When first starting to learn space groups, many people initially stumble on how to read the diagrams found in International Tables for Crystallography, Volume A (hereafter ITA). While the Hermann–Mauguin symbols alone do not reveal the complete picture of a space group, once you learn to read the diagrams, you can see at a glance where the symmetry elements are located within the unit cell and how they map general positions onto one another. The space group pages in International Tables typically display two types of diagrams side by side: the general-position diagram and the symmetry-elements diagram. The former shows the arrangement of equivalent points at the general position, while the latter shows the positions and orientations of the symmetry elements.
On this page, we will organize the essential concepts needed to read the diagrams in International Tables. The meanings of individual symbols themselves were explained on the previous page “3.1. Details of Symmetry Elements,” so here we will focus on “how those symbols are arranged in the diagrams and how they are read.”
First, Where to Look on a Space Group Page
The space group pages in International Tables contain a great deal of information, but beginners need only focus on the following four areas.
The first is the space group symbol and space group number at the top of the page. This tells you which space group that page covers.
The second is the general-position diagram and the symmetry-elements diagram. The geometric content of the space group is almost entirely condensed into these two diagrams.
The third is the coordinate table for general and special positions. Here, the coordinates of equivalent positions, Wyckoff letters, multiplicities, and site symmetries are listed.
The fourth is the description concerning Origin and Asymmetric unit. This is because the appearance of the diagrams and the way coordinates are written depend on where the origin is placed and how the asymmetric unit is chosen.
As you become accustomed, you will find it faster to first look at the diagrams to grasp the arrangement of symmetry elements, and then check the coordinate table to confirm the relationship between general and special positions.
What is a General-Position Diagram?
A general-position diagram illustrates where an arbitrary point that possesses no symmetry whatsoever is mapped when all the symmetry operations of the space group are applied to it after placing it in the unit cell. In other words, a general-position diagram shows “the arrangement of equivalent points generated from a given general point.” In IUCr’s explanations as well, the general-position diagram is described as a figure showing the arrangement of symmetrically equivalent general points under the same coordinate system.
For example, if the coordinate table for general positions lists expressions such as $$(x,y,z),\quad (-x, y+\tfrac{1}{2}, -z+\tfrac{1}{2}), \cdots$$ then the general-position diagram depicts where those points appear within the unit cell. Each point in the diagram is not simply a “dot” but a representative of the equivalent position corresponding to each row of the coordinate table. Therefore, rather than thinking of the diagram and the coordinate table as different things, it is easier to understand them as expressing the same information in two different ways: graphically and algebraically.
Also, the first position appearing in the general-position diagram typically corresponds to the General row in the coordinate table. The multiplicity written there is the number of equivalent points that appear within a single unit cell. The fact that the site symmetry is \(1\) means that the point does not lie on any non-trivial symmetry element. Conversely, if a point is placed on a mirror plane, rotation axis, or similar element, the number of generated equivalent points decreases, and that point becomes a special position.
What is a Symmetry-Elements Diagram?
A symmetry-elements diagram depicts which symmetry elements exist, where they are located, and in what orientation within the unit cell. In IUCr’s diagramming guide as well, the symmetry-elements diagram is described as a figure that shows not only the relative positional relationships of the symmetry elements, but also their absolute positions and orientations with respect to the adopted coordinate system.
The symmetry elements written in the space group symbol are only the representative ones that characterize that space group. The actual space group also contains many “redundant” symmetry elements that arise automatically from combinations of those. Therefore, when you look at the symmetry-elements diagram, mirror planes, screw axes, inversion centers, and other elements that are not directly written in the symbol may appear. This is not an error, but rather a correct reflection of the full picture of the space group. In IUCr’s educational explanations, for example using the diagram of \(Pnma\), it is explained that symmetry elements not explicitly stated in the space group symbol do appear in the diagram.
As we learned on the previous page, rotation axes, screw axes, mirror planes, glide planes, and inversion centers each have their own specific graphical symbols. In the International Tables diagrams, these are distinguished by “line types,” “polygon shapes,” “presence or absence of arrows,” and so on. Therefore, when reading diagrams, two steps are important: first, knowing the meaning of each individual symbol, and second, seeing where that symbol is placed within the unit cell.
