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| In [[crystallography]], '''atomic packing factor''' (APF) or '''packing fraction''' is the fraction of volume in a [[crystal structure]] that is occupied by [[atom]]s. It is dimensionless and always less than unity. By convention, the APF of a crystal structure 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 atom), the APF is represented mathematically by
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| :<math>\mathrm{APF} = \frac{N_\mathrm{atoms} V_\mathrm{atom}}{V_\mathrm{unit cell}}</math>
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| where ''N<sub>atoms</sub>'' is the number of atoms in the unit cell, ''V<sub>atom</sub>'' is the volume of an atom, and ''V<sub>unit cell</sub>'' 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. For multiple-component structures, the APF can exceed 0.74.
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| ==Worked out example==
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| ===Body-centered cubic crystal structure===
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| [[File:CCC crystal cell (opaque).svg|right|thumb|BCC structure]]
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| The [[Unit cell|primitive unit cell]] for the [[body-centered cubic]]
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| (BCC) crystal structure contains several fractions taken from nine
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| atoms: one on each corner of the cube and one atom in the center.
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| Because the volume of each corner atom is shared between adjacent cells,
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| each BCC cell contains two atoms.
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| Each corner atom touches the center atom. A line that is drawn from one
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| corner of the cube through the center and to the other corner passes
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| through 4''r'', where ''r'' is the radius of an atom. By geometry, the
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| length of the diagonal is ''a''√3. Therefore, the length of each side of
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| the BCC structure can be related to the radius of the atom by
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| :<math>a = \frac{4r}{\sqrt{3}}.</math>
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| Knowing this and the formula for the [[volume of a sphere]], it becomes
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| possible to calculate the APF as follows:
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| :<math>\mathrm{APF} = \frac{N_\mathrm{atoms}
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| V_\mathrm{atom}}{V_\mathrm{crystal}} = \frac{2 (4/3)\pi
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| r^3}{(4r/\sqrt{3})^3}</math>
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| <!-- extra blank line for legibility of mathematical notation -->
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| :::<math>= \frac{\pi\sqrt{3}}{8} \approx 0.68.\,\!</math>
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| ===Hexagonal close-packed crystal structure===
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| [[File:Hexagonal close packed.svg|right|thumb|HCP structure]]
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| For the [[Hexagonal crystal system|hexagonal close-packed]] (HCP)
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| structure the derivation is similar. Here the unit cell is a hexagonal
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| prism containing six atoms. Let ''a'' be the side length of its base and
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| ''c'' be its height. Then:
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| :<math>a = 2r</math>
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| <!-- extra blank line for legibility of mathematical notation -->
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| :<math>c = \sqrt{\frac{2}{3}}(4r).</math>
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| It is then possible to calculate the APF as follows:
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| :<math>\mathrm{APF} = \frac{N_\mathrm{atoms}\cdot
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| V_\mathrm{atom}}{V_\mathrm{crystal}} = \frac{6\cdot (4/3)\pi
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| r^3}{[(3\sqrt{3})/2](a^2)(c)}</math>
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| <!-- extra blank line for legibility of mathematical notation -->
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| :::<math>= \frac{6 (4/3)\pi
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| r^3}{[(3\sqrt{3})/2](2r)^2(\sqrt{\frac{2}{3}})(4r)} = \frac{6 (4/3)\pi
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| r^3}{[(3\sqrt{3})/2](\sqrt{\frac{2}{3}})(16r^3)}</math>
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| :::<math>= \frac{\pi}{\sqrt{18}} \approx 0.74.\,\!</math>
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| ==APF of common structures==
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| By similar procedures, the ideal atomic packing factors of all crystal structures can be found. The common ones are collected here as reference, rounded to the nearest hundredth.
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| g৳
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| * [[Hexagonal crystal system|Hexagonal close-packed]]: 0.74
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| * [[Face-centered cubic]]: 0.74
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| * [[Diamond cubic]]: 0.34
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| ==See also==
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| *[[Crystal]]
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| ==References==
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| # {{cite book
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| | author = Schaffer, Saxena, Antolovich, Sanders, and Warner
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| | year = 1999
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| | title = The Science and Design of Engineering Materials
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| | edition = Second Edition
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| | pages = 81–88
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| | publisher = WCB/McGraw-Hill
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| | location = New York
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| }}
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| # {{cite book
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| | author = Callister, W.
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| | title = Materials Science and Engineering
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| | year = 2002
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| | edition = Sixth Edition
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| | publisher = John Wiley and Sons
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| | pages = 105–114
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| | location = San Francisco
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| }}
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| [[Category:Crystallography]]
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Greetings! I am Myrtle Shroyer. California is our birth location. One of the extremely very best things in the globe for me is to do aerobics and I've been performing it for quite a while. Since she was eighteen she's been operating as a meter reader but she's always wanted her own business.
Feel free to surf to my web site std testing at home - hop over to these guys,