The First Cumulant

that non only the ensemble average is needed. In the following we confine ourselves to the first cumulant, i.e. to the initial part of the TCF, and drop the subscript 1 in Eq. (B.37). 

The superscripts Q stands here for quasi-elastic scattering, and the subscript pre means the pre-average approximation. 

Before giving the explicit equations for the various averages, it will be useful to consider the limit of q - 0. Theory on dynamic scattering proves that in this limit 

We now give the formulae for the various averages. For rigid particles the average has to be performed only over all orientations (subscript or ). This yields 

For flexible structures, the function F (z) has to be averaged over the distance distribution. For Gaussian chains one obtains82  

---

C, the fourth parameter, represents the relationship between the first cumulant and the particlescattering factor. For values of 1/F( ) < 10, the double logarithmic plot of the first cumulant against the reciprocal particle-scattering factor yields a straight line, and the exponent v is related to the initial slope C oiF/q D, against by the equation... 

The crossover from 0- to good solvent conditions leads at constant x = (x — 0)/0 to increasing Q(Q,x)/Q3 with decreasing Q. Qualitatively, this effect is well described in the framework of the blob model using the method of the first cumulant, proposed by Akcasu and coworkers... 

At large Q quantitative agreement between experiment and RPA prediction is found. Here the first cumulant is proportional to - we are in the Rouse... 

Figure 6.25 shows the dispersion of the effective diffusion Fq/Q with Fq the first cumulant of the relaxation function. Three branches are visible. A collec-... 

In this section some details of the static and dynamic structure factors and on the first cumulant of the time correlation function are given. Hie quoted equations are needed before the cascade theory can be applied. This section may be skipped on a first reading if the reader is concerned only with the application of the branching theory. 

As was shown by Bixon75, Ackerson79 and Akcasu and Gurol80, the first cumulant can be calculated exactly without knowing the space-time distribution function with the following result... 

In the last chapter, equations were derived for the particle-scattering factor, the mean-square radius of gyration, the diffusion coefficient and the first cumulant of the dynamic structure factor. All these have the common feature that, for homopolymers at least, they can be written in the following form ... 

In Chap. B II.4 we have shown that the angular dependence of the first cumulant of the electric field correlation function can be obtained by integration over the particle-scattering factor. This rule remains valid also for copolymers but is restricted to Gaussian behavior of the subchains. Although the whole q-region can be covered by this integration, which in most cases has to be carried out numerically, it is useful to discuss the... 

Trisilaallene 67 was synthesized as the first stable compound with a formally sp-hybridized silicon atom and also the first cumulative Si = Si doubly bonded... 

The experimental values of [r and were suflSciently close to weight-averaged quantities to be used without correction. The R value was obtained from the first cumulant of the autocorrelation function, which yields the z average of the diffusion coefficient. However, because the diffusion coefficient D is inversely proportional to Ry, the actual value more closely reflects for many synthetic polymers. Although these quantities (and the values of Rg,w obtained previously) are not exactly equal to those that would be obtained for a monodispersed sample with M = M, further correction is within the uncertainty of the limited data. 

Without doing detailed quantitative analysis of the data, it can be stated that the polyion diffusion can be qualitatively described by two theoretical concepts. The first concept capable of qualitative explanation of the polyion diffusion is the concept based on considering polyions as interacting Brownian particles with direct interactions between polyions and hydrodynamic interactions. The short-time collective diffusion coefficient for a system of interacting Brownian particles treated by statistical mechanics is calculated from the first cumulant F of the dynamic structure factor S(q, t) as [15-17]... 

---