Application of x-ray diffraction analysis in molecular biology. X-ray structural analysis

X-rays, discovered in 1895 by V. Roentgen, are electromagnetic oscillations of a very small wavelength, comparable to atomic dimensions, arising when fast electrons act on matter.

X-rays are widely used in science and technology.

Their wave nature was established in 1912 by the German physicists M. Laue, W. Friedrich and P. Knipping, who discovered the phenomenon of X-ray diffraction on the atomic lattice of crystals. Directing a narrow beam of X-rays at a stationary crystal, they registered a diffraction pattern on a photographic plate placed behind the crystal, which consisted of a large number of regularly arranged spots. Each spot is a trace of a diffraction beam scattered by the crystal. A radiograph obtained by this method is called a Lauegram. This discovery was the basis x-ray diffraction analysis.

The wavelengths of X-rays used for practical purposes range from a few angstroms to fractions of an angstrom (Å), which corresponds to the energy of the electrons that cause X-rays from 10³ to 10 5 eV.

X-ray diffraction analysis is a method for studying the structure of bodies using the phenomenon of X-ray diffraction, a method for studying the structure of a substance by distribution in space and intensities of X-ray radiation scattered on the analyzed object. The diffraction pattern depends on the wavelength of the X-rays used and the structure of the object. To study the atomic structure, radiation with a wavelength of ~1Å is used, i.e. about the size of an atom.

Metals, alloys, minerals, inorganic and organic compounds, polymers, amorphous materials, liquids and gases, molecules of proteins, nucleic acids, etc. are studied by the methods of X-ray diffraction analysis. X-ray diffraction analysis is the main method for determining the structure of crystals. When examining crystals, it gives the most information. This is due to the fact that crystals have a strict periodicity in their structure and represent a diffraction grating for X-rays created by nature itself. However, it also provides valuable information in the study of bodies with a less ordered structure, such as liquids, amorphous bodies, liquid crystals, polymers, and others. On the basis of numerous already deciphered atomic structures, the inverse problem can also be solved: the crystalline composition of this substance can be established from the X-ray pattern of a polycrystalline substance, for example, alloy steel, alloy, ore, lunar soil, that is, a phase analysis is performed.

In the course of X-ray diffraction analysis, the sample under study is placed in the path of X-rays and the diffraction pattern resulting from the interaction of the rays with the substance is recorded. At the next stage of the study, the diffraction pattern is analyzed and the mutual arrangement of particles in space, which caused the appearance of this pattern, is established by calculation.

X-ray diffraction analysis of crystalline substances is divided into two stages.

1) Determining the size of the unit cell of a crystal, the number of particles (atoms, molecules) in the unit cell and the symmetry of the arrangement of particles (the so-called space group). These data are obtained by analyzing the geometry of the arrangement of diffraction peaks.

2) Calculation of the electron density inside the unit cell and determination of the coordinates of atoms, which are identified with the position of the electron density maxima. These data are obtained by analyzing the intensity of the diffraction peaks.

Methods of X-ray shooting of crystals.

There are various experimental methods for obtaining and recording a diffraction pattern. In any case, there is an X-ray source, a system for separating a narrow beam of X-rays, a device for fixing and orienting the sample in the beam, and a detector of radiation scattered by the sample. The receiver is a photographic film, or ionization or scintillation counters of X-ray quanta. The registration method using counters (diffractometric) provides a much higher accuracy in determining the intensity of the registered radiation.

It directly follows from the Wulf-Bragg condition that when registering a diffraction pattern, one of the two parameters included in it, ¾l - wavelength or q - angle of incidence, must be variable.

The main X-ray films of crystals are: the Laue method, the powder method (Debyegram method), the rotation method and its variation - the rocking method and various methods of the X-ray goniometer.

In the Laue method a beam of non-monochromatic (“white”) rays falls on a single-crystal sample (Fig.). Diffract only those rays whose wavelengths satisfy the Wulf-Bragg condition. The diffraction spots on the laugram (Fig.) are located along ellipses, hyperbolas and straight lines, necessarily passing through the spot from the primary beam.

