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

# The βBoron (R-105) Structure

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• 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:

ahex = 2 a sin ½ θ     and

chex = 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 Bi(j) to represent the ith vector on the jth 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 ahex X - ½ 12-½ ahex Y + 1/3 chex Z A2 = + 3-½ ahex Y + 1/3 chex Z A3 = - ½ ahex X - ½ 12-½ ahex Y + 1/3 chex Z
• Basis Vectors:

 j = 1-4: Bj(1) = + xj A1 + yj A2 + zj A2 = + ½ (xj - zj) ahex X - 12-½ (xj - 2 yj + zj) ahex Y + 1/3 (xj + yj + zj) chex Z (12i) Bj(2) = + yj A1 + zj A2 + xj A2 = + ½ (yj - xj) ahex X - 12-½ (yj - 2 zj + xj) ahex Y + 1/3 (xj + yj + zj) chex Z Bj(3) = + zj A1 + xj A2 + yj A2 = + ½ (zj - yj) ahex X - 12-½ (zj - 2 xj + yj) ahex Y + 1/3 (xj + yj + zj) chex Z Bj(4) = + zj A1 + yj A2 + xj A2 = + ½ (zj - xj) ahex X - 12-½ (zj - 2 yj + xj) ahex Y + 1/3 (xj + yj + zj) chex Z Bj(5) = + yj A1 + xj A2 + zj A2 = + ½ (yj - zj) ahex X - 12-½ (yj - 2 xj + zj) ahex Y + 1/3 (xj + yj + zj) chex Z Bj(6) = + xj A1 + zj A2 + yj A2 = + ½ (xj - yj) ahex X - 12-½ (xj - 2 zj + yj) ahex Y + 1/3 (xj + yj + zj) chex Z Bj(7) = - xj A1 - yj A2 - zj A2 = - ½ (xj - zj) ahex X + 12-½ (xj - 2 yj + zj) ahex Y - 1/3 (xj + yj + zj) chex Z Bj(8) = - yj A1 - zj A2 - xj A2 = - ½ (yj - xj) ahex X + 12-½ (yj - 2 zj + xj) ahex Y - 1/3 (xj + yj + zj) chex Z Bj(9) = - zj A1 - xj A2 - yj A2 = - ½ (zj - yj) ahex X + 12-½ (zj - 2 xj + yj) ahex Y - 1/3 (xj + yj + zj) chex Z Bj(10) = - zj A1 - yj A2 - xj A2 = - ½ (zj - xj) ahex X + 12-½ (zj - 2 yj + xj) ahex Y - 1/3 (xj + yj + zj) chex Z Bj(11) = - yj A1 - xj A2 - zj A2 = - ½ (yj - zj) ahex X + 12-½ (yj - 2 xj + zj) ahex Y - 1/3 (xj + yj + zj) chex Z Bj(12) = - xj A1 - zj A2 - yj A2 = - ½ (xj - yj) ahex X + 12-½ (xj - 2 zj + yj) ahex Y - 1/3 (xj + yj + zj) chex Z j = 5-13: Bj(1) = + xj A1 + xj A2 + zj A2 = + ½ (xj - zj) ahex X + 12-½ (xj - zj) ahex Y + 1/3 (2 xj + zj) chex Z (6h) Bj(2) = + xj A1 + zj A2 + xj A2 = - 12-½ (xj - zj) ahex Y + 1/3 (2 xj + zj) chex Z Bj(3) = + zj A1 + xj A2 + xj A2 = - ½ (xj - zj) ahex X + 12-½ (xj - zj) ahex Y + 1/3 (2 xj + zj) chex Z Bj(4) = - xj A1 - xj A2 - zj A2 = - ½ (xj - zj) ahex X - 12-½ (xj - zj) ahex Y - 1/3 (2 xj + zj) chex Z Bj(5) = - xj A1 - zj A2 - xj A2 = + 12-½ (xj - zj) ahex Y - 1/3 (2 xj + zj) chex Z Bj(6) = - zj A1 - xj A2 - xj A2 = + ½ (xj - zj) ahex X - 12-½ (xj - zj) ahex Y - 1/3 (2 xj + zj) chex Z j = 14: Bj(1) = + xj A1 + xj A2 + xj A2 = + xj chex Z (2c) Bj(2) = - xj A1 - xj A2 - xj A2 = - xj chex Z j = 15: Bj(1) = + ½ A1 + ½ A2 + ½ A2 = + ½ chex Z (1b)

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 Structures indexed by: This is a mirror of an old page created at theNaval Research LaboratoryCenter for Computational Materials ScienceThe 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).