Steady-state Potential Distributions Research Articles

Analytic models for a spherical satellite charging in sunlight at any spin rate

Abstract. We present analytic models for the steady state potential distributions surrounding a spinning, dielectric-coated, spherical spacecraft charging in sunlight. The sun direction is assumed to lie in the satellite bellyband plane, perpendicular to the spin axes. The models are based on a multipole expansion of Laplacian potentials external to the spacecraft surface. The combination of monopole potentials along with the dipole or quadrupole contributions produce potential barriers which form at the satellite surface. These barriers can block escaping photoelectrons and lead to current balance, allowing sunlight charging to high negative levels. In a previous treatment, analytic models were limited to fast spin relative to differential charging rates so that the solutions had azimuthal symmetry around the spin axes. By introducing an associated Legendre term into the potential expansion, the azimuthal symmetry is removed, and the models can be developed to encompass any spin rate. The analysis turns up three functions of spin rate which are only known at the spin limits, but the characteristics of the charging of a rotating sphere can be explored using approximate forms which represent the basic trends. For finite spin, the sunlit side charges less (negatively) than the shade side which is in contrast to the fast spin case, where these two potentials are equal. Also, for finite spin, differential charging develops perpendicular to the sun and spin axis directions, due to the transverse motion. This transverse charging occurs at all finite spin rates, disappearing only at the zero and infinite spin limits. There is a correlated lag angle between the direction of maximum sheath radius and the sun line. Plots are given to illustrate the potential distributions representing barrier dominated sunlight charging of a spinning dielectric coated spherical satellite.

One-dimensional model of the Debye layer near a dielectric surface

A study is made of the processes occurring in a low-density plasma near a dielectric wall. A one-dimensional non-steady-state model of the electron dynamics is constructed that takes into account secondary electron emission. The Vlasov-Poisson equations are solved numerically. According to the results obtained, the steady-state potential distribution that forms at a low temperature of the incident electrons gives rise to a wall layer whose characteristic thickness is about several Debye lengths and in which the electrons are decelerated. In this case, the electron density is lowest near the wall. The situation in which the temperature of the incident electrons is high is far more complicated: the solution is quasi-periodic in character and the electron density near the wall is the highest.

Levitation and dynamics of charged dust in the photoelectron sheath above surfaces in space

Bodies in space subject to solar ultraviolet flux will emit photoelectrons. At steady state the current of escaping photoelectrons is balanced by an influx of particles from the surrounding plasma. When the photoelectrons dominate the space charge close to the surface, it has previously been shown that two steady state potential distributions can exist, one in which the potential decreases from the surface value to zero monotonically and one in which it decreases to a negative minimum and then increases to zero. It has been suggested that the nonmonotonic distribution is the stable one. By assuming planar geometry and a Maxwellian distribution of the emitted photoelectrons the charging of isolated dust particles in the plasma sheath is calculated for both the monotonic and nonmonotonic potential distribution. By increasing photoemission from the surface from zero a transition from an ordinary Debye sheath above a nonilluminated surface to a photoelectron sheath is simulated. Dynamical properties of the dust particles such as oscillations, damping, stability, and trapping are investigated. After being injected into the sheath or electrostatically levitated, dust may be stably suspended above illuminated surfaces in space, even in the case of zero gravitation. However, the smallest particles may escape completely from the body. For all sheath types an unstable layer exists close to the surface where dust cannot collect. The theory is applied to bodies in the solar wind and to the spokes of Saturn.

Modes in electron-hopping transport over insulators sustained by secondary electron emission

A general formalism for hopping electron transport over insulators sustained by secondary electron emission is presented. Steady-state electron transport takes place when the charging of the insulator, which turns out to be self-stabilizing, is such that the average secondary electron yield becomes equal to unity. The steady-state potential distribution for the electron transport is determined for various insulating geometries with the aid of Monte Carlo calculations and compared with the low-hopping approximation. The Monte Carlo results show that the steady-state potential distribution can exhibit several interesting features such as spontaneous symmetry breaking of statistical origin and the occurrence of local repulsive parts in the geometry. In several cases the numerical results, including the above two features, are found to agree well with the results of experiments.

Notching as an example of charging in uniform high density plasmas

Numerical simulation was used to study both surface charging and ion trajectory distortion during submicron patterning in high density plasma etching. The plasma was assumed uniform and the cause for the surface charging was the directionality difference between ions and electrons. The role of ion transit time effects on the ion energy distribution function was also considered, while the effect of discharging currents such as through insulators was not included. Using a Monte Carlo sheath simulator, a Poisson equation solver, and an ion/electron trajectory simulator, the steady state potential distribution and ion trajectories were calculated for various line-and-space structures and plasma conditions where notching, which is a local sidewall etching, has been observed after the overetching part of polysilicon etching processes. The results show significant positive charging at the bottom of high aspect ratio spaces which depends on the ion energy distribution function. Notching at the bottom of an outermost polysilicon line before a wide space is the result of ion deflection toward the line which has the lower potential from receiving more electrons from a side facing the wide space. The simulator was validated using previously reported electron cyclotron resonance etcher notching results. It was found that the calculated potential difference between the bottom of a space and the adjacent line shows the same qualitative dependence on the wide space width as experimental notch depth results show. In addition, these potential differences are large enough to substantially increase the ion flux at regions where notching occurs for the ion energy distribution function for the plasma conditions used. Thus this shows aspect ratio dependent charging effects which are not dependent on initial plasma nonuniformity.

Monte Carlo calculations of the electron hop transport in various parts of a Zeus display

Electron transport over insulators sustained by secondary electron emission is studied using Monte Carlo calculations. With the aid of these Monte Carlo calculations, the steady-state potential distribution and the properties of the electron trajectories in various insulating structures are determined. Each of these structures represents a part of the Zeus display and together the results show how electrons, generated by the cathode, enter the transport channels and are transported through the Zeus display until they reach the phosphor screen. In the various cases, the results according to the Monte Carlo calculations agree well with the results of experiments.

