Atoms

34 pages, 6203 KB

Open AccessEditor’s ChoiceArticle

An Anomalous Structure in the Critical Screening Parameters of the ECSC Potential

by Grant B. Bunker

Atoms 2026, 14(7), 51; https://doi.org/10.3390/atoms14070051 - 28 Jun 2026

Viewed by 430

The critical binding of quantum states in Screened Coulomb Potentials such as Yukawa/Debye, Hulthén, and ECSC (Exponential Cosine Screened Coulomb) potentials is of perennial interest and relevance in many fields of science, ranging from nuclear and particle physics; plasma physics, astrophysics, cosmology, and [...] Read more.

The critical binding of quantum states in Screened Coulomb Potentials such as Yukawa/Debye, Hulthén, and ECSC (Exponential Cosine Screened Coulomb) potentials is of perennial interest and relevance in many fields of science, ranging from nuclear and particle physics; plasma physics, astrophysics, cosmology, and nuclear fusion; physical chemistry, condensed matter, and materials physics; to synthetic nanostructures and nanophotonics. The purpose of this paper is to heuristically explore two related mysteries, one new, the other more than 50 years old. The solutions to these mysteries have implications for a much broader class of potentials, those addressed by Klaus and Simon. In our recent paper we presented numerical calculations using the Phase Method (PM), which is accurate to 60 digits and to screening lengths

D \leq 10^{3}

au and l = 0–20 of the critical binding parameters for these potentials and, for Yukawa and ECSC, l = 0–12 to

D \leq 10^{5}

au, at 30 digits. In doing so, we discovered an anomalous period-40 sawtooth structure in the critical parameters of the ECSC potential that is not observed for the Yukawa potential. In this second paper, we quantitatively explain the origin and periodicity of this newly discovered structure. To do so, we use two complementary approaches: a “neoclassical” (NC) variant of conventional semiclassical phase-space quantization and the PM for very precise fully quantum calculations. The observed period-40 sawtooth structure is quantitatively explained in terms of a novel “tick-tock” mechanism. The periodicity is calculated in terms of the ratio of phase-space integrals for the primary and secondary potential wells. A quartic double-well potential is used as a simple model to further illustrate the tick-tock mechanism. Using the NC method, an approximate expression is derived to predict the locations of tick-tock glitches from higher-order wells; it is confirmed by a PM calculation up to

D \leq 10^{6}

au. The second mystery is a strangely linear dependence of the total number of bound states vs. screening length for both the Yukawa and ECSC potentials. Using the PM, we confirm and extend these empirical relations. We show, using the PM, that an approximate trivariate linear relation between the square root of the critical screening length

\sqrt{D_{c}}

, state number n, and angular momentum l applies to these potentials. This, plus a geometrical state accumulation argument, solve the second mystery. We show these properties derive from the scaling relation between screening length and coupling constant and, as such, are predicted to be applicable to the whole class of potentials. These results are expected to be of both theoretical interest and experimental relevance when interpreting spectra or calculating thermal properties. The significance of these results, and the applicability of these methods and conclusions to a vast array of related potentials, is briefly discussed. Full article

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24 pages, 9411 KB

Open AccessEditor’s ChoiceArticle

Elastic Electron Scattering from Zn, Cd, and Hg

by Mehrdad Adibzadeh and Constantine E. Theodosiou

Atoms 2026, 14(7), 50; https://doi.org/10.3390/atoms14070050 - 27 Jun 2026

Viewed by 441

Abstract

We present an extensive set of theoretical results for differential, integrated, and momentum-transfer cross sections for the elastic scattering of electrons by zinc, cadmium, and mercury. Our approach is a self-consistent relativistic calculation, with a semi-empirically adjustable cutoff radius of the polarization potential. [...] Read more.

We present an extensive set of theoretical results for differential, integrated, and momentum-transfer cross sections for the elastic scattering of electrons by zinc, cadmium, and mercury. Our approach is a self-consistent relativistic calculation, with a semi-empirically adjustable cutoff radius of the polarization potential. This study further extends the application of our method of calculations, previously employed for stable inert gases and alkaline-earth metals. Based on the satisfactory agreement of our previous investigations with experimental values and other precise theoretical results, we expect to provide a set of accurate data for Zn, Cd and Hg. Full article

(This article belongs to the Special Issue Collisions and Applications: Electron, Positron, and Positronium Interactions with Atoms and Molecules)

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15 pages, 1152 KB

Open AccessEditor’s ChoiceArticle

Liquid Water Ionization by Energetic Electron Impact

by María Laura de Sanctis, Marie-Françoise Politis, Rodolphe Vuilleumier and Omar Ariel Fojón

Atoms 2026, 14(7), 49; https://doi.org/10.3390/atoms14070049 - 27 Jun 2026

Viewed by 397

Abstract

We theoretically study the single ionization of liquid water by impact of fast electrons. A realistic description of the wavefunction for an isolated water molecule in the liquid phase is obtained by means of a Wannier orbital formalism. In this way, we consider [...] Read more.

We theoretically study the single ionization of liquid water by impact of fast electrons. A realistic description of the wavefunction for an isolated water molecule in the liquid phase is obtained by means of a Wannier orbital formalism. In this way, we consider ionization from the most external orbitals 1B₁, 2A₁, 1B₂ and 1A₁ of a single liquid water molecule. Triple, double, single differential and total cross sections are computed through a first order model with proper Coulomb conditions for the ionized electron. We compare our calculations with measurements and other theoretical results for liquid and gaseous phases. An analysis of the main features of the cross sections is performed. Previous theoretical works found almost no discrepancies between these observables in spite of the physical dissimilarities of both phases. In this work, we compare our results with other theories and with the available experiments for vapor. We report interesting differences between the differential and total cross sections of the mentioned phases. Full article

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17 pages, 594 KB

Open AccessEditor’s ChoiceArticle

Modeling Atomic Structure & Behavior Through Electron Configurations

by Stephan Fritzsche, Nishita M. Hosea, Houke Huang, Tianluo Luo and Aloka K. Sahoo

Atoms 2026, 14(7), 46; https://doi.org/10.3390/atoms14070046 - 23 Jun 2026

Viewed by 594

Abstract

Electron configurations are known to provide valuable insights into the electronic structure and behavior of atoms. They specify which and how the electronic (sub-) shells are occupied, and are thus an essential ingredient for most atomic observables. When combined with the shell model [...] Read more.

Electron configurations are known to provide valuable insights into the electronic structure and behavior of atoms. They specify which and how the electronic (sub-) shells are occupied, and are thus an essential ingredient for most atomic observables. When combined with the shell model and the successive filling of shells, these configurations help explain the Periodic Table and much of chemical binding. They also establish a qualitative framework for analyzing excitation, ionization and relaxation processes and may facilitate a wide range of astrophysical and plasma simulations. Here, we review the role of electron configurations for understanding atomic behavior in interactions with particles and radiation. In particular, we identify several central requirements for an efficient treatment of configuration lists and define a domain-specific language in order to generate, manipulate and analyze such lists as well as to extract physically relevant information. We also demonstrate the implementation of this language in Jac, the Jena Atomic Calculator. An efficient handling of configurations will refine the coupling of structure codes with the spectral synthesis of plasma radiation, the setup of ionic cascades or even non-LTE plasma simulations. This common framework for dealing with electron configurations therefore improves consistency, reproducibility and scalability of atomic modeling. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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13 pages, 731 KB

Open AccessEditor’s ChoiceArticle

Electron Emission in Antiproton–Hydrogen Interactions Studied with the One-Centre Basis Generator Method

by Jay Jay Tsui and Tom Kirchner

Atoms 2026, 14(6), 41; https://doi.org/10.3390/atoms14060041 - 24 May 2026

Cited by 1 | Viewed by 575

Abstract

Electron emission from hydrogen atoms induced by antiproton impact at intermediate energies is investigated using the one-centre Basis Generator Method within a semi-classical impact-parameter framework. The formulation employs a single-centre expansion of the time-dependent Schrödinger equation with a pseudostate basis consisting of hydrogenic [...] Read more.

Electron emission from hydrogen atoms induced by antiproton impact at intermediate energies is investigated using the one-centre Basis Generator Method within a semi-classical impact-parameter framework. The formulation employs a single-centre expansion of the time-dependent Schrödinger equation with a pseudostate basis consisting of hydrogenic orbitals acted upon by powers of a Yukawa-regularized potential, providing a compact and effective representation of the electronic continuum. Ionization probabilities are obtained by projecting the time-evolved wavefunction onto Coulomb continuum states, from which energy-differential cross sections (EDCS) are extracted. Exponential piecewise functions are constructed to interpolate between the pseudostate eigenenergies, yielding smooth EDCS profiles for each partial wave. The total EDCS, reconstructed by summing over all partial-wave contributions, exhibits good agreement with results from other pseudostate-based approaches. Full article

(This article belongs to the Special Issue Electronic Dynamics in Atomic and Molecular Collisions)

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40 pages, 755 KB

Open AccessEditor’s ChoiceArticle

Second-Order Rayleigh–Schrödinger Perturbation Theory for the Grasp2018 Package

by Gediminas Gaigalas, Pavel Rynkun and Laima Kitovienė

Atoms 2026, 14(5), 40; https://doi.org/10.3390/atoms14050040 - 21 May 2026

Cited by 1 | Viewed by 725

Abstract

A developed method, based on the stationary second-order Rayleigh–Schrödinger many-body perturbation theory in an irreducible tensorial form, allows us to determine the most important core–valence, core, core–core, and valence–valence correlations for any atom or ion with an arbitrary number of valence and core [...] Read more.

A developed method, based on the stationary second-order Rayleigh–Schrödinger many-body perturbation theory in an irreducible tensorial form, allows us to determine the most important core–valence, core, core–core, and valence–valence correlations for any atom or ion with an arbitrary number of valence and core electrons. This paper presents the Feynman diagrams that describe these correlations. Additionally, it provides the rules for obtaining algebraic expressions in an irreducible tensorial form for any Feynman diagram coming from second-order many-body perturbation theory. Whereas some types of the valence–valence and core–valence correlations are described by the three-particle Feynman diagrams, additional developments to calculate the spin-angular parts of these diagrams have been made to the program library librang of the Grasp2018 As an example of the application of the developed method, the atomic calculations of the energy level structure and transition data for Ar II are presented. Full article

(This article belongs to the Special Issue Future Directions in Atomic Physics Inspired by the Pioneering Work of Charlotte Froese Fischer and Ian Philip Grant)

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23 pages, 1371 KB

Open AccessEditor’s ChoiceArticle

Analytical Study of Electron-Driven Ionization Dynamics and Plasma Formation in Intense Laser Fields

by Hristina Delibašić-Marković, Veljko Vujčić, Vladimir A. Srećković and Violeta Petrović

Atoms 2026, 14(5), 39; https://doi.org/10.3390/atoms14050039 - 20 May 2026

Viewed by 670

Abstract

Laser-induced breakdown in water-rich biological media results from the interplay between primary photoionization processes and avalanche amplification of free electrons. Understanding this competition is essential for predicting ablation thresholds under ultrashort-pulse irradiation. In this work, we develop an analytical rate-equation model for the [...] Read more.

