Theory of the positive column of a gas discharge

An attempt is made to provide a theoretical explanation of the effect of the positive column on the voltage-current characteristic of a glow or an arc discharge. Such theories have been developed before, and all are based on balancing the production and loss of charged particles and accounting for the energy supplied to the plasma by the applied electric field. Differences among the theories arise from the approximations and omissions made in selecting processes that affect the particle and energy balances. This work is primarily concerned with the deviation from the ambipolar description of the positive column caused by space charge, electron-ion volume recombination, and temperature inhomogeneities.

The presentation is divided into three parts, the first of which involved the derivation of the final macroscopic equations from kinetic theory. The final equations are obtained by taking the first three moments of the Boltzmann equation for each of the three species in the plasma. Although the method used and the equations obtained are not novel, the derivation is carried out in detail in order to appraise the validity of numerous approximations and to justify the use of data from other sources. The equations are applied to a molecular hydrogen discharge contained between parallel walls. The applied electric field is parallel to the walls, and the dependent variables—electron and ion flux to the walls, electron and ion densities, transverse electric field, and gas temperature—vary only in the direction perpendicular to the walls. The mathematical description is given by a sixth-order nonlinear two-point boundary value problem which contains the applied field as a parameter. The amount of neutral gas and its temperature at the walls are held fixed, and the relation between the applied field and the electron density at the center of the discharge is obtained in the process of solving the problem. This relation corresponds to that between current and voltage and is used to interpret the effect of space charge, recombination, and temperature inhomogeneities on the voltage-current characteristic of the discharge.

The complete solution of the equations is impractical both numerically and analytically, and in Part II the gas temperature is assumed uniform so as to focus on the combined effects of space charge and recombination. The terms representing these effects are treated as perturbations to equations that would otherwise describe the ambipolar situation. However, the term representing space charge is not negligible in a thin boundary layer or sheath near the walls, and consequently the perturbation problem is singular. Separate solutions must be obtained in the sheath and in the main region of the discharge, and the relation between the electron density and the applied field is not determined until these solutions are matched.

In Part III the electron and ion densities are assumed equal, and the complicated space-charge calculation is thereby replaced by the ambipolar description. Recombination and temperature inhomogeneities are both important at high values of the electron density. However, the formulation of the problem permits a comparison of the relative effects, and temperature inhomogeneities are shown to be important at lower values of the electron density than recombination. The equations are solved by a direct numerical integration and by treating the term representing temperature inhomogeneities as a perturbation.

The conclusions reached in the study are primarily concerned with the association of the relation between electron density and axial field with the voltage-current characteristic. It is known that the effect of space charge can account for the subnormal glow discharge and that the normal glow corresponds to a close approach to an ambipolar situation. The effect of temperature inhomogeneities helps explain the decreasing characteristic of the arc, and the effect of recombination is not expected to appear except at very high electron densities.

Eigenvalue problems associated with Korn's inequalities in the theory of elasticity

Interest in the possible applications of a priori inequalities in linear elasticity theory motivated the present investigation. Korn's inequality under various side conditions is considered, with emphasis on the Korn's constant. In the "second case" of Korn's inequality, a variational approach leads to an eigenvalue problem; it is shown that, for simply-connected two-dimensional regions, the problem of determining the spectrum of this eigenvalue problem is equivalent to finding the values of Poisson's ratio for which the displacement boundary-value problem of linear homogeneous isotropic elastostatics has a non-unique solution.

Previous work on the uniqueness and non-uniqueness issue for the latter problem is examined and the results applied to the spectrum of the Korn eigenvalue problem. In this way, further information on the Korn constant for general regions is obtained.

A generalization of the "main case" of Korn's inequality is introduced and the associated eigenvalue problem is a gain related to the displacement boundary-value problem of linear elastostatics in two dimensions.

