General Form for the Electron Distribution Function for an Inhomogeneous Fully Ionized and Magnetized Plasma Interacting with an Alternating Electric Field

It is the purpose of this paper to give a general form for the isotropic and vectorial part of the distribution function for electrons, in a fully ionized and magnetized plasma, interacting with an alternating electric field. By making the assumption, that the ions have an equilibrium distribution, we obtain the needed solutions as series with respect to time and complex exponentials. Also have indicated a new manner in which the solution may be obtained with any precission we want. A general form for density current is obtained. It should be denoted that these results are obtained, without being done any previous assumption on the temporal dependence of the distribution function.     

Stoßbestimmte Relaxation des Elektronenensembles im Plasma bei zusätzlicher Aufheizung durch ein elektrisches Feld. III. Das periodische Verhalten der Elektronenkomponente im elektrischen Feld großen Modulationsgrades

Collision Dominated Relaxation of the Electron Ensemble in a Plasma with Additional Heating by an Electric Field. III. The Periodic Behaviour of the Electron Component in an Electric Field with a Large Modulation Amplitude With the aid of the non-stationary Boltzmann-equation the periodic behaviour of the isotropic part of the velocity distribution of electrons and thereby determined macroscopic quantities is calculated for periodic electric fields with large modulation amplitude. The investigations concern a weakly ionized column plasma in neon under typical low and medium pressure conditions. Based on the numerical results for typical ranges of field strength and cycle times of the electric field a qualitative physical interpretation for the periodic behaviour of the electron component is obtained. The introduction of special field-dependent adjustment times allows the formalution of conditions which characterize the case of quasi-stationary behaviour and also the case of small amplitudes of modulation in the macroscopic quantities determined by the isotropic distribution function. The periodic states between these two limiting cases can be interpreted as due to two competing processes. The first one is the energy input controlled by the electric field and the second one is the energy loss in binary collisions of the electrons with the atoms.     

Zum Verhalten des Elektronenensembles im anisothermen schwachionisierten Plasma bei Einwirkung eines hochfrequenten elektrischen Feldes

Investigations Concerning the Electron-ensemble in a Nonisothermal Low Ionized Plasma under the Influence of a High Frequency Electric Field The adjustment and the periodic behaviour of the electron-ensemble under the influence of a high frequency electric field were investigated in a nonisothermal low ionized plasma. By direct numerical solution of the nonstationary Boltzmann equation the time behaviour of some macroscopic quantities of the electrons was determined via the calculation of the isotropic distribution function for a neon-plasma. An interpretation of the obtained results is given. The investigations were performed within a range of frequencies of the electric field, where the time behaviour of the isotropic distribution function is essential effected as well as by the electric a.c. field and the collision processes, whilst the alteration of the anisotropic part of the velocity distribution function takes place in a quasistationary way.     

Die Relaxation der Geschwindigkeitsverteilungsfunktion der Elektronen im homogenen elektrischen Feld mit elastischen und anregenden Stößen. II. Das Verhalten des Elektronenensembles bei aperiodischer Feldänderung zwischen zwei stationären Zuständen

Recently the relaxation of the electron component was investigated under field-free conditions after sudden switch-off the electric field and otherwise after sudden changing the electric field to a new value of field strength. To continue these papers we consider now the relaxation process of the electron ensemble in the presence of a monotonous time variable electric field. The investigations are based on a computation of time change in the isotropic part of the velocity distribution function of the electrons and of the macroscopic parameters determined by the distribution function. The start-ing-point is the non-stationary Boltzmann equation with stationary initial states taking into consideration elastic and exciting collisions. Besides the representation of the received numerical results a physical interpretation is obtained for the duration of the whole relaxation process, for its initial stage as well as for the momentary stage of the relaxation by introduction of normalized characteristic time quantities. Further characterizing conditions are found related to the relaxation after quasi-jumplike change of the electric field and in the case of quasistationary field alteration respectively. With the introduced characteristic time quantities statements about the degree of realization of one of this limiting cases are possible for any given monotonous field. The investigations are per-formed in a low ionized non-thermal neon plasma.     

