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Subsections

Coordinate transformations

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

  1. 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.
  2. 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.
  3. 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.
  4. 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.
For these and other reasons either the origin or the basis of the coordinate system or both may have to be changed. The necessary tools for these manipulations are developed in this section.

Origin shift

At first the consequences of an origin shift are considered. We start from Fig. 1.4.1 on p. [*] where $O$ is the origin with the zero column o as coordinates, and $X$ is a point with coordinate column x. The new origin is $O'$ with coordinate column (referred to the old origin) $\mbox{\textit{\textbf{o}}}'=\mbox{\textit{\textbf{p}}}$, whereas $\mbox{\textit{\textbf{x}}}'$ are the coordinates of $X$ with respect to the new origin $O'$. This nomenclature is consistent with that of IT A, see Section 5.1 of IT A.

For the columns, $\mbox{\textit{\textbf{p}}}+\mbox{\textit{\textbf{x}}}'=\mbox{\textit{\textbf{x}}}$ holds, or

\begin{displaymath} \mbox{\textit{\textbf{x}}}'=\mbox{\textit{\textbf{x}}}-\mbox{\textit{\textbf{p}}}. \end{displaymath} (5.3.1)
This can be written in the formalism of matrix-column pairs as
\begin{displaymath} \mbox{$\mbox{\textit{\textbf{x}}}'=(\mbox{\textit{\textbf{I}... ...,\mbox{\textit{\textbf{p}}})^{-1}\,\mbox{\textit{\textbf{x}}}. \end{displaymath} (5.3.2)

[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 \(\mbox{$\mos{x}$}'=\mbox{$\mos{P}$}^{-1}\,\mbox{$\mos{x}$} \) with

\( \mbox{$\mos{P}$}=\left( \begin{array}{ccc\vert c} 1&0&0&p_1\\ 0&1&0&p_2\\ 0&0&1&p_2\\ \hline 0&0&0&1 \end{array}\right).\) A distance vector \(\mbox{$\mos{r}$}=\left( \begin{array}{c} r_1\\ r_2\\ r_3\\ \hline0 \end{array}\right)\) is not changed by the transformation $\mbox{$\mos{r}$}'={\mbox{$\mos{P}$}}^{-1}\,\mbox{$\mos{r}$}$ because the column $\mbox{\textit{\textbf{p}}}$ 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 $\tilde{\mbox{\textit{\textbf{x}}}}=(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}})\,\mbox{\textit{\textbf{x}}}$ holds, in the new one is $\tilde{\mbox{\textit{\textbf{x}}}}'=(\mbox{\textit{\textbf{W}}}',\,\mbox{\textit{\textbf{w}}}')\,\mbox{\textit{\textbf{x}}}'$. By application of equation (5.3.2) one obtains

\begin{displaymath}(\mbox{\textit{\textbf{I}}},\,\mbox{\textit{\textbf{p}}})^{-1... ...,\,\mbox{\textit{\textbf{p}}})^{-1}\mbox{\textit{\textbf{x}}}. \end{displaymath}

Comparison with $\tilde{\mbox{\textit{\textbf{x}}}}=(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}})\mbox{\textit{\textbf{x}}}$ yields

\begin{displaymath} (\mbox{\textit{\textbf{I}}},\,\mbox{\textit{\textbf{p}}})(\m... ...\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}})\ \mbox{ or }\end{displaymath}
\begin{displaymath}(\mbox{\textit{\textbf{W}}}',\,\mbox{\textit{\textbf{w}}}')=(... ...}}})(\mbox{\textit{\textbf{I}}},\,\mbox{\textit{\textbf{p}}}). \end{displaymath} (5.3.3)

This means for the matrix and column parts of the pair $(\mbox{\textit{\textbf{W}}}',\,\mbox{\textit{\textbf{w}}}')$
\begin{displaymath} \mbox{\textit{\textbf{W'}}}=\mbox{\textit{\textbf{W}}},\ \mb... ...W}}}- \mbox{\textit{\textbf{I}}})\,\mbox{\textit{\textbf{p}}}. \end{displaymath} (5.3.4)

