Difference between revisions of "Lattice:HCP"

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(→‎Canonical HCP)
(→‎Particle Positions (basis vectors))
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====Particle B: inner====
 
====Particle B: inner====
 
* <math> 1 \, \mathrm{inner} \, \times \, 1 = 2</math>
 
* <math> 1 \, \mathrm{inner} \, \times \, 1 = 2</math>
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** <math>\left(\frac{1}{3},\frac{1}{3},\frac{1}{2} \right), </math>
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** <math>\left(\frac{1}{3},\frac{1}{3},\frac{1}{2} \right) </math>
  
 +
===Particle Positions (Cartesian coordinates)===
 +
 +
====Particle A: corners====
 +
* <math>\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)</math>
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 +
====Particle B: inner====
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* <math>\left(\frac{a}{3}+\frac{b}{6},\frac{\sqrt{3}b}{6},\frac{c}{2} \right)</math>
  
 
===Examples===
 
===Examples===

Revision as of 11:26, 14 October 2014

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}
  • 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)}

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

See Also

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