Full orientation

In contrast, the second term in (4.6) comprises the full orientation dependence of the nuclear charge distribution in 2nd power. Interestingly, the expression has the appearance of an irreducible (3 x 3) second-rank tensor. Such tensors are particularly convenient for rotational transformations (as will be used later when nuclear spin operators are considered). The term here is called the nuclear quadrupole moment Q. Because of its inherent symmetry and the specific cylindrical charge distribution of nuclei, the quadrupole moment can be represented by a single scalar, Q (vide infra). 

Gamulya and co-workers also concluded on the basis of electron microscopy and micro-electron diffraction that the production of a highly reflective surface occurs when the coefficient of friction reaches a minimum level. The full orientation appears to be limited to a very thin surface layer, which they found to be about 0.1 tjm thick in their tests, while Brudnyi and Karmadonov described it as being only one crystal thick, regardless of the force used for burnishing. On the other hand, a fully densified, fully-oriented layer without discontinuities may not be distinguishable in practice from an extended single crystal. 

For all Azo-PURs, the quantum yields of the forth, i.e., trans—>cis, are small compared to those of the back, i.e., cis—>trans, isomerization—a feature that shows that the azo-chromophore is often in the trans form during trans<->cis cycling. For PUR-1, trans isomerizes to cis about 4 times for every 1000 photons absorbed, and once in the cis, it isomerizes back to the trans for about 2 absorbed photons. In addition, the rate of cis—>trans thermal isomerization is quite high 0.45 s Q 1 shows that upon isomerization, the azo-chromophore rotates in a manner that maximizes molecular nonpolar orientation during isomerization in other words, it maximizes the second-order Legendre polynomial, i.e., the second moment, of the distribution of the isomeric reorientation. Q 1 also shows that the chromophore retains full memory of its orientation before isomerization and does not shake indiscriminately before it relaxes otherwise, it would be Q 0. The fact that the azo-chromophore moves, i.e., rotates, and retains full orientational memory after isomerization dictates that it reorients only by a well-defined, discrete angle upon isomerization. Next, I discuss photo-orientation processes in chromophores that isomerize by cyclization, a process that differs from the isomeric shape change of azobenzene derivatives. 

Liquid crystal phases, or mesophases, are characterized by a partial order, intermediate between the full orientational and translational disorder of the isotropic liquid phase and the full orientational and translational order of the crystalline phase. Thermotropic liquid-crystal phases are obtained for a given compound (or possibly a mixture) as a function of temperature, while the so-called lyotropic liquid-crystal phases are obtained as a function of the concentration of a given solute in a solvent Typical examples of the latter systems are the various types of aggregates formed by amphiphilic molecules either in water or in organic solvents. In this chapter we will be interested only in thermotropic systems. An interesting review on lyotropic ionic liquid crystals can be found in Ref. [2],... 

No evidence of any transition is shown below the melting temperature. In additirai, the measured entropy of fusion, ASj, is that expected for full orientational and positional disordering (see Table 3.1). It is thus clear fltat the torsional libration about the C3 axis must lead to occasional jumps into nei boiing symmetry positions that reach the NMR frequency of 10-100 kHz at about 115 K, M K below the melting temperature. Such rather large-scale jump-motion does not disturb the crystal symme-... 

Setting An = Aw ax when Piicosff)) = 1, i.e. for full orientation, leads... 

The standard, site-site RISM theory performs full orientational reduction of the molecular OZ equation (4.1). As distinct, the 3D-RISM approach averages out just the solvent molecular orientations but keeps the orientational description of the solute molecule. This partial averaging reduces by definition the solute-solvent 6D-TCF h "(ri2, 2)... 

The full orientational averaging applied along similar lines to the molecular OZ equation for pure solvent leads to the site-site OZ equation for the solvent correlations, which can be written in the direct space as... 

Now consider the case of a uniaxially oriented polyethylene film or fibre, with full orientation, where the chain axes are. parallel to the draw direction or fibre axis and consider only the proton pair interaction arising from the CH2 groups. The CH2 groups are now such that the proton pairs lie in a plane perpendicular to the fibre axis. Figure 3 illustrates the situation. The intemuclear vector r makes an angle P with Ho, and an angle <(> with a chosen direction in the plane normal to the fibre axis. The angle between the fibre axis and Ho is y. 

