Ambipolar Diffusion Theory Research Articles

Ambipolar diffusion revisited

An asymptotic theory of ambipolar diffusion (AD) in slightly ionised bounded plasmas, based on utilisation of the small parameter in =rD2/L2(rD is the Debye length and L is a characteristic dimension of the system) and employing general assumptions about the geometry of the system and about the rates of elementary processes, is presented in an easily accessible way. This theory leads in some cases (e.g. in the problem of the Hall effect in diffusion-controlled bounded plasmas) to results which differ essentially from those following from the elementary theory of AD due to Schottky. The problem of spreading of inhomogeneities in an unbounded plasma is also discussed in the framework of this asymptotic theory for a realistic and an exactly solvable case allowing one to formulate a self-similar solution not employing the ambipolar diffusion approximation.

Clarification of some concepts in chemical diffusion, or Darken, Kirkendall, and other sources of difficulties and confusion in diffusion

The assumptions of Kirkendall interdiffusion theory are reviewed and it is shown that this theory does not apply to chemical diffusion experiments in which the stoichiometry changes, but one sublattice is inert. We then summarize the theory of ambipolar chemical diffusion and apply it to the examples CoO, UO 2, CdS, YSZ, and the system BSi. For UO 2 we reevaluate recently published calculations and show that the correct theory gives better agreement with experiments.

Nonneutral and quasi‐neutral diffusion of weakly ionized multiconstituent plasma

A theory is developed for ambipolar diffusion in a multiconstituent weakly ionized plasma. The implications of the presence of negative ions and of differing positive ion and negative ion diffusion coefficients are found. In particular, the presence of numerous negative ions is found to enhance the diffusion of the electrons. Furthermore, this theory of multiconstituent ambipolar diffusion is extended to include the nonneutral case in which the scale length of the inhomogeneity is smaller than or of the order of the electron Debye length.

Ray‐tracing synthesis of HF radar signatures from Gaussian plasma clouds

HF radar signatures have been synthesized from a time‐dependent Gaussian ion cloud using a three‐dimensional ray‐tracing progam that can take into account both electron collisions and the earth's magnetic field. The ion cloud model is based on the theory of ambipolar diffusion and the model parameters are based upon the observations made during a barium cloud test conducted at White Sands, New Mexico. Ray‐tracing results have been generated for both ordinary (O) and extraordinary(X) modes as well as for the “no‐field” case. The results, presented in the form of a time‐history of the Doppler shift and the scattering cross section, are found to be quite similar for all three cases. Relative to the O mode, the X‐mode variations lag in time and obtain a maximum cross section which is greater only by about 2 db. The variations for the no‐field case fall in between that of the O and X, but closer to the O mode. The results show that the cloud is extremely defocussing with an estimated defocussing factor (relative to free space) for the X mode increasing from 7.6 × 103 to 1.85 × 107 over the interval 5 to 190 sec of cloud's life. The ray‐tracing signatures are compared to those synthesized by means of a hard expanding ellipsoid method with a view to evaluating the importance of underlying ionization. The effects are found to be such as to introduce a significant time lag in Doppler history and to cause a cross section reduction by a factor that builds up with time from 5 db to more than 30 db. A comparison drawn between the observed and the ray‐tracing signatures leads to the conclusion that good agreement between the two can be expected only when the clouds experience either slow or no deformation and striation but not otherwise.

Theoretical investigation of the first order magnetic self-striction in a low pressure mercury vapor arc with remote walls

The conditions of magnetic self-striction in an arc of finite length, not confined laterally by walls, are investigated with a recently developed theory of ambipolar diffusion in a magnetic field, including all first order corrections. Indications are given of how numerical predictions can be made; the case of mercury vapor is treated as an example, using earlier measurements of cylindrical arcs.

The electric field in the solar coronal exosphere and the solar wind

The solar coronal exosphere (fixed at a level where the mean free path is comparable with the scale height) is shown to act like the sheath in a laboratory discharge that builds up an excess positive charge and electric field. Equality of flux for the charged particles (protons and electrons) at the exosphere is assumed. The exospheric electric potential (for a coronal temperature of 10 6 ° K ) is found to be about 325 V, which makes the protons virtually weightless. The exospheric sheath potential accelerates the thermal protons to a velocity of 258 km/sec with which they coast to the earth. The proton density in transit varies approximately as the inverse square law, and is 3 cm -3 near the earth. Higher velocities and densities are obtained for higher coronal temperatures. An estimate of the exospheric transition region is obtained by applying the hydrodynamic theory of ambipolar diffusion in a gravitational field to the “observed” coronal density distribution. It is found that the ratio of the electric to the gravitational force on the proton builds up from its hydrostatic value of 1 2 at 2·4R ⊚ to 1 at 8R ⊚ from the solar center.

