# 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: ${\displaystyle n=3}$
• 'inner' particles: ${\displaystyle 2}$
• 'corner' particles: ${\displaystyle 1}$
• Volume of unit cell: ${\displaystyle V_{d}=a^{2}c\sin(60^{\circ })=a^{2}c{\frac {\sqrt {3}}{2}}}$
• Dimensionality: ${\displaystyle d=3}$

### 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.

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

#### Particle B: inner

These are the two inner particles.

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

### Particle Positions (Cartesian coordinates)

#### Particle A: corners

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

#### Particle B: inner

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

### Examples

#### Elemental

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