed of metallic atoms with a high principal quantum number in valence shells are promising materials for applications in state-of-the-art magnetic devices due to the presence of a strong spin-orbit coupling. In this work we experimentally explored the effects of triggering perpendicular magnetic anisotropy (PMA) of ferromagnetic (FM) films by applying a series of $6p$-Pb, Bi and $4d$-Pd (reference) atomic layers with high-level atomic orbital and low-level lattice strain. Our research indicates that, compared with Pd atoms, Pb and Bi atoms can produce higher-strength PMA on adjacent FM films through orbital hybridization at the interface. Moreover, we demonstrate that the PMA induced by Pb and Bi atoms is highly sensitive to the ordering degree of the wetting layer which is determined by the interplay of surface free, cohesive, and strain energies within the grown materials; the Pb(Bi) atoms added on the wetting layer can enhance the PMA of the adjacent FM film only when the wetting layer maintains an ordered structure. This work clarifies the critical interface effects of $6p$-HMs on the FM layer, thus providing important clues to increase control over the spin-orbit interaction engendered by HMs through the interface.

Rydberg Series Excitation of a Single Trapped ^{40}Ca^{+} Ion for Precision Measurements and Principal Quantum Number Scalings.

A complete set of spectroscopic data is indispensable when using Rydberg states of trapped ions for quantum information processing. We carried out Rydberg series spectroscopy for nS_{1/2} states with 38≤n≤65 and for nD_{5/2} states with 37≤n≤50 on a single trapped ^{40}Ca^{+} ion. We determined the ionization energy of 2 870 575.582(15)GHz, 60 times more accurately as compared to the accepted value and contradicting it by 7.5 standard deviations. We confirm quantum defect values of δ_{S_{1/2}}=1.802 995(5) and δ_{D_{5/2}}=0.626 888(9) for nS_{1/2} and nD_{5/2} states, respectively, which allow for unambiguous addressing of Rydberg levels of Ca^{+} ions. Our measurements confirm Rydberg ion scaling properties, e.g., for blackbody induced ionization, linewidths and excitation strengths.

Electron-LO-phonon intrasubband scattering rates in a hollow cylinder under the influence of a uniform axial applied magnetic field

Scattering rates arising from the interactions of electrons with bulk longitudinal optical (LO) phonon modes in a hollow cylinder are calculated as functions of the inner radius and the uniform axial applied magnetic field. Now, the specific nature of electron-phonon interactions mainly depends on the character of the energy spectrum of electrons. As is well known, in cylindrical quantum wires, the application of a parallel magnetic field lifts the double degeneracy of the non-zero azimuthal quantum number states; \(m \not = 0\); irrespective of all electron’s radial quantum number l states. In fact, this Zeeman splitting is such that the \(m < 0\) electron’s energy subbands initially decrease with the increase of the parallel applied magnetic field. In a solid cylinder, the lowest-order; {\(l = 1;\,m = 0\)} subband is always the ground state. In a hollow cylinder, however, as the axial applied magnetic field is increased, the electron’s energy subbands take turns at becoming the ground state; following the sequence \(\lbrace m=0,-1,-2...~ -N\rbrace\) of azimuthal quantum numbers. Furthermore, in a hollow cylinder, in general, the electron’s energy separations between any two subbands are less than the LO phonon energy except for exceptionally high magnetic fields, and some highest-order quantum number states. In view of this, the discussion of the energy relaxation here is focused mainly on intrasubband scattering of electrons and only within the lowest-order {\(l = 1;\,m = 0\)} electron’s energy subband. The intrasubband scattering rates are found to be characterized by shallow minima in their variations with the inner radius, again, for a fixed outer radius. This feature is a consequence of a balance between two seemingly conflicting effects of the electron’s confinement by the inner and outer walls of the hollow cylinder. First; increased confinement of the charge carriers generally leads to the enhancement of the rates. Second; the presence of a hole in a hollow cylinder leads to a significant suppression of the scattering rates. The intrasubband scattering rates also show a somewhat parabolic increase in their variations with the applied magnetic field; an increase which is more pronounced in a relatively thick hollow cylinder.

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.

