Orientation averages parameter

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

The Boltzmann statistical weight for each rotamer is multiplied by (1 + aPt(cos 8)), with a being an adjustable parameter, before a particular rotamer is selected. Values of a, between zero and unity, will give rise to an order parameter for segment orientations. Average of Pt(cos 8) of 0.0 to 0.5. 

Many conformations were sampled by the usual MC procedure. The result is of course that there is no preferred orientation of the molecule. Each conformation can, however, be characterised by an instantaneous main axis this is the average direction of the chain. Then this axis is defined as a director . This director is used to subsequently determine the orientational order parameter along the chain. The order is obviously low at the chain ends, and relatively high in the middle of the chain. It was found that the order profile going from the centre of the molecules towards the tails fell off very similarly to corresponding chains (with half the chain length) in the bilayer membrane. As an example, we reproduce here the results for saturated acyl chains, in Figure 10. The conclusion is that the order of the chains found for acyl tails in the bilayer is dominated by intramolecular interactions. The intermolecular interactions due to the presence of other chains that are densely packed around such a chain,... 

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... 

Furthermore, in all phases studied the first spectral moment Mi of the H NMR spectra can be calculated and the weighted mean splitting of the H NMR spectrum can be obtained, which is proportional to the average chain orientational order parameter of the lipid, using ... 

The order parameter S is the orientational average of the second-order Legendre polynomial P2(a n) (n = the director), and if the orientational distribution function is approximated by the Onsager trial function, it can be related to the degree of orientation parameter ot by... 

Equation 1.10 describes this non-Kramers doublet behavior and its fit to the VTVH MCD data in Figure 1.12a (with orientation averaging for a frozen solution) allows the spin Hamiltonian parameters to be obtained.29,30 These, in turn, can be related to the ligand field splittings of the t2g set of d-orbitals, as described in Ref. 7, which probe the re-interactions of the Fe(II)... 

VqS is the magnitude of the residual tensor. The averaging factor S is the orientational order parameter, i.e., the degree of motional anisotropy, of a given C-2H bond (with respect to its average direction of orientation) ... 

The orientational order parameter is the second Legendre polynomial of cos9 and, where the chemical shift dispersion is reduced by rotational averaging, is given by... 

Figure 3. (a) Two-dimensional, bond orientational order parameter average values in the molecular fluid layers of LI ecu confined in a multi-walled carbon nanotube of diameter D=9norder parameter values for the contact, second, third and fourth layers, respectively. The dotted line represents the bulk solid-fluid transition temperature, (b) Positional and orientational pair correlation functions in the unwraiqred contact layer of U CCU confined in a multi-walled carbon nanotube of diameter D=9.1< (5 nm) showing liquid phase at 7=262 K and crystal phase at 7=252 K. 

The average Ifee-energy density of the NP-LC mixture is expressed in terms of the spatially averaged orientational order parameters S c and Sjqp of the mixture components LC and NP as... 

The local sixfold bond orientational order parameter is defined in Eq. (3.3). g FpFj) is divided out of Eq. (3.15) in order to remove translational correlations from the bond orientational correlation function. In the homogeneous and isotropic liquid phase gl (r,F2) reduces to a function of Fj, only, which we will denote by g r), and a corresponding translation- and rotation-invariant quantity can be defined for the solid phase by performing suitable averages. 

As discussed above, the local sixfold bond orientational order parameter is a sensitive probe of the local geometry of 2D systems. This is illustrated by Fig. 34, which shows that distribution functions for the time-averaged 3584-particle WCA system at various densities. In the solid... 

As for the WCA system, the fraction of sixfold-ordered particles in DRPs (0.547 0.004) is significantly less than the fraction of six-coordinated particles (/g = 0.7861 0.0018), confirming that the sixfold bond orientational order parameter is a more sensitive indicator of local geometrical disorder than is the coordination number. The average size of ordered clusters in DRPs is (s) = 30.4 0.8, and the normalized average number of ordered clusters is Ac/N = 0.0180 0.0005. These values are comparable to those measured in the dense WCA liquid near freezing (see Figs. 52 and 53). 

First we review some typical materials parameters obtained from measurements on randomly oriented ceramics (6-7). Since the YBaCuO structure (5) and electronic properties 81 are highly anisotropic, the orientationally-averaged values obtained from studies of ceramics are only an initial indication until more complete experimental results on single crystals and oriented films and ceramics become available. For material with a resistivity just above the transition of 400 /xficm, a Hall carrier density of 4xl021cm , and dHc2/dT of 2 T/K (6-7). one deduces a BCS coherence length (0) of 1-4 nm, a London penetration depth A(0) of 200 nm, a mean free path t of 1.2 nm, a thermodynamic critical field Hc(0) of 1 T (10000 Oe) and an upper critical... 

Figure 3 Calculated primary ct probability PpTim (upper panel) for all simulated eventB (+) and orientation-averaged (thick line), mean tkel AE and mean vibrational kinetic energy E,b (middle panel), as well as upper and lower limit for the final, i. e. exclusive ct probability obtained along with the limiting fragmentation probabilities (lower panel) as functions of the impact parameter b. The length of the error bars is given by two times the standard deviation of the orientation average.

Figure 22. Potential energy landscape explored by the model calamitic system GB(3, 5, 2, 1) (N = 256) as the system makes a transit through mesophases upon cooling, (a) Temperature dependence of the average inherent structure energy per particle, (< /s), along three isochors at densities p = 0.31,0.32, and 0.33. (b) Evolution of the average second-rank orientational order parameter S with temperature both for the inherent structures (filled) and for the instantaneous configurations (opaque). (Reproduced from Ref. 144.)...

Equation (3.21) implies that the determination of an optical parameter at a given time consists of three steps (1) orientation averaging for a cluster possessing a given particle number(2) statistical averaging over all... 

Figure 14. Extinction and scattering spectra for ballistic RF aggregates with different conjugate numbers N = 1-100. All data are averaged over random orientations (T-matrix method) without statistical averaging. Parameters of conjugates are the shell thickness and refractive index s = 2.5 nm, =1.40, respectively, the gold core diameter <7 =15 nm (a,b) and 60 nm (c,d).

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