Crystallographic defect

A crystallographic defect is a departure from ideal repeated structure. In an ionic crystal, a missing ion can provide an available site for another ion to move into; movement still requires overcoming an activation energy. This is why an ideal immobile-site model is a limit rather than a universal law for solid electrical conductivity.^[ref]

An interface between ordered domains related by a displacement that is a translation of the underlying disordered lattice but not of the ordered chemical structure. Slip by that displacement can leave an antiphase boundary behind a dislocation; a second appropriate displacement can restore the order.

An interface between crystalline regions with different orientations. Its structure affects transport, mechanical response and absorption of mobile point defects.

Resistance to plastic flow associated with grain boundaries obstructing or transmitting dislocations. In the ordinary Hall-Petch regime, smaller grains raise the yield stress, but at high temperature the same boundaries can accelerate diffusion creep.

The empirical scaling ${\sigma }_y={\sigma }_0+k_y{d}^{-1/2}$ of yield stress with average grain size in its applicable regime. It is not a universal law for arbitrarily small grains or high-temperature deformation.

A localized crystal defect such as a vacancy, substitutional impurity or interstitial atom.

A missing carbon-atom inventory in graphene. Local reconstruction changes the nearby ring pattern and electronic states; irradiation and subsequent annealing can change the reconstruction.

A graphene vacancy defect with two carbon atoms removed. A common adjacent-pair reconstruction has a 5–8–5 ring arrangement; other reconstructed arrangements exist and can be selected by energetic processing.

A graphene vacancy defect with one carbon removed. A common reconstruction has a pentagon and nine-membered ring with residual under-coordination. The reconstructed geometry need not retain the symmetry of the unrelaxed missing-atom site.

An extra atom of the host material accommodated outside the ordinary lattice-site inventory. It can migrate, recombine with a vacancy, or join a cluster.

An unoccupied site in an otherwise specified crystal structure. Equilibrium vacancies and an irradiation-driven vacancy supersaturation have different origins.

A vacancy population above its thermal-equilibrium value at the specified temperature. It can drive net vacancy absorption by growing cavities when competing sinks and emission permit.

A vacancy and a displaced atom occupying an interstitial position. Recombination restores the occupied lattice site; spatial separation or trapping can allow the defects to survive.

Notation specifying a defect species, its lattice site and its charge relative to the corresponding ideal lattice site.

A missing carbon atom or group of atoms in a carbon nanotube wall. Local reconstruction can produce a 5–9 monovacancy or 5–8–5 divacancy ring inventory; larger vacancies can interrupt a shell.

A bond rotation that changes four hexagons in a graphitic carbon network into a pentagon-heptagon-heptagon-pentagon ring arrangement without adding or removing atoms.

A line defect in a crystal structure. Its Burgers vector measures the closure failure of a circuit around the line. Motion can produce permanent plasticity without simultaneously breaking every bond across a whole plane.

The dislocation core is the small region near a dislocation line where the smooth linear-elastic continuum description fails. For a screw dislocation, the far-field displacement gradient scales as the Burgers-vector magnitude divided by radial distance, so the predicted elastic stress diverges on the axis. Atomic structure, nonlinear response or local plastic deformation replaces that singular prediction in the core.

A curved dislocation can be approximated as a line with tension $T_{\ell }$ of order $G{b}^2$. The coefficient depends on elastic constants, character and logarithmic length scales. The simple estimate $T_{\ell }\simeq G{b}^2/2$ gives a pinning bypass stress $Gb/\ell$ for a semicircular segment of free span $\ell$.

A dislocation whose Burgers vector is not a full translation of the perfect crystal, so its glide can leave a planar stacking fault. Vector conservation and the actual slip plane constrain reactions and dissociation.

A sessile partial junction associated with intersecting stacking faults. A typical FCC resultant has vector $a⟨110⟩/6$. Its Burgers vector and junction line can define a plane outside the ordinary $\{111\}$ glide family.

An FCC partial dislocation with Burgers vector $a⟨112⟩/6$ lying in a close-packed $\{111\}$ plane. Two suitable Shockley vectors can sum to an FCC perfect vector $a⟨110⟩/2$.

A perfect vector $a⟨110⟩/2$ can separate into two in-plane Shockley partials bounding a stacking-fault ribbon. The sum of squared Burgers magnitudes falls from $a^2/2$ to $a^2/3$, reducing the simplest elastic line-energy estimate, while stacking-fault area costs energy.

An order-preserving total dislocation in an ordered crystal, often dissociated into smaller dislocations separated by an antiphase boundary. Two matching $a⟨110⟩/2$ dislocations can restore order in the gamma prime phase.

