Bragg diffraction

Bragg geometry

For monochromatic radiation and parallel planes separated by d, the path difference is Δ=2d sinθ. The angle θ is measured from the plane, not from its normal. A positive Bragg order n satisfies nλ=2d sinθ. The deflection from the incident beam's forward direction is 2θ. Both d and λ are entered in nanometres.

The sketch shows one incident and one geometrically specular outgoing ray at the selected θ. It does not show several orders being diffracted simultaneously for this one incident geometry. Five horizontal planes are drawn with schematic spacing; their displayed pixel spacing is not a length scale. At θ=0 the rays lie along a plane and the deflection is zero. At θ=90° the incident and outgoing rays share a line but point in opposite directions.

Allowed orders

The Bragg angle is θ_B=asin(nλ/(2d)). Orders with nλ>2d have no real Bragg angle. Equality is allowed and gives θ_B=90°, with deflection 180°. Floating-point equality admits only a tiny arithmetic tolerance: at most 4ε·max(nλ,2d), where ε is JavaScript Number.EPSILON. A ratio just below one retains its actual angle; only exact equality or a rounding-sized excess is set to 90°.

Max. n controls which positive orders are listed and marked, from 1 to 5. It does not restrict the interference calculation or the count of all possible orders, which can reach 20 within the input range. Forbidden rows remain visible. Use angle explicitly selects an allowed row's full-precision angle and recalculates. The residual is Δ−nλ at the selected angle; it is not an instrument tolerance or a measured peak width.

Five-plane interference

The model assumes five coherent, equal, weak scattering amplitudes. With q=Δ/λ, the normalized intensity is I=|Σ_j=0…4exp(2πijq)|²/25. Thus 0≤I≤1, and integer q gives a maximum. Between successive integer orders there are zeros at q=m/5 when m is not divisible by five. The model includes the zero-order maximum at θ=0; the table lists positive orders only.

This is a finite-plane interference model, not a quantitative crystal diffraction intensity prediction. It includes no atomic form factors, crystal basis or systematic absences, absorption, polarization, multiple scattering, mosaicity or instrumental broadening. In particular, its normalized I is not a measured reflectivity. The whole 0…90° angle scan uses the same fixed normalization, independent of the plotted samples and listed orders. The scan is not an experimental rocking curve.

Inputs and state

Use d from 0.08 to 0.5 nm, λ from 0.05 to 0.25 nm, and continuous θ from 0 to 90°. Max. n is an integer from 1 to 5. Calculate is explicit; editing a field removes the old result, and Reset keeps the fields. Returning within the same page session retains a calculated result. Reloading restores valid inputs only and requires Calculate again. Calculations stay in the browser.

Sources

IUCr: Bragg's law explains the path-difference condition. Fifty Years of X-ray Diffraction, §6.1 discusses coherent addition of scattered waves. The five-equal-plane model is the explicit teaching simplification used here.