1.

a) Calculate the packing efficiency of particles in a body centred cube.

Answer»

Solution :The number of atoms per unit cell in bcc structure is two. Each atom is considered as one sphere.
Let the edge length of the unit cell = a
Radius of the sphere = r
Length of the body diagonal = c
Length of the body diagonal = B
In `DeltaEFD`
`FD^(2)=EF^(2)+DE^(2)`
`b^(2)=a^(2)+a^(2)=2a^(2)`
Now in `DeltaAFD`
`AF^(2)=AD^(2)+FD^(2)`
Now in `DeltaAFD`
`AF^(2)=AD^(2)+FD^(2)`
`c^(2)=a^(2)+b^(2)`
`c^(2)=a^(2)+2a^(2)=3a^(2)`
`c =sqrt3a`
`W.K.T c=4r`
Therefore `=4r=sqrt3a`
`a=(4r)/(SQRT3)`

`"Total volume of the unit cell "= a^(3) = ((4r)/(sqrt3))^(3)`
Since bcc LATTICE contains 2 atoms per unit cell
`"Volume of two spheres "=2 xx(4)/(3)pir^(3)`
`"Packing efficiency "= ("Volume of the two spheres in unit cell")/("Total volume of the unit cell")xx100%`
`=(2xx(4)/(3)pir^(3))/(((4r)/(sqrt3))^(3))xx100%`
`=((8)/(3)pir^(3))/((64)/(3SQRT3)r^(3))xx100=68%`


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