Dislocations in Metals — Edge, Screw, and Mixed Types

The plastic deformation of metals — their ability to deform permanently without fracture under applied load — is controlled almost entirely by the motion of crystal line defects called dislocations. Pure theoretical calculations show that a perfect crystal of iron should require a shear stress of approximately G/30 (about 3 GPa) to deform plastically; real iron yields at 50–150 MPa. This factor-of-100 discrepancy, first resolved by Taylor, Orowan, and Polyani in 1934, is explained entirely by the existence of dislocations and the ease with which they move through the crystal compared to the simultaneous displacement of a complete atomic plane. This article provides a rigorous, graduate-level treatment of dislocation geometry, characterisation, energy, dynamics, interactions, and their central role in the four principal strengthening mechanisms used in structural and engineering alloys.

Historical Context and the Theoretical Shear Stress Problem

The theoretical critical resolved shear stress (CRSS) to slide one atomic plane over another in a perfect crystal can be estimated from the sinusoidal restoring force model (Frenkel model, 1926):

Frenkel estimate of theoretical shear strength:
τ_th = G / (2π)  ≈  G/6   to   G/30   (depending on model)

For iron: G ≈ 82 GPa
τ_th (Frenkel) ≈ 13 GPa

Measured CRSS for pure iron single crystal: ~50 MPa
Ratio: τ_theoretical / τ_measured ≈ 260

This ~2½ order-of-magnitude discrepancy was the central unsolved
problem in crystal plasticity until 1934, when Taylor, Orowan, and
Polyani independently proposed the dislocation mechanism.

The resolution is that plastic deformation does not require simultaneous sliding of an entire atomic plane. Instead, a localised region of misfit — the dislocation — propagates sequentially through the crystal at stresses far below the theoretical value, in the same way that a wrinkle in a carpet can be moved with one finger rather than dragging the whole carpet at once.

The Burgers Vector: Defining Dislocation Character

The Burgers vector b is the most important parameter characterising a dislocation. It is determined experimentally by constructing a Burgers circuit: make a closed, clockwise (right-hand convention, RH/FS convention) loop of atomic steps in a plane perpendicular to the dislocation line, taking equal numbers of steps in each direction. If performed in a perfect crystal with identical circuit dimensions, the loop closes. Performed around a dislocation, the loop fails to close; the closure failure vector — drawn from the end of the circuit back to its start — is the Burgers vector b.

Burgers Vectors in Common Crystal Structures

In metals, dislocations are energetically stable only when their Burgers vectors correspond to lattice translation vectors — meaning the crystal returns to an identical atomic arrangement after displacement by b. The minimum energy Burgers vectors correspond to the shortest lattice translations:

FCC (e.g., Cu, Al, Ni, austenitic steel):
  b = a/2⟨110⟩     |b| = a/√2   (a ≈ 0.36 nm for Cu)
  Slip plane: {111}    (close-packed planes)
  Slip system: {111}⟨110⟩  →  12 independent slip systems

BCC (e.g., α-iron, ferritic steel, W, Mo):
  b = a/2⟨111⟩     |b| = a√3/2  (a ≈ 0.286 nm for Fe)
  Slip planes: {110}, {112}, {123}  (pencil glide — no single preferred plane)
  Primary slip: {110}⟨111⟩  →  12 slip systems (preferred)

HCP (e.g., Ti, Zr, Mg, Co):
  Basal: b = a/3⟨11̄20⟩   |b| = a    slip plane: (0001) →  3 systems
  Prismatic: b = a/3⟨11̄20⟩  slip plane: {10̄10}  →  3 systems
  Pyramidal: b = a/3⟨11̄23⟩  |b| = √(a²+c²)/3  →  6 systems (⟨c+a⟩ type)

Self-energy of dislocation per unit length:
  E ≅ ɑ G b²   where ɑ = 1/(4π) to 1/(2π) depending on character
  Frank criterion (energetically favourable dissociation):
  |b₃|² < |b₁|² + |b₂|²

The Frank energy criterion directly governs dislocation reactions. Two dislocations with Burgers vectors b₁ and b₂ will combine to form a product dislocation with b₃ = b₁ + b₂ only if |b₃|² < |b₁|² + |b₂|²; otherwise the reaction is energetically unfavourable and the dislocations repel. This criterion explains why dislocations in FCC metals combine to form sessile (immobile) Lomer-Cottrell locks — a critical mechanism in Stage II work hardening.

