Dielectrics: An Atomic View — Question 6

Question 6

A local-field correction amplifies the response self-consistently

For an isotropic model use $E_{\mathrm{loc}}=E+P/(3\epsilon_0)$, $p=\alpha E_{\mathrm{loc}}$ and P=np. Define $f=n\alpha/(3\epsilon_0)$ and assume $0<f<1$.

1. Solve for susceptibility and relative permittivity.

2. Find kappa for f=.200 and compare with ignoring the local-field correction; explain the limitation as f approaches one.

Question 6 - Solution

A local-field correction amplifies the response self-consistently

(a) Method and calculation

$$P=n\alpha\left(E+\frac{P}{3\epsilon_0}\right)\Rightarrow(1-f)P=3\epsilon_0fE.$$$$\chi=\frac{3f}{1-f},\quad\kappa=\frac{1+2f}{1-f}.$$

(b) Interpretation and further calculation

$$f=.200:\quad\kappa=1.750,\quad\kappa_{\mathrm{no\ correction}}=1+3f=1.600.$$$$f\to1^-:\quad\chi\to\infty.$$This divergence signals a limit of the linear self-consistent model, not a prediction of an unlimited physical response.

Physical check

The result assumes the supplied isotropic local-field model; it cannot be applied indiscriminately to arbitrary molecular arrangements.