1. The problem statement, all variables and given/known data
Show that the ratio of atomic sphere to unit cell volume in HCP (hexagonal close packing) is 0.74.
2. Relevant equations
volumes of spheres, geometry
3. The attempt at a solution
I did the same problem for FCC and BCC and it was fine.
My unit cell structure is that shown below. I labeled the height between the two hexagonal planes by ##h## and the length of the equilateral triangles comprising the hexagon by ##2r##. If we then consider a 1/6 of this structure in the obvious way and orient it suitably so that one of the sides of the lower triangles coincides with the x axis say, then the volume of 1/6 of this structure is $$V = 2 \int_0^{r} \int_0^{\sqrt{3}x} \int_0^h dx dy dz = \sqrt{3}hr^2.$$ Multiply this by 6 to get the whole volume of the unit cell structure shown below.
I would like to try to relate the height of this structure to the radius ##r## of the spheres so that in the ratio, I get cancellation. I am assuming that the three spheres on the middle layer of the structure (labeled B in the sketch) are wholly contained within the structure?
Show that the ratio of atomic sphere to unit cell volume in HCP (hexagonal close packing) is 0.74.
2. Relevant equations
volumes of spheres, geometry
3. The attempt at a solution
I did the same problem for FCC and BCC and it was fine.
My unit cell structure is that shown below. I labeled the height between the two hexagonal planes by ##h## and the length of the equilateral triangles comprising the hexagon by ##2r##. If we then consider a 1/6 of this structure in the obvious way and orient it suitably so that one of the sides of the lower triangles coincides with the x axis say, then the volume of 1/6 of this structure is $$V = 2 \int_0^{r} \int_0^{\sqrt{3}x} \int_0^h dx dy dz = \sqrt{3}hr^2.$$ Multiply this by 6 to get the whole volume of the unit cell structure shown below.
I would like to try to relate the height of this structure to the radius ##r## of the spheres so that in the ratio, I get cancellation. I am assuming that the three spheres on the middle layer of the structure (labeled B in the sketch) are wholly contained within the structure?
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