Long-range retarded interactions

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. 

Under certain circumstances it is possible to utilize physical interactions to maintain surfaces at some minimum distances of separation as a result of an energy maximum in the interaction energy. The practical result of such long-range energy maxima is that, properly utilized, they can prevent or at least retard the natural tendency of surfaces to approach and join spontaneously, thereby reducing interfacial area. This effect is especially important in colloids, foams, emulsions, and similar systems. 

As mentioned in an earlier section, the dispersion interactions exhibit a quantum-mechanical retardation effect at large (on the atomic or molecular scale) distances. Such effects are brought out explicitly by the Lifshitz theory, so that, for example, long range interactions become proportional to r instead of r, where r is the distance of separation, as is observed experimentally. Luckily, however, the effects are seldom of concern in practical systems since their magnitude is extremely small relative to other factors. 

The three most important forces for the long range interaction between macroscopic particles and a surface are steric-polymer forces, electrostatic interactions and Van der Waals forces. If we assume than the Van der Waals interactions between two atoms in a vaccuum are non-retarded and additive, we saw in the previous chapter that the form of the Van der Waals pair potential is w = —CJD where C is the coefficient in the atom-atom pair potential and D is the distance between the two... 

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