There are several reasons to change the coordinate system. Some examples for such reasons are the following:
At first the consequences of an origin shift are considered. We start from Fig. 1.4.1 on p. where is the origin with the zero column o as coordinates, and is a point with coordinate column x. The new origin is with coordinate column (referred to the old origin) , whereas are the coordinates of with respect to the new origin . This nomenclature is consistent with that of IT A, see Section 5.1 of IT A.
For the columns, holds, or
Equation (5.3.2) can be written in augmented matrices with
A distance vector is not changed by the transformation because the column is not effective, see Sections 4.3 and 4.4.
How do the matrix and column parts of an isometry change if the origin
is shifted ? In the old coordinate system
holds, in the new
application of equation (5.3.2) one obtains
Conclusion. A change of origin does not change the matrix part of an isometry. The change of the column w does not only depend on the shift p of the origin, but also on the matrix part W.
How is a screw or glide component changed by an origin shift, i.e. what happens if one replaces in of equation 5.2.4 the column w by ? The answer is simple: the additional term
does not contribute because
An origin shift does not change the screw or glide component of a symmetry operation. The component is the component of p which is vertical to the screw-rotation or rotation axis or to the mirror or glide plane. It causes a change of the location part of the symmetry operation.
A change of basis is mostly described by a matrix
by which the new basis vectors are given as linear combinations of the old
The transformation of an isometry follows from equation (5.3.6)
and from the relation
by comparison with
From this follows
In Fig. 1.5.2 on p. the conventional and a primitive basis are defined for a plane group of the rectangular crystal system. If is the conventional, the primitive basis, then
P = One finds either by trial and error or with equation (2.6.1) on p. .
For the coordinates, holds. The conventional coordinates 1,0 of the endpoint of a become 1,1 in the primitive basis; those of the endpoint 1/2,1/2 of a become 0,1; those of the endpoint 0,1 of b become .
If the endpoints of the lattice-translation vectors of Fig. 1.5.2 on p. , and those of their integer linear combinations are marked with points, a point lattice is obtained.
Suppose, the origin is in the upper left corner of the unit cell of Fig. 1.5.2. Then, the reflection through the line `a' is described by the matrix-column pair
the reflection through the parallel line through the endpoint of the vector
is described by .
The column is the o column because is the o column. According to equation (5.3.8),
the column is obtained from by .
Indeed, this is the image of the origin, expressed in the new basis. All these results agree with the geometric view.
In general both the origin and the basis have to be changed. One can do
this in 2 different steps. Because the origin shift p is referred
to the old basis
, it has to be performed first:
The second equation may be written
In matrices the equation (5.3.10) is written
(with , and )
This shape of equation (5.3.10 ) facilitates the formulation but not the actual calculation. For the latter, the forms 5.3.11 or 5.3.12 are more appropriate.
Fig. 5.3.3 Diagram of `mapping of mappings'.
The formalism of transformations can be displayed by the diagram of Fig. 5.3.3. The points (left) and (right) are represented by the original coordinates und (top) and the new coordinates und (bottom). At the arrows the corresponding transformations are denoted. They describe from left to right a mapping, from top to bottom the change of coordinates. Equation 5.3.10 is read from the figure immediately: On the one hand one reads along the lower edge; on the other hand taking the way up left down one finds
Remark. If there are different listings of the same crystal structure or of a set of related crystal structures, it is often not sufficient to transform the data to the same coordinate system. Even after such a transformation the coordinates of the atoms may be incomparable. The reason is the following:
In IT A for each (general or special) Position the full set of representatives is listed, see the table in Section 4.6. After insertion of the actual coordinates one has a set of triplets of numbers, 24 (including the centering) in the table of Section 4.6. Any one of these representatives may be chosen to describe the structure in a listing; the others can be generated from the selected one. The following Problem shows that different choices happen in reality. For a comparison of the structures it is then necessary to choose for the description corresponding atoms in the structures to be compared.
Problem 3. Change of the coordinate system.
For the solution, see p. .
In R. W. G. Wyckoff, Crystal structures, vol. II, Ch. VIII, one finds the important mineral zircon and a description of its crystal structure under (VIII,a4) on text p. 5, table p. 9, and Figure VIIIA,4. Many rare-earth phosphates, arsenates, and vanadates belong to the same structure type. They are famous for their interesting magnetic properties.
Structural data: Space group , No. 141;
lattice constants a = 6.60 Å; c = 5.88 Å.
The origin choice is not stated explicitly. However, Wyckoff's Crystal Structures started to appear in 1948, when there was one conventional origin only (the later ORIGIN CHOICE 1, i.e. Origin at ).
The parameters and are listed with = 0.20 and = 0.34.
In the Structure Reports, vol. 22, (1958), p. 314 one finds:
`a = 6.6164(5) Å, c = 6.0150(5) Å'
`Atomic parameters. Origin at center () at from .'
`Oxygen: () with = 0.067, = 0.198.'
In order to compare the different data, the parameters of Wyckoff's book are to be transformed to `origin at center 2/', i.e. ORIGIN CHOICE 2.
For a physical problem it is advantageous to refer the crystal structure onto a primitive cell with origin in 2/. The choice of the new basis is
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