In this lecture, we have studied the closest packing of spheres. Layerwise packing of ababab type gives the hexagonal close packing (HCP) arrangement, while abcabc type of layerwise packing gives the cubic closest packing which is equivalent to the face centered cubic (FCC) packing. In both these arrangements, the packing fraction is 74 % and is the highest possible value for spheres. For body centered cubic and simple cubic packing, the packing fraction is smaller. We have also estimated the sizes of the largest spheres (critical radius ratios) that can be accommodated in the various coordination schemes that can be generated with spheres and given examples for systems that fall in each one of these categories. Towards the end of the lecture we have also described in brief the nature of impurities and imperfections in solids and also their role in catalysis and the dynamics of particles in lattices. At high temperatures the lattice develops defects or impurities and the fraction of the defect sites can be calculated using the Boltzmann distribution.
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