How to calculate 2D packing fractions?

In summary, the conversation discusses the calculation of packing fractions for various planes in an FCC structure. The formula for packing fraction is used, but there are some discrepancies due to incorrect assumptions about the cell area. After some discussion and calculations, the correct formula is determined to be pi/4 for the (100) plane, as well as for the (110) and (111) planes using similar reasoning.
  • #1
Talvon
4
0
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 :confused:

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 :)
 
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  • #2
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.
 
  • #3
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 :smile:
 

FAQ: How to calculate 2D packing fractions?

What is 2D packing fraction?

The 2D packing fraction is a measurement of how efficiently particles or objects are arranged in a 2-dimensional space. It is the ratio of the total area covered by the particles to the total area of the space they are contained in.

How do you calculate 2D packing fraction?

The 2D packing fraction can be calculated by dividing the total area covered by the particles by the total area of the space they are contained in. This can be represented by the equation: Packing Fraction = (Total Area Covered by Particles) / (Total Area of Space).

What is the significance of 2D packing fraction in materials science?

2D packing fraction is an important factor in materials science as it can determine the physical properties of materials, such as their strength, density, and porosity. It is also used to study the behavior of particles in different types of materials, such as powders, emulsions, and foams.

How does particle shape affect 2D packing fraction?

The shape of particles can greatly affect 2D packing fraction. For example, regularly shaped particles, such as circles or squares, can achieve higher packing fractions compared to irregularly shaped particles. This is because regular shapes can fit together more closely and efficiently, leaving less empty space in between.

Can 2D packing fraction be greater than 1?

Yes, 2D packing fraction can be greater than 1. This means that the particles are packed in a way that their total covered area is greater than the total area of the space they are contained in. This is possible when particles are arranged in an overlapping or stacked formation, such as in the case of close-packed structures.

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