Laser-induced breakdown in water-rich biological media results from the interplay between primary photoionization processes and avalanche amplification of free electrons. Understanding this competition is essential for predicting ablation thresholds under ultrashort-pulse irradiation. In this work, we develop an analytical rate-equation model for the buildup of electron density in water-like biological tissues. It combines photoionization and chromophore ionization into a single seed-generation term, while avalanche ionization is described through a cascade gain factor. This formulation provides a framework for describing cascade electron-impact ionization processes in liquid-like media under strong-field excitation. Our approach gives an analytical expression for the temporal evolution of electron density driven by a Gaussian laser pulse and makes it possible to separate the contributions of direct ionization of water and ionization of chromophore centers. The analytical results are compared with numerical simulations that include carrier diffusion, bimolecular recombination and trapping. The comparison clarifies the roles of seed formation and cascade amplification in the growth of the electron population. The predicted dependence of threshold fluence on pulse duration agrees well with experimental data reported for water-like tissues such as the corneal tissues at a wavelength of 800 nm. The model provides a simple analytical picture of ultrafast plasma formation and electron-driven energy deposition in water-like biological media. Full article

(This article belongs to the Special Issue Electron Impact Ionization in Atoms and Molecules: Models and Applications)

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35 pages, 2722 KB

Open AccessEditor’s ChoiceReview

Resonant Transfer and Excitation of First-Row Ions Using Zero-Degree Auger Projectile Spectroscopy: Theory and Experiment

by Theo J. M. Zouros and Emmanouil P. Benis

Atoms 2026, 14(5), 38; https://doi.org/10.3390/atoms14050038 - 27 Apr 2026

Viewed by 746

Abstract

Resonant transfer and excitation (RTE) is a correlated two-electron ion–atom collision process mediated by the two-center electron–electron interaction: a projectile electron is excited while a target electron is captured, forming doubly excited states. These states decay via X-ray (RTEX) or Auger (RTEA) emission. [...] Read more.

Resonant transfer and excitation (RTE) is a correlated two-electron ion–atom collision process mediated by the two-center electron–electron interaction: a projectile electron is excited while a target electron is captured, forming doubly excited states. These states decay via X-ray (RTEX) or Auger (RTEA) emission. For sufficiently fast collisions with light targets, RTE becomes analogous to dielectronic capture (DC)—a key plasma process—and is successfully described by the impulse approximation (IA). Early (1983–1992) RTEX and more stringent, state-selective RTEA measurements provided essential indirect DC cross-section information before direct electron–ion measurements became available. A 1992 review by the first author, focusing on zero-degree Auger projectile spectroscopy (ZAPS) of state-selective KLL D states, validated the IA for low-

Z_{p}

(

Z_{p} \leq 9

) projectile ions, yet a puzzling systematic discrepancy remained: IA RTEA cross-sections were consistently larger than experimental, with the disagreement increasing as

Z_{p}

decreased. The present article reviews RTEA progress since 1992, including new refinements to IA calculations, an exact analytic IA formulation, and instrumental ZAPS improvements. A methodical analysis demonstrates impressive agreement across measurements spanning both pre- and post-1992 eras, including new experimental results, effectively eliminating previous systematic discrepancies. IA validity is confirmed down to boron ions, with He⁺ and certain Li-like ions remaining the only notable exceptions. Recently, a rigorous quantum mechanical ion–atom collision treatment has emerged: nonperturbative close-coupling calculations of transfer excitation for He-like carbon ions colliding with He confirm the dominance of RTE via two-center electron–electron interactions at large impact parameters, yielding RTEA results in excellent agreement with experiments. Full article

(This article belongs to the Special Issue X-Ray Spectroscopy in Astrophysics)

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15 pages, 441 KB

Open AccessEditor’s ChoiceArticle

Rci-q: An Improved QED Correction Model for the Grasp2018 Package

by Karol Kozioł

Atoms 2026, 14(5), 35; https://doi.org/10.3390/atoms14050035 - 24 Apr 2026

Viewed by 589

Abstract

The Rci-Q package is an extension to the Grasp2018 suite, improving the model of estimating the quantum electrodynamics corrections to the energy levels. The Flambaum–Ginges radiative potential method is used to estimate the leading self-energy correction to electron energy in many electron [...] Read more.

The Rci-Q package is an extension to the Grasp2018 suite, improving the model of estimating the quantum electrodynamics corrections to the energy levels. The Flambaum–Ginges radiative potential method is used to estimate the leading self-energy correction to electron energy in many electron atoms. The new fitting prefactors to parameterize radiative potential are presented. The correction to self-energy originating from finite nucleus size is included. The Wichmann–Kroll part of the vacuum polarization potential is also implemented. Full article

(This article belongs to the Special Issue Future Directions in Atomic Physics Inspired by the Pioneering Work of Charlotte Froese Fischer and Ian Philip Grant)

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18 pages, 978 KB

Open AccessEditor’s ChoiceArticle

Integral Cross Sections and Transport Properties for Positron–Radon Scattering over a Wide Energy Range (0–1000 eV) and Reduced Electric Field Range (0.01–1000 Td)

by Gregory J. Boyle, Dale L. Muccignat, Joshua R. Machacek and Robert P. McEachran

Atoms 2026, 14(5), 34; https://doi.org/10.3390/atoms14050034 - 23 Apr 2026

Viewed by 613

Abstract

We present fully relativistic calculations of integral cross sections and swarm transport properties for positron–radon scattering over a wide energy range (0–1000 eV) and reduced electric field range (0.01–1000 Td). Elastic (total, momentum-transfer and viscosity-transfer), discrete excitation, direct annihilation, positronium formation and positron-impact [...] Read more.

We present fully relativistic calculations of integral cross sections and swarm transport properties for positron–radon scattering over a wide energy range (0–1000 eV) and reduced electric field range (0.01–1000 Td). Elastic (total, momentum-transfer and viscosity-transfer), discrete excitation, direct annihilation, positronium formation and positron-impact ionization cross sections are obtained using a complex relativistic optical potential method. Owing to the large atomic number of radon and the absence of experimental scattering data, a consistent relativistic treatment is essential. The present work provides the first fully relativistic, internally consistent cross-section dataset for positron swarms in radon gas. Using a multi-term solution of Boltzmann’s equation, steady-state transport coefficients are calculated and found to be strongly influenced by energy-dependent reactive loss, particularly positronium formation. Significant divergence between bulk and flux transport coefficients is observed, including non-monotonic bulk drift velocities and pronounced suppression of longitudinal bulk diffusion at intermediate fields (0.3–1000 Td). Time-dependent field-free calculations further quantify thermalization and annihilation dynamics through the evolution of the mean energy and

⟨ Z_{eff} ⟩ (t)

. These results provide a robust theoretical foundation for modelling positron transport and annihilation in radon and other heavy noble gases where relativistic and reactive effects are crucial. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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13 pages, 281 KB

Open AccessEditor’s ChoiceTutorial

Long-Lived 3d Levels in Highly Charged Iron Ions

by Elmar Träbert

Atoms 2026, 14(4), 32; https://doi.org/10.3390/atoms14040032 - 10 Apr 2026

Viewed by 625

Abstract

In Na- through Ca-like ions of Fe, rather low-lying 3d levels feature level lifetimes in the range from picoseconds to many seconds. This lifetime range is somewhat wider than that of the 3p resonance levels of the same ions. When trying to measure [...] Read more.

In Na- through Ca-like ions of Fe, rather low-lying 3d levels feature level lifetimes in the range from picoseconds to many seconds. This lifetime range is somewhat wider than that of the 3p resonance levels of the same ions. When trying to measure such level lifetimes, the width of the range exceeds the capabilities of a single measurement scheme. However, there also is the fundamental problem of multi-exponential decay curves and the reliability of their analysis. This problem has arisen afresh for the analysis of long-lived 3p levels in the ground configurations of many isoelectronic sequences that are replenished by cascades from long-lived 3d levels that have no

E 1

decay channels, but consequently feature lifetimes of the same order of magnitude as those in the ground configuration. This tutorial addresses the measurement situation for lifetimes of 3d levels in a number of ions of Fe and nearby elements. Full article

(This article belongs to the Special Issue Current Studies and Applications of Radiative and Collisional Processes in Atoms)

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12 pages, 416 KB

Open AccessEditor’s ChoiceArticle

Ionization in C⁶⁺+He Collisions: Singly Differential Cross-Sections

by Sh. U. Alladustov, K. H. Spicer, N. W. Antonio, A. M. Kotian and A. S. Kadyrov

Atoms 2026, 14(4), 31; https://doi.org/10.3390/atoms14040031 - 9 Apr 2026

Viewed by 613

Abstract

Differential ionization in

C^{6 +} +

He collisions is investigated using the single- and two-center wave-packet convergent close-coupling (WP-CCC) method for projectile energies of 1–6 MeV/u. We present three types of singly differential cross-sections (SDCSs) as functions of the ejection angle, ejection [...] Read more.

Differential ionization in

C^{6 +} +

He collisions is investigated using the single- and two-center wave-packet convergent close-coupling (WP-CCC) method for projectile energies of 1–6 MeV/u. We present three types of singly differential cross-sections (SDCSs) as functions of the ejection angle, ejection energy, and projectile scattering angle. The two-center framework incorporates couplings across all channels as well as electron correlations. Overall, both the single- and two-center WP-CCC results agree well with existing experimental and theoretical data (apart from the first Born ones) for the SDCS as a function of electron energy and the SDCS as a function of ejection angle, laying a foundation for investigation of doubly and fully differential ionization cross-sections. The cross-sections differential in the projectile scattering angle are presented for the first time. Full article

(This article belongs to the Special Issue Electronic Dynamics in Atomic and Molecular Collisions)

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9 pages, 1281 KB

Open AccessEditor’s ChoiceArticle

Charge Exchange Studies with n-, l-, and spin-Quantum State Population in Ar⁷⁺-He Collisions

by Yijiao Wu, Hao Yin, Bingsheng Tu, Tianming Meng, Pufang Ma, Xu Tan, Ke Yao, Jun Xiao, Yaming Zou and Baoren Wei

Atoms 2026, 14(4), 30; https://doi.org/10.3390/atoms14040030 - 8 Apr 2026

Viewed by 570

Abstract

The energy-dependent population of fine quantum states in single electron capture (SEC) reflects the intrinsic collision dynamics. Here we report experimental studies of Ar⁷⁺ ions colliding with He in the energy range of 1.05–17.5 keV/u. Owing to the high resolution of a [...] Read more.

