Fourth moment

Since the integral term is the fourth moment of the distribution (m, Eq. (18-35) becomes... 

In the specific case of an MSMPR exponential distribution, the fourth moment of the distribution may be calculated as... 

Gram-Charlier Series This is an infinite series whose coefficients involve the Gaussian distribution and its derivatives (Kendall, Advanced Theory of Statistics, vol. 1, Griffin, 1958). The derivatives, in turn, are expressed in terms of the moments. The series truncated at the coefficient involving the fourth moment is... 

The kurtosis, the fourth moment, is a measure of peakedness or flatness of a distribution. It has no common notation (k is used here) and is given for a continuous random variable by... 

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

Beginning with the fourth moment, all of them are sensitive to the strength of the interaction during collisions. 

Coincidence with the exact expression is caused by the fact that terms of higher order in do not contribute to the second and fourth moments. Correspondingly, for spherical rotators (r = 3) we have... 

Calculation of these correlator and matrix elements of LXl provides the following addition to the fourth moment compared with (2.65b) ... 

The advantage of Raman spectromicroscopy is that very small specimens can be studied while still allowing the determination of the second and fourth moments of the ODF. However, the expressions for the Raman intensities are more complex since the optical effects induced by the microscope objective have to be considered. Although the corrections may be small, they are not necessarily negligible [59]. This problem was first treated by Turrell [59-61] and later by Sourisseau and coworkers [5]. Turrell has mathematically quantified the depolarization of the incident electric field in the focal plane of the objective and the collection efficiency of the scattered light by high numerical aperture objectives. For brevity, only the main results of the calculations will be presented. Readers interested in more details are referred to book chapters and reviews of Turrell or Sourisseau [5,59,61]. The intensity in Raman spectromicroscopy is given by [59-61]... 

The second moment (p ) is twice the electronic kinetic energy, and the fourth moment p ) is proportional to the correction to the kinetic energy due to the relativistic variation of mass with velocity [174—178]. 

Table 2. NaCl canonical moments and averages over the densities of states in section 2 M = second moments L = fourth moments...

The different types of paths are shown in Fig. 4.7 for the tetrahedron. We see that per starting atomic site there are 3 two-atom paths, 12 three-atom paths, and 6 four-membered ring paths, leading to a total of 4(3 + 12 + 6) = 84 paths per tetrahedral molecule as in eqn (4.57). The fourth moment measures the unimodal versus bimodal behaviour of the... 

Fig. 4.7 Examples of nearest-neighbour paths of length four that contribute to the fourth moment of the tetrahedron s eigenspectrum. The solid atom indicates the site from which the path starts and to which it eventually returns. The number under each tetrahedron gives the total number of such paths starting and ending on the solid atom.

Fig, 8.6 The number of different contributions to the fourth moment about a given atom on a Bethe lattice with local coordination,. ... 

The structural sequences in Fig. 8.5 change between the s-valent, p-valent and sp-valent cases because the shape parameter, s, is very sensitive to the angular character of the orbitals. We have already seen in Fig. 4,13 for the triatomic molecule AH2 that the three-atom contributions to the fourth moment are dependent on the bond angle. It follows from eqs (7.14)—(7.17) that hopping along a three-atom path and back again amongst sp-valent orbitals will lead to the fourth-moment contribution... 

Fig. 8.8 The angular dependence of the three-atom fourth moment contribution, eqn (8.24) for the pure s, pure p and sp cases. (From Cressoni and Pettifor (1991).)...

Thus, we expect the puckered graphitic sheet with 90° bond angles to have the smallest normalized fourth moment and shape parameter, s, and hence to be the most stable structure for the half-full p band as is indeed observed in the middle panel of Fig. 8.5. We should also note that if the n bonding is neglected then this three-atom contribution is identically zero for = 90°, so that 5 = 0 and we have the total bimodal behaviour of the p eigenspectrum that is observed in the lower panel of Fig. 8.1 for the arsenic structure type. 

This minimum is responsible for the diamond and graphite lattices with = 109° and 120° respectively having the smallest and second smallest values of the normalized fourth moment, and hence the shape parameter, s, in Fig. 8.7. This is reflected in the bimodal behaviour of their densities of states in Fig. 8.4 with a gap opening up for the case of the diamond cubic or hexagonal lattices. Hence, the diamond structure will be the most stable structure for half-full bands because it displays the most bimodal behaviour, whereas the dimer will be the most stable structure for nearly-full bands because it has the largest s value and hence the most unimodal behaviour of all the sp-valent lattices in Fig, 8.7, We expected to stabilize the graphitic structure as we move outwards from the half-full occupancy because this... 

Fig. 8.10 Upper panel the four-membered ring contribution to the fourth moment of a d band as a function of the bond angle 0. (After Moriarty (1988).) Lower panel A four-membered ring contribution in the fee and bcc lattices respectively. Note that from the upper panel the fee and bcc rings shown contribute positive and negative contributions respectively to the fourth moment.

Calculate the normalized fourth moments / 4// 2 for the dimer, helical linear chain with bond angles of 90°, graphitic sheet with bond angles of 120°, and diamond lattice with bond angles of 109° (assuming = 0). Amongst these structures which do you expect to be the most stable and the least stable for a half-filled p shell ... 

Note that Equation (9) implies that the square of the standard deviation a2 is the second moment of d relative to the mean d. Higher order moments can be used to represent additional information about the shape of a distribution. For example, the third moment is a measure of the skewness or lopsidedness of a distribution. It equals zero for symmetrical distributions and is positive or negative, depending on whether a distribution contains a higher proportion of particles larger or smaller, respectively, than the mean. The fourth moment (called kurtosis) purportedly measures peakedness, but this quantity is of questionable value. 

Broad-line NMR derivative spectra were obtained using a Brucker HFX-90 spectrometer to record the resonance at 84.67 MHz. The specimens, made by compacting granular PTFE into preforms, sintering at 380°C, and cooling slowly at a rate of 0.02 deg/min had a specific gravity of 2.205. The second moment of tire NMR line shape is of interest because the fourth moment of the orientation distribution function is proportional to it. 

Exact methods are developed for calculating second and fourth moments of chain molecules. The RIS approximation is adopted to represent the effects of hinderance potentials affecting bond rotations. IMo other approximations are invoked in deriving the statistical-mechanical averages. 

As explained in Section VI, we must find functions S.p, Qx which minimize J Eq. (5). What we shall do is solve the modified problem containing the fourth moment condition M, and only at the end of the work allow M to approach its canonical value M. Furthermore, we shall use the formalism of the calculus of variations. 

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