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, ..., Tk, Tk+1 = T (k >= 3) of tetrahedral atoms Ti with the property that the Ti's (i = 1, 2, ...., k) are all different and that each Ti is connected to Ti+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,..., cn,... in which each term cn 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.
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.
Cycle sequences have been developed by Prof. W.E. Klee and Georg Thimm, both welcoming questions and comments.
Page maintained by Georg Thimm
[This is a copy of the page http://www.drc.ntu.edu.sg/users/mgeorg/zeolites/cycles.html made on August 28, 2000 in order to avoid problems with broken links.]