The energy-dependent population of fine quantum states in single electron capture (SEC) reflects the intrinsic collision dynamics. Here we report experimental studies of Ar⁷⁺ ions colliding with He in the energy range of 1.05–17.5 keV/u. Owing to the high resolution of a recoil-ion momentum spectrometer, the n-, l-, and spin-state electron capture populations are well resolved, and a strong energy dependence of the SEC cross sections is observed. Most importantly, a clear inversion of the cross-section ratio between the spin-resolved triplet and singlet 3s3d configurations is found, demonstrating a breakdown of spin statistics. Together with recent spin-resolved studies of C³⁺-He collisions (PRL 133, 173002 (2024)), these results suggest that the breakdown of spin statistics is likely a general feature of charge exchange in open-shell highly charged ion systems. Full article

(This article belongs to the Special Issue Electronic Dynamics in Atomic and Molecular Collisions)

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97 pages, 1163 KB

Open AccessEditor’s ChoiceArticle

A Program Library for Computing Pure Spin-Angular Coefficients for One- and Two-Particle Operators in Non-Relativistic Atomic Theory

by Gediminas Gaigalas

Atoms 2026, 14(4), 29; https://doi.org/10.3390/atoms14040029 - 1 Apr 2026

Cited by 2 | Viewed by 965

Abstract

A program library, libang77, for computing pure spin-angular coefficients for any one- and scalar two-particle operator is presented. The method is based on the combination of the second quantization and quasi-spin techniques with angular momentum theory and the method of irreducible tensorial sets. [...] Read more.

A program library, libang77, for computing pure spin-angular coefficients for any one- and scalar two-particle operator is presented. The method is based on the combination of the second quantization and quasi-spin techniques with angular momentum theory and the method of irreducible tensorial sets. A non-relativistic approach is used, in which relativistic corrections may be included in the Breit–Pauli approximation. This program library, libang77, is integrated into the Atomic Structure Package ATSP2K [ATSP2K, C. Froese Fischer, G. Tachiev, G. Gaigalas, and M.R. Godefroid, Comput. Phys. Commun. (2007). DOI: 10.1016/j.cpc.2007.01.006], but it can be implemented in other program packages too. Full article

(This article belongs to the Special Issue Future Directions in Atomic Physics Inspired by the Pioneering Work of Charlotte Froese Fischer and Ian Philip Grant)

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9 pages, 328 KB

Open AccessEditor’s ChoiceArticle

Single-Electron Capture in Intermediate-Energy He⁺ + Ne Collisions

by Hanfeng Yu, Dalong Guo, Xiaolong Zhu, Xuexia Pang, Yong Gao, Dongmei Zhao, Kaizhao Lin, Jinjian Yu, Shaofeng Zhang and Xinwen Ma

Atoms 2026, 14(4), 28; https://doi.org/10.3390/atoms14040028 - 1 Apr 2026

Viewed by 777

Abstract

State-selective single-electron capture in He⁺ + Ne collisions was studied at laboratory He⁺ projectile kinetic energies of 30–100 keV (corresponding to 7.5–25 keV/u) using a reaction microscope. Q-value spectra were obtained through recoil-ion momentum reconstruction, enabling the decomposition of the [...] Read more.

State-selective single-electron capture in He⁺ + Ne collisions was studied at laboratory He⁺ projectile kinetic energies of 30–100 keV (corresponding to 7.5–25 keV/u) using a reaction microscope. Q-value spectra were obtained through recoil-ion momentum reconstruction, enabling the decomposition of the capture yield into three distinct contributions: (i) capture into excited states of the projectile without target excitation, (ii) capture into the projectile ground state accompanied by excitation of the residual Ne⁺ ion, and (iii) capture involving simultaneous excitation of both He and Ne⁺. Across the studied energy range, capture into the projectile ground state accompanied by target excitation is the dominant pathway. With increasing impact energy, the fraction of joint projectile–target excitation increases markedly, whereas the target-excitation-only contribution decreases; the projectile-excitation-only fraction remains at a low, nearly constant level. These findings underscore the significance of multi-electron dynamics in intermediate-energy collisions involving many-electron targets. Full article

(This article belongs to the Special Issue Electronic Dynamics in Atomic and Molecular Collisions)

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20 pages, 819 KB

Open AccessEditor’s ChoiceArticle

Multiplatform Computing of Transition Probabilities in Os V

by Patrick Palmeri, Saturnin Enzonga Yoca, Exaucé Bokamba Motoumba, Alix Niels, Maxime Brasseur and Pascal Quinet

Atoms 2026, 14(3), 22; https://doi.org/10.3390/atoms14030022 - 11 Mar 2026

Viewed by 563

Abstract

Osmium is an element of the Periodic Table with an atomic number Z equal to 76. In Tokamaks with divertors made of tungsten (

Z = 74

), it is produced in the neutron-induced transmutation of the latter. Therefore one can expect that [...] Read more.

Osmium is an element of the Periodic Table with an atomic number Z equal to 76. In Tokamaks with divertors made of tungsten (

Z = 74

), it is produced in the neutron-induced transmutation of the latter. Therefore one can expect that their sputtering may generate ionic impurities of all possible charge states in the fusion plasma. As a consequence, these could contribute to radiation losses in these controlled nuclear devices. The knowledge of radiative rates in all the spectra of osmium is thus important in this field. In this framework, a multiplatform approach has been used to determine the Os V radiative properties and estimate their accuracy. The transition probabilities have been computed for the 2677 electric dipole (E1) transitions falling in the spectral range from 400 Å to 12,000 Å. Three independent atomic structure models have been considered; one based on the fully relativistic ab initio multiconfiguration Dirac–Hartree–Fock (MCDHF) method and two based on the semi-empirical pseudo-relativistic Hartree–Fock (HFR) method. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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28 pages, 491 KB

Open AccessEditor’s ChoiceArticle

Extension of an Efficient Approach for Spin-Angular Integrations in Atomic Structure Calculations

by Gediminas Gaigalas

Atoms 2026, 14(3), 21; https://doi.org/10.3390/atoms14030021 - 9 Mar 2026

Cited by 3 | Viewed by 830

Abstract

In this study, an extension of the general method [G. Gaigalas, Z. Rudzikas, C. Froese Fischer, J. Phys. B, At. Mol. Phys. (1997). DOI: 10.1088/0953-4075/30/17/006] is described for finding algebraic expressions of the spin-angular parts of the reduced matrix elements of any one- [...] Read more.

In this study, an extension of the general method [G. Gaigalas, Z. Rudzikas, C. Froese Fischer, J. Phys. B, At. Mol. Phys. (1997). DOI: 10.1088/0953-4075/30/17/006] is described for finding algebraic expressions of the spin-angular parts of the reduced matrix elements of any one- and two-particle operator for an arbitrary number of shells in an atomic configuration. This extension is related, at first, to a change in the definition of tensor structure, where a non-scalar space with respect to l and s for any two-particle operator acts on four different shells. This leads to more efficient expressions for recoupling matrices and amplitudes, which are presented in the paper. In addition, the paper presents new expressions for some of the recoupling matrices, in which 6j- and 9j-coefficients are summed up algebraically. All this leads to a significantly simpler and faster calculation of the spin-angular parts of any non-scalar two-particle operator. Full article

(This article belongs to the Special Issue Future Directions in Atomic Physics Inspired by the Pioneering Work of Charlotte Froese Fischer and Ian Philip Grant)

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12 pages, 324 KB

Open AccessEditor’s ChoiceArticle

jj to LSJ Transformation for Configuration State Functions with an Arbitrary Number of Open Shells

by Gediminas Gaigalas

Atoms 2026, 14(3), 20; https://doi.org/10.3390/atoms14030020 - 9 Mar 2026

Cited by 1 | Viewed by 1053

Abstract

This paper presents a methodology that allows for calculated energy levels and other atomic characteristics in relativistic atomic theory, i.e., using the

j j

-coupling scheme, to be identified in terms of

L S J

-coupling characteristics. The paper begins with outlining the [...] Read more.

This paper presents a methodology that allows for calculated energy levels and other atomic characteristics in relativistic atomic theory, i.e., using the

j j

-coupling scheme, to be identified in terms of

L S J

-coupling characteristics. The paper begins with outlining the general principles for effectively addressing this problem. Furthermore, it provides a general expression that enables such identification when the atomic state function consists of any number of configuration state functions, each with any number of open shells, and explains how this expression was obtained. The methodology developed in this paper has been successfully implemented in the General Relativistic Atomic Structure Package and can be applied to other similar packages. Full article

(This article belongs to the Special Issue Future Directions in Atomic Physics Inspired by the Pioneering Work of Charlotte Froese Fischer and Ian Philip Grant)

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44 pages, 1063 KB

Open AccessEditor’s ChoiceArticle

Numerical Computation of Critical Binding Parameters of Screened Coulomb Potentials

by Grant B. Bunker

Atoms 2026, 14(3), 18; https://doi.org/10.3390/atoms14030018 - 5 Mar 2026

Cited by 1 | Viewed by 1739

Abstract

For nearly a century, screened Coulomb potentials have been of recognized importance in diverse areas of physics and chemistry. A key feature of interest in these potentials is the phenomenon of critical screening. This paper has three main purposes: to present an extensive, [...] Read more.

For nearly a century, screened Coulomb potentials have been of recognized importance in diverse areas of physics and chemistry. A key feature of interest in these potentials is the phenomenon of critical screening. This paper has three main purposes: to present an extensive, open-access, high accuracy (60 digit) benchmark reference dataset of critical screening parameters, with validation; to confirm excellent past work in the field (to 30 digits), and to correct an historical oversight in its literature; and to present the essentials of our new approach, the “Phase Method” (PM), for computing them. Using the PM, we calculate critical screening parameters, accurate to 60 decimal digits, for the Yukawa/Debye, Hulthén, Pseudo-Hulthén, and Exponential Cosine Screened Coulomb (ECSC)) potentials. The practical feasibility of such calculations on inexpensive hardware opens up new possibilities in research and education. We highlight an apparently overlooked 1989 paper of Demiralp on critical screening parameters of the Yukawa potential, which accurately calculated them to 30 decimal digits. Our main results are computations of the critical screening parameters

μ_{c} = 1 / D_{c}

for screening lengths

D \leq 1000

au and angular momenta

l = 0, \dots, 20

. The claimed accuracy of our results is supported by several independent lines of evidence: comparison with the most accurate (30 digit) values available in the print literature for the Yukawa, Hulthén, and ECSC potentials; comparison to 60 decimal digits accuracy with exactly known eigenvalues and critical binding parameters of the Pseudo-Hulthén potential; consistency tests between computed critical parameters, for various l-values for the Pseudo-Hulthén Potential, and known exact relations between eigenvalues; and application of a novel consistency test between results with different potential parameters, that exploits an exact scaling symmetry of this entire class of potentials. Similar calculations were done for ECSC and Yukawa potentials for screening lengths up to

D \leq 10^{5}

and

l \leq 12

, to 30 digit accuracy, which show interesting (and to our knowledge, not previously reported) periodic structure in

D_{c} (n, l)

for the ECSC potential that is not observed for the Yukawa potential. The asymptotic scaling behavior of critical parameters for the Yukawa and Hulthén potentials is explained quantitatively by simple semiclassical calculations, as is the scaling of circular states for those and other potentials. Full article

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20 pages, 361 KB

Open AccessEditor’s ChoiceArticle

Study of the Hyperfine Structure of Sr II, Ba I and Ba II: An MCDHF Approach for Modeling the Low-Lying Levels

by Lorenzo Nezosi, Lucas Maison, Patrick Palmeri, Per Jönsson and Michel Godefroid

Atoms 2026, 14(3), 17; https://doi.org/10.3390/atoms14030017 - 5 Mar 2026

Cited by 1 | Viewed by 1122

Abstract

Using the Multiconfiguration Dirac–Hartree–Fock method as implemented in the General Relativistic Atomic Structure Package, the magnetic dipole and electric quadrupole hyperfine structure constants were determined for the ground and first excited levels of ^135,137Ba II isotopes, as well as for ¹³⁷Ba [...] Read more.

