X-ray diffraction (XRD)

X-ray diffraction (XRD) is the technique of shining a monochromatic X-ray beam on a crystalline sample and recording the angles and intensities of the beams it scatters. Atomic planes in a crystal are spaced by a few ångströms (0.1 nm), comparable to the wavelength of laboratory X-rays, so the lattice acts as a three-dimensional diffraction grating. The most common source is a copper anode, whose Cu Kα₁ line has a wavelength of 1.5406 Å (1.5418 Å for the weighted Kα₁/Kα₂ mean) and a photon energy of 8.05 keV; molybdenum Kα at 0.7107 Å is the usual alternative for single crystals. The diffraction angles give the lattice spacings, and the pattern of spacings and intensities identifies the phase.

Bragg's law and a worked silicon peak

Waves reflected from successive parallel planes of spacing $d$ add in phase when their path difference is a whole number of wavelengths:

$$2d\sin\theta = n\lambda,$$

where $\theta$ is the angle between the incident beam and the planes (half the scattering angle $2\theta$ that diffractometers report). This is the X-ray form of the Bragg condition that also governs fiber and waveguide gratings. For a cubic crystal of lattice constant $a$, the spacing of the $(hkl)$ planes is

$$d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}.$$

Silicon has $a = 5.431$ Å, so $d_{111} = 5.431/\sqrt{3} = 3.1356$ Å. With Cu Kα, $\sin\theta = 1.5406/(2 \times 3.1356) = 0.2457$, which gives $\theta = 14.22^\circ$ and a peak at $2\theta = 28.44^\circ$. The (220) and (311) reflections follow at 47.30° and 56.12°.

Because $\sin\theta \le 1$, planes with $d < \lambda/2$ cannot diffract at all; with Cu Kα that limit is 0.77 Å. The Kα line is itself a doublet (Kα₁ 1.54056 Å and Kα₂ 1.54439 Å), which splits the Si (111) peak by 0.07° in $2\theta$ and by more at high angles.

Powder and single-crystal measurements

In powder XRD, the sample contains many randomly oriented crystallites, so every set of planes finds some grains at the Bragg angle. The diffracted light forms cones, and a detector scanned in $2\theta$ records a one-dimensional pattern of peaks. Phase identification compares the peak list against reference databases; the same scan also yields lattice parameters, residual strain, texture and, with Rietveld refinement, quantitative phase fractions.

In single-crystal XRD, one crystal, typically 0.05–0.5 mm across, is rotated in the beam and an area detector records thousands of individual reflections. Their positions give the unit cell and their intensities, through the structure factor, give the atomic positions. Thin films are measured with related geometries: high-resolution rocking curves for epitaxial layers, grazing incidence for polycrystalline coatings, and reflectivity for thickness and roughness.

Crystallite size from peak width

Small crystallites broaden the peaks, because too few planes contribute to cancel the intensity just off the Bragg angle. The Scherrer equation estimates the mean size $\tau$ of the coherently diffracting domains:

$$\tau = \frac{K\lambda}{\beta\cos\theta},$$

where $\beta$ is the peak width in radians (FWHM in $2\theta$) after the instrumental contribution is removed, and $K \approx 0.9$ is a shape factor. Worked example: a Si (111) peak measured at 0.25° FWHM on an instrument whose own width is 0.10°. Subtracting in quadrature (appropriate for Gaussian profiles) leaves 0.229°, or $4.00\times10^{-3}$ rad, and

$$\tau = \frac{0.9 \times 1.5406\ \text{\AA}}{4.00\times10^{-3} \times \cos 14.22^\circ}$$

which is about 36 nm. Linear subtraction (appropriate for Lorentzian profiles) leaves 0.15° and gives 55 nm, a spread that shows how much the result depends on the profile model.

Several caveats apply. The Scherrer size is a volume-weighted domain size, which can be smaller than the grain size seen in a microscope; microstrain also broadens peaks and must be separated, for example with a Williamson-Hall plot across several reflections; $K$ ranges from about 0.8 to 1.0 depending on crystallite shape and width definition; and above roughly 100–200 nm the physical broadening becomes comparable to the instrumental width and the method loses precision.

Pitfalls

Sample displacement from the goniometer axis shifts all peaks systematically, and is often mistaken for a lattice-parameter change; an internal standard corrects it. Preferred orientation in pressed powders distorts relative intensities. Fluorescence from iron- or cobalt-rich samples under Cu radiation raises the background. Amorphous material gives only broad humps, so XRD alone underestimates poorly crystalline fractions.

Common questions

What is XRD used for?

Identifying crystalline phases in minerals, pharmaceuticals, cements, catalysts and battery materials; measuring lattice parameters, strain and film quality in semiconductors; and solving crystal structures from single crystals.

Why are X-rays used instead of visible light?

Diffraction requires a wavelength comparable to the spacing being probed. Visible light at 400–700 nm is roughly a thousand times longer than atomic plane spacings, so it cannot satisfy Bragg's law for a crystal lattice. Electrons and neutrons with a suitable de Broglie wavelength can, and electron and neutron diffraction complement XRD.

What does 2θ mean in an XRD pattern?

It is the angle between the incident and diffracted beams. The Bragg angle $\theta$ is half of it, and it is $\theta$ that enters Bragg's law and the Scherrer equation.

References: B. D. Cullity, S. R. Stock, Elements of X-Ray Diffraction, 3rd ed. (Prentice Hall, 2001); J. Als-Nielsen, D. McMorrow, Elements of Modern X-ray Physics, 2nd ed. (Wiley, 2011); E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 10.