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Next: 4. The Diagrams Up: Projections of Cubic Crystals Previous: 2. Clinographic, Orthographic and Perspective Projections

3. Cubic Crystals

The variety of crystal shapes is so great that only crystals built up from cubes are considered in this pamphlet: other structural units have been omitted. Three mutually perpendicular crystallographic axes may be chosen parallel to the edges of the cubic unit cells. These are the familiar Cartesian axes Ox, Oy and Oz of coordinate geometry, where O is the origin. Their directions are represented by [100], [010] and [001]. Directions related by symmetry, such as these three, are written for brevity as $\langle$100$\rangle$. Other directions in the crystal may be referred to these axes; as in the earlier remark that the [621] direction has components in the ratios 6:2:1 along these three axes respectively.

A set of crystal faces, related to one another by symmetry, is called a form ; the symmetry being more obvious in specimens in which the faces are equally developed. For example, the cube, in the most symmetrical class, has three tetrad axes coinciding with the Ox, Oy and Oz axes, and three mirror planes perpendicular to these axes; four triad axes coinciding with the body diagonals $\langle$111$\rangle$; six diad axes parallel to the face diagonals $\langle$110$\rangle$, and six mirror planes (making nine in all) perpendicular to these diad axes. The cube has a centre of symmetry. The underlying symmetry of crystals is much less obvious in specimens for which the faces of individual forms are unequally developed; but may still be deduced by considering the perpendiculars drawn to the faces from a point within the crystal, and their distribution in space. This is best done using a stereographic projection: the subject of another pamphlet2.


 
Figure iii Rhombic dodecahedron made from cubes.
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\includegraphics {figiii.ps}
\end{figure}

In the cubic crystal system there are five distinct classes or point groups : that is, five different combinations of symmetry elements, but all have in common the presence of four triad axes of symmetry. These are listed in Table 1. An axis of symmetry X perpendicular to a mirror plane m is written $\displaystyle\frac{\mbox{X}}{m}$ (or X/m ). The maximum symmetry of a cube, as described above, is represented by the point group symbol $\displaystyle\frac{4}{m}
\overline{3} \frac{2}{m}$ (or 4/m $\overline{3}$ 3/m , often abbreviated to m 3 m ).

The orientation of a plane in crystallography is described in terms of its Miller indices . These are the reciprocals of the relative intercepts, in terms of unit cell edges, that the plane makes with the three crystallographic axes, expressed in whole numbers. In general, if the intercepts are a/h, b/k, c/l, where a, b and c are the cell edges, then the Miller indices of the plane are h, k, l, usually written (hkl). In the cubic crystal system, the unit cells are cubes, so a = b = c. (In the drawings here a = 1.) The face of the cube, perpendicular to the [100] direction (i.e. to the Ox Cartesian axis) has Miller indices (100), since the intercepts on the three axes are a/1, a/0, a/0: the latter two intercepts (on Oy and Oz), being at infinity. (For cubic crystals, the [hkl] direction is perpendicular to the (hkl) plane, but this is not so for crystals of other crystal systems.) The faces of an octahedron make equal intercepts, positive and negative, on all three axes and therefore have Miller indices (111), ($\overline{1}$11), (1$\overline{1}$1), (11$\overline{1}$), ($\overline{111}$), (1$\overline{11}$), ($\overline{1}$1$\overline{1}$) and ($\overline{11}$1). The minus signs are written above the symbols for compactness. All eight symmetrically related faces of the form are written as $\{$111$\}$.


 
Table 1: Cubic crystal classes or point groups

Full symbol Abbreviated symbol   Symmetry elements along the directions  
  
    $\langle$100$\rangle$ $\langle$111$\rangle$ $\langle$110$\rangle$

$\displaystyle\frac{4}{m}\,\=3\,\frac{2}{m}$ $m\,3\,m$ 4-fold axes perpendicular to mirror planes 3-fold rotation-inversion axes 2-fold axes perpendicular to mirror planes
$4\,3\,2$ $4\,3$ 4-fold axes 3-fold axes 2-fold axes
$\=4\,3\,m$ $\=4\,3\,m$ 4-fold rotation-inversion axes 3-fold axes mirror planes perpendicular to these directions
$\displaystyle\frac{2}{m}\,\=3$ $m\,3$ 2-fold axes perpendicular to mirror planes 3-fold rotation-inversion axes -
$2\,3$ $2\,3$ 2-fold axes 3-fold axes -



next up previous
Next: 4. The Diagrams Up: Projections of Cubic Crystals Previous: 2. Clinographic, Orthographic and Perspective Projections

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