States Of Hydrogen Atom Research Articles

The cosmological constant and energy quantization in hydrogen atom in light of a doubly special relativity with a minimum speed

We show the existence of an invariant minimum speed in space–time as a new fundamental constant of nature by forming the basis of a doubly special relativity so-called Symmetrical Special Relativity (SSR). Such an observer-independent minimum speed emerges from the Dirac’s Large Number Hypothesis (LNH), which leads us to build SSR-theory. This allows us to understand that the hydrogen atom represents the most stable bound state in the universe as being a fundamental structure in space–time with two speed limits, i.e., the speed of light c and a minimum speed V. So the symmetry in space–time given by an invariant minimum speed, which has origin in the electrical and gravitational bound states in hydrogen atom is able to provide a deeper understanding of the proton-electron mass ratio. We investigate how the minimum speed plays a fundamental role in the quantization of energy in hydrogen atom. The minimum speed V is related to a preferred background frame (ultra-referential SV) represented by the surface of a dark cone forbidden to the world line of any particle. It is related to the vacuum energy that leads to the cosmological constant. An investigation of SSR is done and a hypothetical experiment of Bose–Einstein condensation is proposed to detect the minimum speed as a possible Lorentz violation at very low energies.

A comparative study of transition oscillator strengths and static polarizabilities of the hydrogen atom confined in Gaussian potential

Abstract The multipole (dipole, quadrupole, and octopole) photon-absorption transition oscillator strengths for the ground state of hydrogen atom confined in Gaussian potential are investigated for a great variety of potential depths and confining radii. It is interestingly found that at fixed potential depth the gradual increase of confining radius shows first destructive and then constructive effect on the multipole oscillator strengths. Such an effect can be understood from the overlap between the initial and final states. Multipole polarizabilities of the system are obtained through the sum-over-states formalism where the contributions from both the bound and continuum spectra of the system are included. Although the separate bound and continuum contributions can not be determined accurately, due to the long-range nature of the Coulomb potential introduced by the nucleus, their summations can be obtained to reasonably good accuracy, leading to fast convergence of numerical calculations of multipole polarizabilities. The present results are compared with previous calculations available in the literature. Although good agreement is observed for the dipole polarizability, significant differences exist in the quadrupole polarizability and orders-of-magnitude differences appear in the octopole polarizability. The possible reason for such large differences is analyzed by comparing the sum rule of corresponding oscillator strengths.

The local state of hydrogen atoms and proton transfer in the crystal structure of natural berborite, Be2(BO3)(OH)‧H2O: Low-temperature single crystal X-ray analysis, IR and 1H NMR spectroscopy, and crystal chemistry and structural complexity of beryllium borates

Beryllium borates are mostly water-free, with the natural mineral berborite, Be2(BO3)(OH)‧H2O, being the only known exception. A sample of berborite from the Vevja quarry, Tvedalen, Larvik, Norway was studied by single-crystal X-ray diffraction, variable temperature 1H nuclear magnetic resonance (NMR), and infrared (IR) spectroscopy. The crystal structure is the trigonal 1T-polytype with space group P321 and is based on electroneutral [BeØ2(BΔO3)]-layers formed by BeO3Ø-tetrahedra and BΔO3-triangles. Due to statistical occupancy of the Ø-ligand by hydroxyl groups (50%) and water molecules (50%), berborite-1T and its related polytypes are characterized by complex systems of hydrogen bonds. Accurate analysis shows that in the structure of berborite there are two independent systems of hydrogen bonds involving mobile protons. Lowering of temperature results in the complete stabilization of protons at the critical “freezing-point” of 243 K. The crystal chemistry and complexity of layered beryllium borates are discussed. Despite the broad structural diversity of these compounds, their crystal structures are classified as simple (20–100 bits per unit cell) or intermediately complex (100–500 bits per unit cell). The complexity of berborite polytypes increases in accordance with the quantity of layers from 29.038 (1T) to 78.077 (2H), which is in good agreement with the negative contribution of complexity to the configurational entropy.

