50 XRD Viva, Interview & GATE/CSIR-NET Questions — With Fully Verified Solutions

nλ = 2d sinθ. Two parallel X-ray beams reflect off adjacent lattice planes separated by distance d. The path difference between the two reflected beams is 2d sinθ, where θ is measured from the plane itself (not the surface normal). Constructive interference — and therefore a detectable diffraction peak — occurs only when this path difference equals an integer number of wavelengths, nλ.

Bragg's Law requires λ to be comparable to d, the interatomic plane spacing (typically 0.1–0.3 nm). Visible light (400–700 nm) is roughly a thousand times too long to diffract off atomic planes. X-rays — CuKα at λ = 0.15406 nm being the standard laboratory source — are the correct order of magnitude. Electrons also satisfy this condition and are used in electron diffraction, but they interact far more strongly with matter, limiting penetration depth; this makes them suited to surface techniques (e.g. RHEED) rather than routine bulk powder phase identification.

Both arise from 2p→1s electron transitions in copper, but from the two spin-orbit-split initial states (2p₃/₂ and 2p₁/₂), giving λ(Kα1) = 0.1540 nm and λ(Kα2) = 0.1544 nm, with an intensity ratio of approximately 2:1. At high 2θ, this doublet causes visible peak splitting or asymmetric broadening unless a monochromator removes Kα2 or it is stripped mathematically during profile fitting.

Unfiltered Kβ (λ = 0.1392 nm) would produce a second, overlapping diffraction pattern from the same sample at different angles, complicating peak indexing. A thin nickel foil filter is used because nickel's K-absorption edge (0.1488 nm) lies precisely between the Cu Kβ and Kα wavelengths — so nickel strongly absorbs the shorter-wavelength Kβ while transmitting most of the longer-wavelength Kα.

An X-ray tube generates a divergent beam; divergence slits shape it; the beam strikes a flat powder sample mounted at the centre of a goniometer circle. The sample rotates at θ while the detector rotates at 2θ (coupled θ–2θ scanning), keeping the sample always bisecting the angle between the incident and diffracted beams. This focusing geometry maximises measured intensity while preserving angular resolution.

d-spacing is the perpendicular distance between adjacent, parallel lattice planes of a given (hkl) family. For a cubic system: d = a / √(h² + k² + l²), where a is the cubic lattice parameter. Other crystal systems require progressively more complex formulas involving multiple lattice constants and, for non-orthogonal systems, the interaxial angles — see Unit Cell and Lattice Parameters in Crystallography for the full derivation across all seven systems.

The structure factor F(hkl) = Σⱼ fⱼ exp[2πi(hxⱼ + kyⱼ + lzⱼ)] sums the scattering contributions of every atom j in the unit cell, weighted by its atomic scattering factor fⱼ and its phase (determined by its fractional coordinates). Diffracted intensity is proportional to |F(hkl)|². If F(hkl) evaluates to zero for a particular (hkl), that reflection is systematically absent — even though it geometrically satisfies Bragg's Law.

For BCC: reflections occur only when (h + k + l) is even. For FCC: reflections occur only when h, k, l are all even or all odd (unmixed parity). These rules follow directly from evaluating the structure factor for the basis atoms of each non-primitive lattice — for the "forbidden" index combinations, the scattering contributions from the extra basis atom(s) destructively cancel the contribution from the corner atom.

Finite beam divergence, slit widths, sample transparency, axial divergence, and detector resolution all contribute a baseline peak width present even in a perfect, strain-free, large-crystallite reference material. This instrumental FWHM is measured at each 2θ using a certified standard (e.g. LaB6 or NIST SRM 640 silicon) and subtracted, typically in quadrature for Gaussian profiles, before applying the Scherrer equation — see FWHM of X-ray Peaks in XRD for a step-by-step walkthrough of measuring and correcting FWHM.

D = Kλ / (β cosθ), where D is the mean crystallite size, K is a shape factor (~0.9–0.94), λ is the X-ray wavelength, β is the FWHM in radians after instrumental correction, and θ is the Bragg angle. Its key limitation is that it assumes broadening arises solely from crystallite size, ignoring microstrain — this typically gives only an approximate, lower-bound estimate, and it measures the coherently diffracting domain, not the physical particle size seen under a microscope. Try the Scherrer XRD Calculator to compute this directly from your own peak data.

It exploits their different θ-dependence: size broadening scales as 1/cosθ, while strain broadening scales as tanθ. Plotting β cosθ against 4 sinθ for several peaks gives a straight line whose y-intercept yields Kλ/D (crystallite size) and whose slope yields the microstrain ε.

Rietveld refinement is a whole-pattern, least-squares fitting method that models the entire diffraction profile — not just individual peak positions — using crystal structure parameters (lattice constants, atomic coordinates, occupancies, thermal parameters), instrumental parameters, and profile shape functions simultaneously, minimising the residual between the observed and calculated intensity at every data point. It enables quantitative phase analysis and precise structure determination in ways single-peak fitting cannot.

