Non-relativistic Case Research Articles

Cholesky Decomposition in Spin-Free Dirac-Coulomb Coupled-Cluster Calculations.

We present an implementation for the use of Cholesky decomposition (CD) of two-electron integrals within the spin-free Dirac-Coulomb (SFDC) scheme that enables to perform high-accuracy coupled-cluster (CC) calculations at costs almost comparable to those of their nonrelativistic counterparts. While for nonrelativistic CC calculations, atomic-orbital (AO)-based algorithms, due to their significantly reduced disk-space requirements, are the key to efficient large-scale computations, such algorithms are less advantageous in the SFDC case due to their increased computational cost in that case. Here, molecular-orbital (MO)-based algorithms exploiting the CD of the two-electron integrals allow us to reduce disk-space requirements and lead to computational cost in the CC step that is more or less the same as in the nonrelativistic case. The only remaining overhead in a CD-SFDC-CC calculation is due to the need to compute additional two-electron integrals, the somewhat higher cost of the Hartree-Fock calculation in the SFDC case, and additional cost in the transformation of the Cholesky vectors from the AO to the MO representation. However, these additional costs typically amount to less than 5-15% of the total wall time and are thus acceptable. We illustrate the efficiency of our CD scheme for SFDC-CC calculations on a series of illustrative calculations for the X(CO)4 molecules with X = Ni, Pd, Pt.

Anatomy of null contractions

We introduce null contractions of the Poincare and relativistic conformal algebras. The longitudinal null contraction involves writing the algebra in lightcone coordinates and contracting one of the null directions. For the Poincare algebra, this yields two non-overlapping co-dimension one Carroll algebras. The transverse contraction is a limit on the spatial dimensions and yields two non-overlapping co-dimension one Galilean algebras. We find, similar to Susskind’s original observation of the non-relativistic case, that the Poincare algebra, written in the lightcone coordinates, naturally contains Carrollian sub-algebras in one lower dimension. The effect of the longitudinal contraction, which essentially focusses on the null direction, is to disentangle the two Carroll algebras that now correspond to the symmetries of the two null boundaries. The transverse contraction similarly separates the overlapping Galilean sub-algebras of the original Poincare algebra. We discuss aspects of the conformal case, where we get lower dimensional Carroll Conformal algebras and Schrödinger algebras.

Exact work distribution and Jarzynski's equality of a relativistic particle in an expanding piston.

We study the nonequilibrium work in a pedagogical model of relativistic ideal gas. We obtain the exact work distribution and verify Jarzynski's equality. In the nonrelativistic limit, our results recover the nonrelativistic results of Lua and Grosberg [J. Phys. Chem. B 109, 6805 (2005)1520-610610.1021/jp0455428]. We also find that, unlike the nonrelativistic case, the work distribution no longer has zeros and the number of collisions in this relativistic gas model is finite. In addition, based on an analysis of the experimental parameters, we conclude that it is difficult to detect the relativistic effects of the work distribution of the ideal gas in a piston system with the current experimental techniques.

Relativistic model-free prediction for neutrinoless double beta decay at leading order

Starting from a manifestly Lorentz-invariant chiral Lagrangian, we present a model-free prediction for the transition amplitude of the process nn→ppe−e− induced by light Majorana neutrinos, which is a key process of the neutrinoless double beta decay (0νββ) in heavy nuclei employed in large-scale searches. Contrary to the nonrelativistic case, we show that the transition amplitude can be renormalized at leading order without any uncertain contact operators. The predicted amplitude defines a stringent benchmark for the previous estimation with model-dependent inputs, and greatly reduces the uncertainty of 0νββ transition operator in the calculations of nuclear matrix elements. Generalizations of the present framework could also help to address the uncertainties in 0νββ decay induced by other mechanisms. In addition, the present work motivates a relativistic ab initio calculation of 0νββ decay in light and medium-mass nuclei.

