Site occupation probabilities

A model for the homogeneous free energy of mixing and B2 ordering can be obtained with site occupation probabilities. In a homogeneous system, the probability, p%, of finding a B-species occupying an a-site is Xg similarly, from Eqs. 17.7 and 17.8,... 

Inspection of Fig. 1(c) reveals that there are a few pairs of atoms with a preferred distance. Analysis of many such images in terms of site occupation probabilities as a function of adatom distances revealed significant deviations from a random distance distribution, and the existence of adsorbate interactions which indeed oscillate with a wave vector of 2kp [16]. The decay followed the l/r2-prediction only for large distances, while significant deviations were observed at distances below 20 A and interpreted as a shortcoming of theory [16]. However, an independent study, carried out in parallel, focused on two body interactions only, i.e., the authors counted only those distances r from a selected atom to a nearby atom where no third scatterer (adatom or impurity) was closer than r [17]. This way, many body interactions were eliminated and the interaction energy E(r) yielded perfect... 

P—>0 as d—>°o is obvious, and one recovers Eq. (75) in the asymptotic limit. By virtue of the 1/ d = 0 expansion, Eq. (114) is exact in high dimensions. Eq. (114) states that the bond percolation threshold, Dc, should go up inversely in proportion to the site-occupation probability, ps, in accord with the observations by Agrawal, Redner and Stanley [60], and by Stauffer [58]. 

It is believed that the site-occupation probability, p5, plays a role of concentration in real systems. If that is the case, the hyperbolic relation between Dc and ps predicted by Eq. (114) is in marked contrast to the corresponding Dc vs. C diagram of real systems, where Dc is known to converge to the ideal value, Dco, with C—o [48,81,86,98,99]. Clearly, the abnormality in the critical behavior of the percolation model is derived from the abnormality in the chemical machinery of the intermolecular reaction (Sect. 5) which is in proportion to the first order of C, more exactly to M0ps2, the same order as the cyclization rate, so that the concentration terms cancel out each other, resulting in the expression of Eq. (114) without the y term. 

Fig. 3-21. Percolation into random uncorrelated media with varying site occupation probabilities (areal porosities). Solids are black. Continuity of major percolation paths from upper boundary indicated by shading.

From Figure 5.1 it is easy to appreciate that the numbers and sizes of the clusters will grow with the site occupation probability, p 1]. The central consequence, however, is that there will be a value of p that is known in the literature as the critical probability, p, such that only for p>Pc there is a spanning cluster. This pc is known as the percdation threshdd of the lattice l-3]. The behavior of the various geometrical or physical properties around p as a function of p — is known as the critkal behav-... 

In our above simple square lattice, we have considered only nearest neighbors and the bonding between them. Let us consider now the case where we also include occupied second nearest neighbors [11]. In that case, under the same site occupation probability p, where farther (second nearest) neighbors are also considered, the percolation onset will be achieved at a p for which no percolation existed in the nearest neighbors-only case. Let us define now the critical pc values for the two... 

Here jUe is the electron field-effect mobility, c is the fullerene site occupation probability (c = Cl Co), p is the bond occupation probability, pc is the threshold bond occupation probability for percolation, a is the wavefunction overlap parameter of the fullerene molecules, and a is given by the following function ... 

Fig. 8.6 Schematic representation of the electrical crarductivily dependence on the site occupation probability when tunnelling is the main conduction mechanism. The dotted lines indicate the local percolation thresholds when all corresponding near neighbours are considered. The overall conductivity behaviour appears to have a staircase dependence of the occupation probability leading to a system of multiple thresholds (refer to the text for more details)...

FIGURE 3.43 A finite size sample with occupied sites (dots) and bounds between them on a square lattice. The samples is shown at different site occupation probability,/ , that is (a) below the percolation threshold, p = 0.2, (b) at the percolation threshold, pc = 0.59, and (c) above the percolation threshold,/ = 0.8. Open symbols in (b) and (c) represent sites that belong to the infinite percolation cluster. 

Site occupation probability in percolation theory (dimensionless) Percolation threshold of site occupation probability (dimensionless) Pdclet number. Equation 1.30 Particle radius distribution function Capillary pressure (Pa)... 

Crl, Grr, A r and A r are the potential parameters of the constituents A and B of the alloy, S r r r, is the structure matrix in the most localized representation, tir are local site-occupation variables which randomly takes value 1 or 0 according to whether the site is occupied by an atom of type A or not, with probabilities proportional to the concentrations of the constituents. According to the prescription of the augmented space formalism, the effective non-random Hamiltonian H in augmented space is then... 

Figure 1. Schematic representation of the NAS1CON structure. The Si04 and P04 tetrahedra arc indicated by light blue, the ZrOfi octahedra by darkblue and the NaO, octahedra by green. The sodium ions are depicted by the red circles. The different radii represent the probability of lattice site occupation large radius 67 percent, small radius 1.1 percent. The Si/P ratio is 0.683 0.317. The a, b, and c axes are indicated.

Occupation Probabilities of the Copper Atomic Sites in CuTeX Compounds... 

Atomic site Number of equivalent positions Occupation probabilities ... 

The reason for preferential axial site occupancy by mono-dentate ligands is probably related to results of recent CNDO calculations on [Co CO) ] (15) which show that the axial carbonyls are least involved in back-bonding. Extrapolation to Rh - and Ir -derivatives seems reasonable and finds some support from n.m.r. measurements, which show that the n.m.r. chemical shift of the axial carbonyl is always at higher field than the radial carbonyl in both rhodium (16) and iridium derivatives. (12)... 

Equivalent v-sites i have the same probability p, to be occupied by a dye molecule. The occupation probability p is equal to the ratio between the occupied and the total number of equivalent sites. The number of unit cells I1C is controlled by the host while ns is determined by the length of the guest, which means that p relies on purely geometrical (space-filling) reasoning and that the dye concentration per unit volume of a zeolite crystal can be expressed as a function of p as follows ... 

The same structure is formed in a number of binary (or ternary) phases, for which a random distribution of the two (or three) atomic species in the two equivalent sites is possible. Typical examples are the (3-Cu-Zn phase (in which the equivalent 0,0,0 A, A, A positions are occupied by Cu and Zn with a 50% probability) and the (3-Cu-Al phase having a composition around Cu3A1 (in which the two crystal sites are similarly occupied, on average by Cu, with a 75% occupation probability, and by Al, with a 25% occupation probability). A number of these phases can be included within the group of the Hume-Rothery phases (see 4.4.5). 

Figure 5 shows the total concentration of dye molecules in the channels of zeolite L [DJtot expressed as occupation probability p versus the dye concentration in solution in units of the total number of available sites, uc. From the results illustrated, it follows that it is easy to prepare materials with low loading, but that sophisticated techniques are needed for high loading. 

The occupation probability p of the sites with a dye is equal to the number of occupied sites divided by the number of sites available. Using this, the Forster-... 

If n-s is the number of unit cells that forms a site and Pdye is occupation probability of dyes, then the number of dyes in the zeolite crystal is. 

---