Rayleigh and Raman scattering cross-sections and phase matrices of the ground-state hydrogen atom, and their astrophysical implications

ABSTRACT We present explicit expressions for Rayleigh and Raman scattering cross-sections and phase matrices of the ground 1s state hydrogen atom based on the Kramers–Heisenberg–Waller dispersion formula. The Rayleigh scattering leaves the hydrogen atom in the ground-state while the Raman scattering leaves the hydrogen atom in either ns (n ≥ 2; s-branch) or nd (n ≥ 3; d-branch) excited state, and the Raman scattering converts incident ultraviolet (UV) photons around the Lyman resonance lines into optical-infrared (IR) photons. We show that this Raman wavelength conversion of incident flat UV continuum in dense hydrogen gas with a column density of NH > 1021 cm−2 can produce broad emission features centred at Balmer, Paschen, and higher level lines, which would mimic Doppler-broadened hydrogen lines with the velocity width of ≳1000 km s−1 that could be misinterpreted as signatures of active galactic nuclei, supernovae, or fast stellar winds. We show that the phase matrix of the Rayleigh and Raman s-branch scatterings is identical to that of the Thomson scattering while the Raman d-branch scattering is more isotropic, thus the Paschen and higher level Raman features are depolarized compared to the Balmer features due to the flux contribution from the Raman d-branch. We argue that observations of the line widths, line flux ratios, and linear polarization of multiple optical/IR hydrogen lines are crucial to discriminate between the Raman-scattered broad emission features and Doppler-broadened emission lines.

The Adsorption Behavior of Hydrogen on the PuO2(111) Surface: A DFT+U Study

Based on density functional theory, a first-principles study of the adsorption behavior of hydrogen atoms on the PuO2(111) surface is carried out in this work. Models for three different surface morphologies of PuO2(111) are established. It is found that the surface with the outermost oxygen atom (sub outer Pu atom) morphology has the best stability. Based on this model, the adsorption energy, bader charge, and electronic density of the states of a hydrogen atom at different adsorption sites are calculated. Finally, we analyzed the process of hydrogen dissociation into hydrogen atoms on the surface using the cNEB method. The results indicate that the top position of the outermost oxygen atom and the bridge position of the second outermost plutonium atom are relatively stable adsorption configurations, where hydrogen atoms lose electrons and release heat, forming O-H bonds with oxygen atoms. The density of states of O-p orbital electrons will undergo significant changes, reflecting the hybridization of O-p and H-s orbital electrons, forming a stable bonding effect. The dissociation of hydrogen molecules into two hydrogen atoms adsorbed on the top of oxygen atoms requires crossing an energy barrier of 1.06 eV. The decrease in total energy indicates that hydrogen tends to exist on the PuO2(111) surface in a hydrogen atom state. The research results lay the foundation for theoretically exploring the hydrogenation corrosion mechanism of the PuO2(111) surface, providing theoretical support for exploring the corrosion aging of plutonium oxide, predicting the material properties of plutonium oxide under extreme and special environments.

The Excitation of the 3D and 4D States of Atomic Hydrogen by Electron Impact

The excitation cross-sections of the 3D and 4D states of atomic hydrogen at low incident energies (from 0.90 to 5.00 Ry) were calculated using the variational polarized orbital method, which is also called the hybrid theory. Up to 12 partial waves (L = 2 to 13) were used to obtain converged cross-sections at high energies. The importance of the long-range forces near the threshold region and the behavior of the cross-sections in that region are indicated. The S, P, and D cross-sections are needed if the total excitation cross-sections are measured in addition to the elastic cross-sections. These cross-sections are also useful if the cascade from the D to the P to the S states is considered in the diagnostics of solar and astrophysical observations.

The Zeldovich Number: A Universal Dimensionless Measure for the Electromagnetic Field

In this paper we extend the Zeldovich formula, which was originally derived for the free electromagnetic field and was interpreted as the number of photons. We show that our extended formula gives a universal dimensionless measure of the overall strength of electromagnetic fields: free fields and fields produced by various sources, in both the classical and the quantum theory. In particular, we find that this number—called here the Zeldovich number—for macroscopic systems is very large, of the order of 1020. For the hydrogen atom in the ground state, the Zeldovich number is equal to 0.025 and for the xenon atom it is around 50.

