Coulomb Systems Research Articles

Hydrogen-, Helium-, and Lithium-like Bound States in Classical and Quantum Plasmas

We study the effective interactions and the mass action constants for pair and triple associations in classical and quantum plasmas. Avoiding double counting, we derive new expressions for the mass action constants. The calculations resulted in values that were substantially smaller than the standard ones in relevant temperature ranges by up to 50 percent. On this basis, we determine the pressure of H, He and Li plasmas and the osmotic coefficient of electrolytes with higher charges such as, e.g., seawater. Classical and quantum Coulomb systems show strong similarities. The contributions in low orders with respect to the interaction e2 are suppressed by thermal and screening effects. The contributions of weakly bound states, near the continuum edge, to the mass action constants are reduced, replacing the exponential functions with cropped exponentials. The new mass action constants are consistent with well-known extended limiting cases of screening effects. We analyze classical examples including the salts CaCl2 and LaCl3, and a model of seawater including multiple associations. In the case of quantum systems, we follow the work of Planck–Brillouin–Larkin for H plasmas and study He and Li plasmas. The equation of state (EoS) for wide-density regions is obtained through the concatenation of the EoS for the low-density region of partial ionization with the EoS of degenerate plasmas, where all bound states are dissolved and Fermi, Hartree–Fock and Wigner contributions dominate.

Electron temperature relaxation in the clusterized ultracold plasmas

Ultracold plasmas are a promising candidate for the creation of strongly coupled Coulomb systems. Unfortunately, the values of the coupling parameter Γe actually achieved after photoionization of the neutral atoms remain relatively small because of the considerable intrinsic heating of the electrons. A conceivable way to get around this obstacle might be to utilize a spontaneous ionization of the ultracold Rydberg gas, where the initial kinetic energies could be much less. However, the spontaneous avalanche ionization will result in a very inhomogeneous distribution (clusterization) of the ions, which can change the efficiency of the electron relaxation in the vicinity of such clusters substantially. In the present work, this hypothesis is tested by an extensive set of numerical simulations. As a result, it is found that despite a less initial kinetic energy, the subsequent relaxation of the electron velocities in the clusterized plasmas proceeds much more violently than in the case of the statistically uniform ionic distribution. The electron temperature, first, experiences a sharp initial jump (presumably, caused by the “virialization” of energies of the charged particles) and, second, exhibits a gradual subsequent increase (presumably, associated with a multi-particle recombination of the electrons at the ionic clusters). As a possible tool to reduce the anomalous temperature increase, we also considered a two-step plasma formation, involving the blockaded Rydberg states. This leads to a suppression of the clusterization due to a quasi-regular distribution of ions. In such a case, according to the numerical simulations, the subsequent evolution of the electron temperature proceeds more gently, approximately with the same rate as in the statistically uniform ionic distribution.

Nuclear cusp and critical nuclear charge

The nuclear cusp relations for the wave function and the density are revisited. A quantity called cusp constant is introduced. It is proved that the ground-state energy of a Coulomb system is equal to the cusp constant. It is shown that the cusp constant is the same constant for any nucleus of a Coulomb system. An equation for the critical nuclear charge is derived. It is an implicit equation containing quantities taken at a nucleus, which can be used to express the necessary and sufficient condition for the N-electronic mononegative atom to be stable in terms of the electron density derived quantities defined at the nuclear position.

Parameter andqasymptotics of Lq‐norms of hypergeometric orthogonal polynomials

AbstractThe three canonical families of the hypergeometric orthogonal polynomials in a continuous real variable (Hermite, Laguerre, and Jacobi) control the physical wavefunctions of the bound stationary states of a great number of quantum systems [Correction added after first online publication on 21 December, 2022. The sentence has been modified.]. The algebraic ‐norms of these polynomials describe many chemical, physical, and information theoretical properties of these systems, such as, for example, the kinetic and Weizsäcker energies, the position and momentum expectation values, the Rényi and Shannon entropies and the Cramér‐Rao, the Fisher‐Shannon and LMC measures of complexity. In this work, we examine review and solve theq‐asymptotics and the parameter asymptotics (i.e., when the weight function's parameter tends towards infinity) of the unweighted and weighted ‐norms for these orthogonal polynomials. This study has been motivated by the application of these algebraic norms to the energetic, entropic, and complexity‐like properties of the highly excited Rydberg and high‐dimensional pseudo‐classical states of harmonic (oscillator‐like) and Coulomb (hydrogenic) systems, and other quantum systems subject to central potentials of anharmonic type (such as, e.g., some molecular systems) [Correction added after first online publication on 21 December, 2022. Oscillatorlike has been changed to oscillator‐like.].

