The Interstices in Closest-packings of Spheres

Inorganic Structural Chemistry, Second Edition Ulrich Muller 2006 John Wiley Sons, Ltd. 

From Fig. 17.1 we can see how adjacent octahedra are linked in a hexagonal closest- 

The bond angles at the bridging X atoms in the common octahedron vertices are fixed by geometry (angles M-X-M, M in the octahedron centers)  

The number of octahedral holes in the unit cell can be deduced from Fig. 17.1(c) two differently oriented octahedra alternate in direction c, i.e. it takes two octahedra until the pattern is repeated. Flence there are two octahedral interstices per unit cell. Fig. 17.1(b) shows the presence of two spheres in the unit cell, one each in the layers A and B. The number of spheres and of octahedral interstices are thus the same, i.e. there is exactly one octahedral interstice per sphere. 

The size of the octahedral interstices follows from the construction of Fig. 7.2 (p. 53). There, it is assumed that the spheres are in contact with one another just as in a packing of spheres. A sphere with radius 0.414 can be accommodated in the hole between six octahedrally arranged spheres with radius 1. 

Inorganic Structural Chemistry, Second Edition Ulrich Muller 

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There are four spheres, four octahedral interstices and eight tetrahedral interstices per unit cell. Therefore, their numerical relations are the same as for hexagonal closest-packing, as well as for any other stacking variant of closest-packings one octahedral and two tetrahedral interstices per sphere. Moreover, the sizes of these interstices are the same in all closest-packings of spheres. 

According to the discussion in Section 17.3, many structures can be derived from the hexagonal closest-packing of spheres by occupying a fraction of the octahedral interstices with other atoms. If the X atoms of a compound MX form the packing of spheres, then the fraction 1 fn of the octahedral interstices must be occupied. The unit cell of the... 

In sodium nitrite the ferroelectric polarization only occurs in one direction. In BaTiOs it is not restricted to one direction. BaTiOs has the structure of a distorted perovskite between 5 and 120 °C. Due to the size of the Ba2+ ions, which form a closest packing of spheres together with the oxygen atoms, the octahedral interstices are rather too large for... 

In order to arrive at the important sodium chloride structure, one could look at and discuss the interstices in the cubic closest sphere packing model [2] interstices that are octahedral and tetrahedral can be found (see E5.8). If one fills the octahedral interstices with smaller spheres, one ends up with the sodium chloride structure (see Fig. 5.13). The larger spheres symbolize the chloride ions, the smaller spheres the sodium ions. The structure can be described as the cubic closest packing of chloride ions, whose octahedral interstices are completely filled by sodium ions. For other salt structures, only part of the octahedral sites are filled, as in aluminum oxide where the ratio of ions 2 3 applies [2],... 

In three-dimensional closest packing, the spherical atoms are located in position 4(a). There are two types of interstices octahedral and tetrahedral holes which occupy positions 4(b) and 8(c), respectively. The number of tetrahedral holes is twice that of the spheres, while the number of octahedral holes is equal to that of the spheres. The positions of the holes are shown in Fig. 10.1.1. 

The great Importance of closest packing in structural chemistry arises from the fact that the anions in many halides, oxides, and sulphides are close-packed (or approximately so) with the metal atoms occupying the interstices between them. As already noted, the polyhedral interstices between c.p. atoms are of two kinds, tetrahedral and octahedral (T and 0 in Fig. 4.7(b)). The number of tetrahedral holes is equal to twice the number of c.p. spheres the number of octahedral holes Is... 

From the fact that the smallest kind of interstice between spheres in contact is a tetrahedral hole it follows that we should expect to find coordination polyhedra with only triangular faces, in contrast to those in, for example, cubic closest packing which have square in addition to triangular faces. Moreover, it seems likely that the preferred coordination polyhedra will be those in which five or six triangular faces (and hence five or six edges) meet at each vertex, since the faces are then most nearly equilateral. It follows from Euler s relation (p. 61) that for such a polyhedron, 1)5 -h Oug = 12, where 1)5 and are the numbers of vertices at which five or six edges meet, so that starting from the icosahedron (vg = 12) we may add 6-fold vertices to form polyhedra with more than twelve vertices. 

The picture of the closest sphere packing for the alum crystal (see Fig. 4.4) correlates to a rough educational reduction. The expert knows that the substance potassium aluminum sulfate dodecahydrate is composed of hydrated potassium ions, whose octahedral and tetrahedral interstices are filled with aluminum ions and with sulfate ions. This model concept may be introduced later and can be used in future lessons. Nevertheless, it is considered reasonable, at first, to introduce the alum particles and to choose reduced concrete models. 

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