The critical angle can be calculated from Snell's law by setting the refraction angle equal to 90°. The exit angle will then approach 90° for some criticalincident angle θ c, and for incident angles greater than the critical angle there will be total internal reflection. Such reflection is commonly called " internal reflection". When light is incident upon a medium of lesser index of refraction, the ray is bent away from the normal, so the exit angle is greater than the incident angle. L.Total Internal Reflection Total Internal Reflection With the order m, the wavelength λ, the period d M of the multi layer system, the angle of incidence θ and the real part of the period-averaged index of refraction of the multi layer system.Ī. The Bragg equation then changes to compensate for the refraction in the layers : 3: X-ray reflection at a multi layer mirror 2: Bragg reflection when die optical path difference Δ=2 d sin θ for a certain wavelength λ is a multiple of λ, the beams will interfere constructively.Īn X-ray mirror can be formed by fabricating a multi layer system consisting of layers of different index of refraction (see fig. elliptically curved crystal mirrors and in monochromators to filter wavelengths and in material science when calculating the atomic lattice constants d from the reflection angles at crystalline samples.įig. This is known as Bragg equation:īragg reflection is used for example to focus monochromatic light with e.g. With the distance d between two adjacent atomic layers and the angle θ of the incoming wave measured to the surface is an integer multiple m of the wavelength λ, constructive interference occurs and consequently the wave is reflected. To understand the physical principle, the path difference of an incoming ray reflected at the surface of a crystal and a neighbour ray being reflected at the next inner atomic layer is regarded (see fig.
These mirrors provide large reflection angles when the reflection condition for a given wavelength is fulfilled. Mirrors using Bragg-reflection to redirect X-rays are called crystal mirrors. Total external reflection is nearly, but not completely loss-less, because the absorption coefficient β is not zero.Ĭrystal surfaces show high reflectivity under special angles depending on the wavelength of the X-rays due to Bragg-reflection. In the example of gold and a photon energy of 12.4 keV, θ critical =0.351°. This approximation results in an error below 0.021% for θ critical<1°. With n 2 = 1- δ and with the Taylor expansion of the cosine
Consequently normally the reflection angles are measured as angles θ between the incoming ray and the mirrors surface. So the reflection angles α are always close to 90°. For X-rays total external reflection occurs only under grazing incidence. So the critical angle α critical isĪs n 2 is only slightly below one (for example for gold for photon energies of 12.4 keV, n 2 = 1-1.88♱0 -5), the maximum angle for TER to occur is close to 90° (for gold at 12.4 keV α critical = 89.65° = 90°-0.351°). Assuming the ray comes from vacuum, n 1=1. Total external reflection occurs for angles α 1, when the angle α 2 reaches 90°. With the angle α 1 of the incoming ray and α 2 of the outgoing ray and the indices of refraction n 2 in matter and n 1 in the medium the ray comes from. From the refraction of X-rays we know that Snell's law is The critical angle can be calculated as follows (see fig. To calculate the critical angle of total external reflection 1: Sketch showing the angles and indices of refraction used When X-rays enter matter under grazing incidence, they will be reflected by Total External Reflection (TER) when the angle of incidence is below the critical angle α critical.įig. Mostly three reflection types are distinguished: Total external reflection X-rays can be reflected under certain conditions when hitting matter.
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