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The geometric meaning of (W,w)

How can one find the geometric meaning of a matrix-column pair ? Large parts of the following recipe apply not only to crystallographic symmetry operations but also to general isometries.

  1. One must know the reference coordinate system of the matrix-column pair. Without this knowledge a geometric evaluation is impossible.

    Example.        The matrix (in IT A shorthand notation) $x-y,\ x,\ z$ describes a 6-fold anti-clockwise rotation if referred to a hexagonal basis. If referred to an orthonormal basis it does not describe an isometry at all but contains a shearing component.

  2. The matrix part is evaluated first.

    In general the coefficients of the matrix depend on the choice of the basis; a change of basis changes the coefficients, see Section 5.3.2. However, there are geometric quantities which are independent of the basis. Correspondingly, there exist characteristic numbers of a matrix from which the geometric features may be derived and vice versa.

One can list this correlation in a table

  $\det(\mbox{\textit{\textbf{W}}})=+1$ $\det(\mbox{\textit{\textbf{W}}})=-1$
tr(W) 3 2 1 0 $-1$ $-3$ $-2$ $-1$ 0 1
type 1 6 4 3 2 $\bar{1}$ $\bar{6}$ $\bar{4}$ $\bar{3}$ $\bar{2}=m$
order 1 6 4 3 2 2 6 4 6 2
.

By this table the type of operation may be found, as far as it is determined by the matrix part. For example, one takes from the table that a specific operation is a two-fold rotation but one does not know if the operation is a rotatation or a screw rotation, what the direction of the rotation axis is and where it is located in space. This characterization will be done in the following list for the crystallographic symmetry operations.

  1. Type 1 or $\bar{1}$:        no preferred direction
    1
    identity (for $\mbox{\textit{\textbf{w}}}=\mbox{\textit{\textbf{o}}}$) or translation for $\mbox{\textit{\textbf{w}}}\neq \mbox{\textit{\textbf{o}}}$.

    The coefficients of w are the coefficients of the translation vector.

    $\bar{1}$
    inversion, coordinates of the inversion center $F$
    \begin{displaymath}
\mbox{\textit{\textbf{x}}}_F=\frac{1}{2}\mbox{\textit{\textbf{w}}}. \end{displaymath} (5.2.2)
  2. All other symmetry operations have a preferred axis (the rotation or rotoinversion axis). The direction u of this axis may be determined from the equation
    \begin{displaymath}
\mbox{\textit{\textbf{W\,u}}}=\pm \mbox{\textit{\textbf{u}}}. \end{displaymath} (5.2.3)

    The $+$ sign is for rotations, the $-$ sign for rotoinversions.

    For type $m$, reflections or glide reflections, u is the direction of the normal of the (glide) reflection plane.

  3. If W is the matrix of a rotation of order $k$ or of a reflection ($k=2$), then \( \mbox{\textit{\textbf{W}}}^k=\mbox{\textit{\textbf{I}}} \), and one determines the intrinsic translation part, also called screw part or glide part $\mbox{\textit{\textbf{t}}}/k$ by

    \begin{displaymath}
(\mbox{\textit{\textbf{W,\,w}}})^k=(\mbox{\textit{\textbf{W...
...+\mbox{\textit{\textbf{w}}})=(\mbox{\textit{\textbf{I,\,t}}})
\end{displaymath} (5.2.4)
    \begin{displaymath}
\ \mbox{ to }\mbox{\textit{\textbf{t}}}/k=\frac{1}{k}\,(\mb...
...}}}+\mbox{\textit{\textbf{I}}})\,\mbox{\textit{\textbf{w}}}.
\end{displaymath} (5.2.5)

    The vector with the column of coefficients $\mbox{\textit{\textbf{t}}}/k$ is called the screw or glide vector. This vector is invariant under the symmetry operation: $\mbox{\textit{\textbf{W\,t}}}/k=\mbox{\textit{\textbf{t}}}/k$: Indeed, multiplication with W permutes only the terms on the right side of equation 5.2.5. Thus, the screw vector of a screw rotation is parallel to the screw axis. The glide vector of a glide reflection is left invariant for the same reason. Therefore, it is parallel to the glide plane.

    If t = o holds, then (W,w) describes a rotation or reflection. For $\mbox{\textit{\textbf{t}}} \neq \mbox{\textit{\textbf{o}}}$, (W,w) describes a screw rotation or glide reflection. One forms the so-called reduced operation by subtracting the intrinsic translation part t/k from (W,w):

    \begin{displaymath}
(\mbox{\textit{\textbf{I}}},\,-\mbox{\textit{\textbf{t}}}/k...
...mbox{\textit{\textbf{W}}},\,\mbox{\textit{\textbf{w}}}_{lp}).
\end{displaymath} (5.2.6)

    The column $\mbox{\textit{\textbf{w}}}_{lp}=\mbox{\textit{\textbf{w}}}-\mbox{\textit{\textbf{t}}}/k$ is called the location part because it determines the position of the rotation or screw-rotation axis or of the reflection or glide-reflection plane in space.

    If W is a diagonal matrix, i.e. if only the coefficients $W_{ii}$ are non-zero, then either is $W_{ii}=+1$ and $\mbox{\textit{\textbf{w}}}_i$ is a screw or glide component, or $W_{ii}=-1$ and $\mbox{\textit{\textbf{w}}}_i$ is a location component. If W is not a diagonal matrix, then the location part $\mbox{\textit{\textbf{w}}}_{lp}$ has to be calculated according to equation 5.2.6.

  4. The fixed points are obtained by solving the equation
    \begin{displaymath}
\mbox{\textit{\textbf{W\,x}}}_F+\mbox{\textit{\textbf{w}}}=\mbox{\textit{\textbf{x}}}_F.
\end{displaymath} (5.2.7)

    Equation (5.2.7) has a unique solution for all rotoinversions (including $\bar{1}$, excluding $\bar{2}=m$). There is a 1-dimensional set of solutions for rotations (the rotation axis) and a 2-dimensional set of solutions for reflections (the mirror plane). For screw rotations and glide reflections, there are no solutions: there are no fixed points. However, a solution is found for the reduced operation, i.e. after subtraction of the intrinsic translation part, by equation 5.2.8
    \begin{displaymath}
\mbox{\textit{\textbf{Wx}}}_F+\mbox{\textit{\textbf{w}}}_{lp}=\mbox{\textit{\textbf{x}}}_F.
\end{displaymath} (5.2.8)

The formulae of this section enable the user to find the geometric contents of any symmetry operation. In reality, IT A have provided the necessary information for all symmetry operations which are listed in the plane-group or space-group tables. The entries of the General position are numbered. The geometric meaning of these entries is listed under the same number in the block Symmetry operations in the tables of IT A. The explanation of the symbols for the symmetry operations is found in Sections 2.9 and 11.2 of IT A.

The section shall be closed with an exercise.


Problem 2B. Symmetry described by matrix-column pairs.
For the solution, see p. [*].

The matrix-column pairs (A,a), (B,b) (C,c), and (D,d) have been listed or derived in Problem 2A, p. [*], which dealt with their combination and reversion.

Question

(viii)
Determine the geometric meaning of the matrix-column pairs

(A,a), (B,b), (C,c), and (D,d).


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