Analytical structure of an apertured vector Gaussian beam in the far field

Based on the angular spectrum representation of the Maxwell’s equations and the complex Gaussian expansion of the aperture function, the structure of an apertured vector Gaussian beam in the far field is presented in the integral form. By means of the method of stationary phase, the analytical vectorial structures are obtained. According to the analytical expressions, the characteristics of vectorial structure of an apertured Gaussian beam are investigated in the far field. The influence of a linearly polarized angle on the vectorial structure is also studied in the far field. This research provides a novel approach to further comprehend the vectorial property of an apertured Gaussian beam.     

Fundamentals of a Technique for Determining Electron Distribution Functions by Multi-Term Even-Order Expansion in Legendre Polynomials. 2. Comprehensive Investigation of a Model Plasma

Applying the new technique for finding the converged solution of the Boltzmann equation in a weakly ionized plasma, which was developed in the first part of this paper, a comprehensive study of the electron velocity distribution function for a model plasma with elastic and exciting collisions is performed by solving the Boltzmann equation with increasing order of approximation. The purpose of this investigation is that of calculating the isotropic distribution f₀, the first contribution f₁ to the anisotropy of the velocity distribution, the important macroscopic quantities and, more generally, that of studying the total anisotropy as well as the changes of all these quantities when the approximation degree is enlarged beyond the 2 terms of the conventional Lorentz approximation. By varying some parameters of the model plasma, that is the electric field strength, the magnitude of the excitation cross section and the excitation threshold, the main features of plasmas in inert as well as molecular gases are modelled and the impact of these parameters on the mentioned quantities is analysed. Some of the converged results are compared with results of corresponding Monte Carlo simulations. The approximation degree required to find the converged values of isotropic distribution, main macroscopic quantities and electron distribution in the velocity space (and thus its real anisotropy) is estimated by solving the Boltzmann equation over wide parameter ranges.     

Scattering from an elliptical cylindrical plasma for electromagnetic waves with wavelength much greater than the dimensions of the plasma cross-section

The distribution of the electric potential inside and outside a cold plasma column with elliptical cross-section when electromagnetic waves scatter from it are obtained. It is assumed that the wavelength of incident wave is much greater than the dimensions of the cross section of the plasma column. The electrical potential inside and outside a magnetized elliptical plasma column for an anisotropic scattering process are obtained. The graphs of the ratio of the electric field inside the plasma and the incident electric field versus the incident angle, the geometrical dimension of the elliptical plasma column and cyclotron frequency are presented separately. Finally, scattering of the same waves from an elliptical cylindrical inhomogeneous isotropic cold plasma is also investigated.     

Analytical vectorial structure of an apertured hollow Gaussian beam in the far field

Based on the vector angular-spectrum and the complex Gaussian expansion of the aperture function, the structure of an apertured hollow Gaussian beam in the far field is given in a integral form. In virtue of the method of stationary phase approximation, the analytical vectorial structures are derived. Starting from the analytical expressions, the propagation properties of apertured hollow Gaussian beams with different order in the far field are illustrated graphically. In addition, the influence of the truncation parameter on far field distribution is studied detailedly. This research can shed light on the further understanding of the vectorial property of an apertured hollow Gaussian beam.     

电偶极子位于均匀介质球中时球外电场的研究

采用分离变量法求解了电偶极子位于均匀介质球中时复连通域的拉普拉斯方程和泊松方程,求出了球内外两种不同介质的电势分布和球面上的极化电荷分布;通过求解二阶非线性微分方程得到了球外的电场线函数;利用计算软件Math-ematica 5.0,作出了相应的相互正交的等势线簇图形和电场线簇图形,并且进行了必要的讨论.     

Die Relaxation der Geschwindigkeitsverteilungsfunktion der Elektronen im homogenen Feld mit elastischen und anregenden Stößen. III. Die Einstellung des periodischen Verhaltens der Elektronen-komponente im elektrischen Feld großen Modulationsgrades

To continue the kinetic investigations concerning the periodic behaviour of the electron component of a low ionized Ne-plasma of a glow discharge column in a periodic electric field [4] with high degree of modulation in this paper the adjustment of the isotropic distribution function of the electrons and by the latter determined macroscopic quantities are calculated and discussed. Starting with the non-stationary Boltzmann equation the adjustment process is considered using different direct components of the field but nearly constant degree of modulation. In the whole range from a nearly momentary adjustment to an adjustment in the high frequency limit of only small modulated distribution function we obtained results concerning the adjustment behaviour and its characteristic times by variation of the cycle time of the electric field starting from the different stationary states. At the other hand the influence of the different degrees of modulation on the adjustment process and its characteristic times was investigated and interpreted. For this interpretation the spectrum of adjustment times of stationary states corresponding to all momentary values of the periodic electric field strength in one cycle plays a dominant role. From the investigations follows that the adjustment times of periodic states can alter their values by some orders of magnitudes in dependence on the parameters of the electric field.     