Systematic derivation of angular-averaged Ewald potential

In this work we provide a step by step derivation of an angular-averaged Ewald potential suitable for numerical simulations of disordered Coulomb systems. The potential was first introduced by E Yakub and C Ronchi without a clear derivation. Two methods are used to find the coefficients of the series expansion of the potential: based on the Euler–Maclaurin and Poisson summation formulas. The expressions for each coefficient is represented as a finite series containing derivatives of Jacobi theta functions. We also demonstrate the formal equivalence of the Poisson and Euler–Maclaurin summation formulas in the three-dimensional case. The effectiveness of the angular-averaged Ewald potential is shown by the example of calculating the Madelung constant for a number of crystal lattices.

Momentum Distribution Functions and Pair Correlation Functions of Unpolarized Uniform Electron Gas in Warm Dense Matter Regime

In this paper we continued our research of the uniform electron gas in a warm dense matter regime, focusing on the momentum distribution functions and pair correlation functions. We use the single–momentum path integral Monte Carlo method, based on the Wigner formulation of quantum statistics to calculate both momentum- and coordinate-depending distributions and average values of quantum operators for many-fermion Coulomb systems. We discovered that the single-particle momentum distribution function deviates from the ideal Fermi distribution and forms the so-called “quantum tails” at high momenta, if non-ideality is strong enough in both degenerate and non-degenerate cases. This effect is always followed by the appearance of the short-range order on pair correlation functions and can be explained by the tunneling through the effective potential wells surrounding the electrons. Furthermore, we calculated the average kinetic and potential energies in the wide range of states, expanding our previous results significantly.

Parabolic quasi-Sturmian approach to proton-impact ionization of helium

A method for calculating ion-atom ionizing collisions is formulated and applied to single ionization of helium induced by energetic proton impact. Within the frozen-core model for the residual helium ion, the four-body problem in the exit channel is recast as an inhomogeneous Schr\"odinger equation for the Coulomb three-body system $({e}^{\ensuremath{-}},{\text{He}}^{+},{p}^{+})$. The asymptotic behavior of its solution contains the transition amplitude. We suggest to solve the driven equation in the representation of so-called parabolic convoluted quasi-Sturmian (CQS) basis functions which are constructed using quasi-Sturmians (QSs) for the $({e}^{\ensuremath{-}},{\text{He}}^{+})$ and $({p}^{+},{\text{He}}^{+})$ subsystems. By applying the corresponding Coulomb Green's function operators, these QSs are generated from orthogonal complements to square-integrable (${L}^{2}$) Sturmian functions in parabolic coordinates. In the proposed parabolic CQS approach explicit asymptotic expressions for the basis functions provide an expansion of the transition amplitude in terms of ``basis amplitudes,'' which we express analytically. The proton-electron interaction is treated as a perturbation and is approximated by a truncated Sturmian basis-set expansion. It is found that, at least in the high-energy limit, the ionization amplitude converges pretty fast as the number of terms in the separable expansion for the proton-electron potential is increased. The calculated fully differential cross sections for singly ionizing 1-MeV $p$-He collisions in several kinematical regimes in the scattering plane are found to be in reasonable agreement with experimental data and with the theoretical results obtained using the first Born approximation, the 3C model, and the wave-packet convergent close-coupling method. The theory-experiment discrepancy over the binary peak position observed by other authors remains unexplained.

Electric double layers with modulated surface charge density: exact 2D results

Electric double layers (EDL) with counterions only, say electrons with the elementary charge −e, in thermal equilibrium at the inverse temperature β are considered. In particular, we study the effect of the surface charge modulation on the particle number density profile and the pressure. The mobile particles are constrained to the surface of a 2D cylinder and immersed in vacuum (no dielectric image charges). An EDL corresponds to the end-circle of the cylinder which carries a (fixed) position-dependent line charge density. The geometries of one single EDL and two EDLs at distance d are considered; the particle density profile is studied for both geometries, the effective interaction of two EDLs is given by the particle pressure on either of the line walls. For any coupling constant Γ ≡ βe 2 = 2 × integer, there exists a mapping of the 2D one-component Coulomb system onto the 1D interacting anticommuting-field theory defined on a chain of sites. Using specific transformation symmetries of anticommuting variables, the contact value theorem is generalized to the EDL with modulated line charge density. For the free-fermion coupling Γ = 2 it is shown that, under certain conditions, the matrix of interaction strengths between anticommuting variables diagonalizes itself which permits one to obtain exact formulas for the particle density profile as well as the pressure. The obtained results confirm the previous indications of weak-coupling and Monte Carlo observations that the surface charge inhomogeneity implies an enhancement of the counterion density at the contact with the charged line and a diminution of the pressure between two parallel lines in comparison with the uniformly charged ones (with the same mean charge densities).

