Atomic Packing Factor of Simple Hexagonal Unit Cell

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SUMMARY

The Atomic Packing Factor (APF) for a simple hexagonal unit cell can be determined by understanding its geometric properties. The unit cell is a right prism with rhombus-shaped top and bottom faces, characterized by angles of 60 and 120 degrees. The relationship between the lattice parameters is defined by the equation c/a = SQRT(8/3), which is crucial for accurate calculations. Additionally, the unit cell contains corner atoms and one fully enclosed atom, which must be accounted for in the APF calculation.

PREREQUISITES
  • Understanding of hexagonal close-packed (HCP) structures
  • Familiarity with unit cell geometry and parameters
  • Knowledge of atomic packing factors
  • Basic skills in geometric calculations
NEXT STEPS
  • Research the geometric properties of hexagonal close-packed (HCP) structures
  • Study the derivation of the c/a ratio in hexagonal unit cells
  • Learn how to calculate the Atomic Packing Factor for different crystal structures
  • Explore visual resources to better understand unit cell configurations
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Students and professionals in materials science, crystallography, and solid-state physics who are studying crystal structures and atomic packing efficiency.

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How would I go about finding the APF for a simple hexagonal unit cell. Which is a rectangle. I know one length is a0(HCP) but I cannot figure out the other side of the rectangle. Also, wouldn't the height be the c?
 
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Okay, there's a few things to keep in mind here :

1. The unit cell is a right prism, the top and bottom faces being rhombuses (of 60 and 120 deg)

2. Besides the corner atoms of this unit cell, there's also an inside atom completely enclosed inside the prism

3. For a simple close-packed hexagonal unit cell, the value of c/a = SQRT(8/3). (This can be proved, if you wish, or used as it is.)

If you're not sure what the unit cell looks like, google it, to find a picture. If you do the calculation with the wrong picture in your head, you'll waste a bunch of time, so make sure you know what the unit cell looks like.
 

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