HIGH ENERGY ELECTRON SCATTERING FROM A TWO CENTER POTENTIAL

The solution of the Dirac equation is obtained for a relativistic electron in the focusing and periodic electrostatic fields modeling the field at the axis or plane of a crystal.

On the spinor representation of surfaces in Euclidean 3-space

Electronic transport through a material depends on the response to local perturbations induced by defects or impurities in the material. The scattering processes can be described in terms of phase shifts and corresponding cross sections. The multiorbital nature of the spinor states in transition metal dichalcogenides would naturally suggest the consideration of a massive Dirac equation to describe the problem, while the parabolic dispersion of its conduction and valence bands would invite a simpler Schr\"odinger equation description. Here, we contrast the scattering of massive Dirac particles and Schr\"odinger electrons, in order to assess different asymptotic regimes (low and high Fermi energy) for each one of the electronic models and describe their regime of validity or transition. At low energies, where the dispersion is approximately parabolic, the scattering processes are dominated by low angular momentum channels, which results in nearly isotropic scattering amplitudes. On the other hand, the differential cross section at high Fermi energies exhibits clear signatures of the linear band dispersion, as the partial phase shifts approach a non-zero value. We analyze the electronic dynamics by presenting differential cross sections for both attractive and repulsive scattering centers. The dissimilar behavior between Dirac and Schr\"odinger carriers points to the limits and conditions over which different descriptions are required for the reliable treatment of scattering processes in these materials.

It is well known that in relativistic quantum theory there is not exact analytical formula for cross section of a charged particle elastic scattering on arbitrary electrostatic potential \(\varphi (\mathbf {r})\). Even in case of Coulomb field for which the exact solution of Dirac equation is known, relativistic “Coulomb wave functions”, nevertheless, because of its complex character it is impossible to obtain exact analytical expression for scattering cross section via these wave functions. The elastic scattering in relativistic domain is mainly described in approximations when the scattering potential can be considered as a perturbation (in opposite limits). These are the well-known Born and eikonal approximations corresponding to quantum perturbation theory by particle wave function (when the condition \(\left| U\right| \ll \hbar \mathrm {v}/a\) is satisfied; U is the potential energy, a is the space size of the range of effective scattering, \(\mathrm {v}\) is the particle initial velocity), and high-momentum approximation (if potential energy in the scattering field is much less than the particle initial energy: \(\left| U\right| \ll p\mathrm {v}\)), respectively. In case of Coulomb field there is also an approximation of large impact parameters (large momenta)—Farry-Sommerfeld-Maue (FSM) approximation , which describes well enough the scattering at the small angles. It is also known that the wave function in the eikonal approximation describes the particle state only in a limited space range of the scattering process (\(z\ll pa^{2}/\hbar \); z is the coordinate along the direction of particle initial momentum), i.e., the eikonal solution is not valid at large distances. The wave function of the Born approximation , in contrast to the eikonal one, describes the particle state at arbitrary point in the scattering range, particularly at asymptotically large distances. Nevertheless, the common region where both approximations under consideration are valid is very restricted. On the other hand, within the (small) potential range where both approximations are applicable, these wave functions describe the scattering by different accuracies. Practically, these wave functions describe the scattering at the opposite conditions: the eikonal wave function describes the particle state when the opposite condition of the Born approximation holds (e.g., for Coulomb field with a charge \(Z_{a}e\)—at the condition: \(Z_{a}e^{2}/\hbar \mathrm {v}\gtrsim 1\)). In these circumstances a natural question arises—is it possible to find out such an approximate solution of Dirac equation beyond the scope both perturbation theory and eikonal approximation—corresponding to more general wave function being applicable in both quantum and quasiclassical limits for relativistic potential scattering (including in particular the Born and eikonal approximations in corresponding limits—under its conditions of applicability)? Here we will try to derive such an approximate solution of the Dirac equation which satisfies the formulated requests. We will call it generalized eikonal approximation (GEA) for a spinor particle scattering on the arbitrary short-range or long-range electrostatic potential. Then we will clear up the relationship of this approximation to the known ones for electron elastic scattering on the short-range (Born and eikonal approximations) and long-range Coulomb potentials (Born, eikonal, and Farry-Sommerfeld-Maue approximations ). This GEA approximation is developed for electron inelastic scattering process too—to describe the relativistic stimulated bremsstrahlung in the field of strong and superstrong laser radiation with electrostatic potential fields of arbitrary form and finite or infinite effective radiuses (atoms, ions, etc.). The significance of GEA type wave function for description of such laser-assisted electron–atom–ion scattering processes, apart from the known restrictions of mentioned above approximations, is also conditioned with the fact that the wave functions in known approximations describe the particle state factorized by elastic scattering and induced radiation-absorption processes. In the result, we lose the phase relations at the description of electron quantum dynamics interacting with the both fields simultaneously, which have important role for induced process, in particular, for coherent part of interaction. Therefore, for description of strong and superstrong laser–matter (plasma) interaction processes, we need the electron wave function that takes into account the simultaneous influence of both scattering and radiation fields including dynamic phase relations. This induced GEA wave function may be applied specifically for the description of the above-threshold ionization (ATI) process of atoms/ions by superstrong laser fields in relativistic theory taking into account the photoelectron rescattering effect on the atomic remainder because of the action of long-range Coulomb field of atomic ion on the photoelectron final state (this process is considered in the next chapter of this book).