Quantification of the interactions in halide-anion-based imidazolium ionic liquids

Non-covalent interactions between constituent ions of ionic liquids (ILs) define their distinctive physicochemical properties and are critical for understanding their micro and bulk structures as well as designing new task-specific ILs. Therefore, the quantification of these interactions can improve our understanding of this unique class of materials. A set of 20 halide-anion-based imidazolium ILs (Cnmim X, X = Cl, Br, I, and BF4; n = 2, 4, 6, 8, and 10) was selected for this study. The variation of the anion (hydrogen-bond-acceptor property and size), presence of water, and the alkyl chain length of the imidazolium ring have all been considered as parameters when attempting to address mainly the following two fundamental questions: How does the strength of hydrogen bonding in these selected Coulomb systems change? How much do dispersion forces play a role in the net attractive interaction energy despite being weak? To this end, dispersion-corrected density functional theory (DFT) computations were executed to acquire the optimized structures of all the considered ILs, followed by an ensemble of NCI and QATIM analyses to explore the strength of non-covalent interactions. Further, quantum mechanical energy decomposition analysis (QM-EDA) based on symmetry-adapted-perturbation-theory (SAPT) has been employed to dissect the total interaction energy into its components. An assessment of QM-EDA demonstrates that the electrostatic interaction dominates the intermolecular attraction, although induction and dispersion components also play a substantial role. The dispersion energies are amplified when water is present, as well as when anion size and alkyl chain length increase. We quantified the non-covalent interactions between ion pairs using NCI-RDG and Bader's QTAIM analyses. Strong hydrogen bonding with a partial covalent character was observed between monoatomic anions and C2-proton of the imidazolium ring, but for the multiatomic anion (BF4−) and interaction between imidazolium ring alkyl groups and anions, a weak electrostatic hydrogen bonding was found. The hydrogen bonding strength decreases with increasing anion size; it is strongest in the C2mim Cl ion pair. NBO analysis gives a clear indication of nX→σC−H*(X=Cl,Br,I,andF) type intermolecular interaction with highest stabilization energy (En→σ*(2)). A clear correlation between E(2) and hydrogen bond lengths were observed. Overall, the current study illuminates the intricate network of covalent-like and non-covalent interactions in selected imidazolium ILs based on halide anions.

Plasma phase transition in historical aspect and the role of bound states

AbstractThis paper considers the problem of the existence of the plasma phase transition (PPT) in Coulomb systems. The main focus is on the role of the bound states. We declare that the existence of bound states and the need to consider them into account is determined largely by the methodology of the PPT search. We consider phase transitions in classical Coulomb models and hydrogen plasma and point out that electron‐ion recombination is the dominant process in numerical simulation.

Erratum: Two-body neutral Coulomb system in a magnetic field at rest: From hydrogen atom to positronium [Phys. Rev. A 103 , 032820 (2021)

Received 4 April 2022DOI:https://doi.org/10.1103/PhysRevA.105.049901©2022 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAtomic & molecular processes in external fieldsElectronic structure of atoms & moleculesZeeman effectAtomic, Molecular & Optical

Edge fluctuations and third-order phase transition in harmonically confined long-range systems

We study the distribution of the position of the rightmost particle x max in a N-particle Riesz gas in one dimension confined in a harmonic trap. The particles interact via long-range repulsive potential, of the form r −k with −2 < k < ∞ where r is the inter-particle distance. In equilibrium at temperature O(1), the gas settles on a finite length scale L N that depends on N and k. We numerically observe that the typical fluctuation of y max = x max/L N around its mean is of . Over this length scale, the distribution of the typical fluctuations has a N independent scaling form. We show that the exponent η k obtained from the Hessian theory predicts the scale of typical fluctuations remarkably well. The distribution of atypical fluctuations to the left and right of the mean ⟨y max⟩ are governed by the left and right large deviation functions (LDFs), respectively. We compute these LDFs explicitly ∀ k > −2. We also find that these LDFs describe a pulled to pushed type phase transition as observed in Dyson’s log-gas (k → 0) and 1d one component plasma (k = −1). Remarkably, we find that the phase transition remains third order for the entire regime. Our results demonstrate the striking universality of the third order transition even in models that fall outside the paradigm of Coulomb systems and the random matrix theory. We numerically verify our analytical expressions of the LDFs via Monte Carlo simulation using an importance sampling algorithm.

