ch. 4+7 : imperfections in solids and defects

What is a point defect?

A 0-dimensional defect involving individual atomic sites, such as vacancies or interstitials.

What is a vacancy defect?

A 0-dimensional point defect where an atom is missing from its normal lattice site.

What is a self-interstitial defect?

An extra atom jammed into a small space (interstice) between normal lattice sites in a crystal.

What is the formula for equilibrium vacancy concentration?

Nv / N = exp(-Qv / (k * T)), where Nv is vacancies, N is total sites, Qv is activation energy, k is Boltzmann's constant, and T is absolute temperature in Kelvin.

How can the activation energy (Qv) of vacancy formation be found graphically?

By plotting ln(Nv/N) vs. 1/T; the slope of the linear fit equals -Qv / k.

What are the four Hume-Rothery rules for substitutional solid solutions?

1. Atomic radius difference Δr < 15%

2. Similar electronegativities

3. Same crystal structure

4. Valence compatibility (higher valence dissolves better).

What is a Schottky defect?

A paired point defect in ionic crystals consisting of one cation vacancy and one anion vacancy to maintain charge neutrality.

What is a Frenkel defect?

A point defect in ionic crystals consisting of a cation vacancy combined with a cation self-interstitial.

What is weight percent (wt%) vs. atom percent (at%)?

wt% is based on the mass fraction of each element; at% is based on the number of moles or atoms fraction of each element.

What is an edge dislocation?

A 1D line defect formed by an extra half-plane of atoms inserted into the crystal lattice, where the Burgers vector is perpendicular to the dislocation line.

What is a screw dislocation?

A 1D line defect produced by shear strain where the atomic planes form a spiral ramp around the dislocation line; the Burgers vector is parallel to the dislocation line.

What is the relationship between the dislocation line and Burgers vector for a mixed dislocation?

The Burgers vector and dislocation line are neither parallel nor perpendicular; they lie at an arbitrary angle between 0° and 90°.

Why is the actual yield shear strength of metals much lower than the theoretical shear strength?

Because metals deform via incremental motion of dislocations (breaking one line of bonds at a time) rather than bodily sliding of entire atomic planes at once.

Why is dislocation motion much easier in metals than in covalent or ionic ceramics?

Metals have non-directional metallic bonding and close-packed planes. Covalent bonds are strong/directional, and ionic crystals resist bringing like-charged ions together.

What is a planar (2D) defect?

An interfacial boundary separating regions of different crystallographic orientations or stacking sequences, such as free surfaces, grain boundaries, twin boundaries, or stacking faults.

Why do free surfaces possess high surface energy?

Surface atoms are not bonded to the maximum number of nearest neighbors, leaving dangling bonds.

What is a low-angle grain boundary?

A grain boundary with a small angle of misalignment (< 10-15°), which can be modeled as an array/wall of aligned dislocations (e.g., edge dislocations for a tilt boundary).

What is the relationship between tilt boundary dislocation spacing D and misalignment angle θ?

D ≈ b / θ, where b is the magnitude of the Burgers vector and θ is the misalignment angle in radians.

What is a twin boundary?

A specific planar defect across which there is a mirror-image lattice orientation symmetry, formed by mechanical shear or thermal annealing.

What is the difference between mechanical twins and annealing twins?

Mechanical twins occur during rapid loading or low temperatures in BCC/HCP metals; Annealing twins form during heat treatment following plastic deformation (common in FCC metals).

What is a stacking fault?

A planar defect caused by an interruption in the regular stacking sequence of close-packed planes (e.g., ABCABABC instead of ABCABCABC in FCC).

How can average grain size be measured using the linear intercept method?

l_bar = L / (N_l * M), where L is line length, N_l is the number of grain boundary intercepts, and M is the magnification.

What is a slip system?

A combination of a preferred slip plane (widely spaced/close-packed plane) and a slip direction (close-packed direction) along which dislocation motion occurs.

What is the slip system for FCC metals and how many independent slip systems exist?

Slip plane: {111}, Slip direction: <110>. There are 12 total independent slip systems in FCC.

What is resolved shear stress (τ_R)?

The shear stress induced on a specific slip plane and along a specific slip direction under an applied tensile or compressive load: τ_R = σ * cos(λ) * cos(ϕ).

What is the Schmid Factor?

The orientation factor cos(λ) * cos(ϕ), where ϕ is the angle between the tensile axis and the slip plane normal, and λ is the angle between the tensile axis and the slip direction.

Under what geometric conditions is resolved shear stress (τ_R) maximized?

When λ = ϕ = 45°, making the Schmid Factor equal to 0.5, so τ_R = σ / 2.

What happens to resolved shear stress (τ_R) if either λ = 90° or ϕ = 90°?

τ_R = 0; no shear stress acts on that slip system, so dislocation motion cannot occur on it.

What is Critical Resolved Shear Stress (τ_CRSS)?

The minimum resolved shear stress required to initiate dislocation motion and cause plastic deformation in a single crystal.

What is the yield condition for a single crystal under tension?

Yielding occurs when the maximum resolved shear stress reaches or exceeds the critical resolved shear stress: τ_R ≥ τ_CRSS.

Why are polycrystalline metals stronger than single crystals of the same composition?

Grain boundaries act as barriers to dislocation motion because neighboring grains have different crystallographic orientations (changing slip direction across boundaries).

What is the Yield-Point Phenomenon (e.g., in low-carbon steel)?

A distinct stress-strain feature where upper yield point (A) occurs due to interstitial atoms (C, N) pinning dislocations. Once unpinned, load drops to lower yield point (a/b) with Lüders band propagation.

What are Lüders bands?

Visible localized shear bands that form and propagate along a specimen during yield-point elongation at a constant average stress.

What causes deformation anisotropy in polycrystalline metals during processes like rolling?

Plastically deforming a metal elongates spherical grains along the working direction, inducing texture and directional variance in mechanical properties.

What is the Plastic Strain Ratio (r-value)?

The ratio of width strain to thickness strain: r = ε_w / ε_t, used to measure normal anisotropy in sheet metals.

How does the c/a ratio of HCP metals affect their deformation anisotropy (r-value)?

High c/a ratio (e.g., Zn = 1.86) restricts slip to basal planes → high thickness strain → small r-value (~0.2).

Low c/a ratio (e.g., Ti = 1.59) enables prismatic/pyramidal slip → high width strain → large r-value (~6.0).