10.3 Disorder in crystals (thermal vibrations, occupational disorder)

Thermal vibrations in crystals

Crystals are never truly static. Even in a "perfect" crystal, atoms constantly oscillate around their equilibrium positions because of thermal energy. These vibrations have real consequences: they blur diffraction data, drive thermal expansion, and influence mechanical and electrical properties. Understanding them is central to interpreting crystallographic results accurately.

Atomic oscillations and displacement

Every atom in a crystal lattice vibrates around its ideal position. The amplitude of these vibrations increases with temperature and decreases with stronger bonding and heavier atomic mass. Over time, each atom occupies a smeared-out cloud of positions rather than a single point, which is why crystallographic models report a mean square displacement rather than a fixed coordinate.

This smearing directly affects diffraction experiments. The Debye-Waller factor quantifies how much thermal motion reduces the intensity of scattered X-rays or neutrons:

$I = I_0 \, e^{-2M}$

Here, $I_0$ is the intensity you'd measure from perfectly stationary atoms, and $M$ is the Debye-Waller factor (which itself depends on the mean square displacement and the scattering vector). As temperature rises, $M$ increases and measured Bragg peak intensities drop.

On a macroscopic level, thermal vibrations increase the apparent size of atoms and drive thermal expansion of the lattice. For example, copper expands by about 1.7% in volume when heated from 0°C to 100°C.

Effects on diffraction and crystal properties

Thermal vibrations show up in diffraction data in two main ways:

• Reduced Bragg peak intensities, especially at high scattering angles where the Debye-Waller factor has the largest effect
• Increased diffuse scattering between Bragg peaks, caused by correlated and uncorrelated atomic displacements

Beyond diffraction, vibrations affect bulk properties:

• Thermal conductivity generally decreases at higher temperatures because increased phonon-phonon scattering shortens the mean free path of heat-carrying phonons.
• Electrical resistivity in metals rises with temperature as lattice vibrations scatter conduction electrons more frequently.
• Elastic constants soften with heating. The Young's modulus of most metals drops measurably as temperature increases, meaning the material becomes easier to deform.

Temperature dependence of vibrations

Vibration amplitude and quantum effects

At high temperatures (above the Debye temperature, $\theta_D$), vibration amplitude increases roughly linearly with temperature, and classical statistical mechanics describes the lattice well. The Debye temperature marks the crossover between classical and quantum regimes. It varies by material: for aluminum, $\theta_D \approx 428 \, \text{K}$; for diamond, it's around 2230 K.

Below $\theta_D$, quantum effects dominate. Vibrations are quantized into phonons, and not all vibrational modes are thermally accessible. A key consequence is zero-point motion: even at absolute zero, atoms still vibrate with a minimum energy of

$E = \frac{1}{2} h\nu$

per mode, where $h$ is Planck's constant and $\nu$ is the mode frequency. This means no crystal is ever perfectly "still."

Impact on material properties

• Thermal expansion arises because real interatomic potentials are anharmonic (asymmetric). Atoms vibrating in an anharmonic well shift their average position outward as amplitude grows. The linear thermal expansion coefficient for steel is roughly $11\text{–}13 \times 10^{-6} \, \text{°C}^{-1}$.
• Debye-Waller factor grows with temperature, so Bragg peak intensities systematically decrease on heating. Crystallographers must account for this during structure refinement.
• Phase transitions can be triggered when vibrations become large enough to destabilize a structure. The $\alpha$-$\beta$ transition in quartz at 573°C, for instance, involves a cooperative change in Si-O-Si bond angles driven partly by increasing thermal motion.
• Heat capacity reflects how vibrational energy is stored. At high temperatures, heat capacity approaches the Dulong-Petit limit of $3R$ per mole of atoms (about 25 J/mol·K), where all vibrational modes are fully excited. At low temperatures, heat capacity drops sharply as quantum effects freeze out higher-energy modes.

