Zur Theorie der Bewegung geladener Teilchen in richtungskonstanten Magnetfeldern exponentieller Zeitabhängigkeit. I. Allgemeine Theorie und Einschaltvorgang

The general solution of the equations of motion for a charged particle in a magnetic field is given for the following case: the spatially homogeneous magnetic field having a constant direction is a superposition of a field constant in time and one decreasing exponentially in time; taken into account is the influence of the electric field induced by the time dependent magnetic field and a friction force proportional to the particle velocity. The higher transcendental functions appearing in the exact solution are approximated in various ways in dependence on the values of the argument and parameters. The important case of a switching process without a friction force is investigated in detail. The higher transcendential functions can be approximated by simplier functions in such a way, that the solutions for the switching process, valid for all times, differ from the solutions in the case of a linear increasing magnetic field only by factors consisting of elementary functions. Approximated formulae of a very simple form are obtained for position, velocity, kinetic energy and magnetic moment of the particle. The particle orbits are classified and their dependence on the initial values and parameters of the magnetic fields is studied. A comparison between our results and a rectangular variation of the field shows that the latter is not a good approximation for a really exponential increasing field. Finally a detailed investigation shows that the electric field induced by the time dependent magnetic field has an important influence on the particle motion.     

Zur Theorie der Bewegung geladener Teilchen in richtungskonstanten Magnetfeldern exponentieller Zeitabhängigkeit. II. Ausschaltvorgang ohne und mit Reibung

The motion of a charged particle in a magnetic field is calculated for the following case: the spatially homogeneous magnetic field having a constant direction decreases exponentially in time (switch-off process); taken into account is the influence of the electric field and a friction force proportional to the particle velocity. The higher transcentendal functions appearing in the exact solution are approximated in various ways in dependence on the values of argument and parameters. In this manner approximated formulae of a very simple form are obtained for position, velocity, kinetic energy and magnetic moment of the particle. The particle orbits are classified and their dependence on the initial values, parameters of the magnetic field and on the magnitude of the friction force is studied. A comparison between our results and a rectangular variation of the field shows that the latter is not a good approximation for a really exponential decreasing field. A detailed investigation shows that the electric field induced by the time dependent magnetic field has an important influence on the particle motion.     

Theorie der Bewegung geladener Teilchen in richtungskonstanten zeitabhängigen Magnetfeldern und elektrischen Zusatzfeldern

The motion of a charged particle in spatially homogeneous electric and magnetic fields is calculated for the case of the magnetic field to have a constant direction and its intensity to vary with an arbitrary power of time. The special case of a linearly increasing magnetic field is treated in detail taking into account a friction force proportional to the particle velocity. Generally, the equations of motion are reduced to a single differential equation of second order which is integrated exactly. The higher transcendental functions appearing in the solution are then approximated by elementary functions. Thus asymptotic approximative equations of a very simple form are obtained for position, velocity, kinetic energy and magnetic moment of the particle. The dependence of the particle orbit on the initial values of position and velocity and on the properties of the magnetic field is studied, and it is shown, how the particle motion is a helical motion superposed by a drift. The influence of the electric field induced by the time dependent magnetic field on the particle motion is considered in detail. For an additional electric field being present a drift formula is derived which is a generalization of the well-known ?? × ?? 93 drift for constant fields.     

Zur Theorie der stationären Bewegung von Ionen im elektrischen Feld

In a static electric field a one-dimensional model of the motion of ions, controlled solely by charge exchange (electron capture from gas moleculs) is considered. Solutions of the kinetic equastions for ions and fast neutrals are derived in several special cases. If the charge exchange cross section is constant, the kinetic equation is easily solvable in a closed form; if the cross section is a function of ion energy, however, the problem is reducible to a simple integral equation for the energy distribution function.     

