Crystal Lattice Structures:	Creation Date: 26 May 2003
	Last Modified: 21 Oct 2004

The βBoron (R-105) Structure

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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)

This is apparently the ground state of Boron, with 105 atoms in the unit cell.
Note the relationship between the icosahedra in this structure, αBoron and T50 Boron.
Donohue gives two possible sets of internal coordinates for the atoms on page 64. We use the first set.
Donohue gives the primitive lattice vectors for this rhombohedral lattice in terms of the length of the primitive lattice vectors, a, and the angle between them, θ. For our purposes it is best to write the primitive vectors in the form derived from the underlying hexagonal lattice. This means we need to define the hexagonal lattice constants for this lattice:
a_hex = 2 a sin ½ θ and

c_hex = a [3 (1 + 2 cos θ)]^½

If you substitute these values into the primitive vectors you will see that they all have length a, and that the angle between any two of them is θ.
As with T50 Boron, this structure has several groups of atoms at the same Wyckoff position. As there, we'll use B_i(j) to represent the i^th vector on the j^th symmetry site.

To summarize the sites:

Index	Wyckoff Notation	Internal Parameters
j = 1	(12i)	(x₁,y₁,z₁)
j = 2	(12i)	(x₂,y₂,z₂)
j = 3	(12i)	(x₃,y₃,z₃)
j = 4	(12i)	(x₄,y₄,z₄)
j = 5	(6h)	(x₅,z₅)
j = 6	(6h)	(x₆,z₆)
j = 7	(6h)	(x₇,z₇)
j = 8	(6h)	(x₈,z₈)
j = 9	(6h)	(x₉,z₉)
j = 10	(6h)	(x₁₀,z₁₀)
j = 11	(6h)	(x₁₁,z₁₁)
j = 12	(6h)	(x₁₂,z₁₂)
j = 13	(6h)	(x₁₃,z₁₃)
j = 14	(2c)	z₁₄
j = 15	(1b)	none

Prototype: βB
Pearson Symbol: hR105
Space Group: R3m (Cartesian and lattice coordinate listings available)
Number: 166

Primitive Vectors:

A₁	=	+ ½ a_hex X - ½ 12^-½ a_hex Y + 1/3 c_hex Z
A₂	=	+ 3^-½ a_hex Y + 1/3 c_hex Z
A₃	=	- ½ a_hex X - ½ 12^-½ a_hex Y + 1/3 c_hex Z

Basis Vectors:

j = 1-4:	B_j(1)	=	+ x_j A₁ + y_j A₂ + z_j A₂	=	+ ½ (x_j - z_j) a_hex X - 12^-½ (x_j - 2 y_j + z_j) a_hex Y + 1/3 (x_j + y_j + z_j) c_hex Z	(12i)
	B_j(2)	=	+ y_j A₁ + z_j A₂ + x_j A₂	=	+ ½ (y_j - x_j) a_hex X - 12^-½ (y_j - 2 z_j + x_j) a_hex Y + 1/3 (x_j + y_j + z_j) c_hex Z
	B_j(3)	=	+ z_j A₁ + x_j A₂ + y_j A₂	=	+ ½ (z_j - y_j) a_hex X - 12^-½ (z_j - 2 x_j + y_j) a_hex Y + 1/3 (x_j + y_j + z_j) c_hex Z
	B_j(4)	=	+ z_j A₁ + y_j A₂ + x_j A₂	=	+ ½ (z_j - x_j) a_hex X - 12^-½ (z_j - 2 y_j + x_j) a_hex Y + 1/3 (x_j + y_j + z_j) c_hex Z
	B_j(5)	=	+ y_j A₁ + x_j A₂ + z_j A₂	=	+ ½ (y_j - z_j) a_hex X - 12^-½ (y_j - 2 x_j + z_j) a_hex Y + 1/3 (x_j + y_j + z_j) c_hex Z
	B_j(6)	=	+ x_j A₁ + z_j A₂ + y_j A₂	=	+ ½ (x_j - y_j) a_hex X - 12^-½ (x_j - 2 z_j + y_j) a_hex Y + 1/3 (x_j + y_j + z_j) c_hex Z
	B_j(7)	=	- x_j A₁ - y_j A₂ - z_j A₂	=	- ½ (x_j - z_j) a_hex X + 12^-½ (x_j - 2 y_j + z_j) a_hex Y - 1/3 (x_j + y_j + z_j) c_hex Z
	B_j(8)	=	- y_j A₁ - z_j A₂ - x_j A₂	=	- ½ (y_j - x_j) a_hex X + 12^-½ (y_j - 2 z_j + x_j) a_hex Y - 1/3 (x_j + y_j + z_j) c_hex Z
	B_j(9)	=	- z_j A₁ - x_j A₂ - y_j A₂	=	- ½ (z_j - y_j) a_hex X + 12^-½ (z_j - 2 x_j + y_j) a_hex Y - 1/3 (x_j + y_j + z_j) c_hex Z
	B_j(10)	=	- z_j A₁ - y_j A₂ - x_j A₂	=	- ½ (z_j - x_j) a_hex X + 12^-½ (z_j - 2 y_j + x_j) a_hex Y - 1/3 (x_j + y_j + z_j) c_hex Z
	B_j(11)	=	- y_j A₁ - x_j A₂ - z_j A₂	=	- ½ (y_j - z_j) a_hex X + 12^-½ (y_j - 2 x_j + z_j) a_hex Y - 1/3 (x_j + y_j + z_j) c_hex Z
	B_j(12)	=	- x_j A₁ - z_j A₂ - y_j A₂	=	- ½ (x_j - y_j) a_hex X + 12^-½ (x_j - 2 z_j + y_j) a_hex Y - 1/3 (x_j + y_j + z_j) c_hex Z

j = 5-13:	B_j(1)	=	+ x_j A₁ + x_j A₂ + z_j A₂	=	+ ½ (x_j - z_j) a_hex X + 12^-½ (x_j - z_j) a_hex Y + 1/3 (2 x_j + z_j) c_hex Z	(6h)
	B_j(2)	=	+ x_j A₁ + z_j A₂ + x_j A₂	=	- 12^-½ (x_j - z_j) a_hex Y + 1/3 (2 x_j + z_j) c_hex Z
	B_j(3)	=	+ z_j A₁ + x_j A₂ + x_j A₂	=	- ½ (x_j - z_j) a_hex X + 12^-½ (x_j - z_j) a_hex Y + 1/3 (2 x_j + z_j) c_hex Z
	B_j(4)	=	- x_j A₁ - x_j A₂ - z_j A₂	=	- ½ (x_j - z_j) a_hex X - 12^-½ (x_j - z_j) a_hex Y - 1/3 (2 x_j + z_j) c_hex Z
	B_j(5)	=	- x_j A₁ - z_j A₂ - x_j A₂	=	+ 12^-½ (x_j - z_j) a_hex Y - 1/3 (2 x_j + z_j) c_hex Z
	B_j(6)	=	- z_j A₁ - x_j A₂ - x_j A₂	=	+ ½ (x_j - z_j) a_hex X - 12^-½ (x_j - z_j) a_hex Y - 1/3 (2 x_j + z_j) c_hex Z

j = 14:	B_j(1)	=	+ x_j A₁ + x_j A₂ + x_j A₂	=	+ x_j c_hex Z	(2c)
j = 14:	B_j(2)	=	- x_j A₁ - x_j A₂ - x_j A₂	=	- x_j c_hex Z	(2c)

j = 15:	B_j(1)	=	+ ½ A₁ + ½ A₂ + ½ A₂	=	+ ½ c_hex Z	(1b)

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