Nonrelativistic Results Research Articles

Relativistic corrections to the electric field gradient given by linear response elimination of the small component formalism

AbstractThis article is concerned with the analysis of relativistic corrections to the electric field gradients (EFGs) via the linear response elimination of the small component scheme (LRESC). Originally developed for magnetic shielding constant, LRESC has been applied in many molecular properties and presented in this work describing EFG for the first time. Within LRESC we obtain relativistic corrections to EFG in terms of 1/c (the speed of light) formally showing that, up to first order (1/c2), there are no virtual pair contributions; recovering the so‐called “no‐pair” approximation. Virtual pair contributions and triplet corrections arise at second order (1/c4). To assess the LRESC description of EFGs at Hartree‐Fock and DFT levels, we applied it to a simple heavy atom containing set of benchmark molecular systems, HX (X = F, Cl, Br, I, and At), and to linear HgX2 (X = Cl, Br, and I) molecules. Fully relativistic four‐component calculations were also done and taken as reference. The most important relativistic correction given by LRESC is a Mass‐velocity related contribution (ΔMv) which represents close to 80% of the nonrelativistic result for At in HAt molecule. For Hg in HgX2 molecular systems, ΔMv is also the most important correction representing close to 60% of the nonrelativistic part. We also describe the overall behavior of LRESC corrections in HgX2 molecules showing low varying results when the weight of the halogen, X, increases. In this kind of molecular system, correlation effects appear in combination to relativity, making them a challenging group to be studied. LRESC results are in very good agreement with previous results for halogen halides, but it shows a need of inclusion of higher order contributions, beyond 1/c2, when applied to Hg in HgX2 set, although LRESC describes accurately At atom, heavier than Hg.

Relativistic Modeling of Ultra-Short Electron Pulse Propagation

The ultrafast electron microscopy, electron diffraction, electron crystallography, and nanocrystallography methods opened the possibility of studying the coherent structural dynamics of matter. The time resolution of the ultrafast electron microscopy and electron diffraction methods determined by the duration of electron pulses is the key parameter of experimental setups. This paper treats electron pulse dynamics in the field free drift region specifically for applications in atomic imaging. The electron beam is modeled as a system of particles (N) with N = 1000 and N = 10 000 electrons. The beam propagates for a certain period of time (1–4 ns); during its propagation, electron distribution parameters (over coordinates and velocities) are calculated to characterize the temporal profile and uncertainty in the electron wavelength at the sample. The results of applying relativistic dynamic equations show that nonrelativistic results are satisfactorily applicable (with 15 per cent or better accuracy) for modeling short electron pulse elongation and broadening at 30 keV and lower energies. However, the results of such modeling may be significantly in error for intermediate energies (300 keV), and for the fast relativistic beams (3 MeV) they become completely wrong. The relative reduction in Coulomb repulsion effects at higher energies is known, however; we give a comprehensive treatment that allows a quantitative picture. Using high-energy electron pulses results in almost complete elimination of the repulsive Coulomb effect. Dispersion of electron velocities becomes much lower at higher energies. For 3 MeV electrons, electron pulse duration as well as its radius does not noticeably change even after traveling for 4 ns (1.2 m). Even at 300 keV, the pulse duration increase is negligible until 1 ns (0.2 m). A simple mean-field model suggested in [13] has been extended to arbitrarily fast relativistic electron pulses with good correspondence to direct dynamic modeling.

Breakdown of agreement between non-relativistic and relativistic quantum dynamical predictions in the non-relativistic regime

To study quantum dynamics in the non-relativistic regime, the standard practice is to use non-relativistic quantum mechanics, instead of the relativistic theory, because it is thought the approximate non-relativistic result is always close to the relativistic one. Here we present a theoretical argument that this expectation is not true in general. In addition, supporting numerical evidence for the free rotor and hydrogen atom also shows the agreement between the two theories can break down quickly. For the radial Rydberg wave packet in hydrogen atom, the breakdown can occur before spontaneous emission and thus could be tested experimentally. Our surprising result shows relativistic quantum mechanics must be used, instead of the approximate non-relativistic theory, to correctly study quantum dynamics in the non-relativistic regime after the breakdown time. This paradigm shift opens a new avenue of research in a wide range of fields from atomic to molecular, chemical and condensed-matter physics.

