Maxwellian Velocity Distribution Function Research Articles

Analytical approximation of cross-section effects on charge exchange spectra observed in hot fusion plasmas

An analytical procedure is presented which enables a fast estimate of collision-energy-dependent cross-section effects on thermal charge exchange spectra. The model is based both on experimental evidence and numerical simulations showing that the observed charge exchange (CX) spectra are essentially Gaussian in their shape. The collision-energy-dependent emission rate leads effectively to a lineshift (apparent velocity), usually to a reduction in linewidth (apparent temperature), and to a change in the effective emission rate averaged over the entire thermal velocity distribution function. It is demonstrated that the cross-section effect can be treated analytically introducing an approximated emission rate factor which retains the characteristics of a Maxwellian velocity distribution function using an exponential expression with only linear and quadratic velocity terms in its exponent. An algebraic deconvolution procedure is described, which enables the reconstruction of true temperature, velocity and intensities from measured CX spectra. Examples taken from a recent JET experimental campaign are used to illustrate the cross-section effects on low-Z impurity CX spectra for a comprehensive variety of neutral beams (deuterium, tritium or helium), target densities, temperatures and toroidal rotation speeds. An overview is given of representative correction factors established for high-power, high-temperature plasmas, as well as for plasmas with combined neutral beam and radiofrequency heating, and for the case of locked modes.

Plane simple shear of smooth inelastic circular disks: the anisotropy of the second moment in the dilute and dense limits

We consider a plane, steady, homogeneous flow of circular disks. The disks are identical, smooth, and inelastic. We adopt the assumption of molecular chaos and introduce an anisotropic Maxwellian velocity distribution function based on the full second moment of the velocity fluctuations. In the limits of dilute and dense flows, we determine approximate analytic solutions of the balance law for the second moment that result in stresses whose qualitative behaviour and magnitudes are in good agreement with numerical simulations.

Fokker-Planck and Langevin equations for arbitrary slip velocities.

An expression for the Fokker-Planck equation governing the velocity distribution function of particles or heavy molecules immersed in a host light gas valid for arbitrary mean velocities of the heavy component is given. This expression generalizes previous results which were limited to small differences between the mean velocities of the heavy and light components compared with the thermal velocity of the light gas. The derivation assumes a Maxwellian velocity distribution function for the light gas, elastic heavy-light collisions, and makes use of integrals computed by Riesco-Chueca, Fern\'andez-Feria, and Fern\'andez de la Mora in Ref. 1. The stochastic Langevin equation associated with this Fokker-Planck collision operator is also obtained. More in general, we derive the Langevin equation corresponding to the general form of the Fokker-Planck collision operator, and particularize it to the present case.

Impedance of plane capacitor in a plasma with a resonance velocity distribution function

The RF impedance characteristics of a plane capacitor, which is immersed in a warm isotropic plasma, are investigated with a kinetic theory. An analytical solution is derived by assuming that the plasma has a resonance velocity distribution function which approximates a Maxwellian one. It is found that the results agree fairly well with those obtained numerically from the Maxwellian velocity distribution function.

Excitation of whistler mode instability by high energy electrons in the magnetosphere

Propagation of whistler mode waves through magnetospheric plasma characterised by a double Maxwellian velocity distribution function has been considered. The presence of a high energy electron tail in the composite velocity distribution function provides the needed free energy which excites instability mechanism for propagating whistler mode waves. The presence of the parallel or the antiparallel electric field governs the growth/decay rate variation with normalised parameter k ̃ = kV e ω H . These results are found to be different from those reported, which have used other forms of electron velocity distribution functions. The dependence of growth/decay rate on magnetospheric parameters plays an important role in the interpretation of wave features recorded by in-situ measurements. The presence of a fraction of high energy tail electrons penetrating through magnetospheric plasma exhibits significant change in the growth/decay rate of whistler mode waves.

Mathematical model of beam intensity dispersion at the detector plane in a molecular-beam apparatus

A mathematical model of intensity dispersion at the detector plane in a molecular-beam apparatus has been developed for the case of a ribbon-shaped beam effusing from a narrow rectangular slit. The model is developed in terms of position probability distribution functions and Maxwellian velocity distribution functions. The relation derived for the axial flux density is nv̄(Δx′)(Δx′)/4π2L2 in units of number of particles per unit area per unit time. A plot of flux density versus off-axis position agrees with the reported distributions obtained experimentally.

Line strengths, collision strengths and excitation rates for multiply-charged silicon ions

Line strengths, collision strengths and excitation rates have been calculated for a variety of transitions in multicharged silicon ions from Si(Vi) to Si(XIV). The collision strengths were evaluated in an LS coupling scheme in the distorted wave approximation neglecting exchange except for the helium-like transitions. Excitation rates were then obtained by integrating the collision strength over a Maxwellian velocity distribution function. These results are then described by a simple two-parameter fit for the rates.

