Chapter 3
Unit Cell Calculations
3.1 Fractional Coordinates
The location of any point within a unit cell (oblique or orthogonal) by means of
three coordinates (x, y, z) and three basis vectors ~ a, ~ b, ~ c may be specified in terms of
position vector ~ r in 2-D and 3-D as
~ r ¼ x~ a þ y ~ b
ð3:1Þ
~ r ¼ x~ a þ y ~ b þ z~ c
ð3:2Þ
They are shown in Fig. 3.1. The coordinates x, y, z being fractional, they are
unitless and the basis vectors ~ a, ~ b, ~ c are usually measured in Å.
When an integer is added or subtracted from a given fractional coordinate, an
equivalent (point) position is obtained in a neighboring unit cell. For example, a
point with fractional coordinates (0.30, 0.25, 0.15) in the reference unit cell will
have an equivalent position (1.30, 0.25, 0.15) in the unit cell just on right and
(−0.70, 0.25, 0.15) in the unit cell just on left. Similar equivalent positions can be
obtained by changing the y and z fractional coordinates. These are the example of
translational symmetry.
The fractional coordinates of equivalent positions can also be obtained by
applying a rotational, mirror, inversion or a compatible combination of these
symmetries. For example, a twofold rotation along the z-axis will change the
fractional coordinates (x, y, z) into (x, y, z), a mirror plane perpendicular to the
z-axis will change (x, y, z) into (x, y, z) and similarly an inversion symmetry
through the reference point will change (x, y, z) into (x, y, z), respectively, as shown
in Fig. 3.2.
The number of equivalent positions is found to increase as we move from low
symmetry crystal system (e.g., triclinic) to high symmetry crystal system (e.g.,
cubic). As a result, the determination of fractional coordinates in higher symmetry
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_3
95
Unit Cell Calculations
3.1 Fractional Coordinates
The location of any point within a unit cell (oblique or orthogonal) by means of
three coordinates (x, y, z) and three basis vectors ~ a, ~ b, ~ c may be specified in terms of
position vector ~ r in 2-D and 3-D as
~ r ¼ x~ a þ y ~ b
ð3:1Þ
~ r ¼ x~ a þ y ~ b þ z~ c
ð3:2Þ
They are shown in Fig. 3.1. The coordinates x, y, z being fractional, they are
unitless and the basis vectors ~ a, ~ b, ~ c are usually measured in Å.
When an integer is added or subtracted from a given fractional coordinate, an
equivalent (point) position is obtained in a neighboring unit cell. For example, a
point with fractional coordinates (0.30, 0.25, 0.15) in the reference unit cell will
have an equivalent position (1.30, 0.25, 0.15) in the unit cell just on right and
(−0.70, 0.25, 0.15) in the unit cell just on left. Similar equivalent positions can be
obtained by changing the y and z fractional coordinates. These are the example of
translational symmetry.
The fractional coordinates of equivalent positions can also be obtained by
applying a rotational, mirror, inversion or a compatible combination of these
symmetries. For example, a twofold rotation along the z-axis will change the
fractional coordinates (x, y, z) into (x, y, z), a mirror plane perpendicular to the
z-axis will change (x, y, z) into (x, y, z) and similarly an inversion symmetry
through the reference point will change (x, y, z) into (x, y, z), respectively, as shown
in Fig. 3.2.
The number of equivalent positions is found to increase as we move from low
symmetry crystal system (e.g., triclinic) to high symmetry crystal system (e.g.,
cubic). As a result, the determination of fractional coordinates in higher symmetry
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_3
95