Coulombic interaction lattice models

Temkin was the first to derive the ideal solution model for an ionic solution consisting of more than one sub-lattice [13]. An ionic solution, molten or solid, is considered as completely ionized and to consist of charged atoms anions and cations. These anions and cations are distributed on separate sub-lattices. There are strong Coulombic interactions between the ions, and in the solid state the positively charged cations are surrounded by negatively charged anions and vice versa. In the Temkin model, the local chemical order present in the solid state is assumed to be present also in the molten state, and an ionic liquid is considered using a quasi-lattice approach. If the different anions and the different cations have similar physical properties, it is assumed that the cations mix randomly at the cation sub-lattice and the anions randomly at the anion sub-lattice. 

The neutral-ionic transition (NIT) at t = 0 occurs abruptly[94] when the Madelung energy M of the ionic lattice exceeds the energy I — A to transfer an electron form D to A. Long-range Coulomb interactions are treated self-consistently as part of A in the modified Hubbard model[95],... 

To conclude, we can draw an analogy between our transition and Anderson s transition to localization the role of extended states is played here by our coherent radiant states. A major difference of our model is that we have long-range interactions (retarded interactions), which make a mean-field theory well suited for the study of coherent radiant states, while for short-range 2D Coulombic interactions mean-field theory has many drawbacks, as will be discussed in Section IV.B. Another point concerns the geometry of our model. The very same analysis applies to ID systems however, the radiative width (A/a)y0 of a ID lattice is too small to be observed in practical experiments. In a 3D lattice no emission can take place, since the photon is always reabsorbed. The 3D polariton picture has then to be used to calculate the dielectric permittivity of the disordered crystal see Section IV.B. 

This model assumes that the Mn ions occupy the sites of a cubic lattice (taken to be of unit lattice parameters), while the dopant A ions occupy an x fraction of the body centre sites of each unit cube formed by the Mn ions. Since the aim is to study the effect of long-range Coulomb interactions on phase separation, we make further simplifying assumptions. It is assumed that the t2g core spins are aligned ferromagnetically and that Jh-> 00 this effectively projects out or b electron spin opposite to that of the t2g core spins—we obtain an effectively spinless model. The above considerations lead us to the following extended b Hamiltonian... 

Recently the coexistence of the 2kp CDW with SDW has been found by a diffuse X-ray scattering study of (TMTSFjjPF [65]. This has been ascribed to a purely electronic CDW involving no lattice distortion. In the conventional model of SDW the charge density should be uniform. Very recently a theory has succeeded to explain the coexistence of the purely electronic 2kp CDW with SDW in terms of the next-nearest-neighbor Coulomb interaction between electrons [66]. It is interesting to find the coexistence also in other materails. 

Equation (6.13) was derived on a model of covalent bonds between nearest neighbors. It is not strictly applicable to ionic solids. The repulsion part of the potential energy must be similar for ionic and covalent cases, but the attraction part for ionic solids must also include the sum of the coulombic interactions with the remainder of the lattice. In effect, the number of bonds is increased. To see the magnitude of this effort, compare Equations (6.7) and (6.8) with their counterparts for a diatomic molecule, or ion-pair. 

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