Zur nichtlinearen Theorie der Strahl-Plasma-Wechselwirkung III. Vorbereitung der numerischen Rechnungen

In order to prepare the system of ordinary differential equations, describing approximately the non-linear behaviour of a beam-plasma system, for numerical solution, at first the dispersion relation is analized. In particular it is shown, that in the case of a three component plasma the complex wave number k = k_r + i_γ can be found without difficulties as a function of frequency ω and the plasma parameters, if the electron densities n_b and n_p are treated as the two dependent quantities. The expressions on the right hand sides of the differential equations are transformed in such a way, that all complex integrals can be found as functions of one basic integral, tabulated in the literature, not only for negative values of γ (amplification) but also for positive values of γ (evanescence). In this way it will be possible, to describe the behavior of the wave amplitude not only for short distances from the input, where amplification of the wave takes place, but also for larger distances, where nonlinear Landau-damping may occur.     

Zur Theorie der nichtstationären stimulierten Raman-Streuung

The theory of transient stimulated Raman scattering is developed taking into account the deplation of the pump pulse. Two coupled differential equations describing the conversion of a laser pulse in a Stokes wave are derived with the aid of a semiclassical nonstationary perturbation calculation without the phonon concept. The equations are solved both analytically by a successive approximation and by computer calculations. Estimations are given of the delay of the Stokes pulse, the gain and a characteristic length concerning the conversion.     

Zur allgemeinen kinetischen Theorie von partiell ionisierten Plasmen in zeitabhängigen Magnetfeldern und elektromagnetischen Zusatzfeldern

A general method is given for the calculation of the electron distribution function of a weakly ionized plasma in external time-dependent magnetic fields and additional electromagnetic fields. The Boltzmann equation of kinetic theory is solved taking into account elastic collisions between electrons and neutrals. The isotropic part f⁰ of the distribution function follows from a general linear integro-differential equation and contains all known standard distributions (Druyvesteyn, Davydov, Margenau and others) as special cases. The direction-dependent part f¹ gives the transport tensors.     

Zur Theorie der Teilchenbewegung in zeitabhängigen elektromagnetischen Feldern

The Langevin equation – i.e. the equation of motion for a charged particle including a collision term proportional to the particle velocity – is solved for arbitrary time-dependent electric and magnetic fields by a new general method. Instead of the usual ansatz: particle velocity = cyclotron velocity + drift velocity the method given makes the ansatz: particle velocity = tensor = cyclotron velocity. The unknown tensor obeys a simple differential equation of the first order which can be generally solved at once. This method is a modification of the variation of constants method for inhomogeneous differential equations. The electromagnetic fields considered must be spatially homogeneous; for (weakly) inhomogeneous fields an iteration procedure of Pytte (1962) may be applied. Some examples are discussed shortly. The Langevin equation treated is completely equivalent to the equation of motion in a magnetohydrodynamic one-fluid theory.     

Zur Druckabhängigkeit der Anodenfallspannung von Hochdrucklichtbögen

The dependence of the anode fall voltage of freely burning high pressure arcs was investigated on the basis of a simplified arc model. It was found that the anode fall voltage does not contribute to the observed increase of arc voltage with pressure. The analysis was based on experimental investigations carried out with an argon arc. Due to the self-magneticly produced plasma flow a stagnation point flow pattern developed at the anode. The experiments were conducted at currents of I = 100 A and I = 150 A. The pressure was varied between p = 1 atm and p = 50 atm. The applied method necessitated the determination of the anode energy balance. As a side result of the investigation it was found that radiation contributes at p = 1 atm with appr. 20% and at p = 50 atm with appr. 50% to the total arc energy balance.     

Zur Theorie der nichtstationären Resonanzfluoreszenz an bewegten Molekülen. I Räumliche Energiedichte der Fluoreszenzstrahlung

Non-stationary Theory of Resonance Fluorescence for Moving Molecules. I. Spatial Energy Density of the Fluorescence Radiation The time dependence of the spatial energy density of the fluorescence radiation of an ensemble of moving molecules generated by resonance scattering of intense laser pulses is calculated. Particulary the influence of the duration and the intensity of short laser pulses upon the coherent and incoherent part of the fluorescence radiation of a Doppler-broadened gas is studied. The validity of the conception ?inhomogeneous relaxation time”? is discussed. Within the framework of resonance fluorescence scattering the phenomenon of photon echo is treated.