Using the Multiconfiguration Dirac–Hartree–Fock method as implemented in the General Relativistic Atomic Structure Package, the magnetic dipole and electric quadrupole hyperfine structure constants were determined for the ground and first excited levels of ^135,137Ba II isotopes, as well as for ¹³⁷Ba I and ⁸⁷Sr II, to assess the robustness of the developed model. This study builds upon and extends previous investigations by examining the levels involved in resonance lines, with the aim of resolving persistent discrepancies in the hyperfine structure of ¹³⁷Ba II and ⁸⁷Sr II. New code developments such as the use of natural orbitals, as well as the addition of polarization effects and Configuration State Function Generators, as implemented in GRASPG, were tested for these heavy elements. The developed strategy allowed us to achieve encouraging results that satisfactorily agree with experiments for all studied levels but

{}^{2}D_{5 / 2}

in the ¹³⁷Ba II isotope. This disagreement was also observed in ¹³⁵Ba II isotope as well as in ⁸⁷Sr II. With two valence electrons, ¹³⁷Ba I is definitely more complex, requiring a multireference approach. Even with the latter, the theory–observation disagreement observed for the hyperfine structure of the low-lying levels remains large in comparison with the alkali-like systems. Possible ongoing developments to remediate this issue are discussed in the conclusions. Full article

(This article belongs to the Special Issue Computational Atomic Physics in Astrophysics)

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13 pages, 1962 KB

Open AccessEditor’s ChoiceArticle

Dielectronic Recombination Strengths and Plasma Rate Coefficients of Lithium-like Argon Ions: Theory and Experiment

by Houke Huang, Zhongkui Huang, Yang Yuan, Hanbing Wang, Zeshan Muhammad, Chang Liu, Weiqiang Wen, Linfan Zhu, Xinwen Ma and Stephan Fritzsche

Atoms 2026, 14(2), 13; https://doi.org/10.3390/atoms14020013 - 13 Feb 2026

Viewed by 1243

Abstract

Dielectronic recombination (DR) is widely recognized as a fundamental atomic process in many astrophysical and laboratory plasmas, where it plays a crucial role in determining ionization balance and level populations over a broad temperature range. Reliable DR resonance strengths and plasma rate coefficients [...] Read more.

Dielectronic recombination (DR) is widely recognized as a fundamental atomic process in many astrophysical and laboratory plasmas, where it plays a crucial role in determining ionization balance and level populations over a broad temperature range. Reliable DR resonance strengths and plasma rate coefficients for such plasma modeling can be computed using the Jena Atomic Calculator (JAC)—a relativistic code based on the multiconfiguration Dirac–Hartree–Fock (MCDHF) method. In this work, we investigate the DR of Li-like Ar¹⁵⁺ ions in their ground state (2s), focusing on resonances associated with the fine-structure core excitations

2 s_{1 / 2} \to 2 p_{1 / 2, 3 / 2}

. The resulting fine-structure-resolved DR resonance strengths and plasma rate coefficients are in good agreement with recent high-resolution DR measurements of Ar¹⁵⁺ ions performed at the Main Cooler Storage Ring (CSRm) in Lanzhou, China. These results provide a stringent benchmark for JAC calculations and support their applicability in plasma modeling. Full article

(This article belongs to the Special Issue Computational Atomic Physics in Astrophysics)

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12 pages, 413 KB

Open AccessEditor’s ChoiceArticle

Two-Atom Superradiance Including Magnetic State Degeneracy

by Paul R. Berman

Atoms 2026, 14(2), 12; https://doi.org/10.3390/atoms14020012 - 6 Feb 2026

Viewed by 783

Abstract

The radiation pattern emitted by two atoms, interacting with each other via the vacuum radiation field, has been calculated, including effects of magnetic state degeneracy for atoms with a ground state having

G = 0

angular momentum and an excited state having [...] Read more.

The radiation pattern emitted by two atoms, interacting with each other via the vacuum radiation field, has been calculated, including effects of magnetic state degeneracy for atoms with a ground state having

G = 0

angular momentum and an excited state having

H = 1

angular momentum. For an initial condition in which both atoms are inverted, the time-integrated radiation pattern is identical to that for non-interacting atoms if the atoms lie on the z-axis, but differs if the atoms lie on the x-axis. The underlying dynamics giving rise to this behavior are examined. Full article

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20 pages, 3504 KB

Open AccessEditor’s ChoiceArticle

Study of the Triplet States in the Autoionizing Electron Spectra of He and Ar Induced by Low-Energy Electrons

by Bratislav P. Marinković, Lorenzo Avaldi and Jozo J. Jureta

Atoms 2026, 14(2), 9; https://doi.org/10.3390/atoms14020009 - 31 Jan 2026

Viewed by 635

Abstract

In this work, the He and Ar triplet autoionizing states have been studied using a non-monochromatic electron beam and a high-resolution electrostatic analyzer at low incident electron energies and three ejection angles: 40°, 90°, and 130°. Low-energy electrons have been used because they [...] Read more.

In this work, the He and Ar triplet autoionizing states have been studied using a non-monochromatic electron beam and a high-resolution electrostatic analyzer at low incident electron energies and three ejection angles: 40°, 90°, and 130°. Low-energy electrons have been used because they have a high probability of exciting triplet states regardless of whether they are discrete isolate states or are embedded in the ionization continuum. Additionally, the He ejected electron spectra have been measured at several ejection angles between 20° and 130° and two incident energies, namely 60.5 eV and 101 eV. The anisotropic angular distributions indicate that orbital angular momentum exchange between the ejected and scattered electrons occurred. The energies of the first triplets 3s3p⁶4s(³S) and 3s3p⁶4p(³P) states of argon are found to be (24.985 ± 0.020) eV and (26.52 ± 0.02) eV, respectively. Full article

(This article belongs to the Special Issue Electronic, Photonic and Ionic Interactions with Atoms and Molecules)

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11 pages, 4878 KB

Open AccessEditor’s ChoiceTechnical Note

NBODYCL: A Program to Generate Regularized Classical Trajectories for the n–Body Coulomb Problem

by Mbuso K. Matfunjwa, Harindranath B. Ambalampitiya and Ilya I. Fabrikant

Atoms 2026, 14(2), 8; https://doi.org/10.3390/atoms14020008 - 23 Jan 2026

Viewed by 1178

Abstract

A program package for calculating regularized classical trajectories for Coulomb n–body problem is developed. The Coulomb singularities from the equations of motion are removed by transformations of variables including the time. This effectively conserves the energy of the time-independent systems to a [...] Read more.

A program package for calculating regularized classical trajectories for Coulomb n–body problem is developed. The Coulomb singularities from the equations of motion are removed by transformations of variables including the time. This effectively conserves the energy of the time-independent systems to a high accuracy for long time propagation. Sample calculations are shown for the cases of 2, 3, 4, and 5 particle systems giving comparisons with the un-regularized trajectories. The program can be used for general purposes including the classical-trajectory Monte-Carlo simulations for charged-particle collisions in free or laser environments. Full article

(This article belongs to the Special Issue Collisions and Applications: Electron, Positron, and Positronium Interactions with Atoms and Molecules)

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18 pages, 799 KB

Open AccessEditor’s ChoiceArticle

Invariant Approach to the Interaction Between Several Fields and an Atom

by Marco A. García-Márquez, Irán Ramos-Prieto and Héctor M. Moya-Cessa

Atoms 2026, 14(1), 4; https://doi.org/10.3390/atoms14010004 - 8 Jan 2026

Viewed by 1025

Abstract

We present a general procedure to describe the dynamics of N degenerate quantized fields interacting resonantly with a two–level atom, all coupled with the same strength, within the rotating–wave approximation. Starting from the analysis of the two and three field cases, we generalize [...] Read more.

We present a general procedure to describe the dynamics of N degenerate quantized fields interacting resonantly with a two–level atom, all coupled with the same strength, within the rotating–wave approximation. Starting from the analysis of the two and three field cases, we generalize the method by identifying dynamical invariants that lead to a factorized form of the time–evolution operator. A unitary transformation reduces the problem to an effective Jaynes–Cummings Hamiltonian, where only one field interacts with the atom and the remaining modes contribute as free fields. Assuming initially coherent fields and an atomic superposition, we compute the atomic inversion and the mean photon number, revealing vacuum Rabi oscillations with a frequency determined by an effective coupling constant that exceeds the individual atom–field coupling, as well as the characteristic collapse–revival behavior. Full article

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16 pages, 1936 KB

Open AccessEditor’s ChoiceArticle

L-Shell Photon Excitation Cross Sections for the Chlorine Isonuclear Sequence Cl^{q+ (q=1−4)}: An Experimental Study

by Jean-Paul Mosnier, Eugene T. Kennedy, Denis Cubaynes, Ségolène Guilbaud and Jean-Marc Bizau

Atoms 2026, 14(1), 3; https://doi.org/10.3390/atoms14010003 - 4 Jan 2026

Viewed by 930

Abstract

We report experimental measurements of the absolute photoionization cross sections for chlorine ions in different stages of ionization, over photon energy ranges corresponding to the L-shell (2s and 2p subshells) excitations. Single, double and triple photoionization channels were investigated for the ions Cl [...] Read more.

