There are often several choices for the basis vectors of a primitive cell (defined in Problem 4). Let $a$ be the length of one edge of a bec Bravais lattice.

(a) One set of possible basis vectors for the bec primitive cell is $$\begin{array}{cc}a& =a\hat{i}\\ b& =a\hat{j}\\ c& =\left(\frac{a}{2}\right)(\hat{i}+\hat{j}+\hat{k}).\end{array}$$
What is the volume of the primitive cell described by these basis vectors?

(b) Another possible set of basis vectors for the bec primitive cell is $$\begin{array}{c}a=\left(\frac{a}{2}\right)(-\hat{i}+\hat{j}+\hat{k})\\ b=\left(\frac{a}{2}\right)(\hat{i}-\hat{j}+\hat{k})\\ c=\left(\frac{a}{2}\right)(\hat{i}+\hat{j}-\hat{k}).\end{array}$$
What is the volume of the primitive cell described by these basis vectors?

Data from Problem 4

The primitive cell of a Bravais lattice is the smallest volume that can translate to fill its Bravais lattice completely. A primitive cell has only one lattice site (counting the sites shared with neighboring lattices).