Ring statistics

Model Studies. The major difficulty associated with separation and analysis of HPD arises from the fact that there are two hydroxyl and two carboxyl groups positioned unsymmetrically on the porphyrin ring. Statistically, there are 3 ether structural isomers and 4 esters possible between two hematoporphyrin molecules and each of these is a diastereomeric mixture. No doubt, this situation has made the isolation and characterization for a specific structure a nightmare. 

Because the randomization process introduces odd rings, one might suppose that when the ring statistics have reached a steady state with only small fluctuations about the steady-state value the structure has been randomized, but it is not so. The steady-state value for ring statistics is reached well before the structure has been randomized [23]. 

WWW algorithm. Ring statistics for the model are presented in Appendix A, along with those for a-Si models generated from the FC-2 structure. 

Here we define and use irreducible rings for ring statistics. 

In Table 1, we list the irreducible ring statistics for two crystal structures, FC-2 and BC-8, and for four models of amorphous silicon. When a bond-pair switch is introduced into the otherwise perfect FC-2 structure, the number of irreducible rings is conserved. Four 5-folds and eight 7-folds are created, but twelve 6-folds are eliminated. This conservation rule holds until the regions of bond-switching overlap, and is not grossly violated even then in the randomization process. Note that the total number of irreducible rings per atom is exactly two for the FC-2 structure and it is almost two for all the amorphous structures. 

Analysing our results in terms of ring statistics we concluded that atomic charge distribution in a-Si is determined primarily by bond angle distortions that are dependent on the number of fivefold and perturbed by the number of sixfold rings attached to the same atom [255]. 

Fig. 16. Ring statistics (a) and possible schemes for fast phase change (b) in a-AIST and a-...

Matsunaga, T. Yamada, N. Tanaka, Y. Suematsu, H. Takata, M. (2006). Structural Basis for the Fast Phase Change of Ge2Sb2Tes Ring Statistics Analogy between the Crystal and Amorphous States. Applied Physics Letters, Vol.89, No.20, (November 2006), pp. 021910-1-3, ISSN 0003-6951 Kolobov, A.V. Fons, P. Frenkel, AL Ankudinov, A.L. Tominaga, J. Uruga, T. (2004). 

It has been reported in the literature that the structures obtained from RMC contain some 3 and 4 membered rings, which are high energy structures and are thus unphysical. These are believed to be an artifact of the RMC method [11]. To test this we calculated the ring statistics of the RMC and the relaxed structures using the method of Franzblau [12]. 

Fig. 5. Ring statistics (rings contain only carbon atoms) comparing RMC, REBO relaxed and TB relaxed models.

Fig. 10. Ring statistics, comparing RMC and REBO relaxed models for CSIOOO.

Fig. 17.5 Ring statistics for a-GST (300 K) and f-GST (900 K). ABAB refers to even-membered rings with bond alternation...

Franzblau D. S., Computation of ring statistics from network models of solids, Phys. Rev. B, 44,4925-4930 (1991). 

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