1.6 X-ray diffraction

Principles of X-ray diffraction

X-ray diffraction (XRD) exploits the fact that X-ray wavelengths are comparable to interatomic spacings in crystals. When X-rays scatter off a periodic array of atoms, the scattered waves interfere constructively only at specific angles, producing a diffraction pattern that encodes the crystal's atomic arrangement. This makes XRD one of the most direct ways to connect microscopic structure to macroscopic material properties.

X-ray properties

X-rays are electromagnetic waves with wavelengths roughly 0.01 to 10 nm. In a typical lab source, electrons are accelerated into a metal target (commonly copper or molybdenum), and the rapid deceleration plus electronic transitions in the target produce characteristic X-ray lines. Copper $K_\alpha$ radiation, for example, has $\lambda \approx 0.154$ nm, which sits right in the range of typical lattice spacings.

• X-rays penetrate deeply into most solids, making XRD a non-destructive technique.
• Scattering occurs primarily off the electron clouds of atoms, so heavier elements (higher atomic number $Z$) scatter more strongly than lighter ones.
• Coherent (elastic) scattering, where the outgoing wave keeps the same wavelength as the incoming wave, is what produces useful diffraction patterns.

Interaction with matter

X-rays interact with matter through several mechanisms, but only elastic (Thomson) scattering contributes to diffraction:

• Elastic scattering preserves the wavelength of the incident X-ray and is the basis of all diffraction analysis.
• Photoelectric absorption ejects core electrons from atoms, reducing transmitted intensity and sometimes producing fluorescence background.
• Compton scattering is an inelastic process where X-rays lose energy to loosely bound electrons, contributing to diffuse background.
• Pair production requires photon energies above 1.02 MeV (far above typical XRD energies of ~8–17 keV) and is not relevant to standard diffraction experiments.

For routine XRD work, you mainly care about elastic scattering (signal) and photoelectric absorption (signal loss and fluorescence background).

Bragg's law

Bragg's law gives the condition for constructive interference from a set of parallel lattice planes:

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

• $n$ is the diffraction order (a positive integer).
• $\lambda$ is the X-ray wavelength.
• $d$ is the spacing between adjacent lattice planes.
• $\theta$ is the angle between the incident beam and the plane (not the surface normal).

The physical picture: waves reflected from successive planes travel an extra path length of $2d\sin\theta$. When that extra path equals a whole number of wavelengths, the waves add constructively and you see a diffraction peak. If you measure $\theta$ and know $\lambda$, you can solve directly for $d$.

Crystal structure analysis

Diffraction patterns contain far more than just plane spacings. By analyzing peak positions, intensities, and systematic absences, you can determine the full atomic arrangement inside the unit cell, including bond lengths, site occupancies, and the space group.

Reciprocal lattice concept

The reciprocal lattice is a mathematical construction that makes diffraction geometry much easier to visualize. Each point in reciprocal space corresponds to a set of lattice planes in real space.

Given real-space primitive vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$, the reciprocal lattice vectors are:

$\mathbf{b}_1 = 2\pi\frac{\mathbf{a}_2 \times \mathbf{a}_3}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)}$

(and cyclic permutations for $\mathbf{b}_2$ and $\mathbf{b}_3$). The denominator is just the unit cell volume $V$.

The Ewald sphere construction ties this to experiment: draw a sphere of radius $1/\lambda$ in reciprocal space centered on the crystal. A diffraction peak appears whenever the sphere intersects a reciprocal lattice point. This geometric picture explains why rotating a crystal (or using a powder with random orientations) is necessary to observe multiple reflections.

Structure factor

The structure factor determines the intensity of each diffraction peak. It accounts for how atoms within the unit cell contribute to scattering at a given reflection $(hkl)$:

$F_{hkl} = \sum_j f_j \, e^{2\pi i(hx_j + ky_j + lz_j)}$

• $f_j$ is the atomic scattering factor of atom $j$ (depends on element and scattering angle).
• $(x_j, y_j, z_j)$ are the fractional coordinates of atom $j$ in the unit cell.
• $(h, k, l)$ are the Miller indices of the reflection.