The complex angular momentum theory of the production of three particles in collisions of two strongly interacting particles at high energy

The problem of the continuation to complex values of the angular momentum of the partial wave amplitude is examined for the simplest production process, that of two particles → three particles. The presence of so-called "anomalous singularities" complicates the procedure followed relative to that used for quasi two-body scattering amplitudes. The anomalous singularities are shown to lead to exchange degenerate amplitudes with possible poles in much the same way as "normal" singularities lead to the usual signatured amplitudes. The resulting exchange-degenerate trajectories would also be expected to occur in two-body amplitudes.

The representation of the production amplitude in terms of the singularities of the partial wave amplitude is then developed and applied to the high energy region, with attention being paid to the emergence of "double Regge" terms. Certain new results are obtained for the behavior of the amplitude at zero momentum transfer, and some predictions of polarization and minima in momentum transfer distributions are made. A calculation of the polarization of the ρ^o meson in the reaction π ^- p → π ^- ρ^op at high energy with small momentum transfer to the proton is compared with data taken at 25 Gev by W. D. Walker and collaborators. The result is favorable, although limited by the statistics of the available data.

Investigations in the theory of electromagnetic scattering from expanding dielectric obstacles

An equation for the reflection which results when an expanding dielectric slab scatters normally incident plane electromagnetic waves is derived using the invariant imbedding concept. The equation is solved approximately and the character of the solution is investigated. Also, an equation for the radiation transmitted through such a slab is similarly obtained. An alternative formulation of the slab problem is presented which is applicable to the analogous problem in spherical geometry. The form of an equation for the modal reflections from a nonrelativistically expanding sphere is obtained and some salient features of the solution are described. In all cases the material is assumed to be a nondispersive, nonmagnetic dielectric whose rest frame properties are slowly varying.

A theory of strong and weak scintillations with applications to astrophysics

The propagation of waves in an extended, irregular medium is studied under the "quasi-optics" and the "Markov random process" approximations. Under these assumptions, a Fokker-Planck equation satisfied by the characteristic functional of the random wave field is derived. A complete set of the moment equations with different transverse coordinates and different wavenumbers is then obtained from the characteristic functional. The derivation does not require Gaussian statistics of the random medium and the result can be applied to the time-dependent problem. We then solve the moment equations for the phase correlation function, angular broadening, temporal pulse smearing, intensity correlation function, and the probability distribution of the random waves. The necessary and sufficient conditions for strong scintillation are also given.

We also consider the problem of diffraction of waves by a random, phase-changing screen. The intensity correlation function is solved in the whole Fresnel diffraction region and the temporal pulse broadening function is derived rigorously from the wave equation.

The method of smooth perturbations is applied to interplanetary scintillations. We formulate and calculate the effects of the solar-wind velocity fluctuations on the observed intensity power spectrum and on the ratio of the observed "pattern" velocity and the true velocity of the solar wind in the three-dimensional spherical model. The r.m.s. solar-wind velocity fluctuations are found to be ~200 km/sec in the region about 20 solar radii from the Sun.

We then interpret the observed interstellar scintillation data using the theories derived under the Markov approximation, which are also valid for the strong scintillation. We find that the Kolmogorov power-law spectrum with an outer scale of 10 to 100 pc fits the scintillation data and that the ambient averaged electron density in the interstellar medium is about 0.025 cm^-3. It is also found that there exists a region of strong electron density fluctuation with thickness ~10 pc and mean electron density ~7 cm^-3 between the PSR 0833-45 pulsar and the earth.

