Some Properties of Atoms — Question 8

Question 8

A displaced electron cloud gives a simple atomic polarizability

In a supplied classical response model, an electron cloud of charge $-e$ is displaced a distance x opposite a uniform electric field of magnitude $\mathcal E$. A restoring force of magnitude kx acts toward the nucleus. Define the dipole magnitude $p=ex$ and $p=\alpha\mathcal E$.

1. Derive the equilibrium displacement and polarizability.

2. Include the restoring energy and field coupling to find the minimum total energy.

Question 8 - Solution

A displaced electron cloud gives a simple atomic polarizability

(a) Method and calculation

$$kx=e\mathcal E\Rightarrow\boxed{x=e\mathcal E/k}.$$$$p=ex=(e^2/k)\mathcal E\Rightarrow\boxed{\alpha=e^2/k}.$$

(b) Interpretation and further calculation

$$U(x)=\tfrac12kx^2-e\mathcal E x,\quad U\prime(x)=kx-e\mathcal E=0.$$$$\boxed{U_{\min}=-e^2\mathcal E^2/(2k)=-\tfrac12\alpha\mathcal E^2}.$$

Physical check

The factor one-half arises because the dipole is induced and requires internal deformation energy. The model is valid only for small displacements in a linear restoring regime.