Centering of the ion trajectory in the cyclotron

In this paper we present the theory of the horizontal motion ions through the acceleration gap in the case of a static magnetic field and time-varying electric field. It also describes the motion of the center of the ion trajectory through the acceleration gap and also during the acceleration process. The result also describes analytically the angle through which the ion passed (so called “flying angle”) between two acceleration gaps. That angle can be measured from the center of the ion trajectory or from the center of the cyclotron. The analysis shows that the “flying angle” of the ions measured in these two reference systems is not same. Namely, the “flying angle” of the ion measured from the center of its trajectory is bigger than in the case when the “flying angle” is measured from the center of the cyclotron. It shows that difference between “flying angle” in these two reference systems becomes less and less during acceleration process.