Showing papers by "Ugo Fano published in 1976"

Electron-optical properties of atomic fields

Abstract: We present a unified discussion and illustrations of the electron-optical aspects of electron penetration into, or escape from, the inner region of atoms Both processes may focus or defocus the amplitudes of wavefunctions and shift their phases, as manifested in countless phenomena ranging from level shifts to $\ensuremath{\beta}$-decay rates A background survey begins by discussing the Fermi-Segr\`e formula for hyperfine splittings and emphasizes the interplay of hydrogenic and WKB approximations The Phase-Amplitude Method, which determines amplitude ratios and phase shifts directly, proves useful for interpreting the systematics of these parameters along the Periodic System We present results of survey calculations, carried through the Periodic System using Hartree-Slater potential fields, of: (a) $\frac{{\ensuremath{\alpha}}_{l}(0)}{{\ensuremath{\alpha}}_{l}(\ensuremath{\infty})}$, the ratio of the wavefunction's amplitude at $r=0$ to that outside the atom; (b) ${\ensuremath{\delta}}_{l}(E=0)$ and ${\frac{d{\ensuremath{\delta}}_{l}}{\mathrm{dE}}|}_{E=0}$, the phase shift and its energy derivative at $E=0$, and (c) the changes in ${\ensuremath{\delta}}_{l}(E=0)$ and $\frac{{\ensuremath{\alpha}}_{l}(0)}{{\ensuremath{\alpha}}_{l}(\ensuremath{\infty})}$ induced by either a unit perturbation localized near $r=0$ or a relativistic correction Thus we provide a broad mapping of certain fundamental parameters based on rather crude but realistic calculations These results are meant to serve as a bench mark in surveying problems and in checking new results, while standard methods are preferable for working out specific applications accurately

Dynamics of electron excitation

Abstract: A key step in any chemical reaction is the transition of bonding electrons from one stable wave‐mechanical state to another. A complete description of the evolution of states in the course of such transitions remains a distant goal—in spite of our rather extensive knowledge of chemical reactions. This article reviews a start on this problem that has been made in recent years.

Post-Adiabatic Analysis of Atomic Collisions

Abstract: The coupling between adiabatic channels can be partially transformed away. The transformation need not induce any transition between channnels; but it correlates the radial wave functions and their gradients with the channel functions and it depresses the lower effective potentials, as the energy increases, in accordance with empirical evidence. (AIP)

Transition matrices for the theory of spectra. Techniques for their construction and calculation

Abstract: Analytical techniques are developed for constructing $n\mathrm{th}$-order (i.e., $n$-electron) tensor matrices pertaining to transitions of an $N$-electron atom between two of its stationary states. These matrices serve to calculate the transition amplitude for the atom under the influence of an external field acting on $n$ electrons (typically $n=1$). Their calculation requires, in turn, the solution of a truncated hierarchy of Schr\"odinger equations introduced in the preceding paper. The techniques presented here are applied to construct the matrices and their Schr\"odinger equation for the example of Ar atoms treated in the preceding paper.

Multipole expansion of the density of states about a crystal cell

Abstract: We construct the expansion of a Bloch wave with energy E into a complete set of multipole waves around a ’’center’’ of a crystal as an analog of the expansion in spherical waves in free space. Crystal point group symmetry is used to classify the set. The density of states in a cell is then analyzed into multipole components whose magnitude depends on the cell’s distance from the ’’center.’’