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Direct space

The direct space (or crystal space) is the point spaceEn, in which the structures of finite real crystals are idealized as infinite perfect three-dimensional structures. To this space one associates the vector spaceVn, of which lattice and translation vectors are elements. It is a 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 En a vector PQ = r of the vector space Vn is attached

(ii) For each point P of En and for each vector r of Vn there is exactly one point Q of En for which PQ = r holds

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

See also

Section  8.1 of International Tables of Crystallography, Volume A