PHYSICS 315
Problem set 1 2004
Solution

2.1-4
The $Na^+Cl^-$ structure is face centered cubic with the oppositely charged ions on the edge centers (see fig 2.6 (a). If $\xi=r_+/r_-$ is small the negative ions will touch and the positive ions will fit in the space between. We then have

$$r_-=a\sqrt(2)/4$$

and the packing fraction is with 4 ions of each kind in each unit cube

when is large the positive and negative ions touch and

or

with the packing fraction

The crossover occurs when the two formulas give the same result or

The resulting packing fraction is plotted in the figure below.
Problem 2.1-5
The structure is simple cubic with the ions of opposite charge in the body centers. Again, it is small the negative ions will touch and we now have and the packing fraction in this regime is with 1 negative ion/unit cube

For large the positive and negative ions touch and

The packing fraction is then given by

The resulting packing fraction is plotted below

Problem 2.1-8 The primitive lattic vectors can be written

Two of the angles are while the third is different. The lenghts of the 3 primitive vectors are different. This makes the lattice monoclinic. [Note: There is an error in figure 2.9 and table 2.1. Base centred monoclinic should be body centered monoclinic. There is no difference between base centered and simple monoclinic. The same error can be founfd in the Books by KIttel andOmar quoted in the references.