# Direct space

### From Online Dictionary of Crystallography

##### Revision as of 13:47, 10 November 2017 by BrianMcMahon (talk | contribs) (Tidied translations and corrected German (U. Mueller))

Espace direct (*Fr*). Direkter Raum (*Ge*). Spazio diretto (*It*). 直空間 (*Ja*). Espacio directo (*Sp*).

## Definition

The direct space (or *crystal space*) is the *point space*, *E ^{n}*, in which the structures of finite real
crystals are idealized as infinite perfect three-dimensional structures. To this space one associates the

*vector space*,

*V*, of which lattice and translation vectors are elements. It is a

^{n}*Euclidean*space where the scalar product of two vectors is defined. The two spaces are connected through the following relations:

(i) To any two points *P* and *Q* of the point space *E ^{n}* a vector

**PQ**=

**r**of the vector space

*V*is attached.

^{n}(ii) For each point *P* of *E ^{n}* and for each vector

**r**of

*V*there is exactly one point

^{n}*Q*of

*E*for which

^{n}**PQ**=

**r**holds.

(iii) If *R* is a third point of the point space, **PQ** + **QR** = **PR**.

## See also

- Direct lattice
*International Tables for Crystallography, Volume A*, 6th edition