The refractive index

The refractive index of a medium is a concept that is heavily used in optics. It relates to the fact that the traveling speed of a wave can vary when it travels through a medium.

The refractive index of the material, \(n_{\rm m}\), is defined by

\begin{equation} n_{\rm m}=\frac{\lambda_0}{\lambda_{\rm m}}=\frac{k_{\rm m}}{k_0}, \label{reflectivityeq:n} \end{equation}

where \(\lambda_0\) is the wavelength in the vacuum, \(\lambda_{\rm m}\) is the wavelength in the medium, and likewise for the wavenumbers.

*Quantum mechanical derivation

The refractive index may be derived by solving a wave equation with the kinetic and potential energies for the wave in the medium. Neutron motion can be described by waves using quantum mechanics, and the relevant wave equation to describe their motion is given by:

\begin{equation} \frac{\hbar^2}{2m_{\rm n}}\frac{d^2\psi}{dr^2}+\left(E-\overline{V}\right)\psi=0, \label{reflectivityeq:WaveEqn} \end{equation}

where \(E\) is the total neutron energy.

The quantity \(\overline{V}\) is the spatial average of the neutron potential, \(V\left(\bf{r}\right)\). A neutron propagating in a medium will see the average scattering length of all the nuclei in the medium,

\begin{equation} \overline{V}=\frac{1}{\text{volume}}\int V\left({\bf{r}}\right)d{\bf{r}}. \label{eq:Vrho} \end{equation}

This quantity is insensitive to the atomistic structure of the medium, hence the refractive index will be the same irrespective of whether the medium is liquid, amorphous, polycrystalline of monocrystalline. This is very different to the topic of neutron diffraction, which measures the arangement of the individual atoms (crystal structure).

Equation \eqref{eq:Vrho} can be shown to rigorously hold, and the proof can be summarized by interpreting the solution of \eqref{reflectivityeq:WaveEqn} as diffraction in the limit \(q = 0\). The length scales probed by a diffraction experiment are proportional to \(q^{-1}\). As \(q \rightarrow 0\), the length scale thus becomes much larger than the distances between atoms. In the limit \(q = 0\), all atoms scatter coherently, and thus \eqref{eq:Vrho} is valid. Equation \eqref{reflectivityeq:WaveEqn} may be rewritten in terms of its wavenumber, \(k\),

\begin{equation} \frac{\hbar^2}{2m}\frac{d^2\psi}{dr^2}+k^2\psi=0, \label{reflectivityeq:WaveEqnk} \end{equation}

where

\begin{equation} k^2=\frac{2m}{\hbar^2}\left(E-\overline{V}\right). \label{eq:k2} \end{equation}

The mean potential for the neutron in a vacuum is equal to zero, \(\overline{V}=0\). The total energy, \(E\), must always be conserved and is therefore always equal to the kinetic energy of the neutron when it travels in a vacuum,

\begin{equation} E=\frac{\hbar^2k_0^2}{2m}, \label{reflectivityeq:E} \end{equation}

and thus the refractive index may be written

\begin{equation} n^2=1-\frac{\overline{V}}{E}. \label{reflectivityeq:n2} \end{equation}

Index of refraction from materials

In the absence of magnetism, the potential for the interaction of a neutron with a nucleus is given by the Fermi pseudo-potential

\begin{equation} V_j({\bf r})=\frac{2\pi\hbar^2}{m_{\rm n}}b_j\delta\left({\bf r}-{\bf r}_j\right). \label{reflectivityeq:Vr} \end{equation}

Integrating equation \eqref{reflectivityeq:Vr} over the volume of the medium gives

\begin{equation} \overline{V}=\frac{1}{\text{volume}}\int V_N\left({\bf{r}}\right)d{\bf{r}}= \frac{2\pi\hbar^2}{m}\rho\overline{b}, \label{reflectivityeq:Vrhob} \end{equation}

where \(\rho\) is the atomic number density and \(\overline{b}\) is the average scattering length of all the nuclei in the medium. The refractive index is therefore sensitive to \(\rho\overline{b}\), the average scattering length density, and is given by

\begin{equation} n^2=1-\frac{4\pi \rho\overline{b}}{k_0^2}=1-\frac{\lambda_0^2}{\pi}\rho\overline{b} \label{reflectivityeq:n2rhob} \end{equation}

The nuclear scattering length is a complex quantity and may be written \(\bar{b} = \bar{b}' - i \bar{b}'' \). The real part is related to t he bound coherent scattering length, \(\bar{b}_c\), and to incoherent scattering due to the interaction between the spin of the neutron and any spin on the nuclei. The imaginary part is given by the average absorption cross-section, \(\bar{\sigma}_a\), through the relation

\begin{equation} \bar{b}'' = \frac{ k \bar{\sigma}_a }{4 \pi} = \frac{\bar{\sigma}_a}{2 \lambda} . \end{equation}

The refractive index is sometimes Taylor expanded and written

\begin{equation} n \approx 1- \frac{\lambda_0^2}{2\pi}\rho\overline{b}_c. \label{reflectivityeq:nTaylor} \end{equation}

It may be noted that equations \eqref{reflectivityeq:n} to \eqref{reflectivityeq:n2} also hold for X-rays, and that the final expression for the X-ray refractive index is very similar equation \eqref{reflectivityeq:n2rhob} with the difference that the scattering length density, \(\rho\overline{b}_c\) for X-rays is given by the electron density multiplied by the classical electron radius, \(\rho_e r_0\). The absorption cross-sections for X-rays is also much larger than those for neutrons and must always be included in the final expression for the refractive index.