Editing Dislocation (section)

== Example in two dimensions (2D) ==

[[File:Dislokation Dissoziation.svg|thumb|Dissociation of a pair of dislocations due to shearing (red arrows) of an hexagonal crystal in 2D. A dislocation in 2D consists of a bound pair of five-folded (green) and seven-folded (orange) coordination number.]]
In two dimensions (2D) only the edge dislocations exist, which play a central role in melting of 2D crystals, but not the screw dislocation.
Those dislocations are [[Topology|topological]] point defects which implies that they cannot be created isolated by an [[affine transformation]] without cutting the hexagonal crystal up to infinity (or at least up to its border). They can only be created in pairs with antiparallel [[Burgers vector]]. If a lot of dislocations are e.&nbsp;g. thermally excited, the discrete translational order of the crystal is destroyed. Simultaneously, the [[shear modulus]] and the [[Young's modulus]] disappear, which implies that the crystal is molten to a fluid phase. The orientational order is not yet destroyed (as indicated by lattice lines in one direction) and one finds - very similar to liquid crystals - a fluid phase with typically a six-folded director field. This so-called [[hexatic phase]] still has an orientational stiffness. The isotropic fluid phase appears, if the dislocations dissociate into isolated five-folded and seven-folded [[disclination]]s.<ref>{{cite journal | last1=Gasser | first1=U. | last2=Eisenmann | first2=C. | last3=Maret | first3=G. | last4=Keim | first4=P. | title=Melting of crystals in two dimensions | journal=ChemPhysChem | volume=11 |issue=5 | date=2010 | doi=10.1002/cphc.200900755 | pmid=20099292 | pages=963–970| url=http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-112435 }}</ref> This two step melting is described within the so-called Kosterlitz-Thouless-Halperin-Nelson-Young-theory ([[KTHNY theory]]), based on two transitions of [[Kosterlitz–Thouless transition|Kosterlitz-Thouless-type]].