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Calculate packing efficiency in body-centred cubic structure. |
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Answer» Solution : In a body-centred unit cell, the spheres LOCATED at the corners donot touch each other but they are in contact with body centred atom. In `triangleEFD` `b^2 = a^2 + a^2` `2a^2` `b = sqrt(2)a`. In `triangleAFD` `c^2 = a^2 + b^2` ` = a^2 + 2a^2` ` = 3a^2` `c = sqrt(3)a` The length of the body diagonal c is equal to 4r, where r is the radius of sphere, as all three spheres along the diagonal touch each other. Therefore,`sqrt(3)a = 4r` `a = (4r)/(sqrt3)` Or `r = (sqrt3)/(4)a` In BCC, the total number of ATOMS per unit cell are two. Hence the total volume of two atoms is`(2 xx 4/3 pi r^3)` Volume of the cube = `a^3 = ((4)/(sqrt3) r)^(3)` % PACKING efficiency =`("Volume occupied by FOUR spheres in unit cell" xx 100)/("Volume of the unit cell")` `= (2 xx (4/3) pi r^3 xx 100)/((4/(sqrt4) r)^3) = 68%`. 
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