1.58: 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 3 N basis vectors \(A_j^*\) such that
\[a_i^{*\nu}~=~\sum_{j=1}^n Z_{ij}^{\nu} A_j^* ~~~(i=1,2,3), \nonumber\]
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.