Nonrelativistic Results Research Articles

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.

Electronic structure of ground and low-lying excited states of BaLi+ molecular ion: spin-orbit effect, radiative lifetimes and Franck-Condon factor

The study of BaLi+ and its reactivity plays a crucial role in advancing our understanding of chemical bonding or reaction mechanisms. The aim of this work is to represent a complete and extended theoretical study of BaLi+ molecular ion including ground and highly excited electronic states of 1,3Σ, 1,3Π and 1,3Δ symmetries, dissociated to the first seven dissociation limits. The corresponding potential energy curves (PECs), permanent and transition dipole moments have been investigated. These calculations were performed using the multireference configuration interaction (MRCI) method in combination with optimized basis sets and non-empirical pseudopotentials (ECP) for both Ba and Li atoms. Afterwards, the spin–orbit (SO) operator is incorporated in valence MRCI calculation using optimized relativistic spin–orbit pseudopotentials and 16 Ω states are generated and splitted into Λ-S states. The SO effect gives rise to a more complicated structure of electronic states presented in PEC and permanent and transition dipole moments. Nonadiabatic coupling matrix elements between the five lowest 1Σ+ states are also presented for the nonrelativistic results. Based on the vibrational radiative lifetime and Franck–Condon calculation, the possibilities of laser cooling of this system have been also discussed.

Local wellposedness of the relativistic Vlasov–Poisson equation in Besov space

We obtain the local existence and uniqueness of solution to the relativistic Vlasov–Poisson system in Besov space, extending the non-relativistic result to the relativistic case. Weaker regularity is required for the initial datum by optimizing the commutator estimates related to the field. Commutator estimates associated with relativistic velocity are proved by dyadic decomposition of the frequency space and the decreasing properties of Bessel potential.

The Casimir Effect between Parallel Plates Separated by Uniaxial Media: The Effects Due to the Orientation of the Optical Axis

The Casimir force is calculated in the configuration of two parallel plates separated by an anisotropic media. The result exhibits a dependence on the orientation of the optical axis of the intervening media. It is possible that the Casimir force switches its direction as the optical axis orientation varies. The greatest magnitude of the force could be achieved at any optical axis orientation, depending on the dielectric properties of the plates and the intervening media. A comparison between the relativistic and nonrelativistic result revealed that the nonrelativistic approximation could significantly underestimate the attraction or overestimate the repulsion. This error was even greater when the optical axis of the intervening media was perpendicular to the surface of the plates. The nonrelativistic approximation might even fail to predict the trends of the Casimir force at small distances.

Relativistic analysis of upper hybrid wave propagation and trapping

We investigate the impact of relativistic effects on upper hybrid (UH) waves in plasmas with thermal electrons, particularly focusing on modifications of the conditions under which UH wave trapping and related low-threshold parametric decay instabilities (PDIs) may occur. A moderately relativistic (MR) dispersion relation for UH waves, valid for electron temperatures up to 25 keV and wave frequencies up to twice the electron cyclotron frequency, is obtained from previous results and shown to reduce to the warm non-relativistic result commonly used for PDI studies at low electron temperatures. The conditions under which MR UH waves propagate are then determined and compared with warm and cold plasma theory, showing a general increase in the electron density and background magnetic field strength at which the UH resonance occurs for finite electron temperatures. We next investigate the impact of the MR corrections on the possibility of UH wave trapping for X-mode electron cyclotron resonance heated (ECRH) plasmas at the ASDEX Upgrade tokamak and scaled versions of the ASDEX Upgrade parameters with core electron temperatures resembling those expected in ITER X-mode ECRH plasmas. The MR UH wave trapping conditions are virtually unchanged for ASDEX Upgrade relative to warm theory, due to the low electron temperatures, while potentially important differences between warm and MR theory exist for ITER-like core electron temperatures; cold theory is found to be insufficient in both cases. Finally, the MR dispersion relation is shown to qualitatively reproduce the PDI thresholds from warm theory for previously studied ASDEX Upgrade cases.