Fig.– Scheme of the X-ray method according to Laue: 1- beam of X-rays incident on a single-crystal sample; 2 - collimator; 3 - sample; 4 - diffracted beams; 5 - flat film;

b – typical Lauegram.

An important property of the Lauegram is that, with the appropriate orientation of the crystal, the symmetry of the arrangement of these curves reflects the symmetry of the crystal. By the nature of the spots on the Laue patterns, one can reveal internal stresses and some other defects in the crystal structure. Indexing individual spots of the Lauegram is very difficult. Therefore, the Laue method is used exclusively to find the desired orientation of the crystal and determine its symmetry elements. This method checks the quality of single crystals when choosing a sample for a more complete structural study.

In the powder method(Fig), as well as in all other X-ray imaging methods described below, monochromatic radiation is used. The variable parameter is the angle q of incidence, since crystals of any orientation with respect to the direction of the primary beam are always present in a polycrystalline powder sample.

Fig - scheme of X-ray photography by the powder method: 1 - primary beam; 2 - powder or polycrystalline sample; 3 - photographic film rolled around the circumference; 4 - diffraction cones; 5 - "arcs" on the film, arising when its surface intersects with diffraction cones;

b – typical powder X-ray pattern (dibayegram).

The rays from all the crystals, in which the planes with the given interplanar distance d hk1 are in the "reflecting position", that is, they satisfy the Wulf-Bragg condition, form a cone with a raster angle of 4q around the primary ray. Each d hk1 corresponds to its own diffraction cone. The intersection of each cone of diffracted X-rays with a strip of photographic film rolled up in the form of a cylinder, the axis of which passes through the sample, leads to the appearance of traces on it that look like arches located symmetrically with respect to the primary beam (Fig.). Knowing the distances between the symmetrical "arcs", it is possible to calculate the corresponding interplanar distances d in the crystal.

The powder method is the simplest and most convenient from the point of view of experimental technique, however, the only information it provides - the choice of interplanar distances - allows one to decipher the simplest structures.

In rotation method(Fig.) the variable parameter is the angle q.

Shooting is done on a cylindrical film. During the entire exposure time, the crystal uniformly rotates around its axis, which coincides with some important crystallographic direction and with the axis of the cylinder formed by the bar. The diffraction rays travel along the generatrices of the cones, which, when crossing with the film, give lines consisting of spots (the so-called layer lines.

The rotation method provides the experimenter with richer information than the powder method. From the distances between the layer lines, one can calculate the lattice period in the direction of the crystal rotation axis.

Rice. – scheme of X-ray survey according to the rotation method: 1 – primary beam;

2 - sample (rotates in the direction of the arrow); 3 – cylindrical film;

b – typical X-ray of rotation.

The method under consideration simplifies the indexing of X-ray spots. So if the crystal rotates around the axis with lattices, then all spots on the line passing through the trace of the primary ray have indices (h,k,0), on layer lines adjacent to it, respectively (h,k,1) and (h,k,1 ¯) and so Further. However, the rotation method does not provide all possible information, so it is never known at what angle of rotation of the crystal around the rotation axis this or that diffraction spot was formed.

In swing method, which is a variant of the rotation method, the sample does not complete a full rotation, but "wobbles" around the same axis in a small angular interval. This facilitates the indexing of spots, since it makes it possible, as it were, to obtain an x-ray pattern of rotation in parts and to determine, to an accuracy of the rocking interval, at what angle of rotation of the crystal to the primary beam certain diffraction spots appeared.

Methods provide the richest information. X-ray goniometer. X-ray goniometer, a device with which you can simultaneously record the direction of X-rays diffracted on the test sample and the position of the sample at the time of occurrence of diffraction. One of them, the Weissenberg method, is a further development of the rotation method. In contrast to the latter, in the Weissenberg X-ray goniometer, all diffraction cones, except for one, are covered with a cylindrical screen, and the spots of the remaining diffraction cone (or, what is the same, a layer line) “unfold” over the entire area of the photographic film by its reciprocating axial movement synchronously with crystal rotation. This makes it possible to determine at what orientation of the crystal each spot of the Wassenbergogram appeared.

