Section 4.7: Exploring Rutherford Scattering

Kinetic energy = MeV | nuclear charge =

| impact parameter =   fm (10−15 m)

The initial animation shows a beam of forty-one alpha particles (test charges) fired at a fixed nucleus. These alpha particles interact via the Coulomb force with the fixed nucleus but not with each other. Because the nucleus is much more massive than any individual alpha particle, modeling this with a fixed nucleus gives a reasonably accurate picture of what happens. Notice that in this case (as opposed to the Thomson model of the atom), the alpha particles can be scattered through large angles if they get close enough to the nucleus and experimentally, alpha particles projected on gold foil do scatter back at large angles.   Restart.

We can use conservation of momentum and conservation of energy to analyze the scattering. This is an elastic collision between a heavy target and a light projectile and the vertical distance the projectile is from the center of the target determines the scattering angle. Because the force is actually the long-range Coulomb force, the collision begins well before the particles would touch. Even so, thinking of this like a classical collision works well. Try different initial kinetic energies (and thus different projectile speeds) and target nuclear charge and note how the scattering pattern changes. The animation depicts non-relativistic objects only.   For a maximum kinetic energy of 30 MeV, what is the maximum speed of an alpha particle (mass = 3750 MeV/c2)?  Is a non-relativistic approach justified?

The animation can also show one alpha particle impinging on a gold nucleus. You can change the impact parameter (vertical distance between the alpha particle and the nucleus) along with the initial kinetic energy and nuclear charge of the target (the deflection angle is given in degrees). The relationship between the deflection angle, θ, impact parameter, b, and the initial kinetic energy, K, of the incident alpha particle is5

b = (zZ /2K) (e2/4πε0) cot(θ/2),                    (4.7)

where z = 2 (charge of alpha particle) and Z is the charge of the nucleus (Z = 79 for gold), and e2/4πε0 = 1.44 eV nm. Although useful for this animation, what is measured in the lab is the fraction of particles scattered through a given angle as given by the Rutherford scattering formula (derived from the relationship above). The Rutherford scattering formula for the probability per unit area for alpha particles scattering into a ring around the angle θ, called N(θ) is

N(θ) = (nt/4r2) (zZ/2K)2 (e2/4πε0)2 (1/sin4(θ/2)),                    (4.8)

where n is the number of atoms or nuclei per unit volume, t is the thickness of the foil, and r is the distance between the detector and the foil. In the laboratory, the number of particles measured at any given angle followed a 1/sin4(θ/2) dependence as predicted by Rutherford's formula. Physicists discarded the Thomson model in favor of the Rutherford model of a positively-charged nucleus that comprised most of the mass of the atom, and is surrounded by orbiting electrons.

5See J. R. Taylor, C. D. Zafiratos, M. A. Dubson, Modern Physics for Scientists and Engineers, Prentice Hall, 2004, pp. 110-118 or S. Thornton and A. Rex, Modern Physics for Scientists and Engineers, 2nd ed, Brooks/Cole, 2002, pp. 123-127 for a full derivation of the Rutherford scattering formula.