High-lying Rydberg atoms surviving intense laser fields

In this work, we have experimentally and theoretically investigated the high-lying Rydberg state excitation for noble gas atoms subject to an intense near-infrared laser field. To obtain the signal of these high-lying Rydberg atoms which are further ionized by a constant electric field $({F}_{c})$, coincident detection of the photoelectrons and photoions with well-chosen arrival times is performed. Based on a fitting to the time dependence of the coincidently measured photofragment yields with a semiempirical formula, we can extract a principal quantum number distribution (PQND) of the population of the excited states in a range closely related to the strength of ${F}_{c}$. The extracted valid PQND is in qualitative agreement with a semiclassical model calculation. Our work thus provides a method to extract the PQND and study the ultrafast high-lying Rydberg state excitation.

Possible high-spin states in hydrogenated C60 molecules

Atoms with principal quantum number smaller than 2 are traditionally regarded to be nonmagnetic, but recent appealing advances in the understanding of ferromagnetism in materials such as graphene nanoribbons with zigzag edges and triangular graphene nanoflakes pave a novel way to investigate magnetism in materials merely with $s$ and $p$ orbitals. Opening the $p$ shell turns out to be essential for the production of unpaired electrons, and the normal chemical treatment is to modify the molecules with various atoms and groups such as hydrogen. In this paper, we combine first-principles calculations with the Hubbard model to investigate magnetic properties in hydrogenated ${\mathrm{C}}_{60}$ (${\mathrm{C}}_{60}{\mathrm{H}}_{n}$). For the enumerated ${\mathrm{C}}_{60}{\mathrm{H}}_{2}$ structures, there are nine configurations with ferromagnetic ground states and one configuration with an antiferromagnetic ground state. We have proposed a pair magnetic interaction model to predict the magnetic order, as well as a general rule to screen the maximum local magnetic moments for given $n$. When $n=3$ and 4, our model can give a good prediction of magnetic order, which is confirmed by the first-principles calculations. We find that the maximum number of unpaired electrons in ${\mathrm{C}}_{60}{\mathrm{H}}_{n}$ is smaller than the number of hydrogen atoms when $n\ensuremath{\ge}6$.

Where are 3P and higher P -wave states in the charmonium family?

How to hunt for higher $P$-wave states of charmonium is still an open topic when $2P$ charmonia were identified. {In this work, we present an unquenched quark model calculation to illustrate the spectroscopy behavior of these $3P$, $4P$ and $5P$ states in charmonium family.} For the $3P$ charmonia, the predicted masses are around 4.2 GeV and their two-body open-charm decay behaviors were given, by which we propose that searching for these $3P$ states via their open-charm decay channels from $\gamma\gamma$ fusion and $B$ decay can be accessible at future experiment like LHCb and Belle II. We continue to calculate the masses of these $4P$ and $5P$ charmonia. Combing with these calculated results of higher $P$-wave states of charmonium, we find that the coupled-channel effect becomes more obvious with increasing the radial quantum number, which can be understood well by the modified Godfrey-Isgur model with screened potential.

Analysis of the Fine Structure of the D‐Exciton Shell in Cuprous Oxide

The exciton states in cuprous oxide show a pronounced fine structure splitting associated with the crystal environment and the resulting electronic band structure. High‐resolution spectroscopy reveals an especially pronounced splitting of the yellow D excitons with one state pushed above any other state with the same principal quantum number. This large splitting offset is related to a strong mixing of these D states with the 1S exciton of the green series, as suggested by previously published calculations. Here, a detailed comparison of this theory with experimental data is given, which leads to a complete reassignment of the experimentally observed D exciton lines. The origin of different amounts of green admixture to D‐envelope states is deduced by analyzing the different terms of the Hamiltonian. The yellow–green mixing leads to level repulsion and induces an exchange interaction splitting to D‐envelope states, from which one of them becomes the highest state within each multiplet. Furthermore, the assignment of D exciton states according to their total angular momentum F is given and corrects an earlier description given in a former study.

