Max Packing Fraction of Hexagonal Unit Cell Volume w/ Hard Spheres

AI Thread Summary
The discussion focuses on calculating the maximum packing fraction of hard spheres in a hexagonal unit cell. The maximum packing occurs when the height of the cell equals the side length, allowing spheres to touch. It is established that one primitive cell contains one lattice point, and a hexagonal unit cell comprises three primitive cells, resulting in three lattice points. The participant seeks confirmation on this approach to determine the number of lattice points. The conversation emphasizes understanding the geometric relationships within the hexagonal structure for accurate calculations.
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Homework Statement



Calculate the maximum packing fraction of the unit cell volume that can be filled by hard spheres in the Hexagonal structure

Relevant eq: Volume of spheres is number of lattice points multiplied with the maximum volume of one sphere.

The Attempt at a Solution



I know maxium is obtained when c = a i.e when height of the cell is as high as one of the sides in the hexagon. Hence, maximum sphere radius is a/2 (I have shown geometrically that the spheres can touch each other).

Now I am to determine the number of lattice points in this structure, I know that one primitive cell contains totally one lattice point, and a unit cell of a hexagonal structure can be made up by exactly three primitve cells, so the number of lattice points is 3. Is that the correct way to do this?

The rest I can figure out by my self, just are unsure how to determine the number of lattice points.
 

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