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The `General Position' in IT A

In IT A the set of all symmetry operations is listed for all space groups. As we have seen, space groups are infinite groups, and there is an infinite number of space groups for each space-group type, see Section 3.4. How can such a listing be done at all ? In this section the principles are dealt with which make the listings possible, as well as the points of view which determine the actual listings.

1. Consider the infinite number of space groups belonging to a certain space-group type. If one refers each space group to its conventional coordinate system, one obtains a set of matrix-column pairs for each space group but all these sets are identical. (This is one way to classify the space groups into space-group types.) Therefore, a listing is necessary only for each space-group type, but not for each space group. This means 230 listings for the space groups and 17 listings for the plane groups, and these listings are indeed contained in IT A. In reality the number of space-group listings is higher by about 20 % because there is sometimes more than one conventional coordinate system: different settings, different bases, or different origins, see IT A.
2. No doubt, there is an infinite number of symmetry operations for each space group. How can they be listed in a book of finite volume ? Let W be a symmetry operation and (W,w) its matrix-column pair. Because the conventional bases are always lattice bases, (W,w) can be decomposed into a translation $(\mbox{\textit{\textbf{I}}},\,\mbox{\textit{\textbf{t}}}_n)$ with integer coefficients and the matrix $(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}}_{\circ})$:
   

   $$(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}})= (... ...textit{\textbf{W}}},\, \mbox{\textit{\textbf{w}}}_{\circ}).$$ (4.6.1)

   
   
   For the coefficients $\mbox{\textit{\textbf{w}}}_{\circ},\ 0 \leq w_{i\circ} < 1$ holds. By this decomposition one splits the infinite set of pairs (W,w) into a finite set of representatives $(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}}_{\circ})$ and an infinite set of translations $(\mbox{\textit{\textbf{I}}},\,\mbox{\textit{\textbf{t}}}_n)$. Clearly, only the representatives need to be listed. Such a list is contained in IT A for each space-group type; the number of necessary entries is reduced from infinite to at most 48.

   For primitive bases, the list is complete and unique. There are ambiguities for centered settings, see the remarks to definition (D 1.5.2). For example, for a space group with an $I$-centered lattice, to each point $x,\,y,\,z$ there belongs a translationally equivalent point $x+1/2,\,y+1/2,\,z+1/2$. Nevertheless, only one entry $(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}}_{\circ})$ is listed. Again, instead of listing a translationally equivalent pair for each entry, the centering translation is extracted from the list and written once for all on top of the listing. For example, the rational translations for the $I$-centered lattice are indicated by `(0,0,0)+ $(\frac{1}{2},\,\frac{1}{2},\,\frac{1}{2})+$'. For each of the matrix-column pairs $(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}}_{\circ})$, listed in the sequel, not only the products $(\mbox{\textit{\textbf{I}}},\,\mbox{\textit{\textbf{t}}}_n) (\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}}_{\circ})$ have to be taken into account, but also the products $(\mbox{\textit{\textbf{I}}},\,\mbox{\textit{\textbf{t}}}_n+\mathbf{\frac{1}{2}})(\mbox{\textit{\textbf{W}}},\, \mbox{\textit{\textbf{w}}}_{\circ})$. (The term $\mathbf{\frac{1}{2}}$ is a symbol for the column with all coefficients $\frac{1}{2}$.) The following example 3 (General position for space-group type $I2_13$, No. 199) provides such a listing.

3. The representatives $(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}}_{\circ})$ could be listed as matrix-column pairs but that would be wasting space. Although one could not save much space with further conventions when listing general matrices, the simple (48 orthogonal + 16 other = 64) standard matrices of crystallography with their many zeroes have a great potential for rationalization. Is there any point to list thousands of zeroes ? Therefore, in crystallography an efficient procedure is applied to condense the description of symmetry by matrix-column pairs considerably. This method works like the shorthand notation for the normal language, when the usual letters are replaced by shorthand symbols.

   The equations (4.1.1) on p. are shortened in the following way:

   1. The left side and the `=' sign are omitted
   2. On the right side, all terms with coefficients 0 are omitted
   3. Coefficients `+1' are omitted, coefficients `-1' are replaced by `-' and are frequently written on top of the variable: $\overline{x}$ instead of $-x$, etc.
   4. The three different rows are written in one line but separated by commas.

