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In this expression η must be restricted to nonzero values in the case of conductors. In these materials the dielectric function has an ω^{-1} singularity at ω = 0 which leads to an ω^{½} behavior in ln ˜γ(ω) near ω = 0. To ensure that *V*(ω) is square integrable, it is therefore necessary to require η ≠ 0 in Eq. (2) and in its analogue for circular modes. In insulators lim_{ω→0} lñγ(ω) is a nonzero constant and no such restriction applies.

The restriction ω ≠ 0 in Eq. (12) also occurs for the derivation from the quotient function Q. It arises from having to avoid improper rearrangement under the integral sign when going from the integral from - ∞ to ∞ one from 0 to ∞.

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A second low-frequency sum rule may be obtained from the expression for ln[γ¯_{+}(ω)/γ¯_{→}(ω)], Eq. (12), if a specific model is assumed. For example, if an insulator consists of a single magnetically active Lorentzian absorption with plasma frequency ω_{p} centered at ω_{0}, the low-frequency limit of ln[γ¯_{+}(ω)/γ¯_(ω)] is[Equation] where *n*_{0} is the static index (1 + ω^{2}_{p}/ω^{2}_{0})^{½}. Combining this with Eq. (12) yields [Equation]. For more complicated models of the absorption spectrum similar forms hold but with ω_{0}, ω_{p}, etc., replaced with average values.

H. Becquerel, C. R. Acad. Sci. (Paris) 125, 679 (1897).

C. G. Darwin and W. H. Watson, Proc. R. Soc. London 114, 474 (1927).

The dielectric function for this system is ∊±(ω) = ∊_{b} - ω^{2}_{p} / (ω^{2} - ω^{2}_{0} ± ωω*c* + *i*ωγ), with the background dielectric function ∊_{b} equal unity for vacuum.

See, for example, R. K. Ahrenkiel, T. H. Lee, S. L. Lyu, and F. Moser, Solid State Commun. 12, 1113 (1973).

An alternative approximation is to employ the inverse-second-moment phase sum rule discussed in Ref. 17 rather than the zeroth moment phase sum rule to evaluating the left-hand side of Eq. (31). This leads to a result similar to Eq. (32), but with ω¯^{2} replaced *n*_{0} ω¯^{2}_{0}, where *n*_{0} is the contribution to the static index from transitions at energies ω >β.

As in the vacuum background case a second sum rule derived from the low-frequency limit holds for insulators, but its form depends on the model assumed. For the single Lorentzian absorption treated in Ref. 17 the low-frequency limit of the quotent yields [Equation] where now ń_{0} = (ε_{∞} + ω^{2}_{p}/ω^{2}_{0})^{½}. This leads to [Equation].