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The Jaynes-Cummings model is the simplest and most fundamental case of interaction between atoms and radiation With a minor modification it can also retain the essential features of material media without burdening us with nonessential features such as local field effects and the nontrivial mode structure that a dielectric gives rise t03 in free space [237] So we adopt what we believe to be the simplest one-dimensional microscopic model of lossless material media: N two-level atoms having the same resonance frequency Wo in a single-mode cavity of resonance frequency w Although we cannot propagate a wave packet in a single-mode cavity to see the effect of dispersion in the packet width, we are able to describe the essential cause of dispersion with this model: the frequency dependence of the dielectric permittivity For simplicity, we deal only with linear media.



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which allow calculation of monomer reactivity ratios for the copolymerization of a pair of monomers from monomer reactivity ratios for each of those monomers copolymerizing separately with styrene and with acrylonitrile and from the monomer reactivity ratios for styrene acrylonitrile copolymerization. The patterns of reactivity parameters, like the Q e parameters, can be used to analyze reactivity in both copolymerization and homopolymerization. Look at the data in Table 6-4 and compare with the parameters in Table 6-8. The highly reactive radicals are those with lower values of r1S . The highly reactive monomers are those with the more positive or less negative values of v. However, v is not the only consideration, polarity is also important. For example, maleic anhydride is a monomer with one of the most positive v values, but it undergoes facile copolymerization only with monomers with which it has a polarity difference. This is the alternation tendency and is given by





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Fano has shown [189] that in the linear-medium approximation, the atoms constituting the medium can be described by harmonic oscillators, just as in the classical Lorentz model [415, 619] The linear-medium approximation works because we assume that these atoms couple only weakly to the cavity mode The essence of Fano's result is, in a nutshell, this: A harmonic oscillator has a ladder of energy levels, but when it is coupled weakly to the field, the upper levels are almost never populated, and it is effectively equivalent to a two-level system To probe the atom-field interaction within this medium, we consider a single guest two-level atom of resonance frequency Wa immersed in the material medium and coupled strongly to the field This guest atom will not, of course, be approximated by a harmonic oscillator (see Figure 71).

log r12 r21 u1 u2 p1 p2 6-66

Knoester and Mukamel [361] adopted a similar model to study impurity molecules in a dielectric host crystal, but in their model the dielectric medium is not placed inside a single-mode cavity Recalling Section 51, the displacement field is given by.

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The alternation tendency is dependent only on the polarity parameters for the monomers and radicals. The patterns of reactivity scheme has also been extended to chain transfer constants. 6-3b-6 Other Quantitative Approaches to Reactivity

Linear-free-energy relationships such as the Hammett and Taft equations [Lowry and Richardson, 1987] have been used to correlate copolymerization behavior with structure, but the approach is limited to considering a series of monomers that are similar in structure. Walling [1957] applied the Hammett equation to copolymerization among various meta- and para-substituted styrenes. The Taft equation in the form

D(x) =

1 log s* r* dES r1 6-67

(~x) ,

has been used to correlate the reactivity of various vinyl esters (M2) toward the radical from ethylene (M1) [Van der Meer et al., 1977b]. The variables r* and d are proportionality constants indicative of the susceptability of the radical monomer reaction to the effect of polar and steric factors, respectively, present in the monomer. Variables s* and ES are constants that indicate the polar and steric factors, respectively, due to the substituents on the double bond of the monomer. For the series of vinyl esters, CH2 CHOCOR, s* and ES are the substituent constants for the various R groups. Similar correlations have been successful for alkyl acrylates and methacrylates [Otsu, 1987]. 6-3c Terminal Model for Rate of Radical Copolymerization

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