Rutherford scattering in the context of Charged particle

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciprocal relationship ${\displaystyle xy=1.}$ In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of a sundial's gnomon, the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle, among others.

Ernest Rutherford, Baron Rutherford of Nelson (30 August 1871 – 19 October 1937), was a New Zealand physicist and chemist who was a pioneering researcher in both atomic and nuclear physics. He has been described as "the father of nuclear physics" and "the greatest experimentalist since Michael Faraday." In 1908, he was awarded the Nobel Prize in Chemistry "for his investigations into the disintegration of the elements, and the chemistry of radioactive substances." He was the first Oceanian Nobel laureate, and the first to perform Nobel-awarded work in Canada.

Rutherford's discoveries include the concept of radioactive half-life, the radioactive element radon, and the differentiation and naming of alpha and beta radiation. Together with Thomas Royds, Rutherford is credited with proving that alpha radiation is composed of helium nuclei. In 1911, he theorised that atoms have their charge concentrated in a very small nucleus. He arrived at this theory through his discovery and interpretation of Rutherford scattering during the gold foil experiment performed by Hans Geiger and Ernest Marsden. In 1912, he invited Niels Bohr to join his lab, leading to the Bohr model of the atom. In 1917, he performed the first artificially induced nuclear reaction by conducting experiments in which nitrogen nuclei were bombarded with alpha particles. These experiments led him to discover the emission of a subatomic particle that he initially called the "hydrogen atom", but later (more precisely) renamed the proton. He is also credited with developing the atomic numbering system alongside Henry Moseley. His other achievements include advancing the fields of radio communications and ultrasound technology.

In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

When two discrete particles interact in classical physics, their mutual cross section is the area transverse to their relative motion within which they must meet in order to scatter from each other. If the particles are hard inelastic spheres that interact only upon contact, their scattering cross section is related to their geometric size. If the particles interact through some action-at-a-distance force, such as electromagnetism or gravity, their scattering cross section is generally larger than their geometric size.