Waves

3D lattice

dynamical diffraction

In the early 20th century, William Lawrence Bragg and his father William Henry Bragg studied the scattering of X-rays by crystals. They discovered unexpected effect (a strange pattern) when X-rays are reflected from crystals (as opposed to, for example, liquids). When irradiated at certain angles, the crystals reflect almost 100% of the incident radiation. When the angles get a little larger or smaller, the reflection drops to zero. For studying and explaining this strange effect, father and son were awarded the Nobel Prize in Physics.

X-ray radiation differs from visible light by a significantly shorter wavelength. This leads to the fact that usually X-rays propagate in accordance with geometric optics. The specifics of the interaction of X-rays with a substance is such that:

There are no materials completely transparent and completely opaque for X-rays;
X-rays gradually absorbed by the material, propagates in a straight line, practically without scattering.

Therefore, it is impossible to create a (transparent) lens for X-rays, and a mirror is very difficult.

However...

Almost ideal periodic structures are found in nature. These are crystals where atoms form a repeating spatial lattice. Moreover, the dimensions of the crystal are many, many, many times greater than the size of the minimum repeating structure, and defects (inclusions) can occur rarely. Such crystals represent almost perfect lattice. Unlike a flat diffraction grating (see above), the crystal structure is three-dimensional. It seems that it will be more difficult to describe the interaction with the incident light in this case... And the "light" should already be not quite ordinary. Since the distances between atoms in crystals are measured in angstroms, the wavelength must be of the same order. And this is ... X-rays.

So what? Wave optics works here too! X-rays are weakly scattered in the materials (on atoms). However, the periodicity of the crystal structure leads to the fact that under certain conditions strong “resonance” may occur.

Consider the incidence and propagation of X-rays in a crystal. To simplify the problem, let's represent the crystal as a (repeating) structure of semitransparent flat "mirrors" (located along the crystal planes at a distance d from each other). Then the incident "light" can partly be reflected from (each) plane, and partly pass further (into the crystal). Obviously, due to reflection (even without taking into account absorption), the intensity of the transmitted "light" will decrease with depth.

But what will happen to the reflected waves?

According to the law of reflection, the incidence and reflection angles are equal to each other. In our case, each crystal plane (which is not necessarily parallel to the surface!) will partially reflect the incident "light" . Recall that if the difference in the path of two rays is an integer number of wavelengths, then the waves add up, amplifying each other. This is condition to get the maximum intensity.

For simplicity, we consider only the first reflections (see figure). The difference in the path of the rays reflected from the first and second planes is equal to AB + BC (AC is perpendicular to BC). Knowing the distance between the planes d and the angle between the beam and crystal plane, for the path difference we get:

Because

for the path difference we have:

The condition of “maximum amplification” is that the path difference is equal to an integer number of wavelengths:

That is, the maximum intensity of the reflected "light" is achieved only at well-defined (but several) angles between the beam of the incident "light" and the crystal planes. In contrast to the reflection by an ordinary mirror of ordinary (visible) light.

The resulting formula expresses Bragg's law for the reflection of X-rays from crystals:

the angle of reflection (from the crystal plane) is equal to the angle of incidence;
the maximum reflection occurs at angles that satisfy the above formula (Bragg angles).

If n=1, for the first-order diffraction maximum, we have:

The formula for the Bragg reflection was derived using the simplest model (semitransparent mirrors). This formula can be obtained more strictly by writing Maxwell's equations for the E-M field in a periodic medium (crystal lattice).

A more rigorous (detailed) solution, taking into account multiple-reflections, leads to another unexpected result: 100% reflection is obtained in some neighborhood of the Bragg angle (not only at a exact value).

And experiments confirm this! Absorption of radiation in real crystals only slightly reduces the intensity reflected near to the Bragg angle.

A rigorous theory that considers the propagation of E-M waves in a periodic three-dimensional structure (crystal) accounting multiple scattering is called the dynamic theory of diffraction (or simply dynamic diffraction).

Note that scattering can occur not only on atoms. For example, interesting results were obtained applying dynamic theory of diffraction when considering resonant scattering in spiral magnetic structures.

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