Coulomb effect on elastic electron scattering by hydrogen atom and hydrogen-like ions in their metastable 2s state in the 511–770 KeV energy range

This paper reports on a study investigating the elastic scattering of hydrogen atoms in their metastable state (2S-2S) when subjected to electron impact, with a focus on the Coulomb effect within an energy range spanning from 511 keV to 770 keV . The analysis employs semi-relativistic wave functions to depict the hydrogen states. The validity of using Darwin wave functions is confirmed by ensuring that the condition Zα << 1 is met, where α represents the structure constant. Both incident and scattered free electrons are described using Dirac spinors. Notably, the semi-relativistic calculations converge with those of the non-relativistic case at low electron velocities. Additionally, in the presence of the Coulomb effect, the electron wave function is approximated by a Dirac function modulated by the confluent hypergeometric function. The paper presents the relativistic differential cross-section (RDCS) as a function of the final scattering angle for various energies of the incident electron, revealing intriguing insights into diffusion probabilities in specific directions. Moreover, the impact of the atomic number Z on the elastic scattering of hydrogen atoms and hydrogen-like ions by electron impact under the influence of the Coulomb effect is thoroughly examined.

Electron scattering on atomic hydrogen and hydrogenoid atoms in their metastable state (2S-2S) in the relativistic regime

We present, in the relativistic regime, the results of the study of elastic scattering of hydrogen atoms in its metastable state (2S-2S) by electron impact in the absence of a laser field. In the first calculation, semi-relativistic wave functions present hydrogen states. Dirac spinor describes the incident and scattered electrons. The condition Zα<<1 of using Darwin wave functions has been verified. The second approach uses the exact relativistic calculation of this process. The relativistic case calculations converge to those of the non-relativistic case within the low electron velocity range. In the high incoming energy range, the Coulomb effect and exchange effect are neglected. We present the relativistic differential cross-section RDCS as a function of the final scattering angle for different energies of the incident electron. The effect of atomic number Z for hydrogenoid atoms on the cross-section is examined.

The normal Casimir–Lifshitz force for laterally moving graphene

We consider the system of two parallel sheets of graphene which are moving with relative parallel velocity v⃗ and calculate the Casimir energy by using the scattering approach. We analyze in detail the normal (perpendicular to the planes) Casimir force for two systems—graphene/graphene and ideal metal/graphene. In the non-relativistic case v ≪ v F, the relative correction to the Casimir energy (Ev−E0)/E0 is proportional to the (v/c)2 (the maximum value is 0.0033 for the gapeless case and v = v F) for the first system, and it is zero up to the Fermi velocity v = v F for system ideal metal/graphene.

Thermodynamic properties of the noncommutative quantum Hall effect with anomalous magnetic moment

In this paper, we study the thermodynamic properties of the noncommutative quantum Hall effect (NCQHE) with anomalous magnetic moment (AMM) for both relativistic and nonrelativistic cases at high temperatures, where the thermodynamic properties are: the Helmholtz free energy, the entropy, the mean energy, and the heat capacity. We also work with the Euler-MacLaurin formula to construct the partition function. Next, we plotted the graphs of properties as a function of temperature for different values of the magnetic field and of the NC parameters. We note that the Helmholtz free energy decreases with the temperature, increases with the NC parameters, and can decrease or increase with the magnetic field, while the entropy increases with the temperature, decreases with the NC parameters, and can decrease or increase with the magnetic field. Besides, the mean energy increases linearly with the temperature and the heat capacity satisfies the Dulong-Petit law, which also verified that the AMM does not influence the thermodynamic properties.

Minimal Action Principle for Gravity and Electrodynamics, Einstein Lambda, and Lagrange Points

The relativistic equations of gravitation and electromagnetism in the form of Vlasov – Einstein – Maxwell equations are proposed and analyzed. For weakly relativistic equations we get an analog of Mealn – McCree solution. We also study Lagrange points in non-relativistic case with Einstein lambda- term.

