Page 106 - Materials Science and Engineering An Introduction

78 • Chapter 3 / The Structure of Crystalline Solids

Using these h, k, and l indices, we want to solve for the values of A, B, and C using rearranged
forms of Equations 3.14a–3.14c. Taking the value of n to be 1—because these three Miller in-
dices are all integers—leads to the following:

na (1)(a)
A = = = a z
h 1
Tutorial Video Intersection with
nb (1)(b) z axis (value of C)
B = = = b
k 0
c
nc (1)(c)
C = = = c
l 1 O y
a
Thus, this (101) plane intersects the x axis at a (be- Intersection with
cause A a), it parallels the y axis (because B b), x axis (value of A)
and intersects the z axis at c. On the unit cell shown x b
next are noted the locations of the intersections for (b)
this plane. z
The only plane that parallels the y axis and in-
tersects the x and z axes at axial a and c coordinates, respectively,
is shown next.
Note that the representation of a crystallographic plane ref- c
erenced to a unit cell is by lines drawn to indicate intersections of
this plane with unit cell faces (or extensions of these faces). The O y
following guides are helpful with representing crystallographic
planes: a

• If two of the h, k, and l indices are zeros [as with (100)], the b
plane will parallel one of the unit cell faces (per Figure 3.11a). x
• If one of the indices is a zero [as with (110)], the plane will (c)
be a parallelogram, having two sides that coincide with
opposing unit cell edges (or edges of adjacent unit cells)
(per Figure 3.11b).
• If none of the indices is zero [as with (111)], all intersections will pass through unit cell faces
(per Figure 3.11c).

Atomic Arrangements
The atomic arrangement for a crystallographic plane, which is often of interest, depends
on the crystal structure. The (110) atomic planes for FCC and BCC crystal structures
are represented in Figures 3.12 and 3.13, respectively. Reduced-sphere unit cells are
also included. Note that the atomic packing is different for each case. The circles repre-
: VMSE sent atoms lying in the crystallographic planes as would be obtained from a slice taken
Planar Atomic through the centers of the full-size hard spheres.
Arrangements
A “family” of planes contains all planes that are crystallographically equivalent—
that is, having the same atomic packing; a family is designated by indices enclosed in
braces—such as 100 . For example, in cubic crystals, the (111), (1 1 1), (111), (1 1 1), (111),
(1 1 1), (111), and (111) planes all belong to the 111 family. However, for tetragonal