11.2 Line defects and dislocations

Line defects, or dislocations, are one-dimensional imperfections in crystal structures that control how materials deform plastically. Nearly all permanent shape change in crystalline solids happens because dislocations move through the lattice, so understanding their geometry, stress fields, and interactions is central to predicting mechanical behavior like yield strength, ductility, and work hardening.

Types of line defects

Dislocations are one-dimensional defects that run along a line through the crystal. They come in several varieties, distinguished by the relationship between the dislocation line direction and the Burgers vector.

Edge dislocations

An edge dislocation forms when an extra half-plane of atoms is inserted into the crystal. The bottom edge of that half-plane is the dislocation line, and the Burgers vector is perpendicular to the dislocation line.

The extra half-plane creates an asymmetric distortion: the lattice is compressed above the dislocation line (where atoms are squeezed together) and in tension below it (where atoms are pulled apart). This distinction matters when you consider how edge dislocations interact with solute atoms of different sizes.

Screw dislocations

A screw dislocation has no extra half-plane. Instead, the lattice is sheared so that atomic planes spiral around the dislocation line like a helical ramp. The Burgers vector is parallel to the dislocation line.

Because there's no inserted plane, the distortion is purely shear with no compressive or tensile component. This gives screw dislocations more geometric freedom: they can cross-slip onto different planes since the Burgers vector lies in every plane that contains the dislocation line.

Mixed dislocations

Most real dislocations are mixed, meaning the dislocation line curves so that the Burgers vector is at some intermediate angle to the line direction. At any point along a curved dislocation, you can decompose the character into an edge component (Burgers vector perpendicular to the local line direction) and a screw component (Burgers vector parallel to it). Pure edge and pure screw are just the two limiting cases.

Partial dislocations

A partial dislocation has a Burgers vector that is a fraction of a full lattice translation vector. Partials typically form in pairs separated by a stacking fault, a planar defect where the normal stacking sequence is disrupted.

In FCC metals, a full dislocation with Burgers vector $\frac{a}{2}\langle110\rangle$ can dissociate into two Shockley partials with Burgers vectors $\frac{a}{6}\langle112\rangle$. This dissociation is energetically favorable because the total elastic energy (proportional to $b^2$) decreases. The width of the stacking fault ribbon between the partials depends on the stacking fault energy of the material: low stacking fault energy metals like gold and silver have wide separations, while high stacking fault energy metals like aluminum have narrow ones.

Burgers vector

The Burgers vector $\vec{b}$ is the single most important quantity characterizing a dislocation. It defines the magnitude and direction of the lattice displacement caused by the defect, and it is conserved along the entire length of a dislocation line.

Definition and the Burgers circuit

You determine $\vec{b}$ using a Burgers circuit:

1. In a perfect reference crystal, draw a closed, atom-to-atom loop (equal steps right, up, left, down, for example).
2. Now draw the same sequence of steps around the dislocation in the real crystal.
3. The circuit will fail to close. The vector needed to complete the loop (the closure failure) is the Burgers vector.

Two conventions exist (FS/RH and SF/RH), so pay attention to which one your course uses, as they differ by a sign.

Burgers vector for edge dislocations

• $\vec{b}$ is perpendicular to the dislocation line.
• Its magnitude equals one lattice spacing in the slip direction.
• It points from the compressed side toward the tensile side of the dislocation.

Burgers vector for screw dislocations

• $\vec{b}$ is parallel to the dislocation line.
• Its magnitude equals one lattice spacing along the line direction.
• The sense (left-hand vs. right-hand screw) is determined by the right-hand rule: thumb along the dislocation line, fingers curl in the direction of the helical distortion.

Stress fields of dislocations

Every dislocation distorts the surrounding lattice, producing a long-range stress field that decays as $1/r$, where $r$ is the distance from the dislocation core. These stress fields govern how dislocations interact with each other and with other defects.

Stress field of an edge dislocation

An edge dislocation produces both normal (hydrostatic) and shear stress components. Above the slip plane the lattice is in compression; below it, in tension. The hydrostatic component means edge dislocations interact with solute atoms that cause volume changes (size misfit), which is the basis of solid solution strengthening.

The stress components for an edge dislocation along the $z$-axis with Burgers vector along $x$ are proportional to:

$\sigma_{ij} \propto \frac{Gb}{2\pi(1-\nu)} \cdot \frac{1}{r}$

where $G$ is the shear modulus, $b$ is the Burgers vector magnitude, and $\nu$ is Poisson's ratio.

Stress field of a screw dislocation

A screw dislocation produces pure shear stress with no hydrostatic component. The shear stress field is:

$\sigma_{\theta z} = \frac{Gb}{2\pi r}$

Because there's no volume change associated with a screw dislocation, it does not interact with solute atoms through size misfit the way an edge dislocation does.

Hydrostatic stress vs. shear stress

• Hydrostatic stress causes volume change without shape change. Only edge dislocations (and the edge component of mixed dislocations) produce it.
• Shear stress causes shape change without volume change. Both edge and screw dislocations produce shear components.