The Influence of Deformation Phase-Space on Spectra of Heavy Quarkonia in Improved Energy Potential at Finite Temperature Model of Shrodinger Equation Via the Generalized Boob’s Shift Method and Standard Perturbation Theory

In this work, we obtain solutions of the deformed Schrödinger equation (DSE) with improved internal energy potential at a finite temperature model in a 3-dimensional nonrelativistic noncommutative phase-space (3D-NRNCPS) symmetries framework, using the generalized Bopp’s shift method in the case of perturbed nonrelativistic quantum chromodynamics (pNRQCD). The modified bound state energy spectra are obtained for the heavy quarkonium system such as charmonium cc- and bottomonium bb- at finite temperature. It is found that the perturbative solutions of the discrete spectrum are sensible to the discreet atomic quantum numbers (j,l,s,m) of the ( QQ- (Q=c,b)) state, the parameters of internal energy potential (T,αs(T), mD (T),β,c), which are the Debye screening mass mD (T), the running coupling constant αs(T) the critical temperature β, the free parameter c in addition to noncommutativity parameters (Θ,θ-). The new Hamiltonian operator in 3D-NRNCPS symmetries is composed of the corresponding operator in commutative phase-space and three additive parts for spin-orbit interaction, the new magnetic interaction, and the rotational Fermi-term. The obtained energy eigenvalues are applied to obtain the mass spectra of heavy quarkonium systems (cc- and bb-). The total complete degeneracy of the new energy levels of the improved internal energy potential changed to become equal to the new value 3n2 in 3D-NRNCPS symmetries instead of the value n2 in the symmetries of 3D-NRQM. Our non-relativistic results obtained from DSE will possibly be compared with the Dirac equation in high-energy physics.

Transfer learning prediction of spin–orbit correction from bond polarizability for electronic properties of group-III monochalcogenides monolayers

Here, it is shown that the spin–orbit correction, Δ SO , for the electronic properties of group-III monochalcogenides is predictable by their corresponding bond polarizabilities. In this regard, the SOC-based correction for the bandgap and Mexican hat have been calculated from the non-relativistic results together with the bond polarizability parameter. The contribution of the spin–orbit interaction in the mentioned parameters of bandgap and Mexican hat is obtained by a normalized root-mean-square error (nRMSE) of 2 . 8 × 1 0 − 4 and 3 . 2 × 1 0 − 4 , respectively. The transfer learning method has been selected to compensate the size of the dataset. The proposed method reduces the high processing cost associated with the SOC correction calculations by the inclusion of the simple parameter of bond polarizability. From the obtained errors, it is found that the case of the deep learning method results in suitable errors. Thus, the underlying elemental characteristics such as the electronegativity of group-III and VI, bond polarizability and the Mexican hat can deliver numerous electronic band properties through the obtained nonlinear deep transfer method. Interestingly, the Mexican hat parameters can solely be described by the corresponding electronegativities in contrast with the SOC bandgap. • It is shown that the spin–orbit correction for the electronic properties of group-III monochalcogenides is predictable by their corresponding bond polarizabilities. • SOC-based correction for the bandgap and Mexican hat have been calculated from the non-relativistic results together with the bond polarizability parameter. • Bandgap and Mexican hat is obtained by a nRMSE of 2.8e−4 and 3.2e−4, respectively. • The transfer learning method has been selected to compensate the size of the dataset. • The proposed method reduces the high processing cost associated with the SOC correction calculations by the inclusion of the simple parameter of bond polarizability. • It is shown that the Mexican hat parameters can solely be described by the corresponding electronegativities in contrast with the SOC bandgap.

Deformed Dirac and Shrödinger Equations with Improved Mie-Type Potential for Diatomic Molecules and Fermionic Particles in the Framework of Extended Quantum Mechanics Symmetries

In this study, the bound-state solutions of the deformed Dirac equation (DDE) have been determined with the improved Mie-type potential including an improved Coulomb-like tensor potential (IMTPICLP) under the condition of the spin or pseudospin symmetry in the extended relativistic quantum mechanics (ERQM) symmetries. The IMTPICLP model includes a combination of the terms 1/r3 and 1/r4 which coupled with the couplings (LΘ and L̃︀Θ) between the physical properties of the system with the topological deformations of space-space. In the framework of the parametric Bopp’s shift method and standard perturbation theory, the new relativistic and nonrelativistic energy eigenvalues for the improved Mietype potential have been found. The new obtained values appeared sensitive to the quantum numbers (j, k, l,̃︀ l, s, s,̃︀ m, m̃︀ ), the mixed potential depths (A, B, C, α), and noncommutativity parameters (Θ, σ, χ). The new energy spectra of the improved Kratzer–Fues potential within an improved Coulomb-like tensor interaction and the improved modified Kratzer potential within the Coulomb-like tensor interaction have been derived as particular cases of IMTPICLP. We recovered the usual relativistic and nonrelativistic results from the literature by applying the three simultaneous limits (Θ, σ, χ) → (0, 0, 0). One can notice that our results are in close agreement with the recent studies.

