Free-electron Wave Research Articles

Numerical methods for free–free radiative transition matrix elements

Increasing interest in multiphoton absorption processes above the ionization threshold has led theorists to reexamine numerical techniques for calculating radiative transition matrix elements between states of a continuum electron moving at large radial distances in the field of an atom or an ion. Here it is shown that accurate free—free radial matrix elements may be obtained using the usual dipole length formula by means of a rotation at finite distance in the complex coordinate plane together with solution of the free-electron wave function's phase and amplitude at finite distance in the complex coordinate plane. The procedure is designed for use with numerically calculated wave functions for many electron atoms and ions. It avoids the use of analytic asymptotic formulas as well as transformation to the dipole acceleration formula and is accurate even for matrix elements between electron stales that are close in energy, which is the case for which the alternative integration-by-parts method is inaccurate. We present comparisons of our numerical procedure with both analytic results and results of the integration-by-parts procedure for the case of free—free electron transitions in a pure Coulomb field.

Effect of the Wigner-Seitz boundary conditions on internal conversion coefficients

Solid state effects are taken into account in an internal conversion coefficients computation by using Wigner-Seitz boundary conditions. Both the bound and free electron wave functions are calculated from an atomic Dirac-Hartree-Fock-Slater self consistent potential. These internal conversion coefficients are compared with those obtained from the usual free atom boundary conditions.

Size dependence of the conduction-electron-spin-resonancegshift in a small sodium particle: Orthogonalized standing-wave calculations

The conduction-electron-spin-resonance (CESR) spectra of small metallic particles are expected to exhibit very different behavior from those of bulk metals and are expected to depend on particle size. In particular, Kawabata has made a comprehensive theoretical study of the size dependence of the CESR line shape and predicts that, for very small diameters, the small-particle $g$ shift contains a contribution which adds to the bulk $g$ shift and which is proportional to the square of the particle diameter. In this paper we develop a formalism which enables us to reexamine the behavior of the $g$ shift as a function of particle size, and we compare the predictions of this formalism with existing experimental data. The formalism we use is based upon a calculation of the $g$ shift in bulk sodium by De Graaf and Overhauser, in which the conduction-electron wave functions are approximated by single orthogonalized plane waves. For a model cubic particle of sodium, we construct orthogonalized standing waves (OSW) by orthogonalizing stationary free-electron waves to the $s$ and $p$ core states. For clusters containing 8, 27, and 64 atoms, using both single and multiple OSW approximations, we first study the effect on the electronic-energy-level spectrum and charge density of altering the (arbitrary in our model) relation between the size of the cubic box in which the conduction electrons are confined and the number of atoms in the cluster. We then calculate in both approximations different contributions to the $g$ shift and show that, in contrast with Kawabata's prediction, the major size-dependent contribution to this quantity can be written $\ensuremath{\delta}g(L)=[1\ensuremath{-}\ensuremath{\alpha}(\frac{a}{L})]\ensuremath{\delta}g(\ensuremath{\infty})$, where $a$ is the lattice constant, $\ensuremath{\alpha}$ is a parameter of the order of unity, $L$ is the length of an edge of the cubic box, and $\ensuremath{\delta}g(\ensuremath{\infty})$ is the bulk $g$ shift. Finally, we show that the term in the $g$ shift that Kawabata calculated for small clusters is an approximation to the term in the bulk $g$-shift formalism which is usually denoted as $\ensuremath{\delta}{g}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}$. We calculate it in the case of a cubic sodium particle and find that it is smaller than Kawabata predicts. Our results, which are qualitatively correct for metals other than sodium, are in good agreement with recent data obtained for small magnesium particles. On the basis of this formalism, we conclude that the major size dependence of the CESR $g$ shift comes from a surface effect.

Resonant impurity states in semiconductors in a strong magnetic field

We present a theory of magnetically induced resonance states in narrow-gap semiconductors and apply it to study the resonance states in InSb. By expanding the impurity wave functions in terms of the free-electron Landau wave functions, we obtain an infinite set of coupled equations. If the magnetic field is sufficiently large, i.e., $\ensuremath{\gamma}>1$ (here, $\ensuremath{\gamma}=\ensuremath{\hbar}{\ensuremath{\omega}}_{\frac{c}{2}}{\mathrm{Ry}}^{*}$), only a small number of Landau levels need be solved self-consistently because for any energy to a good approximation, the coupling need be considered only between the adjacent Landau level and all the lower ones. The calculation of the energy and width can be made using a multicomponent generalization of the Kohn variational method for phase shifts. We have made detailed calculations for the lowest resonant state associated with the $n=1$ Landau level. Screened potentials were used primarily because they simplify the numberical calculation, but the procedure is applicable without modification to any potential $V(z)$ which goes to zero faster than $\frac{1}{z}$. Furthermore, although we have used the parabolic band model, the method can be readily modified to include nonparabolicity.

