3.3 Lattice
41
(a)
(b)
Fig. 3.8 a Construction of a two-dimensional Wigner–Seitz cell, b filling of space with WS cells
Wigner–Seitz cell around R 0 is constructed by drawing lines from R 0 to the next neighbors R j , taking
the point at half distance and erecting a perpendicular plane at (R j + R 0 )/2. The WS cell is the smallest
polyhedron, circumscribed by these planes. A two-dimensional construction is shown in Fig. 3.8.
3.3.4 Point Group
Besides the translations there are other operations under which the lattice is invariant, i.e. the lattice is
imaged into itself. These are:
Identity. The neutral element of any point group is the identity that does not change the crystal. It
is denoted as 1 (E) in international (Schönfließ) notation.
Rotation. The rotation around an axis may have a rotation angle of 2π , 2π /2, 2π /3, 2π /4 or 2π /6 or
their integer multiples. The axis is then called n = 1-, 2-, 3-, 4- or 6-fold, respectively
2 , and denoted
as n (international notation) or C n (Schönfließ). Objects with C n symmetry are depicted in Fig. 3.9.
Mirror operation with respect to a plane through a lattice point. Different mirror planes are discerned
(Fig. 3.10) (after Schönfließ) σ h : a mirror plane perpendicular to a rotational axis, σ v : a mirror plane
that contains a rotational axis, and σ d : a mirror plane that contains a rotational axis and bisects the
angle between two C 2 axes. The international notation is ¯
2.
Inversion. All points around the inversion center r are replaced by −r. The inversion is denoted ¯
1
(i) in international (Schönfließ) notation.
Improper rotation. The improper rotation S n is a rotation C n followed immediately by the inversion
operation i denoted as ¯
n in international notation. There are ¯
3, ¯
4 and ¯
6 and their powers. Only the
combined operation ¯
n is a symmetry operation, while the individual operations C n and i alone are not
symmetry operations. In the Schönfließ notation the improper rotation is defined as S n = σ h C n , with
σ h being a mirror operation with a plane perpendicular to the axis of the C n rotation, denoted as S n .
C 3
C 4
C 6
C 1
C 2
Fig. 3.9 Two-dimensional objects with perpendicular rotation axis C n . Note that the circles do not exhibit σ h symmetry
with respect to the paper plane, i.e. they are different on the top and bottom side
2 5-fold periodic symmetry is geometrically impossible. However, quasicrystals with aperiodic five-fold symmetry exist
[191, 192], some of them possibly being semiconducting [193, 194].
41
(a)
(b)
Fig. 3.8 a Construction of a two-dimensional Wigner–Seitz cell, b filling of space with WS cells
Wigner–Seitz cell around R 0 is constructed by drawing lines from R 0 to the next neighbors R j , taking
the point at half distance and erecting a perpendicular plane at (R j + R 0 )/2. The WS cell is the smallest
polyhedron, circumscribed by these planes. A two-dimensional construction is shown in Fig. 3.8.
3.3.4 Point Group
Besides the translations there are other operations under which the lattice is invariant, i.e. the lattice is
imaged into itself. These are:
Identity. The neutral element of any point group is the identity that does not change the crystal. It
is denoted as 1 (E) in international (Schönfließ) notation.
Rotation. The rotation around an axis may have a rotation angle of 2π , 2π /2, 2π /3, 2π /4 or 2π /6 or
their integer multiples. The axis is then called n = 1-, 2-, 3-, 4- or 6-fold, respectively
2 , and denoted
as n (international notation) or C n (Schönfließ). Objects with C n symmetry are depicted in Fig. 3.9.
Mirror operation with respect to a plane through a lattice point. Different mirror planes are discerned
(Fig. 3.10) (after Schönfließ) σ h : a mirror plane perpendicular to a rotational axis, σ v : a mirror plane
that contains a rotational axis, and σ d : a mirror plane that contains a rotational axis and bisects the
angle between two C 2 axes. The international notation is ¯
2.
Inversion. All points around the inversion center r are replaced by −r. The inversion is denoted ¯
1
(i) in international (Schönfließ) notation.
Improper rotation. The improper rotation S n is a rotation C n followed immediately by the inversion
operation i denoted as ¯
n in international notation. There are ¯
3, ¯
4 and ¯
6 and their powers. Only the
combined operation ¯
n is a symmetry operation, while the individual operations C n and i alone are not
symmetry operations. In the Schönfließ notation the improper rotation is defined as S n = σ h C n , with
σ h being a mirror operation with a plane perpendicular to the axis of the C n rotation, denoted as S n .
C 3
C 4
C 6
C 1
C 2
Fig. 3.9 Two-dimensional objects with perpendicular rotation axis C n . Note that the circles do not exhibit σ h symmetry
with respect to the paper plane, i.e. they are different on the top and bottom side
2 5-fold periodic symmetry is geometrically impossible. However, quasicrystals with aperiodic five-fold symmetry exist
[191, 192], some of them possibly being semiconducting [193, 194].