Relaxationserscheinungen in Elektronenstrahlen

Abstract

In electron beams with high charge density there are observed anomalous shiftings and broadenings as well as symmetrizations of the energy distribution (BoERSCa 1954). Continuing former work 1, where these phenomena were ascribed to a relaxation process changing the energy distribution to a new equilibrium distribution, quantitative expressions are derived for the energy spread and the shape of this new energy distribution. Furthermore a relaxation degree is defined which measures the progress of relaxation. Measurements are described with the highest relaxation degrees reached up to now. In addition to the verification of former results up to these new limits the shape of the measured energy-distributions can be fitted to the derived one with suitable choice of a single parameter. Peculiarities of measurements with superposed axial magnetic fields are discussed.

Solving second order linear differential equations with Klein's theorem

“…For completeness and to set notations, we briefly recall the rational Kovacic algorithm from [27]. Let L = ∂ 2 +A1∂ +A0 where A0, A1 ∈ k. We consider a second order ordinary linear differential equation…”

“…We recall the classification of subgroups of SL ( C) (see e.g [19,25,27,23]) and the invariants and semi-invariants of lowest degree. The group is reducible if there is at least one invariant line in V .…”

“…Since the appearance of [19], many papers have studied and refined the method. The version given in [27] uses invariants instead of the semi-invariants, which is easier to implement especially for differential fields k more complicated than C(x). The paper [15] gives good formulas for computing algebraic solutions (after [25]).…”

“…Our formulas from [4,2,3] rely on computing semi-invariants of the differential Galois groups, which is well-mastered for differential equations with coefficients in C(x). For more general differential fields, however, it may be easier (as noted in [27]) to use algorithms that compute invariants of the differential Galois group instead of semi-invariants. In order to use invariants, we will need to give formulas that are slightly different from those given in [4,2,3].…”

“…The algorithm in section 2 below follows the lines of the rational version of the Kovacic algorithm given in [27]. Section 2 contains the algorithm.…”