This distinction matters for predicting which defects interact with which types of dislocations.

Effect of stress fields on material properties

Dislocation stress fields interact with solute atoms (Cottrell atmospheres), precipitates, grain boundaries, and other dislocations. These interactions are responsible for:

• Solid solution strengthening: solute atoms pin dislocations.
• Dislocation pile-ups: stress concentrations at obstacles.
• Work hardening: overlapping stress fields from many dislocations raise the stress needed for further deformation.

Motion of dislocations

Dislocation motion is the microscopic mechanism behind plastic deformation. There are two fundamentally different modes of motion.

Dislocation glide

Glide is the motion of a dislocation along its slip plane, the plane containing both the dislocation line and the Burgers vector.

• It is conservative: no atoms are added or removed, just rearranged.
• It can occur at any temperature, even near absolute zero.
• The resistance to glide in a perfect lattice is set by the Peierls-Nabarro stress.

For an edge dislocation, the slip plane is uniquely defined. For a screw dislocation, any plane containing the dislocation line is a potential slip plane, which is why screw dislocations can cross-slip onto a different plane.

Dislocation climb

Climb is the motion of an edge dislocation perpendicular to its slip plane. This requires atoms to be added to or removed from the extra half-plane, which means vacancies must diffuse to or from the dislocation core.

• It is non-conservative: the dislocation line moves by absorbing or emitting point defects.
• It is thermally activated, becoming significant only at elevated temperatures (roughly above $0.3\text{–}0.4 \, T_m$, where $T_m$ is the melting temperature).
• Climb allows dislocations to bypass obstacles they cannot pass by glide alone.

Critical resolved shear stress

The critical resolved shear stress (CRSS) is the minimum shear stress resolved onto a slip system needed to start dislocation motion. Schmid's law relates the applied tensile stress $\sigma$ to the resolved shear stress $\tau$:

$\tau = \sigma \cos\phi \cos\lambda$

where $\phi$ is the angle between the stress axis and the slip plane normal, and $\lambda$ is the angle between the stress axis and the slip direction. Yielding begins when $\tau$ reaches the CRSS on the most favorably oriented slip system.

Peierls-Nabarro stress

The Peierls-Nabarro stress is the intrinsic lattice resistance to dislocation glide in a perfect crystal. It arises because the dislocation core must move through the periodic potential energy landscape of the lattice.

The Peierls stress depends strongly on:

• Core width: wider cores (as in close-packed FCC metals) experience a smoother potential and lower Peierls stress.
• Crystal structure: BCC metals have narrow cores and high Peierls stress, making them harder and more temperature-sensitive in their yield behavior. FCC metals have low Peierls stress and are generally more ductile.
• Bonding type: covalent materials (like Si and diamond) have very high Peierls stress, which is why they are brittle at room temperature.

Interactions between dislocations

As dislocations multiply during deformation, their interactions become the dominant factor controlling mechanical behavior.

Dislocation reactions

Dislocations can react when they meet, and the key rule is Frank's rule: a reaction is energetically favorable if the total $b^2$ decreases. The main reaction types are:

• Annihilation: two dislocations with equal and opposite Burgers vectors on the same slip plane meet and cancel out, reducing dislocation density.
• Combination: two dislocations merge to form a new one whose Burgers vector is the vector sum of the originals ($\vec{b}_3 = \vec{b}_1 + \vec{b}_2$).
• Dissociation: a perfect dislocation splits into two or more partials separated by a stacking fault.

Dislocation networks

During deformation, dislocations multiply (via Frank-Read sources, for example) and interact to form complex structures: tangles, cell walls, and subgrain boundaries. These networks act as obstacles to further dislocation motion, which is the physical origin of strain hardening.

Low-angle grain boundaries

A low-angle grain boundary (misorientation typically < 10–15°) can be described as an ordered array of dislocations. A tilt boundary is built from edge dislocations, while a twist boundary is built from screw dislocations. The misorientation angle $\theta$ is related to the dislocation spacing $D$ by:

$\theta \approx \frac{b}{D}$

This relationship, confirmed experimentally, was one of the early successes of dislocation theory.

Dislocation pile-ups

When multiple dislocations on the same slip plane are blocked by an obstacle (grain boundary, precipitate, sessile dislocation), they stack up behind it. The lead dislocation experiences a stress concentration equal to $n$ times the applied shear stress, where $n$ is the number of dislocations in the pile-up.

Pile-ups are central to the Hall-Petch relationship: smaller grains mean shorter pile-ups, higher stress concentrations needed to propagate slip across boundaries, and therefore higher yield strength.

