Question 7.4: IONIC AND ELECTRONIC POLARIZABILITY Consider the CsCl crysta...
IONIC AND ELECTRONIC POLARIZABILITY Consider the CsCl crystal which has one Cs^{+} –Cl^{−} pair per unit cell and a lattice parameter a of 0.412 nm. The electronic polarizability of Cs^{+} \text {and} Cl^{−} ions is 2.7 \times 10^{−40} F m^{2} \text {and} 4.0 \times 10^{−40} F m^{2}, respectively, and the mean ionic polarizability per ion pair is 5.8 \times 10^{−40} F m^{2}. What is the dielectric constant at low frequencies and that at optical frequencies?
The CsCl structure has one cation (Cs^{+}) and one anion (Cl^{−}) in the unit cell. Given the lattice parameter a=0.412 \times 10^{-9} m , the number of ion pairs N_{i} per unit volume is 1 / a^{3}=1 / \left(0.412 \times 10^{-9} m \right)^{3}=1.43 \times 10^{28} m ^{-3}. N_{i} is also the concentration of cations and anions individually. From the Clausius–Mossotti equation,
\frac{\varepsilon_{r}-1}{\varepsilon_{r}+2}=\frac{1}{3 \varepsilon_{o}}\left[N_{i} \alpha_{e}\left( Cs ^{+}\right)+N_{i} \alpha_{e}\left( CI ^{-}\right)+N_{i} \alpha_{i}\right]
That is,
\frac{\varepsilon_{r}-1}{\varepsilon_{r}+2}=\frac{\left(1.43 \times 10^{28} m ^{-3}\right)\left(2.7 \times 10^{-40}+4.0 \times 10^{-40}+5.8 \times 10^{-40} F m ^{2}\right)}{3\left(8.85 \times 10^{-12} F m ^{-1}\right)}
Solving for \epsilon _{r}, we find \epsilon _{r} = 7.18.
At high frequencies—that is, near-optical frequencies—the ionic polarization is too sluggish to allow ionic polarization to contribute to \varepsilon _{r}. Thus, \varepsilon _{rop}, relative permittivity at optical frequencies, is given by
\frac{\varepsilon_{\text {rop }}-1}{\varepsilon_{\text {rop }}+2}=\frac{1}{3 \varepsilon_{o}}\left[N_{i} \alpha_{e}\left( Cs ^{+}\right)+N_{i} \alpha_{e}\left( Cl ^{-1}\right)\right]
That is,
\frac{\varepsilon_{\text {rop }}-1}{\varepsilon_{\text {rop }}+2}=\frac{\left(1.43 \times 10^{28} m ^{-3}\right)\left(2.7 \times 10^{-40}+4.0 \times 10^{-40} F m ^{2}\right)}{3\left(8.85 \times 10^{-12} F m ^{-1}\right)}
Solving for \varepsilon _{rop}, we find \varepsilon _{rop} = 2.69. This very close to the experimental value \varepsilon _{rop} = 2.62. The low frequency experimental value for \varepsilon _{r} is 7.20, but this is normally used to deduce \alpha _{i}.