Free-electron bands

In particular, let us consider the band structure along where kr = (0,0,0) and kx = (2n/a)(l, 0,0) with a the edge length of the face-central cubic unit celL (Note that the X point for fee is In/a not nfa like for simple cubic.) In this direction the two lowest free-electron bands correspond to Ek = (H2/2m)k2 and k+I = (H2/2m)(k + g)2 respectively. The term g is the reciprocal lattice vector (2n/a)(2,0,0) that folds-back5 the free-electron states into the Brillouin zone along so that Ek and k+l... 

Thus, the presence of the periodic potential has opened up a gap in the free-electron band structure with energy separation... 

Table 17.13. Free-electron band energies em for InSb space group 216) at T, A, and X, the BZ of the fee Bravais lattice (Figure 16.12(b)).

MECHANICAL PROPERTIES IN TERMS OF COVALENT BOND AND FREE ELECTRON BAND... 

Covalent bond and free electron band coexist in metals and alloys. 

The ratio of covalent bond / free electron band is a fixed value for a given metal or alloy for a given crystal structure. This ratio may be upset momentarily but the nature of bonding is to restore this ratio as soon as possible. 

Covalent bonded electrons are bosons and a large number of bonded electrons can occupy the same energy level. Free electron band, on other hand, consists of fermions and therefore according to Pauli Exclusion Principle no two electrons can occupy the same energy level. This is the reason they exists in a band... 

Fig. 2. Pictorial illustration of an electron pair transfer from covalent bond to free electron band and back to covalent bond...

For a material without free electron band there would be no ductility, i.e., the material shall have no plasticity and fracture at the yield stress. This type of material includes all organic, inorganic, ceramic material that do not have electronic conduction band. This results in no possibility for creation or annihilation of free radicals. 

For material with good conductivity (i.e. with wide free electron band) shall be very ductile. This is due to their small AE and less resistance to plastic flow. These materials include high conductivity material such as Ag, Cu, Au, Fe, and alkali metals. 

As the name implies, the phenomenon is based on coating a solid metal with a liquid metal. In our theory, liquid metal (being above its melting temperature) has no covalent bonds and the free electrons essentially provide the cohesive energy. It can be recalled that this was the basis for obtaining the correlation (Fig. 11). Thus, by coating a metal that has a distinct ratio of covalent bond over free electron band with a liquid metal that has only free electrons (no covalent bond) can have no effect whatsoever in the AEi (for these notations refer to Fig. 9) which has to do only with covalent bond. This is the observation of 4.1.3. 

Within this understanding, it is easy to see why LME phenomenon does not take place if and when inter-metallic compounds are formed between liquid metal and solid metal (observation 4.1.4.). For, there would be no direct free electron flow between liquid metal and solid metal and subsequently there would be no upsetting of the fundamental Ratio (covalent bond / free electron band) and therefore no LME phenomenon. 

It is well known through our experience that material with conduction electrons suffer from the phenomenon called corrosion i.e., metals turning into metallic oxides in time in air. On the other hand, the materials without conduction electrons do not suffer from corrosion. Technically, the presence of conduction electrons implies the existence of free electrons and conduction band. As pointed out in the mechanical property section these two distinct properties exhibit themselves also in term of plasticity . That is, the existence of free electron band allows plastic deformation whereas in the absence of free electron band the plasticity is nonexistent. It is recalled that the theory we are proposing for metals and alloys requires not only the coexistence of covalent bond and free electron band but also that the ratio of the number of these two type of electrons be maintained at a constant value for a given metal. Within such understanding, we now construct corrosion process in steps ... 

The second step requires as much as 4 electrons per molecule of O2. This supply of free electrons is available only if the material, on which the oxygen molecule come to rest, has free electron band. Thus, organic or inorganic material such as diamond, plastic etc. which do not have free electron band do... 

The process of O2 + 4e" —> 20, abstract free electrons from the free electron band and results in upsetting the ratio of R = N 7 Nc (number of free electrons over number of covalent bonded electrons). In order to restore this ratio, some covalent bond (particularly those near the surface) will have to give up their covalent bonding to become positive ions. In this process free electrons are created ... 

The electron affinity of a semiconductor relates the vacuum level to the conduction band minimum at the surface. An informative way to view the electron affinity is as the conduction band offset between the semiconductor and vacuum. The band structure of the vacuum is simply the parabolic free electron bands, and the minimum energy (or vacuum level) refers to an electron at rest. For most materials, an electron at the bottom of the conduction band is bound to the material by a potential barrier of several volts. This barrier is the electron affinity and is defined as a positive electron affinity. In some instances, the vacuum level can actually align below the conduction band minimum. This means that an electron at the minimum of the conduction band would not see a surface barrier and could be freely emitted into vacuum. This situation is termed a negative electron affinity. 

Fig. 7. Intensity (arbitrary units) as a function of final electron energy for emission from an s orbital adsorbed (a) in the position (0, 0, a) and (b) in the position (a/2, a/2, aj2), relative to the substrate. Single scattering (solid curves), multiple scattering (dotted curves), and no scattering (dashed curves). The arrows indicate the resonance energies specified by the reciprocal lattice vectors, and the corresponding band crossings in the free-electron band structure at the top of the figure.

It is of interest also to compare the true bands with the frce-electron bands, as we did in CsCl frce-clectron bands arc shown in Fig. 3-8,c. It is remarkable how close the relation is between free-electron bands and LCAO bands at low energies. The resemblance between the free-electron bands and the true bands is good at high energies, and clearly, if we wished to study highly excited states, the free-electron basis would provide a much better starting point for this than the. LCAO bands. 

Pantclides (1975c) also discussed the valence bands for the inert-gas solids, indicating that they consist of a narrow p band and an s band, which may be taken as completely sharp. He gave a universal width for the p band, of fi / md), with f/v = 4.2. (Again, his numerical value was different because of a different definition of d.) Presumably the conduction bands, corresponding to electrons added to the crystal, would be quite like free-electron bands. 

D Nearly-Free-Electron Bands and Fermi Surfaces... 

Let us now complete the derivation of formulae for the interatomic matrix elemenfis, which was described in Section 2-D, by equating band energies obtained from LCAO theory and those obtained from nearly-free-electron bands. This analysis follows a study by Froyen and Harrison (1979). The band energies obtained from nearest-neighbor LCAO theory at symmetry points were given in... 

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