Crystal Lattice Structures:	Creation Date: 25 Aug 2002
	Last Modified: 21 Oct 2004

The Paraelectric LiNbO₃ Structure

You can now

• see the structure from several perspectives;
• examine the structure with an external viewer; or
• download the coordinates of the atoms in these pictures in XYZ format.
• visualize the structure (Uses the JMOL Applet)

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• This is the paraelectric phase, which exists above 1430K. There is also a ferroelectric phase.
• We take the positions from the article by Boysen and Altorfer, along with the supporting Acta Cryst. database. Since the position of the Li atoms, particularly, is uncertain, we use the first set of coordinates in the database.
• Note that we use the hexagonal lattice coordinates a and c in a rhombohedral setting.
• Our rhombohedral position coordinates for the oxygen atoms are related to the hexagonal coordinates given in the literature by first transforming the coordinates by a primitive vector of the rhombohedral lattice:
  • x'_hex = x_hex + 1/3
  • y'_hex = y_hex - 1/3
  • z'_hex = z_hex + 2/3
  and then transforming to rhombohedral coordinates:
  • x_rhombohedral = x'_hex - 3^-½ y'_hex + z'_hex
  • y_rhombohedral = 2 (3^-½) y'_hex + z'_hex
  • z_rhombohedral = - x'_hex - 3^-½ y'_hex + z'_hex
• This reduces to a double unit cell version of the cubic perovskite structure in the special case:
  • c/a = 6^½, so that the angle between the primitive vectors is 60°. (In LiNbO₃ the angle is about 57°.)
  • x = ½

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• Prototype: LiNbO₃ (paraelectric)
• Pearson Symbol: hR10
• Space Group: R3c (Cartesian and lattice coordinate listings available)
• Number: 167
• Reference: H. Boysen and F. Altorfer, Acta Cryst. B 50, 405-414 (1994).
• Primitive Vectors:

  A1	=	+ ½ a X - 12-½ a Y + c/3 Z
  A2	=	3-½ a Y + c/3 Z
  A1	=	- ½ a X - 12-½ a Y + c/3 Z

• Basis Vectors:

  B1	=	+ ¼ A1 + ¼ A2 + ¼ A3	=	¼ c Z	(Li)	(2a)
  B2	=	+ ¾ A1 + ¾ A2 + ¾ A3	=	¾ c Z	(Li)	(2a)
  B3	=		0		(Nb)	(2b)
  B4	=	+ ½ A1 + ½ A2 + ½ A3	=	½ c Z	(Nb)	(2b)
  B5	=	+ x A1 + (½ - x) A2 + ¼ A3	=	+ (x - 1/8) a X + ½ 3½ (¼ - x) a Y + ¼ c Z	(O)	(6e)
  B6	=	+ ¼ A1 + x A2 + (½ - x) A3	=	+ ½ (x - ¼) a X + ½ 3½ (x - ¼) a Y + ¼ c Z	(O)	(6e)
  B7	=	+ (½ - x) A1 + ¼ A2 + x A3	=	+ (¼ - x) a X + ¼ c Z	(O)	(6e)
  B8	=	- x A1 - (½ - x) A2 - ¼ A3	=	- (x - 1/8) a X - ½ 3½ (¼ - x) a Y - ¼ c Z	(O)	(6e)
  B9	=	- ¼ A1 - x A2 - (½ - x) A3	=	- ½ (x - ¼) a X - ½ 3½ (x - ¼) a Y - ¼ c Z	(O)	(6e)
  B10	=	- (½ - x) A1 - ¼ A2 - x A3	=	- (¼ - x) a X - ¼ c Z	(O)	(6e)

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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).