Rice. Schematic diagram of the Weissenberg X-ray goniometer: 1 - a fixed screen that passes only one diffraction cone; 2 - a crystal rotating around the X-X axis; 3 – cylindrical photographic film moving forward along the X – X axis synchronously with the rotation of crystal 2; 4 – diffraction cone passed through by a screen; 5 - primary beam.

There are other imaging methods that use the simultaneous simultaneous movement of the sample and photographic film. The most important of these are the method of photographing the reciprocal lattice and the precession method of Burger. All these methods use photographic registration of the diffraction pattern. In an X-ray diffractometer, it is possible to directly measure the intensity of diffraction reflections using proportional, scintillation, and other X-ray photon counters.

Application of X-ray diffraction analysis.

X-ray diffraction analysis makes it possible to objectively establish the structure of crystalline substances, including such complex ones as vitamins, antibiotics, coordination compounds, etc. A complete structural study of a crystal often makes it possible to solve purely chemical problems, for example, establishing or refining the chemical formula, type of bond, molecular weight at a known density or density at a known molecular weight, symmetry and configuration of molecules and molecular ions.

X-ray diffraction analysis is successfully used to study the crystalline state of polymers. Valuable information is also provided by X-ray diffraction analysis in the study of amorphous and liquid bodies. X-ray diffraction patterns of such bodies contain several blurred diffraction rings, the intensity of which rapidly decreases with increasing q. Based on the width, shape, and intensity of these rings, conclusions can be drawn about the features of the short-range order in a particular liquid or amorphous structure.

An important field of application of X-rays is the radiography of metals and alloys, which has become a separate branch of science. The concept of "radiography" includes, along with full or partial X-ray diffraction analysis, also other ways of using X-rays - X-ray flaw detection (transmission), X-ray spectral analysis, X-ray microscopy, and more. The structures of pure metals and many alloys have been determined. The crystal chemistry of alloys based on X-ray diffraction analysis is one of the leading branches of metal science. No state diagram of metal alloys can be considered reliably established if these alloys have not been studied by X-ray diffraction analysis. Thanks to the use of X-ray diffraction analysis methods, it has become possible to deeply study the structural changes that occur in metals and alloys during their plastic and heat treatment.

The method of X-ray diffraction analysis is also characterized by serious limitations. For a complete X-ray diffraction analysis, it is necessary that the substance crystallizes well and gives sufficiently stable crystals. Sometimes it is necessary to conduct research at high or low temperatures. This greatly complicates the experiment. A complete study is very time consuming, time consuming and involves a large amount of computational work.

To establish an atomic structure of medium complexity (~50–100 atoms in a unit cell), it is necessary to measure the intensities of several hundreds and even thousands of diffraction reflections. This very time-consuming and painstaking work is performed by computer-controlled automatic microdensitometers and diffractometers, sometimes for several weeks or even months (for example, in the analysis of protein structures, when the number of reflections increases to hundreds of thousands). In this regard, in recent years, high-speed computers have been widely used to solve problems of X-ray diffraction analysis. However, even with the use of computers, the determination of the structure remains a complex and time-consuming work. The use of several counters in the diffractometer, which can register reflections in parallel, can reduce the time of the experiment. Diffractometric measurements are superior to photographic recording in terms of sensitivity and accuracy.

Allowing you to objectively determine the structure of molecules and the general nature of the interaction of molecules in a crystal, X-ray diffraction analysis does not always make it possible to judge with the required degree of certainty the differences in the nature of chemical bonds within a molecule, since the accuracy of determining bond lengths and bond angles is often insufficient for this purpose. . A serious limitation of the method is also the difficulty of determining the positions of light atoms and especially hydrogen atoms.

The abstract was completed by a 2nd year student of the 2nd group Sapegina N.L.

Ministry of Health of Ukraine

National Pharmaceutical Academy of Ukraine

Department of Physics and Mathematics

Biophysics and physical methods of analysis course

Harkov city

Introduction

X-rays, discovered in 1895 by V. Roentgen, are electromagnetic oscillations of a very small wavelength, comparable to atomic dimensions, arising when fast electrons act on matter.

X-rays are widely used in science and technology.