Role of Exact Exchange and Relativistic Approximations in Calculating 19F Magnetic Shielding in Solids Using a Cluster Ansatz.

Calculations of 19F magnetic shielding in various materials are presented. In calculations on gas-phase molecules, the variation of magnetic shielding with the amount of Hartree-Fock exchange (HFX) in the functional demonstrates that excellent agreement with experiment is obtained with an admixture of 50%, here denoted PBE0 (50%). Calculations at the PBE, PBE0 (25%), and PBE0 (50%) levels on 10 crystalline organofluorines and 15 crystalline inorganic fluorides, in which a cluster ansatz is used to model the lattice environment, were performed. For fluorine-containing aromatics, increasing the admixture of HFX results in the prediction of larger magnetic-shielding spans, whereas increasing the admixture of HFX in calculations for CFCl3 decreases the span. In calculations of 19F magnetic shielding of the inorganic fluorides, the use of sufficiently large clusters of inorganic fluorides results in accuracies similar to those calculated for the organofluorines. Relativistic effects on the magnetic shielding of inorganic fluorides, modeled with ZORA at both the scalar and spin-orbit levels, are dominated by the scalar terms that increase the shielding of most 19F sites over the non-relativistic results. These effects appear to scale with the atomic number of the cation. For most elements of the sixth row (Cs, Ba, La, and Pb), the scalar relativistic contribution to the magnetic shielding is in the range of 20-77 ppm. For elements of group XII (Zn, Cd, and Hg) bonded to fluorine, the scalar relativistic contribution results in deshielding of the 19F site.

Relativistic spinless rotation-vibrational energies of carbon monoxide

We solve the Klein-Gordon equation with the simplified Poschl-Teller potential model by employing the shape invariance technique, and present the relativistic spinless rotation-vibrational energy equation for diatomic molecules with nuclear spin of zero. It is found that the relativistic effects subject to the relative motion of the ions increase slightly the spinless vibrational energies of the ground electronic state of the carbon monoxide molecule in comparison to the nonrelativistic results. We observe that the variation of the relativistic spinless rotation-vibrational energies with respect to the vibrational quantum number in larger rotational quantum numbers holds similar to that with rotational quantum number of zero.

Erratum: Threshold expansion of the three-particle quantization condition [Phys. Rev. D 93 , 096006 (2016)

We recently derived a quantization condition for the energy of three relativistic particles in a cubic box. Here we use this condition to study the energy level closest to the three-particle threshold when the total three-momentum vanishes. We expand this energy in powers of $1/L$, where $L$ is the linear extent of the finite volume. The expansion begins at ${\cal O}(1/L^3)$, and we determine the coefficients of the terms through ${\cal O}(1/L^6)$. As is also the case for the two-particle threshold energy, the $1/L^3$, $1/L^4$ and $1/L^5$ coefficients depend only on the two-particle scattering length $a$. These can be compared to previous results obtained using nonrelativistic quantum mechanics and we find complete agreement. The $1/L^6$ coefficients depend additionally on the two-particle effective range $r$ (just as in the two-particle case) and on a suitably defined threshold three-particle scattering amplitude (a new feature for three particles). A second new feature in the three-particle case is that logarithmic dependence on $L$ appears at $\mathcal O(1/L^6)$. Relativistic effects enter at this order, and the only comparison possible with the nonrelativistic result is for the coefficient of the logarithm, where we again find agreement. For a more thorough check of the $1/L^6$ result, and thus of the quantization condition, we also compare to a perturbative calculation of the threshold energy in relativistic $\lambda \phi^4$ theory, which we have recently presented. Here all terms can be compared and we find full agreement.

Relativistic Approximations to Paramagnetic NMR Chemical Shift and Shielding Anisotropy in Transition Metal Systems.

