In 1911 Ernest Rutherford published a formula which indicated that the number of particles that would be deflected by an angle θ due to scattering from fixed nuclei is inversely proportional to the fourth power of the sine function of one half the angle of deflection; i.e.,

n(θ)Δθ = [κ/sin⁴(θ/2)] Δθ
 

where κ is a constant.

More precisely Rutherford's formula is in terms of the cross section σ per unit solid angle Ω and has the form:

dσ/dΩ = γ²cosec⁴(Θ/2)

 

where γ is a parameter dependent up on a number variables such as the charges of the particles, the force constant for the electrostatic (Coulomb) force, the mass of the particles, etc. Specifically γ=JqQ/(mv) where q and Q are the charges of the two particles, J is the constant for the electrostatic (Coulomb) force, m is the mass of the impinging particle and v is its initial velocity.

Rutherford's analysis has been modified to remove some of the restrictions involved in its derivation. One such modification was by Mott who allowed for relativistic corrections. The Mott scattering formula is of the form

dσ/dΩ = γ²cosec⁴(Θ/2)(1−(v/c)²sin²(Θ/2))
 

where v is the velocity of the impinging particle and c is the speed of light in a vacuum.

A relativistic adjustment can easily be made to the Rutherford formula by allowing for the mass of the impinging particle to be m₀/(1−β²)^{½ where β=v/c.}

(To be continued.)

Sources:
W.S.C. Williams, Nuclear and Particle Physics, Clarendon Press, Oxford, 1991.

For more on the scattering and diffraction of atomic particles see SCATTERING.