X-ray diffraction reads a crystal by listening for angles where scattered wavefronts reinforce rather than cancel. A collimated, monochromatic beam strikes a sample and—whenever the Bragg condition is met for a particular set of lattice planes—a sharp intensity spike emerges at the detector. The resulting 2θ spectrum encodes d-spacings, grain statistics, residual strain, and phase identity in a single measurement that averages over millions of grains without physical sectioning or chemical etching. No technique better balances throughput, depth, and crystallographic specificity for routine thin-film process control in semiconductor fabrication.

Bragg's law defines where peaks appear, but peak shape is where the physics lives. The condition

gives the scattering angle for each family of planes with spacing $d_{hkl}$, and for Cu Kα radiation ($\lambda = 0.1542$ nm) a TiN (111) plane at $d = 0.2449$ nm diffracts at $2\theta = 36.7°$. That angle is well-determined from geometry alone. What the geometry does not supply is the width, height, or asymmetry of the peak—those observables emerge from grain size distributions, microstrain fields, stacking faults, compositional gradients, and the instrument's own resolution function acting in superposition. Treating a diffraction spectrum as a simple peak-position list discards most of its structural information content and renders quantitative extraction of process-relevant parameters impossible.

The instrument broadens every peak before it reaches the detector. A Bruker D8 Advance, Rigaku Ultima IV, or Malvern Panalytical X'Pert³ each imposes an instrumental resolution function (IRF)—the convolution of source divergence, Soller slit acceptance angles, monochromator rocking curves, and detector channel width, wavelength-weighted for the Kα₁/Kα₂ doublet. The IRF is measured on a NIST SRM 660c lanthanum hexaboride standard and subtracted in quadrature from the observed peak width: $\beta_{\rm sample}^2 = \beta_{\rm measured}^2 - \beta_{\rm IRF}^2$. A peak with observed FWHM of 0.60° on an instrument whose IRF is 0.05° has an intrinsic sample contribution of approximately 0.598°—negligible here. On a high-resolution double-axis diffractometer, the IRF narrows to 0.003° and intrinsic widths of 0.010° become physically meaningful for epitaxial quality assessment.

Phase identification anchors every downstream calculation. Before Scherrer widths or lattice parameters mean anything, every peak in the spectrum must be indexed against a known phase. The ICDD PDF-4+ database (International Centre for Diffraction Data) contains d-spacing cards for over 400,000 crystalline phases; automated search-match routines compare measured d-spacings and relative intensities against these cards and rank candidates by figure of merit. For a physical vapor-deposited TiN film on Si(100) probed with Cu Kα radiation, the diagnostic (111) reflection appears at $2\theta = 36.7°$ and the (200) at $42.6°$; the Si substrate contributes a strong (400) reflection at $69.1°$. Confusing TiN (200) with a secondary phase such as Ti₂N or TiO would corrupt every subsequent structural parameter extraction—phase identification is therefore gated by ASTM E915 in any diffraction-based stress or grain-size measurement workflow, and SEMI process qualification records require XRD phase maps alongside sheet-resistance data.

Grain size and microstrain must be separated, not assumed to be independent. The Scherrer equation

with $K \approx 0.94$ for randomly oriented spherical crystallites and $\beta$ the IRF-corrected FWHM in radians, gives the coherence length perpendicular to the diffracting planes. It is exact only if microstrain contributes zero broadening—an assumption that is rarely valid in as-deposited PVD or CVD films under biaxial stress. The Williamson–Hall method corrects this by exploiting the different angular dependences of the two broadening mechanisms: size broadening scales as $1/\cos\theta$ and strain broadening scales as $\tan\theta$, so their combined contribution satisfies $\beta\cos\theta = K\lambda/D + 4\varepsilon\sin\theta$. Plotting $\beta\cos\theta$ against $4\sin\theta$ across at least two reflections yields a straight line whose y-intercept encodes grain size and whose slope encodes microstrain $\varepsilon$. For a 200 nm TiN PVD film on Si(100), the (111) peak at $2\theta = 36.7°$ shows an intrinsic FWHM of 0.59° (0.0103 rad) and the (200) peak at $2\theta = 42.6°$ shows 0.62° (0.0108 rad); the Williamson–Hall regression across these two reflections returns $D = 18$ nm coherent grain size and $\varepsilon = 0.14\%$ microstrain, both consistent with the biaxial compressive stress expected from the 6.8 ppm/K thermal expansion mismatch between TiN ($\alpha = 9.4$ ppm/K) and Si ($\alpha = 2.6$ ppm/K).