Vibrational Corrections to NMR Spin-Spin Coupling Constants from Relativistic Four-Component DFT Calculations.

Zero-point vibrational (ZPV) corrections to the nuclear spin–spin coupling constants have been calculated using four-component Dirac–Kohn–Sham DFT for H2X (where X = O, S, Se, Te, Po), XH3 (where X = N, P, As, Sb, Bi), and XH4 (where X = C, Si, Ge, Sn, and Pb) molecules and for HC≡CPbH3. The main goal was to study the influence of relativistic effects on the ZPV corrections and thus results calculated at relativistic and nonrelativistic approaches have been compared. The effects of relativity become notable for the ZPV corrections to the spin–spin coupling constants for compounds with lighter elements (selenium and germanium) than for the spin–spin coupling constants themselves. In the case of molecules containing heavier atoms, for instance BiH3 and PbH4, relativistic effects play a crucial role on the results and approximating ZPV corrections by the nonrelativistic results may lead to larger errors than omitting ZPV corrections altogether.

Relativistic Two-Photon Matrix Elements for Attosecond Delays

The theory of one-photon ionization and two-photon above-threshold ionization is formulated for applications to heavy atoms in attosecond science by using Dirac–Fock formalism. A direct comparison of Wigner–Smith–Eisenbud delays for photoionization is made with delays from the Reconstruction of Attosecond Beating By Interference of Two-photon Transitions (RABBIT) method. Photoionization by an attosecond pulse train, consisting of monochromatic fields in the extreme ultraviolet range, is computed with many-body effects at the level of the relativistic random phase approximation (RRPA). Subsequent absorption and emission processes of infrared laser photons in RABBIT are evaluated by using static ionic potentials as well as asymptotic properties of relativistic Coulomb functions. As expected, light elements, such as argon, show negligible relativistic effects, whereas heavier elements, such a krypton and xenon, exhibit delays that depend on the fine-structure of the ionic target. The relativistic effects are notably close to ionization thresholds and Cooper minima with differences in fine-structure delays predicted to be as large as tens of attoseconds. The separability of relativistic RABBIT delays into a Wigner–Smith–Eisenbud delay and a universal continuum–continuum delay is studied with reasonable separability found for photoelectrons emitted along the laser polarization axis in agreement with prior non-relativistic results.

Nonrelativistic limits of the relativistic Cucker–Smale model and its kinetic counterpart

We present sufficient frameworks for the uniform-in-time nonrelativistic limits for the relativistic Cucker–Smale (RCS) model and the relativistic kinetic Cucker–Smale (RKCS) equation. For the RCS model, one can easily show that the difference between the solutions to the RCS model and the CS model can be bounded by a quantity proportional to the exponential of time and inversely proportional to some power of the speed of light via a standard Grönwall-type differential inequality. However, this finite-in-time nonrelativistic limit result cannot be used in a uniform-in-time estimate due to the exponential factor of lifespan of solution as it is. For the uniform-in-time nonrelativistic limit, we split the deviation functional between the relativistic solution and the nonrelativistic solution into two parts (finite-time interval and infinite-time interval). In the finite-time interval, the deviation functional is bounded by a finite-in-time nonrelativistic limit result, and then, after a finite time, we use asymptotic flocking estimates with the same asymptotic momentum-like quantity for the RCS model and the CS model to show that the deviation functional can be made as small as possible. In this manner, we can derive a uniform-in-time nonrelativistic limit for the RCS model. For the RKCS equation, we use a uniform-in-time mean-field limit in a measure theoretic framework and a uniform-in-time nonrelativistic limit result for the RCS model to derive a uniform-in-time nonrelativistic limit for the RKCS equation.

Satellite lines of helium-like ions in strongly coupled plasma environment

We report the transition energies and probabilities of satellite lines due to 2p2(3P0,1,2)→1s2p(3P0,1,2) transitions of He-like C,Ne,Al and Ar ions embedded in strongly coupled plasma (SCP) environment. The SCP environment is represented by the ion-sphere (IS) model and the IS radius is varied in such a way that plasma electron densities (∼1020−1026 per cm3) cover a wide range of experimental interest. Both relativistic and non-relativistic energies of 2p2(3P) and 1s2p(3P) states are reported. The relativistic MCDF results are obtained by modified GRASP2K code and the non-relativistic variational results are done by using explicitly correlated Hylleraas basis. The relativistic results are in good agreement with experimental data available in literature. The positions of intercombination lines in presence of dense plasma can be calculated from present results which are comparable with experimental measurements. The partial pressures experienced by the plasma-embedded ions are analyzed in the purview of available experimental observations. An empirical relation is reported to estimate partial pressure on ions in dense plasma environment.

