# Difference between revisions of "Direct space"

### From Online Dictionary of Crystallography

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BrianMcMahon (talk | contribs) (Tidied translations and corrected German (U. Mueller)) |
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− | = | + | <font color="blue">Espace direct</font> (''Fr''). <font color="red">Direkter Raum</font> (''Ge''). <font color="black">Spazio diretto</font> (''It''). <font color="purple">直空間</font> (''Ja''). <font color="green">Espacio directo</font> (''Sp''). |

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− | Espace direct (''Fr''). | ||

== Definition == | == Definition == | ||

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(i) To any two points ''P'' and ''Q'' of the point space ''E<sup>n</sup>'' a vector '''PQ''' = '''r''' of the vector space | (i) To any two points ''P'' and ''Q'' of the point space ''E<sup>n</sup>'' a vector '''PQ''' = '''r''' of the vector space | ||

− | ''V<sup>n</sup>'' is attached | + | ''V<sup>n</sup>'' is attached. |

(ii) For each point ''P'' of ''E<sup>n</sup>'' and for each vector '''r''' of ''V<sup>n</sup>'' there is exactly one point ''Q'' of | (ii) For each point ''P'' of ''E<sup>n</sup>'' and for each vector '''r''' of ''V<sup>n</sup>'' there is exactly one point ''Q'' of | ||

− | ''E<sup>n</sup>'' for which '''PQ''' = '''r''' holds | + | ''E<sup>n</sup>'' for which '''PQ''' = '''r''' holds. |

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

== See also == | == See also == | ||

− | [[ | + | *[[Direct lattice]] |

− | + | *''International Tables for Crystallography, Volume A'', 6th edition | |

[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] |

## Latest revision as of 13:47, 10 November 2017

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