The Diagrams are “Projection Diagrams”
The diagrams for three-dimensional space groups in International Tables are projections of three-dimensional unit cells onto a two-dimensional plane. Therefore, when looking at a diagram, you must always be aware of “from which direction the view is taken.” In IUCr’s explanations, for example regarding the diagram of \(Pnma\), it is explained that the diagram is drawn by projecting along the \(z\) direction onto the page, with the horizontal direction on the page being \(y\) and the vertical direction being \(x\). In other words, the left-right and up-down directions of the diagram do not necessarily correspond directly to the \(a, b, c\) axes; you must reinterpret them according to the projection direction adopted on that page.
This is particularly important when dealing with different settings of monoclinic or orthorhombic systems. If the setting changes, then even for the same space group type, which axis is horizontal on the page and which axis is out of the plane will change. Therefore, rather than judging solely from the shape of the diagram, you must always check the axis labels and setting notes accompanying the diagram. In the International Tables guide, the handling of multiple settings and projections for monoclinic and orthorhombic systems is also explained.
In-Plane Position and Out-of-Plane “Height”
In projection diagrams, two-dimensional coordinates within the page alone are insufficient. Therefore, in International Tables, the position perpendicular to the page, that is, the “height,” is indicated using auxiliary symbols.
In symmetry-elements diagrams, fractions are sometimes appended to mirror planes, glide planes, and similar elements. These indicate how far that symmetry element is from the page plane. According to IUCr’s explanation of graphical symbols, for example, if a fraction \(h\) is appended, that symbol represents symmetry elements at two positions along the projection direction: height \(h\) and \(h+\frac{1}{2}\). If no fraction is attached, the element is normally taken to be at positions \(0\) and \(\frac{1}{2}\).
For points in the general-position diagram as well, auxiliary symbols indicate whether they are above or below the page, or at what height. In IUCr’s symmetry explanations, the symbols \(+\) and \(-\) are shown to represent positions at equal distances above and below the page, respectively. Therefore, when looking at the general-position diagram, you need to read not only the in-plane position of each point but also the auxiliary symbol attached to it as a single piece of information.
The General-Position Diagram and Symmetry-Elements Diagram Always Correspond
These two diagrams are not presenting separate pieces of information. They are in complete correspondence, in the sense that each symmetry element drawn in the symmetry-elements diagram shows how it maps the points in the general-position diagram onto one another. For example, if there is a mirror plane, the points in the general-position diagram have corresponding points on the opposite side of that mirror plane. For a glide plane, since a translation is added to the reflection, equivalent points appear not only on the opposite side but also shifted by a fixed amount. For a screw axis, since a translation along the axis direction accompanies the rotation, although it may look like a rotational correspondence on the page, the height is actually changing as well.
Therefore, the most effective way to practice reading space group diagrams is to look at the symmetry-elements diagram and trace, one by one, which point in the general-position diagram corresponds to which operation. Until you get used to it, looking at only the general-position diagram or only the symmetry-elements diagram can be difficult to understand, but when you go back and forth between the two, the meaning of the symmetry operations suddenly becomes concrete.
Correspondence with the Coordinate Table
Understanding deepens further when you look at the coordinate table alongside the diagrams. The general position section lists the multiplicity, Wyckoff letter, site symmetry, and specific equivalent coordinates. For example, if it reads “8d1,” this means “general position with multiplicity 8, Wyckoff letter \(d\), site symmetry \(1\).” In IUCr’s educational explanations as well, it is explained that an atom at a general position has as many equivalent atoms in the unit cell as the multiplicity indicates, and that this number decreases when the atom moves to a special position.
A special position is one where a point lies on a mirror plane, rotation axis, inversion center, or similar element, so that it is mapped onto itself by a certain symmetry operation. Because of this, the multiplicity is smaller than that of the general position, and the site symmetry becomes a non-trivial symbol such as \(m\), \(2\), \(\bar{1}\), etc. In other words, the general-position diagram is based on “a point that does not lie on any symmetry element,” while the special position table organizes algebraically “the cases where a point does lie on a symmetry element.”