Bremsstrahlung from fully stripped tungsten (W74+) in a Debye-Hückel potential

Based on the exact relativistic multipole calculations using pure Coulomb and Debye-H\"uckel potentials, discussions of bremsstrahlung cross sections and corresponding emitted photon angular distributions from fully stripped tungsten (${\mathrm{W}}^{74+}$) are given for electron scattering at energies of 10--150 keV. Nonrelativistic dipole calculations are also presented, showing that for the above energy region, the nonrelativistic results are completely invalid and the relativistic and multipole effects must be taken into account. The reduction of bremsstrahlung cross sections due to screening of the nuclear charge is studied using the relativistic multipole method at different values of the screening length and photon energy. With an increase of electron scattering energy, the effect of the screening potential on both the dynamics of fast free electrons and the cross sections is reduced. The peak of the angular distribution of bremsstrahlung would shift towards smaller emission angles when the screening strength is reduced.

Analytical Bound State Solutions of the Klein-Fock-Gordon Equation for the Sum of Hulthén and Yukawa Potential within SUSY Quantum Mechanics

The relativistic wave equations determine the dynamics of quantum fields in the context of quantum field theory. One of the conventional tools for dealing with the relativistic bound state problem is the Klein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmount the centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combination of Hulthén and Yukawa potentials. In particular, we show that the relativistic energy eigenvalues and corresponding radial wave functions are obtained from supersymmetric quantum mechanics by applying the shape invariance concept. Here, both scalar potential conditions, which are whether equal and nonequal to vector potential, are considered in the calculation. The energy levels and corresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobi polynomials for arbitrary l states. Beyond that, a closed form of the normalization constant of the wave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive with potential parameters for the quantum states. The nonrelativistic and relativistic results obtained within SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, and it displays that the mathematical implementation of SUSY quantum mechanics is quite perfect.

Two-pion exchange contributions to the nucleon-nucleon interaction in covariant baryon chiral perturbation theory

Employing the covariant baryon chiral perturbation theory, we calculate the leading and next-to-leading order two-pion exchange (TPE) contributions to $NN$ interaction up to order $O(p^3)$. We compare the so-obtained $NN$ phase shifts with $2\leq L\leq 6$ and mixing angles with $2\leq J\leq6$ with those obtained in the nonrelativistic baryon chiral perturbation theory, which allows us to check the relativistic corrections to the medium-range part of $NN$ interactions. We show that the contributions of relativistic TPE are more moderate than those of the nonrelativistic TPE. The relativistic corrections play an important role in F-waves especially the $^3\text{F}_2$ partial wave. Moreover, the relativistic results seem to converge faster than the nonrelativistic results in almost all the partial waves studied in the present work, consistent with the studies performed in the one-baryon sector.

Caloric curves of self-gravitating fermions in general relativity

We study the nature of phase transitions between gaseous and condensed states in the self-gravitating Fermi gas at finite temperature in general relativity. The condensed states can represent compact objects such as white dwarfs, neutron stars, or dark matter fermion balls. The caloric curves depend on two parameters: the system size R and the particle number N. When N < NOV, where NOV is the Oppenheimer–Volkoff limit, there exists an equilibrium state for any value of the temperature T and energy E as in the nonrelativistic case [P.H. Chavanis, Int. J. Mod. Phys. B 20, 3113 (2006)]. Gravitational collapse is prevented by quantum mechanics (Pauli’s exclusion principle). When N > NOV, there is no equilibrium state below a critical energy and below a critical temperature. In that case, the system is expected to collapse toward a black hole. We plot the caloric curves of the general relativistic Fermi gas, study the different types of phase transitions that occur in the system, and determine the phase diagram in the (R, N) plane. The nonrelativistic results are recovered for N ≪ NOV and R ≫ ROV with NR3 fixed. The classical (non quantum) results are recovered for N ≫ NOV and R ≫ ROV with N∕R fixed. We discuss the commutation of the limits c → +∞ and ℏ → 0. We study the relativistic corrections to the nonrelativistic caloric curves and the quantum corrections to the classical caloric curves. We highlight a situation of physical interest where a self-gravitating Fermi gas, by cooling, first undergoes a phase transition toward a compact object (white dwarf, neutron star, dark matter fermion ball), then collapses into a black hole. This situation occurs in the microcanonical ensemble when NOV < N < 3.73 NOV. We also relate the phase transitions from a gaseous state to a core-halo state in the microcanonical ensemble to the onset of red-giant structure and to the supernova phenomenon.

