W. P. Allis

Bio: W. P. Allis is an academic researcher from Massachusetts Institute of Technology. The author has contributed to research in topics: Electron & Magnetic field. The author has an hindex of 11, co-authored 13 publications receiving 1098 citations.

The Transition from Free to Ambipolar Diffusion

Abstract: In gas discharge plasmas with very low charge densities, the charged particles diffuse freely in directions perpendicular to the applied electric field because the space-charge field is negligible. At high charge densities, the space-charge field saturates and gives rise to a combination of diffusive and mobility flow termed ambipolar. The transition between these limits is examined theoretically for the case of plasmas maintained through ionization by electron impact. The ionization frequency per electron, one of the principal parameters of the transition, can be re-expressed in terms of an effective diffusion coefficient; it falls from a high value at the free diffusion limit to a low value at the ambipolar limit as the electron density increases over many orders of magnitude. The trnasition is accompanied by changes in the charge distributions and by the development of a positive ion sheath. The current equations determining the process are examined, and approximate solutions are obtained. Second approximations are obtained for the case where the ratio of electron to ion energies is much greater than unity. Machine solutions are presented both for the above case and for an isothermal plasma in which this ratio equals unity. An application to the afterglow is shown.

Motions of Ions and Electrons

Abstract: This article is divided into five parts according to the mathematical method used, rather than according to the physical situation. In part I we attempt to follow the motion of an individual particle under a Lorentz force including the effects of gradients of the magnetic field. This is no longer possible with the introduction of collisions in part II, but one may yet follow the motion of an average particle. One is then tempted to expect that a swarm of particles will have the behavior of the average particle, and this is remarkably close to the truth for high values of E/p, the ratio of the continuous to the stochastic force, and has the advantage of being more intuitive than the distribution function methods which follow.

The Effect of Exchange on the Scattering of Slow Electrons from Atoms

Abstract: Reasons are given why the Born approximation is incapable of dealing with the scattering of slow electrons. Since this approximation assumes that the sine of the phase angle $\ensuremath{\delta}$ equals $\ensuremath{\delta}$, the results computed by the Born method become completely unreliable when $\ensuremath{\delta}$ is greater than $\frac{\ensuremath{\pi}}{2}$. Exact solutions then must be obtained. Equations are set up, including exchange effects, for the best possible wave function for an electron scattered from hydrogen or helium when the complete wave function is of the separable type usually used in atomic theory. These equations are solved on the differential analyzer to find the best possible curves for the $\ensuremath{\delta}'\mathrm{s}$, for the angle distribution of scattering and for the total cross section, for this type of wave function. The check with experiment for helium is good, the maximum discrepancy in any of the $\ensuremath{\delta}'\mathrm{s}$ being only ten degrees. No data for atomic hydrogen are available. The small error introduced by the use of separable wave functions (neglect of polarization) is discussed. The conclusions are that exchange effects are not important for electron energies greater than 30 volts, and below this energy have an appreciable effect only on the angle distribution curves, and not on the cross section curves. An analytic solution of the equations, valid for any atom having closed electronic shells, is obtained for a simplified form of atomic wave function and potential. The results confirm the above conclusions, and also show that exchange is less important in scattering from heavy atoms than from light ones.

Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models

Abstract: Fluid models of gas discharges require the input of transport coefficients and rate coefficients that depend on the electron energy distribution function. Such coefficients are usually calculated from collision cross-section data by solving the electron Boltzmann equation (BE). In this paper we present a new user-friendly BE solver developed especially for this purpose, freely available under the name BOLSIG+, which is more general and easier to use than most other BE solvers available. The solver provides steady-state solutions of the BE for electrons in a uniform electric field, using the classical two-term expansion, and is able to account for different growth models, quasi-stationary and oscillating fields, electron–neutral collisions and electron–electron collisions. We show that for the approximations we use, the BE takes the form of a convection-diffusion continuity-equation with a non-local source term in energy space. To solve this equation we use an exponential scheme commonly used for convection-diffusion problems. The calculated electron transport coefficients and rate coefficients are defined so as to ensure maximum consistency with the fluid equations. We discuss how these coefficients are best used in fluid models and illustrate the influence of some essential parameters and approximations.

