Crystal retarded interactions

The expression (1.69) for e is quite general in the sense that it gives the response of the crystal to an external field of any wave vector. In particular the poles of s(K, (o) provide, over the whole Brillouin zone, the dispersion curves of the new elementary excitations built up by coulombic and retarded interactions. 

To summarize, the retarded interactions are important only for small wave vectors, of the order of that of the photons. For larger wave vectors the retarded interactions are uncoupled, in the sense that they do not contribute to the local field which describes the interaction between dipoles. This property allows us to understand why in global effects (cohesion energy, dispersion, etc.) retarded interactions make very small contributions, although for small K, the retarded interactions may show very strong effects (such as the quasi-metallic reflection of certain dyes,1 s or of the second singlet of the anthracene crystal). In particular, in all phenomena that involve interactions between excitons and free radiation, the retarded effects are by no means essential. 

Outside of a small region around the center of the Brillouin zone, (the optical region), the retarded interactions are very small. Thus the concept of coulombic exciton may be used, as well the important notions of mixure of molecular states by the crystal field and of Davydov splitting when the unit cell contains many dipoles. On the basis of coulombic excitons, we studied retarded effects in the optical region K 0, introducing the polariton, the mixed exciton-photon quasi-particle, and the transverse dielectric tensor. This allows a quantitative study of the polariton from the properties of the coulombic exciton. 

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. 

In the intermediate domain of values for the parameters, an exact solution requires the specific inspection of each configuration of the system. It is obvious that such an exact theoretical analysis is impossible, and that it is necessary to dispose of credible procedures for numerical simulation as probes to test the validity of the various inevitable approximations. We summarize, in Section IV.B.l below, the mean-field theories currently used for random binary alloys, and we establish the formalism for them in order to discuss better approximations to the experimental observations. In Section IV.B.2, we apply these theories to the physical systems of our interest 2D excitons in layered crystals, with examples of triplet excitons in the well-known binary system of an isotopically mixed crystal of naphthalene, currently denoted as Nds-Nha. After discussing the drawbacks of treating short-range coulombic excitons in the mean-field scheme at all concentrations (in contrast with the retarded interactions discussed in Section IV.A, which are perfectly adapted to the mean-field treatment), we propose a theory for treating all concentrations, in the scheme of the molecular CPA (MCPA) method using a cell... 

The problem is to discuss the generalized polarizability ae(1.49) with the matrix a 1 not commuting with that of the dipolar interactions, 0. To show that the pure retarded interactions may be discarded in the dynamics of mixed crystals, we assume here that the coulombic interactions are suppressed in (ft. The interaction tensor is then reduced to its retarded term (1.74). Then the dispersion is given by (1.35) ... 

The dynamics of the mixed crystal, with purely retarded interactions, is governed by the sums of the type (4.62) ... 

The crystal energy operator in the presence of a retarded interaction... 

In previous chapters we considered elementary crystal excitation taking into account only the Coulomb interaction between carriers. From the point of view of quantum electrodynamics (see, for example, (1)) such an interaction is conditioned by an exchange of virtual scalar and longitudinal photons, so that the potential energy, corresponding to this interaction, depends on the carrier positions and not on their velocity distribution. As is well-known, the exchange of virtual transverse photons leads to the so-called retarded interaction between charges. 

If the retarded interaction is ignored and the operator Hmt is removed from the total Hamiltonian (4.2), then the operator H becomes a sum of two independent Hamiltonians, one of them (Hi) describing the crystal elementary excitations - those which occur when the retardation effects are ignored and the second (H2) giving the elementary excitations - transverse photons in vacuum. The presence of the operator H3 leads to the interaction between carriers with the transverse electromagnetic field. In the case of an atomic gas this interaction causes, in particular, the so-called radiative width of energetic levels of excited states. In the case of an infinite crystal possessing translational symmetry the radiative width of excitonic states vanishes.29... 

It follows from the above relation that the retarded interaction is important only in the vicinity of wavevectors k y/eoQ/c, i.e. in that part of the spectrum, where the frequencies of the Coulomb excitons are near to those of the transverse photons. When the retardation is ignored, the branches of the Coulomb excitons and the transverse photons intersect (Fig. 4.1a). This intersection is removed when the retardation is taken into account (Fig. 4.1b). In a similar way the dependence w(k) for polaritons can be found for crystals with different symmetries. 

To simulate the habit of solution-grown crystals, the interactions of solvent molecnles at the crystal-solution interface conld be considered. In most cases, it is assnmed that the solvent affects crystal habit through preferential adsorption of solvent molecnles on specific faces and that removal of solvent molecules before the deposition of oncoming solnte molecules causes retardation of crystal growth. The extent of solvation of a crystal face could be qualitatively understood from the relative polarities of the varions crystal faces, which can be obtained from electrostatic potential maps calculated at closest approach distances (Berkovitch-Yellin 1985). 

A mixture of lignosulfonates, alkali-treated brown coal, and minor amounts of organic silicon compounds (e.g., ethyl silicone) reduces the permeability of cements [1019]. The additives may interact with the crystallization centers of the cement slurry and form a gel system in its pores and capillaries, thus reducing the permeability of the cement and increasing its isolating capability. Furthermore, it is claimed that the additive retards the setting rate of cement up to 200° C and increases the resistance to corrosive media. 

The interaction of lecithin with starch can also have great functional significance in food systems. Not surprisingly, the structure of the lecithins involved determines their reactivity and hence functionality. Hydrolyzed lecithins have been shown to complex with starch, retarding starch crystallization, and thus slowing staling in yeast-raised baked goods (98, 99). 

MC) and HPMC could significantly inhibit the crystallization of supersaturated hydrocortisone acetate (HA). The mechanism of nucleation retardation was believed to be due to the hydrogen bonding interactions between HA and the polymers. As to the... 

To a large extent, current interest in solid-state polymerization of monoacetylenes derives from the observation of interesting electrical, magnetic, and optical phenomena in polyacetylene, (CH)j (45), a pEutially crystalline material unstable to ambient conditions typically synthesized by Ziegler-Natta techniques. The fundamental study of (CH), and its electron-transferred ( doped ) forms has been retarded by the lack of fully ordered materials. Ftilly ordered polyacetylenes are also of interest because it is conceivable that their crystal structures could allow significant interchain interactions, a situation precluded in most PDA by side chains. 

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