Lattice:Hexagonal diamond

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The hexagonal diamond lattice is an arrangement of tetrahedrally-bonded elements, within a hexagonal unit cell. Whereas conventional diamond (a.k.a. cubic diamond) exists within a cubic unit cell, hexagonal diamond exists within a hexagonal unit cell. In both cases, elements are bonded tetrahdrally. However, in cubic diamond, the six-membered rings are all in the chair conformation, whereas in hexagonal diamond, some six-membered rings are in the boat conformation.

The four distinct positions in the unit cell are each given a different color. From: "Structural and vibrational properties of the 6H diamond: First-principles study" doi: 10.1016/j.diamond.2006.03.013


Canonical Hexagonal Diamond

A canonical hexagonal diamond lattice (single atom/particle type arranged as shown above) has symmetry Fd3m.

Symmetry

  • Crystal Family: Hexagonal
  • Crystal System: Hexagonal
  • Bravais Lattice: hexagonal
  • Crystal class: Hexoctahedral
  • Space Group: P63/mmc
  • 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=4}
  • 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}

Structure

TBD

Particle Positions

There are 14 positions. In total there are 4 particles in the unit cell.

Fractional

Positions are given in terms of fractional coordinates relative to the unit-cell edge-vectors:

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 \mathbf{a}=(a,0,0)}
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 \mathbf{b}=(b/2,\sqrt{3}b/2,0)}
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 \mathbf{c}=(0,0,c)}
  • 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 4 \, \mathrm{bottom\,\, layer}: \, \frac{1}{12} + \frac{1}{6} + \frac{1}{12} + \frac{1}{6} = \frac{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(0,0,0\right),(1,0,0),(0,1,0),(1,1,0)}
  • 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 4 \, \mathrm{mid\,\, layer}: \, \frac{1}{6} + \frac{1}{3} + \frac{1}{6} + \frac{1}{3} = 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,\frac{5}{8}\right),\left(1,0,\frac{5}{8}\right),\left(0,1,\frac{5}{8}\right),\left(1,1,\frac{5}{8}\right)}
  • 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{internal\,\, strut}: \, 1 + 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}{8}\right),\left(\frac{1}{3},\frac{1}{3},\frac{4}{8}\right)}
  • 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 4 \, \mathrm{top\,\, layer}: \, \frac{1}{12} + \frac{1}{6} + \frac{1}{12} + \frac{1}{6} = \frac{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(0,0,1\right),(1,0,1),(0,1,1),(1,1,1)}

Absolute

  • 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 4 \, \mathrm{bottom\,\, layer}: \, \frac{1}{12} + \frac{1}{6} + \frac{1}{12} + \frac{1}{6} = \frac{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(0,0,0\right),(a,0,0),\left(\frac{b}{2},\frac{\sqrt{3}b}{2},0\right),\left(a+\frac{b}{2},\frac{\sqrt{3}b}{2},0\right)}
  • 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 4 \, \mathrm{mid\,\, layer}: \, \frac{1}{6} + \frac{1}{3} + \frac{1}{6} + \frac{1}{3} = 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,\frac{5c}{8}\right),\left(a,0,\frac{5c}{8}\right),\left(\frac{b}{2},\frac{\sqrt{3}b}{2},\frac{5c}{8}\right),\left(a+\frac{b}{2},\frac{\sqrt{3}b}{2},\frac{5c}{8}\right)}
  • 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{internal\,\, strut}: \, 1 + 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{a}{2},\frac{b}{2\sqrt{3}},\frac{c}{8}\right),\left(\frac{a}{2},\frac{b}{2\sqrt{3}},\frac{4c}{8}\right)}
  • 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 4 \, \mathrm{top\,\, layer}: \, \frac{1}{12} + \frac{1}{6} + \frac{1}{12} + \frac{1}{6} = \frac{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(0,0,c\right),(a,0,c),\left(\frac{b}{2},\frac{\sqrt{3}b}{2},c\right),\left(a+\frac{b}{2},\frac{\sqrt{3}b}{2},c\right)}

Distances

For a particle-particle bond-length of 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} :

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 a=\frac{2\sqrt{6}}{3}l}
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 b=a=\frac{2\sqrt{6}}{3}l}
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 c=\frac{8}{3}l}
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{a}{c}=\frac{\sqrt{6}}{4}\approx 0.61237}
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}=\frac{4}{\sqrt{6}}\approx 1.63299}

