Fermi potential

Colloids of more electronegative metals such as cadmium and thallium also act as catalysts for the reduction of water. In the colloidal solution of such a metal, an appreciable concentration of metal ions is present. The transferred electrons are first used to reduce the metal ions, thus bringing the Fermi potential of the colloidal particles to more negative values. After all the metal ions have been reduced, excess electrons are stored as in the case of silver. 

The Self-Consistent-Field (SCF) procedure can be initiated with hydrogenic wave functions and Thomas-Fermi potentials. It leads to a set of solutions w(fj), each with k nodes between 0 and oo, with zero nodes for the lowest energy and increasing by one for each higher energy level. The quantum number n can now be defined asn = / + l + A to give rise to Is, 2s, 2p, etc. orbitals. 

Figure 6 Size-dependence of the redox potential of silver clusters in water ( ) [63] and of the corresponding ionization potential in the gas phase (A) [67,68]. The redox potentials refer to the normal hydrogen electrode whose Fermi potential is at 4.5 eV.

Equation 5 is often used to decribe the interaction between the incoming ion and the target atoms. The interaction between two target atoms generally occurs at low energy where the Thomas-Fermi potential overestimates the interaction. Under this situation a Born-Mayer potential is more appropriate , i.e. ... 

An especially useful approximation for the Thomas-Fermi potential has been developed by Lindhard and co-workers where the screening function is assumed to have the form ... 

Fig. 21. Reduced nuclear stopping power, s (e), as a function of e, bottom scale and of E for Ar —Cu, top scale. Based on Thomas-Fermi potential (see Equation 5). (Based on data presented in ref )...

R is the range and E the incident ion energy. See Sect. 2.2.2 for definitions of symbols. This curve is calculated assuming a Thomas-Fermi potential and with neglect of electronic stopping... 

According to this concept of asymmetric trapping — which is also discussed for inorganic photoconductors by Rose 3.89) the chemical potential of electrons (i.e. the Fermi potential) should be situated close to the conduction band (in -type photoconductors) and valence band (in p-type photoconductors), respectively, as illustrated in Fig. 9. 

Equilibrium was reached in the addition of N (Eq. 26) when the accumulated negative charge in the interior of the silver particles precluded further electron transfer. Addition of N to metallic colloidal silver particles also shifted the Fermi potential to a more negative value (see top part of Fig. 84) [531]. 

There exists a whole number of approximate expressions for Vl(r) (see, for example [139]). The simplest, called the Thomas-Fermi potential, follows from the statistical model of an atom. Unfortunately, it leads to results of very low accuracy. More accurate is the Thomas-Fermi-Dirac model, in which an attempt is made to account for the exchange part of the potential energy of an electron in the framework of the free electron gas approach. Various forms of the parametric potential method are fairly widely utilized, particularly for multiply charged ions. Such potentials may look as follows [16] ... 

The electrochemical potential of an electron in a (solid) phase p is equal to the Fermi potential of that phase. Because at equilibrium, dG = 0,... 

Where the subscript q represents the amount of charge on the Pt electrode, z) and (F) are the un-referenced potential at z and the un-referenced Fermi potential respectively, and UR) is the potential at the reference point in the portion of electrolyte in reference to the vacuum reference point (the first reference). The electrode potential versus a SHE, Uq, is given by... 

By doping a semiconducting catalyst it is possible to alter the electron concentration, which leads of course to a redistribution of the electrons over the quantum states. However, it has been shown by measuring the work function that the Fermi potential at the surface of the solid is not influenced even by drastic shifts of the Fermi potential in the bulk 13-1S>. 

Fig. 7. Comparison of measured activation energies (top) and pre-exponential factors (centre) with the bond strength in the semiconductor surface expressed by the electron concentration in the surface (distance between Fermi potential and band edges, bottom)...

During permanent illumination of the semiconducting oxides without Ag support, a stationary state different from the thermal equilibrium is formed. It is characterized by the appearance of the quasi-Fermi potentials of electrons and holes shifted closer to the band edges. 

We may conclude that the activation energy of the CO-oxidation is proportional to the distance between the Fermi potential and the band edge of the mobile charge carrier at the surface of the catalyst. This proportionality is demonstrated exactly in experiments on NiO in the dark 83>. 

These results show that there is a direct proportionality of the activation energy of a catalytic reaction with the distance between Fermi potential and band edge it may be expressed by the two empirical equations (20, 21), where an and ap are factors specific for the reaction and the catalyst studied. 

Hoffman, G. G. and Pratt, L. R., Optimized Thomas-Fermi potential for discrete propagator electron density functional calculations. In J. D. Doll and J. E. Gubernatis (eds.). Proceedings of the International Workshop on Quantum Simulation of Condensed Matter Phenomena, TeaneckNJ, pp. 105-115. Singapore World Scientific Publishing (1990). 

To locate the approximate redox potentials of the reactive electron-hole pair, the quasi-Fermi levels of the powders were measured as summarized above. The plots of photovoltage versus pH for TH and a series of H2[PtCl6]/TH materials are summarized in Fig. 3. From the inflection point (pHo), the corresponding quasi-Fermi potentials at pH 7, as obtained via Eq. (11) taking = 0.059V, are —0.58V (P25) and —0.54V (TH). The flat-band potential of a single crystal of anatase was reported to be —0.59 V (pH 7) (31). Relative to the value of TH the quasi-Fermi level is shifted anodically by 0.05, 0.09, and... 

Bandgap Energies and Quasi-Fermi Potentials of Electrons... 

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