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B0 groups to react with. As s ! 1 the condition is reached in which [B] remains constant at B 0 during the complete decay of B0 according to Eq. 2-51b. Figure 2-4 shows the decay in a and the corresponding increase in X n with t (since X n 1=a). It becomes progressively more dif cult to achieve high degrees of polymerization in reasonable times for reaction systems of s > 1. At suf ciently large s (small k1 ) the



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Sommerfeld pointed out that for wave motion in an infinite medium, the usual boundary conditions at the finite surfaces involved in the problem are not enough to determine the solution uniquely. He was then led to propose restrictions on the behavior of the wave at infinity to guarantee a unique solution. These restrictions, known as Sommeifeld's radiation condition, amount to an asymptotic boundary condition at infinity. Sommerfeld's argument (see Sec. 28 of [571]) is this: Consider a spherical cavity of radius R and a scalar field ' whose boundary condition is that it vanishes on the surface of the sphere. Then for spherically symmetric radial oscillations, the eigenfunctions ' n of the Helmholtz equation (9.58)





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Fig. 2-4 Decay in a and increase in degree of polymerization with time in polymerization of A with A B B0 . Values of s are noted for each plot. After Ozizmir and Odian [1980] (by permission of Wiley Interscience, New York).

dw { 2w eG'a*(w)

reaction time to obtain high polymer (X n > 50 100) becomes impractical. The extreme situation occurs at s ! 1 k1 ! 0 , where the maximum degree of polymerization is 2. Reaction systems with very small k1 values must be avoided in order to achieve high degrees of polymerization. The desirable system from the viewpoint of polymerization rate and X n is that with s 1. (The exception to this generalization is for systems of s > 1 where both k1 and k2 are large but k2 is larger.) The A plus B 0 system with k1 6 k4 and k2 6 k3 has been treated in a similar man A B ner [Ozizmir and Odian, 1981]. The parameters a, b, g, and s as de ned by Eqs. 2-45a, 2-45b, 2-45c, and 2-45e are retained in the treatment. t is rede ned as

2 ()

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t k1 k2 B 0 t 2-52

(n + 1 )

(7.165)

7-52

For a review of the foundations of macroscopic electrodynamics, see Robinson's book [525] and van Kranendonk and Sipe's review paper [615]. The

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(n + 1 )

statistical ensemble approach to the derivation of the macroscopic Maxwell equations is exemplified by de Groot and Vlieger's paper [148]. See also [539]. For a review on the Huttner-Barnett theory, see [33]. By treating the cavity walls from first principles as a collection of atoms in the vacuum, Saito and Hyuga derived the dynamic Casimir effect (see 4) without having to deal with time-dependent boundary conditions [531]. Another first-principles approach to reflecting walls is Cook and Milonni's derivation of modified spontaneous emission (see 6), Rabi oscillations, and the Ewald-Oseen extinction theorem [121]. It is possible to cancel dispersion using nonlocal quantum effects, see [324, 578,579].

(n + 1 )

in a quantum superposition to enhance the index of refraction and at the same time, cancelabsorption. See [551] for the theory and [665] for the experimental demonstration. A pedagogical discussion of this can be found in [552]. For a pedagogical discussion of the macroscopic quantization approach, where one attempts to quantize the macroscopic Maxwell equations in media directly, see Vogel and Welsch's book [618]. Macroscopic QED has a long history. See [166] for a list of the main historical papers on this subject. Problems

7-53

7.1 In the regime where Wo : ~ : right-hand side of (7.104) in two ways:

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