Edge Dislocations in Detail

An edge dislocation is most simply visualised as the boundary of an extra half-plane of atoms inserted into the crystal. The dislocation line runs along the bottom edge of this half-plane, perpendicular to b. The grain boundary energy analogy is useful: just as a grain boundary is a planar array of dislocations, an edge dislocation is a line of severe lattice misfit.

Strain Field of an Edge Dislocation

The elastic stress field of an edge dislocation in an isotropic continuum (using cylindrical coordinates r, θ measured from the dislocation line) contains both dilatational and shear components:

Stress field of an edge dislocation (isotropic elasticity):

σ_xx = −[Gb / 2π(1−ν)] × y(3x² + y²) / (x² + y²)²

σ_yy = +[Gb / 2π(1−ν)] × y(x² − y²) / (x² + y²)²

τ_xy = +[Gb / 2π(1−ν)] × x(x² − y²) / (x² + y²)²

Hydrostatic stress (dilatation):
p = −(σ_xx + σ_yy + σ_zz)/3
p = +Gb sinθ / [3π(1−ν)r]   (tensile below slip plane, θ>0)

Elastic self-energy per unit length (edge):
E_edge = Gb² / [4π(1−ν)] × ln(R/r₀)

Where: G = shear modulus, b = |b|, ν = Poisson's ratio (~0.3),
R = outer radius (~grain size or spacing), r₀ = core radius (~2b)

The dilatational component of the stress field is what drives Cottrell atmosphere formation: oversized substitutional solute atoms (e.g., C in BCC iron, which occupies octahedral interstices) migrate to the tensile region below the slip plane where the lattice is expanded, reducing the total strain energy. This segregation pinning is responsible for the upper and lower yield point phenomenon observed in low-carbon steels. When stress is sufficient to tear the dislocation free of its Cottrell atmosphere, a large stress drop occurs (upper yield point); subsequent deformation proceeds at a lower stress along Lüders bands.

Glide and Climb of Edge Dislocations

Edge dislocations move by two distinct mechanisms with very different thermodynamic requirements:

• Glide (conservative motion): movement within the slip plane containing both the dislocation line and b. Requires only local bond-breaking and reformation as atoms shuffle around the dislocation core; can occur at any temperature above absolute zero. The stress required is the Peierls-Nabarro stress modified by temperature.
• Climb (non-conservative motion): movement perpendicular to the slip plane by addition or removal of atoms at the dislocation core. Positive climb (dislocation moves away from the half-plane) requires absorption of vacancies (or emission of interstitials); negative climb requires emission of vacancies. Both require atomic diffusion and are therefore significant only at T > 0.4 T_m (homologous temperature). Climb is the rate-controlling mechanism in high-temperature creep and enables recovery by allowing dislocations to leave pile-ups and annihilate at grain boundaries.

Screw Dislocations in Detail

In a screw dislocation, the Burgers vector is parallel to the dislocation line. The atomic planes transform from a flat stack into a helical ramp that spirals around the dislocation line — precisely like a spiral staircase. Each complete circuit around the dislocation line advances one step equal to the Burgers vector magnitude |b|.

Strain Field of a Screw Dislocation

The stress field of a screw dislocation is purely shear in character — there is no dilatational (hydrostatic) component:

Stress field of a screw dislocation (isotropic elasticity):

τ_θz = τ_zθ = Gb / (2πr)

All other stress components = 0 (no volumetric strain)

Elastic self-energy per unit length (screw):
E_screw = Gb² / (4π) × ln(R/r₀)

Note: E_screw < E_edge because edge includes additional
dilatational energy term (factor 1/(1−ν) ≈ 1.43 for ν=0.3)

Consequence: Dislocations seek to be predominantly screw character
when possible to minimise total stored elastic energy.

Cross-Slip

The defining characteristic that distinguishes screw from edge dislocations is the ability to cross-slip. Because the Burgers vector of a screw dislocation is parallel to its line, the dislocation line does not uniquely define a slip plane — any plane containing the dislocation line and b is a valid slip plane. This allows the dislocation to transfer from its current slip plane to an intersecting crystallographic plane of equivalent type without requiring diffusion.

In FCC metals, cross-slip occurs on {111} planes. The cross-slip mechanism explains: (1) the transition from Stage II (linear hardening) to Stage III (parabolic hardening, dynamic recovery) in the tensile work-hardening curve; (2) the temperature and strain-rate dependence of flow stress at high temperatures; (3) wavy slip lines in deformed FCC metals (vs. straight slip lines in low-SFE metals where cross-slip is suppressed). The activation energy for cross-slip is inversely proportional to the stacking fault energy — low-SFE metals (austenitic stainless steel, brass) have widely dissociated partial dislocations that must recombine before cross-slip can occur, making the process thermally activated with a higher barrier.

