This chapter quotes the Clausius-Mossotti relation,

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Dielectric and optical properties of solids · Dielectric and optical properties of solids

What an Instrument Can See, Ch The optical constants of a medium

Problem

This chapter quotes the Clausius-Mossotti relation, ϵ−1ϵ+2=Nαp3ϵ0\dfrac{\epsilon-1}{\epsilon+2}=\dfrac{\mathcal{N}\alpha_p}{3\epsilon_0}, and solves it for n=ϵn=\sqrt\epsilon without showing the algebra. ((a)) Writing x≡Nαp/3ϵ0x\equiv\mathcal{N}\alpha_p/3\epsilon_0, solve the relation for ϵ\epsilon, and hence for the Lorentz-Lorenz index n=ϵn=\sqrt\epsilon, in terms of xx. ((b)) Show that for x≪1x\ll1 your result for ϵ\epsilon reduces to the dilute-gas formula of this chapter, ϵ−1=Nαp/ϵ0\epsilon-1=\mathcal{N}\alpha_p/\epsilon_0. ((c)) Liquid carbon disulfide has a number density N=9.99×1027 m−3\mathcal{N}=9.99\times10^{27}\,\mathrm{m^{-3}} and a molecular polarisability volume αp/4πϵ0=8.22×10−30 m3\alpha_p/4\pi\epsilon_0=8.22\times10^{-30}\,\mathrm{m^3}. Predict its index nn from the Lorentz-Lorenz relation, and compare with the dilute-gas prediction using the same N\mathcal{N} and αp\alpha_p.

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