Zur Theorie der stochastischen Beschleunigung. I

This work contains theoretical investigations on the stochastic acceleration in circular accelerators. The first part deals with the classical problem ofBurshtein, Kolomenskii andVeksler. This problem was formulated in a publication very clearly in 1955, but has not yet found an exact solution. The subject is the calculation of the probability that a charged particle is accelerated from the injection energy to the top energy of the apparatus (“probability of acceleration”), supposing that the particle has an equal chance at every circulation to increase or decrease its energy by a fixed amounteU ₀. The following investigation shows that this problem can be solved if properly formulated as a matrix equation, and that the solution is of a relatively simple closed form. This is the case, too, if the primary problem is extended by permitting particle losses during the acceleration process (“absorption”). The results are discussed, especially regarding the influence of absorption. Under certain conditions (eU ₀ ²/SΣ (E _max?E _min)²] the probability of acceleration becomes independent of absorption and of the gap voltageU ₀.