Bragg's law

The Bragg equation, also called Bragg condition, was developed in 1912 by William Lawrence Bragg. It represents the condition for constructive interference of waves in a three-dimensional lattice scattering at Represent has practical significance in the interpretation of images, resulting in crystalline solids by scattering of X-rays. With their help, the structure of the material at the atomic level determined. The analogue of the Bragg condition in reciprocal space is the Laue condition.

Principle

Does X-rays on a crystal, so this is indeed penetrated unhindered by a large part of the radiation, but it is also observed that radiation components are deflected by the crystal - a phenomenon that is referred to as X-ray diffraction. One mounted behind the crystal a suitable detector, such as a photographic plate to make the deflected radiation components visible result on characteristic patterns.

Cause of the diffraction, the reflection of X-rays to the lattice planes of the crystal, wherein the radiation is only reflected in such directions in which the Bragg equation is satisfied:

The Bragg equation linked:

• The distance d between the parallel lattice planes
• The wavelength of the X-ray radiation, and
• The angle between x-ray and lattice plane, so-called gloss or Bragg angle,
• N is a natural number indicating the order of diffraction.

Each set of parallel lattice planes has a characteristic interplanar spacing d and therefore, the Bragg equation, also a characteristic Bragg angle. For different orientations, under which meets radiation on the crystal, obtained on the detector behind the crystal almost always different images, because always other sets of parallel lattice planes are (in other Bragg angles and with different orientations in the crystal) in a reflection position to the incident beam.

Physical background

In fact, it is a diffraction phenomenon. In the electromagnetic field of the incident radiation, the electrons of atoms are excited and begin to forced vibrations themselves radiation in the form of spherical waves radiate. Since the waves of the individual electrons add up to a first approximation to the associated shafts atoms, and further, the distances in the crystal lattice, and the wavelength of X-rays of a similar magnitude, interference phenomena occur.

Is the Bragg equation is satisfied for a given wavelength for a family of parallel lattice planes, that is, the X-ray is incident at the proper angle onto the crystal, there is constructive interference in the diffraction of the resulting electron clouds of the spherical waves. Macroscopically, the impression of a reflection of the radiation at the crystal.

Derivation

The blue lines (see adjacent diagram: Diagram for the diffraction geometry) correspond to rays that strike parallel lattice planes and thereby form the angle to the solder. The complementary angle is called the Bragg angle or grazing angle. d is the interplanar spacing, the black dots are atoms in the lattice planes.

Due to the high number of atoms in a crystal, there are not only for the case of constructive interference statistically every atom always a second, exactly cancels the diffracted wave of the first, so that no reflection can be observed. This is also the situation in non-crystalline material, regardless of the direction of incidence.

Carrying out the experiment

The rotating crystal arrangement is one possible implementation of the experiment. Since earlier X-ray machines were very heavy and therefore not rotate, the X-ray beam was directed onto a rotating crystal. By rotating the crystal and the recipient of the crystal could then be examined under different angles. A second possibility is the Debye-Scherrer method in which the crystal is pulverized, so that each " rotation" statistically distributed at the same time present.

Importance

• X-ray diffraction experiments on crystals offer the possibility to gain insight into the internal structure of crystals (see: Crystal structure analysis).
• Similarly, the Bragg reflection for the neutron diffraction is significant. Moderated neutrons have comparable wavelengths such as X- rays, so that the same phenomenon occurs at the crystal lattice.
• Diffraction patterns of electron beams may be prepared and observed in electron microscopes. These are basically described by the Bragg law.
• Bragg reflections occur in white-light holograms. There they are responsible for ensuring that the image of the hologram changes color when tilted.
• Bragg reflectors are used for wavelength selection in lasers (see, DBR lasers, DFB lasers ).
• Acousto-optical modulators are based on the principle of Bragg diffraction; Light rays are diffracted at this acoustic waves in crystals.