Radiation of an Electric Dipole from a Small Plasma Spheroid

We obtain and analyze an analytical solution of the boundary-value problem of the field of an electric dipole located in a small spheroid filled with a cold isotropic plasma. The conditions of resonant amplification of the field in vacuum are studied for various shapes and curvatures of the plasma spheroid. It is shown that for a certain spheroid shape, the resonant amplification increases several times compared with the case of spherical plasma environment. The obtained results are compared with the resonant radiation of a dipole from a narrow elliptical plasma cylinder.     

Interpretation of an exactly solvable two-component plasma

The interpretation of the exact calculation of the partition function and correlations of a two-component plasma obtained earlier is considered. The system has species of charge ratio 12 which are constrained to lie on a circle and interact via the two-dimensional Coulomb potential. By studying the exact results we gain an understanding of why the excess thermodynamic quantities of the two component system can be well approximated by the sum of the appropriate excess thermodynamic quantities of the one-component systems.     

The stationary confinement of plasma by a magnetic field at anisotropic pressure

The method of moments of the Boltzmann-Vlasov equation is used to derive relations between the macroscopic quantities of high-temperature plasma (i.e. the density of particles, the mass velocity and the generally anisotropic tensor of the kinetic pressure of particles) and the static electromagnetic field by which the plasma is stationarily confined. The well-known relation between the isotropic pressure of particles and the pressure of the magnetic field is then obtained from this as a special case. Plane geometric configuration of the plasma boundary and negligible collision interaction of the particles are assumed. The results are verified on a simple kinetic model of plasma with known trajectories of the particles.     

Calculation of the vectorial field distribution of an axicon illuminated by a linearly polarized Guassian beam

An analytical expression describing the vectorial field distribution of Gaussian light beams diffracted by an axicon is obtained. The theoretical analysis and simulation calculation show that for the linearly x-polarized light incident on an axicon, the y-component of the diffraction field is very small and the x-component dominates. The intensity of the z-component along the propagation direction is related with the open angle and index of axicon. The open angle plays the more important role in determining the polarization than does the index. For a small open angle, the z-polarized effect can be neglected and the scalar method is simple and valid to evaluating the diffraction field distribution of axicon. However, the vectorial method has to be used for great open angle.     

Statistical description of an ensemble of ultrarelativistic particles in a spatially flat universe

The kinetic equation for massless particles in an unperturbed Friedman universe with gravitational interactions taken into account is obtained by averaging the collisionless kinetic equations over local background gravitational field fluctuations. Uniformly distributed spherically symmetric fluctuations of an ultrarelativistic fluid serve as local inhomogeneities in a spatially flat universe. It is shown that the collision integral of our kinetic equation does not vanish when the distribution function is homogeneous and isotropic.     

Effect of the gravity on the photothermal waves in a semiconducting medium with an internal heat source and one relaxation time

The purpose of the present paper is to study the gravitational effect in a homogeneous isotropic semiconducting medium subjected to an internal heat source, based on Lord–Shulman theory. The problem is formulated in the dimensionless form and then solved analytically using the normal mode analysis method. The physical quantities are obtained and tested by a numerical study using the parameters of silicon as a target and presented graphically. The distribution of these quantities is represented graphically in the presence and absence of the internal heat source as well as gravity.     

Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical propagation expression of a vectorial Laguerre–Gaussian beam beyond paraxial approximation is presented. The far field expression and the scalar paraxial result are obtained as special cases of the general formulae. According to the analytical representation, the light intensity distribution of the vectorial Laguerre–Gaussian beam is depicted in the reference plane. The light intensity distribution of a vectorial Laguerre–Gaussian beam with cos m is also compared with that of a vectorial Laguerre–Gaussian beam with sin m.