The planar low temperature Coulomb gas: separation and equidistribution

We consider planar Coulomb systems consisting of a large number n of repelling point charges in the low temperature regime, where the inverse temperature \beta grows at least logarithmically in n as n\to\infty , i.e., \beta\gtrsim \log n . Under suitable conditions on an external potential, we prove results to the effect that the gas is with high probability uniformly separated and equidistributed with respect to the corresponding equilibrium measure (in the given external field). Our results generalize earlier results about Fekete configurations, i.e., the case \beta=\infty . There are also several auxiliary results which could be of independent interest. For example, our method of proof of equidistribution (a variant of “Landau’s method”) works for general families of configurations which are uniformly separated and which satisfy certain sampling and interpolation inequalities.

Frustrated Coulombic and Cation Size Effects on Nanoscale Boehmite Aggregation: A Tumbler Small- and Ultra-Small-Angle Neutron Scattering Study

To better understand the effects of solution chemistry on particle aggregation in the complex legacy tank wastes at the Hanford (WA) and Savannah River (SC) sites, we have performed a series of tumbler small- and ultra-small-angle neutron scattering experiments on 20 wt % solid slurries of nanoparticulate aluminum oxyhydroxide (boehmite) with M1+ nitrates of various concentrations and radii. The solutes consisted of H, Li, Na, K, and Rb nitrates at 10–5, 10–3, 10–1, 2, and 4 molal (m) concentrations, as well as in pure H2O. Synthetic boehmite nanoparticles were used with a size range from ∼20 to 30 nm. Tumbler cells were used to keep the solids from settling. Although particles initially form individual rhombohedral platelets, once placed in solution, they quickly form well-bonded stacks, primary aggregates, up to ∼1500 Å long, and a second level of aggregates whose concentration and structure vary as a function of cation type and concentration. Aggregation generally increases with increased solute concentration and with cation radius up to a concentration somewhat above 10–1 m, at which point the trend reverses. Primary aggregates become more rodlike and larger. The Kirkwood-like reversal probably reflects a change from Derjaguin–Landau–Verwey–Overbeek (DLVO)/Debye behavior controlled by surface chemistry to a frustrated Coulombic system controlled by the solution structure. These data suggest that an understanding of the effects of salt concentration and chemistry on nanoparticle aggregate structures provides useful physical insights into the microscopic origin of slurry rheology in the Hanford and Savannah River legacy wastes.

Diamagnetic susceptibility from a nonadiabatic path-integral simulation of few-electron systems

Diamagnetism is the response of dynamical compositions of charged particles, electrons, and nuclei, to an incident magnetic field. In this paper, we study how the finite temperature and finite nuclear masses affect the diamagnetic susceptibilities of selected small atoms and molecules, as limiting cases of dilute gas. We use nonrelativistic path-integral Monte Carlo simulation (PIMC), where both electrons and nuclei are treated on equal footing at finite temperatures in sampling exact Coulomb pair density matrices. The PIMC estimator of diamagnetic susceptibility has been briefly introduced in Ceperley [D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995)], but here we present a comprehensive derivation, discussion of practical effects, and proof-of-concept results for selected few-body Coulomb systems. Our results are in perfect agreement with high-accuracy literature references, where available, but also demonstrate additional thermal effects of the diamagnetic response of low-mass systems.

Basis-set correction based on density-functional theory: Rigorous framework for a one-dimensional model.

We re-examine the recently introduced basis-set correction theory based on density-functional theory, which consists of correcting the basis-set incompleteness error of wave-function methods using a density functional. We use a one-dimensional model Hamiltonian with delta-potential interactions, which has the advantage of making easier to perform a more systematic analysis than for three-dimensional Coulombic systems while keeping the essence of the slow basis convergence problem of wave-function methods. We provide some mathematical details about the theory and propose a new variant of basis-set correction, which has the advantage of being suited to the development of an adapted local-density approximation. We show, indeed, how to develop a local-density approximation for the basis-set correction functional, which is automatically adapted to the basis set employed, without resorting to range-separated density-functional theory as in previous studies, but using instead a finite uniform electron gas whose electron-electron interaction is projected on the basis set. The work puts the basis-set correction theory on firmer ground and provides an interesting strategy for the improvement of this approach.

