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

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SUMMARY

The maximum packing fraction of hard spheres in a hexagonal unit cell is determined by the geometric relationship between the cell dimensions and the sphere radius. When the height of the cell (c) equals the side length (a), the maximum sphere radius is a/2. In a hexagonal structure, a unit cell comprises three primitive cells, resulting in a total of three lattice points. This configuration allows for optimal packing of the spheres within the unit cell.

PREREQUISITES
  • Understanding of hexagonal unit cell geometry
  • Familiarity with lattice points in crystal structures
  • Knowledge of packing fractions in solid-state physics
  • Basic principles of sphere volume calculations
NEXT STEPS
  • Research the calculation of packing fractions in cubic and face-centered cubic structures
  • Explore the geometric derivation of lattice points in various crystal systems
  • Study the implications of packing efficiency on material properties
  • Investigate the role of sphere radius in maximizing packing density
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Students and professionals in materials science, solid-state physics, and crystallography who are focused on optimizing packing arrangements of particles in crystalline structures.

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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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