A dislocation that accommodates some lattice mismatch at an interface. An initially coherent epitaxial layer or island can instead store elastic deformation; introducing a misfit dislocation trades some strain energy for the defect's formation energy. A projected image must be indexed before a particular spacing discrepancy is assigned to misfit strain.

Motion of an edge component perpendicular to its glide plane by absorbing or emitting point defects. Unlike conservative dislocation glide, it changes the extra-plane content locally and is coupled to defect transport. Irradiation can drive climb by creating nonequilibrium vacancies and self-interstitials.

Total dislocation line length per material volume, with units ${\mathrm{m}}^{-2}$. Circular loops with common radius $R$ and number density $N_{\ell }$ contribute $2\pi R{N}_{\ell }$.

A closed dislocation line. Vacancy or interstitial clustering can form a platelet bounded by a loop; its Burgers vector, habit plane and possible stacking fault specify its crystallographic character.

A line segment formed by interaction of dislocations from different slip planes. A junction can impede further dislocation glide. Energetic favourability alone does not establish whether the resulting segment is mobile on a particular plane.

A junction produced when suitable partial dislocations on intersecting FCC slip planes react to form a low-energy sessile stair-rod segment. It obstructs further slip and can promote work hardening. Frank energetic favorability alone is not a complete calculation of junction geometry or mobility.

The energy criterion associated with Frederick Charles Frank states that, for dislocations of comparable length with approximately common line-energy factors, a reaction conserving Burgers vector is energetically favourable when the displayed inequality holds. The rule is an energy estimate, not a proof that the geometry or kinetic barriers permit the reaction. Edge/screw character, elastic anisotropy and changed lengths refine it.

Conservative motion within a plane containing both the dislocation line and its Burgers vector. For a nonscrew line their cross product determines the plane normal; a screw line can belong to more than one such plane.

A lattice-friction stress for dislocation glide, associated with Rudolf Peierls and Frank Nabarro. A simple model gives ${\tau }_P\sim 3G\exp (-2\pi w/b)$, where $w$ is core width and $b$ the Burgers vector magnitude. Spreading slip over many atomic rows reduces the periodic energy barrier; narrower cores are harder to move in this model.

A crystallographic plane together with an allowed dislocation glide direction in that plane. Geometry alone does not determine its activation stress.

Plastic flow involving more than one active slip system. Interactions between dislocations on intersecting systems can give substantially faster work hardening than easy single slip.

Dislocation glide in a hexagonal crystal on its basal $(0001)$ plane, ordinarily along a close-packed basal direction in the $⟨11\overline{2}0⟩$ family. Direction indices must be converted using the actual hexagonal metric, including the ratio $c/a$.

For uniaxial stress $\sigma$, projection onto a slip-plane normal and a slip direction gives the displayed shear stress. The angles $\varphi$ and $\lambda$ are measured from the tensile axis to that normal and direction respectively.

The critical resolved shear stress is the shear stress on a specified slip system at which dislocation motion produces slip under stated conditions. For simple uniaxial loading, the resolved component is $\tau =\sigma \cos \varphi \cos \lambda$, with angles to the slip-plane normal and slip direction.

The threshold depends on material state, temperature, obstacles, and deformation history. A crystal can therefore begin to slip at different applied tensile stresses in different orientations even when the slip system has the same critical resolved threshold.

The ratio $M=\sigma /{\tau }_c$ connecting a macroscopic tensile flow stress to a reference critical resolved shear stress in a specified polycrystal model. Texture, slip families and compatibility assumptions matter. An order-three factor is a conventional estimate for randomly oriented cubic grains, distinct from the factor two for a perfectly oriented single slip system.

The ideal activation criterion compares resolved shear stress with a critical resolved shear stress. Non-glide stresses, notably in some body-centred cubic crystals, can modify this ideal criterion.

In an ideal single-slip tensile test neglecting elastic stretch, the specimen end-to-end vector expressed in crystal coordinates changes as $\mathbf{r}={\mathbf{r}}_0+\gamma (\mathbf{n}\cdot {\mathbf{r}}_0)\mathbf{s}$, with unit slip direction $\mathbf{s}$ and slip-plane normal $\mathbf{n}$. It moves towards the slip direction along a great circle. The perpendicular component is unchanged, so $L\sin \lambda =L_0\sin {\lambda }_0$. Secondary slip changes this kinematics; equal Schmid factors determine onset only if the systems have equal current critical stresses.

The orientation factor converting the magnitude of uniaxial stress to resolved shear stress. It lies between zero and $1/2$ because the slip-plane normal and slip direction are perpendicular.

