Atoms

17 pages, 1911 KB

Open AccessArticle

Transformation Between Regular-Irregular Schrödinger Equation Solution Pairs by Means of the Corresponding Milne Function Properties

by Sotirios Danakas and Samuel Cohen

Atoms 2026, 14(8), 71; https://doi.org/10.3390/atoms14080071 - 17 Aug 2026

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Abstract

In scattering and tunnelling-to-continuum problems, the regular and irregular solution pairs to the one-particle radial Schrödinger equation follow various amplitude and phase difference conventions at infinity. The Milne transformation of the Schrödinger equation (a.k.a. the Phase-Amplitude Method) leads to a solution pair whose [...] Read more.

In scattering and tunnelling-to-continuum problems, the regular and irregular solution pairs to the one-particle radial Schrödinger equation follow various amplitude and phase difference conventions at infinity. The Milne transformation of the Schrödinger equation (a.k.a. the Phase-Amplitude Method) leads to a solution pair whose form inherently complies with scattering theory’s requirements of common amplitude and phase difference of

π / 2

at the

r \to \infty

limit. On occasion, applications favoured the adoption of the reverse phase convention. In this work we present a formal framework that permits switching of the regular/irregular solution pairs between representations. To showcase the formalism, two exemplar potential curves are investigated: a repulsive Coulomb barrier and the Coulomb-Stark potential, the latter corresponding to the hydrogen atom in the presence of a static homogeneous electric field. Analysis of the results, assessment of the methodology and a discussion of possible extensions are also included. Ultimately, this work shows that it is the regular solution that acts as the thread that binds together the different representations. Full article

(This article belongs to the Special Issue Quantum and Optical Phenomena in Atomic Systems)

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23 pages, 425 KB

Open AccessArticle

Efficient Methods for Dynamic Correlation in Atoms

by Kenneth G. Dyall

Atoms 2026, 14(8), 70; https://doi.org/10.3390/atoms14080070 - 15 Aug 2026

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Abstract

An algorithm for large-scale correlated calculations on atoms is presented that significantly reduces the scaling of these calculations with the number of single-particle functions used to construct the N-particle states. The reduction is provided in the stage in which the Hamiltonian matrix is [...] Read more.

An algorithm for large-scale correlated calculations on atoms is presented that significantly reduces the scaling of these calculations with the number of single-particle functions used to construct the N-particle states. The reduction is provided in the stage in which the Hamiltonian matrix is contracted with the coefficients or amplitudes of the basis states in an iterative procedure such as the Davidson method. The algorithm relies on the representation of the radial one-particle functions on a grid, and it makes use of the multipole expansion of the electron–electron interaction in a sequence of transformations on the spinors, coefficients, and potentials. It also uses prototyping for the angular integrals and evaluation of recoupling coefficients for the configuration state functions (CSFs) separately rather than for pairs of CSFs. The scaling is verified with calculations on two-electron atoms and analyzed in terms of the number of operations required for each stage of both the proposed algorithm and the conventional methods. 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, 607 KB

Open AccessArticle

Ab Initio Quadrupole-Invariant Analysis of Neutron-Rich Deformed Nuclei Around N = 28

by Liu-Yuan Shen, Xin-Peng Wang, Hong-Hui Li, Hong-Yu Zhu, Wei Zuo and Qi Yuan

Atoms 2026, 14(8), 69; https://doi.org/10.3390/atoms14080069 - 12 Aug 2026

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Abstract

We present an ab initio quadrupole-invariant study of neutron-rich nuclei around

N = 28

. Effective Hamiltonians and consistently evolved

E 2

operators are obtained from chiral NN+3N interactions using the ab initio valence-space in-medium similarity renormalization group, and the resulting

E 2

[...] Read more.

