Editing Dislocation (section)

==Geometry==
[[File:Edge dislocation 2.svg|thumbnail|right|An edge-dislocation (b = [[Burgers vector]])]]
Two main types of mobile dislocations exist: edge and screw. Dislocations found in real materials are typically ''mixed'', meaning that they have characteristics of both.

===Edge===
[[Image:Dislocation edge d2.svg|right|thumb|Schematic diagram (lattice planes) showing an edge dislocation. Burgers vector in black, dislocation line in blue.]]

A crystalline material consists of a regular array of atoms, arranged into lattice planes. An edge dislocation is a defect where an extra half-plane of atoms is introduced midway through the crystal, distorting nearby planes of atoms. When enough force is applied from one side of the crystal structure, this extra plane passes through planes of atoms breaking and joining bonds with them until it reaches the grain boundary.  The dislocation has two properties, a line direction, which is the direction running along the bottom of the extra half plane, and the [[Burgers vector]] which describes the magnitude and direction of distortion to the lattice. In an edge dislocation, the Burgers vector is perpendicular to the line direction.

The stresses caused by an edge dislocation are complex due to its inherent asymmetry. These stresses are described by three equations:<ref name="rhill">{{cite book|first1=R.E.|last1=Reed-Hill|first2=Reza|last2=Abbaschian|date=1994|title=Physical Metallurgy Principles|location=Boston|publisher=PWS Publishing Company|isbn=0-534-92173-6}}</ref>

::<math> \sigma_{xx} = \frac {-\mu \mathbf{b}} {2 \pi (1-\nu)} \frac {y(3x^2 +y^2)} {(x^2 +y^2)^2}</math>

::<math> \sigma_{yy} = \frac {\mu \mathbf{b}} {2 \pi (1-\nu)} \frac {y(x^2 -y^2)} {(x^2 +y^2)^2}</math>

::<math> \tau_{xy} = \frac {\mu \mathbf{b}} {2 \pi (1-\nu)} \frac {x(x^2 -y^2)} {(x^2 +y^2)^2}</math>

where <math>\mu</math> is the [[shear modulus]] of the material, <math>\mathbf{b}</math> is the [[Burgers vector]], <math>\nu</math> is [[Poisson's ratio]] and <math>x</math> and <math>y</math> are coordinates.

These equations suggest a vertically oriented dumbbell of stresses surrounding the dislocation, with compression experienced by the atoms near the "extra" plane, and tension experienced by those atoms near the "missing" plane.<ref name="rhill"/>

===Screw===
A ''screw dislocation'' can be visualized by cutting a crystal along a plane and slipping one half across the other by a lattice vector, the halves fitting back together without leaving a defect. If the cut only goes part way through the crystal, and then slipped, the boundary of the cut is a screw dislocation. It comprises a structure in which a [[Helix|helical]] path is traced around the linear defect (dislocation line) by the atomic planes in the crystal lattice. In pure screw dislocations, the Burgers vector is parallel to the line direction.<ref>{{cite book|author=James Shackelford|title=Introduction to Materials Science for Engineers|year=2009|publisher=Pearson Prentice Hall|location=Upper Saddle River, NJ|isbn=978-0-13-601260-3|pages=110–11|edition=7th}}</ref>
An array of screw dislocations can cause what is known as a twist boundary. In a twist boundary, the misalignment between adjacent crystal grains occurs due to the cumulative effect of screw dislocations within the material. These dislocations cause a rotational misorientation between the adjacent grains, leading to a twist-like deformation along the boundary. Twist boundaries can significantly influence the mechanical and electrical properties of materials, affecting phenomena such as grain boundary sliding, creep, and fracture behavior<ref>{{cite book|author=James Shackelford|title=Introduction to Materials Science for Engineers|year=2009|publisher=Pearson Prentice Hall|location=Upper Saddle River, NJ|isbn=978-0-13-601260-3|pages=110–11|edition=7th}}</ref>
The stresses caused by a screw dislocation are less complex than those of an edge dislocation and need only one equation, as symmetry allows one radial coordinate to be used:<ref name="rhill"/>

::<math> \tau_{r} = \frac {-\mu \mathbf{b}} {2 \pi r} </math>

where <math>\mu</math> is the [[shear modulus]] of the material, <math>\mathbf{b}</math> is the Burgers vector, and <math>r</math> is a radial coordinate.
This equation suggests a long cylinder of stress radiating outward from the cylinder and decreasing with distance. This simple model results in an infinite value for the core of the dislocation at <math>r=0</math> and so it is only valid for stresses outside of the core of the dislocation.<ref name="rhill"/> If the Burgers vector is very large, the core may actually be empty resulting in a [[micropipe]], as commonly observed in [[silicon carbide]].

