# Averaged Second-Rank Tensorial Quantities and Order Parameters

In order to determine the order parameters involved in the relationship between the components of an averaged second rank tensorial quantity in a lab. fixed frame and the components of the tensorial quantity in the molecular fixed frame, we start by considering a traceless and symmetric second rank tensorial quantity in a frame "M” fixed with a rigid molecular segment. We then use the rules of tensor calculus to establish the relation between the components of the tensorial quantity in the "M” frame and in a frame "F,” named phase frame, fixed with the external magnetic induction of the NMR spectrometer and take a time average of the result as the orientation of the "M” frame is changing in time relative to the "F” frame due to molecular motions leading to

Considering the direction cosines defining the orientation of frame "M” in frame "F,” the quantities become

leading to

The order matrix defined by

(de Gennes and Prost, 1993) where the brackets <> indicate an ensemble average, can be used in certain cases to express *RF* (Eq. 5.8) in terms of the order matrix elements. One of these cases corresponds to a constant *RM* tensor, in this case, inverting Eq. 5.9 leads to

and using the ergodic properties of the system the ensemble average replaces the time average in *RF* and Eq. 5.10 is used, leading to

Another case where *RF* can be expressed in terms of the order matrix elements corresponds to the situation where R^ tensor is time dependent due to the presence of molecular conformation changes. Assigning a probability *p _{k}* and an order matrix

*S*to each conformer leads to

^{k}p

The order matrix *Sj* is a symmetric traceless tensor in indices *ij* and *ав*, from its 91 elements only 34 at most are independent (de Gennes and Prost, 1993). The number of independent elements of the order matrix is further reduced by the symmetry of the molecules and the symmetry of the anisotropic fluid phase. When determining order matrix elements using any of the NMR anisotropic observables listed in Eq. 5.2 in the case of constant *R^в*, it is necessary to choose both the phase frame "F" and the molecular frame "M." As the number of NMR observables may be limited, it is convenient to choose for "F" the frame that diagonalizes *Sj* in the lab. related indices *i j* and choose for "M" the frame that diagonalizes

*S^p* in the molecule related indices *ав,* with *i* one of the principal directions of *Sj* in "F." The symmetry axes of the anisotropic fluid phase are good candidates for the principal directions in "F," while the symmetry axis of the molecular segment may also be good candidates for the principal directions of *S"p* in "M." The following example illustrates the method; consider that the principal *z* axis of *Sj* in "F" is along the external magnetic induction field B and that through the analysis of molecular symmetry we have identified the principal "M" frame leading to *Sj =* 0 (i = *j*) in that frame. Then it is possible to calculate the order matrix elements *SZZ* and *SXX — Syy* if two independent values for the NMR observable *RF _{zz }*are determined from the spectral analysis as the following system of equations shows

leading to