Absolute (in terms of particle-particle distance)

  • 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 4 \, \mathrm{bottom\,\, layer}: \, \frac{1}{12} + \frac{1}{6} + \frac{1}{12} + \frac{1}{6} = \frac{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(0,0,0\right),\left(\frac{2\sqrt{6}}{3}l,0,0 \right),\left(\frac{\sqrt{6}}{3}l,\sqrt{2}l,0\right),\left(\sqrt{6}l,\sqrt{2}l,0\right)}
  • 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 4 \, \mathrm{mid\,\, layer}: \, \frac{1}{6} + \frac{1}{3} + \frac{1}{6} + \frac{1}{3} = 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,\frac{5}{3}l\right),\left(\frac{2\sqrt{6}}{3}l,0,\frac{5}{3}l \right),\left(\frac{\sqrt{6}}{3}l,\sqrt{2}l,\frac{5}{3}l\right),\left(\sqrt{6}l,\sqrt{2}l,\frac{5}{3}l\right)}
  • 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{internal\,\, strut}: \, 1 + 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{\sqrt{6}}{3}l,\frac{\sqrt{2}}{3}l,\frac{1}{3}l\right),\left(\frac{\sqrt{6}}{3}l,\frac{\sqrt{2}}{3}l,\frac{4}{3}l\right)}
  • 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 4 \, \mathrm{top\,\, layer}: \, \frac{1}{12} + \frac{1}{6} + \frac{1}{12} + \frac{1}{6} = \frac{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(0,0,\frac{8}{3}l\right),\left(\frac{2\sqrt{6}}{3}l,0,\frac{8}{3}l \right),\left(\frac{\sqrt{6}}{3}l,\sqrt{2}l,\frac{8}{3}l\right),\left(\sqrt{6}l,\sqrt{2}l,\frac{8}{3}l\right)}

Examples

Atomics

  • Lonsdaleite form of carbon (C), also known as hexagonal diamond, 2H diamond, or 'sp3 diamond' (a = 2.51 Å, c = 4.12 Å)

Alternating Hexagonal Diamond

This is the Wurtzite crystal structure, a hexagonal unit cell with alternating species.

Examples

Atomics

  • Wurtzite (Zn,Fe)S (a = b = 3.82 Å, c = 6.26 Å)

Along Connections

This lattice can be thought of as the hexagonal-dimaond analog of the cubic-diamond cristobalite. Here, a four-bonded species occupies all the sites of the canonical hexagonal diamond lattice, and a two-bonded species sits along each of the connections between these tetrahedral sites.

Particle Positions

Particle Type A (bond tetrahedrally)

These are the same positions as the canonical hexagonal diamond.

Particle Type B (two-fold bonded)

There are 11 positions. In total there are 8 particles of this type in the unit cell.

Fractional

Positions are given in terms of fractional coordinates relative to the unit-cell edge-vectors:

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 \mathbf{a}=(a,0,0)}
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 \mathbf{b}=(b/2,\sqrt{3}b/2,0)}
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 \mathbf{c}=(0,0,c)}
  • 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 3 \, \mathrm{lower\,\,tripod}: \, 1 + 1 + 1 = 3}
    • 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}{6},\frac{1}{6},\frac{1}{16}\right),\left(\frac{4}{6},\frac{1}{6},\frac{1}{16}\right),\left(\frac{1}{6},\frac{4}{6},\frac{1}{16}\right)}
  • 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{mid\,\,strut}: \, 1 = 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(\frac{1}{3},\frac{1}{3},\frac{5}{16}\right)}
  • 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 3 \, \mathrm{midlevel\,\,tripod}: \, 1 + 1 + 1 = 3}
    • 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}{6},\frac{1}{6},\frac{9}{16}\right),\left(\frac{4}{6},\frac{1}{6},\frac{9}{16}\right),\left(\frac{1}{6},\frac{4}{6},\frac{9}{16}\right)}
  • 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 4 \, \mathrm{upper\,\,connections}: \, \frac{1}{6} + \frac{1}{3} + \frac{1}{6} + \frac{1}{3} = 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,\frac{13}{16}\right),\left(1,0,\frac{13}{16}\right),\left(0,1,\frac{13}{16}\right),\left(1,1,\frac{13}{16}\right)}

See Also

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