--- title: "Scattering: Cross-Sections and the Born Approximation" description: "A cross-section expresses scattering probability as an effective area. In first Born scattering, the amplitude is proportional to the Fourier transform of the potential at the wave-vector transfer. Me" canonical: https://lightmysky.com/learn/science/scattering-cross-sections-and-the-born-approximation-mt_xmN9KDfSp7 source: https://lightmysky.com/learn/science/scattering-cross-sections-and-the-born-approximation-mt_xmN9KDfSp7.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Scattering: Cross-Sections and the Born Approximation A cross-section expresses scattering probability as an effective area. In first Born scattering, the amplitude is proportional to the Fourier transform of the potential at the wave-vector transfer. Measured intensity constrains the potential but does not generally determine it uniquely. Subject: Science · Area: Quantum & Modern Physics · Ages 23 to 24 Page: https://lightmysky.com/learn/science/scattering-cross-sections-and-the-born-approximation-mt_xmN9KDfSp7 ## Ready when they can - Defines a differential cross-section in terms of counts, beam flux and solid angle - Relates the Born amplitude to the Fourier transform of the potential - States when the approximation is trustworthy and what a partial-wave treatment adds ## Lesson: What scattered particles say about the target A cross-section is an effective interaction area, not necessarily the target's geometric silhouette. For a uniform beam and a thin target of N independent scatterers, a small angular bin gives dσ/dΩ approximately C/(jNtΔΩ). Here C is the event count corrected for background and efficiency, j is incident particles per area per time, t is exposure time and ΔΩ is the bin's solid angle. Solid angle is the area a direction cone cuts out on a unit sphere. The units are area per steradian. For nonrelativistic elastic scattering from local V(r), first Born replaces the full wave in the interaction integral by the incident plane wave. With q=k_f-k_i, f_B(q)=-m/(2pi hbar²) integral exp(-iq·r)V(r)d³r. The spatial Fourier transform sums V(r) weighted by exp(-iq·r). Here m is particle mass, or reduced mass for two bodies. The k vectors are wave vectors; momentum transfer is hbar q. In this convention, dσ/dΩ=|f_B|². Intensity gives |f|² and loses f's complex phase. Limited angles sample only part of the transform, so the potential is not generally reconstructed uniquely. Here patterns are absolute differential cross-sections obtained by dividing corrected counts by jNtΔΩ, not curves scaled to unit total. Different absolute patterns need not imply different spatial shapes: changing the strength of a same-shaped potential can change intensity. Control the experimental conditions. **Example.** Suppose C=30, j=50 particles/(cm² s), N=3, t=2 s and ΔΩ=0.5 sr. Divide by jNtΔΩ=150/(cm²) sr to obtain an average dσ/dΩ=0.2 cm²/sr in that bin. Use corrected counts and keep the target and exposure factors. In perturbative scattering, a golden-rule transition rate divided by incident flux gives a cross-section. *(drawing: This scaling assumes both potentials remain in the first Born regime, with the same beam energy and scattering direction.)* For a central potential V(r), expand the wave in angular-momentum channels and solve each radial equation. The resulting partial-wave phase shifts describe scattering without assuming a weak potential. Include enough contributing channels and impose the radial solutions' scattering boundary conditions. Partial waves are not an automatic replacement for every noncentral or multichannel scattering problem. First Born theory is useful when replacing the interaction-region wave by the incident plane wave changes the result only a little. Strength, range and incident energy all matter; resonances can make a perturbative estimate poor. Comparing a suitable dimensionless parameter, higher Born terms or an exact calculation can test the approximation. A strong-potential result needs a regime-specific check, not more decimal places. **Recap.** Normalize counts by flux, target number, time and solid angle. Born intensity is the squared magnitude of a Fourier-related amplitude. Check its validity; intensity alone does not fix the potential. *Original lesson, questions and flashcards written for this dataset about scattering cross-sections and the first Born approximation.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Particle Accelerators, Detectors and Cross-Sections](https://lightmysky.com/learn/science/particle-accelerators-detectors-and-cross-sections-mt_euJtxtxXyV) - [Time-Dependent Perturbation Theory and Fermi's Golden Rule](https://lightmysky.com/learn/science/time-dependent-perturbation-theory-and-fermis-golden-rule-mt_zDSASGF7SC)