We present an ab initio quadrupole-invariant study of neutron-rich nuclei around

N = 28

. Effective Hamiltonians and consistently evolved

E 2

operators are obtained from chiral NN+3N interactions using the ab initio valence-space in-medium similarity renormalization group, and the resulting

E 2

matrix elements are used to extract the deformation parameters. Along the even–even

N = 28

isotonic chain, the

β

values increase systematically from ⁴⁸Ca to ⁴⁰Mg, providing a direct deformation signature of the weakening of the

N = 28

shell closure. The extracted

β

and

γ

values show a prolate–oblate–triaxial evolution of the

0_{1}^{+}

states from ⁴⁰Mg to ⁴²Si and ⁴⁴S, while the

0_{2}^{+}

states remain mainly in the prolate-to-triaxial region. Extending the analysis to the neighboring

N = 27

and 29 isotones, we find a systematic evolution from strongly prolate Mg isotopes, to oblate-side Si isotopes, and to prolate-to-triaxial S isotopes. These results provide a unified picture of quadrupole deformation, triaxiality, and configuration coexistence around

N = 28

. Full article

(This article belongs to the Special Issue Advances in Nuclear Ab Initio Calculations)

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18 pages, 9081 KB

Open AccessArticle

Reactive Collision Dynamics and Effective Cross-Sections in a Reduced-Dimensional Model Potential

by Sanja Tošić, Vladimir A. Srećković and Veljko Vujčić

Atoms 2026, 14(8), 68; https://doi.org/10.3390/atoms14080068 - 11 Aug 2026

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Abstract

We investigate reactive collision dynamics and effective interaction cross-sections using classical trajectory simulations on a reduced-dimensional reactive potential-energy surface containing reactant and product wells separated by an intermediate barrier region. The simulations are performed over a range of collision velocities for which direct [...] Read more.

We investigate reactive collision dynamics and effective interaction cross-sections using classical trajectory simulations on a reduced-dimensional reactive potential-energy surface containing reactant and product wells separated by an intermediate barrier region. The simulations are performed over a range of collision velocities for which direct scattering, transient trapping, and reactive trajectories coexist within the same interaction landscape. Trajectories are propagated using a velocity-Verlet integration scheme, while reaction probabilities are analyzed as functions of the impact parameter and initial projectile velocity. The calculated probability distributions exhibit strongly localized reactive windows in phase space separated by extended nonreactive regions, indicating pronounced sensitivity of the dynamics to both collision geometry and initial conditions. Probability maps in the (v_x,b) plane reveal a fragmented phase-space structure and highly nonuniform accessibility of the interaction region across the investigated parameter range. The simulations further show substantial variations in the relative importance of reactive, trapped, and back-scattering trajectories with increasing collision velocity, together with non-monotonic behavior of the effective reactive cross-sections. Despite the intentionally reduced dimensionality of the model, the trajectory ensembles reproduce several characteristic features of complex reactive scattering dynamics, including transient trapping, competing dynamical pathways, and broad residence-time distributions. The present results demonstrate that reduced-dimensional classical trajectory approaches can already capture important phase-space mechanisms governing dynamical accessibility and channel competition in reactive molecular collisions. Full article

(This article belongs to the Special Issue Electron-Impact Ionization: Fragmentation and Cross-Section)

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22 pages, 404 KB

Open AccessReview

A Survey on Atomic Clocks in GNSS and Beyond

by Georgios Tzanoulinos, Spiros Makris and Vaios Lappas

Atoms 2026, 14(8), 67; https://doi.org/10.3390/atoms14080067 - 2 Aug 2026

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Abstract

Space-based navigation systems rely on atomic clocks aboard satellites to provide precise time and positioning information through synchronized radio-frequency signals. At the core are Atomic Frequency References (AFRs), which stabilize a local oscillator using atomic transitions to achieve exceptional accuracy and stability. To [...] Read more.

