Lattice:AlB2

Canonical AlB₂

Symmetry

• Crystal Family: Hexagonal
• Space Group: P6/mmm, No. 191
• Particles per unit cell: $n=3$
  • 'inner' particles: $2$
  • 'corner' particles: $1$
• Dimensionality: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \, \mathrm{inner} \, \times \, 1 = 2}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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