Lattice:AlB2

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

Symmetry

• Crystal Family: Hexagonal
• Pearson symbol: hP3
• Space Group: P6/mmm, No. 191
• Particles per unit cell: 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 n=3}
  • '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}
  • 'corner' 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 1}
• Volume of unit cell: 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 V_d=a^2 c \sin(60^{\circ}) = a^2 c \frac{\sqrt{3}}{2}}
• 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.

• 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 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}
  • 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(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

• 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(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

• High-pressure structure of Zr (c/a ~ 0.62)

Atomic

• Aluminium diboride (AlB₂) (a = b = 3.0062 Å, c = 3.2548, c/a = 1.0827)
  • doi: 10.1103/PhysRevB.66.134101
• TiZr ?

Nano

• 4.3 nm and 2.3 nm Fe_xPt_1−x nanoparticles
  • Sra, A.K.; Ewers, T.D.; Xu, Q.; Zandbergen, H.; Schaak, R.E. One-pot synthesis of bi-disperse FePt nanoparticles and size-selective self-assembly into AB2, AB5, and AB13 superlattices Chem. Commun. 2006, 750-752 doi: 10.1039/B515673D
• 15 nm Fe₃O₄ and 6 nm FePt nanoparticles 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