A shadow of the repulsive Rutherford scattering in the laboratory frame

The paper explores the Rutherford scattering shadow in an entire class of comoving frames -- inertial frames moving along the initial projectile direction -- of which the laboratory frame, where the target is initially at rest, is a representative example. The paper is a continuation of the previous work addressing the scattering shadow in the fixed-target and the center-of-mass frame. It is shown that the transition from these frames is technically quite involved, due to the scattering shadow forming at an infinite distance from an initial target position. The central procedure involves solving the $5^\mathrm{th}$ degree polynomial as a part of an associated extremization procedure. The shadow existence itself is subject to certain conditions, dependent on a given comoving frame. A new and unexpected phenomenon is found within a certain set of comoving frames, including the laboratory frame itself. It consists of a phase transition between an entirely smooth type of shadow and the one characterized by a formation of a sharp edge.Comment: 25 pages, 19 figure