Method of cumulants

In this section we consider how to express the response of a system to noise employing a method of cumulant expansions [38], The averaging of the dynamical equation (2.19) performed by this technique is a rigorous continuation of the iteration procedure (2.20)-(2.22). It enables one to get the higher order corrections to what was found with the simplest perturbation theory. Following Zatsepin [108], let us expound the above technique for a density of the conditional probability which is the average... 

Intensity correlation data were routinely processed, using the method of cumulants to provide the average decay rate and the variance V (9)(l0). The latter parameter is a measure of the width of the distribution of decay rates and is given by ... 

Koppel D.E., Analysis of macromolecularpolydispersity in intensity correlation spectroscopy the method of cumulants, J. Chem. Phys., 57, 4814, 1972. 

Another method that has enjoyed a great deal of success is the method of cumulants. In 1972 Koppel (3) showed that the logarithm of the first order AC function was identical to the cummulant generating function for the distribution of decay constants. The coefficients of the cummulant expansion can be related to the moments of the distribution. From the complete set of moments it is theoretically possible to regenerate the exact distribution function. In practice however, even small amounts of noise in the AC data can cause large amounts of error in all but the first two or three moments. For narrow unimodal distributions the first two moments basically define the distribution. However, for multimodal, highly skewed, or very broad distributions, the cumulant expansion is inadequate. 

Thus, given gparticle size distribution. For narrow size distributions, the autocorrelation function is satisfactorily analyzed by the method of cumulants to give the moments of the particle size distribution.(7) However, the analysis of QELS data for samples with polydisperse or multimodal distributions remains an area of active research.(8)... 

The method of cumulants (12) expands g (x) in terms of the moments of the characteristic linewidth distribution... 

By using the method of cumulants, we determined ryv. With values of Z, A2 and M obtained from static light scattering measurements, we obtained DT and f from the slope of a plot of r /K2 versus K2 at different concentrations. By plotting DT against concentration, we estimated Dr 9 70 x 10 ° cm2/sec and kd = -5.10 x 10 g/g. The concentration dependence of the f values suggests that for large particles, internal motions have a measurable effect on translational motions. Thus, we expect that Dg has a concentration dependence. 

Figure 4 shows an example of weighted and unweighted (u j = I) methods of cumulants regressions of PCS data for a Stober [68,69] silica suspension. This example employs a quadratic cumulant (QC) expansion given by... 

Selection of an appropriate delay time range and a scheme for spacing them among correlator channels is necessary to ensure that intensity contributions are sampled from all of the particles in the sample [49,85]. Shaumeyer et al. [62] have explored the effect of the selected value of last delay time on particle size results based on the method of cumulants and single exponential methods. Stepanek [49] discusses the effect of incomplete data sets on Laplace inversion techniques. 

The simplest data analysis technique for dealing with polydisperse systems is the method of cumulants. In this method, Eq. (13) is expanded about an average decay rate r to yield... 

The autocorrelation function can also be analyzed by the method of cumulants. In this approach G(r) is fitted to a low order polynomial. Fora third order cumulants fit ... 

The photon autocorrelation function was analyzed by the method of cumulants (2.) 9 in which the logarithm of the normalized autocorrelation function, ci,t), is fit to a polynomial using... 

Dynamic light scattering data were analyzed by the method of cumulants (24). 

There is no general exact method for extracting size distribution functions n,i dp) from (5.38) and the experimentally measured function G( ) when the form of ttuidp) Is not known. In the method of cumulants, the one most commonly used to estimate hydrosol size distributions from this integral, the logarithm of the autocorrelation function g( ) is expanded in... 

For polydisperse samples of noninteracting particles, the autocorrelation function (or the power spectrum) presents a superposition of the respective functions of the individual species, weighted by the intensities of fight scattered by them. Several procedmes have been employed to analyze the signal from polydisperse samples. The most straightforward procedme is the method of cumulants, in which the log of the measured correlation function is expanded in series ... 

The function (t) can be analysed by the method of cumulants [57] or by inverse Laplace transformation. These methods provide the mean relaxation rate T of the distribution function G(T) (z-average). For the second analysis procedure mentioned above, the FORTRAN program CONTIN is available [97,98]. It is sometimes difficult to avoid the presence of spurious amounts of dust particles or high molecular weight impurities that give small contributions to the long time tail of the experimental correlation functions. With CONTIN it is possible to discriminate these artifacts from the relevant relaxation mode contributing to (t). 

A very simple and widely used method, known as the method of cumulants, was described by D. Koppel [45], This method allows one to obtain mean value of T(0) and the width of distribution, characterized by the polydispersity index, PI. In the method of cumulants the natural logarithm of g(l)(xcorr, 0) is expanded into polynomial series, in which usually only the first two terms are retained, i.e. ... 

Method of cumulants is suitable for characterizing a reasonably narrow uni modal distribution. When G(T, 0) is broad or can not be adequately represented by unimodal distribution, the method of cumulants fails. Indeed, two very different distributions may have the same first two moments, and thus can not be distinguished from each other based on the method of cumulants [46],... 

The alternative is the method of cumulants (31) which considers an unspecified size distribution of diffusing particles corresponding to a distribution (Gr) of relaxation time constants r = Dtq which is normalized so that ... 

Perhaps, by far the greatest feature of the modem dynamic light-scattering (QELS) equipment is the ease of obtaining data. Data interpretadon, relative to the true size and to the real-worid iplicabai is another question. Eariy QELS instrumoits were inflexible in their ability to accept software for any new methods of data analysis and calculated distributions only by the method of cumulants and were inc able of utilizing new histogram or polynomial techniques. 

Dynamic light scattering allows measurement of the translational diffusion coefficient A of particles. For noninteracting particles, the first-order autocorrelation function g r) is directly proportional to A- Polydisperse systems are analyzed using the method of cumulants, in which a distribution of exponential terms C(r) is assumed ... 

The simplest form of postcorrelation analysis is called the method of cumulants and provides for a mean particle size together with an index of polydispersity. If a defined a priori form of a distribution (as recommended in ISO 13321) is now assumed, the cumulants result can now be expressed as a mean size Xpcs and standard deviation of a size distribution. 

QLS has been used routinely to determine the Z-average translational diffusion coefficient and the corresponding hydrodynamic radius of macromolecules in solutions and of colloidal suspensions (11,12) and also to characterize the polydispersity effects the method of cumulants (13). 

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