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:

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

Symmetry

• Crystal Family: Hexagonal
• 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=2}
  • '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 1}
  • '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 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.

• 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

• 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 \, \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) }

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)}

Reciprocal-space Peaks

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

  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 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