We report experimental measurements of the absolute photoionization cross sections for chlorine ions in different stages of ionization, over photon energy ranges corresponding to the L-shell (2s and 2p subshells) excitations. Single, double and triple photoionization channels were investigated for the ions Cl⁺, Cl²⁺, Cl³⁺ and Cl⁴⁺. The measurements were performed on the PLéIADES beamline at the SOLEIL radiation storage ring facility, using the Multi-Analysis Ion Apparatus (MAIA). Resonance energies and line strengths are provided for the isonuclear sequence and the evolution of the inner shell photoionization behaviour is demonstrated for the chlorine ions as the degree of ionization is increased. While dominated by photoionization from the corresponding ground state ions, the photoion yields may also contain contributions from low-lying metastable states. The results provide useful data on these ions for plasma modelling and can serve as benchmarking experimental data for future atomic theoretical calculations. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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15 pages, 10342 KB

Open AccessEditor’s ChoiceArticle

Single Sr Atoms in Optical Tweezer Arrays for Quantum Simulation

by Veronica Giardini, Luca Guariento, Andrea Fantini, Shawn Storm, Massimo Inguscio, Jacopo Catani, Giacomo Cappellini, Vladislav Gavryusev and Leonardo Fallani

Atoms 2026, 14(1), 1; https://doi.org/10.3390/atoms14010001 - 19 Dec 2025

Cited by 1 | Viewed by 2799

Abstract

We report on the realization of a platform for trapping and manipulating individual ⁸⁸Sr atoms in optical tweezers. A first cooling stage based on a blue shielded magneto-optical trap (MOT) operating on the [...] Read more.

We report on the realization of a platform for trapping and manipulating individual ⁸⁸Sr atoms in optical tweezers. A first cooling stage based on a blue shielded magneto-optical trap (MOT) operating on the

|^{1} S_{0} ⟩ \to |^{1} P_{1} ⟩

transition at

461 n m

enables us to trap approximately 4 × 10⁶ atoms at a temperature of 6.8 mK. Further cooling is achieved in a narrow-line red MOT using the

|^{1} S_{0} ⟩ \to |^{3} P_{1} ⟩

intercombination transition at 689 nm, bringing 5 × 10⁵ atoms down to

5 μ K

and reaching a density of 4 × 10¹⁰ cm³. Atoms are then loaded into 813 nm tweezer arrays generated by crossed acousto-optic deflectors and tightly focused onto the atoms with a high-numerical-aperture objective. Through light-assisted collision processes we achieve the collisional blockade, which leads to single-atom occupancy with a probability of about 50%. The trapped atoms are detected via fluorescence imaging with a fidelity of

99.986 (6) %

, while maintaining a survival probability of

97 (2) %

. The release-and-recapture measurement provides a temperature of

12.92 (5) μ K

for the atoms in the tweezers, and the ultra-high-vacuum environment ensures a vacuum lifetime higher than 7 min. These results demonstrate a robust alkaline-earth tweezer platform that combines efficient loading, cooling, and high-fidelity detection, providing the essential building blocks for scalable quantum simulation and quantum information processing with Sr atoms. Full article

(This article belongs to the Special Issue Quantum Technologies with Ultracold Atoms)

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15 pages, 487 KB

Open AccessEditor’s ChoiceArticle

Collective Auger Decay of 4d⁻² Double Inner-Shell Vacancy in Xe

by Jiaolong Zeng, Guoqing Wang, Aihua Deng, Cheng Gao and Jianmin Yuan

Atoms 2025, 13(12), 98; https://doi.org/10.3390/atoms13120098 - 8 Dec 2025

Viewed by 754

Abstract

Auger decay of all levels of the double core-hole states

4 d^{- 2}

of Xe²⁺, including collective Auger decay (CAD) pathways, is investigated using the relativistic distorted-wave approximation. Large-scale configuration interaction calculations were performed to obtain level-to-level Auger decay rates. [...] Read more.

Auger decay of all levels of the double core-hole states

4 d^{- 2}

of Xe²⁺, including collective Auger decay (CAD) pathways, is investigated using the relativistic distorted-wave approximation. Large-scale configuration interaction calculations were performed to obtain level-to-level Auger decay rates. In addition to the typical Auger decay final levels associated with the configurations of

4 d^{- 1} 5 s^{2} 5 p^{4}

,

4 d^{- 1} 5 s^{1} 5 p^{5}

, and

4 d^{- 1} 5 s^{0} 5 p^{6}

, evident contributions are identified from excited channels, leading to configurations such as

4 d^{9} 4 f^{1} 5 s^{2} 5 p^{3}

,

4 d^{9} 5 s^{2} 5 p^{3} 5 d^{1}

,

4 d^{9} 5 s^{2} 5 p^{3} 6 s^{1}

, and

4 d^{9} 5 s^{2} 5 p^{3} 6 p^{1}

. These contributions arise from strong electron correlation between the valence electronic orbitals and the

4 d

inner-shell orbital. The CAD rates and branching ratios (BRs) are determined for each double core-hole level with a minimum CAD BR of 1.28% and a maximum of 4.08% among all CAD channels. The configuration-averaged CAD BR is predicted to be 1.93%, which helps explain recent unexplained experimental findings. The inclusion of CAD processes enriches Auger electron spectroscopy, thereby extending potential applications of this important experimental tool in both fundamental and applied research. Full article

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18 pages, 1413 KB

Open AccessEditor’s ChoiceArticle

Hybrid Basis and Multi-Center Grid Method for Strong-Field Processes

by Kyle A. Hamer, Heman Gharibnejad, Luca Argenti and Nicolas Douguet

Atoms 2025, 13(11), 92; https://doi.org/10.3390/atoms13110092 - 17 Nov 2025

Cited by 1 | Viewed by 1150

Abstract

We present a time-dependent framework that combines a hybrid basis, consisting of Gaussian-type orbitals (GTOs) and finite-element discrete-variable representation (FEDVR) functions, with a multicenter grid to simulate strong-field and attosecond dynamics in atoms and molecules. The method incorporates the construction of the orthonormal [...] Read more.

We present a time-dependent framework that combines a hybrid basis, consisting of Gaussian-type orbitals (GTOs) and finite-element discrete-variable representation (FEDVR) functions, with a multicenter grid to simulate strong-field and attosecond dynamics in atoms and molecules. The method incorporates the construction of the orthonormal hybrid basis, the evaluation of electronic integrals, a unitary time-propagation scheme, and the extraction of optical and photoelectron observables. Its accuracy and robustness are benchmarked on one-electron systems such as atomic hydrogen and the dihydrogen cation (

H_{2}^{+}

) through comparisons with essentially-exact reference results for bound-state energies, high-harmonic generation spectra, photoionization cross sections, and photoelectron momentum distributions. This work establishes the groundwork for its integration with quantum-chemistry methods, which is already operational but will be detailed in future work, thereby enabling ab initio simulations of correlated polyatomic systems in intense ultrafast laser fields. Full article

(This article belongs to the Special Issue Ab Initio Calculations in Atomic, Molecular, and Optical Physics: A Tribute to Barry Irwin Schneider)

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16 pages, 1425 KB

Open AccessEditor’s ChoiceArticle

Combining Physics and Machine Learning: Hybrid Models for Predicting Interatomic Potentials

by Kaoutar El Haloui, Nicolas Thome and Nicolas Sisourat

Atoms 2025, 13(11), 89; https://doi.org/10.3390/atoms13110089 - 10 Nov 2025

Cited by 4 | Viewed by 2422

Abstract

Constructing accurate Potential Energy Surfaces (PES) is a central task in molecular modeling, as it determines the forces governing nuclear motion and enables reliable quantum dynamics simulations. While ab initio methods can provide accurate PES, they are computationally prohibitive for extensive applications. Alternatively, [...] Read more.

Constructing accurate Potential Energy Surfaces (PES) is a central task in molecular modeling, as it determines the forces governing nuclear motion and enables reliable quantum dynamics simulations. While ab initio methods can provide accurate PES, they are computationally prohibitive for extensive applications. Alternatively, analytical physics-based models such as the Morse potential offer efficient solutions but are limited by their rigidity and poor generalization to excited states. In recent years, neural networks have emerged as powerful tools for determining PES, due to their universal function approximation capabilities, but they require large training datasets. In this work, we investigate hybrid-residual modeling approaches that combine physics-based potentials with neural network corrections, aiming to leverage both physical priors and data adaptability. Specifically, we compare three hybrid models—APHYNITY, Sequential Phy-ML, and PhysiNet—in their ability to reconstruct the potential energy curve of the ground and first excited states of the hydrogen molecule. Each model integrates a simplified physical representation with a neural component that learns the discrepancies from accurate reference data. Our findings reveal that hybrid models significantly outperform both standalone neural networks and pure physics-based models, especially in low-data regimes. Notably, APHYNITY and Sequential Phy-ML exhibit better generalization and maintain accurate estimation of physical parameters, underscoring the benefits of explicit physics incorporation. Full article

(This article belongs to the Special Issue Artificial Intelligence for Quantum Sciences)

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15 pages, 1029 KB

Open AccessEditor’s ChoiceArticle

Single Ionization with Dressed Projectiles: An Improved Theory for Both Long- and Short-Range Interactions

by Nicolás J. Esponda, Michele A. Quinto, Roberto D. Rivarola and Juan M. Monti

Atoms 2025, 13(11), 88; https://doi.org/10.3390/atoms13110088 - 9 Nov 2025

Viewed by 752

Abstract

In this work, we present a theoretical model to investigate electron emission in collisions between dressed ions with He atoms and H₂ molecules. The projectile potential is described as the sum of a long- and short-range terms. The last term includes a [...] Read more.

In this work, we present a theoretical model to investigate electron emission in collisions between dressed ions with He atoms and H₂ molecules. The projectile potential is described as the sum of a long- and short-range terms. The last term includes a screening function that has its maximum at short distances. The present model is based on the Continuum Distorted Wave Eikonal Initial State (CDW-EIS) theory, but the Eikonal approximation is only made within the long-range transition amplitude. This now leads to physically correct predictions, whenever dressed projectiles are involved, in the binary-encounter peak. Indeed, double-differential cross-sections spectra is calculated and compared with existing experimental data, finding that this model is capable of reproducing some well-known phenomena depending on the projectile charge state. Namely, the dependence of the binary-encounter peak magnitude with the projectile charge state. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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12 pages, 1196 KB

Open AccessEditor’s ChoiceArticle

The Opacity Project: R-Matrix Calculations for Opacities of High-Energy-Density Astrophysical and Laboratory Plasmas

by Anil K. Pradhan and Sultana N. Nahar

Atoms 2025, 13(10), 85; https://doi.org/10.3390/atoms13100085 - 20 Oct 2025

Cited by 2 | Viewed by 1115

Abstract

Accurate determination of opacity is critical for understanding radiation transport in both astrophysical and laboratory plasmas. We employ atomic data from R-Matrix calculations to investigate radiative properties in high-energy-density (HED) plasma sources, focusing on opacity variations under extreme plasma conditions. Specifically, we analyze [...] Read more.

