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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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- Three '09