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Obtain a relation between radis of anatom and edgelength in the followingcrystalline solids. (1)Body - centred cubic crystal. (2) Face - centred cubiccrystal. |
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Answer» Solution : (1) Body-centred cubic (BCC) structure : In this unit cell, 8 atomsare presentat 8 corners and ONE additional atom is presentat the body center. The atmosare in contactalongthebodydiagonal BF. Let a be theedgelengthand rthe radiusof an atoms. Consider a triangleBCE. `BE^(2) = BC^(2) + CE^(2) = a^(2) +a^(2) =2a^(2)` Consider triangleBEF. `BF^(2)=BE^(2) + EF^(2) = 2a^(2) + a^(2) =3a^(2)` `thereforeBF = sqrt(3)a` From figure , BF = 4r . `therefore 4r = sqrt(3)a` `thereforer = (sqrt(3))/(4)a`  (2) Face-centred cubic (FCC) structure : In the unit cell,there are 8 atomsat 8 coners and 6 atoms at 6 face centre . Theatoms are in contantalongthe facedigaonal BD. Let a be theedgelength and r, theradiusof an atoms. Considera triangleBCD. `BD^(2) = BC^(2) +CD^(2)` `=a^(2) +a^(2) = 2a^(2)` `therefore""BD = sqrt(2a)` From figureBD = 4r `therefore""4r = sqrt(2)a` `therefore ""r=(sqrt(2))/(4)a = (a)/(2sqrt(2))` `r = (a)/(2sqrt(2))` 
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