We apply approximate relativistic methods to calculate the magnetic property tensors, i.e., the g-tensor, zero-field splitting (ZFS) tensor (D), and hyperfine coupling (HFC) tensors, for the purpose of constructing paramagnetic nuclear magnetic resonance (pNMR) shielding tensors. The chemical shift and shielding anisotropy are calculated by applying a modern implementation of the classic Kurland-McGarvey theory ( J. Magn. Reson. 1970 , 2 , 286 ), which formulates the shielding tensor in terms of the g- and HFC tensors obtained for the ground multiplet, in the case of higher than doublet multiplicity defined by the ZFS interaction. The g- and ZFS tensors are calculated by ab initio complete active space self-consistent field and N-electron valence-state perturbation theory methods with spin-orbit (SO) effects treated via quasidegenerate perturbation theory. Results obtained with the scalar relativistic (SR) Douglas-Kroll-Hess Hamiltonian used for the g- and ZFS tensor calculations are compared with nonrelativistically based computations. The HFC tensors computed using the fully relativistic four-component matrix Dirac-Kohn-Sham approach are contrasted against perturbationally SO-corrected nonrelativistic results in the density functional theory framework. These approximations are applied on paramagnetic metallocenes (MCp2) (M = Ni, Cr, V, Mn, Co, Rh, Ir), a Co(II) pyrazolylborate complex, and a Cr(III) complex. SR effects are found to be small for g and D in these systems. The HFCs are found to be more influenced by relativistic effects for the 3d systems. However, for some of the 3d complexes, nonrelativistic calculations give a reasonable agreement with the experimental chemical shift and shielding anisotropy. The influence of scalar relativity is strong for the 5d IrCp2 system. This mixed ab initio/DFT technique, with a fully relativistic method used for the critical HFC tensor, should be useful for the treatment of both electron correlation and relativistic effects at a reasonable computational cost to compute the pNMR shielding tensors in transition metal systems.

Relativistic effects in the non-resonant two-photon K-shell ionization of neutral atoms

Relativistic effects in the non-resonant two-photon K-shell ionization of neutral atoms are studied theoretically within the framework of second-order perturbation theory. The non-relativistic results are compared with the relativistic calculations in the dipole and no-pair approximations as well as with the complete relativistic approach. The calculations are performed in both velocity and length gauges. Our results show a significant decrease of the total cross section for heavy atoms as compared to the non-relativistic treatment, which is mainly due to the relativistic wavefunction contraction. The effects of higher multipoles and negative continuum energy states counteract the relativistic contraction contribution, but are generally much weaker. While the effects beyond the dipole approximation are equally important in both gauges, the inclusion of negative continuum energy states visibly contributes to the total cross section only in the velocity gauge.

Relativistic rotation-vibrational energies for the CP molecule

We solve the Dirac equation with the improved Tietz potential by using the supersymmetric shape invariance approach, and obtain the relativistic rotation-vibrational energy equation for diatomic molecules. We find that the relativistic effects subject to the relative motion of the ions increase slightly the vibrational energies for the X2∑+ state of the CP molecule. The improved Tietz potential energy model does not show any difference in comparison to the nonrelativistic results. It is observed that the behavior of the relativistic rotation-vibrational energies in larger rotational quantum numbers remains similar to that with zero rotational quantum number.

One dimensional PIC simulation of relativistic Buneman instability

Spatio-temporal evolution of the relativistic Buneman instability has been investigated in one dimension using an in-house developed particle-in-cell simulation code. Starting from the excitation of the instability, its evolution has been followed numerically till its quenching and beyond. The simulation results have been quantitatively compared with the fluid theory and are found to be in conformity with the well known fact that the maximum growth rate (γmax) reduces due to relativistic effects and varies with γe0 and m/M as γmax∼32γe0(m2M)1/3, where γe0 is the Lorentz factor associated with the initial electron drift velocity (v0) and (m/M) is the electron to ion mass ratio. Further it is observed that in contrast to the non-relativistic results [A. Hirose, Plasma Phys. 20, 481 (1978)] at the saturation point, the ratio of electrostatic field energy density (∑k|Ek|2/8π) to initial drift kinetic energy density (W0) scales with γe0 as ∼1/γe02. This novel result on the scaling of energy densities has been found to be in quantitative agreement with the scalings derived using fluid theory.

Relativistic corrections to the pair Bc-meson production in e+e− annihilation

Relativistic corrections to the pair Bc-meson production in e+e−-annihilation are calculated. We investigate a production of pair pseudoscalar, vector and pseudoscalar+vector Bc-mesons in the leading order perturbative quantum chromodynamics and relativistic quark model. Relativistic expressions of the pair production cross sections are obtained. Their numerical evaluation shows that relativistic effects in the production amplitudes and bound state wave functions three times reduce nonrelativistic results at the center-of-mass energy s=22 GeV.

