Occupation probability, potential energy

The population coP of the state that pumps the system does not appear in the rate expression (9.88), however it determines the observed flux through Eqns (9.90) or (9.92). In the particular application (Section 17.2.2) to the problem of electronic conduction, when the left and right continua represent the free electron states of metal electrodes coP will be identified as the Fermi-Dirac equilibrium occupation probability at energy Eo, /(Eq) = [exp(( o — ti /kBT) + 1] where is the chemical potential of the corresponding electrode. 

The value of ( — Coo) depends only on the relative probability of occupation of the sites in thermal equilibrium in a constant applied field. Provided that the difference in free energy between the rites, due to the field, is ven only by the potential energy difference gEd, the ratio of the probabilities of occupation of the sites is... 

The above reweighting technique [136] is analogous to the histogram Monte Carlo approach [141-143], but instead of determining the configurational density of states from the canonical potential energy distribution, g, effectively a density of minima, is obtained from the occupation probability of the different basins of attraction. A similar approach has been used to calculate the density of minima as a function of the potential energy for a bulk liquid [144,145]. 

The Fermi level characterizes the electron energy in a solid. In metals the Fermi energy equals the chemical potential of the electrons (see Chapter 2). It corresponds to the highest energy level occupied by electrons at zero Kelvin. At higher temperature, the distinction between occupied and non-occupied levels spreads across an energy interval that lies above and below the Fermi level. At the Fermi level, the occupation probability is just 1/2. 

In the concentration wave (CW) theory [760] the distribution of atoms A in a binary A-B alloy is described by a single occupancy probability function n r). This is the probability to hnd the atom A (La) at the site r of the crystalline lattice. The conhgurational part of the free energy of solid-solution formation (per atom) includes the internal formation energy AU, the function n r), a concentration of particles La(A), the effective interatomic potentials between La atoms (A) and Sr atoms (B), for details see [754]. 

We calculated the free energies of all the minima in order to determine the equilibrium probability distribution (see Section IV.C.2). We found that the several hundred lowest free energy minima have about the same free energy, and that no single minimum has an equilibrium occupation probability which exceeds 0.004. This is in stark contrast with unsolvated tetra-alanine, where the ground state had an equilibrium occupation probability of 0.748, and the lowest three potential energy states accounted for 0.936 of the total equilibrium probability. 

The experimental data presented in Ref. 79 unambiguously prove that the SES are formed in two-electron excitations. According to the measurements of the energy-absorption and ionization probabilities, 17( ) is below unity for CH4 at energies from 27 to 80 eV. The scheme of occupation of energy levels in a CH4 molecule is featured in Fig. 6. Since the highest ionization potential of valence electrons is 23.1 eV, and the next potential, corresponding to ionization of the K shell, is around 290 eV, the SES in this case correspond to excitation of two or more electrons. 

The Fermi energy or Fermi level, Ef, of a solid is that energy at which the probability of electronic energy level occupancy is exactly 0.5. Chemically, the Fermi energy corresponds to the electrochemical potential of electrons in the solid. At equilibrium, all electronically conducting materials in contact have the same Fermi energy. 

Figure 21. Electron tunneling between two metal phases at 0 K (a, b), and temperatures considerably above 0 K [c,d). The dashed lines show the position the electrochemical potential (Fermi-level) in a phase. The length of the arrows indicates the rate of electron transfer in an energy interval. The probability of a single tunneling event and the Fermi-Dirac occupation of the free electron energy levels is taken into account (see section 6).

Many desorption experiments have now shown v values which are well away from the kT/h value. The desorption of CO from Ru 001 [341] gave i 1015s-1. This prompted Ibach et al. [342] to investigate the CO/Ni lll system and they pointed out that a value of 1013 s-1 can only be expected if the adsorbate is mobile and has free rotation or has states of low excitation energy. The model they used to elucidate the kinetics was a modification of an earlier theoretical treatment derived by Landau and Lifshitz [343]. It is to be expected that the chemical potentials of gas phase and adsorbed species will be equal (/ua = pg). From Fermi statistics, the probability of site occupation on the surface is... 

A central concept in describing the interaction of a semiconductor with an electrolyte is the equilibration of the Fermi level of the solution and of the semiconductor. Although the concept of Fermi level has been initially introduced for an electronically conducting phase, as a level for which the probability of electron occupation of a given state is 1/2, in our discussion the Fermi level can be conveniently identified with the electrochemical potential of electrons in the solid and of a given redox couple (0,R) in solution, where the usual free energy relationships hold. Obviously, the solution phase does not contain free electrons, but contains available electronic states [in the form of oxidized (O) and reduced species (R)] which can equilibrate with free electrons in the solid. As a consequence, = (1 + KT n Co/C ), where fl are the electrochemical... 

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