### 5.1.3 Essentials to Chapter 5.1: Dislocations - Basics

| Plastic deformation of crystals=movement of dislocations through the crystal. |  |
|  | The distortion necessary to deform a crystal is localized in a 1-dimensional defect=dislocation that moves through the crystal under the influence of (external shear) forces. |
|  | "Discovery" of dislocations as source of plastic deformation=answer to one of the biggest and oldest scientific puzzles in 1934 (Taylor, Orowan and Polyani). No Noble prize! |
|  | Movement of dislocations produce steps of atomic size characterized by a vector called "Burgers vector". |
|  | Movement of dislocations occurs in a plane (=glide plane) and shifts the upper part of the crystal with respect to the lower part. |

| Dislocations are characterized by |  |
Burgers circuit for a screw dislocation
| 1. Their Burgers vector b= |
|  | vector describing the step obtained after a dislocation passed through the crystal. Vector obtained by a Burgers circuit around a dislocation. Translation vector in the Volterra procedure. |
|  | All definitions of b give identical results for a given dislocations; but watch out for sign conventions! |
|  | By definition, b is always a translation vector T of the lattice. |
|  | For energetic reasons b is usually the shortest translation vector of the lattice; e.g. b=a/2 <110> for the fcc lattice. |
| 2. Their line vector t(x,y,z) describing the direction of the dislocation line in the lattice |
|  | t(x,y,z) is an arbitrary (unit) vector in principle but often a prominent lattice direction in reality |
|  | While the dislocation can be curved in any way, it tends to be straight (=shortest possible distance) for energetic reasons. |
| The glide plane by necessity must contain t(x,y,z) and b and is thus defined by the two vectors |  |
a=90 o: Edge dislocation. a=0 o: Screw dislocation. a=60 o: "Sixty degree" dislocation. a=arbitrary : "Mixed" dislocation.
|  | The angle a between t(x,y,z) and b determines the character or kind of dislocation: |
|  | Note that any plane containing t is a glide plane for a screw dislocation. |
| Dislocations have a large line energy Edis per length and therefore are never thermal equilibrium defects |  |
Edis » 5 eV/|b|

| The formal Volterra definition of dislocations is very useful and extendable to more complex kinds of dislocations |  |
|  | Cut into the lattice with a fictitious "Volterra" knife (make a plane cut to keep it easy), The cut line is a closed loop by necessity. The part of the cut line inside the crystal identifies the dislocation line. |
|  | Move the part of the lattice above (or below - attention, signs change!!) the cut plane by an arbitrary lattice translation vector=Burgers vector of the dislocation. Add or remove lattice points as necessary (=remove or fill in atoms in the crystal going with the lattice). |
|  | Mend the lattice (or crystal) by "welding the upper part to the lower one. There will be a perfect fit by definition everywhere except along the dislocation line. |
|  | Make the best arrangement of the atoms along the dislocation line by minimizing their energy (make best possible bonds). |
| You now have formed a dislocation. |
|  | The procedure is "easily" extended to dislocations in n-dim. lattices, to (special) dislocations with a Burgers vector not defined as translation vector of the lattice and to lattices more complex than a simple crystal lattice. |
| Direct consequences are: |  |
|  | A dislocation cannot just end in the interior of a crystal |
|  | There is a "knot rule" for dislocation knots: S b=0 provided the signs of the line vectors follow a convention (all pointing to or away from the knot) |
|  | There can be all kinds of dislocation loops (just confine your fictitious cut to the lattice interior!) |
| Note: "Simple" geometric considerations allow to deduce a lot about properties of dislocations |  |

© H. Föll (Defects - Script)