Moments, defined

The moments defined by Eq. (7.1) are referred to as the unabridged moments. For moments with / > 2, an alternative, traceless definition is often used. In the traceless definition, the quadrupole moment is given by... 

Some simple characteristics of the momentum density are its moments defined by... 

An elegant solution to analyse the REDOR response in the presence of multiple-spin interactions was introduced by Eckert and co-workers. As shown by the authors, the initial part of the REDOR evolution curves proves to be virtually independent on the exact spin geometry and distribution effects. Thus, when restricting the data analysis to AS/So values <0.3, the data can be analysed in terms of the second moment defined by... 

In the case of multiple-spin interactions, the second moment defined in Equation (2) may be approximated to first order by... 

The alternative is the use of a descriptive mathematical model without any relation with the solution of the transport equation. On the analog of the characterization of statistical probability density functions a peak shape f(t) can be characterized by moments, defined by ... 

The zeroth moment (n=0) gives the total intensity and is related to theory by familiar sum formulae (Chapter 5). For nearly classical systems (i.e., massive pairs at high temperature and not too high frequencies), the first moment (n=l) is very small and actually drops to zero in the classical limit as we will see in Chapter 5. The ratio of second and zeroth moment defines some average frequency squared and may be considered a mean spectral width squared. A complete set of moments (n = 0, 1,... 00) may be considered equivalent to the knowledge of the spectral line shape,... 

Here / = 1/7 in the standard notation. From our general statements in Section in. A, the spinodal criterion derived from the exact free energy (38) must be identical to this this is shown explicitly in Appendix C. Note that the spinodal condition depends only on the (first-order) moment densities p, and the second-order moment densities py of the distribution p(cr) [given by Eqs. (40) and (41)] it is independent of any other of its properties. This simplification, which has been pointed out by a number of authors [11, 12], is particularly useful for the case of power-law moments (defined by weight functions vt>f(excess free energy only depends on the moments of order 0, 1... K — 1 of the density distribution, the spinodal condition involves only 2K— moments [up to order 2(K — 1)]. 

Recalling the expression for the induced dipole moment, defined in terms of the polarizability of the molecule (Equation 12-22), and then differentiating, we obtain our final expression for the electric field arising from the interaction of a small, single molecule with polarized light (Equation 12-25) ... 

We assumed that the magnetic tensors appearing in the spin Hamiltonian and the diffusion tensor have the same principal axis system. The are the Wigner matrices of rank /. The nuclear spin operators are expanded over the set of (21 +1) matrices Af j, I being the nuclear angular moment, defined via... 

The existence of pores and walls implies the existence of two different, mostly independent, CLDs in one and the same sample There is the CLD of the pores A([) (first moment 7) and the CLD of the walls (first moment m). According to the linear integration principle of Rosiwal (1898), these moments define the porosity p via p=V(l+m). Unfortunately, the moments are intermixed in the so-called SAS correlation function y(r), which can be obtained from scattering data 1(h) by solving the inverse problem, [2],... 

Debye in 1912 extended Langevin s procedure to dielectrics, by attributing to the molecules a permanent (natural) electric dipole moment defined as in equation (34), The classical permanent magnetic dipole moment of a molecule is defined as follows (as an axial vector) ... 

Quantitatively, it is characterized by an electric moment, defined by the vector ... 

In practice not only the energy difference A but also the variance a is unknown and must be estimated from the data. Let us assume that, for a given pair of ionic states (S, S ), we generate n statistically uncorrelated estimates of the energy difference y, . .., / each normally distributed with their first and second moments defined in the usual way... 

Figure 17 UV-Vis and fluorescence data for TPP-COOH 19 and Zn-TPP-COOH 20. Bottom bisftetraarylporphyrin) derivative of 5a-cholestane-3a,17/3-diol 23 and 24 a positive helicity between the two effective transition moments defined in direction 5C/15C, the interchromophoric distance R, , UV-Vis and CD spectra of in CH2CI2 UV e, Revalues, and CD amplitudes A of other bischromophoric derivatives of 5a-cholestane-3a,17/3-diol 21 and 22.

We proceed by inserting the nonvanishing multipole moments defined in Eqs. (F.30)-(F.33) into the expansion in Eq. (F.29), which gives... 

In sections 5.22 and 5.24, we have made schematic evaluations of the nature of the vibrational spectrum (see fig. 5.5). At this point, it is convenient to construct approximate model representations of the vibrational spectrum with the aim of gleaning some insight into how the vibrational free energy affects material properties such as the specific heat, the thermal expansion coefficient and processes such as structural phase transformations. One useful tool for characterizing a distribution such as the vibrational density of states is through its moments, defined by... 

Sack Donovan (1971) proposed an alternative approach for the calculation of the coefficient of the recursive formula reported in Eq. (3.5) (and appearing also in the Jacobi matrix) that resulted in higher stability. This approach is based on the idea of using a different set of basis functions naif) to represent the orthogonal polynomials, rather than the usual powers of f. The improved stability results from the ability of the new polynomial basis to better sample the integration interval. The coefficients are calculated from the modified moments defined as follows ... 

These moments define the corresponding cumulants from which fi-... 

The integral can then be expressed in terms of the various second moments defined, for example, by... 

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