Page 83 - Materials Chemistry, Second Edition

70 2 Solid-State Chemistry

npi
Since e ¼ 1or þ1, where n ¼ odd integer or even integer, respectively,
2 2 2
|F| ¼ (2f) when (h þ k þ l) is even, and |F| ¼ 0when (h þ k þ l) is odd. That
is, for a bcc unit cell, all reflections with (h þ k þ l) ¼ 2n þ 1, where n ¼ 0, 1, 2, ...
such as (100), (111), (120), etc. will be absent from the diffraction pattern,
whereas reflections with (h þ k þ l) ¼ 2n such as (110), (121), etc. may be present
if the diffraction satisfies Bragg’s Law. Such missing reflections are referred to as
systematic absences, and are extremely diagnostic regarding the centering present in
the unit cell, as well as the presence of translational symmetry elements such as
screw axes or glide planes (Table 2.8).

Table 2.8. Systematic Absences for X-Ray and Electron Diffraction
Cause of absence Symbol Absences a
Body-centering I h + k + l ¼ 2n + 1 (odd)
A centering A l + k ¼ 2n + 1
B centering B h + l ¼ 2n + 1
C centering C h + k ¼ 2n + 1
Face centering F hkl mixed (not all even or all odd)
Glide plane ⊥ (100) (0kl) b k ¼ 2n þ 1
c l ¼ 2n þ 1
n k þ l ¼ 2n þ 1
d k þ l ¼ 4n þ 1
Glide plane ⊥ (010) (h0l) a h ¼ 2n þ 1
c l ¼ 2n þ 1
n h þ l ¼ 2n þ 1
d h þ l ¼ 4n þ 1
Glide plane ⊥ (001) (hk0) a h ¼ 2n þ 1
b k ¼ 2n þ 1
n h þ k ¼ 2n þ 1
d h þ k ¼ 4n þ 1
Glide plane ⊥ (110) (hhl) b h ¼ 2n þ 1
n h þ l ¼ 2n þ 1
d h þ k þ l ¼ 4n þ 1
Screw axis // a (h00) 2 1 or 4 2 h ¼ 2n þ 1
h ¼ 4n þ 1
4 1 or 4 3
Screw axis // b (0k0) 2 1 or 4 2 k ¼ 2n þ 1
k ¼ 4n þ 1
4 1 or 4 3
Screw axis // c (00l) 2 1 or 4 2 or 6 3 l ¼ 2n þ 1
l ¼ 3n þ 1
3 1 ,3 2 ,6 2 or 6 4
l ¼ 4n þ 1
4 1 or 4 3
l ¼ 6n þ 1
6 1 or 6 5
Screw axis // (110) 2 1 h ¼ 2n þ 1
a
Refers to the Miller indices ((hkl) values) that are absent from the diffraction pattern. For instance, a
body-centered cubic lattice with no other screw axes and glide planes will have a zero intensity for all
reflections where the sum of (h + k + l) yields an odd #, such as (100), (111), etc.; other reflections from
planes in which the sum of their Miller indices are even, such as (110), (200), (211), etc. will be present in
the diffraction pattern. As these values indicate, there are three types of systematic absences: 3-D absences
(true for all hkl) resulting from pure translations (cell centering), 2-D absences from glide planes, and 1-D
absences from screw axes. [40]