of a cloud chamber. Rutherford scattering is now exploited by the materials science community in an analytical technique called Rutherford backscattering. Derivation The differential cross section can be derived from the equations of motion for two charged point particles interacting through a central potential. In general, the equations of motion describing two particles interacting under a central force can be decoupled into the center of mass and the motion of the particles relative to one another. Consider the situation where one particle (labelled 1), with mass and charge with the elementary charge, is incident from very far away with some initial speed on a second particle (labelled 2) with mass and charge initially at rest. For the case of light alpha particles scattering off heavy nuclei, as in the experiment performed by Rutherford, the reduced mass, essentially the mass of the alpha particle and the nucleus off of which it scatters, is essentially stationary in the lab frame. Substituting into the Binet equation, with the origin of coordinate system for particle 1 on the target (scatterer, particle 2), yields the equation of trajectory as where and is the impact parameter. The general solution of the above differential equation is and the boundary condition is Solving the equations using those boundary conditions: and its derivative using those boundary conditions We can obtain At the deflection angle after collision, : Then deflection angle can be expressed as: can be solved to give To find the scattering cross section from this result consider its definition Given the Coulomb potential and the initial kinetic energy of the incoming particles, the scattering angle is uniquely determined by the impact parameter . Therefore, the number of particles scattered into an angle between and must be the same as the number of particles with associated impact parameters between and . For an incident intensity , this implies the following equality For a radially symmetric scattering potential, as in the case of the Coulomb potential, , yielding the expression for the scattering cross section Plugging in the previously derived expression for the impact parameter we find the Rutherford differential scattering cross section This same result can be expressed alternatively as where is the dimensionless fine structure constant, is the initial non-relativistic kinetic energy of particle 1 in MeV, and . Details of calculating maximal nuclear size For head-on collisions between alpha particles and the nucleus (with zero impact