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Calculate the percentage efficiency of packing in case of body centered cubic crystal. |
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Answer» Solution :Packing efficiency `:` Here, the spheres are touching along the leading DIAGONAL of the cube as shown in the figure. In `Delta ABC` `AC^(2) = AB^(2) + BC^(2)` `AC = sqrt( AB^(2) + BC^(2))` `AC = sqrt(a^(2) +a^(2)) = sqrt(2a^(2)) = sqrt(2) a` In `Delta ACG` `AG^(2) = AC^(2) + CG^(2)` `AG= =sqrt( AC^(2) +CG^(2))` `= AG = sqrt((sqrt(2a))^(2) + a^(2))` `AG = sqrt( 2a^(2) + a^(2)) = sqrt( 3a^(2))` `AG =sqrt(3) a` i.e., `sqrt(3) a = 4r` ` r(sqrt(3) )/( 4) a` `:.` Volume of the sphere with radius 'r' `= ( 4)/( 3) pir^(3)` `= ( 4)/( 3) pi ((sqrt(3))/( 4) a)^(3)` `= ( sqrt(3))/(16) pi a^(3)`....(1) Number of spheres belong to a unit cell in bcc arrangement is EQUAL to TWO and hence the total volume of all spheres. `=2 XX ((sqrt(3) pia^(3))/(16))= (sqrt(3) pia^(3))/( 8)` Dividing (2) by (3) Packing fraction `= ((sqrt(3) pi a^(3)))/((a^(3))) xx 100` `= ( sqrt(3)pi)/( 8) xx100` `= sqrt(3) pi xx 12.5 ` `= 1.732 xx 3.14 xx 12.5` `= 68%` i.e., 68% of the available volume is occupied. The available space is used more efficiently than in simple cubic packing. |
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