Non-relativistic Case Research Articles

A sharp stability criterion for the Vlasov–Maxwell system

We consider the linear stability problem for a 3D cylindrically symmetric equilibrium of the relativistic Vlasov–Maxwell system that describes a collisionless plasma. For an equilibrium whose distribution function decreases monotonically with the particle energy, we obtained a linear stability criterion in our previous paper [24]. Here we prove that this criterion is sharp; that is, there would otherwise be an exponentially growing solution to the linearized system. We also treat the considerably simpler periodic $1\frac{1}{2}$ D case. The new formulation introduced here is applicable as well to the non-relativistic case, to other symmetries, and to general equilibria.

Nonrelativistic nuclear energy functionals

In the present volume, several lectures are devoted to the implement- ations of the Density Functional Theory (DFT) in atomic nuclei, either in the non-relativistic or covariant formalism. By restricting ourselves to the non-relativistic case, in this contribution we deal with two aspects of the theory: firstly, its time-dependent extension and the study of the nuclear vibrational excitations, and secondly, the use of effective interactions beyond the mean field framework. Few recent applications, involving both stable and unstable nuclei, are presented.

Poisson-transformed density fitting in relativistic four-component Dirac–Kohn–Sham theory

We present recent developments in the implementation of the density fitting approach for the Coulomb interaction within the four-component formulation of relativistic density functional theory [Belpassi et al., J. Chem. Phys. 124, 124104 (2006)]. In particular, we make use of the Poisson equation to generate suitable auxiliary basis sets and simplify the electron repulsion integrals [Manby and Knowles, Phys. Rev. Lett. 87, 163001 (2001)]. We propose a particularly simple and efficient method for the generation of accurate Poisson auxiliary basis sets, based on already available standard Coulomb fitting sets. Just as is found in the nonrelativistic case, we show that the number of standard auxiliary fitting functions that need to be added to the Poisson-generated functions in order to achieve a fitting accuracy equal or, in some cases, better than that of the standard procedure is remarkably small. The efficiency of the present implementation is demonstrated in a detailed study of the spectroscopic properties and energetics of several gold containing systems, including the Au dimer and the CsAu molecule. The extraction reaction of a H(2)O molecule from a Au(H(2)O)(9) (+) cluster is also calculated as an example of mixed heavy-light-atom molecular systems. The scaling behavior of the algorithm implemented is illustrated for some closed shell gold clusters up to Au(5) (+). The increased sparsity of the Coulomb matrices involved in the Poisson fitting is identified, as are potential computational applications and the use of the Poisson fitting for the relativistic exchange-correlation problem.

Solving the Schrödinger and Dirac equations of hydrogen molecular ion accurately by the free iterative complement interaction method

The nonrelativistic Schrödinger equation and the relativistic four-component Dirac equation of H(2) (+) were solved accurately in an analytical expansion form by the free iterative complement interaction (ICI) method combined with the variational principle. In the nonrelativistic case, we compared the free ICI wave function with the so-called "exact" wave function as two different expansions converging to the unique exact wave function and found that the free ICI method is much more efficient than the exact method. In the relativistic case, we first used the inverse Hamiltonian to guarantee Ritz-type variational principle and obtained accurate result. We also showed that the ordinary variational calculation also gives a nice convergence when the g function is appropriately chosen, since then the free ICI calculation guarantees a correct relationship between the large and small components of each adjacent order, which we call ICI balance. This is the first application of the relativistic free ICI method to molecule. We calculated both ground and excited states in good convergence, and not only the upper bound but also the lower bound of the ground-state energy. The error bound analysis has assured that the present result is highly accurate.

Perpendicularly propagating electromagnetic modes in a strongly magnetized hot relativistic Maxwellian plasma

Using kinetic theory, new dispersion relations based on relativistic Maxwellian velocity distribution are obtained for ordinary and extraordinary modes in a strongly magnetized hot plasma. It is observed that, unlike the non-relativistic Maxwellian distribution case where only real dispersion relation appears for the relativistic case both the real dispersion relations and the damping decrements occur. To illustrate the results graphically, we have plotted the phase velocity squared and the growth rate versus frequency for different values of n. For O-mode, the non-propagation region disappears as we increase the values of the harmonic number n while for the X-mode, the propagation region occurs only for the large harmonics. As regards damping, for O-mode the damping rate is small for small n and increases with n. For X-mode, on the contrary, the damping rate decreases as n increases.

