Understanding HCP Crystal Structure: Confusion with Coordinates of Center Atom

[DollarBill]

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

 

[johng23]

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.

 

[DollarBill]

That definitely cleared it up -- can't believe I didn't get it before. Thanks!