Long range expansion limit

Equations (2.1.68) show that the long-range expansion limit ag(k -+ oo) is finite so that the mean-square size of a large chain is still proportional to N. This confirms that we are indeed in the 0 state, although the chain is somewhat expanded with respect to the phantom state. Considering that the characteristic ratio is given by <(r N) o/Nl [see (2.1.6)] and that r(iV) = T(q = 0) [see (2.1.28)], we have... 

The perturbation theory described in section Al.5.2,1 fails completely at short range. One reason for the failure is that the multipole expansion breaks down, but this is not a fiindamental limitation because it is feasible to construct a non-expanded , long-range, perturbation theory which does not use the multipole expansion [6], A more profound reason for the failure is that the polarization approximation of zero overlap is no longer valid at short range. 

Hence, in the limit A->0, the differential cross-section diverges. This is due to the long range character of the Coulomb field, which entails that the expansion (10-259) of KA does not converge. Alternatively, the asymptotic condition is not valid in this case. 

Evaluation of the long-range part of the electrostatic potential, which results from the long-wavelength limit (fc = 0) of the corresponding Fourier expansion, is the trickiest part in the derivation of the Ewald expression for the electrastatic potential of a Coulombic system. The problem is iminediately apparent from Laplace s equation in Fourier space [sec Eq. (F.21)] which, when solved for k) directly at fc = 0, yields a divergent result because of the factor Fortunately, we are not really... 

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