The scattering by a three-dimensional separable potential of rank 2 is described analytically. The model potential is chosen so that the Lippmann–Schwinger equation can be exactly solved and all calculations are carried out analytically. The explicit expressions for a number of scattering properties, such as phase shifts, cross sections, T- and S-matrices, resonances and poles of the resolvent are calculated. The second virial coefficient is also evaluated in terms of phase shifts and bound state energies for the particular potential model presented.

Stochastic cycles are often incorporated in structural time series models to identify the cyclical components in macroeconomic time series. This paper proposes an extension of the multivariate stochastic cycle to account for phase shifts between individual cyclical components. The properties of the stochastic cycle allow phase shifts to be modelled in an entirely symmetrical way. As a result, cross correlations between cyclical components can be expressed in terms of phase shifts and phase‐adjusted associations. An application demonstrates the role of phase shifts in the business cycle relationships among output, total hours worked and the real wage in the United States.

The nucleus is represented as a Kerr-Newman source, under the assumption that the angular momentum of the source is equal to the intrinsic spin angular momentum of the nucleus. The theoretical values of the hyperfine splitting obtained from a solution of Dirac's equation for this model of the muon, the positron, and the proton agree with the observed values in muonium, positronium, and hydrogen, to within the uncertainty in the respective QED corrections---except for an unaccounted factor of 2. In hydrogen one also has to introduce the observed value of the magnetic moment of the proton. The singularity in Dirac's radial differential equations is shifted from the origin, where it is located in the case of flat space, to a pair of conjugate points on the imaginary axis of the radial coordinate. Consequently, the energy eigenvalues have a trace of an imaginary part, making the bound states of our model unstable.

In quark model calculations of the meson spectrums fully covariant two-body Dirac equations dictated by Dirac`s relativistic constraint mechanics gave a good fit to the entire meson mass spectrum for light quark mesons as well as heavy quark mesons with constituent world scalar and vector potentials depending on just one or two parameters. In this paper, we investigate the properties of these equations that made them work so well by solving them numerically for quantum electrodynamics (QED) and related field theories. The constraint formalism generates a relativistic quantum mechanics defined by two coupled Dirac equations on a sixteen component wave function which contain Lorentz covariant constituent potentials that are initially undetermined. An exact Pauli reduction leads to a second order relativistic Schroedinger-like equation for a reduced eight component wave function determined by an effective interaction -- the quasipotential. We first determine perturbatively to lowest order the relativistic quasipotential for the Schroedinger-like equation by comparing that form with one derived from the Bethe-Salpeter equation. Insertion of this perturbative information into the minimal interaction structures of the two-body Dirac equations then completely determines their interaction structures. Then we give a procedure for constructing the full sixteen component solution to our coupled first-ordermore » Dirac equations from a solution of the second order equation for the reduced wave function. Next, we show that a perturbative treatment of these equations yields the standard spectral results for QED and related interactions.« less

