Some Explanations on Ring Statistics
In the literature, many different definitions of "rings" can be found. In
CRYSTANA we refer to a definition given in
(
Goetzke, K., Klein, H.-J.: Properties and Efficient Algorithmic Determination of Different Classes
of Rings in Finite and Infinite Polyhedral Networks, Journal of
Non-Crystalline Solids, 127 (1991), 215-220).
The ring definition given there is based upon the representation of crystal
structures by graphs. In case of silicates, vertices of such a graph
represent tetrahedra and two vertices are connected by an edge if the corresponding
tetrahedra share a common corner (in this part of CRYSTANA we only consider corner-sharing tetrahedra).
As usual, a cycle of a graph is a closed path with vertices all distinct.
A cycle is a ring if for each pair of vertices of the cycle at least one of
the shortest paths beween these vertices is part of the cycle.
Equivalent definitions are:
A ring is a cycle without short-cuts
(a path between two vertices of a
cycle is called a short-cut of the cycle if it is shorter than both pahts between these
vertices given by the cycle).
A ring is a cycle which is not the sum of two shorter cycles
(the sum of two or more cycles is the set of those edges which are contained in
an odd number of the cycles; in particular, the sum of two cycles is the set
of those edges that are part of one of the cycles but not of both).
In the reference given above an efficient algorithm is presented for computing rings.
This algorithm is used in CRYSTANA.
Remarks:
In a paper by L.W. Hobbs et al. (Philos. Mag. A 78, p. 679, 1998) some
wrong comments on this algorithm can be found. They can easily be refuted by
carefully reading our description of the algorithm.
In a paper by X. Yuan and A.N. Cormack (Comp. Materials Science 24, pp. 343-360, 2002)
it is stated that "a definition of equivalent ring is not provided" in our paper.
This is a wrong comment as well since we explicitly state in that paper (p. 219) that
"ring statistics of each structure contain for each ring length the number of rings and very
strong rings that are not translationally equivalent".
Ring statistics in CRYSTANA have been refined insofar as we consider all symmetries
when building ring statistics. To get the number of rings of a certain size being
not translationally equivalent it is sufficient to count all rings listed for
the different equivalence classes (for an example, see below).
There are four different forms of ring statistics CRYSTANA may be asked for:
- General information on rings of a structure
- Possible coverings of nodes by rings
- Possible covering of edges by rings
- Details of the symmetry classes of rings.
In the following we explain the information content of these statistics using
alpha-quartz as an example.
General information on rings of a structure ("ring statistic")
| ring |diff. rings|
crystal |reference |spgr.| length | per class | symmetry operations
-----------------+----------+-----+--------+-----------+------------------------
| | | | |
ALPHA-QUARTZ | 1/01/79 | 154 | 6 | 3 | 3.2 [001], 3.1 [001]
| | | 8 | 3 | 3.2 [001], 3.1 [001]
| | | | 6 | 2 [110], 3.2 [001],
| | | | | 2 [100], 2 [010],
| | | | | 3.1 [001]
| | | | 3 | 3.1 [001], 3.2 [001]
| | | | 3 | 3.1 [001], 3.2 [001]
| | | | |
Meaning of selected columns:
reference:
An internal key for distinguishing different publications of
the same structure.
spgr.:
The space group number as given in the International Tables.
diff. rings per class:
Number of equivalent rings in a class; in the example,
there are 4 different classes of rings having length 8. The first one has 3 elements
which can be mapped onto each other by applying the symmetry operations given in
the last column.
For alpha-quartz we get three 6-membered rings and fifteen 8-membered rings which
are not translationally equivalent (just add the numbers of rings in the different
classes for each ring size).
symmetry operations:
Printed symbols of the symmetry elements of the corresponding
class are given including their location.
Covering of nodes by rings ("node covering")
When looking for possibilities to cover all nodes (i.e. the tetrahedra) by rings
(i.e. by sets of tetrahedra forming a ring) short rings are preferred to larger rings
and the number of rings in a covering is guaranteed to be minimal. Furthermore, there
are no rings in a covering being symmetrically equivalent.