Steady-State Theory of an Intermediate-Pressure Discharge Column in a Magnetic Field

A steady-state theory of a discharge column in a magnetic field is presented which covers the intermediate pressure range where the ion mean free path is neither much greater than, nor much less than, the discharge radius. With increasing pressure and magnetic field, its predictions tend to those obtainable from ambipolar diffusion theory. With decreasing pressure they approximate those of low-pressure discharge theory based on the use of the third-moment equation for the ions. The analysis in this paper is based on continuity and momentum-transfer equations for the electrons and ions, but in contrast to linear ambipolar diffusion theory, the nonlinear ion inertia term is retained. In the plasma approximation, retention of this term gives rise to a plasma-sheath boundary, where the density is nonzero and the potential finite, located where the ambipolar drift velocity reaches the isothermal sound speed [k(Te+Ti)/mi]1/2. Numerical solutions are presented for the profiles of density, velocity, and potential, for planar and cylindrical discharges. These indicate that potential inversion can occur, and that under such conditions the potential rises to a maximum and then decreases towards the plasma-sheath boundary.

Ambipolar Diffusion of Free Carriers in Insulating CdS Crystals

A new method is described for measuring ambipolar diffusion of free carriers in insulating materials. Strongly absorbed light producing electron-hole pairs is shone on one surface of the material through a transparent noninjecting contact. These carrier pairs set up a concentration gradient extending into the interior due to ambipolar diffusion. To describe this process, the theory of ambipolar diffusion is extended to the insulator case of unequal lifetimes for electrons and holes and significant trapping of both types of carriers. Simultaneously, excess space charge of the same sign as the majority photocarriers is injected through an Ohmic contact on the opposite face and space-charge-limited current (SCLC) flow is established. The point of equality of these two charge distributions establishes an effective thickness ${x}_{0}$. To a good approximation, ${x}_{0}$ can be inserted into the SCLC density formula which, for shallow trapping, is $J=\frac{K{V}^{2}}{{{x}_{0}}^{3}}$ where $K$ is a constant. The thickness ${x}_{0}$ depends on the illumination intensity and therefore introduces an intensity dependence into the SCLC formula. A technique is described with which the ambipolar diffusion length ${L}_{a}$ of free carriers can be calculated from this intensity dependence. The procedure was tested with insulating CdS crystals. The results show that ${L}_{a}\ensuremath{\approx}1\ensuremath{\mu}$ at room temperature from which the minority carrier (hole) recombination lifetime ${\ensuremath{\tau}}_{p}\ensuremath{\approx}5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$ sec is obtained. These values are in good agreement with earlier estimates.

Nonlinear Ambipolar Diffusion of an Isothermal Plasma across a Magnetic Field

Solutions are presented of the first two moment equations, including nonlinear terms, for the ambipolar diffusion of an isothermal plasma across a magnetic field. The two geometries considered are the plane parallel case and the infinite cylinder with axial symmetry. The Bohm criterion is automatically satisfied by the solutions. It is shown that the singularity in the space derivative of the ambipolar drift velocity at the plasma boundary cannot be removed by an axial magnetic field of any strength. Thus the plasma drift velocity and the plasma density remain monotonic functions of the position coordinate. It is also shown, under the assumptions of this theory, that the ambipolar space charge field is always directed outward and does not reverse direction in this isothermal approximation even for extremely high magnetic fields. One is forced to conclude that a realistic theory of ambipolar diffusion requires the consideration of thermal gradients within the plasma.

Plasma Diffusion in a Magnetic Field

The two-dimensional diffusion of a partially ionized plasma in a magnetic field is studied using the macroscopic plasma equations. For a stable steady-state plasma with uniform electron temperature and ionization frequency, it is shown that the density distribution can exist only in the lowest mode by virtue of the internal constraint that the space charge potential is conservative. While the electron and ion diffusion currents in an insulating cavity are everywhere equal, similar to Schottky's ambipolar diffusion theory for one dimension, the plasma in a conducting cavity exhibits an increase in the net particle current to the walls and an unbalance of ion and electron currents in the plasma. Thus an amperian current flows through the plasma and returns through the walls. With adjustable dc potentials applied to the walls of a split cavity, these currents may be continuously varied and even decreased below those given by the ambipolar condition. An experiment with a microwave cavity discharge demonstrates the existence of these unbalanced currents.

