Calculating c/a ratio for h.c.p

[mate1000]

iv been asked to show that the ideal c/a ratio for a hexagonal close packing unit cell is 1.633. The only thing i could come up with was to do with the atomic packing factor. where i get (4/3pi^2r*6) / (3a*2r*c). Am i heading in the right direction or am i way off course. If I am way off would anyone be able to give me a hand?
Cheers

 

[Gokul43201]

The packing factor isn't useful. This is simply a question of 3d geometry. Draw a picture of the HCP unit cell with spherical atoms and start there.

Hint: What do you know about the position of the centroid of an equilateral triangle?

 

[charng]

Your approach using the atomic packing factor is a good start. The c/a ratio in a hexagonal close packing structure can be calculated by considering the coordination number of the atoms. In a hexagonal close packing structure, each atom is surrounded by 12 nearest neighbors in the same plane and 6 nearest neighbors in the adjacent planes. This gives a total coordination number of 18.

The atomic packing factor is defined as the total volume occupied by the atoms in a unit cell divided by the total volume of the unit cell. In a hexagonal close packing structure, the volume occupied by the atoms can be calculated as (18 x volume of one atom). The total volume of the unit cell can be calculated as (a^2c/2), where a is the length of the side of the hexagon and c is the height of the unit cell.

Putting these values into the atomic packing factor equation, we get:

Atomic packing factor = (18 x volume of one atom) / (a^2c/2)

We can also express the volume of one atom in terms of its radius, r, as (4/3 x pi x r^3). Substituting this into the equation, we get:

Atomic packing factor = (18 x 4/3 x pi x r^3) / (a^2c/2)

Now, we know that the ideal atomic packing factor for a hexagonal close packing structure is 0.74. So, we can set this value equal to the atomic packing factor equation and solve for the c/a ratio:

0.74 = (18 x 4/3 x pi x r^3) / (a^2c/2)

Rearranging, we get:

c/a = (18 x 4/3 x pi x r^3 x 2) / (a^2 x 0.74)

Simplifying, we get:

c/a = 1.633 x (r/a)^3

Since the radius of an atom is typically much smaller than the length of the side of the hexagon, we can approximate (r/a)^3 to be very close to 0. So, we can conclude that the ideal c/a ratio for a hexagonal close packing structure is approximately 1.633.

I hope this helps clarify the concept for you. Keep up the good work!