Collision frequencies for use in the continuum momentum equations applied to the lower ionosphere

The collision frequencies for use in the continuum momentum equation are found for low energy electrons in N 2 and O 2 and for the polarization interaction between ions and neutral molecules; consideration is given to the case of low-frequency transport phenomena in a weakly ionized ionospheric plasma. For the electrons, the method of assuming a displaced Maxwellian velocity distribution function is compared with the method of obtaining transport coefficients from the linearized Boltzmann equation. It is found that the method of assuming a displaced Maxwellian distribution for the electrons is not a good approximation, and reasons why this is so are given.

FLOW OF RAREFIED GASES THROUGH PACKED BEDS OF PARTICLES

An analytical expression for the permeability of gases through packed beds of spherical particles in the Knudsen region is obtained from the kinetic theory of gases by using the two-sided Maxwellian velocity distribution functions and Maxwell''s transport equation. The results of the theoretical calculations are in good agreement with the experimental results for beds of spherical particles and also for beds of nonspherical particles, if the equivalent diameters of the particles are properly determined. For other regions, it was found convenient to introduce the Knudsen number, defined as the ratio of the number of molecule-particle collisions to that of intermolecular collisions. By using this Knudsen number, the mechanisms of flow of gases through packed beds can be successfully classified just as are those for circular tubes.

ISOBAR1C DIFFUSION OF GASES IN PACKED BEDS OF FINE PARTICLES

Diffusion of gases through packed beds of particles under uniform total pressure is studied analytically by using the rigorous kinetic theory of gases with the two-sided Maxwellian velocity distribution functions and Maxwell''s transport equation. Attention is paid to the solid parts of the packed beds, although most investigators have concentrated on the space structures to obtain a pore model. With some abbreviations the present results are quite similar to those of Evans et al., which were derived from the macroscopic multicomponent diffusion equation by considering the particles as giant molecules. But in the Knudsen diffusion region their results are about 2.4 times larger than the authors'' or the experimental results. The present treatment is considered to be more adequate and general than that of Evans et al. and others. To obtain experimental confirmation of the analytical results a simple method for measurement of the effective diffusivities and flow permeabilities of gases through porous media is proposed by using the wellknown relation, -NA/NB=(MB/MA)1/2, which is also derived from the present treatment. The experimental results by this method are in relatively good agreement with the analytical ones obtained here. This is perhaps the first direct experimental confirmation of the dependency of the mean effective diffusivities of gases through porous media on composition.

Mean Free Path of Emitted Molecules and Correlation of Sphere Drag Data

The mean free path between molecules emitted from and incident to a surface in rarefied flow is considered. The problem is formulated for the case of two classes of hard sphere molecules with different Maxwellian velocity distribution functions. Approximate analytical solutions are obtained to the integral describing the collision frequency between emitted and incident molecules. These analytical solutions are compared with numerical solutions, and the domain of validity of the analytical results is established. The analytical solutions provide a simple expression for the mean free path which is used to define a correlation parameter. This parameter is used to correlate a great deal of low-density supersonic and hypersonic sphere drag data which have a wide range of freestream and body surface conditions.

Nonlinear Wave Modulation in a Magnetized Collisionless Plasma

Nonlinear modulation of quasi-monochromatic electromagnetic waves propagating parallel to an external magnetic field is investigated with particular attention to contributions of resonant particles at the group velocity. The contributions of these resonant particles give rise to a nonlocal-nonlinear tram and modify the local-nonlinear term of the nonlinear schrodinger equation. An explicit expression of a coefficient of the nonlocal-nonlinear term is given for a plasma characterized by isotropic Maxwellian velocity distribution functions. In a vanishing temperature limit this coefficient becomes zero and the nonlinear Schrodinger equation agrees with that obtained by using a fluid model.

Conversion of Electrostatic Plasma Waves into Electromagnetic Waves: Numerical Calculation of the Dispersion Relation for All Wavelengths

The dispersion curves have been computed for a wide range of wavelengths from electromagnetic waves to electrostatic waves in a magnetoactive warm plasma with a Maxwellian velocity distribution function. The computation was carried out mainly for the perpendicular propagation mode. The upper hybrid resonance is the connection point of the electrostatic waves and the electromagnetic waves. The electrostatic waves not associated with the upper hybrid resonance are subjected to electron cyclotron damping when the wavelength becomes long. Oblique propagation is allowed for the electrostatic waves in a frequency range from the plasma frequency to the upper hybrid resonance frequency in the long‐wavelength region where Landau damping can be neglected and where the electrostatic mode smoothly connects to the electromagnetic X‐mode. In a slightly inhomogeneous plasma, the Bernstein‐mode electrostatic wave can escape by being converted into the O‐mode electromagnetic wave; two reflections take place during this escape process. The frequency range of the escape coincides with the Cerenkov radiation of the electromagnetic waves, but these two mechanisms indicate clear contrast in its origin. The electromagnetic Cerenkov radiation is produced interacting with the high‐energy plasma beam with velocity range l00υth < υ < l000υth (for the nonrelativisitic plasma), where υth is the electron thermal velocity, whereas the origin of the electrostatic waves is in the interaction with the low‐energy plasma beam.

