Crystal Lattice Structures:	Reference Date:  1 Jan 2000
	Last Modified: 21 Oct 2004

Corundum (D5₁)

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• Although corundum has a rhombohedral primitive unit cell, Pearson's Handbook and other references use the equivalent hexagonal lattice, which contains 3 primitive cells. Since we're more interested in the primitive cell we'll use the rhombohedral structure.
• Pearson's Handbook lists a_hcp = 4.758 Å and c_hcp = 12.99Å. In terms of the lattice below, this translates to a = 3.621Å and b = 0.257 Å, with an angle between the vectors of 55.3^o. For the internal parameters, in terms of the hexagonal lattice, x_Al(hex) = 0.35228 and x_O(hex) = 0.3064. On the rhombohedral lattice this translates to z₁ = x_Al(hex) = 0.35228 and x₁ = ¼ - x_O(hex) = -.0564.
• If the angle between the primitive vectors was 60^o instead of 55.3^o, the primitive vectors would be fcc. This occurs when b = 0, below.

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• Prototype: αAl₂O₃
• Pearson Symbol: hR10
• Strukturbericht Designation: D5₁
• Space Group: R3c (Cartesian and lattice coordinate listings available)
• Number: 167
• Reference: Villars and Calvert, Pearson's Handbook, Vol. I, pp. 970-1
• Primitive Vectors:

  A1	=	b X + a Y + a Z
  A2	=	a X + b Y + a Z
  A3	=	a X + a Y + b Z

• Basis Vectors:

  B1	=	+ z1 (A1 + A2 + A3)	=	+ z1 (2a + b) (X + Y + Z)	(Al)	(4c)
  B2	=	- z1 (A1 + A2 + A3)	=	- z1 (2a + b) (X + Y + Z)	(Al)	(4c)
  B3	=	+ (½ + z1) (A1 + A2 + A3)	=	+ (½ + z1) (2a + b) (X + Y + Z)	(Al)	(4c)
  B4	=	- (½ + z1) (A1 + A2 + A3)	=	- (½ + z1) (2a + b) (X + Y + Z)	(Al)	(4c)
  B5	=	+ x1 A1 + (½ -x1) A2 + ¼ A3	=	+ [(¾ - x1) a + x1 b] X + [(¼ + x1) a + (½ - x1) b] Y + [½ a + ¼ b] Z	(O)	(6e)
  B6	=	- x1 A1 - (½ -x1) A2 - ¼ A3	=	- [(¾ - x1) a + x1 b] X - [(¼ + x1) a + (½ - x1) b] Y - [½ a + ¼ b] Z	(O)	(6e)
  B7	=	+ (½ -x1) A1 + ¼ A2 + x1 A3	=	+ [(¼ + x1) a + (½ - x1) b] X + [½ a + ¼ b] Y + [(¾ - x1) a + x1 b] Z	(O)	(6e)
  B8	=	- (½ -x1) A1 - ¼ A2 - x1 A3	=	- [(¼ + x1) a + (½ - x1) b] X - [½ a + ¼ b] Y - [(¾ - x1) a + x1 b] Z	(O)	(6e)
  B9	=	+ ¼ A1 + x1 A2 + (½ -x1) A3	=	+ [½ a + ¼ b] X + [(¾ - x1) a + x1 b] Y + [(¼ + x1) a + (½ - x1) b] Z	(O)	(6e)
  B10	=	- ¼ A1 - x1 A2 - (½ -x1) A3	=	- [½ a + ¼ b] X - [(¾ - x1) a + x1 b] Y - [(¼ + x1) a + (½ - x1) b] Z	(O)	(6e)

See these vectors as LaTeX output.

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Structures indexed by: Strukturbericht Symbol Pearson Symbol Space Group Prototype

This is a mirror of an old page created at the Naval Research Laboratory Center for Computational Materials Science The maintained successor is hosted at http://www.aflowlib.org/CrystalDatabase/ and published as M. Mehl et al., Comput. Mater. Sci. 136 (Supp.), S1-S828 (2017).