Lattice-parameter shifts convert diffraction into a non-contact stress gauge. Once phases are indexed, the d-spacing from each Bragg position gives the strained lattice parameter along the direction normal to those planes. For TiN with rock-salt structure (Fm3m) the bulk reference parameter is $a_0 = 0.4240$ nm; the (111) peak at $36.7°$ yields $d_{111} = 0.1542/(2\sin 18.35°) = 0.2449$ nm and hence $a = d_{111}\sqrt{3} = 0.4242$ nm—a 0.05% out-of-plane tensile shift relative to the unstressed bulk lattice. A $\sin^2\psi$ tilt series extends this to a biaxial stress tensor: $d$ is measured at successive $\psi$ tilts and the slope converted to stress via X-ray elastic constants $S_1$ and $\tfrac{1}{2}S_2$. For TiN, $S_1 = -3.3 \times 10^{-6}$ MPa$^{-1}$ and $\tfrac{1}{2}S_2 = 6.9 \times 10^{-6}$ MPa$^{-1}$; a $\Delta(d/d_0)/\Delta(\sin^2\psi)$ slope of 0.003 yields $-430$ MPa biaxial compressive stress, consistent with the 0.14% microstrain from the Williamson–Hall analysis.

Rocking-curve width ranks epitaxial quality and quantifies dislocation density. In an ω-scan (rocking curve), the detector angle $2\theta$ is fixed at a Bragg peak while the sample tilt ω is swept through several degrees. The resulting FWHM—the mosaicity—reflects the angular spread of coherent diffracting blocks. Device-grade III–V epitaxy achieves FWHM below 30 arcsec; magnetron-sputtered polycrystalline coatings reach 1°–3°. Threading dislocation density $\rho_D$ follows from the rocking-curve width via $\rho_D \approx \omega_{hkl}^2 / (9b^2)$, where $b$ is the Burgers vector magnitude. ASTM E915 governs diffractometer calibration for residual-stress measurement; SEMI M1 specifies wafer-bow limits that XRD stress maps enforce during process qualification.

The critical parameters from the TiN/Si(100) worked example are collected in the reference table below:

Specimen → [Mount on diffractometer stage; verify height alignment with reference flat] → [Select geometry: Bragg–Brentano for bulk polycrystal, parallel-beam for stress, GIXRD for surface layers, ω-scan for epitaxy] → [Collect θ–2θ scan 30°–80°, step 0.02°, Cu Kα] → [Identify peaks above 3σ noise threshold] → [Search-match against ICDD PDF-4+ by d-spacing and intensity] → Phase identified? → No → [Check texture, amorphous halo, secondary phases; adjust incident optics and rescan] → Phase identified?
Phase identified? → Yes → [Fit each peak with pseudo-Voigt profile; extract 2θ₀, FWHM β, integrated area] → [Subtract IRF in quadrature: β_sample² = β_meas² − β_IRF²] → [Williamson–Hall plot: β cosθ vs 4 sinθ across ≥2 reflections] → [D from y-intercept = Kλ/D; ε from slope] → [Optional: sin²ψ series for biaxial stress; ω-scan for mosaicity; Rietveld refinement for phase fractions] → Report: d-spacings, D, ε, biaxial stress σ, phase weight fractions

The choice of geometry determines which structural information is accessible. In standard Bragg–Brentano configuration, only planes parallel to the sample surface diffract, so a strongly (111)-textured TiN film shows that family prominently and the (200) family only through weak mosaic tails—insufficient for Williamson–Hall analysis. A chi-tilt stage or Göbel mirror plus channel-cut analyzer removes this constraint and enables arbitrary (hkl) access. Grazing-incidence XRD (GIXRD) locks the incident beam at 0.3°–1.0° and sweeps only the detector; at these angles penetration depth is 10–50 nm, isolating the film from substrate reflections without cross-sectioning. GIXRD and conventional θ–2θ scans are complementary—the former for near-surface phase identification, the latter for grain-size and stress analysis across the full film.

Rietveld refinement elevates XRD from peak identification to quantitative weight-fraction analysis. A physically parameterized forward model—unit-cell dimensions, site occupancies, Debye–Waller factors, preferred-orientation, and peak-shape descriptors—is fit simultaneously to the full observed pattern by least squares. For a TiN/TiO₂/Ti multilayer stack whose three phases overlap in the 30°–55° window, Rietveld decomposition returns weight fractions at 1–3 wt% precision. FullProf, GSAS-II, and Bruker TOPAS are the standard implementations.

For process control, XRD is deployed after deposition to verify phase and grain size, and after annealing to detect phase transformation or stress relaxation. A TiN barrier converting from FCC to hexagonal ε-TiN above 600°C shows the (111) peak at 36.7° splitting into two new reflections with a drop in (200) intensity—a diagnostic taking under three minutes on a strip-detector diffractometer. The diagram below shows the simulated Cu Kα 2θ spectrum (left) and the Williamson–Hall regression (right) for the TiN/Si(100) worked example.

The shallow positive slope of the fit line confirms that size broadening dominates but microstrain is nonzero; the y-intercept of 0.00805 rad gives $D = K\lambda / 0.00805 = 18$ nm, cross-checking the regression. A purely size-broadened sample would give a horizontal line; a purely strain-broadened sample would pass through the origin—real PVD films sit between these extremes, requiring regression to separate the two contributions without systematic error.

Read x-ray diffraction through a Bragg-condition-and-peak-shape-inversion lens rather than a peak-position-only lens: the angle at which a peak appears tells you the d-spacing, but the width, asymmetry, and relative intensity of that peak encode grain size, microstrain, texture, and phase fractions—all of which require fitting a physical forward model to the observed pattern, not merely locating the maximum of the intensity distribution.