3 examples shall display the procedure.

Example 1.

$\tilde{x}=0\,x+1\,y+0\,z+1/2,\ \tilde{y}=-1\,x+ 0\,y+0\,z+1/2,\ \tilde{z}=0\,x+0\,y+1\,z+1/4$.

The shorthand notation of IT A reads $y+1/2,\, \overline{x}+1/2,\, z+1/4$.

It is found in IT A under space group $P4_32_12$, No. 96 on p. 367. There it is entry (4) of the first block (the so-called General position) under the heading Positions.

Example 2.

is written in the shorthand notation of IT A $y,\, \overline{x}+y,\,z+1/2$; space group $P6_3mc$, No. 186 on p. 575 of IT A. It is entry (5) of the General position.

Example 3. The following table is the actual listing of the General position for space-group type $I2_13$, No. 199 in IT A on p. 603. The 12 entries, numbered (1) to (12), are to be taken as they are (indicated by (0,0,0)+) and in addition with 1/2 added to each element $w_{i\circ}$ (indicated by $(\frac{1}{2},\,\frac{1}{2}, \,\frac{1}{2})+)$. Altogether these are 24 entries, which is announced by the first number in the row, the `Multiplicity'. The reader is recommended to convert some of the entries into matrix-column pairs or $(4\times 4)$ matrices.

Positions Multiplicity, Wyckoff letter, Site symmetry	Coordinates
$(0,0,0)+ \rule{2em}{0ex}(\frac{1}{2},\frac{1}{2},\frac{1}{2})+$

The listing of the `General position' kills two birds with one stone:

(i)

each of the numbered entries lists the coordinates of an image point $\tilde{X}$ of the original point X under a symmetry operation of the space group.

(ii)

Each of the numbered entries of the General position lists a symmetry operation of the space group by the shorthand notation of the matrix-column pair. This fact is not as obvious as the meaning described under (i) but it is much more important. Knowing this way one can extract and make available for calculations the full analytical symmetry information of the space group from the tables of IT A.

Exactly one image point belongs to each of the infinitely many symmetry operations and vice versa. Some of these points are displayed in Figure 3.5.2 on p. .

Definition (D 4.6.2) The set of all points which are symmetrically equivalent to a starting point $X$ (and thus to each other) under the symmetry operations of a space group $\mbox{$\mathcal{R}$}$ is called a point orbit $\mbox{$\mathcal{R}$}\,X$ of the space group.

Remarks.

1. The starting point is a point of the orbit because it is mapped onto itself by the identity operation (I,o) which is a symmetry operation of any space group.
2. The one-to-one correspondence between symmetry operations and points is valid only for the General position, i.e. the first block from top in the space-group tables. In this block the coordinate triplets (shorthand symbols for symmetry operations) refer to points which have site symmetry $\mathcal{I}$, i.e. only the identity operation is a symmetry operation. In all the other blocks, the points have site symmetries $\mbox{$\mathcal{S}$}>\mbox{$\mathcal{I}$}$ with more than one site-symmetry operation. If $\mbox{$\mathcal{S}$}>\mbox{$\mathcal{I}$}$, also $\vert\mbox{$\mathcal{S}$}\vert>1$ holds, where $\vert\mbox{$\mathcal{S}$}\vert$ is the order of $\mbox{$\mathcal{S}$}$. One can show, that the point $X$ is mapped onto its image $\tilde{X}$ by exactly as many symmetry operations of the space group $\mbox{$\mathcal{R}$}$ as is the order of the site-symmetry group $\mbox{$\mathcal{S}$}$ of $X$. Therefore, for such points the symmetry operation can not be derived from the data listed in IT A because it is not uniquely determined.

Definition (D 4.6.2) The blocks with points of site symmetries $\mbox{$\mathcal{S}$}>\mbox{$\mathcal{I}$}$ are called special positions.

Different from the General position, a coordinate triplet of a special position provides the coordinates $\tilde{x},\,\tilde{y},\,\tilde{z}$ of the image point of the starting point $x,\,y,\,z$ only but no information on a matrix-column pair.

Copyright © 2002 International Union of Crystallography