The alpha-Boron (R12) Structure

Last modified 3 May 2001

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.

This is a metastable phase of Boron, and the simplest known phase (the ground state, beta-Boron, has 105 or 320 atoms in the unit cell). Pearson's Handbook gives the lattice constants and positions in terms of the hexagonal unit cell. I've translated everything into the primitive rhombohedral unit cell.

Note the relationship between the icosahedra in this structure and in T50 Boron.

• Prototype: alpha B
• Pearson Symbol: hR12
• Space Group: R(-3)m (Cartesian and lattice coordinate listings available)
• Number: 166
• Primitive Vectors:
  • A₁ = + ½ 3^1/2 a X - ½ a Y + c Z
  • A₂ = a Y + c Z
  • A₃ = - ½ 3^1/2 a X - ½ a Y + c Z
• Basis Vectors. There are two types of atoms, B-I and B-II, both of which occupy (6h) sites. Note that there is nothing special about x_I,II = 0 or x_I,II = ¼.
  • B₁ = + x_I A₁ + x_I A₂ + z_I A₃ = + ½ 3^1/2 (x_I - z_I) a X + ½ (x_I - z_I) a Y + (2 x_I + z_I) c Z   (B-I)
  • B₂ = + x_I A₁ + z_I A₂ + x_I A₃ = - (x_I - z_I) a Y + (2 x_I + z_I) c Z   (B-I)
  • B₃ = + z_I A₁ + x_I A₂ + x_I A₃ = - ½ 3^1/2 (x_I - z_I) a X + ½ (x_I - z_I) a Y + (2 x_I + z_I) c Z   (B-I)
  • B₄ = - x_I A₁ - x_I A₂ - z_I A₃ = - ½ 3^1/2 (x_I - z_I) a X - ½ (x_I - z_I) a Y - (2 x_I + z_I) c Z   (B-I)
  • B₅ = - x_I A₁ - z_I A₂ - x_I A₃ = + (x_I - z_I) a Y - (2 x_I + z_I) c Z   (B-I)
  • B₆ = - z_I A₁ - x_I A₂ - x_I A₃ = + ½ 3^1/2 (x_I - z_I) a X - ½ (x_I - z_I) a Y - (2 x_I + z_I) c Z   (B-I)
  • B₇ = + x_II A₁ + x_II A₂ + z_II A₃ = + ½ 3^1/2 (x_II - z_II) a X + ½ (x_II - z_II) a Y + (2 x_II + z_II) c Z   (B-II)
  • B₈ = + x_II A₁ + z_II A₂ + x_II A₃ = - (x_II - z_II) a Y + (2 x_II + z_II) c Z   (B-II)
  • B₉ = + z_II A₁ + x_II A₂ + x_II A₃ = - ½ 3^1/2 (x_II - z_II) a X + ½ (x_II - z_II) a Y + (2 x_II + z_II) c Z   (B-II)
  • B₁₀ = - x_II A₁ - x_II A₂ - z_II A₃ = - ½ 3^1/2 (x_II - z_II) a X - ½ (x_II - z_II) a Y - (2 x_II + z_II) c Z   (B-II)
  • B₁₁ = - x_II A₁ - z_II A₂ - x_II A₃ = + (x_II - z_II) a Y - (2 x_II + z_II) c Z   (B-II)
  • B₁₂ = - z_II A₁ - x_II A₂ - x_II A₃ = + ½ 3^1/2 (x_II - z_II) a X - ½ (x_II - z_II) a Y - (2 x_II + z_II) c Z   (B-II)

See these vectors in a better notation.

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