A.E., Cer.E., Chem.E, Met.E., And Phys. 377 Fall 2001 Principles of Engineering Materials

Crystallographic Planes and DirectionsCrystallographic Planes and Directions

Reason for Study:Arrangement of atoms is different in various directions & along various planes or surfacesCertain properties will also varyDirectional anisotropyDeformation follows particular planes and directionsNumber and type affect the strength and ductility of materialsCoordinates & Lattice Point LocationsCoordinate SystemBy convention, back bottom corner of a cube is selected as the originCartesian axes are placed along the edges of the cubeThe unit of length in a direction is the lattice parameter in that directionLattice PointsAny point in the lattice may now be specified by (x,y,z) coordinatesMiller Indices:

Miller Indices of Direction:To specify a given direction in the lattice:Determine the coordinate location for two different points lying along the line (start and finish)Subtract the coordinates of the start point from the coordinates of the finish point to determine the distance between the points in terms of lattice parameters in each of the three directions Clear fractions and/or reduce to lowest set of integersEnclose resulting three numbers (without commas) in a square bracket, indicating any negatives by a bar over the numberThe set in the brackets are the Miller Indices of direction: [u v w]Miller Indices of Direction:A direction and its negative are same line, but opposite senseMultiples of a direction are identicali.e. [110] and [220] are identical

x = 1-0 = 1y = 1-0 = 1[u v w] = [1 1 1]z = 1-0 = 1x = 1-1 = 0y = 0-1 = -1[u v w] = [0 ]z = 0-1 = -1Examples:Directions of a FormDirections having the same indices but in a different orderIdentical atomic arrangementIdentical property measurementsCan be lumped together in a single notation using slant brackets[1 0 0] [ 0 0] = [0 1 0] [0 0][0 0 1] [0 0 ]Close Packed DirectionsAtoms touch one another along a close packed directionThese will subsequently relate to metal deformationEasiest to deform if atoms follow one anotherSC cube edgesBCC body diagonalsFCC face diagonalsMiller Indices for PlanesProcedureIf the plane passes through the origin, move the coordinate system (i.e. the origin) to another lattice point within the cellDetermine the intercepts of the plane with respect to the axes (if parallel, use infinity)Take the reciprocals of the interceptsClear fractions (do not reduce to lowest integers)Enclose the three numbers in parentheses without commas: (h k l) --- using bars for negatives. New OriginExamples:x = 11/x = 1y = 11/y = 1(h k l) = (1 1 1)z = 11/z = 1x = 11/x = 1y = -11/y = (h k l) = (1 0)z = 1/z = 0Equivalent Planes:Planes which are parallel and intercept the axes at intervals of one lattice parameter are identicalA plane and its negative are identicalMultiples of Miller Indices designate parallel planes, but not identicalExamples of Planes

Planes of a Form:Planes having the same three numbers, but in different order are indistinguishable from one another (i.e. they have the same atomic structure)

(1 1 1) ( ){1 1 1} = ( 1 1) (1 )(1 1) ( 1 )(1 1 ) ( 1) Close Packed Planes:Planes containing closest possible atomic packingABCABC FCCABABAB HCPAmong the cubic structures, only FCC has close packed planes arrangementMetals deform along close packed planes in close packed directions

*sometimes planes in hexagonal systems are denoted by (h k -i l), where h+k = -i*Perpendicularity:In a cubic system a (h k l) plane is perpendicular to a [u v w] direction if all indices are the same

(1 1 1) [1 1 1]CLOSE PACKED PLANES and DIRECTIONS

the directions and planes where the atoms are all in continuous contact.

Unit Cell

Directions

Planes

SC

None

BCC

None

FCC

{111}

HCP

,

(0001,0002)