Distribution function of clusters

14 Cluster distribution in the region where finite and infinite clusters coexist. 

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The above described lack of smoothness at y = ya is essential. It refers to the characteristic power law distribution functions of cluster sizes in percolation, indicating that the most frequent number Lx of singly connected bonds is unity. This leads to a spontaneous fast decline of G when y exceeds the value yapp> since all L-N-B-chains with Lx=1 break simultaneously at this amplitude. Experimental results show that a smooth transition of G with varying strain amplitude appears that cannot be described by a power law distribution function or the assumed exponential type of/lfl (y). 

The existence of clusters of single-wall carbon nanotubes (SWNTs) in solvents is discussed. The bundlet model for clusters describes the distribution function of clusters by size. The phenomena have an explanation in the bundlet model, in which the free energy of an SWNT is combined from two components a volume one, proportional to the number of molecules n in a cluster, and a surface one proportional to n. ... 

FIGURE 1.32 (continued) Temperature dependences of (c) changes in the Gibbs free energy of SAW and (d) distribution functions of clusters and domains of SAW for hydrated A-300 at /r=50 mg/g placed in different media. 

FIGURE 8.6 Size distribution functions of clusters and domains of water bound in wheat seeds (a) initial and after (b) 1 and (c) 2 days of germinating in (curve 1) pure water or (curve 2) with addition of nanosilica A-300. 

The size distribution functions (/(/ )) of clusters and domains of bound water (Figure 8.23) calculated using TSDC cryoporometry (IGT equation) give more detailed information than PSD based on the DSC data (Figure 8.19b). 

T = (/i + l)ln[iici(t)/i ci(t2)], aiid normalized to the unity distribution function of clusters over dimensions G (u) (/ " G (u)du = 1). Figure 18. illustrates the comparison of distribution function Gi(u) (with d = 3) obtained from approximate solution of (13.1.3), with the experimental one (Hermann uid Rhodin 1966) (for the systems Au-Si02, Au-NaCl, and Ni-KCl). The comparison of experimental data on Sn-C (Blackman and Guzzon 1959) and Au-NaCl (Hermann and Rhodin 1966) with theoretical curve (d = 3, /i = 2) is presented in Fig. 19 and shows a good agreement. 

Fig. 8.20. Metallicity distribution function of globular clusters (crosses indicating error bars and bin widths) and halo field stars (boxes), after Pagel (1991). Copyright by Springer-Verlag.

One such systematic generalization was obtained by Cohen,8 whose method is now given the point of departure was the expansion in clusters of the non-equilibrium distribution functions. This procedure is formally analogous to the series expansion in the activity where the integrals of the Ursell cluster functions at equilibrium appear in the coefficients. Cohen then obtained two expressions in which the distribution functions of one and two particles are given in terms of the solution of the Liouville equation for one particle. The elimination of this quantity between these two expressions is a problem which presents a very full formal analogy with the elimination (at equilibrium) of the activity between the Mayer equation for the concentration and the series... 

It is found that the atomic arrangement, or a vacancy network, in a depleted zone in a refractory metal or a dilute alloy of a refractory metal, created by bombardment of an ion can be reconstructed on an atomic scale from which the shape and size of the zone, the radial distribution function of the vacancies, and the fraction of monovacancies and vacancy clusters can be calculated. For example, Wei Seidman108 studied structures of depleted zones in tungsten produced by the bombardment of 30 keV ions of different masses, W+, Mo+ and Cr+. They find the average diameters of the depleted zones created by these ions to be 18,25 and 42 A, respectively. The fractions of isolated monovacancies are, respectively, 0.13,0.19and0.28,andthe fractions of vacancies with more than six nearest neighbor vacancies (or vacancy clusters) are, respect-... 

Besides the carbon cluster ions, carbide cluster ions are formed in different plasmas with remarkably high ion intensities. The abundance distribution of these carbide cluster ions as a function of cluster size has been observed to be very similar using the different sohd-state mass spectrometric methods, with higher intensities for cluster ions with even numbered carbon atoms than for neighbouring... 

For instance, in the SLSP model, a HSR may be obtained by taking into account both the self-similarity of the percolating cluster and the scale invariance of the cluster size distribution function (71). By utilizing the renormalization procedure [213], in which the size L of the lattice changes to a new size Lc with a scaling coefficient b = Lc/L, the relationship between the distribution function of the original lattice and the lattice with the adjusted size can be presented as w(s, sm) = bd nDw(s, sm), where s = s D and sm = smD. 

For the idealized cubic arrays found as the crystal structures of simple metals, the lattice vector is determined solely as the square root of the sums of squares of the allowed m, n and p coefficient assuming that a = b = c = 1. So the problem to find the distribution of the Oh orbits as a function of cluster radius is reduced to the 3-squares problem in mathematics . 

For a highly symmetric cluster, the radial distances of Eq. (53) will take only some few values, whereas a highly distorted system will have a whole distribution of radial distances. Therefore, plotting the radial distances as a function of cluster size can give information on the construction of atomic shells. However, just as above, it shall be remembered that for a highly symmetric cluster with one extra atom, the center, Eq. (52), will be displaced and, therefore, the occurrence of atomic shells will then not be so easily identified. 

As an illustration, we consider the cluster-size distribution for a monodis-perse Lennard-Jones fluid (particle diameter a, interaction cut-off 2.5(t) at a rather arbitrary density 0.16monotonously decreasing function of cluster size at high temperatures, it becomes bimodal at temperatures around 25% above the critical temperature. The bimodal form is indicative of the formation of large clusters. 

Figure 3 Two-dimensional radial distribution function of Na ions in the active site for the activated precursor simulation without Mg2 present in the active site (dRT-Na). The lower panels show results for cluster A that contains population members that are in active in-line conformations, and the upper panels show results for cluster B that are not in-line (see Table 2). The axes are the distances (in A) to different metal coordination sites. The green lines indicate the regions where Na ions have distances less than 3.0 A to both sites indicated by the axes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this book.)...

Figure 1 The first-order distribution function of the Landau free energy hypersurface, a cluster cluster of size 14.5 A, showing the transformation...

For optimal use of Pt, a preferred size of Pt crystal can be predicted. For catalytic activity of exposed atoms (STY), this would take place around point C (Fig. 6.7 d). The opposite trends of dispersion and catalytic activity (related to metal character of atom in the dispersion) give the resulting curve C for overall catalysis observed as function of cluster size. Catalysts prepared by different methods will, generally, yield different crystallite size distribution and shape of metal crystals. This will result in dissimilarities in sorptive and catalytic behavior, even if allowance is made for the difference in active surface area285. ... 

Another important characteristic of the late stages of phase separation kinetics, for asymmetric mixtures, is the cluster size distribution function of the minority phase clusters n(R,T)dR is the number of clusters of minority phase per unit volume with radii between R and R + dR. Its zeroth moment gives the mean number of clusters at time x and the first moment is proportional to the mean cluster size. 

Therefore, a diffuse ring of scattering centered at the origin of reciprocal space is attached to each particle. Thus, the nematic order parameter which characterizes the distribution function of a single particle is derived from the interferences among a cluster of particles. This assumption is somewhat similar to a mean-field treatment and tends to overestimate S. 

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