Photoionization and Electron-Ion Recombination of n = 1 to Very High n-Values of Hydrogenic Ions

Single electron hydrogen or hydrogenic ions have analytical forms to evaluate the atomic parameters for the inverse processes of photoionization and electron-ion recombination (H I + hν↔ H II + e) where H is hydrogen. Studies of these processes have continued until the present day (i) as the computations are restricted to lower principle quantum number n and (ii) to improve the accuracy. The analytical expressions have many terms and there are numerical instabilities arising from cancellations of terms. Strategies for fast convergence of contributions were developed but precise computations are still limited to lower n. This report gives a brief review of the earlier precise methodologies for hydrogen, and presents numerical tables of photoionization cross sections (σPI), and electron-ion recombination rate coefficients (αRC) obtained from recombination cross sections (σRC) for all n values going to a very high value of 800. σPI was obtained using the precise formalism of Burgess and Seaton, and Burgess. αRC was obtained through a finite integration that converge recombination exactly as implemented in the unified method of recombination of Nahar and Pradhan. Since the total electron-ion recombination includes all levels for n = 1–∞, the total asymptotic contribution of n = 801–∞, called the top-up, is obtained through a n−3 formula. A FORTRAN program “hpxrrc.f” is provided to compute photoionization cross sections, recombination cross sections and rate coefficients for any nl. The results on hydrogen atom can be used to obtain those for any hydrogenic ion of charge z through z-scaling relations provided in the theory section. The present results are of high precision and complete for astrophysical modelings.

Study of Klein Gordon Equation with Minimum Length Effect for Woods-Saxon Potetial using Nikiforov-Uvarov Functional Analysis

The equation of Klein-Gordon for Woods-Saxon potential was discussed in the minimal length effect. We have found the completion of this equation using an approximation by suggesting a new wave function. The Klein-Gordon equation in minimal length formalism for the Woods-Sadon potential is reduced to the form of the Schrodinger-like equation. Then the equation was accomplished by Nikiforov-Uvarov Functional Analysis (NUFA) with Pekeris approximation. This method is applied to gain the radial eigensolutions with chosen exponential-type potential models. The method of NUFA is more compatible by eliminating vanishing the strict mathematical manipulations found in other methods. The energy calculation results showed that angular momentum, radial quantum number, minimal length parameter, and atomic mass influenced it. The higher the radial quantum number and angular momentum, the lower the energg. In contrast to the the minimal length, the energy will increase in value when the minimal length parameter is enlarged. An increase in atomic mass also causes energy to increase as the radial quantum number and angular momentum are held constant.

The very high n Rydberg series of Ar16+ in Alcator C-Mod tokamak plasmas

X-ray transitions of the very high-n Rydberg series in Ar16+ have been observed from Alcator C-Mod tokamak plasmas. Individual emission lines up to 1s16p-1s2 have been resolved and the central chord line brightnesses with principal quantum number n between 7 and 16 are generally found to decay as 1/n α , with α slightly larger than 3. In the plasma periphery, emission from 1s9p-1s2 and 1s10p-1s2 are found to be significantly enhanced relative to this decrease, indicative of selected population of these levels through charge exchange between background neutral deuterium in the ground state and Ar17+. An unresolved feature between the wavelengths of 1s27p-1s2 and 1s30p-1s2 is also present, which arises through charge exchange with neutral deuterium in the n* = 3 excited state. The brightnesses of transitions populated by charge exchange are spatially up/down asymmetric, with an excess on the side of the magnetic surface X-point. The relative brightness of the unresolved very high-n feature compared to 1s7p-1s2 is found to increase with electron temperature and decrease with electron density. Simulations of line emission just on the long wavelength side of the Ar16+ ionization limit indicate that the principal quantum number decay exponent is closer to α = 4 at very high n. The brightness dependence on n below 16 is in excellent agreement with calculations from the flexible atomic code package.