New scale-relativistic derivations of Pauli and Dirac equations

In scale relativity, quantum mechanics is recovered by transcribing the classical equations of motion to fractal spaces and demanding, as dictated by the principle of scale relativity, that the form of these equations be preserved. In the framework of this theory, however, the form of the classical energy equations both in the relativistic and nonrelativistic cases are not preserved. Aiming to get full covariance, i.e., to restore to these equations their classical forms, we show that the scale-relativistic form of the Schrödinger equation yields the Pauli equation, whilst the Pissondes's scale-relativistic form of the Klein–Gordon equation gives the Dirac equation.

Radial position–momentum uncertainties for Klein–Gordon hydrogen-like atoms

We show how the radial position–momentum uncertainty product can be obtained analytically for Klein–Gordon hydrogen-like atoms. Some interesting features of this quantum system are found. First, for the same principal quantum number n, when the angular momentum quantum number l increases, the uncertainties Δpr and ΔrΔpr decrease. However, Δr does not always decrease, which is different from the non-relativistic hydrogen-like atoms case, where it always decreases. Second, for same l both Δr and ΔrΔpr increase as n increases. Third, all uncertainties for same n and l = n − 1 are smallest in comparison with those for same n but l ≠ n − 1. Fourth, ΔrΔpr is independent of the value of charge Z in the non-relativistic case, while it is related to the value of charge Z in the relativistic case. Fifth, the relativistic corrections to the non-relativistic uncertainties are very small when the values of the charge Z are not too big. However, relativistic corrections to them will appear explicitly for large Z.

Bose–Einstein condensation of a relativistic q-deformed Bose gas

The q-deformed Bose–Einstein distribution derived from the q-boson algebra is used to study the low-temperature behavior of an ideal q-deformed Bose gas with relativistic energy spectrum. The effects of q-deformation on the properties relative to Bose–Einstein condensation (BEC) are discussed. It is shown that q-deformation leads to some novel characteristics different from those of an original Bose gas, which include the criteria on the occurrence of BEC, critical temperature and jump of heat capacity at the critical point. The results obtained here provide a unified description for the properties of q-deformed Bose systems from the nonrelativistic case to the ultrarelativistic limit, so that some important conclusions in the literature are included in this paper.

The ideal relativistic spinning gas: Polarization and spectra

We study the physics of the ideal relativistic rotating gas at thermodynamical equilibrium and provide analytical expressions of the momentum spectra and polarization vector for the case of massive particles with spin 1/2 and 1. We show that the finite angular momentum J entails an anisotropy in momentum spectra, with particles emitted orthogonally to J having, on average, a larger momentum than along its direction. Unlike in the non-relativistic case, the proper polarization vector turns out not to be aligned with the total angular momentum with a non-trivial momentum dependence.

Homogenization of the 1D Vlasov-Maxwell equations

In this paper, we investigate the homogenization of the 1D Vlasov–Maxwell system. We indicate the rate of convergence towards the limit solution. In the non-relativistic case, we compute explicitly the limit solution. The theoretical results are illustrated by some numerical simulations.

Resistive-wall impedance of two-layer tube

In contrast to electron machines, beam particles travel in most proton or heavy ion circular accelerators with nonrelativistic speed. The resistive-wall impedance of the cylindrical two-layer tube with finite thickness is studied in this paper for any material in the nonrelativistic case. The nonrelativistic correction for a special case of metallic-type layers is given afterward. Finally, we derive the general expression of the resistive-wall impedance for the multilayer tubes.

