# Incommensurate composite crystal

An incommensurate composite crystal is a compound with two or more (N) subsystems that are themselves modulated structures, with basis structures that are mutually incommensurate. Each subsystem (numbered by ν) has a reciprocal lattice for its basic structure with three basis vectors $a_i^{*\nu}$. There is a basis of the vector module of diffraction spots that has at most 3N basis vectors $A_j^*$ such that
$a_i^{*\nu}~=~\sum_{j=1}^n Z_{ij}^{\nu} A_j^* ~~~(i=1,2,3),$
where $Z_{ij}^{\nu}$ are integer coefficients. If n is larger than the dimension of space (three), the composite crystal is an aperiodic crystal. n is the rank of the vector module.