Dielectrics: An Atomic View — Question 3

Question 3

A two-orientation model saturates through unequal thermal populations

Independent permanent dipoles of magnitude p can point only parallel or antiparallel to E. Their energies are -pE and +pE. At temperature T their probabilities are proportional to $e^{-U/(k_BT)}$. Define $x=pE/(k_BT)$.

1. Find the population ratio and average moment along E.

2. Find weak-field susceptibility for number density n and the strong-field saturation polarization.

Question 3 - Solution

A two-orientation model saturates through unequal thermal populations

(a) Method and calculation

$$\frac{N_+}{N_-}=e^{2x},\quad\langle p_z\rangle=p\frac{e^x-e^{-x}}{e^x+e^{-x}}=p\tanh x.$$$$P=np\tanh x.$$

(b) Interpretation and further calculation

$$x\ll1:\quad P\simeq\frac{np^2}{k_BT}E\Rightarrow\chi=\frac{np^2}{\epsilon_0k_BT}.$$$$x\to+\infty:\quad P\to np.$$

Physical check

This restricted two-state model is not the same as freely rotating three-dimensional dipoles.