Cold ion chemistry within a Rydberg-electron orbit: test of the spectator role of the Rydberg electron in the He(n) + CO → C(n′) + O + He reaction

Recently, a new method has been introduced to study ion-molecule reactions at very low collision energies, down to below k B ⋅ 1 K (Allmendinger et al 2016 ChemPhysChem 17 3596). To eliminate the acceleration of the ions by stray electric fields in the reaction volume, the reactions are observed within the orbit of a Rydberg electron with large principal quantum number n > 20. This electron is assumed not to influence the reaction taking place between the ion core and the neutral molecules. This assumption is tested here with the example of the He(n) + CO → C(n′) + O + He reaction, which is expected to be equivalent to the He+ + CO → C+ + O + He reaction, using a merged-beam approach enabling measurements of relative reaction rates for collision energies E coll in the range from 0 to about k B ⋅ 25 K with a collision-energy resolution of ∼k B ⋅ 200 mK at E coll = 0. In contrast to the other ion-molecule reactions studied so far with this method, the atomic ion product (C+) is in its electronic ground state and does not have rotational and vibrational degrees of freedom so that the corresponding Rydberg product [C(n′)] cannot decay by autoionization. Consequently, one can investigate whether the principal quantum number is effectively conserved, as would be expected in the spectator Rydberg-electron model. We measure the distribution of principal quantum numbers of the reactant He(n) and product C(n′) Rydberg atoms by pulsed-field ionization following initial preparation of He(n) in states with n values between 30 and 45 and observe that the principal quantum number of the Rydberg electron is conserved during the reaction. This observation indicates that the Rydberg electron is not affected by the reaction, from which we can conclude that it does not affect the reaction either. This conclusion is strengthened by measurements of the collision-energy-dependent reaction yields at n = 30, 35 and 40, which exhibit the same behavior, i.e. a marked decrease below E coll ≈ k B ⋅ 5 K.

Two-parameter localization for eigenfunctions of a Schrödinger operator in balls and spherical shells

Here, we investigate the two-parameter high-frequency localization for the eigenfunctions of a Schrödinger operator with a singular inverse square potential in high-dimensional balls and spherical shells as the azimuthal quantum number l and the principal quantum number k tend to infinity simultaneously, while keeping their ratio as a constant, generalizing the classical one-parameter localization for Laplacian eigenfunctions [B.-T. Nguyen and D. S. Grebenkov, SIAM J. Appl. Math. 73, 780–803 (2013)]. We prove that the eigenfunctions in balls are localized around an intermediate sphere whose radius is increasing with respect to the l–k ratio. The eigenfunctions decay exponentially inside the localized sphere and decay polynomially outside. Furthermore, we discover a novel phase transition for the eigenfunctions in spherical shells as the l–k ratio crosses a critical value. In the supercritical case, the eigenfunctions are localized around a sphere between the inner and outer boundaries of the spherical shell. In the critical case, the eigenfunctions are localized around the inner boundary. In the subcritical case, no localization could be observed, giving rise to localization breaking.

Second-harmonic generation of blue series excitons and magnetoexcitons in Cu2O

Nonlinear optical studies of the yellow and green exciton series in ${\mathrm{Cu}}_{2}\mathrm{O}$ have been reported for more than 40 years. Because of the band structure (the two highest even-parity valence and lowest conduction bands), the $S$ excitons of the two lowest exciton series are dipole-forbidden for one-photon absorption and thus dipole-allowed for two-photon absorption. There is an odd-parity higher conduction band that leads with the two even-parity valence bands to the blue and violet exciton series. We report on second-harmonic generation (SHG) of the blue exciton series. The odd-parity $S$-exciton SHG is due to a dipole-quadrupole excitation and a dipole emission process. Because of their high oscillator strength density, polariton effects have to be taken into account, since resonances might be shifted to higher energies by up to $10\phantom{\rule{4pt}{0ex}}\mathrm{me}\mathrm{V}$ compared to the transverse exciton energies. The polariton dispersion for the blue excitons up to $n=4$ is calculated and compared to the experimental results. In magnetic fields up to $10\phantom{\rule{4pt}{0ex}}\mathrm{T}$ applied in a Voigt configuration ($\mathbf{B}\ensuremath{\perp}\mathbf{k}$), SHG of $S$ excitons by a dipole-dipole excitation is observed, which is due to the admixture of dipole-dipole excited $P$ excitons by the effective electric field from the magneto-Stark effect (MSE). From the analysis of the diamagnetic shift and the MSE interaction of the three-level system of $1S, 2S$, and $2P$ excitons, we derive experimental results for the ratio $\ensuremath{\langle}{r}_{n,l}^{2}\ensuremath{\rangle}/{\ensuremath{\mu}}_{X}$ between the average of radius squared for the three states and the reduced exciton mass. For higher principal quantum number states, we observe magnetoexcitons up to $n=8$. We analyze their magnetic field dependence and derive the electron effective-mass values for the crystalline orientations [111], $[1\overline{1}0]$, and [001].

