Assuming that atoms are spherical, calculate the fraction of space which is occupied by...

Question

...atoms(the packing efficiency) in a metal with a face centered cubic unit cell.

Please help! I don't even know where to start! Thank you in advance

Answer 1

im not even going to attempt

Answer 2

go look up face centered cubic.  (first link)

next look up # atoms per unit volume.

there are 2 keys.

1) counting the atoms in the block.  on top, there is 1/2 plus four  1/8 s.  or,  if you put all the pieced on top together, you could make 1 sphere.   in the middle, there are 4 halves, one on each side,  or 2 spheres.  the bottom is the same as the top, so there's 1 there also.  which makes 1+2+1=4 all together.

2) calculating the volume.  for which you need to know a side.  but what you do know is the diagonal across the top.  from which, if you remember good old Pythagoras, you can calculate the length of a side.

so ==========

look at the pic.

there are 4 atoms.

if radius=1, then the total volume = 4(4/3*pi*1^3) = 4(4pi/3) or

(4 * 4 * pi) / 3 = 16.7551608

look at the pic again.

the diagonal across the top = 2 diameters = 4 radii.

so 1 edge = (4^2/2)^0.5  or

((4^2) / 2)^0.5 = 2.82842712

and the volume = ((4^2/2)^0.5)^3 or

(((4^2) / 2)^0.5)^3 = 22.627417

and the packing factor is

16.7551608 / 22.627417 = 0.740480489

alternatively calculated as

(4 * ((4 * pi) / 3)) / ((((4^2) / 2)^0.5)^3) = 0.74048049

and then, you could check the answer.  (2nd link)