Ambipolar Diffusion of a Plasma in a Weak Magnetic Field

A generalization of the phenomenological description of ambipolar diffusion of a plasma is made to include the effects of a weak impressed magnetic field. The assumptions of congruence of the flow vectors (div Γ+ = div Γ−) for ions and electrons and of proportionality [n−grad (n+) = n+ grad (n−)] on the signed charge densities are utilized as in the existing theory of ambipolar diffusion; however, because of the rotational effects of the magnetic field, and additional assumption is made on the curls of Γ±. This assumption is curl Γ+ = curl Γ−. The resulting diffusion is found to be a combination of electronic and ionic diffusion, and the components of the ambipolar diffusion tensor are calculated for a unidirectional magnetic field.

Ambipolar Diffusion In The Magnetic Field

Simon's “short circuit” theory is criticized, which states that the diffusion rate of plasma across the longitudinal magnetic field is larger than that expected from the classical ambipolar diffusion theory, when the ratio of the tube dimension of the field parallel dierction to that of the field parpendicular dierction is rather small.A new measuring method for determining the diffusion rate across the magnetic field quantitatively is introduced to check the short circuit theory.Within the authors' considerations, there is no diffusion mechanism due to the short circuit effect, and the diffusion is ambipolar independent of the tube dimension.

XXXVII. Helix Type Gaseous Discharge Noise Sources at Low Plasma Densities†

ABSTRACT The behaviour of the positive column of a gaseous discharge in helium within a helical line as a noise source is investigated in the range of relatively low plasma densities. The use of a helical line for establishing the h.f. electric field increases considerably the possibilities for the application of the gaseous discharge noise sources and, at the same time. makes it possible to evaluate in advance the optimum design of such devices. To check the derived formulae one only has to measure the direct current conductance of the positive column, the pressure and the effective phase constant of the helical line. Good agreement was found with the measured attenuations depending upon discharge current and pressure. The measured noise figure of the plasma depending upon pressure however resulted in too low a value, giving a maximum deviation of 29% from the values predicted by the ambipolar diffusion theory of the positive column.

Oscillating Glow Discharge Plasma

The plasma potential, electron temperature, and electron concentration have been measured as functions of both time and position in the positive column of an argon glow discharge with moving striations using a plane circular probe 0.25 mm in diameter that could be saturated in the electron collecting region. It is found that (1) the radial distribution of charge is that predicted by the ambipolar diffusion theory, (2) in each cycle, close to the time of the negative striation, the potential at all points in the positive column decreases sharply, (3) no increase in electron concentration occurs at the time of the negative striation. It is concluded that the positive striations move from anode to cathode as a result of ionization in front of the charge maximum combined with diffusion and drift of the charges radially and longitudinally. When the ionization function necessary to produce the observed striation motion is calculated from the observed charge concentrations, it is found to agree closely with the observed light emission of the discharge.

The Positive Column in a Longitudinal Magnetic Field

A magnetic field parallel to the axis of the positive column of a discharge is known to impart a rotary component on the radial motion of electrons. This reduces the flow of both positive ions and electrons to the wall, causing a corresponding decrease in the radial and longitudinal components of the electric field and in the electron temperature. It is shown here that under certain conditions Langmuir probes can be applied to explore a plasma in a magnetic field. A new method of measuring the concentration of charges is used: first relative values of ion concentration are obtained from the ion saturation currents in various magnetic fields. This result together with the electron saturation currents in zero magnetic field gives absolute values of concentration of charges in a magnetic field. Such measurements have been carried out in helium at low pressures. The effect of the magnetic field on the radial distribution of charges, the rate of ionization, the electric field, the electron temperature, the average energy loss in a collision, and the radial potential distribution is discussed. It is concluded that in zero magnetic field the Langmuir theory of free ion fall describes best the properties of the discharge plasma whereas in a magnetic field of sufficient strength Schottky's theory of ambipolar diffusion applies. When the gas becomes highly ionized, the effect of the partial pressure of the electron gas may become important so that finally both the longitudinal component of electric field and the electron temperature should become independent of the magnetic field. Previous observations of a low pressure positive column in caesium at high current densities seem to support this view.

