A shadow of the repulsive Rutherford scattering in the fixed-target and the center-of-mass frame

TL;DR

This paper analyzes the geometric shape of the shadow cast by repulsive Rutherford scattering particles in both fixed-target and center-of-mass frames, revealing universal paraboloidal forms and scale-invariant properties.

Contribution

It provides a detailed geometric analysis of the projectile shadow in different frames, highlighting universal and scale-invariant features of Rutherford scattering shadows.

Findings

Shadow is paraboloidal in both frames

Target is at the focus of the parabola in fixed-target frame

Shadow shape is scale-invariant when properly scaled

Abstract

The paper explores the shadow of the repulsive Rutherford scattering - the portion of space entirely shielded from admitting any particle trajectory. The geometric properties of the projectile shadow are analyzed in detail in the fixed-target frame as well as in the center-of-mass frame, where both the charged projectile and the charged target cast their own respective shadows. In both frames the shadow is found to take an extremely simple, paraboloidal shape. In the fixed-target frame the target is precisely at the focus of this paraboloidal shape, while the focal points of the projectile and target shadows in the center-of-mass frame coincide. In the fixed-target frame the shadow takes on a universal form, independent of the underlying physical parameters, when expressed in properly scaled coordinates, thus revealing a natural length scale to the Rutherford scattering.