X-ray diffraction is the scattering of X-rays, in which secondary deflected beams with the same wavelength appear from the initial beam of rays, resulting from the interaction of primary X-rays with electrons of the substance. The direction and intensity of the secondary beams depend on the structure (structure) of the scattering object.

2.2.1 Scattering of X-rays by electrons

X-rays, which are an electromagnetic wave, directed at the object under study, affect an electron weakly associated with the nucleus and set it into oscillatory motion. When a charged particle oscillates, electromagnetic waves are emitted. Their frequency is equal to the frequency of charge oscillations, and, consequently, to the frequency of field oscillations in the beam of “primary” X-rays. This is coherent radiation. It plays a major role in the study of structure, since it is it that is involved in creating the interference pattern. So, when exposed to X-rays, an oscillating electron emits electromagnetic radiation, thereby “scattering” the X-rays. This is X-ray diffraction. In this case, the electron absorbs part of the energy received from X-rays, and releases part in the form of a scattered beam. These rays scattered by different electrons interfere with each other, that is, they interact, add up and can not only enhance, but also weaken each other, as well as extinguish (the laws of extinction play an important role in X-ray diffraction analysis). It should be remembered that the rays that create the interference pattern and X-rays are coherent, i.e. X-ray scattering occurs without changing the wavelength.

2.2.2 Scattering of X-rays by atoms

Scattering of X-rays by atoms differs from scattering by a free electron in that the outer shell of an atom can contain Z-electrons, each of which, like a free electron, emits secondary coherent radiation. Radiation scattered by the electrons of atoms is defined as a superposition of these waves, i.e. intra-atomic interference occurs. The amplitude of X-rays scattered by one atom A a, which has Z electrons, is equal to

A a = A e F (5)

where F is the structure factor.

The square of the structural amplitude indicates how many times the intensity of scattered radiation by an atom is greater than the intensity of scattered radiation by one electron:

The atomic amplitude I a is determined by the distribution of electrons in the atom of the substance; by analyzing the value of the atomic amplitude, it is possible to calculate the distribution of electrons in the atom.

2.2.3 Scattering of X-rays by a crystal lattice

Of greatest interest for practical work. The theory of interference of X-rays was first substantiated by Laue. It made it possible to theoretically calculate the locations of interference maxima on radiographs.

However, the widespread practical application of the interference effect became possible only after English physicists (father and son Bragg) and at the same time the Russian crystallographer G.V. Wulff created an extremely simple theory by discovering a simpler connection between the location of interference maxima on an x-ray diffraction pattern and the structure of the spatial lattice. At the same time, they considered the crystal not as a system of atoms, but as a system of atomic planes, suggesting that X-rays experience specular reflection from atomic planes.

Figure 11 shows the incident beam S 0 and the beam deflected by the plane (HKL) S HKL .

In accordance with the law of reflection, this plane must be perpendicular to the plane in which the rays S0 and SHKL lie, and divide the angle between them in half, i.e. the angle between the continuation of the incident ray and the deflected ray is 2q.

The spatial lattice is built from a number of planes P 1, P 2, P 3 ...

Let us consider the interaction of such a parallel system; planes with a primary ray using the example of two adjacent planes P and P 1 (Fig. 12):

Rice. 12. To the derivation of the Wolf-Bragg formula

Parallel rays SO and S 1 O 1 fall at points O and O 1 at an angle q to the planes P and P 1 . Moreover, the wave arrives at point O 1 with a delay equal to the difference in the path of the waves, which is equal to AO 1 = d sinq. These rays will be specularly reflected from the planes P and P 1 at the same angle q. The difference in the path of the reflected waves is equal to O 1 B = d sinq . Cumulative path difference Dl=2d sinq. The rays reflected from both planes, propagating in the form of a plane wave, must interfere with each other.

The phase difference of both oscillations is equal to:

(7)

From equation (7) it follows that when the path difference of the rays is a multiple of an integer number of waves, Dl=nl=2d sinq, the phase difference will be a multiple of 2p, i.e. the oscillations will be in the same phase, the “hump” of one wave coincides with the “hump” of the other, and the oscillations reinforce each other. In this case, an interference peak will be observed on the x-ray diffraction pattern. So, we get that the equality 2d sinq = nl (8) (where n is an integer called the order of reflection and determined by the difference in the path of the rays reflected by neighboring planes)

is a condition for obtaining an interference maximum. Equation (8) is called the Wulff-Bragg formula. This formula is the basis for X-ray diffraction analysis. It should be remembered that the introduced term “reflection from an atomic plane” is conditional.

From the Wulff-Bragg formula it follows that if a beam of X-rays with wavelength l falls on a family of plane-parallel planes, the distance between which is equal to d, then there will be no reflection (interference maximum) until the angle between the direction of the rays and the surface corresponds to this equation.

The relationships we have considered reflect the quantitative side of the process of attenuation of X-ray radiation. Let us dwell briefly on the qualitative side of the process, or on those physical processes that cause weakening. This is, firstly, absorption, i.e. the conversion of X-ray energy into other types of energy and, secondly, scattering, i.e. changing the direction of propagation of radiation without changing the wavelength (classical Thompson scattering) and with changing the wavelength (quantum scattering or Compton effect).

1. Photoelectric absorption. X-ray quanta can tear electrons from the electron shells of atoms of matter. They are usually called photoelectrons. If the energy of the incident quanta is low, then they knock out electrons from the outer shells of the atom. Large kinetic energy is imparted to the photoelectrons. With increasing energy, X-ray quanta begin to interact with electrons located in the deeper shells of the atom, whose binding energy with the nucleus is greater than that of electrons in the outer shells. With this interaction, almost all the energy of the incident X-ray quanta is absorbed, and part of the energy given to photoelectrons is less than in the first case. In addition to the appearance of photoelectrons, in this case quanta of characteristic radiation are emitted due to the transition of electrons from higher levels to levels located closer to the nucleus.

Thus, as a result of photoelectric absorption, a characteristic spectrum of a given substance appears - secondary characteristic radiation. If an electron is ejected from the K-shell, then the entire line spectrum characteristic of the irradiated substance appears.

Rice. 2.5. Spectral distribution of absorption coefficient.

Let us consider the change in the mass absorption coefficient t/r due to photoelectric absorption depending on the wavelength l of the incident X-ray radiation (Fig. 2.5). The breaks in the curve are called absorption jumps, and the corresponding wavelength is called the absorption boundary. Each jump corresponds to a certain energy level of the atom K, L, M, etc. At l gr, the energy of the X-ray photon turns out to be sufficient to knock out an electron from this level, as a result of which the absorption of X-ray quanta of a given wavelength increases sharply. The shortest wavelength jump corresponds to the removal of an electron from the K-level, the second from the L-level, etc. The complex structure of the L and M boundaries is due to the presence of several sublevels in these shells. For X-rays with wavelengths somewhat larger than l gr, the energy of the quanta is insufficient to remove an electron from the corresponding shell; the substance is relatively transparent in this spectral region.

Dependence of absorption coefficient on l and Z with the photoelectric effect is defined as:

t/r = Cl 3 Z 3 (2.11)

where C is the proportionality coefficient, Z is the serial number of the irradiated element, t/r is the mass absorption coefficient, l is the wavelength of the incident X-ray radiation.

This dependence describes the sections of the curve in Fig. 2.5 between absorption jumps.

2. Classical (coherent) scattering explains the wave theory of scattering. It occurs when an X-ray quantum interacts with an electron of an atom, and the energy of the quantum is insufficient to remove the electron from a given level. In this case, according to the classical theory of scattering, X-rays cause forced vibrations of the bound electrons of atoms. Oscillating electrons, like all oscillating electrical charges, become a source of electromagnetic waves that spread in all directions.

The interference of these spherical waves leads to the appearance of a diffraction pattern, naturally related to the structure of the crystal. Thus, it is coherent scattering that makes it possible to obtain diffraction patterns, on the basis of which one can judge the structure of the scattering object. Classical scattering occurs when soft X-ray radiation with wavelengths greater than 0.3Å passes through a medium. The scattering power by one atom is equal to:

, (2.12)

and one gram of substance

where I 0 is the intensity of the incident X-ray beam, N is Avogadro’s number, A is the atomic weight, Z– serial number of the substance.

From here we can find the mass coefficient of classical scattering s class /r, since it is equal to P/I 0 or .

Substituting all the values, we get .

Since most elements Z/[email protected] (except for hydrogen), then

those. The mass coefficient of classical scattering is approximately the same for all substances and does not depend on the wavelength of the incident X-ray radiation.

3. Quantum (incoherent) scattering. When a substance interacts with hard X-ray radiation (wavelength less than 0.3Å), quantum scattering begins to play a significant role when a change in the wavelength of the scattered radiation is observed. This phenomenon cannot be explained by wave theory, but it is explained by quantum theory. According to quantum theory, such an interaction can be considered as the result of an elastic collision of X-ray quanta with free electrons (electrons of the outer shells). X-ray quanta give up part of their energy to these electrons and cause their transition to other energy levels. The electrons that gain energy are called recoil electrons. X-ray quanta with energy hn 0 as a result of such a collision deviate from the original direction by an angle y, and will have an energy hn 1 less than the energy of the incident quantum. The decrease in the frequency of scattered radiation is determined by the relationship:

hn 1 = hn 0 - E department, (2.15)

where E rect is the kinetic energy of the recoil electron.

Theory and experience show that the change in frequency or wavelength during quantum scattering does not depend on the ordinal number of the element Z, but depends on the scattering angley. Wherein

l y - l 0 = l = ×(1 - cos y) @ 0.024 (1 - cozy), (2.16)

where l 0 and l y are the wavelength of the X-ray quantum before and after scattering,

m 0 – mass of an electron at rest, c– speed of light.

It is clear from the formulas that as the scattering angle increases, l increases from 0 (at y = 0°) to 0.048 Å (at y = 180°). For soft rays with a wavelength of the order of 1Å, this value is a small percentage of approximately 4–5%. But for hard rays (l = 0.05–0.01 Å), a change in wavelength by 0.05 Å means a change in l by a factor of two or even several.

Due to the fact that quantum scattering is incoherent (l is different, the angle of propagation of the reflected quantum is different, there is no strict pattern in the propagation of scattered waves in relation to the crystal lattice), the order in the arrangement of atoms does not affect the nature of quantum scattering. These scattered x-rays are involved in creating the overall background in the x-ray image. The dependence of the background intensity on the scattering angle can be theoretically calculated, which has no practical application in X-ray diffraction analysis, because There are several reasons why background occurs, and its overall significance cannot be easily calculated.

The processes of photoelectron absorption, coherent and incoherent scattering that we have considered mainly determine the attenuation of X-rays. In addition to them, other processes are possible, for example, the formation of electron-positron pairs as a result of the interaction of X-rays with atomic nuclei. Under the influence of primary photoelectrons with high kinetic energy, as well as primary X-ray fluorescence, secondary, tertiary, etc. may occur. characteristic radiation and corresponding photoelectrons, but with lower energies. Finally, some photoelectrons (and partly recoil electrons) can overcome the potential barrier at the surface of the substance and fly beyond it, i.e. an external photoelectric effect may occur.

All noted phenomena, however, have a much smaller effect on the value of the X-ray attenuation coefficient. For X-rays with wavelengths from tenths to units of angstroms, usually used in structural analysis, all these side effects can be neglected and it can be assumed that the attenuation of the primary X-ray beam occurs on the one hand due to scattering and on the other hand as a result of absorption processes. Then the attenuation coefficient can be represented as the sum of two coefficients:

m/r = s/r + t/r , (2.17)

where s/r is the mass scattering coefficient, taking into account energy losses due to coherent and incoherent scattering; t/r is the mass absorption coefficient, which mainly takes into account energy losses due to photoelectric absorption and excitation of characteristic rays.

The contribution of absorption and scattering to the attenuation of the X-ray beam is not equal. For X-rays used in structural analysis, incoherent scattering can be neglected. If we take into account that the magnitude of coherent scattering is also small and approximately constant for all elements, then we can assume that

m/r » t/r , (2.18)

those. that the attenuation of the X-ray beam is determined mainly by absorption. In this regard, the laws discussed above for the mass absorption coefficient during the photoelectric effect will be valid for the mass attenuation coefficient.

Radiation selection . The nature of the dependence of the absorption (attenuation) coefficient on the wavelength determines to a certain extent the choice of radiation in structural studies. Strong absorption in the crystal significantly reduces the intensity of diffraction spots in the x-ray diffraction pattern. In addition, the fluorescence that occurs during strong absorption illuminates the film. Therefore, it is unprofitable to work at wavelengths slightly shorter than the absorption limit of the substance under study. This can be easily understood from the diagram in Fig. 2.6.

1. If the anode, consisting of the same atoms as the substance under study, radiates, then we obtain that the absorption limit, for example

Fig.2.6. Change in the intensity of X-ray radiation when passing through a substance.

The K-edge of absorption of the crystal (Fig. 2.6, curve 1) will be slightly shifted relative to its characteristic radiation into the short-wave region of the spectrum. This shift is on the order of 0.01–0.02 Å relative to the edge lines of the line spectrum. It always occurs at the spectral position of emission and absorption of the same element. Since the absorption jump corresponds to the energy that must be expended to remove an electron from a level outside the atom, the hardest K-series line corresponds to the transition to the K-level from the most distant level of the atom. It is clear that the energy E required to tear an electron out of the atom is always slightly greater than that which is released when an electron moves from the most distant level to the same K-level. From Fig. 2.6 (curve 1) it follows that if the anode and the crystal under study are one substance, then the most intense characteristic radiation, especially the K a and K b lines, lies in the region of weak absorption of the crystal relative to the absorption boundary. Therefore, the absorption of such radiation by the crystal is low, and the fluorescence is weak.

2. If we take an anode whose atomic number Z 1 larger than the crystal under study, then the radiation of this anode, according to Moseley’s law, will slightly shift to the short-wave region and will be located relative to the absorption boundary of the same substance under study as shown in Fig. 2.6, curve 2. The Kb line is absorbed here, resulting in fluorescence that can interfere with shooting.

3. If the difference in atomic numbers is 2–3 units Z, then the emission spectrum of such an anode will shift even further into the short-wave region (Fig. 2.6, curve 3). This case is even more unfavorable, since, firstly, the X-ray radiation is greatly attenuated and, secondly, strong fluorescence illuminates the film when shooting.

The most suitable, therefore, is an anode whose characteristic radiation lies in the region of weak absorption by the sample under study.

Filters. The selective absorption effect we considered is widely used to attenuate the short-wavelength part of the spectrum. To do this, foil with a thickness of several hundredths is placed in the path of the rays mm. The foil is made of a substance whose serial number is 1–2 units less than Z anode. In this case, according to Fig. 2.6 (curve 2), the edge of the absorption band of the foil lies between the K a - and K b - emission lines and the K b - line, as well as the continuous spectrum, will be greatly weakened. The attenuation of K b compared to K a radiation is about 600. Thus, we have filtered b radiation from a radiation, which almost does not change in intensity. The filter can be foil made of a material whose serial number is 1–2 units less Z anode. For example, when working on molybdenum radiation ( Z= 42), zirconium can serve as a filter ( Z= 40) and niobium ( Z= 41). In the series Mn ( Z= 25), Fe ( Z= 26), Co ( Z= 27) each of the preceding elements can serve as a filter for the subsequent one.

It is clear that the filter must be located outside the chamber in which the crystal is photographed so that the film is not exposed to fluorescence rays.

At work at high voltages, as with radiography at ordinary voltages, it is necessary to use all known methods of combating scattered X-ray radiation.

Quantity scattered x-rays decreases with decreasing irradiation field, which is achieved by limiting the diameter of the working X-ray beam. With a decrease in the irradiation field, in turn, the resolution of the X-ray image improves, i.e., the minimum size of the detail detected by the eye decreases. To limit the diameter of the working X-ray beam, replaceable diaphragms or tubes are still far from being sufficiently used.

To reduce the amount scattered x-rays Compression should be used where possible. During compression, the thickness of the object under study decreases and, of course, there are fewer centers of formation of scattered X-ray radiation. For compression, special compression belts are used, which are included in X-ray diagnostic equipment, but they are not used often enough.

Amount of scattered radiation decreases with increasing distance between the X-ray tube and the film. By increasing this distance and corresponding aperture, a less divergent working beam of X-rays is obtained. As the distance between the X-ray tube and the film increases, it is necessary to reduce the irradiation field to the minimum possible size. In this case, the area under study should not be “cut off”.

To this end, in recent designs X-ray diagnostic devices have a pyramidal tube with a light centralizer. With its help, it is possible not only to limit the area being photographed to improve the quality of the X-ray image, but also to eliminate unnecessary irradiation of those parts of the human body that are not subject to radiography.

To reduce the amount scattered x-rays The part of the object being examined should be as close as possible to the X-ray film. This does not apply to direct magnification radiography. In radiography with direct image magnification, scattered observation practically does not reach the X-ray film.

Sandbags used for fixation the object under study should be located further from the cassette, since sand is a good medium for the formation of scattered X-ray radiation.

With radiography, produced on a table without the use of a screening grid, a sheet of leaded rubber of the largest possible size should be placed under the cassette or envelope with film.
For absorption scattered x-rays screening X-ray gratings are used, which absorb these rays as they exit the human body.

Mastering technology X-ray production at increased voltages on the X-ray tube, this is precisely the path that brings us closer to the ideal X-ray image, that is, one in which both bone and soft tissue are clearly visible in detail.

X-ray diffraction is the scattering of X-rays by crystals or molecules of liquids and gases, in which secondary deflected beams (diffracted beams) of the same wavelength arise from the initial beam of rays, resulting from the interaction of primary X-rays with electrons of the substance. The direction and intensity of the secondary beams depend on the structure of the scattering object. Diffracted beams make up part of the total X-ray radiation scattered by matter. Along with scattering without a change in wavelength, scattering with a change in wavelength is observed - the so-called Compton scattering. The phenomenon of X-ray diffraction, which proves their wave nature, was first experimentally discovered on crystals by German physicists M. Laue, W. Friedrich, and P. Knipping in 1912.

The crystal is a natural three-dimensional diffraction grating for X-rays, since the distance between the scattering centers (atoms) in the crystal is of the same order as the wavelength of X-rays (~1Å=10-8 cm). Diffraction of X-rays by crystals can be considered as the selective reflection of X-rays from systems of atomic planes of the crystal lattice. The direction of the diffraction maxima simultaneously satisfies three conditions determined by the Laue equations.
The diffraction pattern is obtained from a stationary crystal using X-ray radiation with a continuous spectrum (the so-called Lauegram) or from a rotating or oscillating crystal illuminated by monochromatic X-ray radiation, or from a polycrystal illuminated by monochromatic radiation. The intensity of the diffracted beam depends on the structure factor, which is determined by the atomic factors of the atoms of the crystal, their location inside the unit cell of the crystal, and the nature of the thermal vibrations of the atoms. The structure factor depends on the symmetry of the arrangement of atoms in the unit cell. The intensity of the diffracted beam depends on the size and shape of the object and on the perfection of the crystal.
Diffraction of X-rays from polycrystalline bodies results in the formation of cones of secondary rays. The axis of the cone is the primary beam, and the opening angle of the cone is 4J (J is the angle between the reflecting plane and the incident beam). Each cone corresponds to a specific family of crystal planes. All crystals, the family of planes of which are located at an angle J to the incident beam, participate in the creation of the cone. If the crystals are small and there are a very large number of them per unit volume, then the cone of rays will be continuous. In the case of texture, that is, the presence of a preferred orientation of the crystals, the diffraction pattern (x-ray pattern) will consist of unevenly blackened rings.

Unlike many speculations about the structure of the atom that were widespread at that time, Thomson’s model was based on physical facts that not only justified the model, but also gave certain indications of the number of corpuscles in an atom. The first such fact is the scattering of X-rays, or, as Thomson said, the occurrence of secondary X-rays. Thomson views X-rays as electromagnetic pulsations. When such pulsations fall on atoms containing electrons, the electrons, coming into accelerated motion, emit as described by the Larmor formula. The amount of energy emitted per unit time by electrons located in a unit volume will be

where N is the number of electrons (corpuscles) per unit volume. On the other hand, electron acceleration

where E p is the field strength of the primary radiation. Consequently, the intensity of the scattered radiation

Since the intensity of incident radiation according to Poynting’s theorem is equal to

then the ratio of scattered energy to primary

Charles Glover Barcla, who received the Nobel Prize in 1917 for the discovery of characteristic X-rays, was in 1899-1902. as a "research student" (graduate student) with Thomson at Cambridge, and here he became interested in X-rays. In 1902, he was a teacher at University College in Liverpool, and here in 1904, while studying secondary X-ray radiation, he discovered its polarization, which was quite consistent with Thomson's theoretical predictions. In the final experiment of 1906, Barkla caused the primary beam to be scattered by carbon atoms. The scattered beam fell perpendicular to the primary beam and was again scattered by carbon. This tertiary beam was completely polarized.

While studying the scattering of X-rays from light atoms, Barcla in 1904 found that the nature of the secondary rays was the same as the primary ones. For the ratio of the intensity of the secondary radiation to the primary one, he found a value independent of the primary radiation and proportional to the density of the substance:

From Thomson's formula

But density = n A / L, where A is the atomic weight of the atom, n is the number of atoms in 1 cm 3, L is Avogadro's number. Hence,

If we put the number of corpuscles in an atom equal to Z, then N = nZ and

If we substitute the values ​​of e, m, L to the right side of this expression, we will find K. In 1906, when the numbers e and m were not precisely known, Thomson found from Barkle’s measurements for air that Z = A, i.e. the number of corpuscles in an atom is equal to the atomic weight. The value of K obtained for light atoms by Barkle back in 1904 was K = 0.2. But in 1911, Barkla, using Bucherer’s updated data for e / m, the values ​​of e and L obtained Rutherford And Geiger, received K = 0.4, and therefore, Z = 1/2. As it turned out a little later, this relationship holds well in the region of light nuclei (with the exception of hydrogen).

Thomson's theory helped clarify a number of issues, but left even more questions unresolved. The decisive blow to this model was dealt by Rutherford's experiments in 1911, which will be discussed later.

A similar ring model of the atom was proposed in 1903 by a Japanese physicist Nagaoka. He suggested that at the center of the atom there is a positive charge, around which rings of electrons revolve, like the rings of Saturn. He managed to calculate the periods of oscillations performed by electrons with minor displacements in their orbits. The frequencies obtained in this way more or less approximately described the spectral lines of some elements *.

* (It should also be noted that the planetary model of the atom was proposed in 1901. J. Perrin. He mentioned this attempt in his Nobel lecture, given on December 11, 1926.)

On September 25, 1905, at the 77th Congress of German Naturalists and Doctors, V. Wien made a report on electrons. In this report, by the way, he said the following: “The explanation of spectral lines also poses a great difficulty for electronic theory. Since each element corresponds to a certain grouping of spectral lines that it emits while in a state of luminescence, each atom must represent an unchanging system. It would be easiest to think of the atom as a planetary system consisting of a positively charged center around which negative electrons revolve, like planets. But such a system cannot be unchanged due to the energy emitted by the electrons. Therefore, we are forced to turn to a system in which the electrons are in relative rest or have negligible speeds - a concept that contains a lot of doubtful things."

These doubts increased even more as new mysterious properties of radiation and atoms were discovered.