Cycle sequences of crystal structures

Cycle sequences are topological invariants of crystal structures (i.e. when two structures have different cycle sequences, then they are topologically inequivalent). It seems that cycle sequences are more discriminative than other known invariants, at least for zeolites (compare the "Atlas of Zeolite Structure Types" for other invariants).

For purposes of zeolite topology a cycle of a tetrahedral atom T is defined as a sequence T = T₁, T₂, ..., T_k, T_k+1 = T (k >= 3) of tetrahedral atoms T_i with the property that the T_i's (i = 1, 2, ...., k) are all different and that each T_i is connected to T_i+1 via an oxygen bridge (for other structures an atom binding may be more appropriate to be used). The number k is called the length of the cycle. The cycle sequence C(T) of the atom T is then defined as the sequence c₃, c₄,..., c_n,... in which each term c_n is equal to the number of cycles of length n to which the atom T belongs. Note that cycles of lengths 1 and 2 are undefined, as they do not appear for crystal structures.

The cycle sequences of the tetrahedral atoms of more than 100 zeolites are accessible via the accompanying table. The sequences were calculated with the program TOPOLAN.

Cycle sequences are not restricted to zeolites and the above definition can be easily generalized in order to be applicable to all kinds of crystal structures. For example, the cycle sequence of a vertex of the 2-dimensional square net is the sequence 0, 4, 0, 12, 0, 56, 0, 280,.... Here again the program TOPOLAN was used for the calculation of the sequences.