Space groups

Question 1

In which cases does a space group have more than one description?

Answer 1

1. Two choices of origin. Some centrosymmetric space groups (24) has additional points of high symmetry. In these cases description with a high-symmetry points as origin is also presented. Neither are considered as standard.
2. Monoclinic space groups. Here both b- and c-unique axis descriptions are given for all monoclinic space groups (13). In addition, 8 space groups has abbreviated treatments for three different cell choices (labeled ‘cell choice 1-3’)
3. Rhombohedral space groups . The seven rhombohedral space groups are described both with hexagonal axes (triple hexagonal cell) and rhombohedral axes (primitive rhombohedral cell).

Question 2

How many monoclinic space groups are there?

Answer 2

13.

Question 3

What is the name of the short international symbol for space groups?

Answer 3

Hermann-Mauguin.

Question 4

Which setting is the standard for the Hermann-Mauguin symbol for monoclinic space groups?

Answer 4

The b-axis setting (e.g. C2/c).

Question 5

What does the Hermann-Mauguin notation consist of?

Answer 5

A capital letter indicating the centering followed by symbols indicating the symmetry elements of the space group.

Question 6

What is the order of the symmetry elements in the Hermann-Mauguin symbol for orthorhombic space groups?

Answer 6

Primary: a
Secondary: b
Tertiary: c

Question 7

What is the order of the symmetry elements in the Hermann-Mauguin symbol for monoclinic space groups?

Answer 7

Primary: unique axis
Secondary: None
Tertiary: None

Question 8

What is the order of the symmetry elements in the Hermann-Mauguin symbol for triclinic space groups?

Answer 8

No symmetry directions in triclinic space groups. Only identity and inversion centre exist for triclinic space groups.

Question 9

What is the order of the symmetry elements in the Hermann-Mauguin symbol for tetragonal space groups?

Answer 9

Primary: c
Secondary: Square edges
Tertiary: Square diagonal

Question 10

What is the order of the symmetry elements in the Hermann-Mauguin symbol for cubic space groups?

Answer 10

Primary: Square edges
Secondary: Body diagonal
Tertiary: Square diagonal

Question 11

What is a German term for the symmetry directions referring to the entries in the Hermann-Mauguin notation?

Answer 11

Blickrichtung

Question 12

What is the order of the symmetry elements in the Hermann-Mauguin symbol for hexagonal space groups?

Answer 12

Primary: c
Secondary: base edges (e.g. [100], [010])
Tertiary: Across the middle of the rhombus in the base (e.g. [1-10])

Question 13

What is the difference between the full and short Hermann-Mauguin symbols?

Answer 13

In the full symbols, both symmetry axes and symmetry planes are written in full. In the short form, symmetry axes are suppressed as much as possible.

Question 14

What is a symmorphic space group?

Answer 14

A space group with no translation (apart from lattice translation), that is only contains pure point group symmetry (rotation, reflection, inversion).

Question 15

What does the subscript p in the axes indicate?

Answer 15

That the axes are not parallel to the projection plane, and thus is projected onto the plane.

Question 16

What can be said about symmetry elements at height h in a space group diagram?

Answer 16

That they have a corresponding symmetry element at height h + 1/2. This is true for all symmetry elements except for horizontal fourfold axes in cubic space groups).

Question 17

What is the standard setting for orthorhombic space groups?

Answer 17

Projection plane in direction of c, with a axis downward and b axis towards the rights.

Question 18

What happens if you transform two of the axes of an orthorhombic space group in the standard setting?

Answer 18

The right-handedness is turned into left-handedness. Thus, the last axis should be inverted in order to keep right-handedness.

Question 19

How many orthorhombic space groups are there?

Answer 19

59.

Question 20

How many triclinic space groups are there?

Answer 20

2.

Question 21

How many rhombohedral space groups are there?

Answer 21

7.

Question 22

What is the projection direction of tetragonal, trigonal P and hexagonal P space groups?

Answer 22

Along c.

Question 23

What is the projection direction of rhombohedral space groups?

Answer 23

These have two projections superimposed:

1) The primitive rhombohedral cell with a_p, b_p, c_p (projection direction is along the threefold rotation axis which is not in the direction of any of the cell vectors).
2) The triple hexagonal cell along c, with a and b in plane.