Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-03 GTBL042-Callister-v2 September 6, 2007 15:33

64 • Chapter 3 / Structures of Metals and Ceramics

Crystallographic Points,

Directions, and Planes

When dealing with crystalline materials, it often becomes necessary to specify a

particular point within a unit cell, a crystallographic direction, or some crystallo-

graphic plane of atoms. Labeling conventions have been established in which three

numbers or indices are used to designate point locations, directions, and planes.

The basis for determining index values is the unit cell, with a right-handed co-

ordinate system consisting of three (x,y, andz) axes situated at one of the cor-

ners and coinciding with the unit cell edges, as shown in Figure 3.20. For some

crystal systems—namely, hexagonal, rhombohedral, monoclinic, and triclinic—the

three axes arenotmutually perpendicular, as in the familiar Cartesian coordinate

scheme.

3.12 POINT COORDINATES

The position of any point located within a unit cell may be specified in terms of its

coordinates as fractional multiples of the unit cell edge lengths (i.e., in terms ofa,b,

andc). To illustrate, consider the unit cell and the pointPsituated therein as shown in

Figure 3.21. We specify the position ofPin terms of the generalized coordinatesq,r,

andswhereqis some fractional length ofaalong thexaxis,ris some fractional length

ofbalong theyaxis, and similarly fors. Thus, the position ofPis designated using

coordinatesqrswith values that are less than or equal to unity. Furthermore, we

have chosen not to separate these coordinates by commas or any other punctuation

marks (which is the normal convention).

EXAMPLE PROBLEM 3.7

Location of Point Having Specified Coordinates

For the unit cell shown in the accompanying sketch (a), locate the point having

coordinates^14112.

z

x

q r s

P

y

c

rb

qa

sc

a

b

Figure 3.21 The manner in which theq,

r, andscoordinates at pointPwithin the

unit cell are determined. Theq

coordinate (which is a fraction)

corresponds to the distanceqaalong the

xaxis, whereais the unit cell edge length.

The respectiverandscoordinates for the

yandzaxes are determined similarly.