The eigenvalues c ( R ) of the translation operators are related because of the condition: T ^ ^ R 1 T ^ ^ R 2 = T ^ ^ R 2 T ^ ^ R 1 = T ^ ^ R 1 + R 2 We have, T ^ ^ R 1 T ^ ^ R 2 ψ ψ = c ( R 1 ) T ^ ^ R 2 ψ ψ = c ( R 1 ) c ( R 2 ) ψ ψ And, T ^ ^ R 1 + R 2 ψ ψ = c ( R 1 + R 2 ) ψ ψ Therefore, it follows that, c ( R 1 + R 2 ) = c ( R 1 ) c ( R 2 ) Now let the a i 's be the three primitive vector for the Bravais lattice. By a suitable choice of x i , we can always write c ( a i ) in the form c ( a i ) = e 2 π π i x i If R is a general Bravais lattice vector, given by R = n 1 a 1 + n 2 a 2 + n 3 a 3 it follows then, c ( R ) = c ( n 1 a 1 + n 2 a 2 + n 3 a 3 ) = c ( n 1 a 1 ) c ( n 2 a 2 ) c ( n 3 a 3 ) = c ( a 1 ) n 1 c ( a 2 ) n 2 c ( a 3 ) n 3 Substituting c ( a i ) = e 2 π π i x i one gets, c ( R ) = e 2 π π i ( n 1 x 1 + n 2 x 2 + n 3 x 3 ) = e i k ⋅ ⋅ R where k = x 1 b 1 + x 2 b 2 + x 3 b 3 and the b i 's are the reciprocal lattice vectors satisfying the equation b i ⋅ ⋅ a j = 2 π π δ δ i j