The effects of Heisenberg constraint on the classical cross sections in proton hydrogen collision

The interaction between a proton and a ground state hydrogen atom is studied using a standard three-body classical trajectory Monte Carlo (CTMC) and a quasi-classical trajectory Monte Carlo (QCTMC) model where the quantum feature of the collision system is mimicked using the model potential in the Hamiltonian as was proposed by Kirschbaum and Wilets (1980 Phys. Rev. A 21 834). The influence of the choice of the model potential parameters (α, ξ) on the initial radial and momentum distribution of the electron are analyzed and optimized. We found that although these distributions may not be as close to the quantum results as the distribution of standard CTMC results, we can find the combination of the (α, ξ) where the calculated cross sections are closer to the experimental data and closer to the results obtained quantum mechanically. We show that the choice of 3 < α < 5 is reasonable. To validate our observation, we present cross sections for ionization, excitation, charge exchange (CX), and state selective CX to the projectile bound state. Calculations are carried out in the projectile energy range between 10 and 1000 keV amu−1.

Fully-Stripped Beryllium-Ion Collisions with 2ℓm States of Atomic Hydrogen: Target Excitation and Ionisation cross Sections

The wave-packet convergent close-coupling approach is used to calculate integrated target excitation and ionisation cross sections in bare beryllium-ion collisions with the 2ℓm states of atomic hydrogen (where n, ℓ and m are the principal, orbital angular momentum and magnetic quantum numbers, respectively). The calculations are performed at representative projectile energies between 10 keV/u to 1 MeV/u. The calculated cross sections for collisions with H(2s) are compared with recent theoretical results. Generally, good agreement is observed for the n-partial excitation and total ionisation cross sections. However, a significant discrepancy is found for excitation into the dominant n=3 states at 100 keV/u, where the target excitation cross-section peaks. We also present the first calculations of the excitation and ionisation cross sections for Be4+ collisions with H(2p0) and H(2p±1).

Thermally Induced Stark Shifts of Highly Excited States of Hydrogen Atom

Thermally Induced Stark Shifts of Highly Excited States of Hydrogen Atom

Excitation of the 2P State of Atomic Hydrogen by Electron Impact

The excitation cross sections of the 2P state of atomic hydrogen at low incident electron energies (from 0.755 to 3.5 Ry) were calculated using the variational polarized orbital method. Up to 13 partial waves (L = 1, 13) were used to obtain converged cross sections in the above energy range. The importance of the long-range forces is pointed out in the threshold region, and behavior of the cross section is indicated near the threshold. The polarization P of radiation emitted at right angle to the incident electron beam was calculated and the perpendicular cross section was also calculated.

Method for Measuring the Pseudomomentum of Hydrogen Atoms by the Number of Observable Hydrogen Lines Controlled by the Diamagnetism

Hydrogen atoms, being subjected to a strong magnetic field, exhibit an additional, delocalized potential well at almost a microscopic distance from the nucleus. We studied the influence of the delocalized states of hydrogen atoms on the number of observable hydrogen lines in strongly magnetized plasmas. We show that, for sufficiently large values of the pseudomomentum K (K being the integral of the motion controlling the separation of the center of mass and the relative motions), this effect dominates other factors potentially influencing the number of observable hydrogen lines in strongly magnetized plasmas. We provide examples for plasma parameters relevant to edge plasmas of contemporary and future tokamaks, as well as for DA white dwarfs. We demonstrate that our results open up an avenue for the experimental determination of the pseudomomentum K. This is the first proposed method for the experimental determination of the pseudomomentum—to the best of our knowledge.

