# Lattice:Packing fraction

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

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

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

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

For a cubic unit cell of edge-length a:

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

### Examples

For a SC lattice, the packing fraction is 0.524:

• Nearest-neighbor distance: ${\displaystyle d_{nn}=a}$
• Assuming spherical particles of radius R:
• Particle volume fraction: ${\displaystyle \phi =4\pi R^{3}/\left(3a^{3}\right)}$
• Maximum volume fraction: ${\displaystyle \phi _{max}=4\pi /24\approx 0.5236}$ when ${\displaystyle R=a/2}$

For a FCC lattice, the packing fraction is 0.740:

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

For a BCC lattice, the packing fraction is 0.680:

• Nearest-neighbor distance: ${\displaystyle d_{nn}={\sqrt {3}}a/2}$
• Assuming spherical particles of radius R:
• Particle volume fraction: ${\displaystyle \phi =8\pi R^{3}/\left(3a^{3}\right)}$
• Maximum volume fraction: ${\displaystyle \phi _{max}=\pi {\sqrt {3}}/8\approx 0.680}$ when ${\displaystyle R=a{\sqrt {3}}/4}$

For a diamond lattice, the packing fraction is 0.340:

• Nearest-neighbor distance: ${\displaystyle d_{nn}={\sqrt {3}}a/4\approx 0.433a}$
• Assuming spherical particles of radius R:
• Particle volume fraction: ${\displaystyle \phi =32\pi R^{3}/\left(3a^{3}\right)}$
• Maximum volume fraction: ${\displaystyle \phi _{max}=\pi {\sqrt {3}}/16\approx 0.340}$ when ${\displaystyle R=a{\sqrt {3}}/8}$