the scattering cross-section
The cross-section is how physicists quantify 'how big a target looks' to an incoming beam -- an effective area, measured in units of area, that captures the probability of a collision or scattering event. It is the universal currency of scattering experiments, from Rutherford firing alpha particles at gold foil to the LHC smashing protons. A larger cross-section means a more likely interaction; the beauty is that this area encodes the physics of the underlying force, even when the 'target' has no sharp size at all.
Two versions matter. The differential cross-section d sigma / d Omega is defined so that the number of particles scattered per second into a small solid angle d Omega equals the incident flux (particles per area per second) times (d sigma / d Omega) times d Omega. It has units of area per steradian and tells you the angular pattern of scattering. In quantum mechanics it is simply the modulus-squared of the scattering amplitude: d sigma / d Omega = |f(theta, phi)|^2. Integrating over all angles gives the total cross-section sigma = integral |f|^2 d Omega, a single number with units of area (often quoted in barns, 1 barn = 10^-28 m^2) that measures the overall strength of the interaction. Crucially, the cross-section is defined so that it does not depend on your particular beam intensity or detector -- it is an intrinsic property of the interaction.
Because it is intrinsic and directly measurable, the cross-section is the meeting point of theory and experiment across all of physics. Nuclear reactor design lives and dies by neutron-capture cross-sections; particle physicists discover new particles as resonant peaks in a cross-section versus energy; and the Rutherford cross-section proportional to 1/sin^4(theta/2), computed from the Coulomb potential, was the measurement that revealed the atomic nucleus. Whenever a theory predicts a scattering amplitude, the cross-section is how that prediction is put to the test.
A slow neutron passing near a boron-10 nucleus has an absorption cross-section of about 3800 barns -- vastly larger than the nucleus's geometric area of roughly 1 barn. The 'target' looks thousands of times bigger than the nucleus itself because the cross-section reflects the strength of the nuclear reaction, not a physical size, which is exactly why boron is used to soak up neutrons in reactor control rods.
Boron-10's neutron cross-section (~3800 barns) dwarfs its geometric size -- cross-section measures interaction strength, not area.
A cross-section is an effective, not a geometric, area: it can be far larger or smaller than the target's physical size, and near a resonance it can spike by orders of magnitude at a particular energy. The differential cross-section is |f|^2, but this identity holds for the scattering amplitude f as defined for the asymptotic outgoing wave -- keep the conventions consistent when comparing formulas.