Please do all $8$ questions se do all $8$ questions

C$.$ Face$-$Centered Cubic Unit Cell

The face$-$centered cublc unit cell consists of a simple cubic unit cell, with an additional atom at the center of each of the six faces. Aluminum is an example with this type of cell.

Build a model of a single face$-$centered cubic unit cell by placing identical marbles at the corners and in the centers of each face of the plexiglass framework. On the diagram below, list the coordinates of the atom in the center of each of the faces. $($The comer atoms will have the same coordinates as for the simple cubic unit cell in Fig. $1)$

Fig. $4$ Coordinates of face$-$centered cubic unit cell

What fraction of the volume of an atom in the center of a face is contained in a single face$-$centered cubic cell?

What is the NET number of atoms per cell, assuming atoms on both the comers and faces are shared with other unit cells?

What is the coordination number for an atom in a face$-$centered cubic unit cell?

$($A larger model is available for your inspection.$)$ is the coordination number the same tor face atoms as for corner atoms?

In this type of unit cell, atoms along a face diagonal $(d_f)$ touch. What is the mathematical relationship between the radius of the atoms $(r)$ and the unit cell edge $(a_0)?$

$($SHOW WORK$)$

What is the total volume $($in units of $r)$ of the atoms in one of these cells

What is the volume of the face$-$centered unit cell in units of $r?$

What fraction of the total cell volume is occupied by atoms? $($SHOW WORK