Show that the ideal c/a ratio (height of unit cell divided by edge length) for the HCP unit cell is 1.633. (You may wish to refer to Exercise E.2 in the text page GL 1-6) Comment on the fact that real HCP metals display c/a ratios varying from 1.58 (for Be) to 1.89 (Cd). The atomic radius of HCP Mg is 0.1605 nm. Find the lattice constants, c and a, the c/a ratio and theoretical density for Mg.

The structure in the first image file attached below shows the arrangement of atoms in hexagonal close packing

we need to show that the ratio between the height of the unit cell divided by its edge length is 1.633

In the structure, the two atoms are shown apart. But in fact the two atoms are touching. Therefore, the edge length is the sum of the radius of two atoms. If we assume r as radius then the expression for edge length will be as follows.

a = 2r    ..............(1)

other attached files show additional solutions in steps

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