X-ray spectra.

There are two types of radiation: bremsstrahlung and characteristic.

Bremsstrahlung occurs when electrons are decelerated by the anticathode of an X-ray tube. It decomposes into a continuous spectrum with a sharp boundary on the side of short wavelengths. The position of this boundary is determined by the energy of the electrons incident on the substance and does not depend on the nature of the substance. The intensity of the bremsstrahlung spectrum increases rapidly with decreasing mass of bombarding particles and reaches a significant value when excited by electrons.

Characteristic X-rays are formed when an electron is knocked out of one of the inner layers of an atom, followed by a transition to the vacated electron orbit from some outer layer. They have a line spectrum similar to the optical spectra of gases. However, there is a fundamental difference between those and other spectra: the structure of the characteristic spectrum of X-rays (number, relative arrangement and relative brightness of lines), in contrast to the optical spectrum of gases, does not depend on the substance (element) that gives this spectrum.

The spectral lines of the characteristic spectrum of x-rays form regular sequences or series. These series are denoted by the letters K, L, M, N…, and the wavelengths of these series increase from K to L, from L to M, etc. The presence of these series is closely related to the structure of the electron shells of atoms.

Characteristic X-ray spectra emit target atoms, in which an electron escapes from one of the inner shells (K-, L-, M-, ... shells) when colliding with a high-energy charged particle or a photon of primary X-ray radiation. The state of an atom with a vacancy in the inner shell (its initial state) is unstable. An electron from one of the outer shells can fill this vacancy, and the atom in this case passes into a final state with a lower energy (a state with a vacancy in the outer shell).

An atom can emit excess energy in the form of a photon of characteristic radiation. Since the energy E 1 of the initial and E 2 final states of the atom are quantized, an X-ray spectrum line appears with a frequency n=(E 1 - E 2)/h, where h is Planck's constant.

All possible radiative quantum transitions of an atom from the initial K-state form the hardest (short-wavelength) K-series. L-, M-, N-series are formed similarly (Fig. 1).

Rice. 1. Scheme of the K-, L-, M-levels of the atom and the main lines of the K-, L-series

The dependence on the substance manifests itself only in the fact that with an increase in the ordinal number of an element in the Mendeleev system, its entire characteristic X-ray spectrum shifts towards shorter wavelengths. G. Moseley in 1913 showed that the square root of the frequency (or reciprocal wavelength) of a given spectral line is linearly related to the atomic number of the element Z. Moseley's law played a very important role in the physical justification of Mendeleev's periodic system.

Another very important feature of the characteristic X-ray spectra is the circumstance that each element produces its own spectrum, regardless of whether this element is excited to emit X-rays in a free state or in a chemical compound. This feature of the characteristic spectrum of X-rays is used to identify various elements in complex compounds and is the basis of X-ray spectral analysis.

X-ray spectral analysis

X-ray spectral analysis is a branch of analytical chemistry that uses the X-ray spectra of elements for the chemical analysis of substances. X-ray spectral analysis by the position and intensity of the lines of the characteristic spectrum makes it possible to establish the qualitative and quantitative composition of a substance and serves for express non-destructive control of the composition of a substance.

In X-ray spectroscopy, to obtain the spectrum, the phenomenon of ray diffraction on crystals or, in the region of 15-150 Å, on diffraction bar gratings operating at small (1-12°) glancing angles, is used. The basis of high-resolution X-ray spectroscopy is the Wulf-Brag law, which relates the wavelength l of X-rays reflected from a crystal in the direction q to the interplanar spacing of the crystal d.

The angle q is called the slip angle. It is the direction of the rays incident on the crystal or reflected from it with the reflective surface of the crystal. The number n characterizes the so-called reflection order, in which, for given l and d, a diffraction maximum can be observed.

The oscillation frequency of X-rays (n=c/l) emitted by any element is linearly related to its atomic number:

Ö n/R=A(Z-s) (2)

where n is the radiation frequency, Z is the atomic number of the element, R is the Rydberg constant, equal to 109737.303 cm -1, s is the average screening constant, within small limits, depending on Z, A is a constant value for this line.

