Understanding HCP Crystal Structure: Confusion with Coordinates of Center Atom

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SUMMARY

The discussion clarifies the coordinates of the center atom in a hexagonal close-packed (HCP) crystal structure, specifically addressing the confusion surrounding the coordinates (1/3, 2/3, 1/2) in relation to the unit cell defined by vectors (a1, a2, c). The origin's placement is crucial for understanding these coordinates, which derive from geometric principles involving equilateral triangles. The participants also note that the choice of unit cell orientation can affect the coordinate representation, but this is a matter of convention rather than a fundamental difference.

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  • Understanding of hexagonal close-packed (HCP) crystal structures
  • Familiarity with unit cell concepts in crystallography
  • Basic knowledge of trigonometry and geometric principles
  • Ability to interpret crystal coordinate systems
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  • Research the geometric properties of hexagonal close-packed (HCP) structures
  • Study the derivation of coordinates in crystallography
  • Explore the significance of unit cell orientation in crystal structures
  • Learn about the role of symmetry in crystal lattice configurations
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Students and professionals in materials science, crystallography, and solid-state physics who seek to deepen their understanding of crystal structures and their coordinate systems.

DollarBill
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Considering only the 4 lower-right atoms (one unit cell), I'm somewhat confused as to why the coordinates for the center atom (b) is (1/3, 2/3, 1/2) with respect to (a1, a2, c).

c is pretty self explanatory, but I don't quite get the other 2. I think one of my main confusions is where the origin is being taken when considering just one unit cell.

[PLAIN]http://img856.imageshack.us/img856/2272/hexagonalclosepackedstr.gif

Thanks
 
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hcpcoord.png


The triangle drawn has side length a. From simple trigonometry you can find that the center of an equilateral triangle is 1/3 of the way from a side along the perpendicular. The total height is a*cos(30). Then x and y give you the 2/3*a and 1/3*a you are looking for. Actually I chose the unit cell opposite to how you did. So mine is 2/3, 1/3, 1/2 instead of 1/3, 2/3, 1/2. But that's just convention.
 
That definitely cleared it up -- can't believe I didn't get it before. Thanks!
 

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