Defects in Crystals

• Vacancy: Missing atom at a lattice site.
• Interstitial: Extra atom between lattice sites.
• Substitutional: Foreign atom replacing a host atom.
• Frenkel defect: Vacancy-interstitial pair (atom moves to interstitial site).
• Schottky defect: Vacancy pair (in ionic crystals, cation and anion vacancies).

Equilibrium concentration of vacancies:

$n_v = N\,e^{-E_v/(k_B T)}$

where $N$ is the number of lattice sites and $E_v$ is the vacancy formation energy ($\sim 1$ eV).

Derivation. Minimising the free energy $F = n_v E_v - T S_{\mathrm{config}}$ where $S_{\mathrm{config}} = k_B \ln\binom{N}{n_v}$:

$\frac{\partial F}{\partial n_v} = E_v + k_B T \ln\left(\frac{n_v}{N - n_v}\right) = 0$

For $n_v \ll N$: $n_v = N e^{-E_v/(k_B T)}$. $\blacksquare$

• Edge dislocation: Extra half-plane inserted into the lattice. Burgers vector $\mathbf{b}$ is perpendicular to the dislocation line.
• Screw dislocation: The lattice is sheared. $\mathbf{b}$ is parallel to the dislocation line.

Dislocations enable plastic deformation at stresses far below the theoretical shear strength. The Peach-Koehler force on a dislocation:

$\mathbf{F} = (\boldsymbol{\sigma}\cdot\mathbf{b}) \times \hat{\mathbf{t}}$

where $\boldsymbol{\sigma}$ is the stress tensor and $\hat{\mathbf{t}}$ is the unit tangent to the dislocation line.

Defects strongly affect electrical, mechanical, and thermal properties:

• Electrical: Donor and acceptor levels in semiconductors are substitutional defects. Vacancies act as scattering centres, reducing conductivity.
• Mechanical: Dislocations determine yield strength (Hall—Petch relation). Work hardening increases dislocation density.
• Thermal: Point defects scatter phonons, reducing thermal conductivity.

Grain boundaries separate crystalline regions (grains) of different orientation. They are classified by the misorientation angle $\theta$:

• Low-angle grain boundaries: $\theta \lesssim 15^\circ$, composed of dislocation arrays.
• High-angle grain boundaries: $\theta \gtrsim 15^\circ$, have a more disordered structure.

Stacking faults occur when the stacking sequence of close-packed planes is disrupted. For FCC crystals, the normal stacking ABCABC can become ABCABABC (intrinsic fault) or ABCABACABC (extrinsic fault). The fault energy $\gamma_{\mathrm{SF}}$ determines the width of dissociated dislocations.

Twin boundaries are special grain boundaries with mirror symmetry across the interface, with low interfacial energy.

Dislocation motion. Dislocations move by glide (within the slip plane) or climb (perpendicular to the slip plane, requiring mass transport). The Peierls-Nabarro stress $\tau_P$ is the stress required to move a dislocation:

$\tau_P = \frac{2G}{1 - \nu} e^{-2\pi w/b}$

where $G$ is the shear modulus, $\nu$ is Poisson’s ratio, $w$ is the dislocation width, and $b$ is the Burgers vector magnitude.

Dislocation multiplication. Under stress, dislocation sources (Frank-Read sources) generate new dislocation loops, dramatically increasing dislocation density during plastic deformation.

• Precipitates: Second-phase particles formed by supersaturation and nucleation. They can strengthen materials (precipitation hardening) or weaken them (if large and brittle).
• Voids and pores: Agglomerations of vacancies, often formed during solidification or irradiation. They reduce density and can initiate fracture.

• X-ray diffraction: Peak broadening reveals microstrain and crystallite size (Scherrer equation). Diffuse scattering reveals point defect concentrations.
• Transmission electron microscopy (TEM): Direct imaging of dislocations, grain boundaries, and precipitates. Selected area diffraction identifies crystal orientation.
• Scanning electron microscopy (SEM): Surface imaging of grain structure via electron channeling contrast or EBSD.
• Positron annihilation spectroscopy: Sensitive to vacancy-type defects. Positrons become trapped at vacancies, changing their annihilation lifetime.

Problem 1. Calculate the equilibrium vacancy concentration in copper at 1000 K given $E_v = 1.0$ eV and $k_B = 8.62 \times 10^{-5}$ eV/K. How many vacancies per cubic centimeter?

Problem 2. Show that the equilibrium concentration of Frenkel defects (vacancy + interstitial) is $n_F = \sqrt{NN_i}\, e^{-E_F/(2k_B T)}$, where $N_i$ is the number of interstitial sites.

Solution. The free energy for $n$ Frenkel pairs is $F = n E_F - k_B T \left[\ln\binom{N}{n} + \ln\binom{N_i}{n}\right]$. Using Stirling and minimising gives the result. $\blacksquare$

Problem 3. A metal with grain size $d$ has yield strength $\sigma_y = \sigma_0 + k d^{-1/2}$ (Hall-Petch). Explain why smaller grains give higher strength.

Problem 4. Estimate the number of dislocations in a plastically deformed crystal with shear strain $\gamma = 0.1$ and average dislocation slip distance 1 $\mu$m.

Color centers (F-centers) are point defects that absorb light at specific wavelengths, giving color to otherwise transparent crystals. An F-center is an electron trapped at an anion vacancy in an ionic crystal. The energy levels of the trapped electron give characteristic optical absorption bands. F-centers in alkali halides produce vivid colors: NaCl (yellow), KCl (violet), KBr (blue).

High-energy radiation (neutrons, electrons, gamma rays) creates defect cascades in crystals:

• Displacement damage: Atoms knocked from lattice sites, creating Frenkel pairs.
• Ionization damage: Electron-hole pairs that can lead to chemical changes.
• Swelling: Accumulation of voids causes dimensional changes, important in nuclear reactor materials.

Defects are not always undesirable. Defect engineering deliberately introduces controlled defects to tailor material properties:

• Doping: Adding substitutional impurities to control semiconductor conductivity (n-type and p-type).
• Precipitation hardening: Second-phase particles impede dislocation motion, increasing strength.
• Oxygen vacancies in oxides: Used in memristors and solid oxide fuel cells.

• Point defects (vacancies, interstitials, substitutionals) have equilibrium concentrations governed by Boltzmann statistics.
• Dislocations are line defects enabling plastic deformation at stresses below theoretical strength.
• Planar defects include grain boundaries, stacking faults, and twin boundaries.
• Defects affect electrical, mechanical, thermal, and optical properties.
• Defect engineering is used to optimize material performance in applications from semiconductors to structural alloys.

Problem 5. At 300 K, the vacancy concentration in copper is $10^{-12}$ of lattice sites. At 1000 K, it is $10^{-5}$. Estimate the vacancy formation energy $E_v$.

Problem 6. Show that the equilibrium concentration of Schottky defects in an ionic crystal MX is $n_S = N e^{-E_S/(2k_B T)}$, where $E_S$ is the energy to create a cation-anion vacancy pair.