Occupational disorder in crystals

Occupational disorder is a fundamentally different kind of imperfection from thermal vibrations. Instead of atoms moving around their correct positions, occupational disorder means different atomic species (or vacancies) are randomly distributed over crystallographically equivalent sites. The long-range periodicity of the lattice is preserved, but which atom sits at a given site varies from unit cell to unit cell.

Where occupational disorder occurs

Solid solutions are the most common example. In the mineral olivine, $(Mg,Fe)_2SiO_4$, magnesium and iron randomly share the same metal sites. Any given site might hold Mg or Fe, with probabilities set by the overall composition. The crystal structure is the same either way; only the site occupancy changes.

Other important cases:

• Intermetallic alloys: In brass (Cu-Zn), copper and zinc atoms share lattice sites at high temperatures. Cu-Au alloys similarly become disordered above their order-disorder transition temperature.
• Ceramic electrolytes: In yttria-stabilized zirconia (YSZ), $Y^{3+}$ substitutes for $Zr^{4+}$ on cation sites, and charge-compensating oxygen vacancies are distributed randomly across anion sites.
• Nonstoichiometric compounds: Transition metal oxides like $WO_{3-x}$ have oxygen vacancies randomly distributed over oxygen sites.

Types of occupational disorder

• Substitutional disorder: Different atomic species occupy the same type of lattice site. This is the classic case in alloys and mineral solid solutions.
• Vacancy disorder: Some sites are randomly unoccupied, as in the oxygen sublattice of YSZ or in nonstoichiometric oxides.
• Interstitial disorder: Atoms occupy sites that are normally empty in the ideal structure. Carbon atoms in austenitic steel, for example, sit in octahedral interstitial sites within the FCC iron lattice.
• Orientational disorder: In molecular crystals, the molecule's position is fixed but its orientation varies randomly from site to site. Plastic crystals like cyclohexane near room temperature are a classic example: the molecules tumble freely on their lattice sites.

Effects of occupational disorder

Structural and physical property changes

Because each site has a different local atomic environment, occupational disorder introduces local variations in bonding. These local differences average out over the whole crystal but have measurable effects:

• Melting points tend to decrease. Disordered Cu-Au alloys melt at lower temperatures than their ordered counterparts because the mixed bonding environments weaken the average cohesive energy.
• Thermal conductivity is reduced by the mass and force-constant disorder that scatters phonons.
• Electronic properties change significantly. In semiconductors, disorder can create localized energy states within the band gap, affecting conductivity and optical absorption.
• Ionic conductivity can be dramatically enhanced. YSZ is used as a solid electrolyte in fuel cells precisely because the oxygen vacancies introduced by yttria doping allow rapid oxygen ion transport through the lattice.

Mechanical and chemical impacts

• Solid solution strengthening is one of the most practically important effects. Atoms of different sizes on the same lattice sites create local strain fields that impede dislocation motion, making the alloy harder and stronger than either pure component.
• Chemical reactivity and catalytic behavior change because disorder creates sites with unusual coordination or bonding. Defect sites in zeolites, for instance, serve as active catalytic centers with enhanced reactivity compared to the perfect framework.

Effects on diffraction

Occupational disorder complicates crystallographic analysis in specific ways:

• Bragg peak intensities change because the average scattering factor at each site is now a weighted average of the occupying species.
• Diffuse scattering appears between Bragg peaks if the disorder has any short-range correlations (e.g., unlike neighbors preferred over like neighbors).
• Structure refinement becomes harder because you need to refine site occupancy factors alongside positions and displacement parameters. Distinguishing between partial occupancy and high thermal motion requires careful analysis, often combining X-ray and neutron data.

Phase stability

Occupational disorder influences which phases are stable at a given temperature. Entropy favors disorder, so many ordered phases become disordered on heating (like Cu-Au alloys above ~390°C). Conversely, cooling can drive ordering transitions. In shape memory alloys like NiTi, the interplay between ordered and disordered phases underlies the martensitic transformations responsible for the shape memory effect.