# Unit cell problem

1. Oct 22, 2014

### astrophysics12

1. The problem statement, all variables and given/known data
The fraction of volume unoccupied in the unit cell of the body centered cubics lattice is?

2. Relevant equations

3. The attempt at a solution
I got the volume occupied by the atoms as (8/3)πr3. I am not sure if it is right. I just assumed that all the atoms were of same element.
I also don't know about the dimensions of the cube.
Can somebody help me? It is from an old entrance exam question paper.

2. Oct 22, 2014

### ehild

What is r in your result, and how did you get that formula?
The problem asks the fraction of volume unoccupied. What do you think it means?

ehild

3. Oct 22, 2014

### CWatters

You should show/explain your working but I believe that's correct (if r is the radius of the atom).

Assume the atoms are spherical and touching the one in the middle. eg so the diagonal of the cube is 4r.

4. Oct 22, 2014

### Staff: Mentor

5. Oct 22, 2014

### astrophysics12

r is the radius of the atom.
Volume unoccupied is the free volume. It is the difference between total volume of cubic cell and the volume occupied by the atoms. Am I right?

6. Oct 22, 2014

### astrophysics12

Thanks. The side should be 4/√3

7. Oct 22, 2014

### BruceW

yes. Although the OP should also check that the corner spheres do not overlap with each other in this case. (Maybe it is intuitively clear to some people, but I need to check these things with pen and paper, to convince myself).

8. Oct 22, 2014

### BruceW

looks good. but there is an 'r' missing from here. Anyway, keep going, you have the unit cell sides and the occupied volume, so you are pretty close to the final answer now.

9. Oct 22, 2014

### ehild

Correct so far.
You need the ratio of the unoccupied volume to the whole volume of the unit cell. You got that the occupied volume is 8pi/3 r3, and the side of the cubic unit cell is 4r/√3.

ehild