On the interaction of an electron with a nonuniform electric field under the influence of a cut-off point induced by the spiral dislocation topology

We analyse the interaction of an electron with a nonuniform electric field under the influence of a cut-off point induced by the topology of a spiral dislocation. We show that the presence of the cut-off point yields eigenvalues of energy which are infinitely degenerated. Besides, we show that the energy eigenvalues are influenced by the topology of the defect even though there is no interaction between the electron and the defect. Furthermore, we show that the influence of a cut-off point induced by the spiral dislocation topology gives rise to a non-null revival time associated with the radial quantum number.

A crystalline tri-thorium cluster with σ-aromatic metal-metal bonding.

Metal-metal bonding is a widely studied area of chemistry1-3, and has become a mature field spanning numerous d transition metal and main group complexes4-7. By contrast, actinide-actinide bonding, which is predicted to be weak8, is currently restricted to spectroscopically detected gas-phase U2 and Th2 (refs. 9,10), U2H2 and U2H4 in frozen matrices at 6-7K (refs. 11,12), or fullerene-encapsulated U2 (ref. 13). Furthermore, attempts to prepare thorium-thorium bonds in frozen matrices have produced only ThHn (n = 1-4)14. Thus, there are no isolable actinide-actinide bonds under normal conditions. Computational investigations have explored the probable nature of actinide-actinide bonding15, concentrating on localized σ-, π-, and δ-bonding models paralleling dtransition metal analogues, but predictions in relativistic regimes are challenging and have remained experimentally unverified. Here, we report thorium-thorium bonding in a crystalline cluster, prepared and isolated under normal experimental conditions. The cluster exhibits a diamagnetic, closed-shell singlet groundstate with a valence-delocalized three-centre-two-electron σ-aromatic bond16,17 that is counter to the focus of previous theoretical predictions. The experimental discovery of actinide σ-aromatic bonding adds to main group and dtransition metal analogues, extending delocalized σ-aromatic bonding to the heaviest elements in the periodic table and to principal quantum number six, and constitutes a new approach to elaborate actinide-actinide bonding.

Rydberg excitons in synthetic cuprous oxide Cu2O

High-lying Rydberg states of Mott-Wannier excitons are receiving considerable interest due to the possibility of adding long-range interactions to the physics of exciton-polaritons. Here, we study Rydberg excitation in bulk synthetic cuprous oxide grown by the optical float zone technique and compare the result with natural samples. X-ray characterization confirms both materials are mostly single crystal, and mid-infrared transmission spectroscopy revealed little difference between synthetic and natural material. The synthetic samples show principal quantum numbers up to $n=10$, exhibit additional absorption lines, plus enhanced spatial broadening and spatial inhomogeneity. Room temperature and cryogenic photoluminescence measurements reveal a significant excess of copper vacancies in the synthetic material. These measurements provide a route towards achieving \mbox{high-$n$} excitons in synthetic crystals, opening a route to scalable quantum devices.

A Study on Radial Properties of Hydrogenic Ions using Laguerre Polynomials

A preliminary study has been carried out on the radial properties of the hydrogenic systems using the Laguerre polynomial. The aims of this study were to investigate radial properties of hydrogenic ions by evaluating the radial probability density, the most probable radial distance and the average distance of electron from the nuclei of hydrogenic ions. Based on the research results for Z=1 to Z=10 with the principal quantum number (n)=1 to 5, 150 radial function forms, 150 radial probability density, 150 most probable radial distance values and 150 electron average distance values were obtained. The most probable radial distance from the nucleus was compared with the Bohr model and showed the same results as the Bohr model.