For compound (A)_n(B)_m we can expect ionic bonding to predominate when atom A has low electronegativity and atom B has a high electronegativity.  In this case electron transfer from one atom to another leads to the formation of A⁺B^-.  For the main group elements the electron transfer continues until the ions have closed shell configurations.

For ionic compounds the bonding forces are electrostatic and therefore omni-directional.  The bonding forces should be maximized by packing as many cations around each anion, and as many cations around each anion as is possible.  The number of nearest neighbor ions of opposite charge is called the coordination number.  We must realize however that the coordination numbers are constrained by the stoichiometry of the compound and by the sizes of the atoms.

e.g. For sodium chloride, Na⁺Cl^-, there are 6 anions around each cation (coordination number Na = 6); because of the 1:1 stoichiometry there must also be 6 Na cations around each Cl anion.  For Zr⁴⁺O^2-₂ there are 8 anions around each cation, therefore there must be only 4 cations around each anion.

Simple ionic crystal structures can be approached in terms of the close packing procedures developed for metallic structures.  In most (but by no means all) ionic compounds the anions are larger than the cations.  In these cases it is possible to visualize the structures in terms of a close packed arrangement of the larger anions, with the cations occupying the vacant interstices between  the close packed layers.  Recall that although  ccp & hcp are the most efficient ways of packing spheres, only 74% of the available space is filled, the 26% "free space" is in the form of different types of holes or sites which can be occupied by the smaller cations in the ionic structures .

First let us consider the types of holes available in a close packed anion arrangement.

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Types of cations sites available in close packed anion arrays.

As shown below, the stacking of two close packed anion layers produces 2 types of voids or holes.  One set of holes are octahedrally coordinated by 6 anions, the second set are tetrahedrally coordinated by 4 anions.  One octahedral site and two tetrahedral sites are created by each anion in the close packed layer.

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"Stuffing" of the holes by the cations.

Having determined what types of holes are available we must now decide:

(a) Which sites are occupied by a given cation. This determined by the radius ratio (= r_cation/r_anion)
(b) How many sites are occupied.  This is determined by the stoichiometry.

Which sites ?: Radius Ratio rules.

The relative sizes of the anions and cations required for a perfect fit of the cation into the octahedral sites in a close packed anion  array can be determined by simple geometry:
 

Similarly for a perfect fit of a cation into the tetrahedral sites it can be shown that r_cation/r_anion = 0.225.

For these two "ideal fit" radius ratios the anions remain close-packed.
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Stable Bonding Configurations in Ionic solids.

In reality an ideal fit of a cation into the close packed anion arrangement  almost never occurs .  Now consider what would be the consequence of placing a cation that is (a) larger than the ideal, (b) smaller than the ideal, into the cation sites.
 
 

For a stable coordination the bonded cation and anion must be in contact with each other.

If the cation is larger than the ideal radius ratio value the cation and anion remain in contact, however the cation forces the anions apart.  This is not a problem as there is no need for the anions to remain in contact.

If the cation is too small for the site then the cation would "rattle" and would not be in contact with the surrounding anions.  This is an unstable bonding configuration.

Note however in a few rare cases solids do contain cations that are too small for their sites, in these cases the cation moves off the center of the site and adopt a distorted octahedral coordination.  These solids typically exhibit  novel properties, such as, for example, ferroelectricity and piezoelectricity.
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Summary, Radius ratio rules for close packed anion structures.
 

Now we know how to determine which sites will be filled, we place the appropriate number of cations into the structure, making sure that we observe the correct stoichiometry.  The figure below shows a view of the octahedral and tetrahedral interstices that are available in the fcc cell of a ccp anion arrangement .   By filling these to differing degrees a number of very common types of crystal structures can be produced.
 
 

Circles labeled O represent centers of the octahedral interstices in the ccp arrangement of anions (fcc unit cell).  The cell "owns" 4 octahedral sites.	Circles labeled T represent the centers of the tetrahedral interstices in the ccp arrangement of anions.  The cell "owns" 8 tetrahedral sites.

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The table below shows several structure types that result from different site filling schemes.  We will focus on two of them: the rocksalt structure and the ZnS structure.
 

Anions ccp (fcc). Radius Na⁺ = 1.02Å, radius Cl^- = 1.81Å; radius ratio = 0.563.

Therefore Na octahedral.

1 octahedral / anion therefore 100% octahedral sites are filled.

Coordination # Na = 6; coordination # Cl = 6.
 

The figure below shows the filling of the octahedral sites by Na (green) within the ccp (ABCABC) anion (red) array.	This figure shows the face centered cubic unit cell of the NaCl structure.

Note that there is an fcc arrangement of the Na cations and Cl anions.
We can make sense of the cell contents using our knowledge of lattices.  For an fcc lattice there are 4 lattice points per cell, the motif in this case is a Na cation and a Cl anion.  Therefore the cell contents are 4 Na cations + 4 Cl anions.

Lattice parameter and Density calculations.

The cation and its nearest neighbor anions are, by definition in contact.
Therefore the cell edge  = 2r(Na⁺) + 2r(Cl^-) = 2(1.02) + 2(1.81) = 5.66Å

Knowing the size of the cell and its contents, the density of solid NaCl can be calculated using the same procedures used earlier for the metallic structures.

Other examples of solids with the rocksalt structure:

Many halides and oxides have this structure.  Examples include: LiF, LiCl,LiBr,LiI,NaBr,NaI, NaF, KF, KCl,KBr,KI,RbF,RbCl,MgO,CaO,SrO,BaO,TiO,VO,MnO,FeO,CoO, NiO,CdO, etc.
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Zincblende (Zinc sulfide, ZnS) structure.

Anions ccp (fcc). Radius Zn²⁺ = 0.6Å, radius S^2-  = 1.84Å; radius ratio = 0.33 Æ Zn tetrahedral.

Have 2 tetrahedral  sites/ anion, therefore from formula of ZnS only 50% of the tetrahedral sites can be filled.
Coordination # Zn = 4; coordination # S = 4.

Which sites are filled ?: see picture below.
Note the filling of diagonally opposite sites to maximize the cation-cation separations .

Again the lattice is fcc, the motif consists of 1 S and 1 Zn.
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It is also possible to follow the simple methods used above to fill the octahedral and tetrahedral sites in an hcp (ABAB) array of anions.  Again many common structure types can be generated.  These will not be considered in this class.
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In these structures the anions are  not close packed, but occupy just the corners of a cube.  In this case the center of the cube (surrounded by 8 anions) can be occupied by a suitably sized cation.  This site is larger than the tetrahedral or octahedral positions in the close packed structures.  The radius ratio for a perfect fit of a cation in a cubic site can again be calculated using simple geometry and is 0.73.  This structure should therefore be adopted when the r_cation/r_anion is equal to or greater than 0.73.
 
By including the possibility of cubic coordination we can now complete our table for predicting cation coordinations from the radius ratio rules:

CsCl: radius Cs⁺ = 1.74Å, radius Cl^- = 1.81Å: radius ratio = 0.96 Æ predict cubic coordination.
All cubic sites are filled by Cs cations.
Coordination numbers: Cs = 8; Cl = 8.
Note Cs and Cl are in contact along the body diagonal.

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One cubic site per F anion; from stoichiometry only 50% cubic sites filled by Ca cations.
Arrangement of the filled cubic sites is such that the Ca-Ca distances are as large as possible (compare the Ca distribution to that of Zn in ZnS)
Coordination numbers: Ca²⁺ surrounded by 8 F^- 's; F^- surrounded by 4 Ca²⁺'s.
Other examples: ZrO₂,

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