3.3. Site Symmetry and Wyckoff Positions

Concept of Site Symmetry

Site symmetry is a point group that is a subgroup of the space group in question. Let us explain in more detail.

Suppose there is a point $X$ in a space that follows the symmetry of a space group $G$. Let us search within the space group $G$ for symmetry operations that leave the position of point $X$ invariant. Since the position must remain invariant, the operation cannot involve any translation. It must be either a rotation or a rotoinversion operation¹.

The first operation that comes to mind is the identity. All groups, not just space groups, necessarily contain the identity operation. Since it does nothing, $X$ is naturally left invariant. Next, suppose the space group contains a 2-fold rotation operation, and $X$ happens to lie on that rotation axis. A point on the axis does not move under rotation, so this operation also qualifies. Furthermore, if the space group also contains an inversion operation, and $X$ happens to coincide with the center of inversion, then this operation qualifies as well.

In this way, if we collect all operations from space group $G$ that leave $X$ invariant, the resulting set of operations forms a point group with $X$ as the fixed point. Since all elements were selected from within $G$, this point group is also a subgroup of $G$. This point group is called the site symmetry at position $X$.

The concept of site symmetry can also be explained as follows. If a solid body exists centered at position $X$, the shape of that body conforms to the site symmetry (point group) at position $X$. For example, if some molecule or atom occupies a position whose site symmetry is $4/m$, the shape of that molecule or atom must satisfy the symmetry of point group $4/m$.

Site symmetry is a very useful concept for understanding and analyzing crystal structures. And there is a closely related concept that should be understood together: the “Wyckoff position,” described in the next section.

What are Wyckoff Positions?

A Wyckoff position is a grouping of positions whose site symmetries are conjugate to each other. Let us explain using space group $P\bar{1}$ as an example. The operations contained in $P\bar{1}$ can be expressed as 4×4 affine matrices as follows.

$$ \begin{pmatrix}1&0&0&n_1\\ 0&1&0&n_2\\ 0&0&1&n_3 \\ 0&0&0&1\end{pmatrix},
\begin{pmatrix}-1&0&0&n_1\\ 0&-1&0&n_2\\ 0&0&-1&n_3 \\ 0&0&0&1\end{pmatrix}
$$The first is the translation group including the identity, and the second is the set of inversion operations ($n_1,n_2,n_3$ represent all integers). Now, let us denote a certain position in the space governed by $P\bar{1}$ in fractional coordinates as $(x,y,z)$ and examine its site symmetry.

First, even if $x,y,z$ are completely unrestricted real numbers, the identity alone leaves this position invariant. That is, the site symmetry is $1$. Then, where do operations other than the identity map this position? Although we cannot write them all out, with $n_1,n_2,n_3$ representing all integers, they can be expressed as follows.$$(x+n_1,y+n_2,z+n_3) \,\,\, , \,\,\, (\bar{x}+n_1, \bar{y}+n_2, \bar{z}+n_3)
$$These can be said to be a set of equivalent positions mapped by the operations of the space group. The site symmetry at all these positions is naturally also $1$, and they are subgroups of the original space group.

Next, let us consider $(x,y,z)$ to be $(0,0,0)$. In this case, in addition to the identity, the inversion operation located at the origin$$
\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0 \\ 0&0&0&1\end{pmatrix},
\begin{pmatrix}-1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0 \\ 0&0&0&1\end{pmatrix}\,\,\,\,\,\, \cdots (1)
$$—these two operations leave $(0,0,0)$ invariant. The point group (site symmetry) formed by these two is $\bar{1}$. Now, if we apply not just these two operations but all operations, where does the position $(0,0,0)$ get mapped? The answer is immediate:
$$(n_1,n_2,n_3)
$$To be pedantic, $n_1,n_2,n_3$ represent all integers. Then, what is the site symmetry at these positions? You might think “It’s obviously $\bar{1}$!” but in fact, when written out as 4×4 matrices, the content is slightly different. Specifically, the symmetry operations that leave position $(n_1,n_2,n_3)$ invariant are$$ \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0 \\ 0&0&0&1\end{pmatrix},
\begin{pmatrix}-1&0&0&2n_1\\ 0&-1&0&2n_2\\ 0&0&-1&2n_3 \\ 0&0&0&1\end{pmatrix}\,\,\,\,\,\, \cdots (2)
$$So even though they are all $\bar{1}$ in terms of point group notation, there are differences in the details (the 4th column of the second matrix). Now, how should we describe the relationship between these subgroups that are slightly different but clearly similar (i.e., the relationship between $(1)$ and $(2)$)? Here, recall the concept of “conjugate” subgroups. When we choose an element $g$ from space group $G$, the relationship between subgroup $H$ and subgroup $g^{-1}Hg$ is called conjugacy (see also the “2.4. Concepts of Subgroups and Quotient Groups” page). In this case, if we apply, for example, one of the elements of $P\bar{1}$: $$
g=\begin{pmatrix}1&0&0&-n_1\\ 0&1&0&-n_2\\ 0&0&1&-n_3 \\ 0&0&0&1\end{pmatrix}
$$conjugately to the group in $(1)$, it matches the group in $(2)$, so $(1)$ and $(2)$ are conjugate subgroups. And this argument was not restricted to any particular integers for $n_1,n_2,n_3$. In other words, the site symmetries at all positions $(n_1,n_2,n_3)$ are in a conjugate subgroup relationship.