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 $(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}})^k$ of equation 5.2.4 the column w by $\mbox{\textit{\textbf{w}}}'=\mbox{\textit{\textbf{w}}}+ (\mbox{\textit{\textbf{W}}}-\mbox{\textit{\textbf{I}}})\,\mbox{\textit{\textbf{p}}}$ ? The answer is simple: the additional term

$(\mbox{\textit{\textbf{W}}}^{k-1}+\mbox{\textit{\textbf{W}}}^{k-2}+\ldots+ \mbo... ...ox{\textit{\textbf{W}}}-\mbox{\textit{\textbf{I}}})\,\mbox{\textit{\textbf{p}}}$ does not contribute because

$(\mbox{\textit{\textbf{W}}}^{k-1}+\mbox{\textit{\textbf{W}}}^{k-2}+\ldots+\mbox... ...x{\textit{\textbf{W}}}+\mbox{\textit{\textbf{I}}})\,\mbox{\textit{\textbf{I}}}.$

An origin shift does not change the screw or glide component of a symmetry operation. The component $(\mbox{\textit{\textbf{W}}}-\mbox{\textit{\textbf{I}}})\,\mbox{\textit{\textbf{p}}}$ 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 $\mbox{\textit{\textbf{w}}}_{lp}$ of the symmetry operation.

Change of basis

A change of basis is mostly described by a $(3\times 3)$ matrix $\mbox{\textit{\textbf{P}}}$ by which the new basis vectors are given as linear combinations of the old basis vectors:

\begin{displaymath} (\mathbf{a}_1',\,\mathbf{a}_2',\,\mathbf{a}_3')=(\mathbf{a}_... ...f{a})^{^{\mbox{\footnotesize {T}}}}\mbox{\textit{\textbf{P}}}. \end{displaymath} (5.3.5)

For a point $X$, the vector $\stackrel{\longrightarrow}{OX}=\mathbf{x}$ is

\begin{displaymath}\mathbf{x}=\mathbf{a}_1x_1+\mathbf{a}_2x_2+\mathbf{a}_3x_3= \... ...{a}')^{^{\mbox{\footnotesize {T}}}}\mbox{\textit{\textbf{x}}}'.\end{displaymath}

By inserting equation (5.3.5) one obtains
\begin{displaymath}\mathbf{x}=(\mathbf{a})^{^{\mbox{\footnotesize {T}}}}\mbox{\t... ...it{\textbf{x}}}=\mbox{\textit{\textbf{P\,x}}}',\ \quad i.\,e.\ \end{displaymath}
\begin{displaymath} \mbox{\textit{\textbf{x}}}'=\mbox{\textit{\textbf{P}}}^{-1}\... ...,\,\mbox{\textit{\textbf{o}}})^{-1}\mbox{\textit{\textbf{x}}}. \end{displaymath} (5.3.6)

The transformation of an isometry follows from equation (5.3.6) and from the relation $\tilde{\mbox{\textit{\textbf{x}}}}'=(\mbox{\textit{\textbf{W}}}',\,\mbox{\textit{\textbf{w}}}')\mbox{\textit{\textbf{x}}}'$ by comparison with $\tilde{\mbox{\textit{\textbf{x}}}}=(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}})\mbox{\textit{\textbf{x}}}$:
$(\mbox{\textit{\textbf{P}}},\,\mbox{\textit{\textbf{o}}})^{-1}\tilde{\mbox{\tex... ...extit{\textbf{P}}},\,\mbox{\textit{\textbf{o}}})^{-1}\mbox{\textit{\textbf{x}}}$ or $\tilde{\mbox{\textit{\textbf{x}}}}= (\mbox{\textit{\textbf{P}}},\,\mbox{\textit... ...}}},\,\mbox{\textit{\textbf{o}}})^{-1}\mbox{\textit{\textbf{x}}}\ \rightarrow $

\begin{displaymath}( \mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}})=(\... ...tbf{w}}}') (\mbox{\textit{\textbf{P,\,o}}})^{-1}\ \mbox{ or }\ \end{displaymath}
\begin{displaymath} (\mbox{\textit{\textbf{W}}}',\mbox{\textit{\textbf{w}}}')=(\... ...{w}}})(\mbox{\textit{\textbf{P}}},\mbox{\textit{\textbf{o}}}). \end{displaymath} (5.3.7)

