Page 117 - Materials Science and Engineering An Introduction

3.16 X-Ray Diffraction: Determination of Crystal Structures • 89

Bragg’s law— or
relationship among
x-ray wavelength, nl = d hkl sin u + d hkl sin u
interatomic spacing,
and angle of = 2d hkl sin u (3.21)
diffraction for
constructive
interference Equation 3.21 is known as Bragg’s law; n is the order of reflection, which may be
any integer (1, 2, 3, . . .) consistent with sin u not exceeding unity. Thus, we have a simple
Bragg’s law expression relating the x-ray wavelength and interatomic spacing to the angle of the dif-
fracted beam. If Bragg’s law is not satisfied, then the interference will be nonconstruc-
tive so as to yield a very low-intensity diffracted beam.
The magnitude of the distance between two adjacent and parallel planes of atoms
(i.e., the interplanar spacing d hkl ) is a function of the Miller indices (h, k, and l) as well
as the lattice parameter(s). For example, for crystal structures that have cubic symmetry,
Interplanar spacing a
for a plane having d hkl = 2 2 2 (3.22)
indices h, k, and l 2h + k + l
in which a is the lattice parameter (unit cell edge length). Relationships similar to
Equation 3.22, but more complex, exist for the other six crystal systems noted in Table 3.2.
Bragg’s law, Equation 3.21, is a necessary but not sufficient condition for diffrac-
tion by real crystals. It specifies when diffraction will occur for unit cells having atoms
positioned only at cell corners. However, atoms situated at other sites (e.g., face and in-
terior unit cell positions as with FCC and BCC) act as extra scattering centers, which can
produce out-of-phase scattering at certain Bragg angles. The net result is the absence of
some diffracted beams that, according to Equation 3.21, should be present. Specific sets
of crystallographic planes that do not give rise to diffracted beams depend on crystal
structure. For the BCC crystal structure, h k l must be even if diffraction is to occur,
whereas for FCC, h, k, and l must all be either odd or even; diffracted beams for all sets
of crystallographic planes are present for the simple cubic crystal structure (Figure 3.3).
These restrictions, called reflection rules, are summarized in Table 3.5. 9

Concept Check 3.3 For cubic crystals, as values of the planar indices h, k, and l increase,
does the distance between adjacent and parallel planes (i.e., the interplanar spacing) increase
or decrease? Why?
[The answer may be found at www.wiley.com/college/callister (Student Companion Site).]

Table 3.5
Reflection Indices
X-Ray Diffraction Crystal Structure Reflections Present for First Six Planes
Reflection Rules and BCC (h k l) even 110, 200, 211,
Reflection Indices 220, 310, 222
for Body-Centered
Cubic, Face-Centered FCC h, k, and l either 111, 200, 220,
Cubic, and Simple all odd or all even 311, 222, 400
Cubic Crystal Simple cubic All 100, 110, 111,
Structures 200, 210, 211

9 Zero is considered to be an even integer.