Lattice:HCP

HCP (Hexagonal close-packed) is a hexagonal lattice. It is notable (along with FCC) because it achieves the densest possible packing of spheres. It thus arises naturally in many atomic crystals, as well as in colloidal crystals and nanoparticles superlattices.

Canonical HCP

In the canonical HCP, the ratio between the a and c distances is:

${\frac {c}{a}}=2{\sqrt {\frac {2}{3}}}={\frac {2{\sqrt {6}}}{3}}\approx 1.633$

Symmetry

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

Particle Positions (basis vectors)

There are 9 positions, with 2 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

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

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

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

Reciprocal-space Peaks

• Allowed reflections:
  • $l$ even
  • $h+2k\neq 3n$
• Peak positions:

  $q_{hkl}=2\pi \left({\frac {(h^{2}+hk+k^{2})^{2}}{a^{2}}}+{\frac {l^{2}}{c^{2}}}\right)^{1/2}$

Examples

Elemental

Many elements pack into HCP. E.g.:

4. Beryllium (Be) (a = b = 2.290 Å, c = 3.588, c/a = 1.567)

27. Cobalt (Co) (a = b = 2.5071 Å, c = 4.0695, c/a = 1.623)

48. Cadmium (Cd) (a =b = 2.9794 Å, c = 5.6186 Å, c/a = 1.886)

Atomic

• TBD

Nano

• Gold nanoparticles
  • Stoeva et al. Face-Centered Cubic and Hexagonal Closed-Packed Nanocrystal Superlattices of Gold Nanoparticles Prepared by Different Methods J. Phys. Chem. B 2003, 107 (30), 7441-7448 doi: 10.1021/jp030013+

See Also

• Wikipedia: Close-packing of equal spheres
• Lattice:AlB2