Atomic Packing Factor of Simple Hexagonal Unit Cell

[forstajh]

How would I go about finding the APF for a simple hexagonal unit cell. Which is a rectangle. I know one length is a0(HCP) but I cannot figure out the other side of the rectangle. Also, wouldn't the height be the c?

 

[Gokul43201]

Okay, there's a few things to keep in mind here :

1. The unit cell is a right prism, the top and bottom faces being rhombuses (of 60 and 120 deg)

2. Besides the corner atoms of this unit cell, there's also an inside atom completely enclosed inside the prism

3. For a simple close-packed hexagonal unit cell, the value of c/a = SQRT(8/3). (This can be proved, if you wish, or used as it is.)

If you're not sure what the unit cell looks like, google it, to find a picture. If you do the calculation with the wrong picture in your head, you'll waste a bunch of time, so make sure you know what the unit cell looks like.

 

[ravenprp]

The atomic packing factor (APF) is defined as the fraction of the total volume occupied by atoms in a unit cell. For a simple hexagonal unit cell, the APF can be calculated by dividing the total volume of atoms by the total volume of the unit cell.

To find the APF, you will need to know the dimensions of the unit cell. In this case, you have been given that one side of the unit cell is a0, which is the lattice parameter for a hexagonal close-packed (HCP) structure. This means that the length of each side of the hexagonal unit cell is also equal to a0.

To determine the other side of the rectangle, you can use the fact that the hexagonal unit cell is composed of two identical triangles. The height of each triangle is equal to the radius of the atom, which is half of the lattice parameter (a0/2). Therefore, the height of the rectangle is equal to the height of one triangle, or a0/2.

The length of the rectangle can be determined using the Pythagorean theorem. Since the side length of the unit cell is a0, the length of the rectangle can be calculated as follows:

Length of rectangle = √(a0^2 - (a0/2)^2)

Now that you have both dimensions of the rectangle, you can calculate the total volume of atoms in the unit cell. This can be done by multiplying the number of atoms in the unit cell by the volume of each atom. For a simple hexagonal unit cell, there are six atoms, each with a volume of (4/3)πr^3, where r is the radius of the atom (a0/2). Therefore, the total volume of atoms in the unit cell is given by:

Volume of atoms = 6 x (4/3)π(a0/2)^3

Finally, to calculate the APF, you need to divide the volume of atoms by the total volume of the unit cell, which is simply the length and height of the rectangle multiplied together. This gives us the following equation:

APF = (6 x (4/3)π(a0/2)^3) / (a0 x (a0/2))

Simplifying this equation, we get:

APF = (6 x (4/3)π(a0/2)^3) / (a0^2/2)

APF