Cumulants, method

For incremental methods of analysis, the sample is theoretically taken from an extremely thin layer in which h2 is nearly equal to hj for cumulative methods of analysis, the amount settling in a particular plane in a given time, normally at the bottom of the settling vessel, is measured... 

Exercise 14.2 Sort the isotopic groundwater dating methods into decaying and cumulative methods. What are the advantages of each group ... 

When the different approaches to adiabatic elimination are compared, substantial general agreement is found. The approach of the Barcelona group based on a generalization of the cumulant method, the Chapman-... 

BS 3406, British Standard Method for determinig Particle Size Distribution is a comprehensive standard that includes methods for both incremental and cumulative methods of particle size deterination. Part 1, 1984, covers Recommendations for Gravitational Liquid Sedimentation Mehods for Powders and Suspensions. Part 5, 1985 covers... 

The accuracy of the analysis presented in this paper is determined by the validity of two key approximations (1) the description of the energy transfer dynamics by the first order cumulant expansion method, (2) the use of a Gaussian chromophore pair distribution function. Although originally developed for, and successfully applied to, the problem of energy transfer in disordered infinite volume systems, the cumulant method can be modified to provide a highly accurate description of energy transfer in finite volume systems such as polymer coils (2 ). ... 

For a narrow particle size distribution the cumulant analysis is usually satisfactory. The cumulant method is based on the assumption that, for monodisperse suspensions gi(r) is monoexponential. Hence, the log of gi(r) versus t yields a straight line with a slope equal to F,... 

The cumulant method expands the Laplace transform about an average decay rate. 

The main advantage of the cumulant method is that it does not require any assumption about the particular shape of the size distribution. The main drawback of this method is that a variety of rather different distributions may have similar values of and K2. Therefore, we cannot obtain reliable information about the size distribution only from DLS data. The cumulant method is most suitable when the size distribution is known to be monomodal and relatively narrow. 

Cumulative Methods. In cumulative methods the settling rate of suspensions or free-falling diameter are monitored. Cumulative methods have an advantage over incremental methods in that the amount of sample required is small (about 0.5 g), which reduces interaction between... 

In Fig. 14 the correlation function (q (t)q (O) is shown for the nonlinear potential in Eq. (3.85) at /3 = 10. This correlation function presents another nontrivial test of the various approximate methods because, classically, it can have no negative values while, quantum mechanically, it can be negative due to interference effects. Clearly, only the cumulant method can describe the latter effects. The classical result is extremely poor for this low-temperature correlation function. The CMD with semiclassical operators method also cannot give a correlation function with negative values in this case. This feature of the latter method arises because the correlation of the two operators at different times is ignored when the Gaussian averages are performed. Consequently, the semiclassical operator approximation underestimates the quantum real-time interference of the two operators and thus fails to... 

To test the methods outlined in Section III.B.3 for calculating general correlation functions in the phase-space centroid perspective, the correlation function >l(f)B(0)), where A= pq and B = qp, was studied [5], The results of this calculation are shown in Fig. 15 for the nonlinear potential in Eq. (3.85) at /8 = 10. The classical MD result is, as expected, extremely inaccurate for this low temperature. The CMD with semiclassical operators method does not reproduce the amplitude and negative values of this correlation function as well. On the other hand, the cumulant method can describe the quantum interference effects for this correlation function, and it appears to do so quite well. 

An important result of the experience of Baaen c< al. (1964) with the differential method should be mentioned here, namely that isotopio analyses of the early (0 6%) fractions are often the least reliable indicators of the isotope effect the points calculated from these early fractions frequently do not lie on the curve derived from the remaining fractions, and repeated purification of the samples does not usually alter the analyses. This situation could result fromthenearly quantitative removal, with the first fractions, of materials present in trace quantities in the reaction mixture. Such observations indicate that caution must be exercised in the evaluation of isotope effects by the cumulative method. 

In the cumulant method (Koppel, 1972 Pusey, J974), the function ln0i(<) is expanded into its Maclaurin series in powers of < at t —> 0 and is written as... 

Thns, in either homodyne or heterodyne mode, the DLS autocorrelation function can yield D at any given q from an exponential fit of the form in Eqnation 5.63. For polydis-perse systems, DLS provides an average over the diffnsion coefficient distribution of the particles. A nnmber of approaches have been developed for analyzing polydisperse systems, including the robust cumulant method (see Section 8.2.2), histograms, and the inverse Laplace transform method (see Section 8.2.3). To turn the coefficient distribntion into the MWD requires a known relationship between D and M, often of the form D=AM y. 

The widely used cumulant method yields the z-averaged diffnsion coefficient, D. This means that the hydrodynamic radins thus obtained by DLS in the cnmnlant analysis is actually the reciprocal of the z-averaged reciprocal,... 

The average molecular weight of the long chain and short chain is 41,000 and 2500, respectively. When the short chain concentration changed from 20 to 80 wt%, gel concentration increased. The broken line near the solid line is the cooperative diffusion constant obtained by the cumulant method... 

The cumulants method is based on the formalism of the statistical cumulant generating function ... 

The variance is identified by The average linewidth and the width of G(r) distribution can be obtained by this method. In practice, it is difficult to apply the cumulants method when the variance becomes large, say 5. 

Dynamic light scattering has also been used to study ionomer aggregates in solution. It was shown that hydrodynamic radius increases with increasing ion content and ionomer concentration (up to a certain point, above which the hydrodynamic radius becomes almost constant in the case of the THF solution [8]). While Lantman et al. only obtained hydrodynamic radii from scattering data at an angle of 90°, Pedley et al. used the cumulant method to obtain information on the distribution of aggregates and found that the quality factor. 

The cumulants method is a moment expansion method. It compresses the entire multi-exponential decay distribution into the exponent and then expands the exponent F to a polynomial expression of the cumulants (the moments) and is ... 

Because the ACF is a function of the scattering vector K, all cumulants are also a function of K. At small delay times, for particulate systems that have size distributions which are not very broad (P.I. < 0.3), the cumulants method has been proven to be able to provide fairly good mean values of F over a broad range of practice although it will not accurately recover more than the first three cumulants. Although f obtained using the cumulants method is not very sensitive to experimental noise, the P.I. is. It has been concluded that the baseline value must be correctly established better than 0.1% otherwise... 

For monodisperse samples, the reproducibility using the cumulants method to retrieve a mean value and polydispersity index is however relatively good. It can easily achieve a reproducibility of 2-3% for the mean size. Figure 5.15 illustrates repeat PCS measurements of a monodisperse polystyrene latex sample of nominal 300 nm diameter dispersed in n-butanol (the sample had been kept in a sealed cell for seven years.) The measurements were performed using four PCS instruments with 20 repeat measurements on each instrument. The overall standard deviation is just 0.9%. 

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