Since it is not clear how to choose among minimal coverings the "best one" we list
all possible minimal coverings (e.g.: for Clinoptilolite there are three node coverings
possible: one with a 4-membered ring and a 5-membered ring and two with a single
5-membered ring).
| | | || possible ring covers
| | number | complete||number of| ring | number of
structure | date |of nodes| covering|| possib. | size | classes
-----------------+----------+--------+---------++---------+--------+------------
| | | || | |
ALPHA-QUARTZ | 1.01.79 | 3 | y || 1 | 6 | 1
| | | || | |
Meaning of selected columns:
date:
Same meaning as reference column above.
number of nodes:
The number of nodes in the unit cell.
complete covering:
"y" ("n") means that it is (not) possible to cover all nodes by rings.
possible ring covers:
The number of different coverings is given for every possible
covering having the same characteristics (rings sizes and number of different classes for
each ring size in the covering)
For alpha-quartz there is a single possibility to cover all nodes by rings (according to our
rules guaranteeing minimality): a six-membered ring is sufficient, i.e. applying the symmetries
of spacegroup 154 to a representative of the corresponding ring class all nodes of the
structure are "met".
Covering of edges by rings ("edge covering")
The comments given for "node covering" hold as well; the only difference
is that edges (i.e. connections between tetrahedra) have to be covered instead of
nodes (i.e. tetrahedra).
| | | || possible ring covers
| | number | complete||number of| ring | number of
structure | SNUM |of edges| covering|| possib. | size | classes
-----------------+----------+--------+---------++---------+--------+------------
| | | || | |
ALPHA-QUARTZ | 1.01.79 | 12 | y || 1 | 6 | 1
| | | || | |
| | | || | |
Detailed information on single rings ("ring classes")
This statistics gives information on the tetrahedra forming rings.
Since every tetrahedron can be uniquely identified by its central silicon
atom it is sufficient to list all silicon atoms of a ring
in a list reflecting the sequence of the tetrahedra in the ring.
Atoms are referred by their name, number and translation as generated by
CRYSTANA (see
unit cell of alpha-quartz , for example). In the example below, SI 1 2 1 0 0 stands for the atom
obtained by first applying the symmetry operation "(2)" of the space group to
the atom in the asymmetric unit (there is only one atom in the asymmetric unit
of alpha-quartz, hence its number is 1) and then applying the translation (1,0,0).
*****************************************
RINGLENGTH = 6
CLASS = 1
SI 1 2 1 0 0
SI 1 1 0 0 0
SI 1 3 0 0 1
SI 1 1 0 1 0
SI 1 2 0 0 0
SI 1 3 0 0 0
#
SI 1 1 0 1 -1
SI 1 3 0 1 0
SI 1 2 1 1 0
SI 1 1 0 1 0
SI 1 2 0 0 0
SI 1 3 0 0 0
#
SI 1 1 0 0 -1
SI 1 2 1 0 -1
SI 1 1 1 1 -1
SI 1 3 1 0 0
SI 1 2 1 0 0
SI 1 3 0 0 0
*****************************************
RINGLENGTH = 8
CLASS = 1
SI 1 1 0 1 -1
SI 1 2 1 1 -1
SI 1 3 1 1 -1
SI 1 2 2 1 -1
SI 1 1 1 1 -1
SI 1 3 1 0 0
SI 1 2 1 0 0
SI 1 3 0 0 0
#
SI 1 1 0 1 -1
SI 1 3 0 1 0
SI 1 2 1 1 0
SI 1 1 1 2 0
SI 1 3 1 1 1
SI 1 1 1 1 0
SI 1 2 1 0 0
SI 1 3 0 0 0
#
SI 1 1 0 0 -1
SI 1 2 1 0 -1
SI 1 1 1 1 -1
SI 1 3 1 1 0
SI 1 2 1 1 0
SI 1 1 0 1 0
SI 1 2 0 0 0
SI 1 3 0 0 0