Relativistic Terms Research Articles

Reconsideration of relativistic corrections for an electron confined in a two-dimensional quantum dot: I. Spin–orbit coupling and Rashba effect

By implementing the relativistic corrections of Pauli’s spin–orbit (SO) coupling, Darwin’s correction, and mass-velocity (MV) interaction, we computationally study the electronic structure of an electron confined in a two-dimensional (2D) quantum dot (QD), and discuss the influence of these relativistic terms. We further study the influence of the Rashba effect on the electronic structure. In the typical semiconductor QD system, the Rashba effect causes a sizeable shift to the energy eigenstates on the order of a milli-electron volt whereas the other relativistic corrections of the internal SO interaction, MV coupling, and Darwin term amount to a shift in energy eigenstates on the order of less than a micro-electron volt, and are largely negligible. The Rashba coupling hybridizes the opposite spin state, and the second-order perturbation approach reveals that this hybridization causes the interstate interference and a curious distribution appears in the spin density in the degenerated state.

Correlation Energies from the Two-Component Random Phase Approximation.

The correlation energy within the two-component random phase approximation accounting for spin-orbit effects is derived. The resulting plasmon equation is rewritten-analogously to the scalar relativistic case-in terms of the trace of two Hermitian matrices for (Kramers-restricted) closed-shell systems and then represented as an integral over imaginary frequency using the resolution of the identity approximation. The final expression is implemented in the TURBOMOLE program suite. The code is applied to the computation of equilibrium distances and vibrational frequencies of heavy diatomic molecules. The efficiency is demonstrated by calculation of the relative energies of the Oh-, D4h-, and C5v-symmetric isomers of Pb6. Results within the random phase approximation are obtained based on two-component Kohn-Sham reference-state calculations, using effective-core potentials. These values are finally compared to other two-component and scalar relativistic methods, as well as experimental data.

BLACK HOLE TRIPLE DYNAMICS: A BREAKDOWN OF THE ORBIT AVERAGE APPROXIMATION AND IMPLICATIONS FOR GRAVITATIONAL WAVE DETECTIONS

Coalescing black hole (BH) binaries forming in the dense core of globular clusters (GCs) are expected to be one the brightest sources of gravitational wave (GW) radiation for the next generation of ground-based laser interferometers. Favorable conditions for merger are initiated by the Kozai resonance in which the gravitational interaction with a third distant object, typically another BH, induces quasi-periodic variations of the inner BH binary eccentricity. In this paper we perform high precision N-body simulations of the long term evolution of hierarchical BH triples and investigate the conditions that lead to the merging of the BH binary and the way it might become an observable source of GW radiation. We find that the secular orbit average treatment, adopted in previous works, does not reliably describe the dynamics of these systems if the binary is orbited by the outer BH on a highly inclined orbit at a moderate distance. During the high eccentricity phase of a Kozai cycle the torque due to the outer BH can drive the binary to extremely large eccentricities in a fraction of the binary's orbital period. This occurs before relativistic terms become important to the evolution and allows the binary GW signal to reach large GW frequencies (>~10 Hz) at high eccentricities. We show that ~50 % of coalescing BH binaries driven by the Kozai mechanism in GCs will have eccentricities larger than 0.1, with 10 % of them being extremely eccentric, (1-e)<~10^-5, when they first chirp in the frequency band of ground based laser interferometers. This implies that a large fraction of such GW sources could be missed if conventional quasi-circular templates are used for analysis of GW detectors data. The efficient detection of all coalescing BH binaries in GCs will therefore require template banks of eccentric inspiral waveforms for matched-filtering and dedicated search strategies.

Newtonian, Post-Newtonian and Relativistic Cosmological Perturbation Theory

Newtonian cosmological perturbation equations valid to full nonlinear order are well known in the literature. Assuming the absence of the transverse-tracefree part of the metric, we present the general relativistic counterpart valid to full nonlinear order. The relativistic equations are presented without taking the slicing (temporal gauge) condition. The equations do have the proper Newtonian and first post-Newtonian limits. We also present the relativistic pressure correction terms in the Newtonian hydrodynamic equations.

