Interplanar separation

exercise 3_4937

Consider a plane ( hkl ) in a crystal lattice described by primitive translation vectors a 1, a 2 , and a 3 .

( a ) Prove that the reciprocal lattice vector G = h b 1 + k b 2 + l b 3 is perpendicular to this plane where the primitive translation vectors of the reciprocal lattice are defined as

b 1 = 2 p ( a 2 x a 3 ) / ( a 1 a 2 x a 3 )

b 2 = 2 p ( a 3 x a 1 ) / ( a 1 a 2 x a 3 )

b 3 = 2 p ( a 1 x a 2 ) / ( a 1 a 2 x a 3 )

and b i a j = 2 p d ij .

(b) Prove that the distance between two adjacent parallel planes of the lattice is d ( hkl ) = 2 p / | G | .

(c) Show for a simple cubic lattice that d 2 = a 2 / ( h 2 + k 2 + l 2 ).