5.1 Even More Structure

1. Perform calculations involving the theoretical density of a ceramics having the CsCl, ZnS, or CaF₂ crystal structures
2. Explain the arrangement of anions and cations in the CsCl, ZnS, or CaF₂ crystal structures
3. Demonstrate the cation and anion coordination number and coordination geometry (interstitial site) for the CsCl, ZnS, or CaF₂ crystal structures
4. Calculate and describe the geometrically ideal interstitial site size for the tetrahedral interstitial site

For this section, I'd like to try something new and show you the structure first, and then explore and describe its attributes.  Figure 1 shows the CsCl ceramic crystal structure.  

It is likely that it looks familiar.  At first glance, you are likely to ask, "how is this different from the BCC crystal structure?"  This is actually a very good question to ask and it shows that you are making connections between different parts of this course.  It also shows that you are quite familiar with the metallic crystal structures.  Yes, CsCl does look very similar to BCC, in fact, the atom positions are essentially the same.  The only distinction, and this is an important one, is that CsCl represents structures having two atom types: an anion and a cation, while BCC represents a structure in which there is only one atom type.  Of course, from this distinction it then becomes clear that the APF and density will require different calculations, and the relationship between the lattice parameter and the atomic radii must be different.  You'll recall that for BCC, $a=\frac4{\sqrt3}R$, however for CsCl the full cube diagonal would equal $2R_A+2R_C$.  

It's always worthwhile to double check the stoichiometry, as we did for NaCl to discover that we needed a cation at the centre of the unit cell.  The stoichiometry in CsCl is 1:1 and the unit cell has 8 anions in corner positions, each contributing $\frac18$ of an atom for a total of 1 anion.  There is one complete cation at the centre of the unit cell, not cut up.  This gives one anion and one cation, so we are feeling pretty good about ourselves.

If you recognized the similarities between CsCl and BCC it shouldn't be a surprise that the coordination number of the cations in CsCl is eight.  The cation, located at the centre of the unit cell is equidistant from each of the eight corners and makes contact with each of them.

 ${\text{Cation Coordination Number}}_{CsCl}=8$ 

For this section, let's build up the ZnS crystal structure together, as we did for NaCl.  First, we start with anions in "FCC-type" positions, as in Figure 2.  

You will, of course, notice that Figure 2 also has a number of lines drawn on it that are not typically there.  What I have done is sliced the unit cell in half in three planes, the x-z plane, the z-y plane, and the x-y plane.  This yields eight smaller cubes, sometimes I call them sub-cubes.  

These sub-cubes are useful in understanding the ZnS ceramic crystal structure since at the very centre of each of these lies a tetrahedral interstitial site.  One of these interstitial sites is illustrated in Figure 3. 

The tetrahedral interstitial site has a coordination number of four.  That is, an atom located in the centre of this site would touch four nearest neighbour atoms.  However, please remember that interstitial sites are named according to the solid geometry that they create, not their coordination number.  I mention this now, because often the tetrahedral interstitial site creates confusion because it has a coordination number of four and also creates a four sided solid - the tetrahedron.  I have illustrated a tetrahedral interstitial site in Figure 4.

You may be surprised that I have suddenly jumped to checking the stoichiometry without first completing the ZnS structure.  I did this on purpose because I'm hoping that you have identified a problem with what I have described so far.  We have divided the unit cell into eight sub-cubes and have described a tetrahedral interstitial site at the centre of each, however, this would give us 8 cations, while the FCC lattice of anions gives us only 4 anions.  So, the answer to this is that only half of the available tetrahedral interstitial sites are occupied.  Don't fall into the common conceptual trap of believing that every interstitial site somehow must be occupied.  An interstitial site is just a potential space where an atom could be located.  

So, now we are ready to look at the complete ZnS ceramic crystal structure!  

Since only half of the sites will be filled, it would not make sense for all of the interstitial sites, say, at the bottom of the unit cell to be occupied.  There is charge repulsion between like charges and so they will distribute themselves as far apart from one another as possible.  This is illustrated in Figure 4. 

Sometimes, with all of the orange lines drawn, it is difficult to visualize the positions of the occupied interstitial sites.  Several years ago a student showed me a technique that helped her to understand these positions.  

She shaded in the exposed faces of the sub-cubes containing occupied tetrahedral interstitial sites.  I think this is a really nice way to look at this structure so I have re-drawn the ZnS structure in this manner in Figure 5.

Well, after all this talk about the coordination number of the tetrahedral interstitial site, I hope that it will be fairly clear that each cation is coordinated with 4 anions.  

 ${\text{Cation Coordination Number}}_{Zinc\;Sulphide}=4$ 

So far we have seen only ceramic crystal structures with a 1:1 stoichiometric ratio of anions to cations.  How about something new?  

What's more new and fresh than calcium fluorite?  Let's start with the size of the cations relative to the anions.  They are quite large; in fact, the cations occupy the largest of our three interstitial sites: the simple cubic site.  Now, you may well recall that CsCl had cations occupying the simple cubic interstitial sites, and you are correct.  Of course, the difference lies in the stoichiometry.  While CsCl had an equal number of anions and cations, CaF₂ has twice as many anions as cations.  This is achieved in a similar manner to the structure of ZnS.  That is, only half of the available interstitial sites are occupied.  This is illustrated in Figure 6 and Figure 7.  

With all of the coordination directions identified, as they are in Figure 6 the sketch gets very messy and it may be difficult to make sense of what you are looking at.  

Figure 7 attempts to ease this mess by shading in the exposed faces of the sub-cubes containing occupied simple cubic interstitial sites.

It's always good practice to double check our stoichiometry, so let's do that now.  

The cations are fairly straight forward since they are located completely within the unit cell.  There are four cations.  The anions take a little more work and I have illustrated it in partially in Figure 8.  There are eight corners, six faces, twelve edges, and one central anion.

 $\frac188_{(Corners)}+\frac126_{(Faces)}+\frac14{12}_{(Edges)}+1_{(Centre)}=8$ 

You guessed it.  The coordination number for cations in calcium fluorite is 8.

 ${\text{Cation Coordination Number}}_{Calcium\;Fluorite}=8$ 

So far, we have seen three types of interstitial site.  They are, in increasing size order: tetrahedral, octahedral, and simple cubic.  A cation in a ceramic crystal structure will not occupy an interstitial site in which the cation cannot touch the nearest neighbour anions.  Typically, in fact, the cation pushes the anions apart so that the cation is touching the nearest neighbour anions and the anions are not touching one another.

Now that you're well on your way to becoming a veritable expert on octahedral and tetrahedral interstitial sites, let's take a look at another way of understanding them.  

Any time that we stack two close-packed planes we actually create both octahedral and tetrahedral interstitial sites, as shown in Figure 9.  If you look normal to the close-packed planes and peek through the space between three touching atoms you will either see an atom below (green atoms in the figure) or you will see the white color, having peeked also through the green atoms.  When you see the green atom between the three red atoms you are looking into a tetrahedral interstitial site, located exactly in the middle of the red and green planes.  This is because you see a site surrounded by four atoms, or a coordination number of 4.  On the other hand, if you see the white colour between the red atoms this means that you are looking through both the top and bottom planes and into a space with a coordination number of six - an octahedral interstitial site.