Derivation of a modified paraxial formalism for two-dimensional trajectories in electron and ion sources of non-simple geometries

A modified paraxial formalism has been developed which describes two-dimensional charged particle beam dynamics in electron and ion sources with pointed or needle-type geometries. A Hamiltonian formalism, and a more exact treatment of energy conservation is used to derive the modified paraxial equation for two-dimensional trajectories in systems with axially symmetric prolate-spheroidal beams. Calculations have been done for a gallium field emission liquid metal ion source modeled by a hyperboloid of revolution and planar extractor. Two important conclusions emerge from these calculations: i) The dominant effect of space-charge, for source geometries with small radii of curvature, occurs in the region close to the apex (<0.05 n) and ii) beam divergence has a strong dependence on geometry. This latter effect is a consequence of large two-dimensional field gradients near the apex of sources with needle-type or pointed geometry.This work was supported in part by the Division of Materials Research, National Science Foundation, Grant No. DMR-8108829     

Reply to comment on “variational formulation for the equilibrium condition of a conducting fluid in an electric field”

In this comment we respond to the several criticisms of the paper by Sujatha et al. raised by Kingham and Bell. In particular, we demonstrate that, contrary to their assertion, Taylor's solution for the electrostatic fields can never satisfy the boundary conditions for the actual experimental configurations involving field emission liquid metal ion sources and other experiments on electrostatically stressed conducting fluids. It is further argued that a careful analysis of Taylor's experimental procedure and observations suggests that although the observed static structures have a macroscopic axial-symmetry they have not the idealized conical shapes of prescribed angle. Furthermore, the formation of the Taylor cone structure is shown to be inconsistent with the principle of energy minimization.It is concluded that none of the criticism raised by Kingham and Bell invalidate any of the analysis or conclusions presented in the paper by Sujatha et al.This work has been supported in part by the Division of Materials Research, National Science Foundation, Grant No. DMR-81008829     

A review of the theoretical and experimental analyses of electron spin polarization in ferromagnetic transition metals: I. Field emission,photoemission, magneto-optic Kerr effect and tunneling

Until recently studies of solid surfaces have used exclusively photons, atoms, ions, and unpolarized electrons as experimental probes. However, in the last few years polarized electron beams have been used to investigate the electronic and atomic properties of the bulk and surfaces of solids. This has resulted in the development and application of a number of different experimental techniques sensitive to the electron spin polarization (ESP). The experimental results have often been less than definitive and frequently conflicting. For example, the observation of ESP from the ferromagnetic metals by photoemission, field emission, and tunneling experiments lead to contradictory conclusions concerning the validity of the Stoner-Wohl-farth-Slater (SWS) band theory of ferromagnetism. Additional complication of this problem has been introduced by recent magneto-optical Kerr-effect measurements which tend to support the ESP predicted by SWS theory.In this review we present a critical analysis of these experiments and their theoretical interpretation. It is shown that the results of these experiments do not necessarily contradict each other nor any of the current one-electron theories of itinerant ferromagnetism. It is further shown how, based on the use of over-simplified models to describe the physical processes involved in the experiments, the opposite conclusion can be reached. Based on the analyses in this paper, improvements in the theoretical models are suggested. We also discuss, from a theoretical point of view, the relative advantages and the limitations of the different experimental techniques using ESP to probe the electronic and magnetic structures of solids and surfaces.A fundamental objective of this and the following paper is to help develop a theory of spinpolarized electron emission and tunneling phenomena which will provide the basis for a systematic ESP spectroscopy of surfaces, interfaces, and thin films. This has prompted a reexamination of the ferromagnetism of 3d-transition metals and of their surfaces. The present work leads to some observations concerning the current interpretation of itinerant ferromagnetism and possible refinements of the SWS theory to produce better agreement with the ESP sensitive experiments.     

The multiple-image interactions and the mean-barrier approximation in MOM and MVM tunneling junctions

The effect of the multiple image interactions on theI–V characteristics and the reliability of the mean barrier approximation in calculating the current in MOM and MVM tunneling junctions are critically examined. It is demonstrated that the continued use of the uncorrected form of Simmons' original multiple-image force interaction in the analysis of tunneling junctions can lead to serious errors in both the current and the dynamic resistance. An extensive numerical analysis of planar junctions including the image potential suggests that the basic mean barrier approximation formulated by Simmons is essentially a thick barrier approximation. It also is shown that the conventional mean barrier approximation is a relatively poor approximation for the tunneling barriers of interest, and that it is not possible to establish a reliable a priori estimate of its range of validity.This research was supported in part by the NATO Research Grants Program, Grant No. 1902, Scientific Affairs Division, Brussels, Belgium     

Examination of the electronic energy band model for one-dimensional disordered thin films

The nature of the electronic states in a disordered one-dimensional finite system subject to traveling wave boundary conditions is examined. A method is developed whereby these states can be characterized as either “localized” or extended. The concept of localization of states is modified so as to be applicable to amorphous thin films. The extent to which the disorder modifies the band structure of the ordered system is investigated as a function of disorder and the nature of the electronic states is related to the elastic tunneling transmission coefficient for a model metal-semiconductor-metal heterojunction. The results seem to corroborate the Mott-CFO model.     

