Voids closest packing

Atoms are replaced by voids or voids are occupied by atoms. For example hexagonal closest-packing -t Cdl2 type. If the voids are considered to be zero atoms , this can be considered as a substitution of voids by atoms. 

Group-subgroup relations from hexagonal closest-packing of spheres to some MX3 and M2X3 structures. The boxes represent octahedral voids, with the coordinates as given at the top left. The positions of the octahedron centers are labeled by their Wyckoff letters. Gray boxes refer to occupied voids. The dots indicate how the atoms Ru, P and N are shifted from the octahedron centers parallel to c... 

The famous Heusler type structure can be considered as a ternary ordered variant of the BiF3 type. On the other hand, the YPd2Pb structure can also be derived from a cubic closest packing of the lead atoms, where the octahedral voids are filled with yttrium (rocksalt substructure) and the tetrahedral voids are filled by the palladium atoms. 

Figure 28. Central Void of Unit-cell for Spheres in Closest Packing.

Referring now to the shape of the void as shown in Figure 28, the rhombohedral andletrahedral cells are denoted by the letters R and T. The T-spaces correspond to the voids between four spheres in closest packing with one sphere nested in the hollow formed by the other three. In this space there are six points of contact between spheres and four lines or branches. These branches connect one cell with another. 

It is simple to calculate the moisture-holding capacity for spheres in closest packing, since we know the voids amount to approximately 26 percent. If we take the gravity of the spheres to be 2.4, and assume the voids to be completely filled with water,, the moisture-holding capacity Mei ... 

We have established that there are two different types of voids between the spheres of two adjacent layers of closest-packed spheres. We are therefore ready to add another layer of closest-packed spheres to our stack. 

The ideal size of the cation relative to the size of the anion for each of the voids can be determined by trigonometry. For example, a cation will fit snugly into an octahedral void if it is 0.41 times the size of the anion. If it is significantly larger than that, it may not be possible to fit the cation into the closest-packed array. This is the case for compounds such as cesium chloride, where the cation has an ionic radius of 1.67 A and the anion has a radius of 1.84 A (thus, the ratio is 0.91). In this solid, the chloride ions open up to form the simple cubic packing shown in Figure 16. 

Notice that the chloride ions are not closest-packed, but instead are stacked in a square arrangement within a layer and in a neat cubic arrangement in 3-dimensions. This means that the voids between the layers are cubic, with eight chloride ions forming the comers of the cube. The large cesium ion occupies these voids. 

The anions can get closer because the cation in the center of the void is not sufficiently large to keep them opened up. As the anions get closer, the repulsions between them increase and the potential energy goes up (that is, the compound becomes less stable). It then becomes energetically more favorable for the compound to adopt an arrangement that has fewer anions around the cation. Both the repulsions and attractions will decrease, but the decrease in the repulsions will be greater and the new closest-packed arrangement will be more stable. 

In order to analyze the structure of ionic compounds more carefully we need to know the number of tetrahedral and octahedral voids in the closest-packed structures. Figure 18 shows all of the tetrahedral voids surrounding one sphere between two layers of closest-packed spheres. Of course, each of these voids is shared by four spheres. 

Figure 18 shows only two adjacent layers of closest-packed spheres. In order to determine the total nnmber of tetrahedral voids per sphere, we must also consider the layer below the sphere in qnestion. This layer is just like the layer above the sphere (imagine hexagonal packing, so that the layer below looks just like the layer shown with the empty circles) and contains another four voids around the sphere. Thns, the total number of tetrahedral voids per sphere is two. The same sort of analysis shows that there is one octahedral void per sphere. 

Thus, regardless of the type of closest packing, there are two tetrahedral voids and one octahedral void per sphere (anion). 

What is the formula of a compound that has closest-packed cations (not usually the case) and all of the tetrahedral voids filled with anions ... 

MX2. Fluorite (CaF2) is an example of a solid that can be described as closest-packed cations with anions in the tetrahedral voids (see page 78 for another description of the structure of fluorite). There are two anions per cation. 

We return now to the NaCl structure, which contains sodium ions in the voids of a cubic closest-packed array of chloride ions. The cubic closest-packed array of ions was illustrated previously in Figures 14 and 34. The unit cell shown in Figure 38 is based on a face-centered arrangement of sodium ions, which is equivalent to the face-centered cubic structure of the chloride ions. The lines between adjacent ions in Figure 38 are not covalent bonds they merely represent ionic interactions between adjacent ions. 

Fifty percent. Remember that there are two tetrahedral and one octahedral voids per closest-packed sphere. The formnla of zinc snlhde is ZnS and therefore there mnst be one zinc cation per snlhde anion. If there is one cation per anion, bnt two tetrahedral voids per anion, only one of the two voids can be hlled. 

Notice that the layers alternate in the hexagonal closest-packed sequence ABAB. .. The zinc ions occupy, as they did in sphalerite, one-half of the tetrahedral voids. 

Let us return to the cubic closest-packing arranganent and take a look at a mineral that contains ions in all of the tetrahedral voids. The unit ceU of fluorite, CaFj, is shown in Figure 48. 

You should remember that sphalerite has a cubic closest-packed (face-centered) arrangement of sulfide ions with zinc cations occupying half of the tetrahedral voids. 

---