The measured intensity is proportional to $|F_{hkl}|^2$. Because $F_{hkl}$ is complex, you lose phase information in the measurement. This is the famous phase problem in crystallography.

Systematic absences

Some reflections that Bragg's law predicts will have $F_{hkl} = 0$ due to destructive interference among atoms within the unit cell. These missing peaks are called systematic absences, and they're extremely useful for identifying the space group.

Common examples:

• BCC lattice: reflections with $h + k + l = \text{odd}$ are absent.
• FCC lattice: reflections are present only when $h, k, l$ are all even or all odd (i.e., mixed indices are absent).
• Glide planes and screw axes produce absences along specific rows or zones of reflections.

By cataloging which reflections are missing, you can narrow down the crystal's symmetry and space group before doing any detailed refinement.

Experimental techniques

Different sample types call for different diffraction geometries. The choice depends on whether you have a single crystal, a powder, or a thin film, and on what information you need.

Powder diffraction

Powder samples contain many tiny crystallites in random orientations, so all possible Bragg reflections are sampled simultaneously. The result is a pattern of intensity versus $2\theta$.

• Sample preparation is straightforward: grind, pack, and measure.
• Well-suited for phase identification (matching peaks against databases like ICDD/PDF), quantitative phase analysis, and lattice parameter refinement.
• The main drawback is peak overlap, especially for low-symmetry structures with many closely spaced $d$-values. Some 3D structural detail is also lost because orientational information is averaged out.

Single crystal methods

A single crystal gives discrete diffraction spots rather than rings, preserving full 3D information.

• The rotation method (used in modern four-circle diffractometers) collects reflections by systematically rotating the crystal.
• The Laue method uses a broad-spectrum (white) X-ray beam so many reflections satisfy Bragg's law simultaneously without rotation. This is fast and useful for determining crystal orientation.
• Single-crystal data yield the most precise atomic positions, bond lengths, and bond angles.
• The main challenge is growing a crystal of sufficient size and quality (typically > 50 μm for lab sources).

Synchrotron radiation

Synchrotron facilities produce X-rays by bending high-energy electrons in a storage ring. The resulting beam is orders of magnitude brighter than a lab source.

• Tunable wavelength lets you choose $\lambda$ to optimize contrast or exploit anomalous scattering near an element's absorption edge.
• High brilliance enables measurements on very small crystals, weakly scattering samples, or rapid time-resolved experiments.
• Applications range from protein crystallography to in-situ studies of battery materials during charge/discharge cycles.

Diffraction patterns

Interpretation basics

Reading a diffraction pattern involves extracting three things from each peak:

1. Position (the $2\theta$ angle) gives the interplanar spacing $d$ via Bragg's law.
2. Intensity relates to $|F_{hkl}|^2$ and tells you about atomic positions and types within the unit cell. Multiplicity (how many symmetry-equivalent planes contribute to one peak) also affects intensity.
3. Shape and width carry information about crystallite size and microstrain. The Scherrer equation, $\tau = \frac{K\lambda}{\beta\cos\theta}$, estimates average crystallite size $\tau$ from peak broadening $\beta$.

Background signal comes from air scattering, fluorescence, thermal diffuse scattering, and any amorphous content in the sample.

Miller indices

Miller indices $(hkl)$ label sets of parallel lattice planes. To find them for a given plane:

1. Determine where the plane intercepts the three crystal axes (in units of the lattice parameters).
2. Take the reciprocals of those intercepts.
3. Clear fractions to get the smallest set of integers.

For example, a plane intercepting at $a, \infty, \infty$ gives reciprocals $(1, 0, 0)$, so it's the $(100)$ plane. Low-index planes tend to have large $d$-spacings and often produce the strongest reflections.

