Chapter 3: Crystallographic Directions And Planes

Chapter 3: Crystallographic directions and planesOutline Crystallographic directions Crystallographic planes Linear and planar atomic densities Close-packed crystal structuresCrystallographic directions Direction: a line between two pointsand a vector General rules for defining acrystallographic direction pass through the origin of a coordinatesystem determine length of the vectorprojection in the unit cell dimensions a,b, and c remove the units [ua vb wc]---[uvw]e.g [2a 3b 5c]--[2 3 5] uvw are multiplied and divided by acommon factor to reduce them tosmallest integer values1

Crystallographic directions (continue) denote the direction by [uvw] family direction u v w , defined by transformation material properties along any direction in a family are thesame, e.g. [100],[010],[001] in simple cubic are same. for uniform crystal materials, all parallel directions have thesame properties negative index: a bar over the indexDetermine a directionExamplesDetermine the indices of line directions2

ExamplesSketch the following directions : [110], [1 2 1], [ 1 0 2]Hexagonal crystal 4-index, or Miller-Bravais, coordinate system Conversion from 3-index to 4-index systemza2-a3a1[ u'v'w ' ] [ uvtw]1u (2u' - v')31v (2v' - u')3t -(u v)w w'Fig. 3.8(a), Callister 7e.3

Crystallographic planes Orientation representation (hkl)--Miller indices Parallel planes have same miller indices Determine (hkl) A plane can not pass the chosen origin A plane must intersect or parallel any axis If the above is not met, translation of the plane or origin isneeded Get the intercepts a, b, c. (infinite if the plane is parallel to anaxis) take the reciprocal smallest integer rule (hkl) // (hkl) in opposite side of the origin For cubic only, plane orientations and directions with same indices are perpendicular to one anotherCrystallographic planes (continue)Adapted from Fig. 3.9, Callister 7e.4

Crystallographic planes (continue)zExample1. Intercepts2. Reciprocals3.Reductiona11/1114.Miller Indices(110)Example1. Intercepts2. Reciprocals3.4.ReductionMiller Indicesb11/111a1/21/½22c 1/ 00cybaxb 1/ 00zc 1/ 00cyba(100)xCrystallographic planes (continue)In hexagonal unit cells the same idea is a2a3c1111 1/ 00-1-1-1-11111a2a34.Miller-Bravais Indices(1011)a1Adapted from Fig. 3.8(a), Callister 7e.5

Crystallographic planes (continue) Determine Miller indices of planesCrystallographic planes (continue) Determine Miller indices of planes6

Crystallographic planes (continue) Construct planes by Miller indices of planes (0 1 1) and (11 2)Atomic arrangements The atomic arrangement for a crystallographic planedepends on the crystal structureFCC: (a) reduced sphere(b) atomic packing ofan FCC (110) planeBCC: (a) reduced sphere(b) atomic packing ofan BCC (110) plane7

Atomic arrangements A family of planes contains all the planes that arecrystallographically equivalent. In cubic system, planes with same indices,irrespective of order and sign, are equivalent (111), (111), (111) belong to {111} family (100), (100), (010), and (001) belong to {100} family (123), (123), (312) in cubic crystals belong to {123}family In tetragonal, (001) (001) are not as same family as(100), (100)Linear and planar atomic density Linear Density of Atoms LD #atomsUnit length of direction vector Atomic planar density: number of atoms centeredon a plane/area of plane#atomsPlanar Density Area(2D repeat unit)8

Linear density (example)[110]Linear density of Al in [110]directiona 0.405 nm# atomsLD alength2 3.5 nm 12aPlanar Density of (100) IronSolution: At T 912 C iron has the BCC structure.2D repeat unit(100)Planar Density area2D repeat unit1a2 4 3R3Radius of iron R 0.1241 nmAdapted from Fig. 3.2(c), Callister 7e.atoms2D repeat unita 14 3R32 12.1atomsatoms 1.2 x 10192nmm29

Close-packed crystal structures Close-packed plane stackingsequence for HCP Close-packed plane stackingsequence for HCP10

2 Crystallographic directions (continue) denote the direction by [uvw] family direction u v w , defined by transformation material properties along any direction in a family are the same, e.g. [100],[010],[001] in simple cubic are same. for uniform crystal materials, all parallel directions have the same properties negative index: a bar over the index