42
3 Crystals
(a)
v
H
H
O
C 2
v
(b)
F
H
H
F
C 2
(c)
C 2
d
C 4
C 2
Fig. 3.10 Mirror planes: a σ v (at H 2 O molecule), b σ h (at F 2 H 2 molecule) and c σ d
S 6
S 4
S 3
S 1
S 2
Fig. 3.11 Two-dimensional objects with perpendicular improper rotation axis S n . Note that the white and black circles
do not exhibit σ h symmetry with respect to the paper plane, i.e. they are white on the top and black on the bottom. The
circles with a dot in the center exhibit σ h symmetry, i.e. they look the same from top and bottom
There are S 3 , S 4 and S 6 and ¯
3 = S
5
6 , ¯
4 = S
3
4 and ¯
6 = S
5
3 . For successive applications, the S n yield
previously known operations, e.g. S
2
4 = C 2 , S
4
4 = E, S
2
6 = C 3 , S
3
6 = i, S
2
3 = C
2
3 , S
3
3 = σ h , S
4
3 = C 3 ,
S
6
3 = E. We note that formally S 1 is the inversion i and S 2 is the mirror symmetry σ . Objects with S n
symmetry are schematically shown in Fig. 3.11.
These symmetry operations form 32 point groups. These groups are shown (with their different
notations and elements) in Table B.2. The highest symmetry is the cubic symmetry O h = O × i. The
tetraeder group T d (methane molecule) is a subgroup of O h , lacking the inversion operation.
Important for surface symmetries, there are ten two-dimensional point groups (Sect. 11.2 and
Table B.1).
3.3.5 Space Group
The space group is formed by the combination of the elements of the point group with translations.
The combination of a translation along a rotational axis with a rotation around this axis creates a screw
axis n m . In Fig. 3.12a, a so-called 4 1 screw axis is shown. The first index n indicates the rotation angle,
i.e. 2π/n, the second index indicates the translation, i.e. c m/n, c being the periodicity along the axis.
There are eleven crystallographically allowed screw rotations.
3
The combination of the mirror operation at a plane that contains a rotational axis with a translation
along this axis creates a glide reflection (Fig. 3.12b). For an axial glide (or b-glide) the translation
is parallel to the reflection plane. A diagonal glide (or d-glide) involves translation in two or three
directions. A third type of glide is the diamond glide (or d-glide). There are 230 different space groups,
listed in Appendix B. A detailed treatment can be found in [195].
4
Important for surface symmetries, there are 17 two-dimensional space groups (Sect. 11.2).
3 2 1 , 3 1 , 3 2 , 4 1 , 4 2 , 4 3 , 6 1 , 6 2 , 6 3 , 6 4 , 6 5 .
4 One should in particular consider the pitfalls pointed out in Appendix 10 of this reference.
3 Crystals
(a)
v
H
H
O
C 2
v
(b)
F
H
H
F
C 2
(c)
C 2
d
C 4
C 2
Fig. 3.10 Mirror planes: a σ v (at H 2 O molecule), b σ h (at F 2 H 2 molecule) and c σ d
S 6
S 4
S 3
S 1
S 2
Fig. 3.11 Two-dimensional objects with perpendicular improper rotation axis S n . Note that the white and black circles
do not exhibit σ h symmetry with respect to the paper plane, i.e. they are white on the top and black on the bottom. The
circles with a dot in the center exhibit σ h symmetry, i.e. they look the same from top and bottom
There are S 3 , S 4 and S 6 and ¯
3 = S
5
6 , ¯
4 = S
3
4 and ¯
6 = S
5
3 . For successive applications, the S n yield
previously known operations, e.g. S
2
4 = C 2 , S
4
4 = E, S
2
6 = C 3 , S
3
6 = i, S
2
3 = C
2
3 , S
3
3 = σ h , S
4
3 = C 3 ,
S
6
3 = E. We note that formally S 1 is the inversion i and S 2 is the mirror symmetry σ . Objects with S n
symmetry are schematically shown in Fig. 3.11.
These symmetry operations form 32 point groups. These groups are shown (with their different
notations and elements) in Table B.2. The highest symmetry is the cubic symmetry O h = O × i. The
tetraeder group T d (methane molecule) is a subgroup of O h , lacking the inversion operation.
Important for surface symmetries, there are ten two-dimensional point groups (Sect. 11.2 and
Table B.1).
3.3.5 Space Group
The space group is formed by the combination of the elements of the point group with translations.
The combination of a translation along a rotational axis with a rotation around this axis creates a screw
axis n m . In Fig. 3.12a, a so-called 4 1 screw axis is shown. The first index n indicates the rotation angle,
i.e. 2π/n, the second index indicates the translation, i.e. c m/n, c being the periodicity along the axis.
There are eleven crystallographically allowed screw rotations.
3
The combination of the mirror operation at a plane that contains a rotational axis with a translation
along this axis creates a glide reflection (Fig. 3.12b). For an axial glide (or b-glide) the translation
is parallel to the reflection plane. A diagonal glide (or d-glide) involves translation in two or three
directions. A third type of glide is the diamond glide (or d-glide). There are 230 different space groups,
listed in Appendix B. A detailed treatment can be found in [195].
4
Important for surface symmetries, there are 17 two-dimensional space groups (Sect. 11.2).
3 2 1 , 3 1 , 3 2 , 4 1 , 4 2 , 4 3 , 6 1 , 6 2 , 6 3 , 6 4 , 6 5 .
4 One should in particular consider the pitfalls pointed out in Appendix 10 of this reference.