Comparing relativistic three-body wave equations with non-relativistic bound-states results in scalar QFT

We use the variational method in a reformulated model of quantum field theory to compare the three-body relativistic wave equations with non-relativistic equations for our scalar system (consisting of one particle and two antipaticles) interacting via a massless or massive mediating scalar field. Simple Fock-space variational trial states have been employed to derive the equations. Ground state solutions of the relativistic and non-relativistic three-body systems for bound-states equations are obtained and compared for different strengths of coupling, for both massive and massless mediating fields. The relativistic three-body energy solutions are above the non-relativistic results in the present model. In the current formalism we have no negative energy solutions.

The Coherent State of the Landau Hamiltonian and the Relativistic Corrections to the Zeeman Effect in He&amp;lt;sup&amp;gt;+&amp;lt;/sup&amp;gt; Ions

We calculate the energy levels of He+ ion placed in a uniform magnetic field directed perpendicular to the direction of its center of mass (CM) velocity vector, correct to relative order . Our calculations are within the frame work of an approximately relativistic theory, correct to relative order , of a two-particle composite system bound by electromagnetic forces, and written in terms of the position, momentum and spin operators of the constituent particles as proposed by Krajcik and Foldy, and also by Close and Osborn. Since the He+ ion has a net electric charge, the total or the CM momentum is not conserved and a neat separation of the CM and the internal motion is not possible. What is new in our approach is that, for the basis states in a first order degenerate perturbation theory to calculate the effects of the external magnetic field, we use the direct product of the coherent state of the Landau Hamiltonian of the He+ ion in a uniform magnetic field and of the simultaneous eigenstate of the internal Hamiltonian h, j2, l2, s2 and jz, where j, l and s are the internal total, orbital and spin angular moments of the He+ ion. The coherent state is an excellent approximation to the expected classical circular motion of the center of mass (CM) of the He+ ion. In addition to the Z2 a2 corrections to the usual nonrelativistic results, including the small corrections due to the nuclear motion, we also obtain corrections which depend on the kinetic energy (ECM ) of the CM circular motion of the He+ ion, in a nontrivial way. Even though these corrections are proportional to , where M is the mass of the He+ ion, and are small for nonrelativistic CM motion, the results should be verifiable in careful experiments. Our results may also have application in astrophysical observations of the spectral lines of He+ ions in magnetized astrophysical objects.

Testing Modified Gravity Theories by Constraining the Theories Parameter Using Radius and Mass Observations of Brown Dwarfs

Modified Gravity (MOG) theories modify Poisson equation in the Newtonian limit. This MOG is also important as test-bed to analyze whether General Relativity (GR) is consistent with observations. We investigate the mass and radius of Brown Dwarfs predicted by modified Poisson equation from Beyond Horndeski (BH) and Eddington-Inspired-Born-Infeld (EiBI) theories. Brown Dwarfs are sub-stellar objects that are difficult to be detected because of their low luminosity. One of the most important properties of brown dwarfs which recently observed by Johnson and Montet[1] are radii and masses. In this note, we derive the equation of state of the stars analytically, wherein electrons part, we use exact non-relativistic degenerate ideal Fermi gas result. We also derive modified hydrostatic stellar equation predicted by MOG theories and solve them numerically to obtain their profile and mass-radius relation for each parameter on both MOG theories. We compare our theoretical results with the ones obtained Johnson and Montet [1].

Variational relativistic correction to the Thomas-Fermi model of atoms

A relativistic correction to the Thomas-Fermi model of atoms is obtained by utilizing the variational solutions proposed by Csavinszky [Phys. Rev. 166 (1968) 53]. The relativistic effect appears important for heavier elements, and our correction improves the ionization energies more noticeably with an increasing atomic number. The radial expectation values are calculated and compared with Hartree-Fock values. The non-relativistic results for 〈r2〉 are slightly better, and our relativistic correction enhanced the expectation value of the potential energy 〈1/r〉.