Mixed Dislocations and Dislocation Loops

In real deforming crystals, dislocations are not straight infinite lines of pure edge or pure screw character. They are curved, closed loops or segments terminating at free surfaces, grain boundaries, or other dislocations. Along a curved dislocation loop, the character varies continuously: where the local dislocation line direction is perpendicular to b, the segment is edge; where parallel to b, it is screw; at intermediate angles θ between 0° and 90°, the segment is mixed and can be resolved into edge and screw components:

Mixed dislocation character decomposition:

For dislocation line direction t̂ at angle θ to Burgers vector b:

Edge component:   b_edge  = b sinθ   (perpendicular to t̂)
Screw component:  b_screw = b cosθ   (parallel to t̂)

Self-energy of mixed dislocation per unit length:
E_mixed = Gb² / (4π) × [(1 − νcos²θ) / (1−ν)] × ln(R/r₀)

At θ = 0° (pure screw): E = Gb²/(4π) × ln(R/r₀)  [minimum]
At θ = 90° (pure edge): E = Gb²/[4π(1−ν)] × ln(R/r₀)  [maximum]

A prismatic dislocation loop (all edge character, b perpendicular
to loop plane) is emitted from Frank-Read sources or by collapse
of vacancy discs and cannot glide in its plane — it is sessile.

A dislocation loop expanding on a slip plane has edge character at the segments running parallel to b (the leading and trailing edges) and screw character at the segments running perpendicular to b (the side segments). When the screw segments at two sides of an expanding loop meet, they annihilate (opposite sign screws), releasing a complete dislocation loop that can expand further — this is the Frank-Read multiplication mechanism described later.

Dislocation Density and Its Measurement

Dislocation density ρ is defined as the total length of dislocation lines per unit volume of crystal:

Definition:
ρ = total dislocation line length / volume    [m⁻² = m/m³]

Equivalent definition (intersection method):
ρ = number of dislocation intersections per unit area of a sectioning plane × 2
    (factor of 2 accounts for dislocations not perpendicular to section)

Typical values:
  Annealed pure metal:            ρ ≈ 10¹° – 10¹² m⁻²
  Lightly cold-worked metal:      ρ ≈ 10¹² – 10¹² m⁻²
  Heavily cold-worked metal:      ρ ≈ 10¹⁵ – 10¹⁶ m⁻²
  Near-fracture (cold work):      ρ ≈ 10¹⁶ m⁻²
  Average dislocation spacing l = 1/√ρ:
    At ρ = 10¹²: l ≈ 10⁻⁶ m = 1 μm
    At ρ = 10¹⁵: l ≈ 3 nm  (dislocation cores nearly touching)

Measurement Methods

• Transmission Electron Microscopy (TEM): The reference technique. Dislocations appear as dark lines in bright-field TEM through diffraction contrast. Individual dislocations can be imaged; Burgers vectors determined by the invisibility criterion (g·b = 0 condition, where g is the diffraction vector).
• Etch pit counting: Chemical etching reveals dislocation emergence points on a polished surface as pits. Rapid but counts only those reaching the surface; statistically valid for low densities (<10¹² m⁻²).
• X-ray diffraction line broadening (Williamson-Hall plot): Peak broadening from dislocation strain fields; provides a volume-averaged ρ without specimen preparation. EBSD-based kernel average misorientation (KAM) maps give spatially resolved dislocation density information in bulk specimens.
• EBSD-KAM maps: The local misorientation between adjacent EBSD measurement points correlates with the geometrically necessary dislocation (GND) density that accommodates the lattice curvature, providing spatial maps of dislocation accumulation in deformed microstructures.

Dislocation Interactions: Junctions, Jogs, and Frank-Read Sources

The majority of strengthening in metallic alloys arises from dislocation–dislocation interactions rather than from single-dislocation dynamics. Understanding these interactions is essential for predicting work-hardening behaviour and designing thermomechanical processing schedules.