Accurate determination of opacity is critical for understanding radiation transport in both astrophysical and laboratory plasmas. We employ atomic data from R-Matrix calculations to investigate radiative properties in high-energy-density (HED) plasma sources, focusing on opacity variations under extreme plasma conditions. Specifically, we analyze environments such as the base of the convective zone (BCZ) of the Sun (

2 \times 10^{6}

K,

N_{e} = 10^{23}

/cc), and radiative opacity data collected using the inertial confinement fusion (ICF) devices at the Sandia Z facility (

2.11 \times 10^{6}

K,

N_{e} = 3.16 \times 10^{22}

/cc) and the Lawrence Livermore National Laboratory National Ignition Facility. We calculate Rosseland Mean Opacities (RMO) within a range of temperatures and densities and analyze how they vary under different plasma conditions. A significant factor influencing opacity in these environments is line and resonance broadening due to plasma effects. Both radiative and collisional broadening modify line shapes, impacting the absorption and emission profiles that determine the RMO. In this study, we specifically focus on electron collisional and Stark ion microfield broadening effects, which play a dominant role in HED plasmas. We assume a Lorentzian profile factor to model combined broadening and investigate its impact on spectral line shapes, resonance behavior, and overall opacity values. Our results are relevant to astrophysical models, particularly in the context of the solar opacity problem, and provide insights into discrepancies between theoretical calculations and experimental measurements. In addition, we investigate the equation-of-state (EOS) and its impact on opacities. In particular, we examine the “chemical picture” Mihalas–Hummer–Däppen EOS with respect to level populations of excited levels included in the extensive R-matrix calculations. This study should contribute to improving opacity models of HED sources such as stellar interiors and laboratory plasma experiments. Full article

(This article belongs to the Special Issue Electronic, Photonic and Ionic Interactions with Atoms and Molecules)

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14 pages, 865 KB

Open AccessEditor’s ChoiceArticle

Single Electron Capture by Dressed Projectiles Within the Distorted Wave Formalism

by Michele Arcangelo Quinto, Juan Manuel Monti and Roberto Daniel Rivarola

Atoms 2025, 13(10), 84; https://doi.org/10.3390/atoms13100084 - 3 Oct 2025

Cited by 1 | Viewed by 992

Abstract

Single electron capture in collisions involving neutral hydrogen atoms impacted by highly charged dressed projectiles is theoretically investigated using the distorted wave formalism. A series of continuum distorted wave approximations is employed to investigate the electron capture from neutral hydrogen atom impact by [...] Read more.

Single electron capture in collisions involving neutral hydrogen atoms impacted by highly charged dressed projectiles is theoretically investigated using the distorted wave formalism. A series of continuum distorted wave approximations is employed to investigate the electron capture from neutral hydrogen atom impact by boron and carbon projectiles. The projectile potential is described using a two-parameter analytical Green–Sellin–Zachor (GSZ) model potential. The theoretical prediction of total cross sections are compared against other theories and experiments. We looked at a very broad range of collision energies, from 10 keV/u up to 10 MeV/u. In addition, the state-selective cross sections for boron ions are presented. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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28 pages, 3057 KB

Open AccessEditor’s ChoiceArticle

Proton Interactions with Biological Targets: Inelastic Cross Sections, Stopping Power, and Range Calculations

by Camila Strubbia Mangiarelli, Verónica B. Tessaro, Michaël Beuve and Mariel E. Galassi

Atoms 2025, 13(10), 83; https://doi.org/10.3390/atoms13100083 - 24 Sep 2025

Viewed by 2414

Abstract

Proton therapy enables precise dose delivery to tumors while sparing healthy tissues, offering significant advantages over conventional radiotherapy. Accurate prediction of biological doses requires detailed knowledge of radiation interactions with biological targets, especially DNA, a key site of radiation-induced damage. While most biophysical [...] Read more.

Proton therapy enables precise dose delivery to tumors while sparing healthy tissues, offering significant advantages over conventional radiotherapy. Accurate prediction of biological doses requires detailed knowledge of radiation interactions with biological targets, especially DNA, a key site of radiation-induced damage. While most biophysical models (LEM, mMKM, NanOx) rely on water as a surrogate, this simplification neglects the complexity of real biomolecules. In this work, we calculate the stopping power and range of protons in liquid water, dry DNA, and hydrated DNA using semi-empirical cross sections for ionization, electronic excitation, electron capture, and electron loss by protons and neutral hydrogen in the 10 keV–100 MeV energy range. Additionally, ionization cross sections for uracil are computed to explore potential differences between DNA and RNA damage. Our results show excellent agreement with experimental and ab initio data, highlighting significant deviations in stopping power and range between water and DNA. Notably, the stopping power of DNA exceeds that of water at most energies, reducing proton ranges in dry and hydrated DNA by up to 20% and 26%, respectively. These findings provide improved input for Monte Carlo simulations and biophysical models, enhancing RBE predictions and dose accuracy in hadrontherapy. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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12 pages, 541 KB

Open AccessEditor’s ChoiceArticle

Integral Cross Sections and Transport Properties for Electron–Radon Scattering over a Wide Energy Range (0–1000 eV) and a Reduced Electric Field Range (0.01–1000 Td)

by Gregory J. Boyle, Dale L. Muccignat, Joshua R. Machacek and Robert P. McEachran

Atoms 2025, 13(10), 82; https://doi.org/10.3390/atoms13100082 - 23 Sep 2025

Cited by 1 | Viewed by 1031

Abstract

We report calculations for electron–radon scattering using a complex relativistic optical potential method. The energy range of this study is 0–1000 eV, with results for the elastic (total, momentum-transfer and viscosity-transfer) cross section, summed discrete electronic-state integral excitation cross sections and electron-impact ionization [...] Read more.

We report calculations for electron–radon scattering using a complex relativistic optical potential method. The energy range of this study is 0–1000 eV, with results for the elastic (total, momentum-transfer and viscosity-transfer) cross section, summed discrete electronic-state integral excitation cross sections and electron-impact ionization cross sections presented. Here, we obtain our cross sections from a single theoretical relativistic calculation. Since radon is a heavy element, a relativistic treatment is very desirable. The electron transport coefficients are subsequently calculated for reduced electric fields ranging from 0.01 to 1000 Td, using a multi-term solution of Boltzmann’s equation. Full article

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11 pages, 351 KB

Open AccessEditor’s ChoiceArticle

Short–Range Hard–Sphere Potential and Coulomb Interaction: Deser–Trueman Formula for Rydberg States of Exotic Atomic Systems

by Gregory S. Adkins and Ulrich D. Jentschura

Atoms 2025, 13(9), 81; https://doi.org/10.3390/atoms13090081 - 11 Sep 2025

Cited by 3 | Viewed by 1919

Abstract

In exotic atomic systems with hadronic constituent particles, it is notoriously difficult to estimate the strong-interaction correction to energy levels. It is well known that, due to the strength of the nuclear interaction, the problem cannot be solved using Wigner–Brillouin perturbation theory alone. [...] Read more.

In exotic atomic systems with hadronic constituent particles, it is notoriously difficult to estimate the strong-interaction correction to energy levels. It is well known that, due to the strength of the nuclear interaction, the problem cannot be solved using Wigner–Brillouin perturbation theory alone. Recently, high-angular-momentum Rydberg states of exotic atomic systems with hadronic constituents have been identified as promising candidates in the search for new physics in the low-energy sector of the Standard Model. We thus derive a generalized Deser–Trueman formula for the induced energy shift for a general hydrogenic bound state with principal quantum number n and orbital angular momentum quantum number ℓ, and we find that the energy shift is given by the formula

δ E = 2 α_{n, ℓ} β_{ℓ} {(a_{h} / a_{0})}^{2 ℓ + 1} E_{h} / n^{3}

, where

α_{n, 0} = 1

,

α_{n, ℓ} = \prod_{s = 1}^{ℓ} (s^{- 2} - n^{- 2})

,

β_{ℓ} = (2 ℓ + 1) / {[(2 ℓ + 1)!!]}^{2}

,

E_{h}

is the Hartree energy,

a_{h}

is the hadronic radius and

a_{0}

is the generalized Bohr radius. The square of the double factorial,

{[(2 ℓ + 1)!!]}^{2}

, in the denominator implies a drastic suppression of the effect for higher angular momenta. Full article

(This article belongs to the Section Nuclear Theory and Experiments)

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16 pages, 1120 KB

Open AccessEditor’s ChoiceArticle

Simulated Photoabsorption Spectra for Singly and Multiply Charged Ions

by Stephan Fritzsche, Aloka Kumar Sahoo, Lalita Sharma and Stefan Schippers

Atoms 2025, 13(9), 77; https://doi.org/10.3390/atoms13090077 - 3 Sep 2025

Cited by 1 | Viewed by 1415

Abstract

Simulated (or measured) photoabsorption spectra often provide the first indication of how matter interacts with light when irradiated by some radiation source. In addition to the direct, often slowly varying photoabsorption cross-section as a function of the incident photon frequency, such spectra typically [...] Read more.

Simulated (or measured) photoabsorption spectra often provide the first indication of how matter interacts with light when irradiated by some radiation source. In addition to the direct, often slowly varying photoabsorption cross-section as a function of the incident photon frequency, such spectra typically exhibit numerous resonances and edges arising from the interaction of the radiation field with the subvalence or even inner-shell electrons. Broadly speaking, these resonances reflect photoexcitation, with its subsequent fluorescence, or the autoionization of bound electrons. Here, a (relativistic) cascade model is developed for estimating the photoabsorption of (many) atoms and multiply charged ions with a complex shell structure across the periodic table. This model helps distinguish between level- and shell-resolved, as well as total photoabsorption, cross-sections, starting from admixtures of selected initial-level populations. Examples are shown for the photoabsorption of C⁺ ions near the 1s − 2p excitation threshold and for Xe²⁺ ions in the photon energy range from 10 to 200 eV. While the accuracy and resolution of the predicted photoabsortion spectra remain limited due to the additive treatment of resonances and because of missing electronic correlations in the representation of the levels involved, the present implementation is suitable for ions with quite different open-shell structures and may support smart surveys of resonances along different isoelectronic sequences. Full article

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9 pages, 549 KB

Open AccessEditor’s ChoiceArticle

Interfacing the B-Spline R-Matrix and R-Matrix with Time Dependence Computer Codes: An Update

by Juan C. Del Valle, Aaron T. Bondy, Soumyajit Saha, Kathryn R. Hamilton and Klaus Bartschat

Atoms 2025, 13(9), 75; https://doi.org/10.3390/atoms13090075 - 29 Aug 2025

Cited by 2 | Viewed by 1375

Abstract

As a continuation of Schneider et al., Atoms 2022 10, 26, we report recent progress in the development and deployment of the interface between the computational codes B-Spline R-matrix (BSR) and R-Matrix with Time dependence (RMT). These advances have been achieved within [...] Read more.

