Calculating the packing ratio of a crystal structure

Calculating the packing ratio of a crystal structure - r...

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Calculating the packing ratio of a crystal structure (C) Michael Cortie, UTS, 18 th March 2009 Example : body centred cubic If we know the crystal structure, then it is possible to calculate how well packed the atoms are. As mentioned in the lecture, the closest possible packing arrangements are face centred cubic (“fcc”), and hexagonal close packed (“hcp”), both of which give a 74% space filling ratio when the structures are imagined to be made of hard balls. Here we will work out the proportion of the volume that is filled by atoms in a body centred cubic structure (“bcc”). First, note the arrangement of a bcc unit cell actually contains 1 internal atom and 8 one-eighth atoms: Figure 1 Note also that the atoms “touch” through the internal cube diagonals in this structure, i.e. in the drawing below atoms touch along the line AC. Figure 2 Let the radius of an atom = r . From Figure 1 we can see that distance AC = 4 r Let the edge of the cube, e.g. BC, have an unknown length = a We need to work out a in terms of
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Unformatted text preview: r look at triangle ABD: AB 2 =BD 2 + AD 2 = 2a 2 so AB= 2.a Now look at the triangle ABC, angle ABC is 90, and BC (the cube edge) = a because thats what we defined a as. So: AC 2 =AB 2 +BC 2 substituting 4 r for AC, 2. a for AB and a for BC, we get (4 r ) 2 = ( 2. a ) 2 + a 2 (and after some simplification) a = (4/ 3) . r Let r =1 length units (it does not matter what length units we use) so a = 2.309 length units and the volume of the unit cell is a 3 , or 12.31 volume units The volume of 2 spheres (remember that there were an equivalent 2 atoms on the unit cell) is 2x 4/3. . r 3 = 8 /3 volume units for atoms = 8.378 volume units So the ratio of filled volume (atoms!) to total volume (empty space) is 8.378/12.31=0.68 or 68% (notice that the units cancelled out, that is why it did not matter what value we assigned to r in the very beginning. A similar type of argument can be used to calculate the packing ratio of fcc and hcp structures, or indeed any crystal structure....
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Calculating the packing ratio of a crystal structure - r...

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