From this follows

\begin{displaymath} \mbox{\textit{\textbf{W}}}'=\mbox{\textit{\textbf{P}}}^{-1}\... ...}}}'=\mbox{\textit{\textbf{P}}}^{-1}\mbox{\textit{\textbf{w}}}.\end{displaymath} (5.3.8)

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 $(\mathbf{a}){^{\mbox{\footnotesize {T}}}}$ is the conventional, $(\textbf{a}'){^{\mbox{\footnotesize {T}}}}$ the primitive basis, then

P = $ \left(\rule{-0.2em}{0ex}\begin{array}{rr}1/2&1/2 \\ -1/2& 1/2 \end{array} \right). $ One finds $\mbox{\textit{\textbf{P}}}^{-1}= \left( \begin{array}{rr} 1&\bar{1} \\ 1& 1 \end{array} \right)$ either by trial and error or with equation (2.6.1) on p. [*].

For the coordinates, $ \mbox{\textit{\textbf{x}}}'= \mbox{\textit{\textbf{P}}}^{-1}\,\mbox{\textit{\textbf{x}}}\ \mbox{ or }\ x'=x-y,\ y'=x+y$ 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 $-1,\,1$.

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

$(\mbox{\textit{\textbf{W}}}_1,\,\mbox{\textit{\textbf{w}}}_1)=\left( \begin{array}{rr} 1&0\\ 0&\bar{1} \end{array} \right)$, $\left( \begin{array}{r} 0\\ 0 \end{array} \right)$;

the reflection through the parallel line through the endpoint of the vector

$\textbf{b}'$ is described by $(\mbox{\textit{\textbf{W}}}_2,\,\mbox{\textit{\textbf{w}}}_2)=\left( \begin{arr... ...\bar{1} \end{array} \right),\,\left( \begin{array}{r} 0\\ 1 \end{array} \right)$.

\begin{displaymath}\mbox{Then, }\ \mbox{\textit{\textbf{W}}}_1'=\left( \begin{ar... ...right)=\left( \begin{array}{rr} 0&1\\ 1&0 \end{array} \right). \end{displaymath}

The column $\mbox{\textit{\textbf{w}}}'_1$ is the o column because $\mbox{\textit{\textbf{w}}}_1$ is the o column. According to equation (5.3.8),

the column $\mbox{\textit{\textbf{w}}}'_2$ is obtained from $\mbox{\textit{\textbf{w}}}_2$ by $\mbox{\textit{\textbf{w}}}'_2= \left( \begin{array}{rr} 1&\bar{1} \\ 1& 1 \end{... ... 1 \end{array} \right)= \left( \begin{array}{r} \bar{1}\\ 1 \end{array} \right)$.

Indeed, this is the image of the origin, expressed in the new basis. All these results agree with the geometric view.

General coordinate transformations

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 $(\mathbf{a})^{\mbox{\footnotesize {T}}}$, it has to be performed first:

\begin{displaymath} \mbox{\textit{\textbf{x}}}'=(\mbox{\textit{\textbf{P}}},\,\m... ...,\,\mbox{\textit{\textbf{p}}})^{-1}\mbox{\textit{\textbf{x}}}, \end{displaymath} (5.3.9)
\begin{displaymath}\mbox{ using }\left((\mbox{\textit{\textbf{I}}},\,\mbox{\text... ...(\mbox{\textit{\textbf{I}}},\,\mbox{\textit{\textbf{p}}})^{-1}.\end{displaymath}

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

$\tilde{\mbox{\textit{\textbf{x}}}}=(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{... ...xtit{\textbf{W}}}',\,\mbox{\textit{\textbf{w}}}')\,\mbox{\textit{\textbf{x}}}':$

\begin{displaymath}(\mbox{\textit{\textbf{P}}},\,\mbox{\textit{\textbf{p}}})^{-1... ...,\,\mbox{\textit{\textbf{p}}})^{-1} \mbox{\textit{\textbf{x}}}.\end{displaymath}

\begin{displaymath} \mbox{Finally, \ }(\mbox{\textit{\textbf{W}}}',\,\mbox{\text... ...}},\,\mbox{\textit{\textbf{p}}}) \mbox{ \ is obtained, }i.\,e. \end{displaymath} (5.3.10)
\begin{displaymath} \mbox{\textit{\textbf{W}}}'=\mbox{\textit{\textbf{P}}}^{-1}\... ...\mbox{\textit{\textbf{P}}}^{-1}\mbox{\textit{\textbf{W\,p}}}. \end{displaymath} (5.3.11)