As a continuation of Schneider et al., Atoms 2022 10, 26, we report recent progress in the development and deployment of the interface between the computational codes B-Spline R-matrix (BSR) and R-Matrix with Time dependence (RMT). These advances have been achieved within the context of the

L S

-coupling scheme. In its current state, the interface handles atomic target states described by single configurations and supports the Fano–Racah phase convention, as required by RMT. As first example of an application, we use the interface to investigate multiphoton single ionization of helium exposed to a linearly polarized laser field with wavelengths between 280 and 316 nm and a peak intensity of

3 \times 10^{14}

W/cm². As a second example, we consider high-order harmonic generation (HHG) in carbon, driven by an intense 30-cycle laser field at 800 nm and a peak intensity of

1 \times 10^{12}

W/cm². Full article

(This article belongs to the Special Issue Ab Initio Calculations in Atomic, Molecular, and Optical Physics: A Tribute to Barry Irwin Schneider)

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12 pages, 1240 KB

Open AccessEditor’s ChoiceArticle

State-Selective Differential Cross Sections for Single-Electron Capture in Slow He⁺–He Collisions

by Shucheng Cui, Kaizhao Lin, Dadi Xing, Ling Liu, Dongmei Zhao, Dalong Guo, Yong Gao, Shaofeng Zhang, Yong Wu, Chenzhong Dong, Xiaolong Zhu and Xinwen Ma

Atoms 2025, 13(9), 74; https://doi.org/10.3390/atoms13090074 - 28 Aug 2025

Cited by 1 | Viewed by 1521

Abstract

A combined experimental and theoretical study is carried out on the single-electron capture process in He⁺–He collisions at energies ranging from 0.5 keV/u to 5 keV/u. Using cold target recoil ion momentum spectroscopy, we obtain state-selective cross sections and angular differential [...] Read more.

A combined experimental and theoretical study is carried out on the single-electron capture process in He⁺–He collisions at energies ranging from 0.5 keV/u to 5 keV/u. Using cold target recoil ion momentum spectroscopy, we obtain state-selective cross sections and angular differential cross sections. Within the entire studied energy range, the dominant channel is the electron captured into the ground-state, and the relative contribution of the dominant channel shows a decreasing trend with increasing energy. The angular differential cross sections of ground-state capture exhibit obvious oscillatory structures. To understand the oscillatory structures of the differential cross sections, we also performed theoretical calculations using the two-center atomic orbital close-coupling method, which well reproduced the oscillatory structures. The results indicate that these structures are strongly correlated to the oscillatory structures of the impact parameter dependence of electron probability. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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15 pages, 628 KB

Open AccessEditor’s ChoiceArticle

Accurate Nonrelativistic Energy Calculations for Helium 1snp^1,3P (n = 2 to 27) States via Correlated B-Spline Basis Functions

by Jing Chi, Hao Fang, Yong-Hui Zhang, Xiao-Qiu Qi, Li-Yan Tang and Ting-Yun Shi

Atoms 2025, 13(8), 72; https://doi.org/10.3390/atoms13080072 - 4 Aug 2025

Cited by 3 | Viewed by 1814

Abstract

Rydberg atoms play a crucial role in testing atomic structure theory, quantum computing and simulation. Measurements of transition frequencies from the

2^{1, 3} S

states to Rydberg

{}^{1, 3}P

states have reached a precision of several kHz, which poses [...] Read more.

Rydberg atoms play a crucial role in testing atomic structure theory, quantum computing and simulation. Measurements of transition frequencies from the

2^{1, 3} S

states to Rydberg

{}^{1, 3}P

states have reached a precision of several kHz, which poses significant challenges for theoretical calculations, since the accuracy of variational energy calculations decreases rapidly with increasing principal quantum number n. Recently the complex “triple” Hylleraas basis was employed to attain the ionization energy of helium

24 {}^{1}P

state with high accuracy. Different from it, we extended the correlated B-spline basis functions (C-BSBFs) to calculate the Rydberg states of helium. The nonrelativistic energies of

1 s n p

{}^{1, 3}P

states up to

n = 27

achieve at least 14 significant digits using a unified basis set, thereby greatly reducing the complexity of the optimization process. Results of geometric structure parameters and cusp conditions were presented as well. Both the global operator and direct calculation methods are employed and cross-checked for contact potentials. This C-BSBF method not only obtains high-accuracy energies across all studied levels but also confirms the effectiveness of the C-BSBFs in depicting long-range and short-range correlation effects, laying a solid foundation for future high-accuracy Rydberg-state calculations with relativistic and QED corrections included in helium atom and low-Z helium-like ions. Full article

(This article belongs to the Special Issue Atom and Plasma Spectroscopy)

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13 pages, 359 KB

Open AccessEditor’s ChoiceReview

Numerical Methods for the Time-Dependent Schrödinger Equation: Beyond Short-Time Propagators

by Ryan Schneider and Heman Gharibnejad

Atoms 2025, 13(8), 70; https://doi.org/10.3390/atoms13080070 - 28 Jul 2025

Cited by 2 | Viewed by 3548

Abstract

This article reviews several numerical methods for the time-dependent Schrödinger Equation (TDSE). We consider both the most commonly used approach—short-time propagation, which solves the TDSE by assuming that the Hamiltonian is time-independent over sufficiently small (time) intervals—as well as a number of higher-order [...] Read more.

This article reviews several numerical methods for the time-dependent Schrödinger Equation (TDSE). We consider both the most commonly used approach—short-time propagation, which solves the TDSE by assuming that the Hamiltonian is time-independent over sufficiently small (time) intervals—as well as a number of higher-order alternatives. Our goal is to dispel the notion that the latter are too computationally demanding for practical use. To that end, we cover methods whose numerical building blocks are shared by short-time propagators or can be handled by standard libraries. Moreover, we make the case that these methods are best positioned to take advantage of parallel computing environments. One of the alternatives considered is a “double DVR” solver, which applies an expansion in a product basis of functions in space and time to obtain a solution (over all space and at multiple time points simultaneously) with a single linear system solve. To our knowledge, and despite its simplicity, this approach has not previously been applied to the TDSE. Full article

(This article belongs to the Special Issue Ab Initio Calculations in Atomic, Molecular, and Optical Physics: A Tribute to Barry Irwin Schneider)

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16 pages, 1681 KB

Open AccessEditor’s ChoiceArticle

Thermal–Condensate Collisional Effects on Atomic Josephson Junction Dynamics

by Klejdja Xhani and Nick P. Proukakis

Atoms 2025, 13(8), 68; https://doi.org/10.3390/atoms13080068 - 22 Jul 2025

Cited by 1 | Viewed by 2009

Abstract

We investigate how collisional interactions between the condensate and the thermal cloud influence the distinct dynamical regimes (Josephson plasma, phase-slip-induced dissipative regime, and macroscopic quantum self-trapping) emerging in ultracold atomic Josephson junctions at non-zero subcritical temperatures. Specifically, we discuss how the self-consistent dynamical [...] Read more.

We investigate how collisional interactions between the condensate and the thermal cloud influence the distinct dynamical regimes (Josephson plasma, phase-slip-induced dissipative regime, and macroscopic quantum self-trapping) emerging in ultracold atomic Josephson junctions at non-zero subcritical temperatures. Specifically, we discuss how the self-consistent dynamical inclusion of collisional processes facilitating the exchange of particles between the condensate and the thermal cloud impacts both the condensate and the thermal currents, demonstrating that their relative importance depends on the system’s dynamical regime. Our study is performed within the full context of the Zaremba–Nikuni–Griffin (ZNG) formalism, which couples a dissipative Gross–Pitaevskii equation for the condensate dynamics to a quantum Boltzmann equation with collisional terms for the thermal cloud. In the Josephson plasma oscillation and vortex-induced dissipative regimes, collisions markedly alter dynamics at intermediate-to-high temperatures, amplifying damping in the condensate imbalance mode and inducing measurable frequency shifts. In the self-trapping regime, collisions destabilize the system even at low temperatures, prompting a transition to Josephson-like dynamics on a temperature-dependent timescale. Our results show the interplay between coherence, dissipation, and thermal effects in a Bose–Einstein condensate at a finite temperature, providing a framework for tailoring Josephson junction dynamics in experimentally accessible regimes. Full article

(This article belongs to the Special Issue Quantum Technologies with Ultracold Atoms)

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19 pages, 764 KB

Open AccessEditor’s ChoiceArticle

Subradiance Generation in a Chain of Two-Level Atoms with a Single Excitation

by Nicola Piovella

Atoms 2025, 13(7), 62; https://doi.org/10.3390/atoms13070062 - 1 Jul 2025

Cited by 1 | Viewed by 1374

Abstract

Studies of subradiance in a chain N two-level atoms in the single excitation regime focused mainly on the complex spectrum of the effective Hamiltonian, identifying subradiant eigenvalues. This can be achieved by finding the eigenvalues N of the Hamiltonian or by evaluating the [...] Read more.

Studies of subradiance in a chain N two-level atoms in the single excitation regime focused mainly on the complex spectrum of the effective Hamiltonian, identifying subradiant eigenvalues. This can be achieved by finding the eigenvalues N of the Hamiltonian or by evaluating the expectation value of the Hamiltonian on a generalized Dicke state, depending on a continuous variable k. This has the advantage that the sum above N can be calculated exactly, such that N becomes a simple parameter of the system and no longer the size of the Hilbert space. However, the question remains how subradiance emerges from atoms initially excited or driven by a laser. Here we study the dynamics of the system, solving the coupled-dipole equations for N atoms and evaluating the probability to be in a generalized Dicke state at a given time. Once the subradiant regions have been identified, it is simple to see if subradiance is being generated. We discuss different initial excitation conditions that lead to subradiance and the case of atoms excited by switching on and off a weak laser. This may be relevant for future experiments aimed at detecting subradiance in ordered systems. Full article

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19 pages, 857 KB

Open AccessEditor’s ChoiceArticle

An Explicit Numerical Scheme for Milne’s Phase–Amplitude Equations

by Robin Piron and Mikael Tacu

Atoms 2025, 13(6), 57; https://doi.org/10.3390/atoms13060057 - 18 Jun 2025

Viewed by 1039

Abstract

We present an explicit numerical method to solve Milne’s phase–amplitude equations. Existing methods directly solve Milne’s nonlinear equation for amplitude. For that reason, they exhibit high sensitivity to errors and are prone to instability through the growth of a spurious, rapidly varying component [...] Read more.