Potential distribution in three-dimensional periodic myocardium. I. Solution with two-scale asymptotic analysis

The use of two-scale asymptotic analysis allows development of a model of the steady-state potential distribution in three-dimensional cardiac muscle preserving the underlying cellular network. The myocardium is modeled as a periodic structure consisting of cylindrical cells embedded in extracellular fluid and connected by longitudinal and side junctions. The method is applicable to cardiac muscle of arbitrary extent since the periodicity of the tissue is dealt with analytically, and thus numerical computations require no more resources than a continuous volume conductor problem. The asymptotic analysis approach reveals that the potential in a periodic myocardium consists of two components. The large-scale component provides the baseline for the total solution and can be determined from the anisotropic monodomain model associated with the original periodic problem. The method provides the formula for calculating the conductivity of the equivalent monodomain model on the basis of cell geometry and conductivity distribution in the cardiac tissue. The small-scale component reflects the periodicity of the underlying structure and oscillates with periods determined by the dimensions of cardiac cells. The magnitude of these oscillations depends upon the gradient of the large-scale component. During stimulation with extracellular electrodes, the small-scale component determines both the shape and the magnitude of the transmembrane potential, while the influence of the large-scale component is negligible. Hence, the small-scale component merits closer attention in pacing and defibrillation studies, especially since the model based on two-scale asymptotic analysis provides an effective means of its computation.

Thermodynamic and kinetic considerations for crystal growth of complex molecules from solution

A general thermo-kinetic picture of crystallization from a nutrient phase is presented in terms of the driving force available in both the far field and interface regions. The necessary and sufficient boundary value problems that must be solved to correct the far field thermodynamic state variables to the interface state variables is sketched and the material parameters needed to describe the processes for small molecules identified. Extension of this modeling approach to protein crystallization is made with the recognition that small crystallites and large molecules in aqueous solution may behave like colloid particles with an accompanying electrical double layer. Thus, the concepts and numerics of colloid stability theory should probably be considered when dealing with protein crystallization. Key constraints on controlled crystallization in this case involve (a) the controlled passage of the molecules from the secondary potential minimum to the primary potential minimum of the crystal interface double layer, (b) the simultaneous partial electrolyte/solvent stripping and protein particle incorporation into the crystal, and (c) the bulk solvent neutralization of the incremental electrolyte and pH changes due to crystallization so that a steady state potential distribution may be maintained in the interface region.

Determination of axial steady-state potential distributions in collisionless single-ended Q-machines

A method is presented which permits one to reliably determine axial steady-state potential distributions in collisionless single-ended plasmas. The method requires standard measurements of plasma parameters at negative cold-plate biases, whence the (usually unstable) steady-state conditions at positive biases can be predicted by means of a simple theoretical extrapolation. In contrast to some models used in the literature, only two types of potential distributions are found to be possible at large positive cold-plate biases. A qualitative discussion of grid effects leads to the conclusion that a grid is equivalent to a massive cold plate only in the saturation regime of its current-voltage characteristics.

Surface potential of a photoemitting plate

Earlier work [Guernsey and Fu, 1970] studied the potential distribution in the neighborhood of a photoemitting plate immersed in plasma. The result indicates that two steady state potential distributions can exist, namely one in which the potential decreases from its plate value to zero monotonically and one in which it decreases from its plate value to a negative minimum and then increases to zero (‘overshoot’ type). In the present work a Maxwellian distribution is assumed for the emitting photoelectrons instead of a Fermi shape distribution. Furthermore the difference of the surface potential for these two types of distributions is investigated in more detail. The result shows that this difference can exist only if 1<Te/Tν<nν/ne, where Te, Tν, ne, nν, are the temperatures and number densities of plasma electrons and photoelectrons.

Potential distribution surrounding a photo-emitting, plate in a dilute plasma

Steady state potential distributions near photoemitting plate immersed in plasma, noting over-shoot effects role for sheaths around satellites in interplanetary plasma

CARRIER GENERATION AND SWITCHING PHENOMENA IN n—GaAs DEVICES

By measuring the steady-state potential distribution in n—GaAs samples with ohmic contacts, very high electric fields have been found near the anode contacts. In samples with n · L (carrier concentration × sample length) products greater than 1012 cm-2, switching from the Gunn mode to a high current, lower (constant) voltage state takes place along with the appearance of a high field at the anode. In samples with n · L products less than 1012 cm-2 the buildup of the high field is accompanied by a sudden increase in current. These high currents are due to carrier generation in a narrow region near the anode.

Distribution of potential in cylindrical thermionic valves

This paper presents a derivation of the steady-state potential distribution in a space bounded by coaxial cylinders, between which a homogeneous electron current flows. The problem is not formally soluble, but the solution is represented by twenty-four particular solutions covering the whole range of values of the variables likely to occur in practice. The potential equation, derived through the equation of motion of an electron, is reduced to a non-dimensional second-order form. The complete solution of this is obtainable by evaluating a singly infinite set of solutions, as the independent variable has an arbitrary zero. The physical meaning of this form of solution is expressed in similarity theorems. The behaviour of the solution in limiting cases, and the approximate graphical solution of the first-order form, suggested the procedure adopted in solving the second-order equation, which was suitable for integration by means of the differential analyser. In some solutions, a singularity in the potential equation was avoided by numerical integration of the equation of motion, to give initial conditions for the mechanical integration away from the singularity. The solutions are tabulated. Some applications of the results to thermionic valve problems are illustrated by numerical examples.