Rutherford differential cross-section ($\frac{d\sigma }{d\Omega }=\frac{{\kappa }^2}{16E^2}{\csc }^4\frac{\theta }{2}$)

For an incident beam uniform across its relevant impact parameters, an incident annulus has area $d\sigma =2\pi b|db|$. Rotational symmetry maps it to outgoing solid angle $d\Omega =2\pi \sin \theta d\theta$. The central-potential classical differential cross-section and the Rutherford impact-parameter relation therefore giveThis is a differential scattering cross-section, with units of area per steradian, conventionally ${\mathrm{m}}^2\mathrm{s}{\mathrm{r}}^{-1}$, not a detector count and not an area per degree. It applies for $0<\theta <\pi$, with a finite limiting value at $\pi$. Doubling incident kinetic energy divides it by four at fixed angle; doubling either electric charge number multiplies it by four.

A useful independent check integrates over all azimuths between two positive angles. For $0<{\theta }_1<{\theta }_2\le \pi$,This equals the incident annulus area $\pi (b_1^2-b_2^2)$ directly. A detector covering only a small patch at one angle does not collect this full azimuthal band.

At fixed incident flux, projectile energy and detector geometry, the Rutherford differential cross-section gives count rate $S\propto \mathcal{N}{Z}^2$, where $\mathcal{N}$ is the target nuclei per unit area. Equal $\mathcal{N}$ gives a $Z^2$ rate ratio. Equal foil mass per area instead gives $Z^2/A$, with $A$ the atomic mass number, while equal physical thickness also requires the material mass density. An empirical log-log plot with slope $s$ supplies a different, experimentally normalized relation $S\propto A^s$. These normalizations must be specified before comparing different target elements.