The forgoing list is meant to be illustrative not exhaustive.  Only the most common  .
elements or minerals possessing structural types that can be modeled with Polymorf .
are included. .
     The fact that many of these mineral groups are isostructural indicates that their .
underlying structural framework is as important in determining the group's physical .
and chemical properties as is their individual chemical composition.  The topological .
approach and the compositional approach to describing matter therefore should be .
seen as both sides of the same coin.  The one cannot be fully appreciated without the .
Topological classification system
     Another classification scheme emphasizes the internal structure of minerals and
the coordination of its constituent atom species.  It arranges minerals by the type and
percent occupation of interstitial voids present in their crystal structures, that is by the
way that the atoms pack together.
     Factors that affect how atoms of different species pack together include their
relative sizes, the direction and strength of the bonding forces of each atom type, and
the necessity for electronic neutrality of the structure.
     Atoms that pack together as the result of ionic or metallic bonds between them
tend to generate structures that are closely packed together.  Covalently bonded
atoms are oriented in specific directions relative to each other which affects how they
pack together to form solids.
     There are a few compounds formed from different elements with nearly equal
atomic volumes, and therefore can be modeled like the pure elements as one type of
close packing or another.  But the constituent atomic species of most compounds
have different volumes and therefore pack together according to the ratio of the
different atoms' volumes.  Given the wide range of atomic radii exhibited by the
elements and the thousands of different combinations they can achieve with each
other, compounds can exhibit a wide range in the radius ratios of their constituent
atom species.
     However, certain specific ratios of atomic radii give rise to general types of
mineral structures based on how the different sized atoms pack together.  Typically
the relatively larger non-metallic atoms, or anions (X), in a compound pack together
so as to provide interstitial voids between them which can accommodate the
relatively smaller metal ions, or cations (A).  In the following figure the largest size
sphere represents the anion with a radius of one and the smaller spheres are cations
of the various critical sizes.  The smaller A atoms can be accommodated in holes

Figure 56 -  Critical radius ratio values for
      cubic, octahedral, and tetrahedral
       coordination of interstitial atoms
formed by packing the largest X atom with a radius of 1.0 in the shape of a cube (void
size .732), an octahedron (void size .414), and a tetrahedron (void size .225).

click to enlarge

click to enlarge

click to enlarge

Figure 57 -  Cubic,
  tetrahedral and
 octahedral voids
 click image to enlarge

cubic (AX8)

octahedral (AX6)

tetrahedral (AX4)


     Thus these ratios result in eight large X atoms being coordinated around the
smaller A atom in the cubic arrangement, six X atoms in the octahedral arrangement,
and four X atoms in the tetrahedral arrangement.  Other, less common coordinations
are trigonal prismatic with coordination six and a radius ratio of .528, square anti-
prismatic with coordination eight and a radius ratio of .645, and triangular with
coordination three and a radius ratio of .155 .  And as already demonstrated for the
elements, compounds with radius ratios approaching one, that is with equal radius
atoms, usually pack with coordination twelve.  Simply put, the coordination number,
or ligancy, increases with an increase in the radius ratio.

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Page  39 -  Structure matters - Interstitial voids

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