Coulomb scattering is the elastic scattering of charged particles by the Coulomb interaction. The physical phenomenon was used by Ernest Rutherford in a classic 1911 paper that eventually led to the widespread use of scattering in particle physics to study subatomic matter. The details of Coulomb scattering vary with the mass and properties of the target particles, leading to special subtypes and a variety of applications. Rutherford scattering refers to two nuclear particles and is exploited by the materials science community in an analytical technique called Rutherford backscattering. Electron on nuclei are employed in electron polarimeters and, for coherent electron sources, in many different kinds of electron diffraction.
History Couloumb scattering, especially as a means for studying atomic, nuclear and subatomic systems, began with Ernest Rutherford's alpha particle scattering model of 1911. The ingredients in Rutherford's approach were well known at the time. The Coulomb force acts as central force along a line between two particles and varies with the inverse square, matching a detailed theory developed under the name of the Kepler problem. JJ Thomson had applied some of the ideas in his 1910 paper on beta particle scattering. Scattering theory and atomic models
In a 1909 experiment, Geiger and Marsden discovered that the metal foils could scatter some alpha particles in all directions, sometimes more than 90°. This should have been impossible according to the then current Plum pudding model of the atom. According to this model, by JJ Thomson, the atom consists of a sphere of positive charge filled with circulating electrons.
The extreme scattering observed forced Rutherford to revise the model of the atom. The issue in Thomson's model was that the charges were too diffuse to produce a sufficiently strong electrostatic force to cause such repulsion. In Rutherford's new model, the positive charge does not fill the entire volume of the atom but instead constitutes a tiny nucleus at least 10,000 times smaller than the atom as a whole. All that positive charge concentrated in a much smaller volume produces a much stronger electric field near its surface. The nucleus also carried most of the atom's mass. This meant that it could deflect alpha particles by up to 180° depending on how close they pass. The electrons surround this nucleus, spread throughout the atom's volume. Because their negative charge is diffuse and their combined mass is low, they have a negligible effect on the alpha particle.
To verify his model, Rutherford developed a scientific model to predict the intensity of alpha particles at the different angles they scattered coming out of the gold foil, assuming all of the positive charge was concentrated at the centre of the atom. This model was validated in an experiment performed in 1913. His model explained both the beta scattering results of Thomson and the alpha scattering results of Geiger and Marsden.
Rutherford's scattering model
The historical and pedological starting point for Coulomb scattering is Rutherford's nuclear particle scattering. Rutherford's 1911 paper has become a classic and his approach is repeated in modern references. He uses conservation of energy and of momentum to set the parameters of particle trajectories. An alternative approach based on Newtonian force diagrams is described in
Rutherford begins his paper with a discussion of Thomson's results on scattering of beta particles, a form of radioactivity that results in high velocity electrons. Thomson's model had electrons circulating inside of a sphere of positive charge. Coulomb scattering for Thomson's model is described in
Rutherford highlights that this model requires compound or multiple scattering events: the deflections predicted for each collision are much less than one degree. Thomson's simple model for the effect of multiple scattering is discussed in .
Rutherford then proposes a model which will produce large deflections on a single encounter: place all of the positive charge at the centre of the sphere and ignore the electron scattering as insignificant. The concentrated charge will explain why most alpha particles do not scatter to any measurable degree – they fly past too far from the charge – and yet particles that do pass very close to the centre scatter through large angles.
Maximum nuclear size estimate Rutherford begins his quantitative analysis by considering a head-on collision between the alpha particle and atom. This will establish the minimum distance between them, a value which will be used throughout his calculations.
Assuming there are no external forces and that initially the alpha particles are far from the nucleus, the inverse-square law between the charges on the alpha particle and nucleus gives the potential energy gained by the particle as it approaches the nucleus. For head-on collisions between alpha particles and the nucleus, all the kinetic energy of the alpha particle is turned into potential energy and the particle stops and turns back.
Where the particle stops at a distance from the centre, the potential energy matches the original kinetic energy:"Electrons (+ and -), Protons, Photons, Neutrons, Mesotrons and Cosmic Rays" By Robert Andrews Millikan. Revised edition. Pp. x+642. (Chicago: University of Chicago Press; London: Cambridge University Press, 1947.)Cooper, L. N. (1970). "An Introduction to the Meaning and Structure of Physics". Japan: Harper & Row.
where
Rearranging:
For an alpha particle:The convention in Rutherford's time was to measure charge in electrostatic units, distance in centimeters, force in dynes, and energy in ergs. The modern convention is to measure charge in coulombs, distance in meters, force in newtons, and energy in joules. Using coulombs requires using the Coulomb constant in certain equations. In this article, Rutherford and Thomson's equations have been rewritten to fit modern notation conventions. (mass) = = (for the alpha particle) = 2 × = (for gold) = 79 × = (initial velocity) = (for this example) The distance from the alpha particle to the centre of the nucleus () at this point is an upper limit for the nuclear radius. Substituting these in gives the value of about , or 27 fm. (The true radius is about 7.3 fm.) The true radius of the nucleus is not recovered in these experiments because the alphas do not have enough energy to penetrate to more than 27 fm of the nuclear centre, as noted, when the actual radius of gold is 7.3 fm.
