A structural view of crystal symmetry bottom-up crystallography

1 A structural view of crystal symmetry bottom-up crystallography 

The study of molecular crystals from a structural point of view requires neither abstract lattices nor group theory. A molecular crystal in flesh and bones is an aggregate of molecules that, at least within finite domains of size much larger than the size of one molecule, arrange themselves according to a very small number of periodic symmetry conditions under the action of an intermolecular potential. An organic crystal can therefore be studied by defining just a handful of such symmetry conditions and 

Since translation is so important in crystals, consider first what happens if a molecular object is translated in two directions in space to form a periodic structure defined by two translational periods (Fig. 5.1). There are many different choices of the unit cell the cell defined by periods a and b is preferable over the cell defined by a and b because the cell angle (aOb) is closer to 90° than the cell angle (a Ob). The unit cell defined by a and b is called a non-primitive (centered) unit cell, because it encloses one molecule related by pure translation by a sub-multiple of the cell periodicity. The choice of such a cell introduces a quite unnecessary complication in this case. 

Extension to the third dimension, with periods a, b, and c, is obvious the cell angles are called a (the be angle), f) (the ac angle), and y (the ab angle). The collection of points at the corners of the unit cell parallelepiped, repeated by translation, form what is called the crystal lattice. There is no preferred location and there are no restrictions on the position of the origin there are no restrictions to the length of the three ttans-lational periods, nor to the angles between the unit cell ttanslation vectors. Such a crystal system is called triclinic. 

Consider now a molecular pair formed over a center of symmetty, symbolized here by the capital letter I, and apply the same translation procedure to the molecular pair (Fig. 5.2). It is now intuitively obvious that the origin of the cell system must be chosen in the one very unique point in the whole structure, the center of symmetty. It turns out that with this choice not only does the visual picture of the suiicture becomes clearer and simpler, but also all the mathematical manipulations pertaining to crystal structure analysis become much easier. The space-group symbols for the pure translation of a single object or of a centrosymmetric pair are PI and PI respectively, where the letter P stands for primitive cell and the symbol 1 is the traditional crystallographic symbol for the inversion operator. The contents of the unit cell is the portion of crystalline matter that is repeated in space by pure translation. 

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