Orientational order Saupe matrix

In the molecular orientation system of axes x, y, z, the second-order Saupe matrix is diagonal... 

One of the characteristic features of mesogenic molecules in uniaxial mesophases is their orientational order specified by a set of order parameters, which forms a symmetric and traceless order matrix S. For rigid molecules, the elements of order matrix Sa/Y defined by Saupe,52 are given by... 

Note 3 Molecules which constitute nematogens are not strictly cylindrically symmetric and have their orientational order given by the Saupe ordering matrix which has elements 5 aP = (3< a P> - 5ap)/2, where la and ip are the direction cosines between the director and the molecular axes a and P, 5ap is the Kronecker delta and a, p denote the molecular axes X, Y, Z. 

Note 4 The constituent molecules of a nematogen are rarely rigid and their orientational order is strictly defined, at the second-rank level, by a Saupe ordering matrix for each rigid sub-unit. 

The orientational order of molecules in nematics is often described in terms of the so-called Saupe matrix [1], whose elements are defined as ... 

Equations [11] and [12] separate the information on anisotropic molecular optical properties (0(0/) and (4)0(0/) from that on orientation distribution and polarizer orientation ( D and ((4)p5T[JV ) which can be expressed through the parameters chosen to describe the orientation distribution the orientation factors [3], [4], the Saupe matrix elements [8], or the order parameters [9]. 

Although the irreducible spherical tensor approach is valuable fix the definition of the (xdo ing tensors and for their manipulation under rotation it does not always provide a ready undostanding of the physical significance of the various components. This is sometimes available from a Cartesian representation of the ordering tensor. The most familiar example is the Saupe ordering matrix which represents the orientational ordering at the second rank level [8]. It is defined by... 

Here the bar indicates an ensemble average and /(cu) is the normalized singlet orientational distribution function for finding the director at an orientation co in a molecular frame. Of this infinite set only the second rank quantities 2, can readily be measured and these five components, corresponding to m =0, 1, 2, form the ordering tensor which is the irreducible analogue of the more familiar Saupe ordering matrix. This matrix is defined by... 

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