Retarded interaction

Flame-retardant additives are capable of significant reduction in the ha2ard from unwanted fires, and techniques are now available to quantify these improvements. Combined with an understanding of fire-retardant mechanisms, polymer-retardant interactions, and reuse technology, formulations optimi2ed for pubHc benefit and manufacturing practicaUty can be selected. 

As a next step we also need to specify the magnetic and retardation interactions experienced by an electron i and generated by all other electrons. In a first approximation retardation is neglected and we assume that electron i experiences the electromagnetic field immediately. For the scalar potential j,unret, and the vector potential A/ Unret created by electron j and felt by electron i the classical expression reads ... 

The ABS component and its flame retardant interact with both PVC and PET in the course of thermal decomposition (135,139). 

The most complicated issue is the transformation of the operator of retarding interactions Hr to the irreducible form. The final result is as follows ... 

Finally, a similar consideration of the energy operator of retarding interactions (19.67) leads to the equality... 

Let us consider in a similar manner the matrix elements of the operators of magnetic Hm and retarding Hr interactions. For them also, a formula of the sort (20.6), where the necessary matrix elements in the case of one subshell of equivalent electrons are defined by equations (19.78) and (19.84), is valid. Retarding interactions exist only between subshells, while inside them they, according to (19.84), vanish. 

The direct part of the matrix element of the energy operator of the retarding interactions vanishes... 

This expression holds both for electrostatic interaction energy operator (2.5) and for magnetic and retardation interactions (2.6) and (2.7). 

Equations (3.86) and (3.88) give some examples of the nonretarded van der Waals forces for ideal contact geometries. For retarded interactions, the exponent for the distance of separation increases by 1 with the change of the corresponding numerical coefficients. The preceding theory, assuming complete additivity of forces between individual atoms, is known as the microscopic approach to the van der Waals forces. 

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. 

Umklapp process In the interaction of a continuous wave (photon, electron, etc.) with the lattice, the quasi-momentum of the wave is conserved, modulo a vector in the reciprocal lattice. The introduction of these quanta of momentum leads to the Umklapp process. In many macroscopic treatments the matter is treated as a continuous medium and Umklapp processes are neglected. In our treatment, Umklapp processes are included in the coulombic interactions (calculation of the local field), but implicitly omitted in the retarded interactions, since we dropped the term (cua/c)2 in (1.64). 

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. 

When the point-dipole approximation is no longer valid, the exact distribution of transition charges on the molecule is introduced. The difference between this distribution and that of the point dipoles is important only in short-range interactions and modifies only the analytic part of the dispersion. In particular, the retarded interactions (and the associated properties) are not modified. 

We must make a remark on the exciton-phonon coupling Rigorously speaking the hamiltonian H (2.96) should contain the retarded interactions,... 

As indicated in Section II.C. I, in calculating the contribution of the exciton motion in the presence of phonons to the exciton self-energy, retarded interactions may be neglected because they operate only in the optical region (middle of the Brillouin zone). This approximation is valid for short times,... 

In what follows, we present in Section IV.A a theory of the effects of weak disorder on the retarded interactions of 2D strong dipolar excitons, and in Section IV.B we analyze the effects of stronger disorders on the coulombic interactions, calculating the density of states and absorption spectra in 2D lattices, in the framework of various approximations of the mean-field theory. 

Figure 4.3. Schematic transition to coherence (Fig. 4.2) for a square distribution of domains as the disorder from A = T to < nr. The extended disorder is opposed by retarded interactions ( R,(z) ->) For A T all the domains have transferred their oscillator strength ( Im z - 00) to the coherent state at = 0, while the strong homogeneous width dominates the emission line.

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... 

---