Quantity to measure order in crystals Imagine a cubic lattice for simplicity and put some atomic configuration inside of each unit cell. Lattice is of course a highly ordered entity (long range order). Please, how could I quantitatively describe this via some correlator function? I assume no up/down spins are present on the lattice.
This is most probably a very basic question, but I come from a different field of physics and could not find the right references.
 A: There may be a few options for describing the crystalline order. Personally, I love the one used by Anderson in his book 'Basic Notions of Condensed Matter Physics'. Following him, one writes the atomic density as $\rho(\vec{r}) = \sum_{\vec{G}}\rho_{\vec{G}}e^{i\vec{G}\vec{r}}$. The appearance of Crystalline order is signified by a set of finite $\rho_{\vec{G}}$, with $\vec{G}$ describing the direction of crystal planes. Suppose you seek a correlator to describe this thing. A proper one is $C(\vec{r},\vec{r}') = \langle \rho(\vec{r})\rho(\vec{r}')\rangle$. In the high-symmetry phase (e.g. liquid), the system is homogeneous, and you can write $C(\vec{r},\vec{r}') = C(\vec{R})$, with $\vec{R} = \vec{r} - \vec{r}'$. The appearance of crystalline order is now signified by the divergence of a set of Fourier components (at some $\vec{G}$s) of $C(\vec{R})$. 
A: You need to define an appropriate "order parameter" for your system, one that takes into account the symmetries in the configuration as well (rotational, transnational, etc). There are many ways you could define such "correlator" as you call it, it depends on the system. For example the nematic order parameter in liquid crystals is taken with respect to a vector denoted "the director $\mathbf{n}$", which can e.g. be the average orientation of the anisotropic rods in the nematic phase (to simplify here), this way the order parameter for each arbitrary rod can be correlated to the director as: $$S \propto \left<cos^2\theta^{\alpha}-1 \right>$$
Where $\theta$ would be the angle between the direction of the main axis of the rod denoted $\alpha$ and the director. 
Now this was an example for a uniaxial degree of order, but it can be biaxial as well and so on...More rigourosly you'd define a distribution function for the orientation of rods in your system, and from which you'd calculate the correlation function (usually an average) that you look for. Finally based on the symmetries in the system (inversion symmetry etc) the tensor degree of the order parameter is defined (order 0 a scalar, 1 a vector etc.).
But this was for liquid crystals, which give a very straightforward idea to how order parameters are defined.
Now for crystals, you would be more interested in the local deviation of each atom from its lattice position. So the correlation function you're looking for in this case could be the average displacement needed in each degree of freedom of the system to bring back the atom to its lattice position defined by the crystalline structure. In contrast to the earlier example, here we work with the center of mass vectors of each rod or atom instead of the vector along their main axis (if they are anisotropic i.e.).
For more references, look into any textbook of Solid state physics if you are studying crystals, and for example De Gennes' book for liquid crystals.
Finally keep in mind that there can be many different forms of order in a system, so the order parameter is defined accordingly (system specific), but they are all referred to as order parameters.