Relativistic Modeling Capabilities in PERSEUS Extended-MHD Simulation Code for HED Plasmas

A relativistic version of the PERSEUS extended magnetohydrodynamic simulation code for high-energy-density (HED) plasmas is developed and validated. Nonrelativistic PERSEUS solves the two-fluid equations, formulated in terms of a generalized Ohm’s law, so as to model about nine orders of magnitude in density variation using a local semi-implicit method. Relativistic PERSEUS preserves this structure and, therefore, retains these advantageous properties, enabling it to model a broader range of relativistic HED phenomena than the existing particle and fluid codes. The relativistic code is validated against a published particle-in-cell (PIC) simulation of the penetration of a laser into a supercritical hydrogen plasma with applications to fast ignition. PERSEUS recovers the penetration ahead of the laser of magnetized relativistic jets. As in the PIC simulation, multiple jets form and subsequently coalesce into larger centralized jets. The relativistic code also recovers expected nonrelativistic results for a problem in which the smallest electron length and time scales are under-resolved, and which is, therefore, inaccessible to PIC codes and conventional (explicit) fluid codes. The code can now be applied to relativistic phenomena with under-resolved electron dynamical scales, e.g., $X$ -pinches and electrode surface plasmas.

Threshold expansion of the three-particle quantization condition

We recently derived a quantization condition for the energy of three relativistic particles in a cubic box. Here we use this condition to study the energy level closest to the three-particle threshold when the total three-momentum vanishes. We expand this energy in powers of $1/L$, where $L$ is the linear extent of the finite volume. The expansion begins at ${\cal O}(1/L^3)$, and we determine the coefficients of the terms through ${\cal O}(1/L^6)$. As is also the case for the two-particle threshold energy, the $1/L^3$, $1/L^4$ and $1/L^5$ coefficients depend only on the two-particle scattering length $a$. These can be compared to previous results obtained using nonrelativistic quantum mechanics and we find complete agreement. The $1/L^6$ coefficients depend additionally on the two-particle effective range $r$ (just as in the two-particle case) and on a suitably defined threshold three-particle scattering amplitude (a new feature for three particles). A second new feature in the three-particle case is that logarithmic dependence on $L$ appears at $\mathcal O(1/L^6)$. Relativistic effects enter at this order, and the only comparison possible with the nonrelativistic result is for the coefficient of the logarithm, where we again find agreement. For a more thorough check of the $1/L^6$ result, and thus of the quantization condition, we also compare to a perturbative calculation of the threshold energy in relativistic $\lambda \phi^4$ theory, which we have recently presented. Here all terms can be compared and we find full agreement.

Dispersion relation of low-frequency electrostatic waves in plasmas with relativistic electrons

AbstractThe dispersion relation of electrostatic waves with phase velocities smaller than the electron thermal velocity is investigated in relativistic temperature plasmas. The model equations are the electron relativistic collisionless hydrodynamic equations and the ion non-relativistic Vlasov equation, coupled to the Poisson equation. The complex frequency of electrostatic modes are calculated numerically as a function of the relevant parameters kλDe and ZTe/Ti where k is the wavenumber, λDe, the electron Debye length, Te and Ti the electron and ion temperature, and Z, the ion charge number. Useful analytic expressions of the real and imaginary parts of frequency are also proposed. The non-relativistic results established in the literature from the kinetic theory are recovered and the role of the relativistic effects on the dispersion and the damping rate of electrostatic modes is discussed. In particular, it is shown that in highly relativistic regime the electrostatic waves are strongly damped.

Current Density Scaling Expressions for a Bipolar Space-Charge-Limited Cylindrical Diode

Exact analytical solutions for the currents in a bipolar space-charge-limited (SCL) diode in cylindrical geometry do not exist. Expressions for the electron and ion current densities as functions of the diode cathode and anode radii and the applied voltage are developed and fit to numerical solutions. The results are provided for nonrelativistic and relativistic electrons. These provide scaling relations for the diode, which were not previously available. They also converge to known planar diode expressions in the appropriate limit. The relativistic scaling relations are then combined with nonrelativistic results using a fit function technique to provide a general expression for the current densities valid for all values of the diode voltage. The results are presented for both negative and positive polarities. For completeness, single-species SCL flow results in cylindrical geometry are also developed using the same techniques presented here for bipolar SCL flow and are included in the Appendix.