Ecological study of the River Lambourn. Report for the period October 1970 to March 1972

A small investigation of the ecology of the river Lambourn was carried out during 1967-70. During this period it became clear that detailed and reliable information could only be obtained by a much larger investigation. The present study was planned to meet the minimum requirements. No pumping is expected during this period and, since the pumping carried out during the pilot scheme was on a small scale, it is reasonable to assume that the river is still relatively unaffected by pumping.The present investigation has two main objectives both of which depend on obtaining a detailed picture of the ecology of the river at the present time. First, it will provide basic information on the state of the river prior to the development of the pumping scheme which will be available for comparison at any later date. Secondly, it may be possible to use some of the data to predict the ecological changes which may occur if the flow of the river is altered by the pumping scheme. This report is part of a series of fives studies on the River Lambourn which were undertaken between 1970 and 1979. [PDF contains 83 pages]

Ecological study of the River Lambourn. Report for the period October 1970 to September 1973

Few detailed studies have been made on the ecology of the chalk streams. A complex community of plants and animals is present and much more information is required to achieve an understanding of the requirements and interactions of all the species. It is important that the rivers affected by this scheme should be studied and kept under continued observation so that any effects produced by the scheme can be detected. This need has been recognised by the Thames Conservancy and the Water Resources Board who jointly sponsored this investigation on one of the chalk streams involved in the scheme. The present investigation has two main objectives both of which depend on obtaining a detailed picture of the ecology of the river at the present time. First, it will provide basic information on the state of the river prior to the development of the pumping scheme which will be available for comparison at any later date. Secondly, it may be possible to use some of the data to predict the ecological changes which may occur if the flow of the river is altered by the pumping scheme. The study covers suspended solids, invertebrates, fish.This report is part of a series of fives studies on the River Lambourn which were undertaken between 1970 and 1979. [PDF contains 24 pages]

Ecological study of chalk and limestone streams. Annual Report October 1974 to September 1975

Few detailed studies have been made on the ecology of the chalk streams. A complex community of plants and animals is present and much more information is required to achieve an understanding of the requirements and interactions of all the species. It is important that the rivers affected by this scheme should be studied and kept under continued observation so that any effects produced by the scheme can be detected. The report gives a brief synopsis of work carried out during the second year of a four year ecological study sponsored jointly by the Thames Water Authority and the Central Water Planning Unit. It assumes some familiarity with the investigations carried out on the River Lambourn during the preceding three years which was sponsored jointly by the Thames Conservancy and Water Resources Board (immediate predecessors of the present sponsoring organisations). (PDF contains 31 pages)

Ecological study of chalk and limestone stream. Annual report October 1975 to September 1976

Few detailed studies have been made on the ecology of the chalk streams. A complex community of plants and animals is present and much more information is required to achieve an understanding of the requirements and interactions of all the species. It is important that the rivers affected by this scheme should be studied and kept under continued observation so that any effects produced by the scheme can be detected. The report gives a brief synopsis of work carried out during the third year of a four year ecological study sponsored jointly by the Thames Water Authority and the Central Water Planning Unit. It assumes . It assumes some familiarity with the investigations carried out on the River Lambourn during the preceding three years which was sponsored jointly by the Thames Conservancy and Water Resources Board (immediate predecessors of the present sponsoring organisations). (PDF contains 35 pages)

Ecological study of chalk streams. Lambourn project. Interim report October 1977 to April 1979

Few detailed studies have been made on the ecology of the chalk streams. A complex community of plants and animals is present and much more information is required to achieve an understanding of the requirements and interactions of all the species. It is important that the rivers affected by this scheme should be studied and kept under continued observation so that any effects produced by the scheme can be detected. The report gives a synopsis of work carried out between 1971 and 1979 focusing on the present phase 1978-1979. It assumes some familiarity with the investigations carried out on the River Lambourn during the preceding years. The aims of the present phase of the project may be divided into two broad aspects. The first involves collecting further information in the field and includes three objectives: a continuation of studies on the Lambourn sites at Bagnor; comparative studies on other chalk streams; and a comparative study on a limestone stream. The second involves detailed analyses of data previously collected to document the recovery of the Lambourn from operational pumping and to attempt to develop simple conceptual and predictive models applicable over a wide range of physical and geographical variables. (PDF contains 43 pages)

Ecological study of the River Lambourn. Report for the period October 1973 to September 1977