Collisional effect on the Weibel instability in a semi-relativistic anisotropic plasma

The detailed properties of the classical electromagnetic Weibel instability in a semi-relativistic anisotropic plasma are investigated for Maxwellian distribution. In this article, the effects of one particular factor affecting the growth rate of Weibel instability, Coulomb collision effect of electrons and ions, is studied and discussed based on the equilibrium semi-relativistic Maxwellian distribution function, in a dense and unmagnetized anisotropic plasma. An analytical expression is derived for the growth rate of the Weibel instability. The two limiting cases ( $\left| \xi\right| \ll1$ and $\left| \xi\right| \gg1$ ) are considered. It is shown that in the limit $\left| \xi\right| \ll 1$ , the quantity η, which is due to the collision term, will appear in the growth and in the conditions of the rate of the Weibel instability. The quantity χ symbolizes the contribution from relativistic terms which becomes unity as we approach the non-relativistic Maxwellian case, leading to the standard Weibel instability scenario. The increasing of η leads to decreasing of the growth rate, and with the decreasing of η the growth rate will increase.

Newtonian hydrodynamics with general relativistic pressure

We present the general relativistic pressure correction terms inNewtonian hydrodynamic equations to the nonlinear order: these are equations (1.1)–(1.3). Thederivation is made in the zero-shear gauge based on the fully nonlinear formulation of cosmological perturbation in Einstein's gravity. The correction terms differ from many of the previously suggested forms in the literature based on hand-waving manners. We confirm our results by comparing with (i) the nonlinear perturbation theory, (ii) the first order post-Newtonian approximation, and (iii) the special relativistic limit, and by checking (iv) the consistency with full Einstein's equation.

Contracted Lorentz invariance for gravity with a preferred foliation

In canonical gravity, the choice of a local time direction is not obviously compatible with local Lorentz invariance. One way to address this issue is to view gravity as a gauge theory on observer space, rather than spacetime. In a Lorentz covariant theory observer space is the space of unit future-directed timelike vectors; picking such a vector locally breaks the symmetry to a subgroup SO(D) of SO(D,1), so that on observer space the local symmetry group is SO(D). Observer space geometries naturally describe any gravitational theory that only assumes local invariance under SO(D). Here we construct nonrelativistic observer spaces for gravity with a fixed foliation, applicable to preferred foliation theories such as Horava-Lifshitz gravity. Different Horava-Lifshitz observers at a point are related by a change in the shift vector, leaving the preferred foliation invariant. Gravity is formulated on a non-relativistic frame bundle with structure group ISO(D); the shift vector acts as a symmetry breaking field breaking ISO(D) symmetry spontaneously to SO(D). A natural torsion-free connection is constructed, including the usual SO(D,1) connection of relativistic gravity plus terms depending on derivatives of the shift vector. This observer space geometry provides a novel geometric framework for the study of preferred foliation theories.

Relativistic differential-difference momentum operators and noncommutative differential calculus

The relativistic kinetic momentum operators are introduced in the framework of the Quantum Mechanics (QM) in the Relativistic Configuration Space (RCS). These operators correspond to the half of the non-Euclidean distance in the Lobachevsky momentum space. In terms of kinetic momentum operators the relativistic kinetic energy is separated as the independent term of the total Hamiltonian. This relativistic kinetic energy term is not distinguishing in form from its nonrelativistic counterpart. The role of the plane wave (wave function of the motion with definite value of momentum and energy) plays the generating function for the matrix elements of the unitary irreps of Lorentz group (generalized Jacobi polynomials). The kinetic momentum operators are the interior derivatives in the framework of the noncommutative differential calculus over the commutative algebra generated by the coordinate functions over the RCS.

Nonrelativistic limit of the Dirac-Schwarzschild Hamiltonian: GravitationalZitterbewegungand gravitational spin-orbit coupling

We investigate the nonrelativistic limit of the gravitationally coupled Dirac equation via a Foldy-Wouthuysen transformation. The relativistic correction terms have immediate and obvious physical interpretations in terms of a gravitational Zitterbewegung and a gravitational spin-orbit coupling. We find no direct coupling of the spin vector to the gravitational force, which would otherwise violate parity. The particle-antiparticle symmetry described recently by one of us [Jentschura, Phys. Rev. A 87, 032101 (2013)] is verified on the level of the perturbative corrections accessed by the Foldy-Wouthuysen transformation. The gravitational corrections to the electromagnetic transition current are calculated.

