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 } ).