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As D has no noise contribution from the medium up to second order, the contribution Rosewame found can come only from the polarization P. But the polarization (7.2) vanishes whenever one is not at one of the positions occupied by an atom of the medium. So except for these positions, there should be no noise contribution from the medium just as there is not for D. Rosewame's E had no such discontinuities, because he assumes a continuous distribution of atoms. This means that the dynamical variables he uses are supposed to be macroscopic averages from the beginning. How can we define a macroscopic average Lorentz introduced macroscopic averages to show how Maxwell's equations in material media can be obtained from more fundamental equations for the electromagnetic fields in the vacuum and the constituent charged particles that make up the media. This average was over "physically infinitesimal" volume elements, which were yet large enough to contain many atoms. There are some problems with Lorentz's definition of macroscopic average. Perhaps the most important for us is that in many physical situations there is no physically infinitesimal volume that is both macroscopically small and yet contains a large number of atoms. In optics, for example, the volume element cannot exceed 109 A3~ otherwise, optical oscillations would be averaged to zero. But the addition of a single extra electron to such a volume would cause an enormous change in the charge density: about 160 C/m 3 -by no means an infinitesimal change! Many alternative definitions of macroscopic averaging have been proposed in the literature.6 A well-established trend is to regard macroscopic averaging as just a



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For such a system the copolymer will consist of copolymerized A and B groups with pendant, unreacted C groups until the later stages of reaction. Crosslinking does not occur until relatively late in the reaction due to the low reactivity of the C group. The critical extent of reaction at gelation in this case is given by





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sort of ensemble averaging. Robinson [524, 525] points out, however, that the aim of doing macroscopic averages is to discard unwanted microscopic detail. This is different from the ensemble averages of statistical physics, for there we do an average only because our knowledge of the macroscopic state of the system is incomplete. Robinson showed that the best way to get rid of the unwanted microscopic detail from a given dynamical variable is to eliminate high-frequency components from its spatial Fourier expansion. We will adopt his idea here, as it seems to have "hit the nail on the head" as far as macroscopic averaging is concerned. As the cavity has a finite length, we can expand the delta functions in (7.2) into a Fourier series. Moreover, as the atoms of the medium can be everywhere inside the cavity except at the positions of the cavity mirrors, this Fourier series will be a sine series; that is,

pc 1 exp 1 2qX w r 6-97

(7.60)

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where q is the mole fraction of the diene in the initial comonomer feed. In selecting the makeup of a crosslinking system one has a number of variables that can be used to control the process. Thus gelation can be delayed with the production of a highconversion uncrosslinked product, which is amenable to subsequent crosslinking. Gelation can be delayed by reducing the amount of the diene, the degree of polymerization by using chaintransfer agents, or the reactivity of one of the double bonds of the diene by proper choice of the diene reactant. The extent of crosslinking in the nal product is also controlled by these variables. Extensive crosslinking, with the formation of tight network structures, is obtained by avoiding chain transfer and using increased amounts of dienes whose double bonds have similar reactivities. A special situation that is often encountered is that where the reaction of one of the double bonds of the diene results in a decrease in the reactivity of the remaining double bond. If the decrease in reactivity is large, the effect is to markedly delay the crosslinking reaction. This case then becomes very similar to the last one where the C group has a much lower reactivity than the A and B groups. The most notable case in which there is a large drop in reactivity of one group on reaction of the other is in the copolymerization of 1,3-dienes where 1,4polymerization leads to residual 2,3-double bonds that have lowered reactivity. These are subsequently used to bring about crosslinking by reactions discussed in Sec. 9-2b.

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