Solids consist of a crystalline lattice of atoms-a unit cell that has a certain configuration of atoms that is repeated over and over. The picture that I can't post here, shows a pyramidal structure of metal spheres. The base is 8 spheres by 8 spheres with a height of 8 spheres. The metals spheres represent a lattice configuration called face centered cubic (fcc). Calculate the packing fraction for this case, e.g., the amount of volume occupied by the metal spheres divided by the total volume of the pyramidal structure.
I have no idea how to figure out or approach this problem. I did my best with what I have below. Please show me how to figure it out and walk me through it. I am just returning to math from a 15 year absence. I need to see how to walk through it and the answer in order for it to click.
The Attempt at a Solution
Let a be the A the side length of the unit cell of FCC lattice and R the diameter of the atoms.
The FCC unit cell is formed by 8 atoms:
- 8 times one eighth of an atom at the corners of the cube
- 4 times a half of an atom at the center of the of the faces.
At the faces the atoms at the corners and the center atom touch, so that the perfectly fill the face. Hence the length of the face diagonal is
D = R + 2R + R = 8R
From Pythagorean theorem you get
A² + A² = D²
A = √8 · R = √2 · 2·R
The volume of the cube cell is
Vc = A³ = √2 · 16·R
The volume of the atoms in the cell is
Va = 8 · (8·π·R³ /3) = 64·π·R³ /3
The packing density is
p = Va / Vc
= (64·π·R³ /3) / (√2 · 64·R)
= π / (3·√2)