Saupe orientation tensor

The latter is calculated as a tensor product of the spin-spin coupling tensor J l expressed in the molecular fixed coordinate frame and the Saupe orientation tensor S ... 

The residual anisotropy A describes the orientation assumed by heavy water molecules. It is expressed as a function of the Saupe order tensor and of the Euler angles describing the orientation of the D20 water molecule with respect to the local director (29) ... 

In eq. (4.25), is the orientational order parameter (relative to the external magnetic field) of the vector mn, and < > indicates averaging over internal vibrations. The coefficients sfp in eq. (4.26) are Saupe orientational order tensor elements relative to the external magnetic field in a molecular-fixed frame. Consequently, it is necessary to know the tensor in order to be able to determine experimentally the direct dipolar contribution from... 

The chirality interaction tensor Wy is responsible for the interaction of the chiral guest, the chirality of which is described by the chirality tensor Cy, with the anisotropic host Wy = ikLkj- Ly covers the anisotropic host properties. W = Tr JT), is different from zero. The gyki are the orientational distribution coefficients of the guest in the molecular ensemble of the guest-host phase. From the two possible representations to avoid non-diagonal elements in (3.19), the representation in the system of principal axes of the order tensor is chosen instead of the principal axes of the chirality interaction tensor Wy. With Saupe s order parameters the HTP is also a sum of three terms... 

Saupe matrix elements T = two-photon absorption tensor U V = photon electric field vector direction (polarization) x, y, = molecular orientation axes x y, z - arbitrary molecular axes X, Y, Z = laboratory axes a, P, / = Euler angles ... 

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... 

It is clearly the second term which allows for the molecular biaxiality in the theory. As in the Maio--Saupe theory the tensor components depend on the orientational order and the Luckhurst-Zannoni-Nordio-Segre theory [24] gives this dependence as... 

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... 

---