Algebraic tools for dealing with the atomic shell model. I. Wavefunctions and integrals for hydrogen-like ions

Abstract

Today, the 'hydrogen atom model' is known to play its role not only in teaching the basic elements of quantum mechanics but also for building up effective theories in atomic and molecular physics, quantum optics, plasma physics, or even in the design of semiconductor devices. Therefore, the analytical as well as numerical solutions of the hydrogen-like ions are frequently required both, for analyzing experimental data and for carrying out quite advanced theoretical studies. In order to support a fast and consistent access to these (Coulomb-field) solutions, here we present the Dirac program which has been developed originally for studying the properties and dynamical behaviour of the (hydrogen-like) ions. In the present version, a set of Maple procedures is provided for the Coulomb wave and Green's functions by applying the (wave) equations from both, the nonrelativistic and relativistic theory. Apart from the interactive access to these functions, moreover, a number of radial integrals are also implemented in the Dirac program which may help the user to construct transition amplitudes and cross sections as they occur frequently in the theory of ion-atom and ion-photon collisions. * To whom correspondence should be addressed (surz@physik.uni-kassel. Keywords: Analytical solution, Coulomb-Green's function, Coulomb problem, Dirac equation, energy level, expectation value, hydrogen-like ion, hydrogenic wavefunction, matrix element, radial integral, special functions. Nature of the physical problem:Analytical solutions of the hydrogen atom are widely used in very different fields of physics [2,3]. Despite of the rather simple structure of the hydrogen-like ions, however, the underlying 'mathematics' is not always that easy to deal with. Apart from the well-known level structure of these ions as obtained from either the Schrödinger or Dirac equation, namely, a great deal of other properties are often needed. These properties are related to the interaction of bound electron(s) with external particles and fields and, hence, require to evaluate transition amplitudes, including wavefunctions and (transition) operators of quite different complexity. Although various special functions, such as the Laguerre polynomials, spherical harmonics, Whittaker functions, or the hypergeometric functions of various kinds can be used in most cases in order to express these amplitudes in a concise form, their derivation is time consuming and prone for making errors. In addition to their complexity, moreover, there exist a large number of mathematical relations among these functions which are difficult to remember in detail and which have often hampered quantitative studies in the past. Method of solution:A set of Maple procedures is developed which provides both the nonrelativistic and relativistic (analytical) solutions of the 'hydrogen atom model' and which facilitates the symbolic evaluation of various transition amplitudes. 2 Restrictions onto the complexity of the problem:Over the past decades, a large number of representati...

Electron capture into few-electron heavy ions: Independent particle model

“…Similar to before, this parameter has been calculated by using the computer code DIRAC [38]. In contrast to the results of Section 3.1, however, besides the direct electron capture into the 2p 3/2 state, the cascade feeding from the high-lying levels has also been taken into account.…”

“…The evaluation of these one-electron parameters within the framework of the relativistic Dirac theory has been discussed previously at a number of places [28,33]. Below, we calculate the statistical tensors A 20 (2p 3/2 ) by using the Dirac program [38]. Since these (one-particle) tensors are used later also to explore the alignment of the helium-like ions, the screening of the nucleus by the core electron should be taken into account right from the beginning by a proper choice of the effective charge Z eff .…”

“…where the one-electron decay rates have been calculated within the Dirac's relativistic framework [38] and by using the effective charges Z E1 eff = 91.7 and Z M2 eff = 91.2 as estimated from the "true" transition energies. From equation (20), one may see that a strong 1s 2p 3/2 J f = 2 → 1s 2s J 0 = 1 transition leads to the fact that only about 70% of the population of the triplet 1s 2p 3/2 J f = 2 state contributes to the Kα 1 decay.…”

Angular and polarization analysis of x-rays emitted from highly-charged, few-electron ions

“…These cross sections were calculated for projectile energies from 1 to about 500 MeV/u and are shown above for the two selected energies 2.18 MeV/u and 218 MeV/u, respectively. Results from multiconfigurational computations in Coulomb and Babushkin gauge are compared with the data from an independent-particle approximation (IPA), in which a single Slater determinant was utilized together with hydrogen-like orbitals and by making use of the Dirac program [20]. In these IPA calculations, the effective charge was set to Z eff = Z − 1 to ensure a good agreement with the MCDF computations for 218 MeV/u and at all larger projectile energies.…”

Photon polarization in the two-photon decay of heavy hydrogen-like ions

“…Apart from the direct summation as implemented, for example, by the B-spline finite basis set approach [10], the amplitudes (1) can be evaluated also by means of the Green's function of the ion. This latter method is particularly helpful for hydrogen-like ions, since the CoulombGreen's functions are known analytically, both within the nonrelativistic as well as relativistic theory [19,20].…”