In an ideal powder, crystallites are randomly oriented, so all (hkl) planes are statistically equally represented. Preferred orientation arises when crystallites have a non-random orientation distribution — common in rolled sheets, thin films, or platy/needle-shaped particles — causing some peaks to appear anomalously intense and others suppressed relative to the ICDD reference pattern, which can be mistaken for a phase-purity issue if not recognised.

Crystalline materials have long-range periodic order, so Bragg's Law is satisfied sharply at discrete angles for each (hkl), producing sharp peaks. Amorphous materials lack long-range order (retaining only short-range order over a few atomic distances), so there is a continuous distribution of interatomic spacings — no single d satisfies Bragg's Law exactly, producing one or two broad, diffuse humps instead of sharp peaks. See Amorphous vs. Crystalline XRD Patterns Explained for worked examples of both pattern types.

A crystallite is a coherently diffracting domain — a region of essentially perfect, continuous lattice. A physical particle (as seen by TEM or SEM) can contain many crystallites separated by grain boundaries, twin boundaries, or stacking faults. XRD-measured crystallite size is therefore always less than or equal to the physically observed particle size; the two coincide only for single-crystalline particles with no internal boundaries.

Indexing assigns each observed peak to a specific (hkl) plane family, which — once matched to the correct crystal system's d-spacing formula — yields the lattice parameters (a, b, c, α, β, γ). It also confirms phase identity when cross-checked against ICDD or COD reference patterns and reveals the crystal system and symmetry — see Introduction to Crystal Structure for a refresher on how the seven crystal systems relate to these indices.

Powder XRD uses a polycrystalline sample with randomly oriented crystallites, producing a 1D intensity-versus-2θ pattern in which each peak sums contributions from all crystallites satisfying Bragg's Law at that angle — this collapses 3D orientation information. Single-crystal XRD instead rotates one crystal systematically to collect a full 3D reciprocal-space dataset, allowing complete structure solution (atomic positions, bond lengths and angles) rather than only phase identification and lattice parameters.

The reciprocal lattice is a mathematical construction in which each point corresponds to a real-space set of lattice planes (hkl); the vector to that point has magnitude 1/d(hkl) and is perpendicular to those planes. It reframes Bragg's Law geometrically through the Ewald sphere construction: a diffraction peak occurs whenever a reciprocal lattice point intersects a sphere of radius 1/λ. This makes diffraction geometry far easier to visualise and compute than working directly in real space.

The atomic scattering factor f(θ) quantifies how strongly a single atom scatters X-rays, computed as the Fourier transform of the atom's electron density. It decreases with increasing sinθ/λ because at larger angles, the finite spatial extent of the electron cloud causes destructive interference between waves scattered from different parts of the same atom — unlike at θ = 0, where all electrons scatter exactly in phase. Heavier atoms, having more electrons, systematically show higher f at every angle.

These are certified, highly crystalline, strain-free, large-grain materials with precisely known lattice parameters. They are used to calibrate 2θ zero-offset and sample-displacement errors, and to measure the instrumental peak-broadening function required for accurate Scherrer or Williamson-Hall analysis.

GIXRD keeps the incident beam at a fixed, very shallow angle (often at or below the critical angle for total external reflection) while the detector scans 2θ. This confines the X-ray penetration depth to a few nanometres to microns, making it ideal for characterising thin films without an overwhelming substrate signal — unlike a conventional θ–2θ scan, which probes much deeper into the sample.

X-ray scattering power scales roughly with atomic number Z (the number of electrons). Hydrogen has only one electron, so its contribution to diffracted intensity is extremely weak next to heavier atoms and is often effectively invisible in an X-ray structure refinement. Neutron diffraction — which scatters off atomic nuclei rather than electron density — is used instead when hydrogen positions must be determined precisely.

Residual microstrain is non-uniform elastic lattice distortion — from dislocations, grain-boundary stress, quenching, or cold-working — that produces a distribution of d-spacings around the mean, broadening peaks without necessarily shifting their centroid. This distinguishes it from macrostress, which shifts peak position uniformly. It is quantified using Williamson-Hall analysis, or more rigorously via the sin²ψ method for residual stress measurement.

By comparing the experimental pattern, peak-by-peak (both position and relative intensity), against the ICDD or COD reference pattern for the target phase. Absence of any unindexed or extra peaks indicates single-phase material; presence of secondary peaks suggests a secondary phase — for example, a pyrochlore secondary phase is a commonly reported impurity in donor/acceptor-doped bismuth sodium titanate (BNT) ceramics.

The key sources are sample displacement from the diffractometer's focusing circle, 2θ zero-offset misalignment, and axial divergence — all angle-dependent, systematic errors. They are minimised by using an internal standard mixed with the sample, by extrapolation methods that weight high-angle peaks more heavily (since these errors approach zero as θ → 90°, e.g. the Nelson-Riley function), or by full-pattern Rietveld refinement, which treats these corrections as refinable instrumental parameters.