Lattice Structures $(15$ P$)$

The unit cell of an important metallic lattice structure is shown below.

a$)$ What does a "unit cell" generally describe $?(2$ P$)$

b$)$ Which two different names are commonly used for the shown structure $?(2$ P$)$

c$)$ What is meant by the $$APF$$ of a lattice structure $?$ Which of the two above

shown visual displays of the lattice is better suited for a calculation of APF $?$

Briefly explain. $(4$ P$)$

d$)$ What is the mathematical relation between atomic radius $($r$)$ and edge length

of the unit cell $($a$)$ in above shown structure $?(2$ P$)$

e$)$ Please calculate APF of the shown lattice structure. $(5$ P$)$

f$)$ Bonus question: Do you see a possibility to increase APF of above shown lattice $?$ Briefly explain.