Forest Hardening and Dislocation Cutting

When a moving dislocation on one slip plane intersects a stationary (forest) dislocation on a different slip plane, the two dislocations cut through each other, each acquiring a jog equal to the Burgers vector of the other. This cutting reaction requires energy to create the jog and is the primary mechanism of Stage II work hardening. The forest dislocation density on all slip systems acts as an obstacle field; the flow stress increment from forest hardening is captured by the Taylor equation:

Taylor (Bailey-Hirsch) equation:
Δσ = M × α × G × b × √ρ

Where:
  M = Taylor factor = 3.06 (polycrystalline FCC, von Mises)
              ≈ 2.75 (BCC)
  α = dimensionless constant  ≈ 0.2–0.4
      (depends on dislocation arrangement: low α for random array,
       high α for regular forest or low-energy structures)
  G = shear modulus (GPa)
  b = Burgers vector magnitude (m)
  ρ = dislocation density (m⁻²)

Example: cold-drawn ferritic steel wire
  G = 82 GPa, b = 0.248 nm, α = 0.3, ρ = 5×10¹⁵ m⁻²
  Δσ = 3.06 × 0.3 × 82×10³ × 0.248×10⁻¹ × √(5×10¹⁵)
       = 3.06 × 0.3 × 82×10³ × 2.48×10⁻¹° × 7.07×10⁷
       ≈ 1250 MPa   (consistent with heavily cold-drawn wire strength)

Dislocation Locks (Lomer-Cottrell Lock)

When two dislocations on different {111} slip planes in an FCC metal react, their partial dislocations may combine to form a stair-rod dislocation with Burgers vector a/6⟨110⟩ lying along the intersection of the two planes. This product dislocation is sessile — its Burgers vector does not lie in any slip plane — so it cannot glide and acts as a strong barrier (Lomer-Cottrell lock) that stops further dislocation motion on both original slip planes. The accumulation of Lomer-Cottrell locks is the microscopic explanation for the linear Stage II work-hardening region, where the flow stress increases approximately as σ ≈ σ₀ + Gε/100 — a nearly constant hardening rate governed by the increasing density of sessile locks.

Frank-Read Source

The Frank-Read source resolves the paradox of how dislocation density can increase by five orders of magnitude during plastic deformation. A dislocation segment of length L, pinned at both ends (by Lomer-Cottrell nodes, precipitate particles, or point defects), bows out under applied shear stress τ. The critical stress to maintain the expanding arc is:

Critical stress for Frank-Read source operation:
τ_FR = αGb / L

Where L = pinned segment length (m)
      α ≈ 0.5–1.0 (geometry factor)

Operation cycle:
1. Segment bows out, reaching maximum curvature (semicircle, r = L/2)
   at τ = τ_FR (the critical maximum stress)
2. Arms elongate, wrap around pinning points
3. Opposite-signed screw arms meet behind source → annihilate
4. Complete loop released, segment restored → repeat indefinitely

Smallest loop emitted when L is smallest (fewest precipitates);
largest loops when L is largest (coarser obstacle spacing).

Dislocation density increase per strain increment:
dρ/dε = M/bL (Kocks-Mecking model storage rate)
           − k₂ρ (dynamic recovery term)

At steady state (dρ/dε = 0):  ρ_ss = (M/bLk₂)²

The Four Dislocation Strengthening Mechanisms

All strengthening in structural metals and alloys is achieved by increasing the resistance to dislocation motion. Four principal mechanisms exist, and their contributions to yield stress are approximately additive in the absence of cross-interactions:

Total yield stress (additive rule of mixtures, approximate):
σ_y = σ_0 + Δσ_WH + Δσ_SS + Δσ_GB + Δσ_PH

Where:
  σ_0   = Peierls-Nabarro lattice friction stress (intrinsic)
  Δσ_WH = work hardening (MαGb√ρ)
  Δσ_SS = solid solution hardening
  Δσ_GB = Hall-Petch grain boundary hardening
  Δσ_PH = precipitation / dispersion hardening

Note: Linear additivity overestimates when mechanisms compete;
Pythagorean summation (σ = √(Δσ_A² + Δσ_B²)) better for
mechanisms operating on similar obstacle spacing scales.

Peierls-Nabarro Stress and Crystal Structure Dependence

The intrinsic lattice friction stress — the stress required to move a dislocation through a perfect crystal with no other obstacles — is described by the Peierls-Nabarro (P-N) model. This stress represents the fundamental lower bound on dislocation-mediated plastic flow and governs temperature-dependent yielding behaviour, particularly the strong temperature dependence of yield stress in BCC metals.