We present an explicit numerical method to solve Milne’s phase–amplitude equations. Existing methods directly solve Milne’s nonlinear equation for amplitude. For that reason, they exhibit high sensitivity to errors and are prone to instability through the growth of a spurious, rapidly varying component of the amplitude. This makes the systematic use of these methods difficult. On the contrary, the present method is based on solving a linear third-order equation which is equivalent to the nonlinear amplitude equation. This linear equation was derived by Kiyokawa, who used it to obtain analytical results on Coulomb wavefunctions. The present method uses this linear equation for numerical computation, thus resolving the problem of the growth of a rapidly varying component. Full article

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12 pages, 819 KB

Open AccessEditor’s ChoiceArticle

Ionization of CF₃CH₂F by Protons and Photons

by Carlos E. Ferreira, Jorge A. de Souza-Corrêa, Alexandre B. Rocha and Antônio C. F. Santos

Atoms 2025, 13(6), 58; https://doi.org/10.3390/atoms13060058 - 18 Jun 2025

Viewed by 1396

Abstract

(1) Background: Ionizing radiation in the Earth’s atmosphere drives key chemical transformations affecting atmospheric composition. Despite their environmental relevance, experimental data on proton collisions with hydrofluorocarbons remain limited, and theoretical models for total cross-sections and stopping power are still underdeveloped. (2) Methods: This [...] Read more.

(1) Background: Ionizing radiation in the Earth’s atmosphere drives key chemical transformations affecting atmospheric composition. Despite their environmental relevance, experimental data on proton collisions with hydrofluorocarbons remain limited, and theoretical models for total cross-sections and stopping power are still underdeveloped. (2) Methods: This study applies Rudd’s semiempirical model to calculate proton impact ionization cross-sections for the CF₃CH₂F molecule, considering contributions from both outer and inner electron shells. The model enables the estimation of differential cross-sections and the average energy of secondary electrons. In addition, we calculate the photoionization cross-sections using a discretized representation of the continuum—the so-called pseudo-spectrum—obtained through TDDFT with PBE0 as an exchange–correlation functional and compare it with the cross-section obtained for proton impact in the high-energy limit. (3) Results: The Rudd model proves highly adaptable and suitable for numerical applications. However, its validation is hindered by the scarcity of experimental data. Existing models, such as SRIM and Bethe–Bloch, show significant discrepancies due to their limited applicability at intermediate energies and lack of molecular structure consideration. (4) Conclusions: A comparison between the Rudd and BEB models reveals strong agreement in the analyzed energy range. This consistency stems from both models accounting for the molecular structure of the target, as well as from the fact that protons and electrons possess charges of the same magnitude, supporting a coherent description of ionization processes at these energies. Full article

(This article belongs to the Special Issue Electronic, Photonic and Ionic Interactions with Atoms and Molecules)

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15 pages, 875 KB

Open AccessEditor’s ChoiceArticle

Multi-Configuration Dirac–Hartree–Fock Calculations of Pr⁹⁺ and Nd¹⁰⁺: Configuration Resolution and Probing Fine-Structure Constant Variation

by Songya Zhang, Cunqiang Wu, Chenzhong Dong and Xiaobin Ding

Atoms 2025, 13(6), 54; https://doi.org/10.3390/atoms13060054 - 16 Jun 2025

Cited by 1 | Viewed by 1591

Abstract

We present high-precision multi-configuration Dirac–Hartree–Fock (MCDHF) calculations for the metastable states of

\Pr^{9 +}

and

{Nd}^{10 +}

ions, systematically investigating their energy levels, transition properties, Landé

g_{J}

factors, and hyperfine interaction constants. Our results show excellent agreement with available experimental [...] Read more.

We present high-precision multi-configuration Dirac–Hartree–Fock (MCDHF) calculations for the metastable states of

\Pr^{9 +}

and

{Nd}^{10 +}

ions, systematically investigating their energy levels, transition properties, Landé

g_{J}

factors, and hyperfine interaction constants. Our results show excellent agreement with available experimental data and theoretical benchmarks, while resolving critical configuration assignment discrepancies through detailed angular momentum coupling analysis. The calculations highlight the significant role of Breit interaction and provide the first theoretical predictions of electric quadrupole hyperfine constants (

B_{hfs}

). These findings deliver essential atomic data for the development of next-generation optical clocks and establish lanthanide highly charged ions as exceptional candidates for precision tests of fundamental physics. Full article

(This article belongs to the Special Issue Atomic and Molecular Data and Their Applications: ICAMDATA 2024)

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64 pages, 2080 KB

Open AccessEditor’s ChoiceReview

Triaxial Shapes in Even–Even Nuclei: A Theoretical Overview

by Dennis Bonatsos, Andriana Martinou, Spyridon K. Peroulis, Dimitrios Petrellis, Polytimos Vasileiou, Theodoros J. Mertzimekis and Nikolay Minkov

Atoms 2025, 13(6), 47; https://doi.org/10.3390/atoms13060047 - 31 May 2025

Cited by 7 | Viewed by 2998

Abstract

Triaxial shapes in even–even nuclei have been considered since the early days of the nuclear collective model. Although many theoretical approaches have been used over the years for their description, no effort appears to have been made for grouping them together and identifying [...] Read more.

Triaxial shapes in even–even nuclei have been considered since the early days of the nuclear collective model. Although many theoretical approaches have been used over the years for their description, no effort appears to have been made for grouping them together and identifying regions on the nuclear chart where the appearance of triaxiality might be favored. In addition, over the last few years, discussion has started on the appearance of small triaxiality in nuclei considered so far as purely axial rotors. In the present work, we collect the predictions made by various theoretical approaches and show that pronounced triaxiality appears to be favored within specific stripes on the nuclear chart, with low triaxiality being present in the regions between these stripes, in agreement with parameter-free predictions made by the proxy-SU(3) approximation to the shell model, based on the Pauli principle and the short-range nature of the nucleon–nucleon interaction. The robustness of triaxiality within these stripes is supported by global calculations made in the framework of the Finite-Range Droplet Model (FRDM), which is based on completely different assumptions and possesses parameters fitted in order to reproduce fundamental nuclear properties. Full article

(This article belongs to the Section Nuclear Theory and Experiments)

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12 pages, 9743 KB

Open AccessEditor’s ChoiceArticle

by Peter Van Reeth and John W. Humberston

Atoms 2025, 13(6), 46; https://doi.org/10.3390/atoms13060046 - 28 May 2025

Viewed by 1594

Abstract

Scattering differential cross-sections (DCSs) are important tools, both experimentally and theoretically, in the investigation of scattering processes in lepton–atom collisions. In the present work, the elastic scattering differential cross-sections (EDCSs) for e⁺-H and e⁺-He below the first excitation threshold [...] Read more.

Scattering differential cross-sections (DCSs) are important tools, both experimentally and theoretically, in the investigation of scattering processes in lepton–atom collisions. In the present work, the elastic scattering differential cross-sections (EDCSs) for e⁺-H and e⁺-He below the first excitation threshold of the target were evaluated using the Kohn variational method and found to be very similar. In both cases, the EDCS below the positronium formation threshold, i.e., for pure elastic scattering, had minimum valley features in which significant minima close to 90 degrees were found at ≈2.8 eV for H and ≈2 eV for He. These minima were shown to be linked to the zero in the s-wave phase shift, which gives rise to the Ramsauer minimum in the elastic integrated cross-sections. They were not vortices, but the overall EDCS structure was found to be related to the structures and vortices found in the Ps formation differential cross-sections just above the Ps formation threshold. The valley-type structure in the EDCS went smoothly through the Ps formation threshold, where it linked up with a similar valley structure in both the EDCS above the threshold and the Ps formation DCS. A comparison with the EDCS for e⁻-H and e⁻-He scattering over the same energy range revealed similarities with the positron EDCS, however, with less pronounced structures that had different angular and momentum dependences. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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14 pages, 1754 KB

Open AccessEditor’s ChoiceArticle

The Single-Active-Electron Approximation with Angular-Momentum-Dependent Potentials: Application to the Helium Atom

by Juan Carlos del Valle and Klaus Bartschat

Atoms 2025, 13(5), 43; https://doi.org/10.3390/atoms13050043 - 14 May 2025

Cited by 1 | Viewed by 2966

Abstract

We discuss an extension of the Single-Active-Electron (SAE) approximation in atoms by allowing the model potential to depend on the angular-momentum quantum number ℓ. We refer to this extension as the ℓ-SAE approximation. The main ideas behind ℓ-SAE are illustrated [...] Read more.

We discuss an extension of the Single-Active-Electron (SAE) approximation in atoms by allowing the model potential to depend on the angular-momentum quantum number ℓ. We refer to this extension as the ℓ-SAE approximation. The main ideas behind ℓ-SAE are illustrated using the helium atom as a benchmark system. We show that introducing ℓ-dependent potentials improves the accuracy of key quantities in atomic structure computed from the Time-Independent Schrödinger Equation (TISE), including energies, oscillator strengths, and static and dynamic polarizabilities, compared to the standard SAE approach. Additionally, we demonstrate that the ℓ-SAE approximation is suitable for quantum simulations of light−atom interactions described by the Time-Dependent Schrödinger Equation (TDSE). As an illustration, we simulate High-order Harmonic Generation (HHG) and the three-sideband (3SB) version of the Reconstruction of Attosecond Beating by Interference of Two-photon Transitions (RABBITT) technique, achieving enhanced accuracy comparable to that obtained in all-electron calculations. One of the main advantages of the ℓ-SAE approach is that existing SAE codes can be easily adapted to handle ℓ-dependent potentials without any additional computational cost. Full article

(This article belongs to the Special Issue Ab Initio Calculations in Atomic, Molecular, and Optical Physics: A Tribute to Barry Irwin Schneider)

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13 pages, 1058 KB

Open AccessEditor’s ChoiceArticle

A Novel Approach to Calculate the Range of High-Energy Charged Particles Within a Medium

by Ioannis Psychogios, Stylianos Vasileios Kontomaris, Anna Malamou and Andreas Stylianou

Atoms 2025, 13(5), 38; https://doi.org/10.3390/atoms13050038 - 27 Apr 2025

Viewed by 1017

Abstract

The determination of energy loss of charged particles as they pass through a medium and consequently the calculation of their range within the medium are of tremendous importance in various areas of physics from both theoretical and practical perspectives. Previous works have derived [...] Read more.

The determination of energy loss of charged particles as they pass through a medium and consequently the calculation of their range within the medium are of tremendous importance in various areas of physics from both theoretical and practical perspectives. Previous works have derived approximate equations regarding the range of ions within a medium, focusing on providing simple solutions for practitioners in the radiotherapy field that do not require significant computational cost, unlike traditional Monte Carlo methods. These solutions focus on radiotherapy and are limited to specific ions’ initial speeds, which should be up to 0.65c (where c is the speed of light in vacuum). In this paper, solutions for significantly larger initial velocities are explored. A new analytical equation for determining the range of charged particles within a medium for initial velocities between 0.6c and 0.9c is presented. This equation provides excellent results when compared to the accurate numerical solution. Beyond its theoretical and mathematical interest, this solution is also reliable for radiotherapy applications. It provides excellent results for protons with initial energies between 200 MeV and 350 MeV and has the major advantage of being expressed in terms of elementary functions, making its use more straightforward compared to other approaches. Full article

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