Rutherford Scattering Cross Section

\[\frac{d\sigma}{d\Omega}=\left(\frac{Z_1Z_2e^2}{16\pi\epsilon_0E}\right)^2\frac{1}{\sin^4(\theta/2)}\]

Variables

dsigma[dσ/dΩ]differential scattering cross section

Z1projectile atomic number

Z2target atomic number

eelementary charge

epsilon0[ε₀]vacuum permittivity

Eincident particle kinetic energy

theta[θ]scattering angle

Description

What is this formula?

The Rutherford scattering cross section formula calculates the angular distribution of charged particles scattered by the Coulomb field of a target nucleus. It describes how likely particles are to scatter into a given solid angle.

When to use it

Use this formula for classical Coulomb scattering of charged particles, especially alpha particles or ions interacting with atomic nuclei when nuclear forces and quantum corrections are not dominant.

Example

An alpha particle is scattered by a gold nucleus.

Data:

Z1 = 2

Z2 = 79

E = 5 MeV

theta = 60°

Formula:

dσ/dΩ = ((Z1*Z2*e²)/(16π*epsilon0*E))² * 1/sin⁴(theta/2)

Substitution:

dσ/dΩ = ((2*79*e²)/(16π*epsilon0*5 MeV))² * 1/sin⁴(30°)

Result:

The differential cross section gives the probability density for scattering at 60°.

Applications

Nuclear scattering experiments, alpha particle scattering, atomic nucleus studies, accelerator physics, detector calibration, and Coulomb interaction analysis.