Answer
See the answer below.
Work Step by Step
a) fcc, atoms are present in the all faces
b) tetrahedral holes, all of the atoms in "holes" are surrounded by four atoms each.
c) $8\times1/8=1$ in the vertexes, $6\times1/2=3$ in the center, 4 wholly within, 8 in total.
d) Mass: $8\times 28.085\ g/mol\div 6.022\times 10^{23}\ atoms/mol=3.731\times10^{-22}\ g/cell$
Volume: $(543.1\times 10^{-10}\ cm)^3=1.602\times10^{-22}\ cm^3/cell$
Density: $3.731\times10^{-22}\ g/cell\div 1.602\times10^{-22}\ cm^3/cell=2.329\ g/cm^3$
e) Projecting the point of a Si in a tetrahedral hole into the line connecting two Si, one in the corner and one in the face, we get a right triangle whose Face-Hole-Corner angle is half of the internal angle of a tetrahedron and the opposite side is half of the face diagonal:
$sin(109.5^{\circ}/2)=l/2.\sqrt2/2.r$
$r=235.13\ pm$
