The reason crystals form is the attraction between the atoms. Because they attract one another it is often favorable to have many neighbors. Thus, the coordination number, or number of adjacent atoms, is important.


For a square lattice as shown, the coordination number is 4 (the number of circles touching any individual).  The coordination geometry is square, as shown below.


The empty spaces between the atoms are interstitial sites. Since each is touched by 4 circles, the interstitial sites are also 4-coordinate, and the geometry is also square.  In this example there are the same number of interstitial sites as circles.

Also because the atoms attract one another, there is a tendency to squeeze out as much empty space as possible. The packing efficiency is the fraction of the crystal (or unit cell) actually occupied by the atoms. It must always be less than 100% because it is impossible to pack spheres (atoms are usually spherical) without having some empty space between them.

 P.E. = (area of circle) / (area of unit cell)

Regarding our square lattice of circles, we can calculate the packing efficiency (PE) for this particular lattice as follows:

 P.E. =  pi r2 / (2r)2 = pi/4 = 78.54%.

The interstitial sites must occupy 100% - 78.54% = 21.46%.

 Let us compare this with a hexagonal lattice of circles. We note that :

By examining the diagram you can see that in this packing there are twice as many of  these 3-coordinate interstitial sites as circles - for each circle there is one pointing left and another pointing right.  Thus, these sites are much smaller than those in the square lattice.

The larger coordination number (more bonds) and greater packing efficiency suggest that this would be a more stable lattice than the square one.

If we compare the square and hexagonal lattices, we see that both are made of columns of circles. However, in the hexagonal lattice every other column is shifted allowing the circles to nestle into the empty spaces. Thus, the higher packing efficiency.
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Structure of Crystals
Crystal Lattices
Unit Cells
From Unit Cell to Lattice
From Lattice to Unit Cell
Packing & Geometry
Simple Cubic Metals
Close Packed Structures
Body Centered Cubic
Cesium Chloride
Sodium Chloride
Rhenium Oxide
Niobium Oxide
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