International
Tables for Crystallography Volume A Space-group symmetry Edited by Th. Hahn © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A. ch. 5.1, pp. 78-85
https://doi.org/10.1107/97809553602060000510 Chapter 5.1. Transformations of the coordinate system (unit-cell transformations)a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany Transformations of the coordinate system are useful when nonconventional descriptions of a crystal structure are considered, for instance in the study of relations between different structures, of phase transitions and of group–subgroup relations. Unit-cell transformations occur particularly frequently when different settings or cell choices of monoclinic, orthorhombic or rhombohedral space groups are to be compared or when `reduced cells' are derived. In this chapter, matrix notation is used to describe a general transformation. Selected frequently occurring transformation matrices are tabulated and illustrated by diagrams. Keywords: coordinate systems; unit-cell transformations; transformation matrices. |
There are two main uses of transformations in crystallography.
Throughout this volume, matrices are written in the following notation:
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Here the crystal structure is considered to be at rest, whereas the coordinate system and the unit cell are changed. Specifically, a point X in a crystal is defined with respect to the basis vectors a, b, c and the origin O by the coordinates x, y, z, i.e. the position vector r of point X is given by The same point X is given with respect to a new coordinate system, i.e. the new basis vectors a′, b′, c′ and the new origin O′ (Fig. 5.1.3.1)
, by the position vector
In this section, the relations between the primed and unprimed quantities are treated.
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General affine transformation, consisting of a shift of origin from O to O′ by a shift vector p with components |
The general transformation (affine transformation) of the coordinate system consists of two parts, a linear part and a shift of origin. The matrix P of the linear part and the
column matrix p, containing the components of the shift vector p, define the transformation uniquely. It is represented by the symbol (P, p).
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For a pure origin shift, the basis vectors do not change their lengths or orientations. In this case, the transformation matrix P is the unit matrix I and the symbol of the pure shift becomes (I, p).
Also, the inverse matrices of P and p are needed. They are and
The matrix q consists of the components of the negative shift vector q which refer to the coordinate system a′, b′, c′, i.e.
Thus, the transformation (Q, q) is the inverse transformation of (P, p). Applying (Q, q) to the basis vectors a′, b′, c′ and the origin O′, the old basis vectors a, b, c with origin O are obtained.
For a two-dimensional transformation of a′ and b′, some elements of Q are set as follows: and
.
The quantities which transform in the same way as the basis vectors a, b, c are called covariant quantities and are written as row matrices. They are:
the Miller indices of a plane (or a set of planes), (hkl), in direct space and the coordinates of a point in reciprocal space, h, k, l.
Both are transformed by Usually, the Miller indices are made relative prime before and after the transformation.
The quantities which are covariant with respect to the basis vectors a, b, c are contravariant with respect to the basis vectors of reciprocal space.
The basis vectors of reciprocal space are written as a column matrix and their transformation is achieved by the matrix Q: The inverse transformation is obtained by the inverse matrix
:
These transformation rules apply also to the quantities covariant with respect to the basis vectors
and contravariant with respect to a, b, c, which are written as column matrices. They are the indices of a direction in direct space, [uvw], which are transformed by
In contrast to all quantities mentioned above, the components of a position vector r or the coordinates of a point X in direct space x, y, z depend also on the shift of the origin in direct space. The general (affine) transformation is given by
Example
If no shift of origin is applied, i.e. , the position vector r of point X is transformed by
In this case,
, i.e. the position vector is invariant, although the basis vectors and the components are transformed. For a pure shift of origin, i.e.
, the transformed position vector r′ becomes
Here the transformed vector r′ is no longer identical with r.
It is convenient to introduce the augmented matrix
which is composed of the matrices Q and q in the following manner (cf. Chapter 8.1
):
with o the
row matrix containing zeros. In this notation, the transformed coordinates x′, y′, z′ are obtained by
The inverse of the augmented matrix
is the augmented matrix
which contains the matrices P and p, specifically,
The advantage of the use of
matrices is that a sequence of affine transformations corresponds to the product of the corresponding matrices. However, the order of the factors in the product must be observed. If
is the product of n transformation matrices
,
the sequence of the corresponding inverse matrices
is reversed in the product
The following items are also affected by a transformation:
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Monoclinic centred lattice, projected along the unique axis. Origin for all cells is the same. (a) Unique axis b. Cell choice 1: C-centred cell |
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Body-centred cell I with a, b, c and a corresponding primitive cell P with a′, b′, c′. Origin for both cells O. A cubic I cell with lattice constant |
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Face-centred cell F with a, b, c and a corresponding primitive cell P with a′, b′, c′. Origin for both cells O. A cubic F cell with lattice constant |
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Tetragonal lattices, projected along |
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Unit cells in the rhombohedral lattice: same origin for all cells. The basis of the rhombohedral cell is labelled a, b, c. Two settings of the triple hexagonal cell are possible with respect to a primitive rhombohedral cell: The obverse setting with the lattice points 0, 0, 0; |
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Hexagonal lattice projected along |
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Hexagonal lattice projected along |
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Rhombohedral lattice with a triple hexagonal unit cell a, b, c in obverse setting (i.e. unit cell a1, b1, c in Fig. 5.1.3.6 |
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Rhombohedral lattice with primitive rhombohedral cell a, b, c and the three centred monoclinic cells. (a) C-centred cells |