← Part VIII/Cross Sections

1. Scattering Cross Sections

Reading time: ~28 minutes | Pages: 7

Quantifying collision outcomes: fundamental observable in particle physics and nuclear reactions.

The Scattering Experiment

Setup:

Beam of particles with flux $\Phi$ (particles per unit area per unit time)
Target (single scattering center or thin target)
Detector at angle $(\theta, \phi)$ with solid angle $d\Omega$

Question: How many particles scatter into detector?

Differential Cross Section

Rate of scattering into solid angle $d\Omega$ at angles $(\theta, \phi)$:

$$\frac{dN}{dt} = \Phi \frac{d\sigma}{d\Omega}(\theta, \phi) d\Omega$$

Differential cross section:

$$\frac{d\sigma}{d\Omega} = \frac{\text{scattered particles per unit time per unit solid angle}}{\text{incident flux}}$$

Units: area per steradian (e.g., barn/sr, where 1 barn = 10⁻²⁸ m²)

Total Cross Section

Integrate over all scattering angles:

$$\sigma_{tot} = \int \frac{d\sigma}{d\Omega} d\Omega = \int_0^{2\pi}\int_0^\pi \frac{d\sigma}{d\Omega} \sin\theta\, d\theta\, d\phi$$

Total probability of scattering (any direction)

Physical interpretation: Effective area presented by target

Larger $\sigma$ → more likely to scatter
Not necessarily equal to geometric cross section

Quantum Mechanical Formula

Asymptotic wave function for scattering:

$$\psi(\vec{r}) \xrightarrow{r\to\infty} e^{i\vec{k}\cdot\vec{r}} + f(\theta,\phi)\frac{e^{ikr}}{r}$$

Incident plane wave + outgoing spherical wave

Scattering amplitude: $f(\theta,\phi)$

Differential cross section:

$$\frac{d\sigma}{d\Omega} = |f(\theta,\phi)|^2$$

Elastic vs Inelastic Scattering

Elastic: Kinetic energy conserved

$E_i = E_f$ (no internal excitation)
$|k_i| = |k_f|$
Example: Rutherford scattering at low energy

Inelastic: Internal energy changes

Target excited or deexcited
Particle production/absorption
Example: Compton scattering, resonance excitation

Scattering Length

Low-energy limit $(k \to 0)$:

$$f(\theta) \xrightarrow{k\to 0} -a_s$$

where $a_s$ is scattering length (constant, independent of angle)

Cross section:

$$\sigma = 4\pi a_s^2$$

$a_s > 0$: Repulsive potential dominates
$a_s < 0$: Attractive potential (may have bound state)

Classical vs Quantum

Classical (Rutherford):

$$\frac{d\sigma}{d\Omega} = \left(\frac{Z_1 Z_2 e^2}{16\pi\epsilon_0 E}\right)^2 \frac{1}{\sin^4(\theta/2)}$$

Point charges, Coulomb potential

Quantum corrections:

Diffraction effects when $\lambda \sim$ target size
Identical particle effects (symmetrization)
Spin effects
Resonances (not in classical theory)

Optical Theorem

Fundamental relation between forward scattering and total cross section:

$$\sigma_{tot} = \frac{4\pi}{k}\text{Im}[f(0)]$$

Imaginary part of forward scattering amplitude determines total cross section

Physical meaning: Interference between incident and scattered waves

Conservation of probability → optical theorem

Experimental Considerations

Thin target approximation: Single scattering events only
Detector acceptance: Finite solid angle $\Delta\Omega$
Energy resolution: Distinguish elastic from inelastic
Background subtraction: Multiple scattering, beam contamination
Luminosity: $\mathcal{L} = \Phi \times n_t$ (flux × target density)

Applications

Nuclear physics: Neutron cross sections for reactors, nucleon-nucleon scattering
Particle physics: Electron-positron annihilation, proton-proton collisions at LHC
Atomic physics: Electron-atom scattering, photoionization
Condensed matter: Neutron scattering from crystals, X-ray diffraction
Astrophysics: Stellar opacity, neutrino cross sections

Part VIII Overview Born Approximation →

Quantum Mechanics Course