Unit cell
=========

- A *crystal* is a solid body which consists of *identical* unit
  cells (following J.H.).

- Unit cells fill the whole space *without* gaps (following J.H.). Each
  unit cell is mapped into another (identical) unit cell by
  a translation vector. Unit cells do *not* *overlap* when translated
  (i.e. unit cells are partition of space into equivalence classes);

  -- In a "classical" crystal, unit cells fill a 3D space; therefor
     translation vectors are mathematically vectors in a 3D space.

  -- yet, interestingly enough, the same model can be applied also to
     quasi-periodic structures like modulated structures and
     quasicrystals; in that case the structure is *modelled* as a
     periodic space filling array of unit cells in *higher*
     dimensional space (4D, 5D, 6D – more for quasicrystals?); The
     observed 3D structure is obtained by an irrational 3D cut through
     this higher dimensional periodic space.

     Thus, the same definition of unit cell and the same requirements
     to unit cell apply also in the case of modulated structures and
     quasicrystals, just the dimensionality differs.

     NB: I am not sure if this is the only way to model quasicrystals
     and if every quasicrystal can be modelled in this way; need to
     consult advances in the field.

  -- This definition seems to include stuff like Fotonic crystals
     (https://en.wikipedia.org/wiki/Photonic_crystal), which is
     intended.

- Potentially, unit cells can fill space of arbitrary size (so a
  crystal is not limited in size by its definition and by its periodic
  nature); we can even talk about "infinite ideal" crystals, but we
  do not need to: the infinity is *potential* here, not
  *actual*. Event then, some mathematical formalism *will* assume
  infinite crystals (e.g. the theory that Fourier transform yields
  delta-functions);

- It is an interesting mathematical fact that such arrays of unit
  cells *diffract*, i.e. their Fourier transforms have sharp
  (delta-function shaped?) peaks;

- Unit cells *must* have some non-homogeneity in them (atoms,
  molecules); otherwise we get homogeneous "gas" filling the whole
  space (that, as we know, does not diffract :) (S.G.)

- Aside from being non-homogeneous, unit cells do not need to have any
  specific structure: they *can* contain atoms (as in the definition
  of R.A.) at fixed sites (90% of cases), but they can also contain
  positionally disorder groups (alternative conformations),
  occupational disorder at fixed sites, disordered regions
  (e.g. filled with solvent); the cells can contain atoms that vibrate
  (move scholastically) around they average positions.

  This disorder, although it (potentially) makes every unit cell
  different in a physical sample, still allows to speak about
  *identical* unit cells since, in this model, we assume that disorder
  is the same in every unit cell, and so unit cells are identical *on*
  *average* (average in time; average in space). Thus our model of
  a unit cell is the *average* cell that fills the space.

- (H.J. requirements): parameters. The unit cell is defined by three
  (in 3D space) or N (in N-dim space) linearly independent
  (non-coplanar) vectors. In the 3D case we most often call them
  \vec{a}, \vec{b} and \vec{c}.

  These vectors:

  a) define translations (a translation group) that map one unit cell
  in a crystal into some other identical unit cell in the same
  crystal.

  b) define the *shape* of a unit cell as a parallelepiped spanned by
  these 3 vectors;

- Additional parameters might be needed: additional parameters are
  needed to model internal structure of the unit cells can be atom
  sites, atom thermal displacement factors, atom kinds, atom
  occupancies, atom shapes, electron density (a 3D array of values)

  -- NB: *some* parameter is *necessary* since, as mentioned above,
     the unit cell *must* be non-homogeneous; which parameters are
     used, however, is not so important.

- The periodicity (vectors \vec{a}, \vec{b}, \vec{c}, ...) of unit
  cells, their identity and their non-homogeneity is what
  distinguishes the crystal from other forms (models) of matter like
  glass, liquid, gas, plasma, vacuum, etc.

- Additional parameters might be necessary for a crystal model
  (e.g. temperature), which can be added as needed as long as they are
  compatible with the model of identical periodically placed unit
  cells that do not overlap and fill the space without gaps.

This seems to capture essential properties and parameters of the unit
cell and the crystal...