Lattice:Packing fraction - Revision history

KevinYager: /* Examples */

2015-02-11T23:23:57Z

‎Examples

KevinYager at 15:02, 18 June 2014

2014-06-18T15:02:53Z

68.194.136.6 at 00:56, 5 June 2014

2014-06-05T00:56:27Z

68.194.136.6: /* Examples */

2014-06-05T00:55:56Z

‎Examples

68.194.136.6 at 00:55, 5 June 2014

2014-06-05T00:55:01Z

KevinYager at 22:03, 3 June 2014

2014-06-03T22:03:23Z

KevinYager at 21:53, 3 June 2014

2014-06-03T21:53:45Z

KevinYager at 21:44, 3 June 2014

2014-06-03T21:44:44Z

KevinYager: Created page with "The '''packing fraction''' (or particle volume fraction) for a lattice is given by: : $\phi = \frac{ N V_{\mathrm{particle}} } { v_{\mathrm{cell}} }$ Where ''N'' is ..."

2014-06-03T21:43:11Z

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 ..."

New page

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 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>
For a cubic cell:
:<math>\phi = \frac{ N 4 \pi R^3 } { 3 a^3 }</math>
===Examples===
For a [[Lattice:FCC#Symmetry|FCC lattice]], the packing fraction is 0.740:
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{2}a/2</math>
* Assuming spherical particles of radius ''R'':
** Particle volume fraction: <math>\phi=16 \pi R^3/\left(3a^3\right)</math>
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{2}/6\approx0.740</math> when <math>R=a/(2\sqrt{2})</math>
For a [[Lattice:BCC#Symmetry|BCC lattice]], the packing fraction is 0.680:
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/2</math>
* Assuming spherical particles of radius ''R'':
** Particle volume fraction: <math>\phi=8 \pi R^3/\left(3a^3\right)</math>
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{3}/8\approx0.680</math> when <math>R=a\sqrt{3}/4</math>
For a [[Lattice:Diamond#Symmetry|diamond lattice]], the packing fraction is 0.340:
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/4 \approx 0.433 a</math>
* Assuming spherical particles of radius ''R'':
** Particle [[Lattice:Packing fraction|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>

@@ Line 1: / Line 1: @@
-The '''packing fraction''' (or particle volume fraction) for a [[lattice]] is given by:
+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>
 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:

@@ Line 6: / Line 6: @@
 :<math>\phi = \frac{ N 4 \pi R^3 } { 3 a^3 }</math>
 ===Examples===
 For a [[Lattice:FCC#Symmetry|FCC lattice]], the packing fraction is 0.740:
 * Nearest-neighbor distance: <math>d_{nn}=\sqrt{2}a/2</math>

@@ Line 3: / Line 3: @@
 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>
-For a [[Lattice:Cubic|cubic]] [[unit cell]] of edge-length ''a'':
+For a cubic [[unit cell]] of edge-length ''a'':
 :<math>\phi = \frac{ N 4 \pi R^3 } { 3 a^3 }</math>
 ===Examples===

@@ Line 19: / Line 19: @@
 * Nearest-neighbor distance: <math>d_{nn}=\sqrt{3}a/4 \approx 0.433 a</math>
 * Assuming spherical particles of radius ''R'':
-** Particle [[Lattice:Packing fraction|volume fraction]]: <math>\phi=32 \pi R^3/\left(3a^3\right)</math>
+** 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>