The T-50 Boron (A_g) Structure

Last modified 5 April 2001

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• Prototype: Boron
• Pearson Symbol: tP50
• Strukturbericht Designation: A_g
• Space Group: P4₂/nnm (Cartesian and lattice coordinate listings available)
• Number: 134
• Primitive Vectors:
  • A₁ = a X
  • A₂ = a Y
  • A₃ = c Z
• Basis Vectors:

  The fifty Boron atoms in the primitive cell occupy five symmetry sites in the P4₂/nnm space group. We'll number the types by the index j, and basis vector B_i(j) represents the i^th atom on the j^th symmetry site. To summarize the sites first:

  Index	Wyckoff Notation	Internal Parameters
  j = 1	(2b)	None
  j = 2	(8m)	[x2,z2]
  j = 3	(8m)	[x3,z3]
  j = 4	(16n)	[x4,y4,z4]
  j = 5	(16n)	[x5,y5,z5]

  Note that we will use the second setting of space group #134, so that the origin is at an inversion site. This is not the definition used by Donohue or Pearson's Handbook. These references place one of the (2a) atoms on the inversion site.

  Also note that we only list the Cartesian coordinates of the atoms in the basis, leaving the transformation to lattice coordinates as an exercise for the reader.

  • j = 1 (2b)

    B1(1)	=	+ ¾ a X	+ ¼ a Y	+ ¼ c Z
    B2(1)	=	+ ¼ a X	+ ¾ a Y	+ ¾ c Z

  • j = 2 and j = 3 (8m)

    B1(j)	=	+ xj a X	- xj a Y	+ zj c Z
    B2(j)	=	- xj a X	+ xj a Y	- zj c Z
    B3(j)	=	- xj a X	+ (½ - xj) a Y	+ (½ + zj) c Z
    B4(j)	=	+ xj a X	+ (½ + xj) a Y	+ (½ - zj) c Z
    B5(j)	=	+ (½ + xj) a X	+ xj a Y	+ (½ + zj) c Z
    B6(j)	=	+ (½ - xj) a X	- xj a Y	+ (½ - zj) c Z
    B7(j)	=	+ (½ - xj) a X	+ (½ + xj) a Y	+ zj c Z
    B8(j)	=	+ (½ + xj) a X	+ (½ - xj) a Y	- zj c Z

  • j = 4 and j = 5 (16n)

    B1(j)	=	+ xj a X	+ yj a Y	+ zj c Z
    B2(j)	=	- xj a X	- yj a Y	- zj c Z
    B3(j)	=	+ (½ + yj) a X	+ (½ + xj) a Y	+ zj c Z
    B4(j)	=	+ (½ - yj) a X	+ (½ - xj) a Y	- zj c Z
    B5(j)	=	- xj a X	+ (½ + yj) a Y	+ (½ + zj) c Z
    B6(j)	=	+ xj a X	+ (½ - xj) a Y	+ (½ - zj) c Z
    B7(j)	=	(½ - yj) a X	+ xj a Y	+ (½ + zj) c Z
    B8(j)	=	(½ + yj) a X	- xj a Y	+ (½ - zj) c Z
    B9(j)	=	(½ + xj) a X	- yj a Y	+ (½ + zj) c Z
    B10(j)	=	(½ - xj) a X	+ yj a Y	+ (½ - zj) c Z
    B11(j)	=	+ yj a X	+ (½ - xj) a Y	+ (½ + zj) c Z
    B12(j)	=	- yj a X	+ (½ + xj) a Y	+ (½ - zj) c Z
    B13(j)	=	+ (½ - xj) a X	+ (½ - yj) a Y	+ zj c Z
    B14(j)	=	+ (½ + xj) a X	+ (½ + yj) a Y	- zj c Z
    B15(j)	=	- yj a X	- x j a Y	+ zj c Z
    B16(j)	=	+ yj a X	+ x j a Y	- zj c Z

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