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Voids in a Close-Packing

In case of close-packed inorganic compounds, the larger atoms or ions occupy positions approximately corresponding to those of equal spheres in a close- packing while the smaller atoms are distributed among the voids. Three- dimensional close-packings of spheres have two kinds of voids:

(i) If the triangular void in a close-packed layer has a sphere directly over it, there results a void with four spheres around it, as shown in Fig. 2a. Such a void is called a tetrahedral void since the four spheres surrounding it are arranged on the corners of a regular tetrahedron (Fig. 2b). If R denotes the radius of the four spheres surrounding a tetrahedral void, the radius of the spheres that would just fit into this void is given2,5 by 0.225 R.

(ii) If a triangular void pointing up in one close-packed layer is covered by a triangular void pointing down in the adjacent layer, then a void surrounded by six spheres results (Fig. 2c). Such a void is called an octahedral void since the six spheres surrounding it lie at the corners of a regular octahedron (Fig. 2d). The radius of the sphere that would just fit into an octahedral void in a close-packing is given2,5 by 0.414 R.


Figure 2: Voids in a close-packing:
(a):Tetrahedral void (b): Tetrahedron
\begin{figure} \includegraphics {fig2a.ps} \end{figure} \begin{figure} \includegraphics {fig2b.ps} \end{figure}
(c): Octahedral void (d): Octahedron formed by the centres of squares
\begin{figure} \includegraphics {fig2c.ps} \end{figure} \begin{figure} \includegraphics {fig2d.ps} \end{figure}

To determine the number of tetrahedral and octahedral voids in a three-dimensional close-packing of spheres, we note that a sphere in a hexagonal close-packed layer A is surrounded by three B voids and three C voids (Fig. 1). When the next layer is placed on top of this, the three voids of one kind (say B) are occupied and the other three (say C) are not. Thus the three B voids become tetrahedral voids and the three C voids become octahedral voids. A single sphere in a three-dimensional close-packing will have similar voids on the lower side as well. In addition, the particular sphere being considered covers a triangular void in the layer above it and another in the layer below it. Thus two more tetrahedral voids surround the spheres. This results in 2 $\times$ 3 + 1 + 1 = 8 tetrahedral voids and 2 $\times$ 3 = 6 octahedral voids surrounding the sphere. Since a tetrahedral void is shared by four spheres, there are twice as many tetrahedral voids as there are spheres. Similarly, since an octahedral void is surrounded by six spheres, there are as many octahedral voids as there are spheres.

In an actual crystal structure a particular atom can best fit into one or the other kind of void depending on its size relative to that of the close-packed atoms. Thus the radius ratio of the atoms present in a crystal imposes limitations on the coordination that they can have in real structures. Conversely, the coordination number of an atom imposes a limitation on the radius ratio. In effect this means that the size and coordination number of a central atom may require that its close-packed neighbours do not touch each other.


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Next: Symmetry and Space Group of Close-Packed Up: Close-Packed Structures Previous: Close-Packed Structures

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