Page 75 - Materials Chemistry, Second Edition

62 2 Solid-State Chemistry

For screw axes, the nomenclature is of the form n x , indicating a 360 /n rotation,
followed by a x=n translation along one of the unit cell vectors, a, b,or c. For
example, the 6 1 and 6 3 screw axes would imply sixfold axes of rotation followed by
1=6 and 1=2 translations, respectively.
It is noteworthy to point out that two sequential screw-axis or glide-plane
operations will yield the original object that has been translated along one of
the unit cell vectors. For example, a 6 3 axis yields an identical orientation of the
molecule only after 6 repeated applications – 3 unit cells away (i.e.,6 1=2 ¼ 3).
Since glide planes feature a mirror plane prior to translation, the first operation will
cause a change in handedness of the molecule. By contrast, screw axis operations do
not alter the stereoisomerism of the molecule.
Both glide and screw axes are not point group operations because they involve
translations. That is, one cannot distinguish between analogous rotation and screw
axes, or between glide and mirror planes, by simply looking at the crystal faces. You
may notice that of the symmetry elements discussed, both glide planes and screw
axes are absent from the list of point group symbols, listed in Table 2.5. For the
purposes of determining the crystallographic point group, screw axes are treated as
rotation axes (e.g.,6 3 6), and glide planes treated as mirror planes (e.g.,b m).
When the symmetry elements are applied to species arranged periodically on a
crystal lattice, the result is a space group. The combination of the 32 crystallo-
graphic point groups with the 14 Bravais lattices yields a total of 230 possible space
groups for crystals, designated by the Hermann-Mauguin (H-M) space group sym-
bol. The 73 different space groups that can be generated from point groups only,
without using glide planes and/or screw axes, are called symmorphic space groups.
The first letter of the H-M symbol is a single letter that refers to the Bravais
centering, L. The letters used are P (primitive), A ((100) face centered), B ((010)
face centered), C ((001) face centered), F (face centered), and I (body centered). The
remaining three letters refer to the crystal system as well as symmetry elements
contained in the lattice. Table 2.6 lists the symmetry elements corresponding to
each of the primary, secondary and tertiary terms of the space group symbol, L(1 )

(2 )(3 ). Both rotation and screw axes are parallel, whereas mirror/glide planes are
perpendicular, to the directions listed in Table 2.6. Note that the only space groups
Table 2.6. Space Group Symmetry Element Symbolism
a
b
Crystal system Symmetry direction (symbol: L 1 2 3 )

1 2 3
Triclinic N/A N/A N/A
Monoclinic [010] (b-unique) N/A N/A
Orthorhombic [100] [010] [001]
Tetragonal [001] [100]/[010] [110]
Hexagonal/Trigonal [001] [100]/[010] [120]/[110]
Cubic [100]/[010]/[001] [111] [110]
a
Mirror and glide planes will be perpendicular to the indicated directions, whereas rotation and screw
axes will be aligned parallel to the directions.
b
L refers to the Bravais lattice centering (i.e.,P ¼ primitive, I ¼ body-centered, F ¼ face-centered,
C ¼ c-centered).