Influence of the boundary conditions on the fermi energy and density of states in a free-electron solid of sub-micron dimensions

Asymptotic expressions for the distribution of the eigenvalues of the Helmholtz-Schrödinger equation are used to anlyze the dependence of the Fermi energy, E_F, and the density of states, ρ(E), on sample size, shape, and electron density, in a free-electron model with Dirichlet boundary conditions. It is found that for very small samples E_F is increased relative to its asymptotic (i.e., bulk) value and ρ(E) is decreased relative to its bulk value. These effects are more pronounced for samples with low electron density and with a large surface-to-volume ratio. In general E_F and ρ(E_F) deviate significantly from their bulk values only for systems with fewer than 50,000 electrons and/or with linear dimensions of 100 Å or less. The use of smoothing functions to represent the density of states obtained from the exact eigenvalue distribution is also discussed. It is shown that an oscillating density of states leads to small cusps in the plot of E_F as a function of sample size. This is in qualitative agreement with the results of experiments on size-dependent oscillations in field emission from thin metallic films. Comparison is also made between photoemission experiments from thin films and other results obtained in this study.     

A Green's function solution to the image and multiple image interactions for hyperboloidal geometry: Application to metallic pointcontact infrared detectors

Using the Mehler-Fock transformation to solve Poisson's equation in prolate spheroidal coordinates, we have obtained an exact Green's function solution for all multiple image corrections to the vacuum tunneling barrier for a hyperboloidal tip-planar-anode model of a point-contact junction consisting of identical or dissimilar metals.These calculations show that the image corrections significantly modify both the form and area of the barrier, producing an enhancement in the rectification and tunneling currents at low bias.I–V characteristics have also been obtained for the hyperboloidal tip model using estimates of the emission and collection regions based on field emission experiments for whisker tips of comparable dimensions. These results are compared with earlier calculations whichThis research was supported in part by the NATO Research Grants Program, Grant No. 1902, Scientific Affairs Division, Brussels, BELGIUM.     

Generalized gauge independence and the physical limitations on the von Neumann measurement postulate

An analysis is presented of the significance and consequent limitations on the applicability of the von Neumann measurement postulate in quantum mechanics. Directly observable quantities, such as the expectation value of the velocity operator, are distinguished from mathematical constructs, such as the expectation value of the canonical momentum, which are not directly observable. A simple criterion to distinguish between the two types of operators is derived. The non-observability of the electromagnetic four-potentials is shown to imply the non-measurability of the canonical momentum. The concept of a mechanical gauge is introduced and discussed. Classically the Lagrangian is nonunique within a total time derivative. This may be interpreted as the freedom of choosing a mechanical (M) gauge function. In quantum mechanics it is often implicitly assumed that the M-gauge vanishes. However, the requirement that directly observable quantities be independent of the arbitrary mechanical gauge is shown to lead to results analogous to those derived from the requirement of electromagnetic gauge independence of observables. The significance of the above to the observability of transition amplitudes between field-free energy eigenstates in the presence (and absence) of electromagnetic fields is discussed. E- and M-gauge independent transition amplitudes between field-free energy eigenstates in the absence of electromagnetic fields are defined. It is shown that, in general, such measurable amplitudes cannot be defined in the presence of externally applied time-dependent fields. Transition amplitudes in the presence of time-independent fields are discussed. The path dependence of previous derivations of E-gauge independent Hamiltonians and/or transition amplitudes in the presence of electromagnetic fields are related to the inherent M-gauge dependence of these quantities in the presence of such fields.     

Gauge invariance and gauge independence of the S-matrix in nonrelativistic quantum mechanics and relativistic quantum field theories

The gauge independence of transition rates as opposed to the gauge invariance of the equations of motion and gauge dependence of operators and state vectors is critically examined and explicitly demonstrated, both in nonrelativistic quantum mechanics and quantum field theory. Time independent as well as time dependent gauge transformations are explicitly analyzed using several techniques in order to clarify the physical content and significance of gauge independence and the conditions for its applicability.