Occupancy probability

Experimentally, phases can be obtained by measurements of occupation probabilities of states using Eq. (9). (We have calculationally verified this for the case treated in [264].)... 

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

Atomic site Number of equivalent positions Occupation probabilities ... 

Orientational disorder is also present if a molecule or part of a molecule occupies two or more different orientations in the crystal, even without performing unusual vibrations. For example, tetraethylammo-nium ions often occupy two orientations that are mutually rotated by 90°, in such a way that the positions of the C atoms of the methyl groups coincide, but the C atoms of the CH2 groups occupy the vertices of a cube around the N atom, with two occupation probabilities. 

A recent breakthrough in molecular theory of hydrophobic effects was achieved by modeling the distribution of occupancy probabilities, the pn depicted in Figure 4, rather than applying a more difficult, direct theory of po for cavity statistics for liquid water (Pohorille and Pratt, 1990). This information theory (IT) approach (Hummer et al., 1996) focuses on the set of probabilities pn of finding n water centers inside the observation volume, with po being just one of the probabilities. Accurate estimates of the pn, and po in particular, are obtained using experimentally available information as constraints on the pn. The moments of the fluctuations in the number of water centers within the observation volume provide such constraints. 

Before we proceed to obtain rough estimates for the occupation probabilities, we note a few points. 

Table 18.2 Occupation probability of the valence orbital of a few alkali and halide ions adsorbed on mercury ( = 4.5 eV). For alkali atoms eo denotes the ionization energy for halide atoms, the electron affinity.

Equations (18.26) and (18.28) are implicit equations for the occupation probability n, since it appears both on the left- and on the right-hand side. They can be solved by simple numerical procedures. In the cases considered here there is always one unique solution. 

For a temperature T and chemical potentials fit (relative to the nucleon masses) the nucleon occupation probability reads... 

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 visual proof of energy transfer in this experiment is based on the observation that Py+ and Ox+ are incorporated into zeolite L from an aqueous solution with about equal rates. It is therefore possible to control the mean distance between donors D and acceptors A by varying the occupation probability. The main processes are energy transfer and luminescence as illustrated in Figure 1.27. Energy... 

This equation shows that the ratio between the acceptor and donor fluorescence quantum yields is directly proportional to the energy-transfer rate constant kET. We have shown that this leads to the following linear relation between the fluorescence intensity of the acceptor 70x and that of the donor 7py, and the occupation probability pQx of the acceptor [3, 77] ... 

Different kinds of modulations can be considered the displacive modulation is related to the shift of the atom position from the average structure the occupational (or substitutional) modulation is related to the changes of the atomic occupation probability depending on the position. Generally, in alloys, displacive modulation is small but not negligible (Yamamoto 1996) whereas substitutional modulation often occurs. 

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). 

Note that the important system parameters such as the occupation probability interaction with polar solvent polarization ) strength of interaction of the reactant with the metal A, and the electronic energy of reactants are the functions of position x of the reactant from the electrode surface. These parameters are given below as a function of x. 

The occupation probability can be obtained from the self-consistency equation,... 

Fig. 2-20. Electron state density and ranges of Fermi energy where electron occupation probability in the conduction band of an electron ensemble of low electron density (e.g., semiconductor) follows Boltzmann function (Y i)or Fermi function (y > 1) y = electron activity coeffident ET =transition level from Y 4= 1 to Y > 1 0(t) = electron energy state density CB = conduction band. [From Rosenberg, I960.]...

Figure 3 Dependence of the equilibrium constant K for insertion of a dye in zeolite L as a function of the occupation probability Pr, calculated for Ki = 7.75 X 10. ...

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. 

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