Rutherford's 1911 paper started with a slightly different formula suitable for head-on collision with a sphere of positive charge:
In Rutherford's notation, e is the elementary charge, N is the charge number of the nucleus (now also known as the atomic number), and E is the charge of an alpha particle. The convention in Rutherford's time was to measure charge in electrostatic units, distance in centimeters, force in dynes, and energy in ergs. The modern convention is to measure charge in coulombs, distance in meters, force in newtons, and energy in joules. Using coulombs requires using the Coulomb constant (k) in the equation. Rutherford used b as the turning point distance (called rmin above) and R is the radius of the atom. The first term is the Coulomb repulsion used above. This form assumes the alpha particle could penetrate the positive charge. At the time of Rutherford's paper, Thomson's plum pudding model proposed a positive charge with the radius of an atom, thousands of times larger than the rmin found above. Figure 1 shows how concentrated this potential is compared to the size of the atom. Many of Rutherford's results are expressed in terms of this turning point distance rmin, simplifying the results and limiting the need for units to this calculation of turning point.
Single scattering by a heavy nucleus From his results for a head on collision, Rutherford knows that alpha particle scattering occurs close to the centre of an atom, at a radius 10,000 times smaller than the atom. The electrons have negligible effect. He begins by assuming no energy loss in the collision, that is he ignores the recoil of the target atom. He will revisit each of these issues later in his paper. Under these conditions, the alpha particle and atom interact through a central force, a physical problem studied first by Isaac Newton. A central force only acts along a line between the particles and when the force varies with the inverse square, like Coulomb force in this case, a detailed theory was developed under the name of the Kepler problem. The well-known solutions to the Kepler problem are called orbits and unbound orbits are hyperbolas. Thus Rutherford proposed that the alpha particle will take a hyperbolic trajectory in the repulsive force near the centre of the atom as shown in Figure 2.
To apply the hyperbolic trajectory solutions to the alpha particle problem, Rutherford expresses the parameters of the hyperbola in terms of the scattering geometry and energies. He starts with conservation of angular momentum. When the particle of mass and initial velocity is far from the atom, its angular momentum around the centre of the atom will be where is the impact parameter, which is the lateral distance between the alpha particle's path and the atom. At the point of closest approach, labeled A in Figure 2, the angular momentum will be . Therefore
Rutherford also applies the law of conservation of energy between the same two points:
The left hand side and the first term on the right hand side are the kinetic energies of the particle at the two points; the last term is the potential energy due to the Coulomb force between the alpha particle and atom at the point of closest approach (A). qa is the charge of the alpha particle, qg is the charge of the nucleus, and k is the Coulomb constant.
The energy equation can then be rearranged thus:
For convenience, the non-geometric physical variables in this equation can be contained in a variable , which is the point of closest approach in a head-on collision scenario which was explored in a previous section of this article:
This allows Rutherford simplify the energy equation to:
This leaves two simultaneous equations for , the first derived from the conservation of momentum equation and the second from the conservation of energy equation. Eliminating and gives at a new formula for :
The next step is to find a formula for . From Figure 2, is the sum of two distances related to the hyperbola, SO and OA. Using the following logic, these distances can be expressed in terms of angle and impact parameter .
The eccentricity of a hyperbola is a value that describes the hyperbola's shape. It can be calculated by dividing the focal distance by the length of the semi-major axis, which per Figure 2 is . As can be seen in Figure 3, the eccentricity is also equal to , where is the angle between the major axis and the asymptote.Casey, John, (1885) "A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples" Therefore:
As can be deduced from Figure 2, the focal distance SO is
and therefore
With these formulas for SO and OA, the distance can be written in terms of and simplified using a trigonometric identity known as a half-angle formula:
Applying a trigonometric identity known as the cotangent double angle formula and the previous equation for gives a simpler relationship between the physical and geometric variables:
The scattering angle of the particle is and therefore . With the help of a trigonometric identity known as a reflection formula, the relationship between θ and b can be resolved to:
which can be rearranged to give
Rutherford gives some illustrative values as shown in this table: + Rutherford's angle of deviation table 10 5 2 1 0.5 0.25 0.125 5.7° 11.4° 28° 53° 90° 127° 152°