Page 101 - Materials Science and Engineering An Introduction

3.9 Crystallographic Directions • 73

the vector head (a 1 , a 2 , a 3 , and z ) may be determined using rearranged forms of Equations
3.12a–3.12d, taking the value of n to be unity. Thus,
ua (1)(a) a
a 1 = + a 1 = + 0a =
3n (3)(1) 3
va (1)(a) a
a 2 = + a 2 = + 0a =
3n (3)(1) 3
ta (-2)(a) -2a
a 3 = + a 3 = + 0a =
3n (3)(1) 3
wc (3)(c)
z = + z = + 0c = c
3n (3)(1)
In constructing this direction vector, we begin at the origin (point o), and first proceed
a
3
units along the a 1 axis to point p; next, from this point parallel to the a 2 axis units to point
a
3
q; then parallel to the a 3 axis - units to point r; and finally we continue parallel to the z
2a
3
axis c units to point s. Thus, the [1123] direction is represented by the vector that is directed
from point o to point s, as noted in the sketch.
(c) Of course, it is possible to draw the equivalent [111] direction using a three-coordinate-axis
(a 1 -a 2 -z) technique, as shown in (b). In this case, we modify Equations 3.10a–3.10c to read
as follows:
a 1 - a 1
U = na b (3.13a)
a
a 2 - a 2
V = na b (3.13b)
a
z - z
W = na b (3.13c)
c
where again, single and double primes for a 1 , a 2 , and z denote head and tail coordinates,
respectively. When we locate tail coordinates at the origin (i.e., take a 1 = a 2 = 0a and
z = 0c) and make the vector head (i.e., single-primed) coordinates of the above equations
dependent parameters (while assuming n 1), the following result:
Ua (1)(a)
a 1 = + a 1 = + 0a = a
n (1)
Va (1)(a)
a 2 = + a 2 = + 0a = a
n (1)
Wc (1)(c)
z = + z = + 0c = c
n (1)
To locate the vector head, we begin at the origin (point O), then proceed a units along
the a 1 axis (to point P), next parallel to the a 2 axis a units (to point Q), and finally parallel
to the z axis c units (to point R). Hence, the [111] direction is represented by the vector that
passes from O to R, as shown.
It may be noted that this [111] direction is identical to [1123] from part (b).

The alternative situation is to determine the indices for a direction that has been drawn
within a hexagonal unit cell. For this case, it is convenient to use the a 1 -a 2 -z three-coordinate-
axis system and then convert these indices into the equivalent set for the four-axis scheme.
The following example problem demonstrates this procedure.