# Packing efficiency of particles in solids-space lattices

I had to do an experiment in which I built examples of hexagonal closest packing, face-centered closest packing and a body-centered space lattice. I had to find the volume of a box that would fit tightly around them, and then calculate the density in units/cm3 (assuming a mass of 1 unit per sphere).

I know the packing efficiencies are 74%, 74%, and 68% respectively (based on online research).

However, my results do not remotely reflect those numbers. There are 13 units for both hexagonal and face-centered, but the size of my "imaginary box" around them is different (and there is no possible way they can be the same). I assume I'm supposed to get the same density for those, but that is not possible when you have the same number of units but a different size of box. (Body-centered has 9 units...and according to my measurements has a greater density than hexagonal- and I measured REPEATEDLY to check).

I have spent 2 days trying to figure this out and it is getting frustrating. Any guidance would be greatly appreciated!!!

## The Attempt at a Solution

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Mapes
Homework Helper
Gold Member
There's going to be an error if you want to fit a box around the spheres, because the actual (well, imaginary) unit cell enclosure passes through the center of the atoms. The error will be smaller the more spheres/atoms you consider. Does this answer your question?

So I'm getting the wrong answers because the experiment is poorly designed, not because I am doing it wrong?

I thought it would make more sense to find the total volume of the spheres, then divide it by the volume of the "box".

I just don't know how I am supposed to answer the questions for the lab based on information that is incorrect. For example, it asks "What is the relationship between coordination number, packing, and density (as mass/volume)?"

And even "Which type of packing has the least efficient arrangement of atoms? Justify your answer." Based on the densities I obtained, hexagonal closest packing is the least efficient...however I know for a fact that is incorrect.

I'm sure I can figure out some fake numbers in order to make the results work as they should. I just don't understand how this experiment is supposed to work.

Mapes
Homework Helper
Gold Member
I find it hard to believe that your estimate of bcc packing fraction, if calculated correctly, is higher than that of hcp packing fraction. Can you describe your calculations?

Also, every experiment is an approximation. Again, the estimate will converge to the correct value as the number of spheres is increased.

These are my original calculations:

HCP:
13 units: 3 layers- 3 in bottom layer, 7 in middle layer, 3 in top layer
Volume of "box" that fits tightly around the layers: l x w x h = 13 x 14.5 x 12.5 = 2356.25
Density in units/cm3: 13/2356.25 = 0.005517

FCC:
13 units: 3 layers- 4 in bottom layer, 5 in middle layer, 4 in top layer
Volume of "box" that fits tightly around the layers: 11.5 x 11.5 x 11.5 = 1520.88
Density in units/cm3: 13/1520.88 = 0.008548

BC:
9 units: 3 layers- 4 units in bottom (slight spaces btw them), 1 in middle, top layer is the same as the bottom layer
Volume of "box" that fits tightly around the layers: 10.1 x 10.1 x 10.9 = 1111.91
Density in units/cm3: 9/1111.91 = 0.008094

Mapes
Homework Helper
Gold Member
OK, got it. This is a very small number of spheres. Try repeating the calculation with ten times as many, then a hundred times as many. You should see the answers converge to the numbers you found online.

I understand what you mean. But is there any way I can make this experiment work doing it exactly as directed? It is a correspondence course, so I have no one to go to about it. At this point I am about to make up fake data to make the numbers work for me.

It even says you can use marshmallows instead of balls- anything "reasonably spherical" in shape. I'm sure it would be quite interesting to see what kind of results are found using marshmallows...which are cylindrical!

Mapes