On the free electron state in the internal conversion and on the relation ICC?|?(0)|2 for Tc-99m

The final continuum electron states in the internal conversion (IC) process are considered under two points of view:1) Regarding the potential for the free electron we use neither the unchanged atomic potential (no-hole model) nor the Rose- or the Band-hole model, but we apply another model which is introduced and discussed. The differences between the models are significant for low kinetic energies (a few keV).2) Changes of the chemical or ionization state of the atom cause variations of the IC coefficients (ICC) not only via the bound states (proportionality relation ICC∼|Ψ(0)|2), but in principle via the continuum states, too, especially for low kinetic energies or for inner shells. This effect is examined for the transition of Tc-99m (2.17keV;E3) with variation of the transition energy. For the non-s-subshells it turns out that the simple proportionality relation holds for the Tc-99m transition, but for lower energy it has to be extended to the free electron wave functions. In thes-shells there appear discrepancies with decreasing energy even from the extended proportionality relation. A short description of the used ICC code is included in the paper.

Electrically pumped relativistic free-electron wave generators

Stimulated scattering induced by the longitudinal electric field of a pump wave is studied theoretically for the case of dense relativistic electron beams traveling in cylindrical metal waveguides. Two processes are examined. In one, the pump wave decays parametrically into a slow and a fast space-charge wave. In the other, it decays into a slow space-charge wave and a TM wave of the guide. The frequency characteristics and stimulated growth rates are given for each process, as a function of beam diameter, velocity, and density.

Model calculation of dynamical spin susceptibility of paramagnetic nickel

The dynamical spin susceptibility of paramagnetic nickel is investigated using a noninteracting isotropic band model based on the energy-band calculations of Hanus. The free-electron wave function and the simple tightbinding wave functions are used for $s$ and $d$ conduction electrons, respectively. Explicit expressions are obtained for the intraband and interband parts of the susceptibility function. Numerical computations are carried out for paramagnetic nickel for the atomic configuration ${(3d)}^{9.4}{(4s)}^{0.6}$ for field wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ along the [100] direction and energy-transfer range 0.01-0.4 eV. The intraband and interband parts are compared and the intraband part is found to be dominating. The exchange enhancement of the susceptibility function is also studied and the results are compared with other existing theoretical calculations and the experimental measurements. The agreement is found to be reasonably good.

Cotton–Mouton constant of benzene calculated by free electron wave functions

Cotton–Mouton constant of benzene calculated by free electron wave functions

Photoionization Cross-Sections for Atoms and Ions of Aluminum, Silicon, and Argon

Photoionization cross sections for all levels belonging to the configurations of atoms and ions of aluminum, silicon, and argon have been calculated using Hartree-Fock bound-electron wave functions and close-coupling approximation free-electron wave functions. The results are presented in the form of a computationally convenient interpolation formula and should find wide astrophysical application.

Excitation of semi-forbidden 2s21S-2s2p3P lines observed in quasars and nebulae

Collisional excitation of C+2 2s2p 3P by free electrons determines the strength of the lambda 1909 emission line observed in many quasars. Values of the collision strength for excitation of this term, and also for excitation of 2s2p 1P, are calculated for C+2 and other members of the isoelectronic sequence from B+ to Ne+6, by solving the coupled integro-differential equations for the free-electron radial wave functions. The solutions are carried out by an iterative procedure. The averaged effect of the resonances converging on the 1P threshold is taken into account through the Gailitis (1963) formula.

Photo-Ionization of the Hydrogen Molecule

One-center wave functions are employed to investigate the photo-ionization of the hydrogen molecule from its ground state $X(1s\ensuremath{\sigma}{^{1}\ensuremath{\Sigma}_{g}}^{+})$, from 700 to 300 \AA{}. It is assumed that the residual ion is left in its ground state, and the free electron is in a $p\ensuremath{\sigma} or p\ensuremath{\pi}$ orbital. Using the one-center wave functions for $\mathrm{H}_{2}^{}{}_{}{}^{+}$ for the internuclear distance equal to $1.4{a}_{0}$, the free-electron wave functions are obtained by solving the integrodifferential equations in exchange and polarized-orbital approximations. The oscillator strengths obtained in the polarized-orbital approximation are found to be in satisfactory agreement with the experimental data.

Low-Energy Electron Scattering From Atoms and Molecules: A Model

A simple model is proposed for the calculation of free-electron wave functions for low-energy electrons incident upon neutral atoms. The region of interaction of the electron with the atom may be separated into three important regions of potential: Coulomb plus exchange, Coulomb, and far-field induced polarization. The division of regions is based on mathematical and physical reasons for the characteristic behavior of wave functions in the scattering problem. Examples are given for the elastic scattering of electrons from nitrogen, oxygen, and argon. The agreement with experiment and effective-range theory is good. The application of the method to the prediction of low-energy electron scattering from homonuclear diatomic molecules is discussed.