Intensity analysis

Extracting accurate structure factors from measured intensities requires several corrections:

• Lorentz-polarization factor: corrects for geometric effects (how long a reflection stays in the diffracting condition) and the partial polarization of the scattered beam.
• Debye-Waller (temperature) factor: atoms vibrate about their equilibrium positions, which reduces peak intensities and increases diffuse background. The effect grows with temperature and with $\sin\theta/\lambda$.
• Absorption correction: important for strongly absorbing samples or transmission geometry.
• Extinction: in nearly perfect crystals, the diffracted beam can be re-diffracted, reducing observed intensities for the strongest reflections.

After all corrections, the integrated intensity of a reflection is proportional to $|F_{hkl}|^2$.

Advanced applications

Thin films vs. bulk materials

Thin film XRD uses grazing incidence geometry (very small incident angles, often < 1°) to keep the X-ray path within the film rather than penetrating into the substrate.

• Grazing incidence XRD reveals film texture, phase composition, and epitaxial relationships with the substrate.
• X-ray reflectivity (XRR) measures the intensity of specularly reflected X-rays as a function of angle, giving film thickness, density, and interface roughness with sub-nanometer precision.
• Thin films often show residual strain and preferred orientation that bulk samples of the same material do not.

Strain and texture analysis

Deviations from ideal peak positions and intensities reveal mechanical and microstructural information:

• Lattice strain shifts peaks to higher or lower $2\theta$ (compressive strain decreases $d$, shifting peaks to higher angles; tensile strain does the opposite).
• Williamson-Hall analysis plots peak broadening ($\beta\cos\theta$) versus $\sin\theta$ to separate contributions from small crystallite size (intercept) and microstrain (slope).
• Texture (preferred orientation) means crystallites aren't randomly oriented, so some peaks are anomalously strong or weak. Pole figure measurements quantify this by mapping reflection intensity as a function of sample orientation.
• The $\sin^2\psi$ method measures how $d$-spacing varies with tilt angle $\psi$ to determine residual stress magnitude and direction.

In-situ diffraction studies

Collecting diffraction data while the sample is being heated, compressed, or electrochemically cycled lets you watch structural changes in real time:

• High-temperature XRD tracks phase transitions, thermal expansion coefficients, and decomposition reactions.
• High-pressure cells (diamond anvil cells) reveal pressure-induced phase transitions.
• Operando battery studies monitor electrode crystal structures during charge and discharge, linking structural changes to electrochemical performance.
• Time-resolved experiments at synchrotrons can capture transient phases with sub-second resolution.

Data analysis methods

Peak fitting

Accurate peak fitting is the foundation of quantitative diffraction analysis.

1. Select a peak profile function. Common choices are Gaussian (dominates when instrumental broadening is large), Lorentzian (dominates for small crystallite sizes), or Pseudo-Voigt (a weighted mix of both).
2. Fit the function to the observed data using least-squares minimization, adjusting peak position, height, width, and shape parameters.
3. For overlapping peaks, fit multiple profiles simultaneously (deconvolution).
4. Extract the integrated area under each peak, which is proportional to $|F_{hkl}|^2$ after applying corrections.

Rietveld refinement

Rietveld refinement fits the entire powder diffraction pattern at once, rather than peak by peak. You start with a structural model and iteratively adjust parameters until the calculated pattern matches the observed one.

Refined parameters typically include:

• Lattice parameters
• Atomic positions and site occupancies
• Thermal displacement parameters (Debye-Waller factors)
• Peak profile parameters and background coefficients

The quality of fit is judged by residual factors like $R_{wp}$ (weighted profile R-factor). A good Rietveld refinement also yields quantitative phase fractions in multi-phase samples.

Phase identification

Phase identification matches your measured pattern against a database of known structures (most commonly the ICDD Powder Diffraction File).

