I think, posed in those terms, the question is quite tricky and any answer is in princuple right or wrong! If your system is defect free (i.e. 3D periodic) then you know that asymmetric unit + space group are sufficient to describe the position of any atom in your crystal. Knowing the unit cell or the whole crystal makes no difference from the point of view of the information content, so knowing the local order gives you access to the global as well (you can't say the same e.g. for a glass or a defected crystal).

If you go below the size of a unit cell then you start seeing the local environment and the local order. What it seems you want to do is to compare e.g. the local environment of Si in quartz (tetrahedron) with the average long range structure: not sure this has some meaning though

PS by the way, a "defect free" crystal cannot exist from a thermodynamically point of view. You'll always have vacancies: however the quantity is usually insufficient to give an appreciable diffuse scattering contribution to be sensed by diffraction.

Not in the closest sense, because (in thermodynamic equilibrium) a crystal is an infinite stacking of identical unit cells in all 3 dimensions of space. And from this any local difference in structure is excluded. But practically the story is occasionally a bit different: Sometimes one can grow even bulk crystals with a concentration gradient. Si_{1-x}Ge_x is a good example (http://www.sciencedirect.com/science/article/pii/0168900296001994). Here the crystal structure is everywhere diamond type, but the concentration has a gradient over the crystal length. And consequently, also the lattice constant (as one component of crystal structure) is locally different, following the concentration gradient.

Thank you(prof. Matteo Leoni and prof. Detlef Klimm) very much for your nice answer.

Now coming to next question, How a crystal will retain long range Centro-symmetry even after break local Centro-symmetry (e.g case of Multiferroic LnCrO3)?

Typically, during ferroelectric/multiferroic phase transitions domains are created. Each single domain has a lower symmetry, compared with the high-T phase. This means that some symmetry elements (such as center of symmetry) are no longer present in them. But the system tries to maintain its higher previous symmetry by arranging the many domains in such a way, that the "lost" symmetry elements are represented by this arrangement.

what will be difference between diffraction pattern of Centro-symmetric and non-Centro-symmetric space group (Taken an example of BiFeO3 crystal have R3c or R-3c space group)

Friedel law can give you an answer... the XRD pattern will be the same (if you don't have enough anomalous effects)

Jeppe, as far as I know there is still no "perfect" or unique definition of a crystal (http://www.oldenbourg-verlag.de/sites/default/files/filebase/files/1_Steurer.pdf) and Occam's razor would go for the one that creates less confusion.

The one you refer to is that given by the commission on aperiodic crystals (http://journals.iucr.org/a/issues/1992/06/00/es0177/es0177.pdf) that considers under the same umbrella also those structures that are periodic in superspace or that are quasiperiodic (ordered but not periodic). I think modulated structures are already in the definition of the ITVA, except for those that are incommensurately modulated.

If I would have to use the "internationally recognised" definition I mostly use, I'd then go for that of Dornberger-Schiff (page 127, "Crystallography of Modular Materials, Ferraris et al..).

The definition of the commission on aperiodic crystals is probably still not "perfect": are multiply twinned particles (decahedra and icosahedra) in or out? And superlattices made of nanoparticles? And any possible man-made superlattice (not necessarily made of atoms or molecules), expressly designed to give Bragg scattering? and modular materials?

That's why (as non crystallographer) and to avoid misunderstanding (as in this case), I prefer to use some "layman definition":

- "crystal" something that has order and 3D periodicity and that can be described using asymmetric unit and 3D space group

- "aperiodic crystal" something where 3D periodicity is no longer a prerequisite

In the paper of Walter Steurer we have several examples of crystals).

Unfortunately those that often pass on my desk are powders of "imperfect crystals" (I still find this definition too wide), i.e. those that are often