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 a crystal is equivalent to the nature of its constituents and their actual spatial arrangement, including the absolute distances between those constituents.

This

It is partly implied by the chemical composition and partly described by the total symmetry content (the Space Group) of the atomic arrangement.

A 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

With respect to

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

As has been said, the

A completely

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

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

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