Tetrahedral and Cubic Three-Dimensional Free-Electron Networks

In the boron cage of B4Cl4 and in the carbon cage of C8H8, valence orbitals hybridize to form an easy path along the edges of a symmetric polyhedron. On each structure, free-electron waves were constructed corresponding to the conventional delocalized molecular orbitals. Four different kinds of branching were observed. Energy levels were calculated without and with a delta-function singularity on each cage nucleus.

Free electron model and photoionization of ?-electrons from aromatic hydrocarbons

Using the free electron wave function it has been shown that the propability of photoionization of a π-electron from aromatic hydrocarbons should be maximum if the incident photon has energy sufficient to ionize and to leave an additional momentum p=√5 Kħ2/2 to the escaping electron. This momentum further appears to be independent of the ring size.

Strong-Field Magnetoresistance of Impurity Conduction inn-Type Germanium

A simple theory is given of the strong-field magnetoresistance in the phonon-induced hopping region of impurity conduction in $n$-type germanium. It is shown to be convenient to consider separately the effect of a magnetic field in three cases: the weak, the moderately strong, and the extremely strong field case. In a moderately strong field the shrinking of a donor wave function and the phase difference produced by the magnetic field between two neighboring donors lead to a formula qualitatively identical with the empirical formula found experimentally by Sladek and Keyes. The magnetoresistance in an extremely strong magnetic field is also discussed, in which the wave function in the plane perpendicular to the field becomes similar to the free-electron wave function in a magnetic field.

Wave Functions for the Free Electron. II. The Inclusion of Polarization and Exchange

The effect of the polarization of the atomic core by the free electron on the free-electron wave function and the effect of the exchange of the free electron with the bound orbitals on this wave function are treated by perturbation theory. Polarization must be considered first. Its effect on the atomic charge cloud is introduced through an expansion over the bound wave functions for the atom in terms of the free-electron separation as a parameter. This parametric treatment of electron separation means we cannot accept the solution at small separations from the nucleus although this is not a serious restriction. From this wave function we obtain a polarized Coulomb potential from which a solution for the free-electron function may be obtained using our old programs. Having solved the free-electron wave equation with the exchange potential terms supposed zero, we use this solution to compute the exchange integrals. The equation including these integrals is then solved to obtain approximate wave functions for free electrons containing both exchange and polarization.

Deionization Cross Section for Oxygen

Our atomic- and free-electron wave functions are applied to the calculation of the deionization cross section for O II. Our s- and d-wave numerical solutions to the free-electron wave equation is fitted to Coulomb functions for normalization. This result is used to determine the cross section for the transitions s wave to 2p orbital and d wave to 2p orbital. We carried out an approximate calculation for the transition to the 3p orbital with the resulting indication that the hydrogen result may reasonably be used. For the contributions from transitions to 3d and higher orbitals the hydrogen cross sections have been adopted. Our final result is 198×10—21 cm2 leading to a rate constant for radiative deionization of 218×10—14 cm3/sec.

Fermi Momentum for Free Electron Metals Perturbed by Localized Imperfections

The Fermi momentum k m and the interior density ρ i are calculated for free electron metals perturbed by localized imperfections, and the Thomas-Fermi relation ρ i = k m 3 /3π 2 is verified up to the higher order. Then it is shown that the Friedel theorem can be based on the above relation. When the Fermi momentum k m is expressed in terms of the phase shift η of the free electron wave function, the charge neutrality condition in the interior leads to the Friedel sum rule for the case of the plane surface as well as of the point imperfection.

The bound free continuum for C −

The wave functions for the ground state of carbon and the negative carbon ion are determined using our small variation program. The Coulomb potential for the free electron is then computed using the carbon wave function thus obtained, and the subsequent obtention of the s- and d-wave portions of the free electron wave function is detailed. The dipole length form of the matrix elements is then determined for the various desired values of the free electron momenta under the assumption that the lower state is the complete negative carbon ion wave function while the upper state is the antisymmetric product made up of the free electron plus the complete neutral carbon atom wave functions. Finally a simple calculation using the matrix element results provides the photo-ionization cross section.

Wave Function for the Free Electron. I. The Coulomb Potential

The wave function for a free electron in the presence of an atom is considered for the case of the Coulomb potential, a potential which ignores both the effects of electron exchange and core polarization. A sample calculation involving the neutral oxygen atom is carried through. From the single-determinant wave function for the $^{3}P$ ground state of this atom the Coulomb potential is developed, and the important parts of the potential for the $s$- and $d$-wave portions of the free-electron wave function are considered. The relevant Schr\odinger equations are solved by means of an IBM 704 program which is written so as to be readily adaptable to electrons in the presence of other atoms or ions having electrons through $2p$. In our oxygen example the wave function calculations were carried out for values of the free-electron linear momentum ranging from 0.01 through 0.80 atomic unit.