Nl-Selective Classical Charge-Exchange Cross Sections in Be4+ and Ground State Hydrogen Atom Collisions

Charge-exchange cross sections in Be4+ + H(1s) collisions are calculated using the three-body classical trajectory Monte Carlo method (CTMC) and the quasi-classical trajectory Monte Carlo method of Kirschbaum and Wilets (QCTMC) for impact energies between 10 keV/amu and 300 keV/amu. We present charge-exchange cross sections in the projectile n = 2 and nl = 2s, 2p states. Our results are compared with the previous quantum-mechanical approaches. We found that the QCTMC model is a powerful classical model to describe the state-selective charge-exchange cross sections at lower impact energies and the QCTMC results are in good agreement with previous observations.

Experiments on the Electron Impact Excitation of the 2s and 2p States of Hydrogen Atoms Confirm the Presence of Their Second Flavor as the Candidate for Dark Matter

For the excitation of the n = 2 states of hydrogen atoms due to electron impact, we compared the experimental and theoretical ratios of the cross-sections σ2s/σ2p. We found this theoretical ratio to be systematically higher than the corresponding experimental ratio by about 20%—far beyond the experimental error margins. We suggest that this discrepancy can be explained by the presence of the Second Flavor of Hydrogen Atoms (SFHA) in the experimental hydrogen gas. The explanation is based on the fact that, in the experiments, the cross-section σ2s was determined by using the quenching technique—by applying an electric field that mixed the 2s and 2p states, followed by the emission of the Lyman-alpha line from the 2p state. However, the SFHA only had the s-states, so the quenching technique would not count the excitation of the SFHA in the 2s state and, thus, lead to the underestimation of the cross-section σ2s. We estimates the share of the SFHA in the experimental hydrogen gas required for eliminating the above discrepancy and found this share to be about the same as the share of the usual hydrogen atoms. Thus, our results constitute the third proof from atomic experiments that the SFHA does exist, the first proof being related to the experimental distribution of the linear momentum in the ground state of hydrogen atoms, and the second proof being related to the experimental cross-section of charge exchange between hydrogen atoms and low-energy protons.

A new method for theoretical calculation of atomic hyperfine structure

Schrödinger equation is a nonrelativistic wave equation, which does not have Lorentz invariance. Therefore, this equation has a large theoretical error in the precise calculation of hydrogen-like system. So the commonly used method is Dirac-Hartree-Fock approximation in the calculation of atomic system. However, we have found a new eigen equation, whose eigenvalue of the hydrogen-like system approximates the calculation of quantum electrodynamics. Hence, we propose a new calculation scheme for the atomic hyperfine structure based on the eigen equation and the basic principle of Hartree-Fock variational method, and come to our conclusion through the correlation calculation of excited single states of hydrogen atom, U 91+ ion, helium atom and lithium atom as well as the comparison with NIST, that is, our method is a better improved model of the stationary Schrödinger equation. Meanwhile, we list the correlation algorithms of energy functional, two-electron coupling integral and radial generalized integral in the appendix.

Charge transfer of potassium by the impact of proton particle

The total cross section for electron capture is being calculated in this paper for 2s state of hydrogen atom in proton-potassium collision. The energy range chosen for the calculation is 50 keV to 10,000 keV and we have used Coulomb projected Born (CPB) approximation method. Also the differential cross section at different scattering angle has been calculated. The result obtained are compared with available theoretical data and found it to be in good agreement.

State-selective electron capture cross sections in collision between fully striped ions and ground state hydrogen atom—Classical treatment of the collision

We present a state-selective electron capture cross sections database from the ground state hydrogen atom. A standard three-body classical trajectory Monte Carlo (CTMC) and quasi-classical trajectory Monte Carlo (QCTMC) models were employed for impact energies between 10 and 200 keV/amu. The projectile ions are H+, He2+, Li3+, Be4+, B5+, C6+, N7+, and O8+. The cross sections are tabulated for each value of the final quantum numbers n, l, m, summed over m for the given n and l and over l and m for the given n. The maximum quantum number nmax is set to 4 for H+, He2+ and Li3+, as well as to 5 for Be4+, B5+, C6+, N7+ and O8+, respectively.