The explanation has been lengthy, but now we can finally give a clear definition of Wyckoff positions. First, a Wyckoff position does not refer to a single position, but is a term that distinguishes sets of equivalent positions. And if the site symmetry at one position is conjugate to the site symmetry at another position, those positions are considered to belong to the same Wyckoff position. When we take any two positions from a set of equivalent positions, the site symmetries at those two positions are necessarily conjugate, so a set of equivalent positions belongs to a single Wyckoff position. For example, the set of positions $(n_1,n_2,n_3)$ all belong to the same Wyckoff position.

Then, do the equivalent position set $$(x+n_1,y+n_2,z+n_3) \,\,\, , \,\,\, (\bar{x}+n_1, \bar{y}+n_2, \bar{z}+n_3)$$and the equivalent position set obtained by adding $0.1, 0.2, 0.3$ to $x,y,z$ respectively: $$(x+0.1+n_1,y+0.2+n_2,z+0.3+n_3) \,\,\, , \,\,\, (\bar{x}-0.1+n_1, \bar{y}-0.2+n_2, \bar{z}-0.3+n_3)$$belong to the same Wyckoff position? In both cases, the site symmetry is $1$ (identity only). A subgroup consisting only of the identity is necessarily conjugate to itself, so these two sets of equivalent positions do belong to the same Wyckoff position. In other words, there is only one Wyckoff position with site symmetry $1$, and it is sufficient to list just one.

Incidentally, while there are infinitely many sets of equivalent positions belonging to a given Wyckoff position, it is customary to express them within a single unit cell. Since adding integers to each coordinate component obviously gives the expression for a neighboring unit cell, the letters $n_1, n_2, n_3$ that appeared in the previous examples can be deleted, and we write expressions like “Wyckoff position $(x,y,z) \, ,\, (\bar{x}, \bar{y}, \bar{z})$” or “Wyckoff position $(0,0,0)$.”

Let us continue a little further. In fact, space group $P\bar{1}$ has Wyckoff positions that have site symmetry notation $\bar{1}$ but must be distinguished from “Wyckoff position $(0,0,0)$.” For example, the site symmetry at position $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ is the point group $\bar{1}$ consisting of the following two operations: $$
\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0 \\ 0&0&0&1\end{pmatrix},
\begin{pmatrix}-1&0&0&1\\0&-1&0&1\\0&0&-1&1\\0&0&0&1\end{pmatrix}
$$However, this point group can never be made conjugate to the site symmetry of any position belonging to Wyckoff position $(0,0,0)$ (such as the point group in $(1)$)². Therefore, $(0,0,0)$ and $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ are regarded as different Wyckoff positions. There are other such Wyckoff positions as well, but it is difficult to avoid overlooking them. The straightforward approach is to consult ITA (International Tables for Crystallography, Volume A). According to ITA, $P\bar{1}$ has the following nine Wyckoff positions.

Multiplicity	Wyckoff Letter	Site Symmetry	Coordinates
2	$i$	$1$	$(1)\, x,y,z\,\,\,\,\, (2)\, \bar{x},\bar{y},\bar{z},$
1	$h$	$\bar{1}$	$ \frac{1}{2}, \frac{1}{2}, \frac{1}{2}$
1	$g$	$\bar{1}$	$ 0, \frac{1}{2}, \frac{1}{2}$
1	$f$	$\bar{1}$	$ \frac{1}{2},0, \frac{1}{2}$
1	$e$	$\bar{1}$	$ \frac{1}{2}, \frac{1}{2}, 0$
1	$d$	$\bar{1}$	$ \frac{1}{2}, 0,0$
1	$c$	$\bar{1}$	$0, \frac{1}{2}, 0$
1	$b$	$\bar{1}$	$ 0,0, \frac{1}{2}$
1	$a$	$\bar{1}$	$0,0,0$

Multiplicity: Indicates how many equivalent points exist within a single unit cell
Wyckoff Letter: Labeled with lowercase letters starting from $a$ in alphabetical order, from smallest multiplicity³
Site Symmetry: The point group at this Wyckoff position (for how to read the notation, see the next section)
Coordinates: The coordinates of equivalent positions contained in a single unit cell (compound lattice translations are usually omitted)

Every space group has exactly one Wyckoff position with site symmetry $1$, and it has the largest multiplicity. This Wyckoff position is called the general position. All other Wyckoff positions are called special positions. Special positions have some restrictions imposed on the coordinates $x,y,z$, and their site symmetry is denoted by a symbol other than $1$. Since the same symbol may appear multiple times, Wyckoff letters are used to distinguish them.

