Second moment relations

The principal axis transformation requires that the off-diagonal elements (the deviation moments or inertial products) vanish in the PAS, the resulting three conditions are known as the second moment relations ... 

However, there are two common operationally defined types of structure that are determined from effective moments of inertia. The more common, the so-called effective or r0 structure, is somewhat loosely defined. In practice, any structural parameter that requires for its determination fitting one or more of the second moment relations is designated as r0. r0 structures are not uniquely defined since, for any over-determined system, the value of structural parameters obtained depends somewhat on the manner in which the data are treated and the values are isotopically dependent. This problem is examined in more detail by Schwendeman (this volume). 

There are several different definitions relating to second moments in the literature, and it is necessary to take care when applying seemingly simple formulae that involve such quantities. Not only that, authors often use the same name for the different definitions. 

The second order central moment is used so frequently that it is very often designated by a special symbol o2 or square root of the variance, o, is usually called the standard deviation of the distribution and taken as a measure of the extent to which the distribution is spread about the mean. Calculation of the variance can often be facilitated by use of the following formula, which relates the mean, the variance, and the second moment of a distribution... 

The set of central second moments is related to the autocorrelation function by means of the following simple formula,... 

Van Vleck 8) has shown how the second moment and fourth moment interactions between the magnetic nuclei. 

Moment Analysis. The zeroth moment is the molar concentration of polymer and is expressed by Equation 5. The first moment is proportioned to the mass of polymer formed and is related to monomer concentration Equation 3, The second moment WA(t) is expressed by... 

The same relation is found for the end-linking of molecules of low functionality if = 3 or 4) and for the vulcanization of long molecular chains. The second-moment average number of cross-linking sites along the chain, f2, is defined as... 

If we were to have an isolated polymer chain with a single nuclear spin attached to each segment (the marked chain) crosslinked into an unmarked network, the second moment of the NMR line of that spin species would carry information relating to the separation of chain segments, and to their relative orientation with respect to the field direction. If the network were to be subjected to a bulk deformation, these geometrical parameters would be altered, and hence we would expect a corresponding change in the value of the experimentally measured... 

From these recurrent relations, one obtains the following expression for the second moment in the case c = —oo (c < xq < d) ... 

This involves obtaining the mean-residence time, 0, and the variance, (t, of the distribution represented by equation 19.4-14. Since, in general, these are related to the first and second moments, respectively, of the distribution, it is convenient to connect the determination of moments in the time domain to that in the Laplace domain. By definition of a Laplace transform,... 

Pope, S.B. 1994. On the relation between stochastic Lagrangian models of turbulence and second-moment closures. J. Physics Fluids 6(2) 973-85. 

The data provided by these measurements are presented In this paper In terms of a parameter that measures the extent and degree of molecular mobility. M2J is an empirical second moment of the frequency spectrum of the NMR signal (25), truncated here at 16 kHz (26), and Is Inversely related to the average molecular mobility of the specimen. Thus the relative decrease In 2 Is a sensitive measure of the extent and degree of mobility acquired by rigid molecular structures as a result of their destabilization by thermal or solvent treatment. 

In the nonzero trace definition, this equality is no longer valid. To illustrate the relation for the diagonal elements of the second-moment tensor, we rewrite the xx element as... 

The relation between the second moments pxP of the deformation density and the traceless moments QxP can be illustrated as follows. From Eq. (7.2a), we may write... 

One should also mention that the NMR timescale tnmr is related to the second moment M2 via... 

If a pulse of tracer is injected into a flowing stream, this discontinuity spreads out as it moves with the fluid past a downstream measurement point. For a fixed distance between the injection point and measurement point, the amount of spreading depends on the intensity of dispersion in the system, and this spread can be used to characterize quantitatively the dispersion phenomenon. Levenspiel and Smith (L16) first showed that the variance, or second moment, of the tracer curve conveniently relates this spread to the dispersion coefficient. 

The heats of formation of equiatomic AB transition-metal alloys may be predicted by generalizing the rectangular d band model for the elements to the case of disordered binary systems, as illustrated in the lower panel of Fig. 7.13. Assuming that the A and transition elements are characterized by bands of width WA and WB, respectively, then they will mix together in the disordered AB alloy to create a common band with some new width, WAB. The alloy bandwidth, WAB may be related to the elemental bond integrals, hAA and , and the atomic energy level mismatch, AE — EB — EAt by evaluating the second moment of the total alloy density of states per atom ab( ), namely... 

Formulae for the classical zeroth and second moments of collision-induced absorption of the rare gases [148, 165, 333] have been reported the classical second moment y2 is related to the first quantum moment, Mi [317, 177],... 

The dipolar broadening parameter AG, independent of the type of solvent used, is related to the residual second moment (AM2b) ... 

Fig. 23. Plot of the dipolar broadening parameter AG (left scale) and its relation to the square root of rigid lattice second moment, AM2 = 15 kHz (right scale), for polystyrene networks crosslinked with DVB (solid symbols) and EDM (open symbols) swollen to equilibrium, vs l/n, the reciprocal nominal number of C—C bonds between crosslinks points. Solvents CC14 ( ), CDC13 (A),...

Since crosslinked polymer networks exhibit anisotropic internal motions, the effect of MAS on the line narrowing is explained assuming that the rigid lattice second moment AM2 consists of two terms, analogous to Eq. (34). Then, the residual line width in the NMR MAS experiment, Acomas, can be described by the following relation ... 

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