The concept of cross section, as its name suggests, is that of effective area for collision. The cross section of a spherical target is
The units of cross section are then area units, but for nuclear scattering the effective area is on the order of the cross sectional area of a nucleus. For a gold nucleus of mass number A=197, the radius determined from the nuclear radius relationship is about 7 fermis.
The most common unit for cross section for nuclear scattering interactions is the barn. The cross section in barns for alpha scattering above a selected angle is a standard part of the analysis of Rutherford scattering. In the case of 6 MeV alpha particles scattered from a gold foil, for example, you don't know the impact parameter for any given alpha particle, so the calculation of the scattered fraction takes on a statistical character. For alpha particles bombarding a gold foil of thicknes 1 micrometer, less than 1 in 100,000 would meet the conditions for scattering at over 140°. The cross section for scattering at smaller angles will be larger, because the alpha doesn't have to come as close to be scattered through the smaller angle. For example, the cross section for scattering through 90° or above is 11.3 barns.
For neutron scattering, there is no coulomb repulsion, so there is no appreciable scattering unless the impact parameter is less than the effective radius for the strong interaction. Therefore for a given target geometry the cross sections are smaller, typically measured in millibarns.
Since the probability for interaction depends upon the strength and range of the interaction, the cross-sections for neutrino interactions are very small since they interact only by the weak interaction.
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