Reciprocal space

Espace réciproque (Fr). Reziprokes Raum (Ge). Espacio reciproco (Sp). Spazio reciproco (It). 逆空間 (Ja).

Definition

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,

${\bold r*} = {\bold r_1} \times {\bold r_2} = V (v_1 w_2 - v_2 w_1) {\bold a*} + V (w_1 u_2 - w_2 u_1) {\bold b*} + V (u_1 v_2 - u_2 v_1) {\bold c*}.$

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 coordinate system change, 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 space transform like the coordinates in direct space and are called contravariant.

Geometrical relationships

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 β).

History

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].