A relativistic spin zero particle in a spherical cavity

The problem of a relativistic massive scalar particle trapped in an infinite potential spherical well is pedagogically addressed in this paper. The wave function solutions and probability density of the Klein–Gordon equation in spherical coordinates are derived, as well as the energy levels. The results are compared with the non-relativistic solutions of the Schrödinger equation for different values of the particle’s mass. As expected, for very large masses the non-relativistic results are recovered. For illustration, these results are discussed in the specific case of the standard model Higgs field constrained inside a proton, in the quadratic approximation of the Higgs potential around the expectation value.

Accurate determination of the free–free Gaunt factor – II. Relativistic Gaunt factors

When modelling an ionised plasma, all spectral synthesis codes need the thermally averaged free-free Gaunt factor defined over a very wide range of parameter space in order to produce an accurate prediction for the spectrum. Until now no data set exists that would meet these needs completely. We have therefore produced a table of relativistic Gaunt factors over a much wider range of parameter space than has ever been produced before. We present tables of the thermally averaged Gaunt factor covering the range log10(gamma^2) = -6 to 10 and log10(u) = -16 to 13 for all atomic numbers Z = 1 through 36. The data were calculated using the relativistic Bethe-Heitler-Elwert (BHE) approximation and were subsequently merged with accurate non-relativistic results in those parts of the parameter space where the BHE approximation is not valid. These data will be incorporated in the next major release of the spectral synthesis code Cloudy. We also produced tables of the frequency integrated Gaunt factor covering the parameter space log10(gamma^2) = -6 to 10 for all values of Z between 1 and 36. All the data presented in this paper are available online.

Limiting energy spectrum of an electron radiation belt

AbstractTo determine the Kennel‐Petschek limiting particle flux in a planetary radiation belt in a fully relativistic regime, without assuming a predetermined form for the particle energy distribution, has been a long‐standing challenge in space physics. In this paper, for the case of whistler mode wave‐electron interaction, we meet this challenge. The limiting flux is determined by a steady state marginal stability criterion in which a convective wave gain condition is applied over all frequencies for which wave growth occurs. This condition produces an integral equation for the trapped flux. We find that in the relativistic regime the limiting electron energy spectrum varies asymptotically as 1/E, for large energy E, just as in the nonrelativistic case. However, the scaling coefficient in the relativistic case is twice that in the nonrelativistic result. We compare numerical solutions for the limiting spectra with measured energetic electron spectra at Jupiter.

Applying Monte Carlo configuration interaction to transition metal dimers: Exploring the balance between static and dynamic correlation

We calculate potential curves for transition metal dimers using Monte Carlo configuration interaction (MCCI). These results, and their associated spectroscopic values, are compared with experimental and computational studies. The multireference nature of the MCCI wavefunction is quantified and we estimate the important orbitals. We initially consider the ground state of the chromium dimer. Next we calculate potential curves for Sc2 where we contrast the lowest triplet and quintet states. We look at the molybdenum dimer where we compare non-relativistic results with the partial inclusion of relativistic effects via effective core potentials, and report results for scandium nickel.

Relativistic corrections to the pair double heavy diquark production ine+e−annihilation

On the basis of perturbative QCD and a relativistic quark model we calculate relativistic and bound state corrections in the production processes of a pair of double heavy diquarks. Relativistic factors in the production amplitude connected with the relative motion of heavy quarks and the transformation law of the bound state wave function to the reference frame of the moving diquark $S$-wave bound states are taken into account. For the gluon and quark propagators entering the production vertex function we use a truncated expansion in the ratio of the relative quark momenta to the center-of-mass energy $s$ up to the second order. Relativistic corrections to the quark-quark bound state wave functions in the rest frame are considered by means of the Breit-like potential. It turns out that examined effects change essentially nonrelativistic results of the cross sections. The estimate of the yield of pairs of double heavy baryons $(ccq)$ at the $B$ factory is presented.