Random orientation fluctuations

The case of random orientation fluctuations for oriented systems has been treated by Stein and Hotta whose equations reduce, for example, to ... 

The preceding theory is for the case of random orientation fluctuations, where in the unoriented state, the oriented domains have spherical symmetry (Fig. 25). In many systems, this is not so and the domains may be rod or disclike in shape. In fact, in the case of more complicated morphology such as that of spherulites. 

If we select the z axis to be normal to the plane of the clay film, the angle 0 can be imagined to fluctuate rapidly as the probe tumbles in the interlayer. This time-averaged system is symmetric about the z axis because of the random orientation of the a and b axes of the clay platelets in the x y plane. 

In a system of nanoparticles, thermal fluctuations of their magnetic moments severely reduce the anisotropy of the resonance magnetic field, resulting in superparamagnetic spectra narrowing. This reduction is the more pronounced the smaller is the particle size. Therefore, the SPR spectra of macroscopically isotropic nanoparticle systems characterised by a distribution in size usually maintain a distinct shape asymmetry characteristic of powder patterns of randomly oriented anisotropic particles. From an inspection of such spectra, one can conclude that the angular dependence of the resonance magnetic field of individual particles is not completely reduced. 

In the simplest possible model for an Ideal chain, the bonds between atoms In the backbone are treated as vectors connecting volumeless points which do not Interact. Such a model chain is depicted in fig. 5.2 the (fluctuating) distance between the end points is denoted as r. If, moreover, any orientation between two consecutive bonds is assumed to have the same probability, the conformational properties of long chains can be described by the universal random-Jlight model, first introduced by Kuhn l. Let the chain have N randomly oriented bonds, each of length t. Such a model chain contains IV + 1 backbone atoms. When these bonds are assumed to be fully Independent of each other, the conformation resembles the trajectory of a particle diffusing under the action of a random force, for which the solution is well known -S- ). The mean square displacement [Pg.614]

Here, it is assumed that the distance r is a constant independent of time. After the time-consuming calculations for Equations (3.3)-(3.5) and (3.11), Ti, T2, and NOE can finally be expressed by the auto-correlation functions G (t) of the orientation functions F, which describe the random time fluctuation of the C—H vector, or by the spectral densities Jq(o)) that are the Fourier transforms of G,(t) with frequency <0 as follows ... 

The parameters PM(r) associated with the scattered radiation may fluctuate in time for instance, as a result of a random rotation motion of the polarizable element. In general, the experimentalist observes the time average of the parameters P. Then, he has to extract from this average, the contribution of the orientation fluctuations, which he considers either as a nuisance or as a studydeserving effect. We shall perform the calculation by using the formula [see (6.1.13) and (6.3.58)]... 

Tc is a measure of the time interval during which any two spins in the system maintain a given orientation with respect to each other, and is related to the time of rotation of a molecule, or the time of diffusion into a neighbouring position. For most liquids is roughly the time of rotation of a molecule and is generally of the order of 10 s. It occurs in the relaxation expressions because the random frequency fluctuations of molecular motion contain all frequencies below that appropriate to Tc of the Brownian motion, and except in viscous liquids this includes v. 

The arrangement of optic axes directions for random rodlike and disclike orientation fluctuations. (From Stein, R. S. and Wilson, P. (1962). J. Appl. Phys., 33,1914.)... 

Fig. 18 End-view diagram of the triple-layer structure model for the injection-molded NCH bar 3 mm thick. The flow direction caused by injection-molding is normal to the paper plane. Curved arrows with one head mean random orientation round the axis normal to the plane containing the curve. Arrows with two heads indicate fluctuation...

---