1.91: Reciprocal Space
The basis vectors a* , b* , c* of the reciprocal space are related to the basis vectors a , b , c of the direct space (or crystal space) through either of the following two equivalent sets of relations:
(1)
a* . a = 1; b* . b = 1; c* . c = 1;
a* . b = 0; a* . c = 0; b* . a = 0; b* . c = 0; c* . a = 0; c* . b = 0.
(2)
a* = ( b × c )/ ( a , b , c );
b* = ( c × a )/ ( a , b , c );
c* = ( b × c )/ ( a , b , c );
where ( b × c ) is the vector product of basis vectors b and c and ( a , b , c ) = V is the triple scalar product of basis vectors a , b and c and is equal to the volume V of the cell constructed on the vectors a , b and c .
The reciprocal and direct spaces are reciprocal of one another, that is the reciprocal space associated to the reciprocal space is the direct space. They are related by a Fourier transform and the reciprocal space is also called Fourier space or phase space .
The vector product of two direct space vectors, \({\bold r_1} = u_1 {\bold a} + v_1 {\bold b} + w_1 {\bold c}\) and \({\bold r_2} = u_2 {\bold a} + v_2 {\bold b} + w_2 {\bold c}\) is a reciprocal space vector,
Reciprocally, the vector product of two reciprocal vectors is a direct space vector.
As a consequence of the set of definitions (1), the scalar product of a direct space vector r = u a + v b + w c by a reciprocal space vector r* = h a* + k b* + l c* is simply:
r . r* = uh + vk + wl .
In a change of coordinate system , The coordinates of a vector in reciprocal space transform like the basis vectors in direct space and are called for that reason covariant . The vectors in reciprocal transform like the coordinates in direct space and are called contravariant .