Problem: Snell's Law

Question 1

Knowing that momentum must be conserved across the interface, derive Snell's Law.

Solution

The momentum of the neutron is given by the wavevector through the de Broglie relation \begin{equation} {\bf{p}}=\hbar\bf{k}. \end{equation} The only components of the momentum to change on reflection/refraction are those normal to the interface, i. e. \( q_z\). The components of the momentum perpendicular to the interface normal are unchanged across the interface. Thus, \begin{equation} k_i\cos\theta_i = k_r\cos\theta_r = k_t\cos\theta_t. \end{equation} or more generically: \begin{equation} k_1\cos\theta_1 = k_2\cos\theta_2. \end{equation}

This equation from the Neutron reflectivity page, \(n_m=\frac{\lambda_0}{\lambda_m}=\frac{k_m}{k_0}\), gives the relation between the refractive index and the wavenumber. Substituting this into the equation above gives Snell's Law: \begin{equation} n_1\cos\theta_1 = n_2\cos\theta_2. \end{equation}