Particle size covariance

Temperature control is an important factor in determining particle size by sedimentation methods. During a typical run changes in spin fluid temperature of 2-4°C were common. This temperature change (DELTEMP) was used as the covariate in the analysis of covariance. 

Analyses of covariance for both turbidity weight-average particle size (Dw) and polydispersity (Dw/5n) (Dn, number-average particle size) were carried out. The Tables I and II show the results for the three samples combined. 

The covariate, DELTEMP, is highly significant in the determination of the Particle Size. This indicates that temperature control, from the statistical point of view, is more important than has been previously considered. This is not surprising physically, in view of the temperature dependence of the spin fluid viscosity and the density terms contained in Stokes Law. The cooling capacity of the two DCPs were known to be different. Both instruments had been physically recalibrated prior to running these experiments. 

Lumme and Rahola [53] considered cometaiy particles as stochastically shaped, i.e., particles whose shape can be described by a mean radius and the covariance function of the radius given as a series of Legendre polynomials. They made computations for a variety of particle shapes and size parameters (x = 16) using the refractive index m = 1.5 + i0.005. They found that the particles should have size parameters x > 1 to provide the negative polarization and low maximum polarization. Ensembles of particles with a power-law size distribution showed phase functions of intensity and polarization similar to the cometaiy ones. No information of the spectral characteristics was presented. 

The coupled cluster (CC) approach is the most powerful and accurate of generally applicable electron correlation methods. This has been shown in many benchmark applications of 4-component relativistic CC methods to atoms [11-18] and molecules [19-31]. The CC method is an all-order, size-extensive, and systematic many-body approach. Multireference variants of relativistic 4-component CC methods capable of handling quasidegeneracies, which are important for open-shell heavy atomic and molecular systems, have been developed in recent years [15,17-19,21,31]. In particular, the multireference FSCC scheme [32,33] is applicable to systems with a variable number of particles, and is an ideal candidate for merging with QED theory to create an infinite-order size-extensive covariant many-body method applicable to systems with variable numbers of fermions and bosons [6,7]. 

Although the number of degrees of freedom has been minimized, this approach is computationally intensive, and imposes severe limitations on the size of the system that can be studied. Since every particle interacts with every other particle, the calculation of the mobility matrix scales as 0 N ), where N is the number of Brownian particles. In addition, the covariance matrix for the random displacements requires a Cholesky decomposition of the mobility matrix, which scales as 0 N ) [27]. The computational costs of Brownian dynamics are so large that even today one cannot treat more than a few hundred Brownian particles [28]. 

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