Dislocation density

Dislocation density ($\rho$) is the total length of dislocation lines per unit volume:

$\rho = \frac{L}{V}$

It has units of $\text{m}^{-2}$ (length per volume simplifies to inverse area). Typical values span a huge range:

Measuring dislocation density

1. TEM: Direct imaging of dislocation lines in thin foils. Most detailed but limited to small sample volumes.
2. X-ray diffraction (XRD): Peak broadening analysis (Williamson-Hall or Warren-Averbach methods) gives an average $\rho$ over a larger volume.
3. Etch pit technique: Chemical etching reveals where dislocations intersect the surface. Simple and inexpensive, but only gives surface density and limited type information.
4. Electrical resistivity: Dislocations scatter conduction electrons, so resistivity increases with $\rho$. Requires calibration but can be done on bulk samples.

Work hardening and dislocation density

As plastic deformation proceeds, dislocations multiply and $\rho$ increases. The average spacing between dislocations decreases as $\sim 1/\sqrt{\rho}$, so their stress fields overlap more strongly and the stress required to move additional dislocations rises.

This is captured by the Taylor hardening equation:

$\sigma = \sigma_0 + \alpha G b \sqrt{\rho}$

where:

• $\sigma_0$ is the friction stress (lattice resistance plus solute effects)
• $\alpha$ is a dimensionless constant (typically 0.2–0.5)
• $G$ is the shear modulus
• $b$ is the Burgers vector magnitude
• $\rho$ is the dislocation density

The $\sqrt{\rho}$ dependence is one of the most well-established results in physical metallurgy.

Dislocations and plastic deformation

Plastic deformation in crystalline solids occurs overwhelmingly by dislocation glide. The theoretical shear strength of a perfect crystal ($\sim G/30$) is far higher than the experimentally observed yield stress, and dislocations explain this discrepancy: they allow slip to occur one atomic row at a time rather than simultaneously across an entire plane.

Slip systems and slip planes

A slip system is defined by a slip plane and a slip direction. Dislocations glide most easily on the closest-packed planes in the closest-packed directions, because these have the largest interplanar spacing (lower Peierls stress) and the shortest Burgers vector (lower elastic energy).

FCC metals are generally ductile because they have 12 independent slip systems with low Peierls stress. BCC metals have many possible systems but high Peierls stress, making their yield behavior strongly temperature-dependent. HCP metals have few easy slip systems, which limits ductility.

Critical resolved shear stress and yield strength

In a polycrystalline material, grains are oriented randomly, so different grains reach their CRSS at different applied stresses. Yielding of the bulk requires enough slip systems to be activated to accommodate arbitrary shape changes. The von Mises criterion states that five independent slip systems are needed for general plastic deformation of a polycrystal.

Yield strength can be increased by any mechanism that hinders dislocation motion:

• Solid solution strengthening: solute atoms create local stress fields that pin dislocations.
• Precipitation hardening: second-phase particles force dislocations to cut through or bow around them (Orowan mechanism).
• Grain boundary strengthening: grain boundaries block dislocation transmission (Hall-Petch effect).
• Work hardening: accumulated dislocations impede further dislocation motion.

Strain hardening and dislocation interactions

During deformation, dislocations multiply, interact, and form increasingly complex structures: jogs, kinks, Lomer-Cottrell locks, dislocation forests, and cell structures. Each of these acts as an obstacle to further glide.

The strain hardening rate ($d\sigma/d\varepsilon$) typically decreases with strain as recovery processes (cross-slip, climb) begin to annihilate dislocations and rearrange them into lower-energy configurations. The competition between dislocation storage and dynamic recovery determines the shape of the stress-strain curve.

Observation of dislocations

Directly observing dislocations confirms theoretical predictions and provides quantitative data on dislocation densities, types, and arrangements.

Transmission electron microscopy (TEM)

TEM is the most powerful tool for studying individual dislocations. Electrons transmitted through a thin foil (< 100 nm thick) are diffracted by the crystal lattice, and the strain field around a dislocation alters the local diffraction conditions, producing contrast.

A key technique is the invisibility criterion: a dislocation becomes invisible when $\vec{g} \cdot \vec{b} = 0$, where $\vec{g}$ is the diffraction vector. By imaging with different $\vec{g}$ vectors and finding the conditions where a dislocation disappears, you can determine its Burgers vector.

X-ray topography

X-ray topography images dislocations over large areas (up to several $\text{cm}^2$) with micron-scale resolution. It's non-destructive and works on bulk single crystals, making it useful for studying dislocation distributions during in-situ deformation or thermal processing. The trade-off is much lower spatial resolution compared to TEM.

Etch pit technique

Chemical etchants attack the crystal surface preferentially where dislocations emerge, because the strain energy at the core makes those sites more reactive. The resulting pits can be counted under an optical microscope to estimate surface dislocation density. The technique is simple and cheap but tells you little about dislocation type or subsurface structure.

Decoration technique

Solute atoms or small precipitates can be made to segregate preferentially to dislocation lines, "decorating" them so they become visible in TEM or SEM. Copper decoration in silicon is a classic example. This technique reveals dislocation arrangements in bulk material, but the decoration process itself (which requires heating) can alter the dislocation structure you're trying to study.