Ring statistics have been developed mainly to study the connectivity of amorphous materials.

However, the wide variety of sequences and structural defects that may be encountered in these materials make this type of analysis particularly challenging.

Some scenarios should therefore be reviewed, to lay the groundwork to look for rings in these compounds. In particular we need to evaluate the influence of different bond defects in the amorphous network.

However, in each case both the impact on the connectivity, and the impact on the physical properties which can be deduced from the analysis, have to be studied.

Furthermore it is necessary to determine if the influence of the bond defects depends on the criteria used to define a ring.

### ABAB and BABA rings

The ring statistics of amorphous networks are often focused on finding rings made of a succession of atoms with an alternation of chemical species, called ABAB rings.

The most common examples come from the alternation of Si and O atoms (in silica polymorphs) or Ge and S (in GeS_{2} polymorphs). These solids are usually built with tetrahedra (SiO_{4} or GeS_{4} ) therefore we study the network distribution of tetrahedra.

The ideal technique to setup the analysis of such systems is to choose the atoms of highest coordination to initiate the search, respectively Si in SiO_{2} and Ge in GeS_{2} . In most cases all rings can be found using this method. Nevertheless we can demonstrate that some solutions, so some rings, can be ignored by this analysis. This is highlighted in figure [Fig. 1] which represents a cluster of atoms isolated from an AB_{2} amorphous network.

**Figure 1**: Cluster of atoms isolated from an AB

_{2}amorphous network. A bond defect is located on an atom of the chemical species B (blue square). When looking for King's shortest paths [a, b] using the chemical species A to initiate the search the central ring with 10 nodes is ignored. However among the solutions of the analysis (with the initial nodes circled in green) other rings with 10 nodes are found in the network.

We can see that this piece of network is characterized by a bond defect. An atom of the B species appears to be over-coordinated by three atoms of the A species. When looking for rings, using the King's criterion [a, b] and initiating the search using the A atoms, the central ring with 10 nodes is ignored. Nevertheless other rings with 10 nodes are found and stored as solutions of the analysis. In order to find the central ring the search has to be initiated from the over-coordinated B atom.

By analogy with the terminology ABAB this ring can be called a BABA ring. Indeed the alternation of chemical species is well respected. Therefore it is legitimate to question the relevance of the analysis without this result.

In other words we have to check out if this BABA ring is, or not, an ABAB ring.

The properties of this ring meet the definition and can therefore improve the description of the connectivity of the network. This kind of coordination defect [Fig. 1] is uncommon in vitreous silica [c, d], nevertheless it is frequent in chalcogenide glasses [e, f]. Therefore it is justified to discuss the impact of such BABA rings in the analysis of these compounds.

Homopolar bonds

In amorphous materials the homopolar bond defects can have a significant influence on the ring statistics. This is true in particular for AB_{2} chalcogenide glasses. Figure [Fig. 2] illustrates standard cases that may be encountered when looking for rings in an AB_{2} system which contains homopolar bonds.

**Figure 2:**Illustration of the influence of homopolar bonds in ABAB rings: in both examples the smallest rings found when initiating the search using the circled nodes (green color) contain an homopolar bond A-A or B-B.

The smallest rings found when initiating the search using the circled nodes (green color) are not ABAB rings. Therefore their size must be given using the total number of nodes.

In figure [Fig. 2] the smallest rings are a ring with 9 nodes and a ring with 11 nodes containing respectively an A-A and a B-B homopolar bond. These rings are significantly smaller than the shortest ABAB ring with 18 nodes that may be found when looking for rings using the ame green-circled nodes to initiate the analysis [Fig. 2].

Therefore it is once again justified to discuss the impact of such bond defects when analyzing the connectivity of amorphous chalcogenides using ring statistics.

### In the R.I.N.G.S. code

The R.I.N.G.S. code has been developed in order to let the user define very precisely the ring statistics he/she wants to perform, including a full analysis of the impact of the bond defects.

Furthermore the R.I.N.G.S. code does not focus only on the analysis of the connectivity, i.e. finding the rings, but also on the physical properties which can be deduced from the ring statistics.

### => More informations on features of the R.I.N.G.S. code

**=> More informations on ring statistics in the R.I.N.G.S. code**

**=> More informations on ring statistics**

- a
- S. V. King.
*Nature*,**213**:1112 (1967). - b
- D. S. Franzblau.
*Phys. Rev. B*,**44**(10):4925-4930 (1991). - c
- W. Jin, P. Vashishta, R. K. Kalia, and J. P. Rino.
*Phys. Rev. B*,**48**(13):9359-9368 (1993). - d
- J. P. Rino, I. Ebbsjö, R. K. Kalia, A. Nakano, and P. Vashishta.
*Phys. Rev. B*,**47**(6):3053-3062 (1993). - e
- S. Blaineau and P. Jund.
*Phys. Rev. B*,**69**(6):064201 (2004). - f
- S. Le Roux and P. Jund.
*J. Phys.: Cond. Mat.*,**19**(19):169102, 2007.

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