Above-threshold ionization of atoms in linearly polarized laser fields of relativistic intensities

Within the framework of the relativistic quantum theory the multiphoton above-threshold ionization (ATI) of hydrogenlike atoms in the field of super-strong electromagnetic radiation of linear polarization, taking into account the photoelectron stimulated bremsstrahlung (SB) on the residual ion, is investigated. The latter is considered in the Born approximation by the Coulomb field which allows us to simplify the previous results in the generalized eikonal approximation upon the scattering potential and obtain an analytic formula for the relativistic ATI rate summed over the intermediate transitions of a photoelectron through the virtual states of the atomic continuum. The energetic spectra and angular distributions of photoelectrons are obtained with the secondary SB of photoelectrons. Due to the latter the partial ATI rates are 4 times larger than the ATI rates of the direct process. In contrast to the nonrelativistic case, the main peaks in the angular distributions of photoelectrons are shifted toward the wave propagation direction and have a tendency to be aligned about this direction with the increase of the field intensity.

Observational Appearance of Relativistic, Spherically Symmetric Massive Winds

Abstract The photon mean free path in a relativistically moving medium becomes long in the downstream direction, while short in the upstream direction. As a result, the observed optical depth, $\tau$, becomes small in the downstream direction while large in the upstream direction. Hence, if a relativistic spherical wind blows off, the optical depth depends strongly on its speed and the angle between the velocity and the line-of-sight. Abramowicz, Novikov, and Paczynski (1991, ApJ, 369, 175) examined such a relativistic wind, and found that the shape of the photosphere at $\tau=1$ appears to be convex in the nonrelativistic case, but concave for relativistic velocities. We further calculated the temperature distribution and luminosity of the photosphere both in the comoving and inertial frames. We found that the limb-darkening effect would be strongly modified in the relativistic regime. We also found that the luminosities of the photosphere become large as the wind speed increases due to relativistic effects. In addition, the luminosity in the inertial frame is higher than that in the comoving frame. These results suggest that the observed temperature and brightness in luminous objects may be overestimated when there are strong relativistic winds.

Relativistic ray-tracing of electron Bernstein waves in a spherical tokamak reactor

Electron Bernstein waves (EBW) can propagate between electron cyclotron harmonics in hot, dense plasmas with no high-density cutoff (unlike electron cyclotron waves). ARIES-ST is a spherical tokamak (ST) reactor study. A promising method of heating and driving current in such an ST reactor would be with electron Bernstein waves. These could be mode converted from microwave vacuum modes (X-mode and O-mode) launched from the edge. In this paper, fully relativistic ray-tracing calculations (both damping and propagation) of EBWs in ARIES-ST are presented, using the relativistic dispersion solver R2D2 and the ray-tracing code GENRAY. The ray paths, damping locations, and polarizations are compared with the more commonly used non-relativistic EBW ray-tracing. A range of frequencies are shown for EBWs with small n∥ that could be produced by the X-B mode conversion process, and EBWs with large n∥ ≈ 0.6, produced by the O-X-B mode conversion process. With the density and temperature profile chosen for this paper, the greatest depth into the core that could be reached with mode-converted EBWs is a radial location of approximately ρ = 0.4. Although the radial location of the damping in most cases was not significantly different between the relativistic and non-relativistic cases, there are differences in the poloidal locations, as well as in the polarizations of the wave along the ray path, especially in the damping region.

Dispersion in a relativistic degenerate electron gas

AbstractRelativistic effects on dispersion in a degenerate electron gas are discussed by comparing known response functions derived relativistically and non-relativistically. The main distinguishing feature is one-photon pair creation, which leads to logarithmic singularities in the response functions. Dispersion curves for longitudinal waves have a similar tongue-like appearance in the relativistic and non-relativistic case, with the main relativistic effects being on the Fermi speed and the cutoff frequency. For transverse waves the non-relativistic treatment has a non-physical feature near the cutoff frequency for large Fermi momenta, and this is attributed to an incorrect treatment of the electron spin. We find (with two important provisos) that one-photon pair creation is allowed in superdense plasmas, implying relatively strong coupling between transverse waves and pair creation.

