Consider a hexagonal close-packed unit cell as shown here. How many corner atoms

Question

Consider a hexagonal close-packed unit cell as shown here. How many corner atoms (orange) are shown in this image? What fraction of each comer atom is inside the boundaries of the cell? How many face atoms (red) are shown in this image? What fraction of each face atom is inside the boundaries of the cell? Additionally there are 3 blue atoms that are 100% inside the boundaries of the unit cell. If you sum all the whole and fractions of atoms, how many atoms are actually inside a hexagonal close-packed unit cell?

Explanation / Answer

No of corner atoms: 12

fraction of each corner atom inside the one unit cell : 1/6 (because one corner atom is shared by 6 unit cells)

No of face atoms: 2

fraction of each face atom inside the one unit cell: 1/2 (because one face atom is shared by 2 unit cells)

No of atoms in one cell: (1/6)*12 + (1/2)*2 + 1*3 = 6

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