Rutherford scattering

Rutherford scattering. It is the elastic scattering of charged particles by the Coulomb interaction with an atomic nucleus. The phenomenon was famously investigated in the Geiger–Marsden experiments conducted at the University of Manchester, which led to the discovery of the atomic nucleus and the downfall of the plum pudding model. The theoretical explanation, provided by Ernest Rutherford, established the modern picture of the atom and represents a cornerstone of nuclear physics.

Introduction

The phenomenon was first observed in 1909 when Hans Geiger and Ernest Marsden, under the direction of Ernest Rutherford, bombarded a thin foil of gold with alpha particles from a radium source. Contrary to expectations based on J. J. Thomson's model, a small fraction of particles were deflected at very large angles, including directly backwards. This surprising result was famously described by Rutherford as "almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you." The analysis required a new atomic model where most mass and positive charge were concentrated in a tiny, dense core.

Experimental setup

The key apparatus involved a radioactive source, typically radon or radium, placed inside a lead shield to produce a collimated beam of alpha particles. This beam was directed onto a very thin metal foil, often gold or platinum, mounted in a vacuum chamber. A movable zinc sulfide scintillation screen or a Geiger counter was used to detect scattered particles at various angles relative to the incident beam. The entire setup was meticulously engineered to minimize background noise and allow precise angular measurements, with crucial work conducted in the laboratories of the University of Manchester.

Scattering formula

Rutherford derived a formula predicting the angular distribution of scattered particles. The differential cross section, which gives the probability of scattering into a given solid angle, is proportional to the square of the product of the atomic numbers of the projectile and target nuclei, and inversely proportional to the fourth power of the sine of half the scattering angle and the square of the kinetic energy of the incident particle. This famous relationship, known as the Rutherford formula, successfully explained the experimental data from the Geiger–Marsden experiments and implied an inverse-square law force, confirming the Coulomb nature of the interaction.

Derivation

The derivation assumes a point-like, massive, positively charged nucleus and treats the scattering as a two-body Coulomb interaction in a central force field, neglecting nuclear forces and relativistic effects. Using classical mechanics, the trajectory of an incoming alpha particle is a hyperbola, with the nucleus at one focus. Conservation of angular momentum and energy are applied, and the relationship between the impact parameter and the scattering angle is established. Integrating over all impact parameters leads to the differential cross section, a result that was later shown to be identical to the solution obtained from the Born approximation in non-relativistic quantum mechanics.

Significance and impact

The analysis of the phenomenon directly led to the Rutherford model of the atom, which posited a small, dense nucleus surrounded by orbiting electrons. This was a pivotal development in atomic physics, superseding the plum pudding model and paving the way for Niels Bohr's model of the atom. It provided the first direct evidence for the existence of the atomic nucleus and established the scale of nuclear dimensions. The success of the classical derivation also demonstrated the applicability of Newtonian mechanics to atomic-scale events under certain conditions, influencing the development of early quantum theory.

Applications

The principle serves as the basis for Rutherford backscattering spectrometry, a vital analytical technique in materials science for determining the composition and thickness of thin films. It is also fundamental to the design and understanding of particle accelerators like the Large Hadron Collider at CERN, where Coulomb scattering between beam particles must be managed. Furthermore, the technique is used in ion implantation studies and in nuclear physics experiments to probe nuclear sizes and charge distributions, forming a foundational concept in the study of nuclear reactions and particle physics. Category:Scattering Category:Nuclear physics Category:Physics experiments