unit cells on the positive x, y, z axes and the unit cells on the negative x, y, z axes,
respectively):
ð1:6; 0:5; 0:25Þ; ð0:6; 1:5; 0:25Þ; ð0:6; 0:5; 1:25Þ
and ðÀ0:4; 0:5; 0:25Þ; ð0:6; À0:5; 0:25Þ; ð0:6; 0:5; À0:75Þ
In a similar manner, the fractional coordinates in other 3-D lattices can be obtained.
Example 3 Determine the set of fractional coordinates of equivalent positions
related to a 3-fold rotational symmetry in 2-D and 3-D.
Solution: Given: A 3-fold rotational symmetry (means 120° rotations).
Select crystallographic axes and consider a point with fractional coordinates
(x, y, z) and then apply the given symmetry (Fig. 3.4).
The position vector of the point at (x, y, z) is
~ r ¼ x~ a þ y ~ b þ z~ c
Application of 120° rotation changes the position vector to
~ r 0 ¼ x ~
a 0 þ y ~
b 0 þ z ~ c 0
where a′ = b, b' = –a–b and c′= c. Therefore,
r
0
¼ xb þ yðÀa À bÞ þ zc
¼ Àya þ ðx À yÞ b þ zc
Fig. 3.4 Fractional
coordinates of three
equivalent positions
3.1 Fractional Coordinates
99