# Reciprocal space

### From Online Dictionary of Crystallography

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

## 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, [math]{\bold r_1} = u_1 {\bold a} + v_1 {\bold b} + w_1 {\bold c}[/math] and [math]{\bold r_2} = u_2 {\bold a} + v_2 {\bold b} + w_2 {\bold c}[/math] is a reciprocal space vector,

[math] {\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*}. [/math]

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].

## See also

- Reciprocal lattice
*The Reciprocal Lattice*(Teaching Pamphlet No. 4 of the International Union of Crystallography)- Chapter 1.1 of
*International Tables for Crystallography, Volume B* - Chapter 1.1 of
*International Tables for Crystallography, Volume C* - Chapter 1.1.2 of
*International Tables for Crystallography, Volume D*