Scattering: Cross-Sections and the Born Approximation · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

What scattered particles say about the target

Science · Quantum & Modern Physics · ages 23-24
Name ______________________   Date ____________

  1. In first Born theory, what is the elastic scattering amplitude proportional to, with q=k_f-k_i?

    • The potential evaluated at position r=q, without an integral
    • The squared magnitude of the potential's transform at every q
    • The potential's Fourier transform evaluated at q

  2. What does a scattering cross-section represent?

    • The target's geometric silhouette in every scattering experiment
    • An effective interaction area for the incident beam
    • The raw detector count before accounting for beam and exposure

  3. Two potentials giving the same measured differential cross-section must be identical.

    Circle one:   True   False

  4. Replace V by 2V while keeping beam energy and direction fixed. Both cases remain in first Born theory. By what factor does dσ/dΩ change?

    • 2
    • 4
    • 8

  5. A thin target gives 60 corrected events in ΔΩ=0.2 sr. The uniform flux is 100 particles/(cm² s), N=2 and t=5 s. What is the bin-average differential cross-section?

    • 0.3 cm²/sr
    • 1.5 cm²/sr
    • 3.0 cm²/sr

  6. A strong central potential is unsuitable for first Born theory. Which approach can describe its scattering without a weak-potential expansion?

    • Discard every angular-momentum channel except one without checking.
    • Use the first Born amplitude again with extra numerical digits.
    • Solve radial partial-wave equations and obtain the phase shifts.

  7. A report uses first Born theory for a strong potential without discussing its range, incident energy or scattering corrections. What check is missing?

    • A check that printing more digits makes the perturbation expansion valid.
    • A regime-specific check that neglected scattering corrections are small.
    • A check that all central potentials scatter only in one angular channel.

  8. At the same beam energy, two first Born patterns of absolute dσ/dΩ differ. Counts were divided by flux, target number, time and solid angle, not scaled to unit total. Does this alone prove different potential shapes?

    • No; different strengths of the same spatial shape can change intensity.
    • Yes; any intensity difference determines a different spatial shape uniquely.
    • Yes; intensity contains every phase needed to recover the shape uniquely.
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Answer key

For grown-ups. Fold this page away before handing over the rest.

What scattered particles say about the target W1-mt_xmN9KDfSp7-s1

  1. The potential's Fourier transform evaluated at q · The Born interaction integral contains exp(-iq·r)V(r), so f_B is proportional to the Fourier transform at q.
  2. An effective interaction area for the incident beam · A cross-section expresses interaction probability as an effective area, which need not equal a geometric target area.
  3. False · Measured intensity loses the amplitude's complex phase and may sample only a limited angular range. It does not generally identify a unique potential.
  4. 4 · The Born amplitude is linear in V. Doubling it multiplies its squared magnitude by four.
  5. 0.3 cm²/sr · 60/(100×2×5×0.2)=0.3 cm²/sr. Omitting t gives 1.5 numerically; omitting both N and t gives 3.0. Both omissions lose required exposure factors.
  6. Solve radial partial-wave equations and obtain the phase shifts. · For a central potential, radial partial-wave solutions and enough contributing channels can describe scattering without assuming weak V.
  7. A regime-specific check that neglected scattering corrections are small. · Born validity depends on the scattering regime. Strength alone is not enough; assess range, energy and the size of neglected corrections.
  8. No; different strengths of the same spatial shape can change intensity. · In first Born theory, replacing V by cV changes f_B to cf_B and absolute dσ/dΩ to |c|²dσ/dΩ. Strength can therefore change the pattern without changing spatial shape.
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