There are several reasons to change the coordinate system. Some examples for such reasons are the following:

- If one and the same crystal structure is described in different
coordinate systems by different authors, then the structural data
*e.g.*lattice constants, atomic coordinates, or displacement parameters (thermal parameters) have to be transformed to the same coordinate system in order to be comparable. The same holds for the comparison of related crystal structures. - In phase transitions frequently the phases are related by their
symmetries,
*e.g.*in phase transitions of the second order. Often the conventional setting of the new phase is different from that of the original one. Then a change of the coordinate system may be necessary in order to find the structural changes which are connected with the phase transition. - In the physics of macroscopic crystals (thermal expansion,
dielectric constant, elasticity, piezoelectricity,
*etc.*) the properties are described mostly relative to an orthonormal basis. Therefore, for physical calculations the direction and Miller indices*etc.*have to be transformed from the conventional crystallographic to an orthonormal basis. - In IT A, 44 space-group types are described in more than one conventional coordinate system. The transition from one description to the other may be necessary and needs coordinate transformations.

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

For the columns, holds, or

This can be written in the formalism of matrix-column pairs as[It may look strange to write the simple equation (5.3.1) in the complicated form of (5.3.2). The reason will become apparent later in this section].

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
one is
. By
application of equation (5.3.2) one obtains

Comparison with
yields

(5.3.3) |

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

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
basis vectors:

By inserting equation (5.3.5) one obtains

The transformation of an isometry follows from equation (5.3.6)
and from the relation
by comparison with
:

or

From this follows

Example

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

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:

In the usual way one concludes from equation (5.3.9) together with

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

Both ways start at the same point and end at the same point. Therefore, the one way can be equated to the other, and herewith equation 5.3.10 is derived in a visual way.

*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.

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.

Questions

**(i)**- What are the new coordinates of the atoms ?
**(ii)**- What are the new coordinates of the atoms ?
**(iii)**- What are the new coordinates of the atom at ?
**(iv)**- What are the new coordinates of the other atoms ?

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

).

Questions

**(v)**- What are the new coordinates of the first atom ?
**(vi)**- What are the new coordinates of the first atom ?
**(vii)**- What are the new coordinates of the atom originally at 0, ?

**Copyright © 2002 International Union of
Crystallography**