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

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

3.15 Linear and Planar Densities • 75

a 1

a

a

c

H

F

E

G

D

A

a 2

a 3

z

C

B

Solution

To determine these Miller–Bravais indices, consider the plane in the figure

referenced to the parallelepiped labeled with the letters A through H at its

corners. This plane intersects thea 1 axis at a distanceafrom the origin of the

a 1 - a 2 - a 3 - zcoordinate axes system (point C). Furthermore, its intersections with

thea 2 andzaxes are –aandc, respectively. Therefore, in terms of the lattice

parameters, these intersections are 1, –1, and 1. Furthermore, the reciprocals of

these numbers are also 1, –1, and 1. Hence

h= 1

k=− 1

l= 1

and, from Equation 3.8

i=−(h+k)

=−(1−1)= 0

Therefore the (hkil) indices are (1101).

Notice that the third index is zero (i.e., its reciprocal=∞), which means

that this plane parallels thea 3 axis. Upon inspection of the above figure, it may

be noted that this is indeed the case.

3.15 LINEAR AND PLANAR DENSITIES

The two previous sections discussed the equivalency of nonparallel crystallographic

directions and planes. Directional equivalency is related tolinear densityin the sense

that, for a particular material, equivalent directions have identical linear densities.

The corresponding parameter for crystallographic planes isplanar density, and planes

having the same planar density values are also equivalent.

Linear density (LD) is defined as the number of atoms per unit length whose

centers lie on the direction vector for a specific crystallographic direction; that is,

LD=

number of atoms centered on direction vector

length of direction vector

(3.9)

Of course, the units of linear density are reciprocal length (e.g., nm−^1 ,m−^1 ).