Phase matching of electromagnetic wave on moving interface

Due to the boundary conditions of electromagnetic fields and phase matching of electro magnetic waves on interface being the basis to drive the Snell’s laws and Fresnel’s laws, they are also crucial for the analysis of electromagnetic wave propagation in a moving medium. There are mainly two methods to derive the boundary conditions of electromagnetic fields on moving interface. One of them is to use the kinematic integral form, yet this method is based on the classical time-space, and the other is based on the relativistic transformation, the boundary conditions are derived from the scaling effect with geometric method, or from the principle of relativity directly. However, the first one has a form the same as the form obtained by using the kinematic integral form, while the second one obtains a different form. At the same time, the phase matching of electromagnetic wave on moving interface is only discussed by Galileo transformation, however this is unreasonable, because of the relativistic effect cannot be ignored here. Therefore, it is necessary to reexamine the boundary conditions of electromagnetic fields and phase matching of electromagnetic wave on moving interface. Herein, firstly, the relativistic transformation formula of the unit normal vector of moving surface is derived from the surface equation and principle of relativity. Secondly, the boundary conditions of electromagnetic fields on moving interface are given based on the relativistic transformation formula and the non-relativistic transformation formula of the unit normal vector and electromagnetic fields, which show that the boundary conditions of electromagnetic fields on moving interface under the relativistic case and the non-relativistic case have the same form. This is not accidental but definite, because the change of flux of electromagnetic fields, like the change of magnetic flux, from the induction of electromagnetic filed is the same as that from the variation of surface element. Thirdly, the phase matching of electromagnetic wave on moving interface is given based on the relativistic transformation formula of the unit normal vector and the phase matching of electromagnetic wave on resting interface. In the problem of light incident on a homogeneous medium moving at a constant velocity in vacuum or air, using the phase matching of electromagnetic wave on moving interface, the same results can be easily obtained through other methods. The discussion in this study belongs to classical electrodynamics with no quantum effects considered, but the results will provide some conveniences for theoretically analyzing electromagnetic communication, remote sensing and telemetering.

Relativistic hydrodynamics from the single-generator bracket formalism of nonequilibrium thermodynamics

Abstract We employ the generalized bracket formalism of nonequilibrium thermodynamics by Beris and Edwards to derive Lorentz-covariant time-evolution equations for an imperfect fluid with viscosity, dilatational viscosity, and thermal conductivity. Following closely the analysis presented by Öttinger (Physica A, 259, 1998, 24–42; Physica A, 254, 1998, 433–450) to the same problem but for the GENERIC formalism, we include in the set of hydrodynamic variables a covariant vector playing the role of a generalized thermal force and a covariant tensor closely related to the velocity gradient tensor. In our work here, we derive first the nonrelativistic equations and then we proceed to obtain the relativistic ones by elevating the thermal variable to a four-vector, the mechanical force variable to a four-by-four tensor, and by representing the Hamiltonian of the system with the time component of the energy-momentum tensor. For the Poisson and dissipation brackets we assume the same general structure as in the nonrelativistic case, but with the phenomenological coefficients in the dissipation bracket describing friction to heat and viscous effects being properly constrained for the resulting dynamic equations to be manifest Lorentz-covariant. The final relativistic equations are identical to those derived by Öttinger but the present approach seems to be more general in the sense that one could think of alternative forms of the phenomenological coefficients describing friction that could ensure Lorentz-covariance.

An approach without partial wave expansion to calculate scattering of spin-0 and spin- 1 2 $\frac{1}{2}$ particles in high energy regions and those governed by long range interactions

Abstract Scattering of spin-0 and spin- 1 2 $\frac{1}{2}$ particles is formulated in momentum space based on basis states being not expanded in partial waves. No sequential calculations with increasing angular momentum are performed to reach physical convergence, which depends on the scattering energy and the interaction range. Both nonrelativistic and relativistic cases are described. We put forward for consideration the utilization of this approach. By taking some simple interaction models we show some advantages in calculations representing those of high energy scattering as well as those of scattering governed by long range interactions.

