Dirac Eigenfunctions Research Articles

DWBA theory for elastic scattering of polarized electrons from heavy unpolarized nuclei

Differential cross sections and spin asymmetries for the elastic scattering of spin-polarized electrons from inert spin- nuclei are calculated within the distorted-wave (DW) Born approximation. This is achieved by means of a coherent superposition of the electric and magnetic (M1) transition amplitudes, using Dirac eigenfunctions for the electronic scattering states throughout. For heavy nuclei (Z ≳ 40), high collision energies and backward scattering angles considerable deviations occur from previous DW plane-wave calculations where the magnetic transition is calculated within the Born approximation. For electrons spin-polarized perpendicular to the beam direction the spin asymmetry is quenched when magnetic scattering becomes dominant. Predictions are made for 50–200 MeV electrons scattering from 15N, 89Y and 207Pb nuclei.

Convergence of expansions in Schrödinger and Dirac eigenfunctions, with an application to the R-matrix theory

Expansion of a wave function in a basis of eigenfunctions of a differential eigenvalue problem lies at the heart of the R-matrix methods for both the Schrödinger and Dirac particles. A central issue that should be carefully analyzed when functional series are applied is their convergence. In the present paper, we study the properties of the eigenfunction expansions appearing in nonrelativistic and relativistic R-matrix theories. In particular, we confirm the findings of Rosenthal [J. Phys. G 13, 491 (1987)] and Szmytkowski and Hinze [J. Phys. B 29, 761 (1996); Szmytkowski and Hinze J. Phys. A 29, 6125 (1996)] that in the most popular formulation of the R-matrix theory for Dirac particles, the functional series fails to converge to a limit claimed by other authors.

Exact Solutions of Effective Mass Dirac Equation with Non-PT-Symmetric and Non-Hermitian Exponential-type Potentials

By using a two-component approach to the one-dimensional effective mass Dirac equation, bound states are investigated under the effect of two new non-PT-symmetric and non-Hermitian exponential type potentials. It is observed that the Dirac equation can be mapped into a Schrödinger-like equation by rescaling one of the two Dirac wave functions in the case of the position-dependent mass. The energy levels and the corresponding Dirac eigenfunctions are found analytically.

On the eigenfunctions of the Douglas–Kroll operator

The matrix elements of the quasirelativistic Douglas–Kroll operators up to the fourth order for hydrogen-like ions is constructed with as few additional approximations as possible, to investigate the behaviour of its 1s eigenfunctions in the vicinity of the nucleus. Because Douglas–Kroll is a momentum space theory, we use a basis set of spherical waves which are eigenfuntions of the square of the momentum operator. While this avoids the most serious approximation of the standard Douglas–Kroll–Hess protocol, namely that the basis functions used to construct the Douglas–Kroll operator are eigenfunctions of the (squared) momentum operator, it also makes the convergence of this expansion very slow, because spherical waves are not well suited to represent the (weak) singularities of the eigenfunctions at the position of the point-like nucleus. On the other hand, the convergence is quite monotonic, and information on the behaviour close to the nucleus can be extracted from the convergence rate. Starting with the second-order, the eigenfunctions of the Douglas–Kroll operator are not “more singular” than the Dirac eigenfunctions, and the occurence of an additional error when using regular basis sets, as postulated in the literature, can not be observed. The resolution of the identity, which is involved in any practical approach to construct the matrix elements of the Douglas–Kroll operators beyond the first order, is a minor problem for heavy nuclei.

Transition Probabilities for Hydrogen-Like Atoms

E1, M1, E2, M2, E3, and M3 transition probabilities for hydrogen-like atoms are calculated with point-nucleus Dirac eigenfunctions for Z=1–118 and up to large quantum numbers l=25 and n=26, increasing existing data more than a thousandfold. A critical evaluation of the accuracy shows a higher reliability with respect to previous works. Tables for hydrogen containing a subset of the results are given explicitly, listing the states involved in each transition, wavelength, term energies, statistical weights, transition probabilities, oscillator strengths, and line strengths. The complete results, including 1 863 574 distinct transition probabilities, lifetimes, and branching fractions are available at http://www.fisica.unam.mx/research/tables/spectra/1el

Test of pseudospin symmetry in deformed nuclei

Pseudospin symmetry is a relativistic symmetry of the Dirac Hamiltonian with scalar and vector mean fields equal and opposite in sign. This symmetry imposes constraints on the Dirac eigenfunctions. We examine extensively the Dirac eigenfunctions of realistic relativistic mean field calculations of deformed nuclei to determine if these eigenfunctions satisfy these pseudospin symmetry constraints.

Salient Features of Electric and Magnetic Multipole Transition Probabilities of Hydrogen-Like Systems

Several aspects of E1, M1, E2, M2, E3 and M3 transition probabilities, lifetimes and branching fractions for hydrogen-like atoms evaluated with point-nucleus Dirac eigenfunctions for Z = 1–118 are discussed.

Pseudospin symmetry and relativistic mean field eigenfunctions

Pseudospin symmetry imposes conditions on the Dirac eigenfunctions independent of the potentials, from triaxial to spherical potentials. The general conditions on the relativistic mean field eigenfunctions in the pseudospin symmetry limit are derived. For the upper components, these conditions include differential realtions. These differential realtions are tested for realistic eigenfunctions in the spherical symmetry limit.

A simple solution for the Dirac hydrogenlike atom

Transformed, radial Dirac–Coulomb equations are solved for bound-state energy eigenvalues and coordinate-space eigenfunctions by a method Schrödinger used for the nonrelativistic case. The calculations use first-order differential operators and are simpler to perform than in the nonrelativistic limit because of a normalization property of the transformed Dirac eigenfunctions.

Scalar and dirac eigenfunctions on the squashed seven-sphere

The spectrum of massless and massive states in a Klauza-Klein theory is determined by the eigenvalues of certain differential operators in the internal space. We calculate the eigenvalues of the scalar laplacian and Dirac operator for the squashed seven-sphere, which has recently been proposed as a ground state solution of eleven-dimensional supergravity. The techniques used are applicable also to the other operators which appear in the formulate for the mass matrices.

Incoherent scattering of gamma rays in heavy atoms

Exact relativistic differential cross sections for the incoherent scattering of gamma rays by the inner shell electrons of heavy atoms are developed from the second order S matrix element for the electron-photon interaction in the Furry (1951) picture. Electron binding is accounted for completely by using the bound electron propagator and Dirac eigenfunctions for the initial and final electron states. A pure Coulomb potential is used to calculate the energy spectrum and the differential cross section for 662 keV gamma rays scattered by the K electrons of lead. In contrast with the results of Di Lazzaro and Missoni (1966) the energy spectrum maximum is displaced by 10% from the Compton energy towards higher photon energies, and there is no increase in the spectrum at the low-energy end. The relativistic results for the ratio of the bound electron to feed electron cross sections are larger than the incoherent scattering function calculations and exceed unity for scattering angles above 52 degrees .