How to calculate 2D packing fractions?

[Talvon]

I have a question regarding various planes in an FCC, and determine their packing fractions.

I searched but couldn't find anything :)

For example, one of the planes is the (100) plane, and I have said there are 2 full atoms (1 in the middle, and 4 quarters from each side), distance 'a' apart. Using the usual packing fraction equation ((Volume of atom x number of atoms)/Volume of cell), but replacing the volume with area, I calculate this to be pi/2, which is wrong because it can't be higher than one

My exact numbers were:
Area of a circle = pi x r², where r=a/2
-> (pi x a²/4) x 2 (-# of atoms) / a²
The a² cancels, as does the 4 with the 2 to leave pi over 2.

Any help would be appreciated, cheers :)

 

[Cthugha]

Why do you assume the area of the cell to be a²?

If you have a look at the diagonal line of your cell area, you get the maximum packing fraction if the atoms exactly touch each other. So the diagonal line must have a length of:
a (full atom) + 2* a/2 (quarter of 2 other atoms) =2 a.

So using Pythagoras you will get that the side lines of your area will have a length of sqrt(2) a, which leads to a cell area of 2 a².

 

[Talvon]

Ace, cheers :D Didn't make sense initially but then it clicked :P

Managed to work it through to get pi/4, and managed to solve the (110) and (111) plane in a similar way