The refractive index and Snell's law

The interaction of an electromagnetic wave with matter can be described by introduction of a refractive index $n$. The refractive index depends on the frequency $\omega$ and reflects the resonant behaviour at frequencies corresponding to the electronic and nuclear transitions of atoms. The nuclear transitions will play an important role in one of the following chapters about nuclear resonant scattering. The refractive index for X-rays can be in general written as

$$n=1-\delta+i\beta$$ (7)

with

$$\delta=\frac{2 \pi r_{0} \rho}{\vert k\vert^{2}}$$ (8)

$$\beta=\frac{\mu}{2 \vert k\vert}$$ (9)

where $\rho$ is the electronic density, $r_{0}=2.818~10^{-5} \textnormal{\AA}$ is the classical electron radius, $\mu$ is the linear absorption coefficient and $\vert k\vert$ is the length of the incident wave vector of the used X-rays.

As shown in equation the refractive index $n$ is imaginary. The wave vector of an X-ray entering a medium with the refractive index changes from k into $n$k. It means the propagating wave in the medium can be described as $e^{inkr}$. The comparison of the amplitude with the usual definition of the absorption length ( $\frac {1}{\mu}$) leads immediately to equation as will be explained later.

$$e^{ik_{T}z}=e^{inkz}=e^{i(1-\delta)kz} e^{-ik\beta z r}=e^{i(1-\delta)kz} e^{-i \frac{\mu z}{2}}$$ (10)

It is evident that the real part of the index of refraction for X-rays is smaller than unity ($\delta$ is in the order of about $10^{-5}$ for solid materials). The consequence is the total external reflection of X-rays which occurs bellow an incident angle $\alpha$ smaller than the critical angle $\alpha_{C}$. The critical angle $\alpha_{C}$ can be calculated from Snell's law which describes the propagation of X-rays in matter

$$n_{V} \cos \alpha=n \cos \alpha^{'}$$ (11)

where $n_{V}$ is the refractive index of vacuum ($n_{V}=1$), $n$ is the refractive index of the investigated layer, $\alpha$ is the incident angle, $\alpha^{'}$ is the angle between the vacuum/layer interface and the transmitted wave vector as defined in figure .

Figure: The incident radiation ( $\Psi_{I}, k_{I}, \alpha$) is reflected ( $\Psi_{R}, k_{R}, \alpha$) and transmitted ( $\Psi_{T}, k_{T}, \alpha^{'}$). The propagation of X-rays is described by Snell's law in equation

The total external reflection occurs if the angle $\alpha^{'}$ is equal to 0. From Snell's law (equation ), neglecting absorption, follows in this case

$$\cos \alpha_{C}=n\approx1-\delta.$$ (12)

For small angles $\alpha$ the cosines and sines can be written as

$$\cos \alpha \approx 1-\frac{\alpha^{2}}{2}$$

$$\sin \alpha \approx \alpha$$ (13)

and the critical angle is therefore

$$\alpha_{C}\approx\sqrt{2\delta}.$$ (14)

Assuming the typical value of $\delta=10^{-5}$ for hard condensed matter the critical angle $\alpha_{C}$ is in the order of $mrad$, e.g. $\alpha_{C}=$ 3.8 $mrad$ for the 14.4 $keV$ radiation and an iron layer.