Qualitative analysis of the anisotropic two-body problem with relativistic potential

In this paper we study the two-body problem that describes the motion of two-point masses in an anisotropic space under the influence of a Newtonian force-law with two relativistic correction terms. We will show that the set of initial conditions leading to collisions and ejections have positive measure and study the capture and escape solutions in the zero-energy case using the infinity manifold. We will also apply the Melnikov method to show that the flow on the zero-energy manifold of another potential which is the sum of the classical Keplerian potential and two anisotropic perturbation which also take into account two relativistic correction terms is chaotic.

Relativistic neoclassical radial fluxes in the 1/ν regime

The radial neoclassical fluxes of electrons in the 1/ν-regime are calculated with relativistic effects taken into account and compared with those in the non-relativistic approach. The treatment is based on the relativistic drift-kinetic equation with the thermodynamic equilibrium given by the relativistic Maxwell–Jüttner distribution function. It is found that for the range of fusion temperatures, Te < 100 keV, the relativistic effects produce a reduction of the radial fluxes which does not exceed 10%. This rather small effect is a consequence of the non-monotonic temperature dependence of the relativistic correction caused by two counteracting factors: a reduction of the contribution from the bulk and a significant broadening with the temperature growth of the energy range of electrons contributing to transport.The relativistic formulation for the radial fluxes given in this paper is expressed in terms of a set of relativistic thermodynamic forces which is not identical to the canonical set since it contains an additional relativistic correction term dependent on the temperature. At the same time, this formulation allows application of the non-relativistic solvers currently used for calculation of mono-energetic transport coefficients.

Relativistic corrections to [formula omitted]-pair production in high energy proton–proton collisions

On the basis of perturbative QCD and the relativistic quark model we calculate relativistic corrections to the double ηc meson production in proton–proton interactions at LHC energies. Relativistic terms in the production amplitude connected with the relative motion of heavy quarks and the transformation law of the bound state wave functions to the reference frame of moving charmonia are taken into account. For the gluon and quark propagators entering the amplitude we use a truncated expansion in relative quark momenta up to the second order. Relativistic corrections to the quark bound state wave functions are considered by means of the Breit-like potential. It turns out that the examined effects decrease total non-relativistic cross section more than two times and on 20 percents in the rapidity region of LHCb detector.

Jahn-Teller theory beyond the standard model

In the standard model of Jahn-Teller (JT) theory, the spin-free electronic Hamiltonian is expanded up to second order in normal-mode displacements, while the spin-orbit (SO) coupling operator is approximated in zeroth order. In this article, the systematic extension of JT theory beyond the standard model is outlined. For the classic E × e and T2 × t2 JT problems in C3v and Td symmetry, respectively, it is shown how the conventional Taylor expansion of the spin-free Hamiltonian can be replaced by an expansion in invariant polynomials up to arbitrarily high orders. The theory of SO coupling in JT systems is extended beyond the standard model by an expansion of the Breit-Pauli SO-coupling operator up to first order in normal-mode displacements. The spectroscopic relevance of the higher-order electrostatic and the linear relativistic JT coupling terms is illustrated for the vibronic structures of the photoelectron spectra of tetrahedral P4 and Sb4.

Relativistic gas in a Schwarzschild metric

A relativistic gas in a Schwarzschild metric is studied within the framework of a relativistic Boltzmann equation in the presence of gravitational fields, where Marle’s model for the collision operator of the Boltzmann equation is employed. The transport coefficients of the bulk and shear viscosities and thermal conductivity are determined from the Chapman–Enskog method. It is shown that the transport coefficients depend on the gravitational potential. Expressions for the transport coefficients in the presence of weak gravitational fields in the non-relativistic (low temperature) and ultra-relativistic (high temperature) limiting cases are given. Apart from the temperature gradient the heat flux has two relativistic terms. The first one, proposed by Eckart, is due to the inertia of energy and represents an isothermal heat flux when matter is accelerated. The other, suggested by Tolman, is proportional to the gravitational potential gradient and indicates that—in the absence of an acceleration field—a state of equilibrium of a relativistic gas in a gravitational field can be attained only if the temperature gradient is counterbalanced by a gravitational potential gradient.