For line tangent $\mathbf{t}$, a screw dislocation has Burgers vector parallel to $\mathbf{t}$, an edge dislocation has it perpendicular, and a mixed dislocation has both components.

A mixed dislocation has a line direction neither parallel nor perpendicular to its Burgers vector, combining screw and edge character. Along a curved dislocation line, the local character can change continuously while the Burgers vector remains fixed.

A dislocation loop therefore need not be assigned one character everywhere. Decomposing a local segment into edge and screw contributions helps analyze its stress field, mobility, and interaction with obstacles.

An edge dislocation is a crystal line defect whose Burgers vector is perpendicular to the dislocation line. It is often pictured as the termination of an extra half-plane of atoms inside the crystal.

Motion of the line allows a slip displacement to advance through the material without every bond across a whole plane changing simultaneously. Its surrounding elastic distortion includes compressive and tensile regions, connecting an atomic-scale defect with macroscopic plastic deformation.

A screw dislocation is a crystal line defect whose Burgers vector is parallel to the dislocation line. Atomic planes around it have a helical arrangement, rather than the extra-half-plane geometry associated with an edge dislocation.

Its motion permits crystal slip. Because several planes can contain the same line and Burgers vector, screw segments can sometimes change their glide plane by cross-slip, with the actual possibility depending on the crystal and defect structure.

The antiplane field $w=b\arg (x+iy)/(2\pi )$ has a smooth local gradient on the punctured plane and satisfies the differential strain compatibility condition there. Its circulation around the defect axis is nevertheless $b\ne 0$, excluding a globally single-valued displacement on that domain. A radial cut permits a branch with an axial jump $b$ between the two cut faces; joining faces with that mismatch is the Volterra construction of a screw dislocation with Burgers vector parallel to the axis. The stress has a singular core where linear elasticity requires microscopic or plastic regularization. This example separates a local differential integrability test from global closure around a hole.

Motion of a screw segment from one allowed glide plane to another containing the same Burgers vector.

The lattice translation represented by closure failure of a circuit around a dislocation, with a consistent circuit convention. It is a vector, so dislocation reactions conserve its direction as well as its magnitude.

A missing atom at a lattice site that would be occupied in the reference crystal. A vacancy defect is a point defect and can contribute configurational entropy because many choices of empty sites can have the same macroscopic energy. Formation energy and pressure-volume work oppose this entropy gain. Vacancies can be present in thermal equilibrium; they need not all be preparation mistakes. Their allowed number and creation mechanism depend on whether the lattice size, atom number and surface reservoirs are fixed.

An electron trapped at an anion vacancy in an ionic crystal.

An unoccupied oxygen site in a reference crystal structure. Removing an oxide ion gives an effective charge $+2e$ relative to the perfect lattice; the charge of the actual missing ion is $-2e$. A physical sample must satisfy overall charge neutrality through other defects, electronic carriers or changed cation valences.

Consider fixed $N$ atoms and variable $n$ vacancies among $N+n$ sites, each of volume $v$, with vacancy energy $ϵ$. The configuration count is $(N+n)!/(N!n!)$. At fixed pressure and temperature, leading Stirling's approximation gives the excess Gibbs free energy $n(ϵ+pv)-k_{B}T[(N+n)\ln (N+n)-N\ln N-n\ln n]$. Its stationary point is the displayed $n_{\ast }$, and its second derivative is $k_{B}TN/[n(N+n)]>0$. Relative to the vacancy-free Gibbs free energy, the minimum is $N{k}_{B}T\ln (1-e^{-(ϵ+pv)/(k_{B}T)})$. The resulting mean-field volume is $v(N+n_{\ast })$. Normalizability requires $ϵ+pv>0$, and large vacancy fractions lie outside a realistic dilute-defect crystal model.

If the variable-site model counts every $n$ sector with weight $\left(\genfrac{}{}{0ex}{}{N+n}{n}\right)q^n$, its exact relative partition sum is $(1-q{)}^{-N-1}$. Differentiating its logarithm gives the displayed exact mean. Minimizing the leading extensive Gibbs free energy instead gives $Nq/(1-q)$, the same result to relative order $1/N$. Its exact relative equilibrium free energy is $(N+1)k_{B}T\ln (1-q)$. This correction distinguishes a thermodynamic saddle from an exact finite-system average; changing the microscopic site-creation convention can also change subextensive factors.

An interstitial defect is an atom occupying a position between the regular atomic positions in a crystal structure. Small solute atoms or an extra host atom can occupy such positions. Different interstitial orientations can induce anisotropic local strain; an applied stress changes their relative energies. Thermally activated hopping among those positions can then produce delayed elastic strain and the Snoek effect.