Theory of Dielectric Waveguides

Abstract: Dielectric waveguides are the structures that are used to confine and guide the light in the guided-wave devices and circuits of integrated optics. This chapter is devoted to the theory of these waveguides. Other chapters of this book discuss their fabrication by such techniques as sputtering, diffusion, ion implantation or epitaxial growth. A well-known dielectric waveguide is, of course, the optical fiber which usually has a circular cross-section. In contrast, the guides of interest to integrated optics are usually planar structures such as planar films or strips. Our discussion will focus on these planar guides even though most of the fundamentals are applicable to all dielectric waveguide types.

The Kinetic Theory of Gases

Abstract: THE difficulty of reconciling line spectra with the kinetic theory of gases, has been referred to by Prof. Fitzgerald (NATURE, January 3, p. 221). The following considerations show that it is possible under certain suppositions to have a number of spectral rays with a very restricted number of degrees of freedom. Most of us, I believe, now accept a definite atomic charge of electricity, and if each charge is imagined to be capable of moving along the surface of an atom, it would represent two degrees of freedom. If a molecule is capable of sending out a homogeneous vibration, it means that there must be a definite position of equilibrium of the “electron.” If there are several such positions, the vibrations may take place in several periods. Any one molecule may perform for a certain time a simple periodic oscillation about one position of equilibrium, and owing to some impact the electron may be knocked over into a new position. The vibrations under these circumstances would not be quite homogeneous, but if the electron oscillates about any one position sufficiently long to perform a few thousand oscillations, we should hardly notice the want of homogeneity. Each electron at a given time would only send out vibrations which in our instruments would appear as homogeneous. Each molecule could thus successively give rise to a number of spectral rays, and at any one time the electron in the different molecules would, by the laws of probability, be distributed over all possible positions of equilibrium, so that we should always see all the vibrations which any one molecule of the gas is capable of sending out. The probability of an electron oscillating about one of its positions of equilibrium need not be the same in all cases. Hence a line may be weak not because the vibration has a smaller amplitude, but because fewer molecules give rise to it. The fact that the vibrations of a gas are not quite homogeneous, is borne out by experiment. If impacts become more frequent by increased pressure, we should expect from the above views that the time during which an electron performs a certain oscillation is shortened; hence the line should widen, which is the case. I have spoken, for the sake of simplicity, as if an electron vibrating about one position of equilibrium could only do so in one period. If the forces called into play, by a displacement, depend on the direction of the displacement, there would be two possible frequencies. If the surface is nearly symmetrical, we should have double lines.

Momentum transfer cross sections for slow electrons in he, ar, kr, and xe from transport coefficients,

Abstract: Momentum-transfer cross sections for electrons in He, Ar, Kr, and Xe are obtained from a comparison of theoretical and experimental values of the drift velocities and of the ratio of the diffusion coefficient to the mobility coefficient for electrons in these gases. The theoretical transport coefficients are obtained by calculating accurate electron-energy distribution functions for energies below excitation using an assumed energy-dependent momentum-transfer cross section. The resulting theoretical values are compared with the available experimental data and adjustments made in the assumed cross sections until good agreement is obtained. The final momentum cross sections for helium is 5.0\ifmmode\pm\else\textpm\fi{}0.1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ for an electron energy of 5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}3}$ eV and rises to 6.6\ifmmode\pm\else\textpm\fi{}0.3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ for energies near 1 eV. The cross sections obtained for Ar, Kr, and Xe decrease from 6\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$, 2.6\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$, and ${10}^{\ensuremath{-}14}$ ${\mathrm{cm}}^{2}$, respectively, at 0.01 eV to minimum values of 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}17}$ ${\mathrm{cm}}^{2}$ at 0.3 eV for Ar, 5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}17}$ ${\mathrm{cm}}^{2}$ at 0.65 eV for Kr, and 1.2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ at 0.6 eV for Xe. The agreement of the very-low-energy results with the effective-range theory of electron scattering is good.