===Mixed===
In many materials, dislocations are found where the line direction and Burgers vector are neither perpendicular nor parallel and these dislocations are called ''mixed dislocations'', consisting of both screw and edge character. They are characterized by  <math>\varphi</math>, the angle between the line direction and Burgers vector, where <math>\varphi=\pi/2</math> for pure edge dislocations and  <math>\varphi=0</math> for screw dislocations.

===Partial===
{{main article|Partial dislocations}}

Partial dislocations leave behind a stacking fault. Two types of partial dislocation are the ''Frank partial dislocation'' which is sessile and the ''Shockley partial dislocation'' which is glissile.<ref name="radwan"/>

A Frank partial dislocation is formed by inserting or removing a layer of atoms on the {111} plane which is then bounded by the Frank partial. Removal of a close packed layer is known as an ''intrinsic'' stacking fault and inserting a layer is known as an ''extrinsic'' stacking fault. The Burgers vector is normal to the {111} glide plane so the dislocation cannot glide and can only move through ''climb''.<ref name="hull01"/>

In order to lower the overall energy of the lattice, edge and screw dislocations typically disassociate into a [[stacking fault]] bounded by two Shockley partial dislocations.<ref name=Föll>{{cite web|title=Defects in Crystals|first=Helmut|last=Föll|url=http://dtrinkle.matse.illinois.edu/MatSE584/index.html |access-date=2019-11-09}}</ref> The width of this stacking-fault region is proportional to the [[stacking-fault energy]] of the material. The combined effect is known as an ''extended dislocation'' and is able to glide as a unit. However, dissociated screw dislocations must recombine before they can [[cross slip]], making it difficult for these dislocations to move around barriers. Materials with low stacking-fault energies have the greatest dislocation dissociation and are therefore more readily cold worked.

=== Stair-rod and the Lomer–Cottrell junction ===
If two glide dislocations that lie on different {111} planes split into Shockley partials and intersect, they will produce a stair-rod dislocation with a [[Lomer-Cottrell junction|Lomer-Cottrell dislocation]] at its apex.<ref>{{cite web |title=Reaction Forming a Stair-Rod Dislocation |url=https://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/illustr/i5_4_2.html |access-date=26 Nov 2019}}</ref> It is called a ''stair-rod'' because it is analogous to the rod that keeps carpet in-place on a stair.

=== Jog ===
[[File:Jog-Kink.gif|thumb|right|Geometrical differences between jogs and kinks]]
A '''Jog''' describes the steps of a dislocation line that are not in the [[glide plane]] of a [[crystal structure]].<ref name=Föll/> A dislocation line is rarely uniformly straight, often containing many curves and steps that can impede or facilitate dislocation movement by acting as pinpoints or nucleation points respectively. Because jogs are out of the glide plane, under shear they cannot move by glide (movement along the glide plane). They instead must rely on vacancy diffusion facilitated climb to move through the lattice.<ref>{{cite book|first=W.|last1=Cai|first2=W. D.|last2=Nix|title=Imperfections in crystalline solids|location=Cambridge, UK|publisher=Cambridge University Press|year=2016}}</ref> Away from the melting point of a material, vacancy diffusion is a slow process, so jogs act as immobile barriers at room temperature for most metals.<ref name="courtney00">{{cite book|first=T. H.|last=Courtney|title=Mechanical behavior of materials|location=Long Grove, IL|publisher=Waveland|year=2000}}</ref>

Jogs typically form when two non-parallel dislocations cross during slip. The presence of jogs in a material increases its [[yield strength]] by preventing easy glide of dislocations. A pair of immobile jogs in a dislocation will act as a [[Frank–Read source]] under shear, increasing the overall dislocation density of a material.<ref name="courtney00" /> When a material's yield strength is increased via dislocation density increase, particularly when done by mechanical work, it is called [[work hardening]]. At high temperatures, vacancy facilitated movement of jogs becomes a much faster process, diminishing their overall effectiveness in impeding dislocation movement.

=== Kink ===
{{main article|Kink (materials science)}}

Kinks are steps in a dislocation line parallel to glide planes. Unlike jogs, they facilitate glide by acting as a nucleation point for dislocation movement. The lateral spreading of a kink from the nucleation point allows for forward propagation of the dislocation while only moving a few atoms at a time, reducing the overall energy barrier to slip.