Visible Light‐Mediated Dearomative Hydrogen Atom Abstraction/ Cyclization Cascade of Indoles

AbstractThe photochemical synthesis of yet unknown 2‐oxospiro[azetidine‐3,3′‐indolines] (17 examples, 80–95 % yield), 2,4‐dioxospiro[azetidine‐3,3′‐indolines] (eight examples, 87–97 % yield), and 1‐oxo‐1,3‐dihydrospiro[indene‐2,3′‐indolines] (17 examples, 85–97 % yield) is described. Starting from readily accessible 3‐substituted indoles, a dearomatization of the indole core was accomplished upon irradiation at λ=420 nm in the presence of thioxanthen‐9‐one (10 mol%) as the sensitizer. Based on mechanistic evidence (triplet energy determination, deuteration experiments, by‐product analysis) it is proposed that the reaction proceeds by energy transfer via a 1,4‐ or 1,5‐diradical intermediate. The latter intermediates are formed by excited state hydrogen atom transfer from suitable alkyl groups within the C3 substituent to the indole C2 carbon atom. Subsequent ring closure proceeds with pronounced diastereoselectivity to generate a 4‐ or 5‐membered spirocyclic dearomatized product with several options for further functionalization.

Visible Light-Mediated Dearomative Hydrogen Atom Abstraction/ Cyclization Cascade of Indoles.

The photochemical synthesis of yet unknown 2‐oxospiro[azetidine‐3,3′‐indolines] (17 examples, 80–95 % yield), 2,4‐dioxospiro[azetidine‐3,3′‐indolines] (eight examples, 87–97 % yield), and 1‐oxo‐1,3‐dihydrospiro[indene‐2,3′‐indolines] (17 examples, 85–97 % yield) is described. Starting from readily accessible 3‐substituted indoles, a dearomatization of the indole core was accomplished upon irradiation at λ=420 nm in the presence of thioxanthen‐9‐one (10 mol%) as the sensitizer. Based on mechanistic evidence (triplet energy determination, deuteration experiments, by‐product analysis) it is proposed that the reaction proceeds by energy transfer via a 1,4‐ or 1,5‐diradical intermediate. The latter intermediates are formed by excited state hydrogen atom transfer from suitable alkyl groups within the C3 substituent to the indole C2 carbon atom. Subsequent ring closure proceeds with pronounced diastereoselectivity to generate a 4‐ or 5‐membered spirocyclic dearomatized product with several options for further functionalization.

Distinctive Features of Charge Exchange Involving the Second Flavor of Hydrogen Atoms—The Candidates for Dark Matter

The second flavor of hydrogen atoms (SFHA) refers to the kind of hydrogen atoms that have only the states of the zero orbital angular momentum (the S-states), both in the discrete and continuous spectra. They were first discovered theoretically in one of my earlier papers, where a proof of their existence was also provided by analyzing atomic experiments concerning the high-energy tail of the linear momentum distribution in the ground state of hydrogen atoms. From a theoretical point of view, the discovery was based on the standard Dirac equation for hydrogen atoms without changing the existing physical laws. Recently, the existence of the SFHA was seemingly also confirmed by two types of astrophysical observations: the allowance for the SFHA explained the puzzling results concerning both the anomalous absorption of the redshifted 21 cm spectral line from the early Universe, and the observations by the Dark Energy Survey (DES) team where it was found that the distribution of dark matter in the Universe is noticeably smoother than predictions employing Einstein’s relativity. In the present review, we exhibit results from two recent papers where attention was brought to a visible difference in the cross-sections of the resonant charge exchange for collisions of the SFHA with incoming protons, compared to collisions of the usual hydrogen atoms with incoming protons. It was shown that, after taking into account the SFHA, there is a better agreement with the corresponding experimental cross-section. Coupled with the previous evidence of the existence of the SFHA, deduced from the analysis of the other kind of atomic experiments, and evidenced by two different kinds of astrophysical observations, this strengthens the standing of the SFHA as the most probable candidate for all or a part of dark matter.