This colloquium examines the theoretical modeling of nonequilibrium low-temperature (tens of thousands of degrees) plasmas, which involves a juxtaposition of three distinct fields: atomic and molecular physics, for the input of scattering cross sections; statistical mechanics, for the kinetic modeling; and electromagnetic theory, for the simultaneous solution of Maxwell's equations. Cross sections come either from single-scattering beam experiments or, at very low energies (<0.5 eV), from multiple-scattering experiments on swarms in gases—the free diffusion or large Debye length limit of a plasma, where they are embedded in transport coefficient data. The same Boltzmann kinetic theory that has been developed to a high level of sophistication over the past 50 years, specifically for the purpose of unfolding these transport data, can be employed for low-temperature plasmas with appropriate modification to allow for self-consistent rather than externally prescribed fields. A full kinetic treatment of low-temperature plasmas is, however, a daunting task and remains at the developmental level. Fortunately, since the accuracy requirements for modeling plasmas are generally much less stringent than for swarms, such a sophisticated phase-space treatment is not always necessary or desirable, and a computationally more efficient but correspondingly less accurate macroscopic theoretical model in configuration space at the fluid level is often considered sufficient. There has been a proliferation of such fluid modeling in recent times and this approach is now routinely used in the design and development of a large variety of plasma technologies, ranging from plasma display panels to plasma etching reactors for microelectronic device fabrication. However, many of these models have been developed empirically with specific applications in mind, and rigor and sophistication vary accordingly. In this colloquium, starting from the governing Boltzmann kinetic equation, a unified, general formulation of fluid equations is given for both ions and electrons in gaseous media with transparent and internally consistent approximations, all benchmarked against established results. Thereby a fluid model is obtained that is appropriate for practical application but at the same time is based on a firmer physical foundation.

Maxwell's equations, Lorentz's equations of motion, and Dirac's equations are investigated on a cosmological background. Maxwell's equations are conform invariant and the equations of motion can easily be made conform invariant. Then, without introducing any new assumptions, Dirac's equations are seen to be conform invariant. In open universes the solutions of Maxwell's and Dirac's equations are the same as in flat Minkowski space. The behavior of these equations is different in closed universes by reasons of topology. In closed universes both Maxwell's and Dirac's equations provide new eigenvalue problems. Dirac's equations exhibit a marked difference between an elliptic and a spherical closed universe.

Two partially coherent and perpendicularly polarized vibrations induce, in a polarization sensitive photographic emulsion H, an anisotropy that is continuously varying across the plate. When the two incident amplitudes are assumed to be constant all over H, those variations are shown to be periodic functions of the effective phase shift phi. After exposure H is observed between crossed polarizers. It reconstructs an interferogram whose contrast is maximum whatever may be the degree of partial coherence and the relative amplitudes of the two vibrations. In plane rotation of the plate, H causes changes in the spatial frequency of the interferogram and localization of its dark fringes. When the direction of the polarizer is that of one of the original vibrations, the distance in terms of phase shift between any two consecutive dark fringes reaches pi, and these fringes characterize the lines of equal phase phi = Kpi. Similar phenomena may be retrieved from that last setting by rotation of the analyzer initially initially crossed with the polarizer.

On the interpretation of the field theoretical potential in terms of phase shifts

The Hamiltonian in Supersymmetric Quantum Mechanics is defined in terms of charges that obey the same algebra as that of the generators of supersymmetry in field theory. The consequences of this symmetry for the spectra of the component parts that constitute the supersymmetric system are explored. The implications of supersymmetry for the solutions of the Schrodinger equation, the Dirac equation, the inverse scattering theory and the multi‐soliton solutions of the KdV equation are examined. Applications to scattering problems in Nuclear Physics with specific reference to singular potentials which arise from considerations of supersymmetry will be discussed.

The $n$-fold Darboux transformation $T_{n}$ of the focusing real mo\-di\-fied Kor\-te\-weg-de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the $n$-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues $\lambda_{j}$ and the corresponding eigenfunctions of the associated Lax equation. The nonsingular $n$-positon solutions of the focusing mKdV equation are obtained in the special limit $\lambda_{j}\rightarrow\lambda_{1}$, from the corresponding $n$-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the $n$-positon solution into $n$ single-soliton solutions, the trajectories, and the corresponding "phase shifts" of the multi-positons are also investigated.

Amplitudes for scattering of electrons by hydrogenic and alkali-like atomic systems