X-Ray Beamline Optics

What is modelled

A crystal reflects X-rays only at the angle where the waves scattered by successive atomic planes add up. Two parallel crystals pick one energy and send the beam on in the original direction, shifted up. A mirror at a very small grazing angle reflects X-rays by total external reflection, but only below an energy set by its density, so it removes the higher harmonics that the crystal also passes.

Equations

• Bragg's law n λ = 2 d sinθ, λ = 12.3984 / E (Å, keV).
• Refractive index n = 1 - δ + iβ, δ = re λ² ne / 2π; critical angle θc = √(2δ), so it scales as 1/E.
• Fresnel reflectivity at grazing angle; above θ_c approximately [(θ - √(θ² - θc²)) / (θ + √(θ² - θc²))]².
• Bandwidth ΔE/E = √(Λ² + (Δθ cotθ)²) with Λ the intrinsic width of the crystal reflection.
• Fixed exit: the beam offset is 2 g cosθ, so the gap is g = h / (2 cosθ).

Checks run in the page

• Bragg's law solved by bisection against the closed form, including the third-order harmonic.
• The exact Fresnel reflectivity against the grazing-angle formula and total reflection below the critical angle.
• The bandwidth from a finite difference of E(θ) against the cotθ formula.
• A ray traced through two parallel crystals by the law of reflection against the fixed-exit offset.

Honest simplifications

The intrinsic crystal width is a single number you set, not computed from the structure factor, the mirror is a perfect flat with one electron density and a small fixed absorption, and polarisation, heat load, roughness and source size are left out. Mirror angles and beam height are drawn exaggerated; the crystal angles are true. The 3D lab is a fictional beamline.

• Raise the photon energy past the mirror cut-off and watch the reflectivity collapse.
• Choose Si(220): the bandwidth narrows but the Bragg angle is larger at the same energy.
• Choose Wide divergence: the angular spread, not the crystal, then sets the energy resolution.