The investigations described in this report were carried out to fulfill three distinct but interrelated objectives. In 1973 the Thames Conservancy were making plans for a second stage of their groundwater scheme which would take water from the chalk aquifers in the valley of the Kennet and they wanted basic information on the ecological state of this river and its upper tributaries. Little appeared to be known about limestone streams and a preliminary study of one of the streams in this area was desirable as a basis for planning more detailed studies if these were needed later. At a progress meeting held in March 1976 the problems and opportunities presented by the developing drought conditions were considered. It was concluded that the ecological effects of the exceptionally low natural flows should be studied and that it would be important to assess the ecological impact of the groundwater scheme if it was brought into operation that year. This could only be done on the Lambourn and the Winterbourne and it was decided that considerable effort should be diverted there for this purpose and that the field observations should be extended to cover any recovery period after the end of the drought. To make this possible it was agreed that the studies of invertebrates and detritus on the Kennet should be reduced considerably and that the proposed study of the limestone stream should be abandoned. The revised objectives were as follows: A detailed ecological study of several sites on the Kennet and its tributaries above Kintbury, extending over at least two years and involving observations on wate r weeds , invertebrates, fish, detritus and the trophic relationships within the river community. Quantitativ e and qualitative sampling of water weeds and invertebrates during one year at a number of sites on several chalk streams to determine whether the patterns and relationships found in the Lambourn are also found at the other sites. Observations on the Lambourn at Bagnor were to continue for most of the period to look for long-term fluctuations and to enable these sites to act as controls with which the other sites could be compared. Further detailed studies on the Lambourn and the Winterbourne to assess the impact of low flows, trial pumping and the operation of the groundwater scheme.

Ecological effects of trial pumpings in the Upper Lambourn: Stage 1 group pumping test Upper Lambourn group, September - November 1975

This report to the Thames Water Authority and Central Water Planning Unit is on research carried out in conjunction with the Stage 1 Group Pumping Test of five boreholes in the upper Lambourn Group for a period of three months in September, October and November 1975. The aim of the study was to assess the ecological effects of the pumpin g of five bore-holes in the upper Lambourn. That is, to determine how the seasonal sequence of ecological events in the river differed from what would hav e occurred had no pumping taken place. Since this 'experiment' has no control it is not possible to make a direct assessment. Nevertheless, by careful monitoring of ecological events before, during and after the pumping it is possible to document changes in th e river and by reference to the data already available for the Rive r Lambourn, normal seasonal changes in the flora and fauna can be separated from changes which may be attributable to the pumping and subsequent events.

The resolution of the thermodynamic paradox and the theory of guided wave propagation in anisotropic media

The resolution of the so-called thermodynamic paradox is presented in this paper. It is shown, in direct contradiction to the results of several previously published papers, that the cutoff modes (evanescent modes having complex propagation constants) can carry power in a waveguide containing ferrite. The errors in all previous “proofs” which purport to show that the cutoff modes cannot carry power are uncovered. The boundary value problem underlying the paradox is studied in detail; it is shown that, although the solution is somewhat complicated, there is nothing paradoxical about it.

The general problem of electromagnetic wave propagation through rectangular guides filled inhomogeneously in cross-section with transversely magnetized ferrite is also studied. Application of the standard waveguide techniques reduces the TM part to the well-known self-adjoint Sturm Liouville eigenvalue equation. The TE part, however, leads in general to a non-self-adjoint eigenvalue equation. This equation and the associated expansion problem are studied in detail. Expansion coefficients and actual fields are determined for a particular problem.

Ecological indicators of water quality in small rivers

At the present time hydrobiological indicators are widely used for the control of surface water quality. Results of the applying of methods suggested at the 1st Soviet-American seminar (1975), development of improved methods and estimation of their usefulness for various conditions are presented in this report. Among the criteria permitting an estimation of the degree and character of changes in water quality and their connection with the functioning of river ecosystems in general, the biological tests of natural waters appears to be the most universal one and is being carried out in two main directions — ecological and physiological. This study summarises approaches in both directions.