Editing Liquid crystal (section)

===Order parameter===

[[File:Nematic-Director.png|thumb|The ''local nematic director'', which is also the ''local optical axis'', is given by the spatial and temporal average of the long molecular axes.]]

The description of liquid crystals involves an analysis of order. A second rank symmetric traceless tensor order parameter, the [[Q tensor]] is used to describe the orientational order of the most general [[biaxial nematic]] liquid crystal. However, to describe the more common case of uniaxial nematic liquid crystals, a scalar order parameter is sufficient.<ref>{{cite book |last1=Chaikin |first1=P. M. |last2=Lubensky |first2=T. C. |title=Principles of condensed matter physics |date=1995 |location=Cambridge |isbn=9780521794503 |page=168 |publisher= Cambridge University Press }}</ref> To make this quantitative, an orientational order parameter is usually defined based on the average of the second [[Legendre polynomial]]:

:<math>S = \langle P_2(\cos \theta) \rangle = \left \langle \frac{3 \cos^2(\theta) - 1}{2} \right \rangle</math>

where <math>\theta</math> is the angle between the liquid-crystal molecular axis and the ''local director'' (which is the 'preferred direction' in a volume element of a liquid crystal sample, also representing its ''[[optical axis|local optical axis]]''). The brackets denote both a temporal and spatial average. This definition is convenient, since for a completely random and isotropic sample, ''S''&nbsp;=&nbsp;0, whereas for a perfectly aligned sample S=1. For a typical liquid crystal sample, ''S'' is on the order of 0.3 to 0.8, and generally decreases as the temperature is raised. In particular, a sharp drop of the order parameter to 0 is observed when the system undergoes a phase transition from an LC phase into the isotropic phase.<ref>{{cite journal|title = A model for the orientational order in liquid crystals| doi =10.1007/BF02453342|journal=Il Nuovo Cimento D|volume = 4| issue =3|date =1984| page = 229|bibcode = 1984NCimD...4..229G| vauthors = Ghosh SK | s2cid =121078315}}</ref> The order parameter can be measured experimentally in a number of ways; for instance, [[diamagnetism]], [[birefringence]], [[Raman scattering]], [[Nuclear magnetic resonance|NMR]] and [[Electron Paramagnetic Resonance|EPR]] can be used to determine S.<ref name=b5/>

The order of a liquid crystal could also be characterized by using other even Legendre polynomials (all the odd polynomials average to zero since the director can point in either of two antiparallel directions). These higher-order averages are more difficult to measure, but can yield additional information about molecular ordering.<ref name=b2/>

A positional order parameter is also used to describe the ordering of a liquid crystal. It is characterized by the variation of the density of the center of mass of the liquid crystal molecules along a given vector. In the case of positional variation along the ''z''-axis the density <math>\rho (z)</math> is often given by:

:<math>\rho (\mathbf{r}) = \rho (z) = \rho_0 + \rho_1\cos(q_sz - \varphi) + \cdots \, </math>

The complex positional order parameter is defined as <math>\psi (\mathbf{r}) = \rho_1 (\mathbf{r})e^{i\varphi(\mathbf{r})}</math> and <math>\rho_0</math> the average density. Typically only the first two terms are kept and higher order terms are ignored since most phases can be described adequately using sinusoidal functions. For a perfect nematic <math>\psi = 0</math> and for a smectic phase <math>\psi</math> will take on complex values. The complex nature of this order parameter allows for many parallels between nematic to smectic phase transitions and conductor to superconductor transitions.<ref name=b1/>