When expressing Wyckoff positions, it is common to combine the multiplicity and Wyckoff letter. For example, one might say “at Wyckoff position $4f$ of space group $Pnnm$, …”

Finally, here we have only used space group $P\bar{1}$ as an example. Wyckoff positions for all space groups are listed on separate pages, organized by crystal system (Triclinic & Monoclinic, Orthorhombic, Tetragonal, Trigonal & Hexagonal, Cubic). Please refer to them if interested. Additionally, for those who wish to apply Wyckoff position calculations in computer programs, the source code of “spglib,” a software developed primarily by Atsushi Togo at NIMS (for example, https://github.com/spglib/spglib/blob/develop/database/Wyckoff.csv), serves as a useful reference.

How to Read Site Symmetry Notation

In a nutshell, if you remove the dots “$ .$”, you get the standard Hermann–Mauguin symbol. However, the dots “$ .$” also carry important meaning. Below, we explain how to correctly read site symmetry notation with examples. (Note that this content is a paraphrased translation of ITA Section 2.1.3.12. For a precise understanding, please read the original text.)

Example 1: Site symmetry “$.\,.\,2$” of Wyckoff position $4f$ in $P4_22_12$

In $P4_22_12$ (tetragonal system), the site symmetry of Wyckoff position $4f$ is “$.\,.\,2$”. Inserting separators before and after the dots “$ .$” makes the meaning clearer. That is, think of it as “$ .\,|\, . \,|\, 2$”. The dot “$ .$” means “there is no symmetry operation along that principal (or secondary) axis.” As described on another page, the axis settings for the tetragonal system are:

Primary axis: $c$ axis direction
Secondary axis 1: $a$ axis and $b$ axis directions
Secondary axis 2: $[110]$ and $[1\bar{1}0]$ directions

Therefore, “$.\,.\,2$” means there is nothing along the primary axis and secondary axis 1, and there is a $2$ along secondary axis 2. That is, there is a 2-fold rotation operation in either the $[110]$ or $[1\bar{1}0]$ direction, which is the monoclinic point group $2$.

Example 1′: Site symmetry “$2\, .\, 22$” of Wyckoff position $2b$ in $P4_22_12$

Similarly in $P4_22_12$, the site symmetry of Wyckoff position $2b$ is “$2\, .\, 22$”. The first “$2$” means there is a $2$ along the primary axis ($c$ axis), the next “$ . $” means there is nothing along secondary axis 1. The final “$22$” should be taken as a single unit, meaning that $2$ exists in both directions corresponding to secondary axis 2: $[110]$ and $[1\bar{1}0]$. Since $c$, $[110]$, and $[1\bar{1}0]$ are all mutually orthogonal, this site symmetry is the orthorhombic point group $222$.

Example 2: Site symmetry “$222\,.\,.$” of Wyckoff position $6f$ in $I23$

In $I23$ (cubic system), the site symmetry of Wyckoff position $6f$ is “$222\,.\,.$”. This means the symmetry operation along the primary axis is “$222$”, and there is nothing along secondary axis 1 or secondary axis 2. The axis settings for the cubic system are:

Primary axis: $a$, $b$, and $c$ axis directions
Secondary axis 1: $\langle111\rangle$ ( $[111]$, $[\bar{1}11]$, $[1\bar{1}1]$, and $[11\bar{1}]$) directions
Secondary axis 2: $\langle110\rangle$ ( $[110]$, $[1\bar{1}0]$, $[011]$, $[01\bar{1}]$, $[ 101]$, and $[\bar{1}01]$) directions

Therefore, this means there is a $2$ operation along each of the primary axes $a$, $b$, and $c$, which is the orthorhombic point group $222$.

Example 3: Site symmetry “$42\,.\,2$” of Wyckoff position $6b$ in $Pn\bar{3}n$

In $Pn\bar{3}n$ (cubic system), the site symmetry of Wyckoff position $6b$ is “$42\,.\,2$”. As before, the primary axis corresponds to “$42$”, secondary axis 1 to “$.$”, and secondary axis 2 to “$2$”, but some care is needed in reading the notation.