Near Earth Asteroids with measurable Yarkovsky effect

We seek evidence of the Yarkovsky effect among Near Earth Asteroids (NEAs) by measuring the Yarkovsky-related orbital drift from the orbital fit. To prevent the occurrence of unreliable detections we employ a high precision dynamical model, including the Newtonian attraction of 16 massive asteroids and the planetary relativistic terms, and a suitable astrometric data treatment. We find 21 NEAs whose orbital fits show a measurable orbital drift with a signal to noise ratio (SNR) greater than 3. The best determination is for asteroid (101955) 1999 RQ36, with an SNR∼200. In some cases it is possible to constrain physical quantities otherwise unknown. Furthermore, the distribution of the detected orbital drifts shows an excess of retrograde rotators that can be connected to the delivery mechanism from the most important NEA feeding resonances and allows us to infer the obliquity distribution of NEAs. We discuss the implications of the Yarkovsky effect for impact predictions. In particular, for asteroid (29075) 1950 DA our results favor a retrograde rotation, which may have implications for the 2880 impact threat.

Properties of bottomonium in a semi-relativistic model

Using a semi-relativistic potential model we investigate the spectra and decays of the bottomonium (bb̄) system. The Hamiltonian of our model consists of a relativistic kinetic energy term, a vector Coulomb-like potential and a scalar confining potential. Using this Hamiltonian, we obtain a spinless wave equation, which is then reduced to the form of a single particle Schrodinger equation. The spin dependent potentials are introduced as a perturbation. The three-dimensional harmonic oscillator wave function is employed as a trial wave function and the bb̄ mass spectrum is obtained by the variational method. The model parameters and the wave function that reproduce the the bb̄ spectrum are then used to investigate some of their decay properties. The results obtained are then compared with the experimental data and with the predictions of other theoretical models.

Electronic structure and relativistic terms of one-electron spherical quantum dot

We calculated the energy eigenvalues and eigenfunctions of the ground and excited states of a hydrogenic impurity located at the center of a spherical quantum dot using the Quantum Genetic Algorithm (QGA) and Hartree–Fock Roothaan (HFR) method. In addition, we carried out the relativistic effects such as the relativistic correction to the kinetic energy, spin–orbit and Darwin terms by using the perturbation method. The electronic charge density for the Darwin term is computed from the Hiller, Sucher and Feinberg formulation instead of the traditional delta function operator. The results show that impurity, dot radius and confining potential have a great influence on the relativistic effects. In addition, as the absolute value of confining potential increases, the peak values of relativistic corrections increase and move toward lower dot radii.

Antilensing: The Bright Side of Voids

More than half of the volume of our Universe is occupied by cosmic voids. The lensing magnification effect from those underdense regions is generally thought to give a small dimming contribution: objects on the far side of a void are supposed to be observed as slightly smaller than if the void were not there, which together with conservation of surface brightness implies net reduction in photons received. This is predicted by the usual weak lensing integral of the density contrast along the line of sight. We show that this standard effect is swamped at low redshifts by a relativistic Doppler term that is typically neglected. Contrary to the usual expectation, objects on the far side of a void are brighter than they would be otherwise. Thus the local dynamics of matter in and near the void is crucial and is only captured by the full relativistic lensing convergence. There are also significant nonlinear corrections to the relativistic linear theory, which we show actually underpredicts the effect. We use exact solutions to estimate that these can be more than 20% for deep voids. This remains an important source of systematic errors for weak lensing density reconstruction in galaxy surveys and for supernovae observations, and may be the cause of the reported extra scatter of field supernovae located on the edge of voids compared to those in clusters.

Relativistic corrections to double charmonium production in high energy proton-proton interaction

On the basis of pertubative QCD and the relativistic quark model we calculate relativistic corrections to the process of pair $J/\psi$ production in proton-proton collisions at LHC energy $\sqrt S=7$ TeV. Relativistic terms in the production amplitude connected with the relative motion of heavy quarks and the transformation law of the bound state wave functions to the reference frame of moving $J/\psi$ mesons are taken into account. For the gluon and quark propagators entering the amplitude we use a truncated expansion in relative quark momenta up to the second order. Relativistic corrections to the quark bound state wave functions are considered by means of the Breit-like potential. It turns out that the examined effects decrease initial nonrelativistic cross section more than two times. The final result lies below the experimental value measured by LHCb.

Holographic fermions in charged dilaton black branes

By imposing the relativistic boundary term and Lorentz violating that in the dilatonic black brane with a Lifshitz like IR geometry and AdS4 boundary, we study the properties of the spectral functions of the fermions. We find that in the two fixed points, there are emergent Fermi-surface structures and many properties seem to be in agreement with that of Fermi liquid. Especially, the low energy behavior exhibits a linear dispersion relation. In addition, we also find that a holographic flat band also emerges in this background of the dilatonic black brane.