Space-based navigation systems rely on atomic clocks aboard satellites to provide precise time and positioning information through synchronized radio-frequency signals. At the core are Atomic Frequency References (AFRs), which stabilize a local oscillator using atomic transitions to achieve exceptional accuracy and stability. To maintain 1 m positioning precision, timing uncertainties below 3 ns are required, achievable only with high-quality atomic clocks. This paper surveys the principles, architectures, and performance of AFRs in satellite navigation, including the types deployed in major Global Navigation Satellite System (GNSS) constellations such as GPS, Galileo, GLONASS, and BeiDou, and discusses current practices and emerging trends in satellite timing technologies, including the nascent Low-Earth-Orbit Positioning, Navigation, and Timing (LEO-PNT) paradigm. Full article

(This article belongs to the Special Issue Ultra-Precise Atomic Clocks)

12 pages, 1054 KB

Open AccessArticle

Semirelativistic BSR–RMT Interface: Photoionization of Highly Charged Two-Electron Ions

by Aaron T. Bondy and Klaus Bartschat

Atoms 2026, 14(8), 66; https://doi.org/10.3390/atoms14080066 - 1 Aug 2026

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Abstract

We outline an intermediate step toward a semirelativistic BSR–RMT interface by using inner-region structure information generated with the B-spline R-matrix (BSR) method as the input to the Seaton/Badnell STGF/STGBF outer-region asymptotic codes used in R-matrix photoionization calculations. The long-term goal is to exploit [...] Read more.

We outline an intermediate step toward a semirelativistic BSR–RMT interface by using inner-region structure information generated with the B-spline R-matrix (BSR) method as the input to the Seaton/Badnell STGF/STGBF outer-region asymptotic codes used in R-matrix photoionization calculations. The long-term goal is to exploit the compact, nonorthogonal, term-dependent target descriptions available in BSR for time-dependent R-matrix calculations with the R matrix with time dependence (RMT) method, especially for processes sensitive to semirelativistic and spin-orbit effects. To probe these effects, we consider the ground-state photoionization of Ne⁸⁺, Ar¹⁶⁺, Fe²⁴⁺, and Kr³⁴⁺, focusing on resonance structures and the singlet-triplet separation of the predominantly

{2 s 2 p}^{3} P^{o}

and

{2 s 2 p}^{1} P^{o}

autoionizing states and their spin-orbit mixing. For Fe²⁴⁺, we also analyze higher resonances and the region between the ionic thresholds, with R-matrix I (RM-I) calculations using Badnell’s version for comparison. The BSR results agree well overall with the available Iron Project data and with the NIST separations between the predominantly

{2 s 2 p}^{3} P^{o}

and

{2 s 2 p}^{1} P^{o}

levels. Since semirelativistic RMT currently uses RM-I input; the successful interfacing of BSR inner-region data with STGF/STGBF codes that likewise use RM-I input represents a direct precursor to semirelativistic BSR–RMT capability. 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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22 pages, 455 KB

Open AccessArticle

“Square-Root” Klein–Gordon Equation: The Harmonic and Morse Potentials

by Luis A. Poveda, Bill Poirier and Arthur R. B. de Magalhães

Atoms 2026, 14(8), 65; https://doi.org/10.3390/atoms14080065 - 1 Aug 2026

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Abstract

Quantum relativistic solutions of a “square-root” version of the Klein–Gordon equation, for a particle in a one-dimensional Morse potential, are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, [...] Read more.

Quantum relativistic solutions of a “square-root” version of the Klein–Gordon equation, for a particle in a one-dimensional Morse potential, are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, with the latter allowing smooth variation of the system parameters from non-relativistic to ultra-relativistic limits. Analytical expressions for the energy levels and wavefunctions are obtained, as solutions to a Schrödinger-type equation, including relativistic effects through a state-dependent rescaled mass. The eigenstates of the Morse potential exhibit suitable and smooth behavior and approach the corresponding harmonic oscillator solutions as the depth of the Morse potential well increases, as expected. A comparison is also presented between the relativistic harmonic oscillator obtained with this method and the so-called “Klein–Gordon oscillator”. Full article

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

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16 pages, 2050 KB

Open AccessArticle

Configuration Assignments to K = 1/2 Three-Quasiparticle Bands

by Manpreet Kaur, Sushil Kumar, Sukhjeet Singh and A. K. Jain

Atoms 2026, 14(8), 64; https://doi.org/10.3390/atoms14080064 - 1 Aug 2026

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Abstract

In the present work, configuration assignments to K = 1/2 bandheads of given three-quasiparticle (3qp) quadruplets experimentally observed and yet to be observed in odd-A nuclei lying in the rare-earth mass region are carried out based on an improved version of an earlier [...] Read more.