To fix our ideas consider again the simple model for the polethylene fibre with full orientation, but where there is rotation of the chain around its axis (at a frequency greater than the line width in Hz, i.e. lO -lO Hz for proton resonance with Hq 10 kg). The length of the internuclear vector r is unchanged by the motion and we have for the proton pair interaction only ... 

The anisotropy of T, and Ti, have been little studied. The only comprehensive treatment of this has been given by McCall and co-workers,where results for tetrafluoroethylene-hexafluoropropylene copolymer fibres are presented, together with a theoretical treatment for the full orientation case. Any anisotropy in Ti was found to be minimal the anisotropy in Ti was comparable to that in Tj or the second moment, as would be expected on theoretical grounds. 

A qualitative fit to experimental data was found for polyethylene, and by later workers for nylon,polyethylene terephthalate, and polypropylene, but the predicted birefringence at small draw ratios tended to be low, with the approach to full orientation being more gradual than found experimentally (Fig. 1). 

One of the main attractions of the X-ray method is that the full orientation function /(0) can be obtained in the case of uniaxial orientation and the function /([Pg.313]

The full orientation-dependent pair correlation function cannot be obtained from experiments. Instead, only information on the spatial correlation function is obtainable. The spatial correlation function is defined as... 

As already obvious in example structure 4.2c, the full orientation function is overly complex for larger molecules. So we generally restrict our considerations to a few relevant orientations by selecting a subset S of the set of atom quadruples. Recall that we have altogether ( ) sets of 4 of the altogether n atoms, and that we can form 24... 

The amount of infonnation stored is drastically reduced compound with the full orientation function, while different conformations eue still distinguished, as we will see in the next sections. 

The cyclohexane example was presented in more detail in earlier articles (103, 104]. However, zero orientations were not considered in these. This led to 13 abstract POFs containing positive and negative orientations only, the three chemically realizable ones among them. Additional restrictions discussed in [103] are described in the following sections. For the example of cyclohexane, they reduce the number of abstract full orientation functions as reported there. However, the restrictions do not reduce the number of candidates when considering abstract POFs with respect to the given selection S. 

Example (A candidate for structure b in Figure 4.7) Consider the following abstract full orientation function as a candidate ... 

Example (The constitution of cyclohexane) Consider the abstract full orientation function x of cyclohexane, structure b in Figure 4.2 ... 

Here we applied the test to an abstract full orientation function and successfully excluded one that was not chemically realizable. The test is applicable only in situations where hve quadruples a,b, c, d), (a, b, c, e), a,b,d,e), ( , c, d,e) and (h.c.d.e) are either all selected or their function values can be obtained by known minimal Radon partitions. We can only determine the abstract Radon partition corresponding to the 5-set a, b, c, d, e in such cases and decide whether or not it is a forbidden one. However, if we select quadruples of type i) and ii), then no such cluster of five quadruples is available as a rule, and Test 5 and Test 6 do not exclude any candidate abstract POP. 

Perhaps some of the most important information on the mode of molecular packing of water in the liquid state is contained in the radial distribution function, which, in principle, can be obtained by processing X-ray or neutron scattering data. There are, however, several difficulties in extracting the proper information from the experimental data. First, it should be kept in mind that the full orientation-dependent pair correlation function cannot be obtained from such an experiment. Instead, only information on the spatial pair correlation function is accessible. We recall the definition of this function,... 

These techniques of electromagnetic absorption spectrometry readily give the orientation parameter (P2(cosar)) with respect to external axes. Combined with the appropriate form of microscopy the methods can, in principle, give orientational information on the scale of the microstructure. However, the absorption property is a second-rank tensor, which means that it is only able to give the orientational information contained in the first spherical harmonic component, Pzicos or)). The higher-order components and hence the full orientation function are inaccessible. The techniques do have the advantage of speed of measurement and provide a real-time evaluation procedure. 

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