Order parameters fourth rank

Like Raman scattering, fluorescence spectroscopy involves a two-photon process so that it can be used to determine the second and the fourth rank order parameters. In this technique, a chromophore, either covalently linked to the polymer chain or a probe incorporated at small concentrations, absorbs incident light and emits fluorescence. If the incident electric field is linearly polarized in the e direction and the fluorescent light is collected through an analyzer in the es direction, the fluorescence intensity is given by... 

The values of these autocorrelation functions at times t = 0 and t = 00 are related to the two order parameters orientational averages of the second- and fourth-rank Legendre polynomial P2(cos/ ) and P4 (cos p). respectively, relative to the orientation p of the probe axis with respect to the normal to the local bilayer surface or with respect to the liquid crystal direction. The order parameters are defined as... 

Lee (1988) further connected the viscosities of liquid crystalline polymers (in the unit of f]) with concentration and molecular length as Equation (6.37), where A is a constant less than unity which is associated with the stability of simple shear flow, is a dimensionless parameter associated with the interaction in the fluid, R = (/A(a n) is the second order parameter where a is the molecular long axis and P4 is the fourth rank Legendre polynomial. 

The second rank-order parameter S can be derived from measurements of the macroscopic tensor properties such as birefringence and diamagnetic susceptibility. It varies typically between 0.4 at the clearing temperature to 0.7 at T ni- r= 20K in nematic phase. The fourth rank-order parameter (P ) may play an important role for a subtle analysis of the orientational distribution function and can be determined using polarized Raman spectroscopy. ... 

Humphries et al. have considered in addition to the order parameter S the fourth rank-order parameter (P4), which modifies the shape of the potential. They approximated the volume dependence of the interaction energy by... 

For D D , the reciprocals of these correlation times are raised by an amount [(D /D ) — 1] as indicated in Eq. (7.55). As noted previously, K rriL, ttim) values at a particular temperature are computed from (P2) and (P4). The fourth-rank order parameter P4) cannot be directly measmed from a NMR spectrmn, but may be derived from measurements of the mean square value of a second-rank quantity [7.19-7.22]. In the Raman scattering technique [7.21], the second-rank molecular quantity is the differential polarizability tensor of a localized Raman mode. In fluorescence depolarization [7.19], the average of the product of the absorption and emission tensors is used to determine (P4). Since there is a lack of experimental determination of (P4) in liquid crystals, this may be calculated based on the Maier-Saupe potential... 

FIGURE 2 The variation of the fourth tank orientational order parameter, < 4>, with the second rank . 

The values of the scaled transition temperature, l TNi/e2oo, together with the transitional values of the ord parameters ni, ni and ni are listed in TABLE 1 for several values of the biaxiality parameter, X. As the molecular biaxiality increases so the major order parameter at the transition decreases indeed when X is 0.3 ni has decreased to about half that for a uniaxial molecule. In contrast although as expected the biaxial ordo- parameter ni increases with X the value is extremely small. It would seem, therefore, that the molecular biaxiality has the greatest effect on Ni and that ni is essentially negligible. The influence of X on the second rank order parameter is mimicked by its fourth rank counterpart ni indeed when X is 0.3 this order parameter is four times smaller than that for uniaxial molecules. For X of 0.4 all of the transitional order parameters are extremely small, hinting at the approach of a second ord transition. This occurs when X is 1 / >/6 but now the transition from the isotropic phase is directly to a biaxial and not a uniaxial nematic phase, as we shall see. 

In FIGURE 3 we show the second and fourth rank order parameters , obtained for a system of N = 10 GB(3,5,1,3) particles using MC at a scaled density p of 0.30 [25]. The temperature dependence of is well-represented, after subtraction of the small residual order due to the finite sample size, by the Haller type expression often used to fit experimental data ... 

The determination of the sixth-order crystal-field parameters for Gd + is really problematic, because none of the levels which can be determined by spectroscopic measurements have sufficiently high values for the IT reduced matrix element. The crystal-field splitting is in a good approximation determined by the second- and fourth-rank parameters only. 

The above rules for an extraction of the 5 crystal-field parameters directly from experimental data caimot be applied for most symmetries to J levels for which J >1, because of the presence of a large number of off-diagonal matrix elements. Every level is in this case a function of at least three parameters. It is more difficult to determine the parameters than the second- or fourth-rank parameters. One can first determine the parameter and subsequently the 5 parameters 0). In the end both and 5 are varied in order to remove discrepancies. An alternative approach is to restrict 5 /5g to the PCEM ratio, because in this way only one sixth-rank parameter will remain. 

Symmetry restrictions for third- and fourth-order anharmonic temperature parameters are Used in the International Tables for X-ray Crystallography Vol. IV (1974). A more complete list for elements up to rank eight has been derived by Kuhs (1984). 

The fine structure interaction involves the interaction between the magnetic dipole moments of electrons on an atom containing more than one unpaired electron. The magnitude of this interae-tion is described by the second rank D tensor. Second-order terms, D and EID correspond to the axial zero field splitting (Z)) and the asymmetry parameter EID, which varies from 0 (axial symmetry) to 1/3 (rhombic symmetry) (S.D.S). Fourth- and sixth-order corrections to the fine structure interaction tensor D may also be necessary to adequately interpret the spectrum,... 

---