Analysis of the Plasma of Fluorescent Lamps

An attempt has been made to explain the properties of the ``plasma'' of fluorescent lamps in considerable detail, using the ambipolar diffusion theory of Schottky, modified to include ionization by multiple electron impacts. Ionization is assumed to occur by electron collisions with excited and metastable atoms. The cross section for this process is one of two inaccurately known parameters, which we treat as adjustable constants, occurring in the calculations. The other is the effective diffusion constant for quanta of the resonance line, in the presence of a high-temperature electron gas. Qualitative agreement of the foregoing theory with experiment is demonstrated for all the derived quantities as functions of the several parameters of the discharge. The magnitudes of the calculated electron temperature, electric field, total light output, and efficiency are also in sufficiently good agreement with experimental values for the lamps of greatest commercial interest to permit extrapolations to be made with reasonable confidence. Optimum agreement of theory with experiment for T-12 lamps is obtained using a mean life time of ultraviolet photons in the discharge 3.6 times that employed by Kenty for diffusion in un-ionized vapor, and with a value 3.3 for the ratio of cross section for ionization of excited state to that of the ground state.

Diffusion Coefficients from the Rate of Decay of Meteor Trails

The effective diffusion coefficient for a meteor trail is calculated from the theory of ambipolar diffusion and the physical constants of the upper atmosphere. The absolute value of the diffusion coefficient so calculated, and also its gradient with height, are confirmed by measurement of the rates of decay of a large number of meteor echoes of known heights.

Experiments on the Behavior of an Ionized Gas in a Magnetic Field

Measurements with probes in a plasma of helium ions and electrons in the afterglow of an induction-excited discharge in a toroidal tube with a toroidal magnetic field reveal an oscillatory probe current which is presumably indicative of fluctuations in the ion density. The space distribution of these fluctuations and the dependency of their amplitude and frequency on the pressure and magnetic field are consistent with the assumption that plasma waves, similar to the magnetohydrodynamic type described by Alfv\`en, exist in this toroid. Measurements of electron density by a microwave method in another toroidal tube, which is in the form of a microwave resonant cavity, show that in this tube the degree of diffusion control in helium at low pressures is apparently orders of magnitude less than one would calculate from the classical theory of ambipolar diffusion of ions and electrons in a magnetic field. The measurements show a marked minimum in the diffusion coefficient as a function of magnetic field at about 600 gauss. An attempt is made to explain this minimum and the lack of diffusion control in terms of plasma waves. Experimental evidence that plasma waves exist in this toroid in a magnetic field has been seen in oscillations which appear on the rf envelope of the exciting pulse from the magnetron.

Sub-normal discharges

When very small currents are passed through mercury vapour in cylindrical tubes, the characteristic current versus voltage gradient curves for the positive column have negative slope. A previous analysis of conditions in the column based on ambipolar diffusion theory is restated and extended somewhat in terms of plasma-sheath theory. It appears that the plasma-sheath boundary region of high current discharges expands to fill the cross-section when the current is low. Probe experiments on positive columns in mercury vapour and other gases indicate the probable presence of positive space charges throughout the tube when the current density is small. It is suggested that this may be a common property of many discharges with small current density. References are given to literature on the subject.

Probe Studies of Energy Distributions and Radial Potential Variations in a Low Pressure Mercury Arc

Probe measurements were made in the plasma of a low pressure mercury arc. The electron-energy distributions showed depletions from a Maxwellian distribution in the high energy range. Coupling effects between adjacent probes were investigated and were found to be quite small but in the proper direction to agree with the Langmuir-Tonks theory. Drift-current distortion of the random electron-energy distributions was measured with a bidirectional probe and compared with theory. A multisection probe extending from tube axis to tube wall allowed a determination of radial potential and density variations. Results over a pressure range from 3.4 microns to 35 microns showed good agreement with the ambipolar diffusion theory based on cumulative ionization. A direct calculation of ionization rate in the plasma was made from the ionization probability for a one-step ionizing process; comparison of this calculation with the observed ionization rate at 1.7 microns indicated that at that pressure the ionization is half direct, half cumulative. For higher arc pressures cumulative ionization evidently predominates.