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Chemistry LibreTexts

Vector module

vector module is the set of vectors spanned by a number n of basis vectors with integer coefficients. The basis vectors should be independent over the integers, which means that any linear combination


with mi integers is equal to zero if, and only if, all coefficients mi are zero. The term Z-module is sometimes used to underline the condition that the coefficients are integers. The number of basis vectors is the rank of the vector module.


An n-dimensional lattice in an n-dimensional vector space is an example of a vector module, with rank n. In reciprocal space, the reciprocal lattice corresponding to a crystallographic structure is a special case of a vector module. The Bragg peaks for the crystal fall on the positions of the reciprocal lattice. More generally, the Bragg peaks of an m-dimensional aperiodic crystal structure belong to a vector module of rank n, larger than m. To indicate that this module exists in the reciprocal space, it is sometimes called Fourier module.