Ensemble of clusters

One way to see that a transition is discontinuous is to detect a coexistence of two phases, in this case the orientationally ordered and disordered phases, in a temperature interval. This is revealed by time variation of the potential energy of the cluster. In the temperamre region of phase coexistence, each cluster dynamically transforms between the phases, and its potential energy fluctuates around two different mean values (Fig. 4). In an ensemble of clusters, the coexistence of different phases is observable insofar as a fraction of the clusters (e.g., in a beam [17]) can exhibit the structure of one phase, while another fraction takes on the stmcture of another phase. 

FIGURE 14.12 Polydispersity factor Cp for the large qR, power law regime of the structure factor S q) = cCp(qRg for an ensemble of clusters as a function of the width parameter T of the scahng size distribution for three values of the fractal dimension D. (From Sorensen, C.M. and Wang, G.M., Phys. Rev. E, 60, 7143, 1999. With permission.)... 

From (6.54) it follows that the ensemble average of the cluster radius grows at N > N t, the critical size N (( being finite at T < T. On the other hand, each cluster melts with the overwhelming probability at T> Tm. The fact that T > Tm suggests at first sight a paradox situation, namely, in the temperature range the liquid is metastable with respect to the ensemble of clusters... 

We are now in a position to derive our first important result for evaporative ensembles of clusters. The logarithm of the RRKM molecular rate constant, k(E) (without the reaction degeneracy) is given by... 

To test the proposed new procedure, we generate monodisperse statistical ensembles of clusters with N = const and calculate two sets of averaged... 

Magnetic properties have also been investigated for clusters deposited on a crystal surface and for an ensemble of clusters embedded in a matrix or in a thin film [27-29]. In these cases the hybridizations, distortions and bond-length changes induced by cluster-matrix interactions may result in magnetic properties that are considerably... 

Figure 6.7 Magnetization m T, H) and the corresponding magnetoresistance Ap(7 , //) as a function of the external magnetic field H of an ensemble of clusters each having N atoms in a matrix with random orientation of the anisotropy easy axes. The cluster temperature is chosen to be T = O.lNKi, where K2 is the anisotropy constant. The magnetization is calculated for statistical equilibrium (7 7bi) and for strong blocking (7 7bi), where 7bi is the blocking temperature. A...

The energy of the vdW vibrational states w> is approximately 180 cm. The average density of vdW levels with the right symmetry in this energy region is of the order of 1 per cm and therefore we think that the number of interacting states is rather low. Nevertheless we could not observe any oscillatory decay of the fluorescence. The reason for the absence of quantum beats must be that so many rovibronic levels 6a > 0> J,K> are excited in the ensemble of clusters because of the relatively large spectral width... 

We considered the nonequilibrium size distribution function/(N, t), the number of new phase droplets consisting of N stractural units at time t. The evolution of the ensemble of clusters formed by nucleation and growth processes is described by the kinetic equation of the Fokker-Plank type [94,95]. Here we present the case of particles of equal size. The main task of such a kinetic model is to describe the volume fraction p of the new phase 1 during the temperature cycling of the isolated nanoparticle ensemble (Figure 13.18). The volume fraction of the new phase p is obtained as a function of T and No. In this section, we report the obtained kinetic result size-induced hysteresis. 

Class 111-type behavior is the consequence of this impossibihty to create step-edge-type sites on smaller particles. Larger particles wiU also support the step-edge sites. Details may vary. Surface step directions can have a different orientation and so does the coordinative unsaturation of the atoms that participate in the ensemble of atoms that form the reactive center. This wiU enhance the activation barrier compared to that on the smaller clusters. Recombination as well as dissociation reactions of tt molecular bonds will show Class 111-type behavior. 

A structural entity is a particle or an ensemble of arranged (i.e., correlated) particles that causes a distinct scattering pattern upon irradiation. Sometimes we call a structural entity made from several particles a cluster - not meaning that such particles are touching each other. 

Model of a supramolecular structure of polymolecular ensembles or clusters, determined by interaction and mutual arrangement of the forming molecules. At this level, the specific mechanisms of supramolecular chemistry, including molecular recognition, self-assembly, etc. [4] can be allocated. In most cases, it is possible to limit this area to objects with the sizes under 1 to 2 nm, since further increase in the sizes admits application of statistical concepts like phase and interphase surface. 

The model of clusters or ensembles of sites and bonds (secondary supramolecular structure), whose size and structure are determined on the scale of a process under consideration. At this level, the local values of coordination numbers of the lattices of pores and particles, that is, number of bonds per one site, morphology of clusters, etc. are important. Examples of the problems at this level are capillary condensation or, in a general case, distribution of the condensed phase, entered into the porous space with limited filling of the pore volume, intermediate stages of sintering, drying, etc. 

The approaches considered allow modeling of the primary texture of PS and the processes, limited by individual PBUs that mainly correspond to level III and partially to level IV in the hierarchical system of models (see Section 9.6.3). PBUs are identical in regular PSs, and simulation of numerous processes may be reduced to analysis of a process in a single PBU/C or PBU/P. An accurate modeling of the processes in irregular PSs requires the studies of the properties of structure and properties of the ensembles (clusters) of particles and pores (level IV of the system of models) and the lattices of such clusters (levels V to VII of the system of models). Let us consider the composition of clusters on the basis of fractal [127], and the lattices on the basis of percolation [8] theories. 

When n > 2, one can draw the reducible contributions made up of sequences of binary kernels and where states k = 0 between these kernels exist. Thus, the class associated with the skeleton of Fig. 3b contains a state k = 0 and contributes, not to Eq. (56), but to Eq. (70). In the following we shall need the relation which expresses Yg,- n) as the difference between ) and the ensemble of reducible contributions to (70) (of the type of Fig. 3b for n = 3, for example). It is necessary for us now to study systematically the points k = 0 of Eq. (70) so as to extract the reducible contributions. A study of the selection rules will permit us to solve this problem. We shall associate the appearance of the points k = 0 with the structure of the skeletons that we have introduced we shall see that the reduci-bility will be a dynamical translation of certain topological properties of the equilibrium clusters. 

It is also interesting to consider the ensemble of frequencies produced by the map [98], To this end we rerun the simulation and calculate the electric field at every putative H atom from all the point charges out to half the box length. This generates a distribution of fields, which through the map leads to a distribution of frequencies. This distribution can be compared to the histogrammed distribution of actual ab initio frequencies, now from 999 clusters. This... 

Multicenter bonding is the key to understanding carboranes. Classical multicenter n bonding gives rise to electron-precise structures characteristic of Hiickel aromatics, which are planar and have 4n + 2 n electrons. Clusters are defined here as ensembles of atoms connected by non-classical multicenter bonding , i.e., all... 

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