1. Measure peak positions and relative intensities.
2. Run a search-match algorithm against the database, optionally constraining by known chemical composition.
3. Evaluate candidate matches by comparing all peaks, not just the strongest ones.
4. For multi-phase samples, subtract identified phases and repeat the search for remaining unmatched peaks.

Difficulties arise with solid solutions (peaks shift continuously with composition), nanocrystalline phases (broad peaks), and unknown compounds not yet in the database.

Limitations and challenges

Peak overlap

When two or more reflections have nearly the same $d$-spacing, their peaks merge. This is especially common in low-symmetry crystals (monoclinic, triclinic) and multi-phase mixtures. High-resolution diffractometers and synchrotron sources help by producing narrower peaks. Complementary techniques like neutron diffraction, which has different relative peak intensities, can also resolve ambiguities.

Preferred orientation

If crystallites in a powder sample aren't randomly oriented, relative peak intensities will be distorted. Plate-like crystals (e.g., clays, graphite) tend to lie flat, and needle-shaped crystals align along their long axis.

Mitigation strategies:

• Careful sample preparation: side-loading, spray-drying, or mixing with an amorphous binder.
• Spinning the sample during measurement.
• Modeling preferred orientation in Rietveld refinement using March-Dollase or spherical harmonics corrections.

Sample preparation effects

• Over-grinding can introduce microstrain, reduce crystallite size, or even trigger phase transformations in mechanically sensitive materials.
• Surface roughness in reflection geometry alters peak intensities and background levels.
• Poor particle statistics (too few crystallites in the beam) cause spotty Debye rings and irreproducible intensities. Spinning the sample or using finer powders helps.
• Absorption in highly absorbing samples (e.g., those containing heavy elements) requires correction, especially in transmission geometry.

Complementary techniques

Neutron diffraction vs. X-ray

Neutrons scatter off atomic nuclei rather than electron clouds, which gives neutron diffraction several distinct advantages:

• Light-element sensitivity: hydrogen, lithium, and oxygen scatter neutrons strongly, whereas they're nearly invisible to X-rays.
• Isotopic contrast: different isotopes of the same element can have very different neutron scattering lengths, enabling contrast variation experiments.
• Magnetic structure: neutrons have a magnetic moment and interact with unpaired electrons, making neutron diffraction the primary tool for determining magnetic ordering.
• Bulk penetration: neutrons penetrate centimeters into most materials, so you can study large samples or samples inside complex environments (furnaces, pressure cells).

The trade-off is that neutron sources (reactors, spallation sources) are far less accessible than lab X-ray instruments, and experiments typically require larger sample volumes.

Electron diffraction

Electron diffraction uses high-energy electrons (typically 100–300 keV in a transmission electron microscope) instead of X-rays.

• Electrons interact much more strongly with matter than X-rays, so you can get diffraction patterns from nanometer-scale crystals or thin film regions.
• You can combine diffraction with real-space imaging in the same instrument.
• 3D electron diffraction (including rotation electron diffraction) now enables full structure determination of crystals too small for X-ray methods.
• The strong interaction also means multiple scattering (dynamical diffraction) is common, complicating intensity analysis. Beam damage is another concern for organic and biological samples.

Spectroscopic methods

Diffraction tells you where atoms are, but spectroscopy tells you about their chemical environment and bonding:

• X-ray absorption spectroscopy (XAS) probes the local coordination environment and oxidation state of a specific element, even in amorphous or disordered materials.
• X-ray photoelectron spectroscopy (XPS) measures surface composition and chemical states (top ~5–10 nm).
• Raman spectroscopy detects molecular vibrations and is sensitive to local symmetry, phase, and strain.
• Solid-state NMR reveals local atomic environments in both crystalline and amorphous phases, complementing the long-range order information from diffraction.

Combining diffraction with one or more spectroscopic methods gives a much more complete picture of a material's structure than either approach alone.