A certain metal can crystallise in simple cubic (sc), body-centred cubic (bcc), and face-centred cubic (fcc) lattices. Assuming the atomic radius r of the metal remains constant across all three forms, let a₁ , a₂ , and a₃ be the unit cell edge lengths for the sc, bcc, and fcc lattices, respectively. The value of the expression 3 ( a₂ a₃ a₁^2 )^2 is _________.

Step-by-step solution

The relationship between the edge length a and the atomic radius r for the three cubic lattices is as follows: For simple cubic (sc): a₁ = 2r For body-centred cubic (bcc): 3 a₂ = 4r a₂ = 4r 3 For face-centred cubic (fcc): 2 a₃ = 4r a₃ = 2 2 r Substitute these into the given expression a₂ a₃ a₁^2 : a₂ a₃ a₁^2 = ( 4r 3 ) (2 2 r) (2r)^2 a₂ a₃ a₁^2 = 8 2 r^2 3 4r^2 = 2 2 3 Now, square this term: ( a₂ a₃ a₁^2 )^2 = ( 2 2 3 )^2 = 8 3 Finally, multiply by 3: 3 8 3 = 8 Answer: 8