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→∞ 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.
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.
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 (vx,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.
We present an ab initio quadrupole-invariant study of neutron-rich nuclei around N=28. Effective Hamiltonians and consistently evolved E2 operators are obtained from chiral NN+3N interactions using the ab initio valence-space in-medium similarity renormalization group, and the resulting E2 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 01+ states from 40Mg to 42Si and 44S, while the 02+ 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.
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”.
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 2s2p3Po and 2s2p1Po 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 2s2p3Po and 2s2p1Po 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.
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.
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.
Precision tests of the Standard Model at low energy are increasingly limited by nuclear-structure theory rather than by experiment. We review two such cases: the two-photon-exchange correction to the Lamb shift in muonic atoms, and the gamma-W box radiative correction to superallowed beta decays. Although they probe different physics, both are governed by the same generalized hadronic tensor, so that the chiral effective field theory Hamiltonians and currents, Lanczos-based response methods, and Bayesian uncertainty quantification developed for one carry over directly to the other. We summarize recent ab initio progress in light nuclei and its impact on nuclear charge radii, on the helium isotope-shift puzzle, and on the extraction of Vud for the top-row CKM unitarity test, and state a future outlook.
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.
Total, momentum-transfer, and differential cross sections, together with spin-polarization (Sherman) functions, are reported for elastic scattering of low-energy electrons from neutral strontium atoms over the energy range 0.001-15 eV. The calculations are performed within a fully relativistic Dirac framework for the continuum states. The target structure is described using multi-configuration Dirac-Hartree-Fock wavefunctions obtained with the GRASP2018 package, while continuum orbitals are generated using the recently developed GRASPC extension. Long-range target polarization effects are incorporated using a dipole model potential, and exchange interactions are treated explicitly for the large and small components of the continuum wavefunctions. Particular attention is given to the ultralow-energy regime, where reliable cross section data for Sr remain limited. The calculated total cross section exhibits a broad maximum near 1 eV, while the momentum-transfer cross section shows a shallow minimum near 0.05-0.06 eV. The differential cross sections are in good agreement with earlier static-exchange-plus-polarization calculations over much of the 1-5 eV range, whereas at lower energies, visible differences appear, especially at forward angles where the results are most sensitive to the polarization interaction. In the ultralow-energy region, the present differential cross sections remain smooth and show no indication of additional low-lying shape resonances within the adopted model. The calculated Sherman functions follow the general trends of earlier theoretical studies at higher energies and decrease rapidly in the sub-eV range. Overall, the present results provide a consistent relativistic dataset for elastic e-Sr scattering at low energies, with emphasis on the near-threshold region.
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-Zp (Zp≤9) projectile ions, yet a puzzling systematic discrepancy remained: IA RTEA cross-sections were consistently larger than experimental, with the disagreement increasing as Zp 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.
In this work, state-selective electron-capture processes in collisions of Ar16+ with ground-state hydrogen are analyzed in classical terms by means of the classical trajectory Z-CTMC method. Oscillations in the n-state-selective charge-exchange cross-sections are observed in the impact-energy range 1-10 keV/u for n-values greater than the nmax value at which charge exchange maximizes. The oscillations are ascribed to an electron-swap mechanism between centers previously identified in ion-Rydberg and ion-alkali charge-exchange collisions. A detailed analysis of the structures in the perpendicular momentum-transfer distributions and their association with the different numbers of swaps is developed. Their dynamics in terms of the collisional impact parameters are also presented.
We study exotic orbits in the photoassociation process by considering both the vibrational and rotational motions in a classical model. In the presence of rotational motion, the exotic orbits possess regular bound segments even when the total energy exceeds the threshold energy of the interaction potential. Such features of the exotic orbits are interpreted by introducing an effective potential. We employ Lagrangian descriptors and escape time to characterize the phase-space structure and show that exotic orbits are distributed around the stable region in the short-range phase space. We further calculate the photoassociation probability. Our work provides new insights into the dynamical mechanisms of photoassociation processes.
Ian Philip Grant (see Figure 1) was a monumental figure in relativistic atomic and molecular atomic physics [...]
