Structure of Three-dimensional Crystals

Chemical Lattice Types

In the above expositions we considered the total SYMMETRY of three-dimensional (single) crystals. This symmetry is one of the aspects of the crystal's INTERNAL STRUCTURE.
The other aspect concerns the actual chemical structure of the lattices, which boils down to a replacement of the (abstract) motifs by actual chemical units (the just mentioned motifs only display the (point) symmetry of the chemical units. Only when the lattice (initially a point lattice, but later provided with motifs to give the Space Group) is provided with the actual chemical units, we have to do with the actual complete internal STRUCTURE of the given (single) crystal.
In order to understand this actual structure we will start with a recapitulation of some considerations already laid down earlier (In the first Part of this website, namely in : The Structure of Crystals revisited, Special Series, accessible via the back to homepage link). These concern SYMMETRY and STRUCTURE (of 3-D single crystals).

Structure, Dynamical Law, Space Group, Chemical Composition and Thermodynamic Conditions


The direct manifestation of the whatness of a (certain) crystal is its actual  s t r u c t u r e.
The Structure of a crystal is equivalent to the nature of its constituents and their actual spatial arrangement, including the absolute distances between those constituents.
This structure has the form of (1) a concrete lattice, i.e. a chart of the locations of repeated atomic groupings, and (2) the precise chemical content of those groupings.
It is partly implied by the chemical composition and partly described by the total symmetry content (the Space Group) of the atomic arrangement.


The  s y m m e t r y  of an object is described by indicating the complete set of coincidence operations that are allowed by that object.
c o i n c i d e n c e  o p e r a t i o n  is an operation that, performed on the object, will transform this object into itself.
P o i n t  G r o u p  and  S p a c e  G r o u p  describe this total of coincidence operations. They are accordingly descriptions (in contradistinction to causes) of the symmetry of an object (See also The Essay Crystals and the Substance-Accident Metaphysics).

Point Group

A Point Group describes the symmetry of an object in sofar as only those coincidence operations are involved in which at least one point of the object remains in place during the execution of such a coincidence operation. Examples are Reflection and Rotation, in contradistinction to a translation (= linear shift) in which not any point remains in place. So a Point Group of an object describes the total of non-translative symmetry operations that the object allows [ NOTE 1 ].
With respect to Crystals there are only 32 Point Groups at all possible (the 32 Crystal Classes).

Space Group

The Space Group also describes the symmetry of an object, i.e. describes the total of symmetry operations allowed by the object, but now including all the executable translative coincidence operations (with respect to that object) as well. In the case of a translative coincidence operation not any point remains in place, all the points are displaced, but the object, when considered infinitely extended in space, will coincide with itself. An example is a (linear) shift (of a regular point pattern in a certain direction and along a certain distance).
If we imagine a crystal to be indefinitely extended in space, then crystals exhibit several translational symmetries, meaning that when we subject it to certain translational operations, then the result is a coincidence of the (internal) crystal pattern before and after the operation.
This translative symmetry (aspect) is however only microscopically demonstrable, because the translation distances (i.e. the repeat distances of atoms) are very small [ NOTE 2 ]. For crystals there are in principle 230 Space Groups possible. If we subtract the translative symmetries from a Space Group, then we end up with the corresponding Point Group.
The Space Group (and by implication also the Point Group) all by itself consequently does not directly describe the complete structure of the crystal lattice but exclusively its symmetry, namely the complete symmetry of the crystal lattice considered to be provided with motifs, motifs, representing the (point) symmetry of the constituent atomic configurations that are repeated throughout the crystal. The Space Group does not describe the constituent atoms, but it does describe the total lattice symmetry caused by them. So these atoms, in the form of the Chemical Composition of the crystal, cause (in contradistintion to describe) this symmetry, but not only the crystal's symmetry, they also cause the complete structure (including the relevant distances between the lattice points) of the crystal lattice [ NOTE 3 ]. But the conditions during growth, like temperature and pressure, are also determining, which implies that one chemical substance can nevertheless give rise to several different crystal forms.

Implicit and Explicit Description

The characterization of a crystal, i.e. the (potentially) complete description of all per se features with respect to that crystal, can be done by means of its Chemical Composition + Space Group. Such a description is to be sure complete, but not wholly explicit.
As has been said, the Space Group exclusively specifies the (complete) symmetry aspect of the relevant lattice as it is (when) provided with motifs, and it does so explicitly. It does, however, not describe the different absolute distances between the lattice points or between the atoms for that matter, neither does it describe the absolute angles between the relevant chemical bonds within the crystal (The distances and angles are implied by the crystal's chemical composition). So the Space Group does not describe structure. The distances are directly dependent on the chemical composition (namely on the action radii of the relevant atoms), but are - by mentioning just the chemical composition - only implicitly given.
A completely explicit description of all per se features of the crystal should specify the crystal lattice (under consideration) quantitatively, and then indicate which lattice points are occupied by (or associated with) which atoms (ions or molecules).
The significatum (= that to which reference is made) of both types of descriptions, taken together, is the 'fixed kernel' of the crystal, its actual structure, a kernel which doesn't change during the growth of the crystal. So it is 'concurrent' with its Dynamical Law.
The Dynamical Law is the most implicit (and so most compact) characterization of the crystal by means of a very implicit description of all per se features of the crystal. These per se features are the per se determinations, and these are implied by the Dynamical Law, in the sense of :   generated. They are however not identical to the Dynamical Law. The latter is the relevant crystallization law, prevailing during the origin and growth of the crystal.
A precise formulation of the Dynamical Law of one or another crystal is generally not directly possible, but the importance of the introduction of this concept consists in its emphasizing the regular course of real processes, and this means that such a process is deterministic and in principle repeatable. So the concept of Dynamical Law emphasizes the fact that the crystal is a product (a stadium or state) of a dynamical system, and as such it denotes the Essence of the crystal.

Chemical Lattice Types

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