Once we have found that many different topological structures
may appear in closely related crystals, it becomes very important
to c$determine the reasons behind such variability.
In doing this analysis, we will concentrate into **C** and **R**
schemes, the **B10** one being rare as it was described before.

We can define topological ionic radii as the
distance from the nucleus to the surface of its attraction basin. It
is evident from Fig. 5.2
that such topological
radii do depend strongly on the direction being analyzed. Not all
directions are equally important from the chemical point of view:
the directions that contain
bonds are the most significant ones for the crystal stability. In
this way, the b1 CP, that represents a *M*-*X* bond, provides
estimated ionic radii for *M* and *X*, that we will designate
as R(M) and R1(X), respectively. Similarly, the b2 CP
represents an *A*-*X* bond, and provides the R(A) and R2(X) radii.
The b3 CP, found in many crystals, corresponds to a *X*-*X* bond
and gives R3(X).

We have found, after some initial surprise, that the ratio of ionic
radii along the *M*-*X* and *A*-*X* bond lines
completely determines the topological scheme exhibited by the crystal.
To show this we
have prepared Fig. 5.3, in which we have plotted the ratio
R(M)/R1(X) * versus* the ratio R(A)/R2(X), every crystal being
represented by a symbol different for each type of topological scheme.

It is observed that the **C** and **R**
topological families are decided according to the value of R(A)/R2(X).
If the *A* ion is larger than about 0.88 times the
*X* ion in the *A*-*X* bond line, the crystal belongs to the
**C** family, and it belongs to the **R** family if it is smaller.

The R(M)/R(X) ratio, on the other hand, is in control of the
topological complexity within each family: the larger *M* is
in comparison to *X*, the simpler the topological scheme appears.

It is interesting to notice that topological radii can be compared and behave similarly to Shannon's empirical radii However, only the topological radii provide an exact criterion to predict the topological family.

Fig. 5.3 The topological schemes are determined by the ratio of topological radii R(A)/R2(X) and R(M)/R1(X).Admin.: