# Diffraction of X Rays

## Diffraction of X Rays

the scattering of X rays by crystals (or molecules of liquids and gases), during which secondary deflected beams of uniform wavelength, which appear as a result of the interaction between the primary X rays and the electrons of the given substance, arise from the initial beam of rays; the direction and intensity of the secondary beams depend on the structure of the dispersive medium. Diffracted beams constitute part of all X-radiation scattered by a substance. In addition to scattering without a change in wavelength, scattering with a change in wavelength—the so-called Compton effect—is also observed. The phenomenon of X-ray diffraction, which proves the wave nature of the rays, was first detected experimentally in crystals by the German physicists M. Laue, W. Friedrich, and P. Knipping in 1912.

A crystal is a natural three-dimensional diffraction grating for X rays, since the distance between the scattering centers (atoms) in a crystal is of the same order as the wavelength of X rays (~lA = 10”8 cm). The diffraction of X rays in crystals may be considered a selective reflection of X rays from systems of the atomic planes of a crystal lattice (Bragg-Wolf condition). The direction of the diffraction maxima simultaneously satisfies three conditions:

*a* (cos *α* − cos *α*_{0}) = *H*λ,

*b* (cos *β* − cos *β*_{0}) = *K*λ,

*c* (cos *γ* − cos *γ*_{0}) = *L*λ,

where *a, b*, and *c* are the periods of the crystal lattice along its three axes: *α*_{0}*β*_{0} and *γ*_{0} are the angles formed by the incident beam; *α*, *β*, and *γ* are the angles formed by the scattered beams with the axes of the crystal; λ is the wavelength of X rays; and *H, K*, and *L* are whole numbers (Laue equations). A diffraction pattern results either from an immobile crystal through X-radiation with a continuous spectrum (a so-called Lauegram) or from a rotating or oscillating crystal (the angles *α*_{0} and *β*_{0} change while *γ*_{0} remains constant) that is subjected to monochromatic X-radiation (λ is constant), or from a semicrystal illuminated by monochromatic radiation. In the latter case, because of the fact that some crystals in the sample are oriented randomly, the angles *α*_{0}, *β*_{0} and *γ*_{0} change.

The intensity of a diffracted beam depends above all on the so-called structural factor, which is determined by the atomic factors of the crystal’s atoms, their arrangement within a unit cell of the crystal, and the character of the thermal oscillations of the atoms. The structural factor depends on the symmetry of arrangement of the atoms in the unit cell. The intensity of the diffracted beam also depends on the size and shape of the object and the perfection of the crystal.

The diffraction of X rays by semicrystalline bodies leads to the marked appearance of cones in the secondary beams. The primary beam is the axis of the cone; the apex angle of the cone is equal to 4 *θ* (*θ* being the angle between the reflecting plane and the incident beam). Every cone corresponds to a specific family of crystalline planes. All crystalline particles whose family of planes lies at an angle *θ* to the incident beam help form the cone. If the crystalline particles are small and are present in very large numbers per unit volume, then the cone of beams will be continuous. In the case of a texture, that is, when there is preferred orientation of the crystalline particles, the diffraction pattern (X-ray photograph) will consist of unevenly darkened rings (Debye-Scherrer method).

The method of X-ray diffraction in crystals has provided the possibility of determining the wavelength of X rays when the structure of the crystal lattice is known; X-ray spectroscopy, which played an important role in establishing the structure of the atom, developed as a result of this. Observations of the diffraction of X rays of known wavelength in a crystal of unknown structure make it possible to establish the nature of this structure (the arrangement of the ions, atoms, and molecules that make up the crystal), which has served as the basis for X-ray structural analysis.

The diffraction of X rays is also observed on scattering by amorphous solids, liquids, and gases. In this case broad rings of a halo type appear around the central spot on the curve of the ratio of intensity to scattering angle. The position of these rings (angle *θ*) is determined by the average distance between the molecules or by the distances between the atoms within the molecule. The density distribution of the substance can be determined from the ratio of intensity to scattering angle.

X-ray diffraction can also be observed in an ordinary optical diffraction grating when the incidence of the X rays on the grating is glancing (less than the angle of complete reflection). By using this method it is also possible to measure X-ray wavelengths directly and with a high degree of accuracy.

### REFERENCES

Landsberg, G. S.*Optika*, 4th ed., Moscow, 1957 (

*Obshchii kurs fiziki*, vol. 3).

Borovskii, I. B.

*Fizicheskie osnovy rentgenospektral’nykh issledovanii*, Moscow, 1956.

V. I. IVERONOVA