Order parameter

Question

Order parameter:

Suppose, for simplicity, that the molecules composing a nematic or cholesteric liquid crystal are rigid and rodlike in shape. Then we can introduce a unit vector v") along the axis of the ith molecule which describes its orientation. This vector should not be confused with the direcior n which gives the average preferred direction of the molecules. Since liquid crystals possess a center of symmetry, the average of v") vanishes. It is thus not possible to introduce a vector order parameter for a liquid crystal analogous to the magnetization in a ferromagnet, and it is necessary to consider higher harmonics or tensor@. A natural order parameter to describe the ordering in a nematic or cholesteric is the second rank tensor:

$S_{\alpha\beta}(r)=\frac{1}{N} \sum_{i} \left(v_\alpha ^{i} v_\beta ^{i} -\frac{1}{3}\delta_{\alpha\beta}\right)$

what is beta and alpha ?

Reference : Physics of liquid crystals ,Michael J. Stephen and Joseph P. Straley*

Answer 1

$\alpha$ and $\beta$ are tensor components. If you have never seen this, try to get a feeling for tensors, you will get to know them when discussing rigid bodies or special relativity. To get started, you may want to just think of them as matrices, making $\alpha$ and $\beta$ matrix components (this picture is, however, incomplete. I'll end this answer with a book recommendation).

So $\alpha$ and $\beta$ both run over your coordinate system's basis vectors. Most commonly, one works in a three dimensional Cartesian coordinate system with basis vectors $\mathbf{e}_x$, $\mathbf{e}_y$ and $\mathbf{e}_z$, so the components of $S_{\alpha\beta}$ are $S_{xx}$, $S_{xy}$, $S_{xz}$, $S_{yx}$, ... They are usually summarized in a matrix, $$ \textbf{S} = \begin{pmatrix} S_{xx} & S_{xy} & S_{xz}\\ S_{yx} & S_{yy} & S_{yz}\\ S_{zx} & S_{zy} & S_{zz} \end{pmatrix} = \frac{1}{N} \sum_i \begin{pmatrix} v_x^i v_x^i - \frac{1}{3} & v_x^i v_y^i & v_x^i v_z^i\\ v_y^i v_x^i & v_y^i v_y^i - \frac{1}{3} & v_y^i v_z^i\\ v_z^i v_x^i & v_z^i v_y^i & v_z^i v_z^i - \frac{1}{3}\\ \end{pmatrix}. $$ If you want a good understanding of tensors, I can highly recommend the first half of Nadir Jeevanjee's book "An Introduction to Tensors and Group Theory for Physicists" to get you started. It's quite beginner-friendly.