12.2.1. Alpha-particle trajectory

The trajectory traced by an a-particle depends on the impact parameter, b of collision. The impact parameter is the perpendicular distance of the initial velocity vector of the a-particle from the centre of the nucleus.

The impact parameter (b) is a critical concept in understanding how alpha-particles interact with atomic nuclei. It is defined as the shortest distance from the center of the nucleus to the path of the alpha-particle at a right angle. This parameter helps predict the scattering angles: smaller impact parameters mean closer approaches to the nucleus and hence larger scattering angles, while larger impact parameters result in less deflection.

Consider throwing a basketball at a hoop. If you aim directly at the center of the hoop, the ball will go straight in, analogous to a small impact parameter which represents a more direct collision. However, if you throw the ball from the side, it may miss the hoop entirely or just graze it, representing a larger impact parameter and resulting in a smaller chance of a significant interaction.

A given beam of a-particles has a distribution of impact parameters b, so that the beam is scattered in various directions with different probabilities. It is seen that an a-particle close to the nucleus (small impact parameter) suffers large scattering. In case of head-on collision, the impact parameter is minimum and the a-particle rebounds back (q @ p). For a large impact parameter, the a-particle goes nearly undeviated and has a small deflection (q @ 0).

When alpha-particles are directed toward a target nucleus, the scattering angle they undergo after colliding with the nucleus varies according to the impact parameter. If the impact parameter is small, the alpha-particle is more likely to have a significant interaction with the nucleus, leading to a high scattering angle (often referred to as a 'large deflection'). Conversely, if the impact parameter is large, the alpha-particle passes further away and only experiences minor deflection, resembling a glancing blow.

Imagine a game of pool. When the cue ball strikes the eight ball head-on (small impact parameter), it sends the eight ball off at a sharp angle and likely towards a pocket. If the cue ball hits the eight ball from the side (large impact parameter), the eight ball moves away gently, barely changing direction. This similarity illustrates how small and large impact parameters lead to drastically different outcomes in scattering.

The fact that only a small fraction of the number of incident particles rebound back indicates that the number of a-particles undergoing head-on collision is small. This, in turn, implies that the mass and positive charge of the atom is concentrated in a small volume. Rutherford scattering therefore, is a powerful way to determine an upper limit to the size of the nucleus.

The results of the scattering experiment reveal that most alpha-particles pass through the gold foil without significant interaction, while a small fraction bounce back after close encounters with the nucleus. This observation supports the idea that the atomic nucleus, which contains most of the atom's mass and positive charge, is very small compared to the overall size of the atom. Understanding how this scattering works enables scientists to estimate the dimensions of the nucleus accurately.

Think of shooting a BB gun at a dartboard. If most of the BBs pass cleanly through the dartboard (representing the atom) and a few stick to the bullseye (the nucleus), it tells you that the bullseye occupies a tiny proportion of the dartboard's area. Similarly, the scattering data tells us how tiny and dense the nucleus is relative to the rest of the atom, emphasizing the vast amounts of empty space within an atom.

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