Difference between revisions of "Lattice:Packing fraction"

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(Created page with "The '''packing fraction''' (or particle volume fraction) for a lattice is given by: :<math>\phi = \frac{ N V_{\mathrm{particle}} } { v_{\mathrm{cell}} }</math> Where ''N'' is ...")
 
(→‎Examples)
 
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The '''packing fraction''' (or particle volume fraction) for a lattice is given by:
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The '''packing fraction''' (or particle volume fraction) for a [[Lattices|lattice]] is given by:
 
:<math>\phi = \frac{ N V_{\mathrm{particle}} } { v_{\mathrm{cell}} }</math>
 
:<math>\phi = \frac{ N V_{\mathrm{particle}} } { v_{\mathrm{cell}} }</math>
 
Where ''N'' is the number of particles per unit cell (which has volume <math>v_{\mathrm{cell}}</math>). For a sphere, the volume is <math>V=4\pi R^3/3</math> so:
 
Where ''N'' is the number of particles per unit cell (which has volume <math>v_{\mathrm{cell}}</math>). For a sphere, the volume is <math>V=4\pi R^3/3</math> so:
 
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 v_{\mathrm{cell}} }</math>
 
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 v_{\mathrm{cell}} }</math>
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For a cubic cell:
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For a cubic [[unit cell]] of edge-length ''a'':
 
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 a^3 }</math>
 
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 a^3 }</math>
 
===Examples===
 
===Examples===
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For a [[Lattice:SC#Symmetry|SC lattice]], the packing fraction is 0.524:
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* Nearest-neighbor distance: <math>d_{nn}=a</math>
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* Assuming spherical particles of radius ''R'':
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** Particle volume fraction: <math>\phi=4 \pi R^3/\left(3a^3\right)</math>
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** Maximum volume fraction: <math>\phi_{max}=4\pi/24\approx0.5236</math> when <math>R=a/2</math>
 
For a [[Lattice:FCC#Symmetry|FCC lattice]], the packing fraction is 0.740:
 
For a [[Lattice:FCC#Symmetry|FCC lattice]], the packing fraction is 0.740:
 
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{2}a/2</math>
 
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{2}a/2</math>
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* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/4 \approx 0.433 a</math>
 
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/4 \approx 0.433 a</math>
 
* Assuming spherical particles of radius ''R'':
 
* Assuming spherical particles of radius ''R'':
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** Particle [[Lattice:Packing fraction|volume fraction]]: <math>\phi=32 \pi R^3/\left(3a^3\right)</math>
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** Particle volume fraction: <math>\phi=32 \pi R^3/\left(3a^3\right)</math>
 
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{3}/16\approx0.340</math> when <math>R=a\sqrt{3}/8</math>
 
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{3}/16\approx0.340</math> when <math>R=a\sqrt{3}/8</math>

Latest revision as of 19:23, 11 February 2015

The packing fraction (or particle volume fraction) for a lattice is given by:

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 \phi = \frac{ N V_{\mathrm{particle}} } { v_{\mathrm{cell}} }}

Where N is the number of particles per unit cell (which has volume 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_{\mathrm{cell}}} ). For a sphere, the volume 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 V=4\pi R^3/3} so:

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 \phi = \frac{ N 4 \pi R^3 } { 3 v_{\mathrm{cell}} }}

For a cubic unit cell of edge-length a:

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 \phi = \frac{ N 4 \pi R^3 } { 3 a^3 }}

Examples

For a SC lattice, the packing fraction is 0.524:

  • Nearest-neighbor 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 d_{nn}=a}
  • Assuming spherical particles of radius R:
    • Particle volume fraction: 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 \phi=4 \pi R^3/\left(3a^3\right)}
    • Maximum volume fraction: 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 \phi_{max}=4\pi/24\approx0.5236} when 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 R=a/2}

For a FCC lattice, the packing fraction is 0.740:

  • Nearest-neighbor 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 d_{nn}=\sqrt{2}a/2}
  • Assuming spherical particles of radius R:
    • Particle volume fraction: 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 \phi=16 \pi R^3/\left(3a^3\right)}
    • Maximum volume fraction: 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 \phi_{max}=\pi\sqrt{2}/6\approx0.740} when 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 R=a/(2\sqrt{2})}

For a BCC lattice, the packing fraction is 0.680:

  • Nearest-neighbor 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 d_{nn}=\sqrt{3}a/2}
  • Assuming spherical particles of radius R:
    • Particle volume fraction: 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 \phi=8 \pi R^3/\left(3a^3\right)}
    • Maximum volume fraction: 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 \phi_{max}=\pi\sqrt{3}/8\approx0.680} when 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 R=a\sqrt{3}/4}

For a diamond lattice, the packing fraction is 0.340:

  • Nearest-neighbor 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 d_{nn}=\sqrt{3}a/4 \approx 0.433 a}
  • Assuming spherical particles of radius R:
    • Particle volume fraction: 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 \phi=32 \pi R^3/\left(3a^3\right)}
    • Maximum volume fraction: 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 \phi_{max}=\pi\sqrt{3}/16\approx0.340} when 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 R=a\sqrt{3}/8}
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