Electrons coming from Fermi level

The equilibrium situation can be achieved with the reacting electrons coming from the Fermi level Ep of a metal electrode in contact with the solution of the redox couple. The free energy change in the respective redox reaction is then zero. 

The electrons in the Fermi level in a metal—those that undergo the Fermi distribution law—are mobile and that is where the difference comes from electrons in solution which are, in fact, in the bound levels of ions. Such electrons are not mobile and the statement that they have a Fermi energy may therefore be misleading, for they do not obey the same distribution law as the electrons with which they are in equilibrium.4... 

Calculation shows that the contributions of the above or below Fermi-level electrons to the cathodic current are negligible compared to those of the electrons coming from the Fermi level. After this further simplification,... 

For the hydrogen evolution reaction, dark current electrons come from the Fermi level, because, even if the potential is as high as 2 V (vs. the p.z.c.), the Fermi level is still below the ground state of the acceptor ions, such that the electrons go to it and not to the solvent state. As the acceptor states... 

In case (3), both orbital ligand and anti-ligand are below the Fermi level, that is, the anti-ligand orbital is occupied by electrons coming from the metal, and hence there is total repulsion between the electrons of the metal and of the atomic molecule. The adsorption is very low. 

When we speak of a preferable transition from some level, we mean a narrow band of width of the order of -RT near this level. Gerischer[93] was the first to draw the conclusion that not only the electrons from the Fermi level but also from the nearby levels participate in an ordinary discharge (energy spread of the order of several RT). He also showed for the first time that similar considerations explain why the main contribution in reactions at a semiconductor electrode comes from the levels lying near the bottom of the conduction band or near the top of the valence band. 

Typically the contributions of the two bands to the current are of rather unequal magnitude, and one of them dominates the current. Unless the electronic densities of states of the two bands differ greatly, the major part of the current will come from the band that is closer to the Fermi level of the redox system (see Fig. 7.6). The relative magnitudes of the current densities at vanishing overpotential can be estimated from the explicit expressions for the distribution functions Wled and Wox ... 

This electrochemical potential has a name. It is called the Fermi energy of the electron in solution, and the name arises because if one expels an electron into a vacuum, the electrons come (overwhelmingly, if not entirely) from the Fermi level of electrons in... 

An objection to consideration of vibrational states above the ground state (Levich, 1970) was that if such states were indeed involved in electron transfer, then there would be no smooth Tafel lines because as a change in overpotential altered the energy of the Fermi level (from which, e.g., cathodic electrons come), matching levels in the solution species would not be available until the next vibrational state, perhaps 1/2 eV away, was reached by the change in the electrode potential. Hence, in this picture, the... 

All this material about the Fermi-Dirac equation for the probability of filling of the electron states comes down in practice to one approximation Electrons taking part in electrode processes are from the Fermi level and hence have the Fermi level energy. 

Electrochemical reactions involving semiconductors occur in a more varied way than with metals. For example, if a semiconductor, like n-doped silicon, is put in a circuit and used as a cathode in a normal way (e.g., driven by an outside power source), the available electrons come, not from around the Fermi level as with metals, but from the conduction band [Fig. 10.1(a), Fig. 10.2] of the semiconductor. Correspondingly, when one wants to oxidize a redox ion such as Fe2+ at p-Si by using an outside power source, the electrons emit from the ion in solution in the interfacial region and enter holes in the valence band of p-Si. In a metal, they would enter around the Fermi level. 

Although electrochemistry has much in cotmnon with surface science, the apphcation of the principles of catalytic activity to the reactions taking place in an electrochemical enviromnent is not straightforward. All electrochemical reactions of practical interest imply at least one step where an electron is transferred between species coming from the solution side or the electrode surface. Therefore electrochemical reactions occurring at the interfaces are governed by the interaction of the reactant both with the solvent and with the electrode. There is also an additional effect produced by the external applied potential, so that the Fermi level of the reactant can be easily tuned relative to the Fermi level of the electrode. 

A further term, which has no analogue in hydrogen, arises in the fine structure of positronium. This comes from the possibility of virtual annihilation and re-creation of the electron-positron pair. A virtual process is one in which energy is not conserved. Real annihilation limits the lifetimes of the bound states and broadens the energy levels (section 12.6). Virtual annihilation and re-creation shift the levels. It is essentially a quantum-electrodynamic interaction. The energy operator for the double process of annihilation and re-creation is different from zero only if the particles coincide, and have their spins parallel. There exists, therefore, in the triplet states, a term proportional to y 2(0). It is important only in 3S1 states, and is of the same order of magnitude as the Fermi spin-spin interaction. Humbach [65] has given an interpretation of this annihi-... 

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