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Atomic Packing Factor – APF

In crystallography, atomic packing factor (APF), packing efficiency, or packing fraction is the sum of the sphere volumes of all atoms within a unit cell (assuming the atomic hard-sphere model) divided by the unit cell volume.

apf - atomic packing factor

crystal structure - lattice
Source: U.S. Department of Energy, Material Science. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

By convention, the APF is determined by assuming that atoms are rigid spheres, and the radius of the spheres is taken to be the maximum value such that the atoms do not overlap. In materials science, the atomic packing factor of a unit cell explains many properties of materials. For example, metals with a high atomic packing factor will have a higher malleability or ductility, similar to how a road is smoother when the stones are closer together, allowing metal atoms to slide past one another more easily.

For example, in an fcc arrangement, a unit cell contains (8 corner atoms × ⅛) + (6 face atoms × ½) = 4 atoms. This structure, along with its hexagonal relative (hcp), has the most efficient packing (74%).

References:
Materials Science:
  1. U.S. Department of Energy, Material Science. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
  2. U.S. Department of Energy, Material Science. DOE Fundamentals Handbook, Volume 2 and 2. January 1993.
  3. William D. Callister, David G. Rethwisch. Materials Science and Engineering: An Introduction 9th Edition, Wiley; 9 edition (December 4, 2013), ISBN-13: 978-1118324578.
  4. Eberhart, Mark (2003). Why Things Break: Understanding the World by the Way It Comes Apart. Harmony. ISBN 978-1-4000-4760-4.
  5. Gaskell, David R. (1995). Introduction to the Thermodynamics of Materials (4th ed.). Taylor and Francis Publishing. ISBN 978-1-56032-992-3.
  6. González-Viñas, W. & Mancini, H.L. (2004). An Introduction to Materials Science. Princeton University Press. ISBN 978-0-691-07097-1.
  7. Ashby, Michael; Hugh Shercliff; David Cebon (2007). Materials: engineering, science, processing, and design (1st ed.). Butterworth-Heinemann. ISBN 978-0-7506-8391-3.
  8. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.

See above:

Crystal Structure