Apparent Frequency Dependence of the Microwave Power Radiated from Gas Discharge Noise Sources (Correspondence)

Historically it has been assumed that the microwave power per unit bandwidth radiated from the plasma of typical argon and neon gas discharge noise sources is frequency independent. There are now both a priori basis and experimental evidence for questioning this assumption. For such frequency independence to exist over a frequency range of 100 MHz to 100 GHz the following conditions must be met: (1) /spl planck/ /spl omega/ / kT<<1; (2) proper matching of the discharge to the guide; (3) an appropriate level of absorptivity of the plasma for these frequencies; (4) a Maxwellian electron velocity distribution function; and (5) absence of collective oscillations. The first three conditions are shown to be met. The fourth condition involves the concept of a radiation temperature. Because the appropriate electron velocity distributions are not Maxwellian this radiation temperature does depend somewhat on frequency. This dependence is limited to a small region around the electron-atom collision frequency. For the discharges of concern it is shown that these collision frequencies are 10/sup 9/ to 10/sup 10/s/sup -1/ and therefore contained in the frequency range of interest. The fifth condition relates to the fourth and recently has been shown in theory to be untrue. Finally there is now available experimental evidence which appears to show a small (0.2-0.3 dB) variation of excess noise ratio with frequency on existing tubes. Suggestions for future experiments and interim considerations are made.

SOME CONSIDERATIONS OF MASS TRANSFER UNDER VACUUM

Mass transfer between infinite parallel plates under vacuum was treated by analyzing the Boltzmann equation with the two-sided Maxwellian velocity distribution function of molecules. The equation obtained from this analysis coincides completely with that derived from the series-resistance model'' of the resistances of evaporation, condensation and diffusion. The results were compared with the theoretical results obtained by Uyeha et al. in 1960.

Effect of magnetic perturbation on the magneto-active plasma

The effect of magnetic perturbation on the magneto-active plasma has been considered. The collisionless Boltzmann transport equation has been solved for an assumed perturbation in the Maxwellian velocity distribution function. An expression for energy transfer has been derived in terms of perturbed distribution function. Using the magnitude of perturbation for certain astrophysical and laboratory conditions, the heating or cooling of the magneto-active plasma has been studied.

Constitutive Relationships For InhomogeneousDense Rapid Granular Flows Of Highly InelasticSpheres

Constitutive relationships are developed in this present study for inhoinogeneous dense rapid granular flows of identical smooth highly inelastic spheres. The important mean fields in this flow are the solid fraction, mean velocity, and full second moment of fluctuation velocity. By extending constitutive theories for homogeneous granular flows of highly inelastic spheres by Richman Chou, and Chou & Richman^, the constitutive relationships, which consist of collisional source of second moment, collisional flux of second moment, and collisional flux of momentum, are based upon an anisotropic Maxwellian velocity distribution function. To apply this constitutive theory to inhomogeneous flows of highly inelastic spheres would require appropriate boundary conditions that ensure the balance of momentum and second moment at the boundary.

Bulk Negative Differential Conductivity in Stressed n‐Type Germanium

AbstractThe velocity‐field characteristics of stressed n‐type germanium at 300 °K have been theoretically obtained assuming a Maxwellian velocity distribution function. It is found that the experimentally observed bulk negative differential conductivity cannot be explained if the carriers are assumed to be confined to the 〈111〉 valleys only. The experimental results on the other hand may be fitted if carrier scattering to the 〈100〉 valleys is taken into account and the mobility of the carriers in these valleys is taken to be 690 cm2 V−1 s−1.

Propagation of microwaves through a magneto-plasma, and a possible method for determining the electron velocity distributions

Sagdeyev and Shafranov have shown that the absorption of microwaves in a hot plasma in a steady magnetic field can be calculated in simple closed form with the help of theBoltzmann equation, provided that the effect of collision can be ignored. The special case of propagation of circularly polarized waves parallel to the magnetic field, and the extraordinary ray, in magneto-ionic terminology, were studied. It is shown that the formula given by Sagdeyev and Shafranov for this case can be deduced by considering the motions of individual electrons by elementary dynamical methods, using the concepts of Doppler shift and velocity distribution functions to obtain a macroscopic conductivity formula for a high-temperature plasma. From this, the absorption is calculated. It is emphasized that the calculation in no way depends upon the assumption of a Maxwellian velocity distribution function. The absorption can in fact be obtained in closed form for any arbitrary velocity distribution function. This suggests that a diagnostic technique for the determination of velocity distribution could be based on measurements of absorption of the extraordinary ray, and the potentialities and limitations of this proposal are briefly discussed. (auth)