In the present work, configuration assignments to K = 1/2 bandheads of given three-quasiparticle (3qp) quadruplets experimentally observed and yet to be observed in odd-A nuclei lying in the rare-earth mass region are carried out based on an improved version of an earlier empirical formulation. The important contributions of the irrotational component, residual n-p interactions, and rotor–particle coupling terms are explored and highlighted. The magnitude of the RPC term is found to be comparatively smaller (≈5–10 keV) than the irrotational and residual interactions and can affect the 3qp bandheads with small energy separation. We confirm the tentative configuration assignments of three K = 1/2 bandheads having configurations 1/2[411]_π⊗7/2[523]_π⊗5/2[523]_ν and 1/2[411]_π⊗9/2[514]_π⊗7/2[514]_ν in ¹⁷⁵Yb and 9/2[514]_π⊗7/2[514]_ν⊗1/2[521]_ν in ¹⁷⁷Ta nuclides. The average deviation between experimental data and present model calculations for ¹⁷⁵Yb and ¹⁷⁷Ta is 37.9 keV and 163.2 keV, as compared to the earlier model estimates, which yielded a deviation of 249.7 keV and 222.3 keV, respectively. We also resolved some issues pertaining to ambiguous configuration assignments of 3qp quadruplets observed in ¹⁷⁷Lu, ¹⁷³Ta, and ¹⁷⁹Ta nuclides. Encouraged by the good overall agreement among experimental and calculated bandhead energies, we have extended these calculations to predict the locations of 18 K = 1/2 bandheads, which are members of 3qp quadruplets of ^171,175,177Lu, ^173,179,181Ta, and ^179,183Re nuclides but have not been observed experimentally and will be useful for predicting bandheads associated with various 3qp states. Full article

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10 pages, 532 KB

Open AccessArticle

First Direct Mass Measurement of ¹⁸⁴Ta and Surrounding Nuclei

by Jinn Ming Yap, Peter Schury, Sarah Naimi, Gemma Bartram, Alison M. Bruce, Navodhayan Chandrakumar, Jiulong Chen, James G. Cubiss, Siddharth Doshi, Sunil Dutt, Jilehu Gada, Song Guo, Yoshikazu Hirayama, George Hudson-Chang, Sota Kimura, Filip G. Kondev, Gregory Lane, Jenny Lee, Guangshun Li, Yury Litvinov, Zhong Liu, Pengcong Ma, Sorin Pascu, Zsolt Podolyák, Aiko Takamine, Michiharu Wada, Philip Walker, Hiroshi Watanabe and Yutaka X. Watanabe Show full author list Hide full author list

Atoms 2026, 14(8), 63; https://doi.org/10.3390/atoms14080063 - 1 Aug 2026

Viewed by 427

Abstract

Mass measurements of isotopes around the

A = 184

region near stability were performed at the KEK Isotope Separation System within the RIKEN Nishina Center, Japan. The isotopes were produced via multi-nucleon transfer reactions with a ¹³⁶Xe beam of 7.2 MeV/u impinging [...] Read more.

Mass measurements of isotopes around the

A = 184

region near stability were performed at the KEK Isotope Separation System within the RIKEN Nishina Center, Japan. The isotopes were produced via multi-nucleon transfer reactions with a ¹³⁶Xe beam of 7.2 MeV/u impinging on a natural tungsten target. We report the first direct mass measurement of ¹⁸⁴Ta, showing a 62(27) keV deviation compared to previous indirect measurements. Full article

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