Lattice:Packing fraction

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

$\phi ={\frac {NV_{\mathrm {particle} }}{v_{\mathrm {cell} }}}$

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

$\phi ={\frac {N4\pi R^{3}}{3v_{\mathrm {cell} }}}$

For a cubic unit cell of edge-length a:

$\phi ={\frac {N4\pi R^{3}}{3a^{3}}}$

Examples

For a FCC lattice, the packing fraction is 0.740:

• Nearest-neighbor distance: $d_{nn}={\sqrt {2}}a/2$
• Assuming spherical particles of radius R:
  • Particle volume fraction: $\phi =16\pi R^{3}/\left(3a^{3}\right)$
  • Maximum volume fraction: $\phi _{max}=\pi {\sqrt {2}}/6\approx 0.740$ when $R=a/(2{\sqrt {2}})$

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: $\phi _{max}=\pi {\sqrt {3}}/8\approx 0.680$ when $R=a{\sqrt {3}}/4$

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: $\phi =32\pi R^{3}/\left(3a^{3}\right)$
  • 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}