Electron in solids

Pfann, W.G. (1958, 1966) Zone Melting, 1st and 2nd editions (Wiley, New York). Pippard, B. (1994) Obituary of John Bardeen, Biograp. Mem. Fellows R. Soc. 39, 21. Pippard, B. (1995) Electrons in solids, in Twentieth Century Physics, vol. 3, ed. Brown, L.M., Pais, A. and Pippard, B. (Institute of Physics Publications, Bristol and Amer. Inst, of Physics, New York) p. 1279. 

The most important driving forces for the motion of ionic defects and electrons in solids are the migration in an electric field and the diffusion under the influence of a chemical potential gradient. Other forces, such as magnetic fields and temperature gradients, are commonly much less important in battery-type applications. It is assumed that the fluxes under the influence of an electric field and a concentration gradient are linearly superimposed, which... 

ENERGY LEVEL OF ELECTRONS IN SOLID STATE ELECTROCHEMISTRY... 

Before discussing the experimental results, which by themselves suggest a unique choice of the reference (zero) state of electrons in solid state electrochemistry, which is the same with the choice of Trasatti for aqueous electrochemistry,14 16 it is useful to discuss some of the similarities and differences between aqueous and solid electrochemistry (Fig. 7.3). 

All models of this type have become known colloquially by the misnomer free-particle model. Diverse objects with formal resemblance to chemical systems are included here, such as an electron in an impenetrable sphere to model activated atoms particle on a line segment to model delocalized systems particle interacting with finite barriers to simulate tunnel effects particle interacting with periodic potentials to simulate electrons in solids, and combinations of these. 

The band structure that appears as a consequence of the periodic potential provides a logical explanation of the different conductivities of electrons in solids. It is a simple case of how the energy bands are structured and arranged with respect to the Fermi level. In general, for any solid there is a set of energy bands, each separated from the next by an energy gap. The top of this set of bands (the valence band) intersects the Fermi level and will be either full of electrons, partially filled, or empty. 

The situation described here is based on a simple one-electron model which can hardly be expected to predict the behaviour of complex many-electron systems in quantitative detail. There can be no doubt however, that the qualitative picture is convincing and probably that the broad principles of electronic behaviour in solids have been identified. The most significant feature of the model is the band structure that makes no sense except in terms of the electron as a wave. Important, but largely unexplored aspects of solid-state reactions and heterogeneous catalysis must also relate to the nearly-free models of electrons in solids. 

For high density electron ensembles such as free valence electrons in solid metals where electrons are in the state of degeneracy, the distribution of electron energy follows the Fermi function of Eqn. 1-1. According to quantum statistical dynamics [Davidson, 1962], the electrochemical potential, P., of electrons is represented by the Fermi level, ep, as shown in Eqn. 1-10 ... 

Figure 1-6 shows schematically the relationship between Pi, ii, and a,. In the case of electrons in solids, the real potential a, corresponds to the negative work function - >(= a,) work function is the differential energy required for the emission of electrons from solids. 

Fig. 2- State density distribution curve of electrons in solid ZXe) = electron state density tu = uiq>er band edge = lower band edge.

Heine, V. (1963). On the general theory of surface states and scattering of electrons in solids. Proc. Phys. Soc. 81, 300-310. 

Figure 8 shows the attenuation length of electrons in solids as a function of their kinetic energy. The few theoretical calculations available cire in good agreement with these empirical data Only unscattered electrons convey useful information, while scattered electrons contribute to a structureless background (secondary electrons). From Fig. 8, it is clear that photoelectron spectroscopy probes at most a few tens of Angstroms. 

The first application of quantum mechanics to electrons in solids is contained in a paper by Sommerfeld published in 1928. In this the free-electron model of a metal was introduced, and for so simple a model, it was outstandingly successful. The assumptions made were the following. All the valence electrons were supposed to be free, so that the model neglected both the interaction of the electrons with the atoms of the lattice and with one another, which is the main subject matter of this book. Therefore each electron could be described by a wave function j/ identical with that of an electron in free space, namely... 

This picture, by the way, finds a most important application in the description of bonding in metals. Take, for instance, sodium, with one electron per atom for four orbitals. It is quite clear that here there is a vast excess of orbitals over electron pairs, and electrons in solid sodium, as in all metals, are effectively delocalized over the whole metal crystal. This idea can account satisfactorily for the electrical conductivity of metals. A more detailed discussion of metals would go beyond the space available here. 

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