The basis vectors a*, b*, c* of the
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.
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, band 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
The vector product of two direct space vectors, and 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
r . r* = uh + vk +wl.
In a change of
The volume V* = (a*, b*, c*) of the cell constructed on the reciprocal vectors a*,b* and c* is equal to 1/V.
The lengths a*, b*, c* of the reciprocal basis vectors and the angles, α*, β*, γ*, between the pairs of reciprocal vectors (b*, c*), (c*, a*), (a*, b*), are related to the corresponding lengths and angles for the direct basis vectors through the following relations:
a* = b c sin α/V; b* = c a sin β/V; c* = a b sin γ/V;
cos α* = (cos βcos γ - cos α)/|sin β sin γ|; cos β* = (cos γcos α - cos β)/|sin γ sin α|; cos γ* = (cos αcos β - cos γ)/|sin α sin α|.
The notion of reciprocal vectors was introduced in vector analysis by J. W. Gibbs (1881 - Elements of Vector Analysis, arranged for the Use of Students in Physics. Yale University, New Haven).
Section 5.1, International Tables of Crystallography, Volume A
Section 1.1, International Tables of Crystallography, Volume B
Section 1.1, International Tables of Crystallography, Volume C
Section 1.1.2, International Tables of Crystallography, Volume D