First, “primary axis is $42$” means that along one of the primary axes $a$, $b$, $c$, there is a 4-fold rotation axis, and along the remaining two directions there is a $2$. One might think this should be written as “$422$”. However, writing “$422$” could cause confusion with the tetragonal point group $422$⁴, so it is written as “$42$”.

The next secondary axis 1 is “$.$”, so there is nothing there. The final secondary axis 2 is written as “$2$”, which means that among the six secondary axis 2 directions $\langle110\rangle$, there are two directions perpendicular to the 4-fold rotation axis, and 2-fold rotation axes exist in those two directions⁵. The question of why it is not written as “$22$” is resolved similarly by considering the tetragonal notation convention.

To summarize, the site symmetry “$42\,.\,2$” in this space group is the tetragonal point group $422$, with one of the $a$, $b$, $c$ directions as the principal axis and the remaining two as secondary axis 1.

Example 4: Site symmetry “$\bar{4}\,2\,m$” of Wyckoff position $2a$ in $P4_2/nnm$

In $P4_2/nnm$ (tetragonal system), the site symmetry of Wyckoff position $2a$ is “$\bar{4}\,2\,m$”. This is an example of notation that does not contain any dots “$.$”. “$\bar{4}$”, “$2$”, and “$m$” correspond to the primary axis, secondary axis 1, and secondary axis 2, respectively. In other words, it is the point group $\bar{4}2m$ with the same axis settings as the space group.

Summary

As demonstrated in the examples above, site symmetry notation has some quirks, but there is a rule that removing the dots “$.$” gives the point group notation. To achieve this, when the crystal system of the site symmetry is lower than that of the space group, two or more characters may be used for a single axis.

The dots “$.$” also play an important role. Specifically, by using dots “$.$”, the relationship between the axis settings of the space group and the axis settings of the point group indicated by the site symmetry is made explicit. For example, in a cubic space group, the two site symmetries “$222\,.\,.$” and “$2\,.\,22$” are both the orthorhombic point group $222$, but note that their orientations are different⁶.

Footnotes

The inversion operation ( $ \bar{1} $) and mirrors ( $m=\bar{2}$ ) are also types of rotoinversion operations. ↩︎
If an operation such as$$g=\begin{pmatrix}1&0&0&\frac{1}{2}\\ 0&1&0&\frac{1}{2}\\ 0&0&1&\frac{1}{2} \\ 0&0&0&1\end{pmatrix}$$existed, then$$g^{-1} \begin{pmatrix}-1&0&0&1 \\ 0&-1&0&1\\ 0&0&-1&1 \\ 0&0&0&1\end{pmatrix} g=\begin{pmatrix}-1&0&0&0 \\ 0&-1&0&0\\ 0&0&-1&0 \\ 0&0&0&0\end{pmatrix}$$would hold. However, in this space group, the components of the 4th column of any element are always integers, so an element like $g$ cannot exist. ↩︎
When all 26 lowercase letters are exhausted, it returns to uppercase A (only space group $P2/m\ 2/m\ 2/m$ falls into this case). ↩︎
When $4$, $2$, $2$ operations exist along three orthogonal $c$, $a$, $b$ axis directions respectively, the point group is $422$. However, the last “$2$” in the point group symbol means that “$2$” exists in the $[110]$ and $[1\bar{1}0]$ directions, not the $b$ axis direction. ↩︎
For example, if $4$ is along the $c$ axis, the two directions perpendicular to it are $[110]$ and $[1\bar{1}0]$. ↩︎
The former has $2$ in each of the $a$, $b$, $c$ axis directions. The latter has $2$ in one of the $a$, $b$, $c$ axis directions, and additionally $2$ in two of the $\langle110\rangle$ directions perpendicular to it. The two are related by a 45° angle. ↩︎

Multiplicity	Wyckoff Letter	Site Symmetry	Coordinates
2	\(i\)	\(1\)	\((1)\, x,y,z\,\,\,\,\, (2)\, \bar{x},\bar{y},\bar{z},\)
1	\(h\)	\(\bar{1}\)	\( \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\)
1	\(g\)	\(\bar{1}\)	\( 0, \frac{1}{2}, \frac{1}{2}\)
1	\(f\)	\(\bar{1}\)	\( \frac{1}{2},0, \frac{1}{2}\)
1	\(e\)	\(\bar{1}\)	\( \frac{1}{2}, \frac{1}{2}, 0\)
1	\(d\)	\(\bar{1}\)	\( \frac{1}{2}, 0,0\)
1	\(c\)	\(\bar{1}\)	\(0, \frac{1}{2}, 0\)
1	\(b\)	\(\bar{1}\)	\( 0,0, \frac{1}{2}\)
1	\(a\)	\(\bar{1}\)	\(0,0,0\)

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