Energies ionic

Next, we consider metal ion M in a metal electrode. In the same way as we considered the electron level, the ionic energy level is represented by the real potential aM"[Pg.101]

Table 3-0.—Extra Ionic Energy of Bonds and Electronegativity Differences of thf. Bonded Atoms...

The differences in electronegativity of atoms for the bonds in Table 3-6 are given in the columns headed Za. — %b. If the extra ionic energy A (A— B) were given accurately by the equation... 

Camus et a/.34 explained their observations by a picture which has sometimes been called the frozen planet model. Qualitatively, the relatively slowly moving outer electron produces a quasi-static field at the inner electron given by l/rc2, and this field leads to the Stark effect in Ba+. The field allows the transitions to the n >n0Z and ,f 0 states and leads to shifts of the ionic energies. The presence of the njpn0f and n in0t resonances in the spectrum of Fig. 23.12 is quite evident. Camus et al. compared the shifts to those calculated in a fashion similar to a Bom-Oppenheimer calculation. With the outer electron frozen in place at ra they calculated the Ba+ energies, W,(rQ), and wavefunctions. They then added the energy W0(r0) to the normal screened coulomb potential seen by the outer electron. This procedure leads to a phase shift in the outer electron wavefunction... 

The f electrons within the 4fn configuration of the lanthanides have only weak interactions with the crystal field because they are shielded by outer 5s and 5p electrons. Consequently the spectra are dominated by transitions between the atomic states of the lanthanide ion. The left side of Figure 1 shows the ionic energy levels of Eu3+ in the gas phase. When the ion enters a crystal lattice, there will be additional crystal field interactions. The interactions cause small crystal field splittings on the order of 200 cm-1 that are superimposed on the atomic transitions and are easily observable. 

Atomic and ionic energy levels are characterized by a term symbol of the general form 25+1L/. The values of S, L, and J of lanthanide ions Ln3+ in the ground state can be deduced from the arrangement of the electrons in the 4f subshell, which are determined by Hund s rules and listed in Table 18.1.4. 

We may call the electrode potential defined by the ionic energy level the ionic electrode potential, and the electrode potential defined by the electronic energy level may be called the electronic electrode potential. In the case in which the electrode has no electronic level in the energy range of our interest such as certain membrane electrodes, it is convenient to describe the system in terms of the ionic electrode potential rather than the electronic electrode potential [Refs. 4 and 5.]. 

Although the above equation works well for IA cations it fails somewhat for IIA cations, Fig. 3, and much more significantly for IB and IIB cations. A series of modified equations have been proposed in which new terms are added to the ionic energy. 

Table 4. Calculation of modified ionic energies in electron volts (Pearson and Gray (13))...

The simplest model for a solvent is one of a continuous dielectric which effectively reduces charge-charge interactions but does not influence energies due to covalence. No matter how naive this model may be it stresses a very simple point. As the dielectric constant increases and ionic energies diminish then the effect of covalence in bringing about association becomes dominant. Consider the equilibrium... 

Note that generally bands correspond to hybrids, and attributing bands to certain elements is an approximation. Unlike the electronic energy level distribution, the distribution of ionic energy levels in crystals (the meaning of which we will consider in the next section) is discrete. In the electronic case we face bands comprising a manifold of narrowly neighbored levels, so that we better speak of a continuous density of states. 

The density description focused attention on the total ionic energy E(Z, N) and led to the Z-1/3 expansion (48), when combined with the 1/Z series (43). Two further developments of E(Z, N) will be recorded here, following the work... 

Fig. 1. Schematic energy level diagram for MgO (a) free ion energies (b) ionic energy levels in point charge Madelung potential (c) ionic energy levels after allowance for polarisation (d) after allowing development of bandwidths due to interactions amongst ionic level. Adapted from ref. 1.

Lower central panel, schematic ionic energy level diagram showing how the Sn levels found in a centrosymmetric environment evolve on switching on a perturbation V due to a non-centrosymmetric field which allows 5s-5p mixing. 

For application to nonmolecular solids, the bond description is similar but certain modifications are needed. First, the covalent energy must be multiplied by the equivalent number n of two electron covalent bonds per formula unit that must be broken for atomization. The evaluation of n will be discussed in detail presently. Second, the ionic energy must be evaluated as the potential energy over the entire crystal, corrected for the repulsions among adjacent electronic spheres. This is done by using the Born-Mayer equation for lattice energy, multiplying this expression by an empirical constant, a, which is 1 for the halides and less than 1 for the chal-cides, as follows ... 

The Madelung constant for corundum, a-AbOa, is 24.24, and k is estimated as 0.84. The ionic energy then is ... 

The occupied valence band states, and empty conduction band states for nanocrystalline Ti02 films with a physical thickness >4nm, and annealed at a temperature of at least 700 °C (a) in qualitative and quantitative agreement with the ionic energy level approach of Cotton in [13] using SALC s of atomic states as a basis set, and (b) display a complete removal of J-T d-state degeneracies. 

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