Questions: X-Ray Diffraction and Crystal Identification

Question 1

An XRD scan of copper (FCC crystal structure) produces no peak at the 2θ position predicted by Bragg's law for the {100} planes. The most accurate explanation is:

• A. The X-ray wavelength is too long to satisfy Bragg's law for the large {100} d-spacing at achievable angles
• B. Copper atoms are too light to scatter X-rays effectively from {100} planes
• C. Destructive interference between waves scattered by corner atoms and face-center atoms in the FCC unit cell eliminates the {100} reflection — a systematic absence
• D. The {100} reflection is present but overlaps with a stronger adjacent peak and cannot be resolved

Answer: (C)

Bragg's law gives a necessary condition for diffraction (correct path length difference between planes), but the structure factor determines whether waves scattered by different atoms within the same unit cell add constructively or destructively. For FCC metals, planes with mixed h, k, l Miller indices (like {100}: indices 1, 0, 0 — mixed) experience destructive interference between rays from corner atoms and face-center atoms. This systematic absence is not a measurement artifact but an intrinsic consequence of the FCC arrangement. Only planes with all-odd or all-even Miller indices ({111}, {200}, {220}, {311}…) produce peaks. This pattern of absences fingerprints the FCC structure before a single peak position is measured.

Question 2

An engineer measures the XRD pattern of a shot-peened steel surface (which introduces compressive residual stress) and compares it to an unstressed reference. How will the diffraction peaks shift?

• A. Peaks shift to lower 2θ (larger d-spacing), because compressive in-plane stress pushes lattice planes apart in the normal direction
• B. Peaks shift to higher 2θ (smaller d-spacing), because compressive in-plane stress squeezes lattice planes closer together in the direction normal to the surface via the Poisson effect
• C. Peak positions do not change; only peak widths increase with compressive residual stress
• D. Peaks shift to lower 2θ because shot-peening increases the lattice parameter uniformly in all directions

Answer: (B)

Compressive residual stress in the surface plane causes the lattice to contract in the direction perpendicular to the surface via the Poisson effect — in-plane compression leads to through-thickness tension, reducing the spacing between planes parallel to the surface. XRD typically measures d-spacings of planes parallel to the surface (perpendicular to the normal direction). A smaller d-spacing means a larger sinθ by Bragg's law (nλ = 2d sinθ: smaller d → larger θ → larger 2θ). Thus compressive stress shifts peaks to higher 2θ. This is the basis for non-destructive residual stress measurement by XRD in engineering components.

Question 3

The pattern of systematic absences in an XRD diffractogram can distinguish an FCC crystal from a BCC crystal of the same element, independently of the lattice parameter value.

• T. True
• F. False

Answer: True

Systematic absences are determined by crystal structure type, not by lattice parameter. FCC crystals allow only reflections with all-odd or all-even Miller indices: {111}, {200}, {220}, {311}… BCC crystals allow only reflections where h+k+l is even: {110}, {200}, {211}, {220}… These different selection rules produce distinct peak patterns. FCC shows a peak at {111} but not {110}; BCC shows {110} but not {111}. By examining which peaks are present and absent, you can identify the crystal structure type before any quantitative analysis of lattice parameter. This is routinely used to identify phase transformations in steel (FCC austenite → BCC ferrite produces a completely different peak set).

Question 4

Any crystal planes that satisfy Bragg's law — for which the equation nλ = 2d sinθ has a solution at an achievable angle — will produce an observable diffraction peak.

• T. True
• F. False

Answer: False

Bragg's law is a necessary but not sufficient condition for diffraction. The structure factor — which accounts for interference between waves scattered by all atoms within the unit cell — can produce complete destructive interference (systematic absence) for planes that are geometrically permitted by Bragg's law. For FCC metals, the {100} planes satisfy Bragg's law geometrically, but the structure factor is zero: face-center atoms produce waves exactly out of phase with corner atom waves, eliminating the peak. Bragg's law predicts where a peak could appear; the structure factor determines whether it actually does. This is one of the key subtleties of XRD: geometry and atomic arrangement are both required for a complete prediction.

Question 5

Why do amorphous materials produce broad, diffuse XRD patterns rather than sharp peaks, and what does this reveal about the relationship between crystal order and diffraction sharpness?

Model answer: Sharp XRD peaks arise from long-range periodic order: when thousands of identical planes are arranged periodically, their scattered X-rays add constructively only at precise Bragg angles, producing narrow peaks. Amorphous materials lack long-range periodicity — atomic positions are only correlated over a few bond lengths, not across extended crystallites. Without periodic repetition extending over many unit cells, scattered X-rays add constructively over a broad range of angles rather than at a precise angle, producing a diffuse hump. Diffraction sharpness is therefore a direct measure of long-range order: the narrower the peak, the larger the coherently scattering crystallite domain, quantified by the Scherrer equation L = Kλ/(β·cosθ).

This principle has practical consequences: nanocrystalline materials with 5–10 nm crystallites produce noticeably broad peaks even though they are crystalline; truly amorphous materials have no crystallite size at all and produce only a broad background hump. XRD can therefore distinguish crystalline from amorphous phases in a mixture and estimate crystallite size from peak breadth — two capabilities that come directly from understanding the relationship between periodic order and diffraction sharpness.