For now it is impossible to compare the results of ring statistics to experimental data. Therefore we have decided to improve the description of the both the network and the material on the basis proposed by the ring statistics.
The main goal is to allow to link this analysis method to other physical properties of the material that can be measured experimentally.
The following article illustrates features implemented in the R.I.N.G.S. code in order to improve the results of the ring statistics.
Number of "potentially not found" paths
One of the first information possible to extract from ring statistics, except the number of rings, is the number of rings not found by the analysis. Indeed calculations times do strongly depend on the maximum search depth, ie. the maximum size of a ring. To carry out the analysis this value has to be chosen to get the best possible compromise between CPU time and quality of the description.
Nevertheless whatever this limiting value is, some rings of a size bigger than the maximum search depth may not be found by the analysis. In the King [a, b] and the Guttman [c] shortest paths criteria it is possible to evaluate the number of "potentially not found" paths during the search. Thus for a given atom At we can consider that a closed path exists and is not found:
- If the atom At has at least 2 nearest neighbors
- If no closed path is found:
- If the 2 nearest neighbors of the atom At have at least 2 nearest neighbors (to avoid non bridging atoms)
Thus if during the analysis these 3 conditions are full filled (1, 2-1, 3 for the Guttman's criterion, and 1, 2-2, 3 for the King's criterion) then we can say that this analysis has potentially missed a ring between the neighbors of atom At. The smaller this number of "potentially" missed rings will be the better this analysis will be and the better the description of the connectivity of the material studied will be. The term "potentially" has been chosen because this method allows only to avoid the first neighbor non bridging atoms.
In a second time we focused on the evaluation of standard values which are often computed for crystalline or amorphous systems. For each physical property values are averaged and classified based upon the size of rings.
In the R.I.N.G.S. code the physical properties illustrated in figure [Fig. 1] can be evaluated for each size of ring:
- The average coordination number for a particle in a ring with n nodes
- The average inter-atomic distances between particles in a ring with n nodes:
- Distances between first neighbor atoms in a ring with n nodes
- Distances between second neighbor atoms in a ring with n nodes
- The average angular distributions:
- Bond angles in a ring with n nodes
- Dihedral angles in a ring with n nodes
- The average "Vis à vis" distance in a ring with n nodes
it is usual to consider rings as structural units having a circular geometry. This is easy when imagining the nodes of ring as the vertices of cyclic polygon. Following this idea we have implemented the evaluation of the average distance separating 2 particles in "Vis à vis" on the ring. We have call this distance "pseudo-diameter", we also define the "pseudo-radius" as the half of the "pseudo-diameter"
For a ring with n nodes this "pseudo-radius" can be compared to 2 other values obtained thru calculation. On the one hand to the radius of the inscribed circle of a regular polygon with n vertices, Ri, also called the apothem of the polygon. On the other to the radius of the circumscribed circle of a polygon with n vertices, Rc. The edge of this regular polygon with n vertices is the average distance between first neighbor atoms in a ring with n nodes [Fig. 2].
Ri and Rc can be calculated using:
|θ = Rc = Ri =||(1)|
where n is the number of vertices of the polygon, θ is the angle at the intersection of the straight lines from the center of the circumscribed circle and passing by 2 first neighbor vertices, and
dAB is the average distance between first neighbor vertices.
Following this reasoning on rings geometry towards obviously to study their 3D properties. Thus it is easy to associate the ideas either of occupation or vacancy of space with the presence of rings in the material. In the R.I.N.G.S. code we have chosen to obtain more informations on the correlation between the positions of rings and voids in the material.
To achieve this goal we first have to calculate the position of the barycenters of rings with n nodes:
|Baryc(P) = P = x, y, z||(2)|
where m(i) represents the mass of particle i.
Then it is possible to use the positions of these fictious objects to solve the equations usually applied to the particles of the system. In particular we have implemented the study of the distribution of barycenters (and so of rings) thru the evaluation of the pair correlation functions gn(r). The function gn(r) relates to the distribution of the rings with n nodes.
Afterward it is necessary to define a method to determined the positions of voids in the material. The first step was to choose the limit between zones filled by particles and void zones. To do that we discretize space in the material to give to each point a value corresponding to the minimum distance separating this point from the closest particle in the network. Then we have to choose a maximal distance above which a point would be far enough from the particles and could therefore be considered in a void zone. In the R.I.N.G.S. code this particular distance defining voids is given by:
|dvoids(max) = Rmoy + and Rmoy =||(3)|
where Rcov(i) is the covalent radius of chemical species i, Rcov(min) is the smallest covalent radius of the different chemical species of the material and NS represents the number of chemical species in the material. This relation is arbitrary, and can be easily adapted to the compounds you are studying.
Following this method it is possible to isolate the positions of voids in the structure. Furthermore this method allows to study the distribution of voids according to their sizes. Thus it is possible to consider voids as spheres of radius more or less important which correspond to the distance separating the center of the sphere to the closest particle. Then as for the barycenters of rings we have implemented the study of the distribution of voids using the pair correlation functions gv(r). The functions gv(r) relates to the distribution of voids of size v in the material.
The informations obtained studying on the one hand the average positions of rings using their barycenters and on the other hand the distribution of voids discretizing space can be analyzed and compared.
The study of the correlation between positions of rings and voids was a first step in the analysis of the relation between the structural properties of the material and the ring statistics. The second step directly comes from this first analysis and consists in the evaluation of several neutron structure factors related to rings. In the calculation of the structure factor we simulate the effect of a beam radiation on the material, and in particular on the particles of the material which are therefore scattering centers. However it is possible to modify the nature of the scattering centers used in the calculation and then to compute neutron structure factors for the rings with n nodes.
The following cases of scattering centers have been implemented in the R.I.N.G.S. code:
- Utilization of the barycenters of the rings: this is the logical evolution of the study of the distribution of the rings barycenters
The main idea is to study the effect of the correlations between rings with n nodes.
- Utilization of the particles at the origin of the rings with n nodes, ie. the particles where looking for rings allow find ring(s) with n nodes
- Utilization of the all particles involved in rings with n nodes, ie. all the particles in ring(s) with n nodes
- The particles of each ring separately to obtain the average structure factor for a ring with n nodes
- S. V. King.
Nature, 213:1112 (1967).
- D. S. Franzblau.
Phys. Rev. B, 44(10):4925-4930 (1991).
- L. Guttman.
J. Non-Cryst. Solids., 116:145-147 (1990).
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