The symmetry-preserving mean field condition for electrostatic correlations in bulk.

Accurate simulations of a condensed system of ions or polar molecules are concerned with proper handling of the involved electrostatics. For such a Coulomb system at a charged planar interface, the Coulomb interaction averaged over the lateral directions with preserved symmetry serves as a necessary constraint in building any accurate handling that reconciles a simulated singlet charge density with the corresponding macroscopic charge/dielectric response. At present, this symmetry-preserving mean field (SPMF) condition represented in the reciprocal space is conjectured to be necessary for a simulated bulk system to reproduce correctly the charge structure factor of the macroscopic bulk as well. In this work, we further examine analytically the asymptotic behavior of the charge structure factor at small wavenumbers for an arbitrary charge-charge interaction. In light of our theoretical predictions, simulations with lengths of nearly 0.1μm are carried out to demonstrate that typical efficient methods violating the SPMF condition, indeed, fail to capture the exact charge correlations at small wavenumbers for both ionic and polar systems. However, for both types of systems, these existing methods can be simply amended to match the SPMF condition and subsequently to precisely probe the electrostatic correlations at all length scales.

Solitonic natural orbitals in Coulombic systems.

High-accuracy electronic structure calculations on the members of the helium isoelectronic series and the H2 molecule with a stretched bond reveal that the ground-state wavefunctions of these Coulombic systems give rise to natural orbitals (NO) with unusual properties. These solitonic NOs (SoNOs) possess fewer nodes than expected from their small occupation numbers, exhibit substantial spatial localization, and respond (with approximate retention of their shapes) in a paradoxical manner (e.g., by moving away from nuclei upon an increase in the nuclear charge or decrease in the internuclear distance) to changes in the underlying Hamiltonian. An efficient tool for the identification of the SoNOs is provided by an index constructed from two expectation values pertaining to a given NO and the corresponding occupation number. In the case of the helium-like species, the rapid decay of the occupation numbers of the SoNOs with increasing nuclear charge Z is governed by an asymptotic expression that involves the radial positions and spreads of the orbitals. Three s-type SoNOs (with the occupation numbers amounting to only ∼7.9·10-67, 6.8·10-92, and 9.0·10-113 in the case of the helium atom) are predicted to turn into unoccupied NOs at Z equal to ∼2.673, 2.587, and 2.536, respectively. The persistence of the analogous p-type orbital beyond Z = 2 is consistent with the computed properties of the σu-type SoNO of the H2 molecule with a stretched bond. In particular, the profiles of this SoNO along two perpendicular lines bear great resemblance to the radial profiles of its p-type counterpart.

Characterization of the continuous transition from atomic to molecular shape in the three-body Coulomb system

We present an alternative, univocal characterization of the continuous transition from atomic to molecular shape in the Coulomb system constituted by two identical particles and a third particle with the opposite charge, as the mass ratio of the particles varies. Applying a marginal-conditional exact factorization to a variationally optimized wavefunction, we construct a nonadiabatic potential energy surface for the relative motion between the single particle and each of the identical particles in the ground state. The transition is revealed through the evolution with the mass ratio of the topography of such surface and of the shapes of the associated marginal and conditional distributions. Our approach unifies and extends to the nonadiabatic regime the Born-Oppenheimer and charge-distribution pictures of molecular shape.

Temperature relaxation in strongly-coupled binary ionic mixtures

New facilities such as the National Ignition Facility and the Linac Coherent Light Source have pushed the frontiers of high energy-density matter. These facilities offer unprecedented opportunities for exploring extreme states of matter, ranging from cryogenic solid-state systems to hot, dense plasmas, with applications to inertial-confinement fusion and astrophysics. However, significant gaps in our understanding of material properties in these rapidly evolving systems still persist. In particular, non-equilibrium transport properties of strongly-coupled Coulomb systems remain an open question. Here, we study ion-ion temperature relaxation in a binary mixture, exploiting a recently-developed dual-species ultracold neutral plasma. We compare measured relaxation rates with atomistic simulations and a range of popular theories. Our work validates the assumptions and capabilities of the simulations and invalidates theoretical models in this regime. This work illustrates an approach for precision determinations of detailed material properties in Coulomb mixtures across a wide range of conditions.