The continuity equation in the phase space quantum mechanics

A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The 1-D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free motion and of scattering in a nonrelativistic and relativistic case for setups with internal degrees of freedom is discussed and illustrated. Properties of Wigner functions of unbound states are analysed.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model has been used to investigate the new bound state solutions of the deformed Dirac equation under the condition of spin symmetry and pseudospin symmetry in the 3D-relativistic noncommutative space (3D-RNCS) symmetries. The combination of spatially dependent mass Coulomb potential with Coulomb-like tensor interaction (ISDM(CP-CLTI)) and the terms (1/r^3 and 1/r^4 ) which are connected with the induced couplings (LΘ and L ̃Θ) produced from the topological defects of space-space gives ISDM(CP-CLTI) model. The new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms (H-atoms) such as He+, Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived within the parameters of the parametric Bopp shift method and standard perturbation theory. The new values E_nc^sp(n,C,m_0,m_1,H,Θ,τ,χ,j,l,s,m) and E_nc^ps(n,C,m_0,m_1,H,Θ,τ,χ,j,l,l ̃,s ̃,m ̃) that we obtained appeared to be sensitive to the quantum atomic discrete quantum numbers (j,k,l,s,m,l ̃,s ̃,m ̃), the mixed potential depths (C/q,m_0,m_1,H), and noncommutativity parameters (NP) (Θ,σ,χ). Within the context of relativistic extended quantum mechanics, we have obtained certain significant special cases that, in our view, will be interesting to the specialized researcher. When we applied the three simultaneous limitations (Θ,σ,χ)→(0,0,0), NP we were able to retrieve the typical results of relativistic and non-relativistic cases in the literature. Our new model presented new physical features compared to other models known in the literature.

Semianalytical findings for the dynamics of the charged particle in the Störmer problem

In this semianalytical research, we present a new ansatz in solving the Störmer problem with numerical findings in graphical representations of solutions where dynamics of the charged particle in the classical dipole magnetic field (here, Earth's magnetic field) was investigated in detail. We have fully solved the Störmer problem for partial class of the charged particle's motion close to the equatorial plane of Earth; this result is the main important theoretical finding presented here with prescribed symmetry. Aforementioned motions, determined by Lorentz force in nonrelativistic case, thereby are explored in polar coordinates to present them in a quasi‐periodic motions in a general case (in equatorial plane of Earth). Thus, the system of momentum equations has been successfully solved numerically or semianalytically in each case and the resulting semianalytical solving algorithm can clarify the quasi‐periodic structure of such family of solutions (with reduction of their geometric symmetry around the Earth in an equatorial plane).

FEATURES OF NEW CYCLOTRON RESONANCES, AS WELL AS CONDITIONS FOR RESONANT ACCELERATION OF CHARGED PARTICLES IN A VACUUM WITHOUT A MAGNETIC FIELD

It is shown that the known conditions for cyclotron resonances are strictly valid only under autoresonance conditions or in the nonrelativistic case. In other cases, it is necessary to use the conditions established in the present work. The main features of charged particle dynamics under new resonant conditions are presented. Conditions are found for infinite acceleration of electrons by a transverse electromagnetic wave in a vacuum without a magnetic field.

A numerical investigation on electron runaway threshold at the initial stage of atmospheric streamer development