Laser cooling of electrons and X-ray generation in a relativistic quantum heat engine

We consider a relativistic quantum heat engine that goes through a thermodynamical cycle consisting of stages involving laser-assisted cooling of electrons and the generation of X-ray radiation. Quantum treatment of the processes makes it possible to obtain the necessary condition and the amount of work extracted from the interaction ingredients, as well as the efficiency of the heat engine. We have also found that the efficiency of the relativistic engine is less than the one for the nonrelativistic case for the same momenta. The obtained results set the limits to the cooling, as well as the intensity of X-ray radiation, in the quantum regime of the interaction of electrons with laser fields.

Coulomb Potential from Lorentz Invariance in N Dimensions

Although Maxwell theory is O(3,1)-covariant, electrodynamics only transforms invariantly between Lorentz frames for special forms of the field, and the generator of Lorentz transformations is not generally conserved. Berard, Grandati, Lages, and Mohrbach have studied the O(3) subgroup, for which they found an extension of the rotation generator that satisfies the canonical angular momentum algebra in the presence of certain Maxwell fields, and is conserved by the classical motion. The extended generator depends on the field strength, but not the potential, and so is manifestly gauge invariant. The conditions imposed on the Maxwell field by the algebra lead to a Dirac monopole solution. In this paper, we study the generalization of the Berard, Grandati, Lages and Mohrbach construction to the full Lorentz group in N dimensions. The requirements can be maximally satisfied in a three-dimensional subspace of the full Minkowski space; this subspace can be chosen to describe either an O(3)-invariant space sector, or an O(2,1)-invariant restriction of spacetime. The field solution reduces to the Dirac monopole found in the nonrelativistic case when the O(3)-invariant subspace is selected. When an O(2,1)-invariant subspace is chosen, the field strength can be associated with a Coulomb-like potential of the type A μ(x) = n μ/ρ, where ρ = (x μ x μ)1/2, similar to that used by Horwitz and Arshansky to obtain a covariant generalization of the hydrogen-like bound state. In the presence of these fields, which are determined entirely by symmetry considerations, without reference to a source equation, the extended generator is conserved under classical relativistic system evolution.

Dynamics of colliding branes and black brane production

We study the dynamics of colliding domain walls including self-gravity. The initial data is set up by applying a Bogomol'nyi-Prasad-Sommerfield (BPS) domain wall in five-dimensional supergravity, and we evolve the system determining the final outcome of collisions. After a collision, a spacelike curvature singularity covered by a horizon is formed in the bulk, resulting in a black brane with trapped domain walls. This is a generic consequence of collisions, except for nonrelativistic weak field cases, in which the walls pass through one another or multiple bounces take place without singularity formation. These results show that incorporating the self-gravity drastically changes a naive picture of colliding branes.

Numerical solution of the linear dispersion relation in a relativistic pair plasma

We describe an algorithm that computes the linear dispersion relation of waves and instabilities in relativistic plasmas within a Vlasov–Maxwell description. The method used is fully relativistic and involves explicit integration of particle orbits along the unperturbed equilibrium trajectories. We check the algorithm against the dispersion curves for a single component magnetized plasma and for an unmagnetized plasma with counter-streaming components in the nonrelativistic case. New results on the growth rate of the Weibel or two-stream instability in a hot unmagnetized pair plasma consisting of two counter-streaming relativistic Maxwellians are presented. These are relevant to the physics of the relativistic plasmas found in gamma-ray bursts, relativistic jets and pulsar winds.

Hamilton-Jacobi approach for power-law potentials

The classical and relativistic Hamilton-Jacobi approach is applied to the one-dimensional homogeneous potential, V(q) = alphaq n, where alpha and n are continuously varying parameters. In the non-relativistic case, the exact analytical solution is determined in terms of alpha, n and the total energy E. It is also shown that the non-linear equation of motion can be linearized by constructing a hypergeometric differential equation for the inverse problem t(q). A variable transformation reducing the general problem to that one of a particle subjected to a linear force is also established. For any value of n, it leads to a simple harmonic oscillator if E > 0, an "anti-oscillator" if E < 0, or a free particle if E = 0. However, such a reduction is not possible in the relativistic case. For a bounded relativistic motion, the first order correction to the period is determined for any value of n. For n >> 1, it is found that the correction is just twice that one deduced for the simple harmonic oscillator (n = 2), and does not depend on the specific value of n.