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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 spacegroup 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.
 Consider the infinite number of space groups belonging to a
certain spacegroup type. If one refers each space group to its
conventional coordinate system, one obtains a set of
matrixcolumn pairs for each
space group but all these sets are identical. (This is one way
to classify the space groups into spacegroup types.)
Therefore, a listing is necessary only for each spacegroup 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 spacegroup
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.
 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
matrixcolumn pair. Because the conventional bases are always lattice
bases, (W,w) can be decomposed into a translation
with integer coefficients and the matrix
:

(4.6.1) 
For the coefficients
holds.
By this decomposition one splits the infinite set of pairs
(W,w) into a finite set of representatives
and an infinite set of translations
. Clearly, only the representatives need to be
listed. Such a list is contained in IT A for each spacegroup 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 centered
lattice, to each point there belongs a translationally
equivalent point
. Nevertheless, only one
entry
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 centered lattice are indicated
by `(0,0,0)+
'.
For each of the matrixcolumn pairs
,
listed in the sequel, not only the products
have to be taken into account, but also
the products
. (The term
is a symbol for
the column with all coefficients .) The following example 3
(General position for spacegroup type , No. 199) provides such a listing.
 The representatives
could be
listed as matrixcolumn 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 matrixcolumn 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:
 The left side and the `=' sign are omitted
 On the right side, all terms with coefficients 0 are omitted
 Coefficients `+1' are omitted, coefficients `1' are replaced
by `' and are frequently written on top of the variable:
instead of , etc.
 The three different rows are written in one line but separated
by commas.
3 examples shall display the procedure.
Example 1.
.
The shorthand notation of IT A reads
.
It is found in IT A under space group , No. 96 on p. 367.
There it is entry (4) of the first block (the socalled General position)
under the heading
Positions.
Example 2.
is written in the shorthand notation of IT A
;
space group , 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 spacegroup type , 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 (indicated by
. 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 matrixcolumn pairs or
matrices.
Positions
Multiplicity,
Wyckoff letter,
Site symmetry

Coordinates


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 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 matrixcolumn 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 (and thus to each other) under the symmetry
operations of a space group
is called a point orbit
of the space group.
Remarks.
 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.
 The onetoone correspondence between symmetry operations and
points is valid only for the General position, i.e. the first
block from top in the spacegroup tables. In this block the coordinate
triplets (shorthand symbols for symmetry operations) refer to points
which have site symmetry , i.e. only the identity
operation is a symmetry operation. In all the other blocks, the points
have site symmetries
with more than one sitesymmetry
operation. If
,
also
holds, where
is the order of
.
One can show, that the point is mapped onto its image by
exactly as many symmetry operations of the space group
as is
the order of the sitesymmetry group
of . 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
are called special positions.
Different from the General position, a coordinate triplet of a special
position provides the coordinates
of the image point of the starting point only but no
information on a matrixcolumn pair.
Next: Special aspects of the matrix formalism
Up: The description of mappings by ...
Previous: The matrixcolumn pairs of crystallographic symmetry
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