Pre-ionization caused by runaway electrons is an important mechanism for negative streamer development. The aim of this paper is to investigate the runaway criteria and overvoltage threshold of electrons at the initial stage of streamer development in air, with the self-developed 3D particle-in-cell with Monte Carlo Collision code. First, numerical simulations are performed with fixed number of electrons to study the runaway criteria in nonrelativistic cases. This method takes the stochastic fluctuations of collisions into account and solves the major shortcomings of theoretical approach. The simulated critical electric field is less than that of the theoretical approach, and the amplitude of the difference increases with electron energy, due to the “tunneling effect” caused by the stochastic fluctuations of collisions. Then, simulations of negative streamers at various applied voltages are performed to investigate the overvoltage threshold. A more intuitive method, searching energetic electrons in front of the negative streamer head, which corresponds to the nature of runaway electrons, is applied to determine the generation of runaway electrons. Electrons that escaped a certain distance ahead of the streamer can be observed at 30 kV. Thus, the overvoltage threshold for runaway electrons can be roughly estimated as 3.3 in our simulations, which is about three times less than the previously published one. At last, with the redefined overvoltage threshold, the figure of regions of breakdown development for various mechanisms depending upon the overvoltage in air is updated.

Numerical approximation of 3D particle beams by multi-scale paraxial Vlasov-Maxwell equations

Even now, solving numerically the 3D time-dependent Vlasov-Maxwell equations is a challenging problem. Hence, it is always interesting to develop simpler but accurate approximate models. By introducing a small parameter, we derived paraxial asymptotic models that approximate these equations, allowing us to treat relativistic cases, much slower beams or even non-relativistic cases. These models are static or quasi-static and with a nth order accuracy that may be chosen as required. Here, we propose a 3D numerical approximation of this multi-scale model. It is based, for the paraxial Maxwell model, on a finite element method coupled with a Particle-In-Cell approximation of the paraxial Vlasov model. Numerical results illustrated the efficiency of the method.

Quantitative analysis of (3He,t) charge exchange reactions at 140 MeV/u beam energy

Composite particle, [Formula: see text], charge exchange reactions on targets [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] at 140 MeV/u beam energy have been analyzed by employing distorted wave impulse approximation (DWIA). Specifically, unit cross-section and angular distribution have been calculated using normal optical model potential (NOMP) and single folding optical model potential (SFOMP) for both relativistic and non-relativistic cases. The sensitivity of present results on exchange terms has also been examined and it is pertinent to report here that the inclusion of these effects reduces the cross-section in magnitude up to 60%, which in turn brings it closer to the data except for [Formula: see text].

Новая схема интегрирования уравнений движения частиц в методе “частиц-в-ячейках” для задач физики плазмы

Представлен новый метод решения релятивистских уравнений движения заряженных частиц в электромагнитных полях, учитывающий условие постоянства их значений на каждом временном шаге. Проведено сравнение точности и эффективности вычислений при решении тестовых задач в дву- и трехмерной постановках на основе нового метода, метода Бориса и его модификаций. В каждом случае рассматривались варианты аналитически и дискретно заданных значений электрического и магнитного полей. At present, the Boris method is mostly common for the numerical solution of problems of the dynamics of charged particles in electromagnetic fields. In recent years, new modifications of the Boris method have appeared that are capable to simplify, refine, or speed up calculations. However, despite the rather large number of publications which consider algorithms for calculating the trajectories of charged particles, new, more accurate and high-performance algorithms are desired. The article presents a detailed analysis of some explicit methods for solving the problem of the motion of charged particles in electromagnetic fields, which differ in how they set the average values of velocities. A new scheme based on an analytical solution, which has not been studied before, is proposed. For the Boris, Vay, Higuera – Cary schemes and the new VD1 scheme, a comparative analysis of the accuracy, convergence and counting time for the relativistic and nonrelativistic cases is carried out. In the nonrelativistic case, the new scheme allows accurate solving of the equations for motion of charged particles, and in the relativistic case, the accuracy of the new scheme remains higher compared to other schemes. As the relativistic factor increases, the accuracy of all considered schemes for calculating the trajectory of particles in an electromagnetic field decreases. For large values of the relativistic parameter, the new scheme is more accurate than the other schemes and retains the second order. In practical calculations, when particles can have different speeds, the scheme automatically adjusts to the given speed of the particle. The presented new scheme for calculating trajectories can be important for solving a wide range of problems in astrophysics and thermonuclear fusion, where high accuracy in determining particle trajectories in inhomogeneous fields for large moments of time is required.