Dielectrics: An Atomic View — Question 4

Question 4

Free rotation reduces the weak-field alignment by an angular factor

Independent permanent dipoles rotate freely in three dimensions. With $x=pE/(k_BT)$, their angular normalization is $Z(x)=\int e^{x\cos\theta}d\Omega=4\pi\sinh x/x$. Use $\langle\cos\theta\rangle=d\ln Z/dx$.

1. Find the polarization at arbitrary positive x.

2. Use $\coth x=1/x+x/3+O(x^3)$ to find the weak-field susceptibility and compare it with the two-orientation model.

Question 4 - Solution

Free rotation reduces the weak-field alignment by an angular factor

(a) Method and calculation

$$\langle\cos\theta\rangle=\frac d{dx}[\ln(\sinh x)-\ln x]=\coth x-1/x.$$$$P=np(\coth x-1/x).$$

(b) Interpretation and further calculation

$$x\ll1:\quad P\simeq np(x/3)=\frac{np^2E}{3k_BT}.$$$$\chi_{\mathrm{3D}}=\frac{np^2}{3\epsilon_0k_BT}=\chi_{\mathrm{two\ state}}/3.$$

Physical check

The angular phase space, including the sine-theta solid-angle weight, changes the response even when each dipole has the same p.