World Library  
Flag as Inappropriate
Email this Article

Atomic packing factor

Article Id: WHEBN0003436583
Reproduction Date:

Title: Atomic packing factor  
Author: World Heritage Encyclopedia
Language: English
Subject: Crystallography, APF, Cubic crystal system, Glossary of physics, Glossary of engineering
Collection: Crystallography
Publisher: World Heritage Encyclopedia

Atomic packing factor

In crystallography, atomic packing factor (APF), packing efficiency or packing fraction is the fraction of volume in a crystal structure that is occupied by constituent particles. It is dimensionless and always less than unity. In atomic systems, by convention, the APF is determined by assuming that atoms are rigid spheres. The radius of the spheres is taken to be the maximal value such that the atoms do not overlap. For one-component crystals (those that contain only one type of particle), the packing fraction is represented mathematically by

\mathrm{APF} = \frac{N_\mathrm{particle} V_\mathrm{particle}}{V_\mathrm{unit cell}}

where Nparticle is the number of particles in the unit cell, Vparticle is the volume of each particle, and Vunit cell is the volume occupied by the unit cell. It can be proven mathematically that for one-component structures, the most dense arrangement of atoms has an APF of about 0.74 (see Kepler conjecture), obtained by the close-packed structures. For multiple-component structures, the APF can exceed 0.74.


  • Single component crystal structures 1
    • Body-centered cubic 1.1
    • Hexagonal close-packed 1.2
  • See also 2
  • References 3
  • Further reading 4

Single component crystal structures

Common sphere packings taken on by atomic systems are listed below with their corresponding packing fraction.

The majority of metals take on either the hcp, ccp or bcc structure.[2]

Body-centered cubic

BCC structure

The primitive unit cell for the body-centered cubic crystal structure contains several fractions taken from nine atoms: one on each corner of the cube and one atom in the center. Because the volume of each of the eight corner atoms is shared between eight adjacent cells, each BCC cell contains the equivalent volume of two atoms (one central and one on the corner).

Each corner atom touches the center atom. A line that is drawn from one corner of the cube through the center and to the other corner passes through 4r, where r is the radius of an atom. By geometry, the length of the diagonal is a√3. Therefore, the length of each side of the BCC structure can be related to the radius of the atom by

a = \frac{4r}{\sqrt{3}}.

Knowing this and the formula for the volume of a sphere, it becomes possible to calculate the APF as follows:

\mathrm{APF} = \frac{N_\mathrm{atoms} V_\mathrm{atom}}{V_\mathrm{crystal}} = \frac{2 (4/3)\pi r^3}{(4r/\sqrt{3})^3}
= \frac{\pi\sqrt{3}}{8} \approx 0.68.\,\!

Hexagonal close-packed

HCP structure

For the hexagonal close-packed structure the derivation is similar. Here the unit cell (equivalent to 3 primitive unit cells) is a hexagonal prism containing six atoms. Let a be the side length of its base and c be its height. Then:

a = 2r
c = \sqrt{\frac{2}{3}}(4r).

It is then possible to calculate the APF as follows:

\mathrm{APF} = \frac{N_\mathrm{atoms}\cdot V_\mathrm{atom}}{V_\mathrm{crystal}} = \frac{6\cdot (4/3)\pi r^3})(4r)} = \frac{6 (4/3)\pi r^3})(16r^3)}
= \frac{\pi}{\sqrt{18}} \approx 0.74.\,\!

See also


  1. ^ a b c d Ellis, Arthur B. [et al.] (1995). Teaching general chemistry : a materials science companion (3. print ed.). Washington: American Chemical Society.  
  2. ^ Moore, Lesley E. Smart; Elaine A. (2005). Solid state chemistry : an introduction (3. ed.). Boca Raton, Fla. [u.a.]: Taylor & Francis, CRC. p. 8.  

Further reading

  1. Schaffer, Saxena, Antolovich, Sanders, and Warner (1999). The Science and Design of Engineering Materials (Second ed.). New York: WCB/McGraw-Hill. pp. 81–88. 
  2. Callister, W. (2002). Materials Science and Engineering (Sixth ed.). San Francisco: John Wiley and Sons. pp. 105–114. 
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.