Peierls-Nabarro stress:
τ_PN = (2G/(1−ν)) × exp(−2πw/b)

Where:
  w = dislocation width = 2a/(1−ν)  (a = interplanar spacing of slip plane)
  b = Burgers vector magnitude

The critical parameter is w/b:
  High w/b (wide dislocation, close-packed planes) → low τ_PN
  Low w/b (narrow dislocation, open planes) → high τ_PN

Approximate τ_PN / G values:
  FCC (Al, Cu) on {111}: τ_PN ≈ 10⁻⁵ G  (very easy glide)
  BCC (α-Fe) on {110}: τ_PN ≈ 10⁻³ G  (moderate friction)
  HCP (Mg) on (0001):   τ_PN ≈ 10⁻⁵ G  (basal; easy)
  HCP (Ti) on {10̄10}:  τ_PN ≄ 10⁻³ G  (prismatic; harder)
  Diamond cubic (Si):   τ_PN ≈ 10⁻² G  (brittle at RT)

The strong temperature dependence of BCC yield stress — the basis of the ductile-to-brittle transition (DBT) in ferritic steels — arises because at low temperatures the P-N stress dominates yield: as temperature falls, the thermal energy available to assist dislocations over the P-N barrier decreases, and yield stress rises sharply. This is the metallurgical foundation of the Charpy impact transition temperature curve, described in detail in the Charpy impact testing article.

Comparative Summary: Edge, Screw, and Mixed Dislocations

Slip Systems in Common Metal Crystal Structures

Observing Dislocations: TEM Characterisation

Transmission electron microscopy (TEM) remains the primary technique for direct dislocation observation. Dislocations appear in bright-field TEM as dark, meandering lines through contrast arising from the bending of diffracting lattice planes by the dislocation strain field. The Burgers vector is determined using the invisibility criterion:

TEM invisibility criterion:
g · b = 0    →  dislocation invisible (no contrast)
g · b ≠ 0    →  dislocation visible (strong contrast)

Where g = diffraction vector (reciprocal lattice vector
         connecting transmitted to diffracted beam)

Burgers vector determination procedure:
1. Tilt specimen to two-beam condition with known g vector
2. If dislocation invisible: g · b = 0  (b is perpendicular to g)
3. Repeat with a second g' at different tilt
4. b lies in the plane perpendicular to both g and g'
   →  b is parallel to g × g'

Partial dislocation: g · b_partial = 0 or ±1/3 (fractional values
  confirm partial; full dislocation gives integer g·b)

Stacking fault fringe contrast:
  Visible when g · R ≠ integer (R = fault displacement vector)
  SFE estimation: measure equilibrium partial separation in weak-beam DF

Weak-beam dark-field (WBDF) TEM imaging — imaging with a weakly excited diffracted beam — sharply reduces dislocation image width from ~10 nm in bright-field to ~1.5 nm, enabling clear resolution of closely spaced dislocations and accurate measurement of partial dislocation separation for stacking fault energy determination. For bulk, large-volume dislocation density mapping, electron backscatter diffraction (EBSD) kernel average misorientation (KAM) analysis is now the standard complementary technique to TEM, providing spatially resolved geometrically necessary dislocation (GND) density information at the grain scale without specimen thinning.

Engineering Significance: Dislocations in Design and Failure Analysis

Dislocation theory underlies every aspect of structural metal performance:

• Steel heat treatment design: Quenching to produce martensite — which contains a dislocation density of ~10¹⁵ m⁻² from the diffusionless transformation — then tempering to allow partial recovery and carbide precipitation on dislocation tangles, achieves the optimal combination of strength and toughness in high-strength steel components.
• Cold-forming and wire drawing: Each drawing pass increases ρ and flow stress. The Taylor equation predicts the achievable strength; the Kocks-Mecking model predicts when dynamic recovery will limit further hardening. Wire break analysis uses TEM to confirm the deformation mode (dislocation cell structure vs. shear band).
• Welding and HAZ softening: The heat input of a weld pass triggers recovery and recrystallisation in cold-worked parent material within the HAZ, locally reducing ρ and yield strength. This HAZ softening in work-hardened pipe steels can reduce the weld joint efficiency below unity unless the weld procedure is engineered to minimise peak HAZ temperature. See the HAZ microstructure article for details.
• Creep and high-temperature service: Dislocation climb rate determines the minimum creep rate in power-law creep: ṣ = Aσⁿexp(−Q_c/RT), where Q_c is the activation energy for climb (approximately equal to the self-diffusion activation energy). Dispersion-strengthened alloys (ODS steels, MA957) use Y₂O₃ particles to impede climb at elevated temperatures.
• Fatigue crack initiation: Cyclic plastic deformation produces persistent slip bands (PSBs) — channels of intense dislocation activity with a characteristic wall-and-channel microstructure — that concentrate slip at the surface, producing extrusions and intrusions that initiate fatigue cracks. The PSB mechanism is the basis of fatigue life prediction in smooth specimens.