Tailoring the spatial and angular-momentum structure of high-harmonic radiation is an important route toward controllable extreme-ultraviolet structured light. Here, we introduce an orbital angular momentum (OAM)-resolved photon-channel framework to elucidate spatial-mode formation in high-harmonic generation driven by bichromatic vortex fields. Combined with numerical solutions of the time-dependent Schrödinger equation for argon, this framework shows that the spatial structure of the emitted harmonics is governed by spectral broadening and interference among dominant OAM-carrying photon channels. This mechanism drives a continuous transition from clean vortex beams at harmonic peaks, characterized by high OAM purity, nearly pure circular polarization, and strong transverse coherence, to disordered speckle-like patterns in harmonic valleys, where multi-channel interference induces pronounced phase distortion. These results identify photon-channel interference as a useful mechanism for controlling the OAM, polarization, and spatial coherence of harmonic beams. The proposed analysis provides a field-control and OAM-channel framework that may be extended in future studies to structured-light probes of ultrafast dynamics and topological phases in quantum materials.
Earlier work by some of the present authors identified power-law statistical regularities across the periodic table: a near-Zipf relationship W∝Z1.09 between atomic weight and atomic number in neutral atoms, and a fine-structure splitting scaling ΔE∝Z1.78 in the 3d2+ (Cr-like) isoelectronic sequence of highly charged ions (HCIs). The apparent tension between this superlinear empirical exponent and the linear-in-Z scaling derived analytically by Lyu, Keitel, and Harman for relativistic clock-state transitions in nd6 ions is resolved by writing the scaling law in the form ΔE=A(Z−Z∗)γ, which gives γ=1.03 and Z∗=25.83 for the 3d2+ data and restores a near-Zipf interpretation. We extend the thresholded model from the single 3d2+ sequence to fourteen isoelectronic sequences spanning C-like (Z≥6) through Mo-like (Z≥42), drawing on the NIST Atomic Spectra Database, evaluated compilations of strontium, copper, and tungsten ion data, and digitization of GRASP MCDHF Mo-like results. Comparing the fitted threshold Z∗ against the classical Slater shielding σSlater of the valence shell reveals three regimes: (i) Coulomb-LS sequences (C/N/O/Ne-like) with Z∗ fixed at zero by construction; (ii) a mainstream cluster of nine sequences (a mix of spin–orbit fine-structure and, for the d2 Ca-/Sr-like members, Coulomb term separations) for which Z∗ tracks σSlater within ±1.5 units; (iii) a pair of nd6D05 sequences (Cr-like 3d6 and Mo-like 4d6) with Z∗−σSlater=+7.20 and +10.36, scaling linearly with Z in the jj regime (after resolving a low-Z LS–jj crossover for the Mo-like member). The slope ratio AMo/ACr≈0.48 is of the order of the hydrogenoid radial-extent ratio ⟨r2⟩(3d)/⟨r2⟩(4d)≈0.32, exceeding it by ∼50% as expected from relativistic 4d contraction. A falsifiable extrapolation for the W-like 5d6D05 sequence is offered: Δ∈[+7,+11], AW∈[0.12,0.18] eV/Z, with explicit failure modes specified for an independent GRASP MCDHF test.
Molecular tunnel ionization sets the stage for the subsequent attosecond processes. Therefore, a comprehensive understanding of its dependence on various molecular properties, such as binding energy, wave function shape, dipole moment, and polarizability, is crucial for tailoring attosecond science concepts for scientific and technological applications. While the first three properties have been relatively well understood, the effects of polarizability have not been explored to the same extent in studies of molecular tunnel ionization. To address this gap, in this work we reveal the peak-shifting effect of permanent polarizability in tunnel ionization rates as the ionizing field varies. By analyzing a specially defined rate gradient, we determine the characteristic directions of the shift from the sign of the local slope of the gradient curve. Furthermore, we show how ionization into higher parabolic channels can transform minima of orientation-dependent rates into maxima. These findings demonstrate how the common assumption that the orientation dependence of tunneling rates resembles orbital shape can break down. We benchmark the method employed, the weak-field asymptotic theory (WFAT), and demonstrate good agreement with ab initio results, underscoring the efficiency and accuracy of WFAT in simulating strong-field ionization at low-to-moderate field strengths.
Single-electron emission from the coronene (C24H12) molecule induced by the impact of 100 keV H+ projectiles is investigated using the continuum distorted wave–eikonal initial state approximation within a semi-classical impact-parameter framework. Multi-center orbitals in the initial channel are derived using the Gaussian quantum chemistry software package, while wave functions of the emitted electron in the exit channel are evaluated using a spherically averaged static potential of the molecular core. The evaluated total ionization cross sections and probabilities are compared and discussed with results obtained from a simplified description of the molecule in which the molecular geometry is ignored. The ionization cross sections of the molecular orbitals generally increase monotonically with decreasing binding energy; however, several deviations from this overall trend are observed. Both the overall trend and the deviations are interpreted in terms of information-theoretical quantities.
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.