The second equation may be written

\begin{displaymath} \mbox{\textit{\textbf{w}}}'=\mbox{\textit{\textbf{P}}}^{-1}(... ...f{W}}}-\mbox{\textit{\textbf{I}}})\mbox{\textit{\textbf{p}}}). \end{displaymath} (5.3.12)

From equation 5.3.11 one obtains the equations (5.3.4) and (5.3.8) as special cases.

In $(4\times 4)$ matrices the equation (5.3.10) is written

(with $(\mbox{\textit{\textbf{P}}},\,\mbox{\textit{\textbf{p}}})^{-1}\rightarrow\mbox{... ...x{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}})\rightarrow\mbox{$\mos{W}$}$, and $(\mbox{\textit{\textbf{P}}},\,\mbox{\textit{\textbf{p}}})\rightarrow\mbox{$\mos{P}$}$)

\begin{displaymath} \mbox{$\mos{W}$}\,'=\mbox{$\mos{P}$}^{-1}\mbox{$\mos{WP}$}.\end{displaymath} (5.3.13)

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]

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 $X$ (left) and $\widetilde{X}$ (right) are represented by the original coordinates $\mbox{\textit{\textbf{x}}}$ und $\tilde{\mbox{\textit{\textbf{x}}}}$ (top) and the new coordinates $\mbox{\textit{\textbf{x}}}'$ und $\tilde{\mbox{\textit{\textbf{x}}}}'$ (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 $\tilde{\mbox{\textit{\textbf{x}}}}'= (\mbox{\textit{\textbf{W}}}',\mbox{\textit{\textbf{w}}}')\mbox{\textit{\textbf{x}}}'$ along the lower edge; on the other hand taking the way up $\rightarrow$ left $\rightarrow$ down one finds

\begin{displaymath}\tilde{\mbox{\textit{\textbf{x}}}}'=(\mbox{\textit{\textbf{P}... ...{P}}},\,\mbox{\textit{\textbf{p}}})\mbox{\textit{\textbf{x}}}'.\end{displaymath}

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 $(\mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}}_{\circ})$ 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 $ZrSiO_4$ 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 $I4_1/amd=D_{4h}^{19}$, 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 $\bar{4}m2$).

$ \begin{array}{rcl} Zr:& (a)&\ 0,0,0;\ 0,\frac{1}{2},\frac{1}{4};\ \frac{1}{2},... ...x{and the same with}\ (\frac{1}{2},\frac{1}{2},\frac{1}{2})+\,]. \end{array} $

The parameters $u$ and $v$ are listed with $u$ = 0.20 and $v$ = 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 ($2/m$) at $0,\bar{\frac{1}{4}},\frac{1}{8}$ from $\bar{4}m2$.'

`Oxygen: ($0,\,y,\,z$) with $y$ = 0.067, $z$ = 0.198.'

In order to compare the different data, the parameters of Wyckoff's book are to be transformed to `origin at center 2/$m$', i.e. ORIGIN CHOICE 2.

Questions

(i)
What are the new coordinates of the $Zr$ atoms ?
(ii)
What are the new coordinates of the $Si$ atoms ?
(iii)
What are the new coordinates of the $O$ atom at $0,u,v$ ?
(iv)
What are the new coordinates of the other $O$ atoms ?

For a physical problem it is advantageous to refer the crystal structure onto a primitive cell with origin in 2/$m$. The choice of the new basis is

${\bf a}' = {\bf a};\ {\bf b}' = {\bf b};\ {\bf c}' = \frac{1}{2}({\bf a}+{\bf b}+{\bf c}$).

Questions

(v)
What are the new coordinates of the first $Zr$ atom ?
(vi)
What are the new coordinates of the first $Si$ atom ?
(vii)
What are the new coordinates of the $O$ atom originally at 0,$u,v$ ?

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Next: Solution of the exercises Up: Special aspects of the matrix formalism Previous: The geometric meaning of (W,w)

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