Lattice:AlB2

AlB₂ is a hexagonal lattice, with two distinct kinds of particles.

Symmetry

• Crystal Family: Hexagonal
• Space Group: P6/mmm, No. 191
• Particles per unit cell: $n=3$
  • 'inner' particles: $2$
  • 'corner' particles: $1$
• Dimensionality: $d=3$

Particle Positions (basis vectors)

There are 10 positions, with 3 particles in the unit cell

Particle A: corners

These are the corners of the hexagonal frame. There are 8 corner positions, which contributes a total of 1 particle.

• $8\,\mathrm {corners} :{\frac {1}{12}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{6}}=1$
  • $\left(0,0,0\right),\,(0,0,1),\,(0,1,0),\,(1,0,0),\,(0,1,1),\,(1,0,1),\,(1,1,0),\,(1,1,1)$

Particle B: inner

These are the two inner particles.

• $2\,\mathrm {inner} \,\times \,1=2$
  • $\left({\frac {1}{3}},{\frac {1}{3}},{\frac {1}{2}}\right),\,\left({\frac {2}{3}},{\frac {2}{3}},{\frac {1}{2}}\right),$

Particle Positions (Cartesian coordinates)

Particle A: corners

• $\left(0,0,0\right),\,(0,0,c),\,\left({\frac {b}{2}},{\frac {{\sqrt {3}}b}{2}},0\right),\,(a,0,0),\,\left({\frac {b}{2}},{\frac {{\sqrt {3}}b}{2}},c\right),\,(a,0,c),\,\left(a+{\frac {b}{2}},{\frac {{\sqrt {3}}b}{2}},0\right),\,\left(a+{\frac {b}{2}},{\frac {{\sqrt {3}}b}{2}},c\right)$

Particle B: inner

• $\left({\frac {a}{3}}+{\frac {b}{6}},{\frac {{\sqrt {3}}b}{6}},{\frac {c}{2}}\right),\,\left({\frac {2a}{3}}+{\frac {b}{3}},{\frac {{\sqrt {3}}b}{3}},{\frac {c}{2}}\right),$

Examples

Elemental

None

Atomic

• Aluminium diboride (AlB₂)

Nano

• 15 nm Fe₃O₄ and 6 nm FePt assembled at liquid interface
  • Dong, A., Chen, J., Vora, P. M., Kikkawa, J. M. & Murray, C. B. Binary nanocrystal superlattice membranes self-assembled at the liquid–air interface Nature 2010 466, 474–477 doi: 10.1038/nature09188
  • Korgel, B.A. Nanocrystal superlattices: Assembly at liquid interfaces, Nature Materials 2010, 9, 701-703 doi: 10.1038/nmat2846