X-ray spectral analysis is based on the dependence of the emission frequency of the lines of the characteristic spectrum of an element on their atomic number and the relationship between the intensity of these lines and the number of atoms participating in the emission.

X-ray excitation of atoms of a substance can occur as a result of bombardment of the sample with high-energy electrons or when it is irradiated with x-rays. The first process is called direct excitation, the last is called secondary or fluorescent. In both cases, the energy of an electron or a quantum of primary X-ray radiation bombarding the radiating atom must be greater than the energy required to pull the electron out of a certain inner shell of the atom. Electron bombardment of the substance under study leads to the appearance not only of the characteristic spectrum of the element, but also, as a rule, of sufficiently intense continuous radiation. Fluorescent radiation contains only a line spectrum.

In the course of the primary excitation of the spectrum, an intense heating of the substance under study occurs, which is absent during the secondary excitation. The primary method of excitation of rays involves the placement of the test substance inside an X-ray tube evacuated to a high vacuum, while to obtain fluorescence spectra, the samples under study can be located in the path of the primary X-ray beam outside the vacuum and easily replace each other. Therefore, devices using fluorescence spectra (despite the fact that the intensity of the secondary radiation is thousands of times less than the intensity of the rays obtained by the primary method) have recently been almost completely replaced from practice by installations in which X-rays are excited using a stream of fast electrons.

Equipment for x-ray spectral analysis.

An X-ray fluorescence spectrometer (Fig. 2) consists of three main units: an X-ray tube, the radiation of which excites the fluorescence spectrum of the sample under study, a crystal analyzer for decomposing rays into a spectrum, and a detector for measuring the intensity of spectral lines.

Rice. Fig. 2. Scheme of an X-ray multichannel fluorescence spectrometer with flat (a) curved (b) crystals: 1 – X-ray tube; 2 – analyzed sample; 3 - Soller diaphragm; 4 - flat and curved (radius - 2R) crystal - analyzers; 5 – radiation detector; 6 - the so-called monitor, an additional recording device that allows measuring the relative intensity of spectral lines in the absence of stabilization of the intensity of the X-ray source; R is the radius of the so-called image circle.

In the spectrometer design most often used in practice, the radiation source and detector are located on the same circle, called the image circle, and the crystal is in the center. The crystal can rotate around an axis passing through the center of this circle. When the glide angle changes by q, the detector rotates through an angle of 2q

Along with flat-crystal spectrometers, focusing X-ray spectrometers operating "for reflection" (Kapitza-Johann and Johanson methods) and for "transmission" (Koush and Du-Mond methods) have become widespread. They can be single or multi-channel. Multichannel, so-called x-ray quantometers, autometers and others, allow you to simultaneously determine a large number of elements and automate the analysis process. usually they are equipped with special x-ray tubes and devices that provide a high degree of stabilization of x-ray intensity. The wavelength region in which the spectrometer can be used is determined by the interplanar spacing of the crystal-analyzer (d). In accordance with equation (1), the crystal cannot "reflect" rays whose wavelength exceeds 2d.

The number of crystals used in X-ray spectral analysis is quite large. The most commonly used are quartz, mica, gypsum and LiF.

As X-ray detectors, depending on the region of the spectrum, Geiger nets, proportional, crystal and scintillation quantum counters are successfully used.

Application of X-ray spectral analysis.

X-ray spectral analysis can be used for the quantitative determination of elements from Mg 12 to U 92 in materials of complex chemical composition - in metals and alloys, minerals, glass, ceramics, cements, plastics, abrasives, dust and various products of chemical technology. The most widely used X-ray spectral analysis is used in metallurgy and geology to determine macro- (1-100%) and micro-components (10 -1 - 10 -3%).

Sometimes, to increase the sensitivity of X-ray spectral analysis, it is combined with chemical and radiometric methods. The limiting sensitivity of X-ray spectral analysis depends on the atomic number of the element to be determined and the average atomic number of the sample to be determined. Optimal conditions are realized when determining the elements of the average atomic number in a sample containing light elements. The accuracy of X-ray spectral analysis is usually 2-5 relative percent, the weight of the sample is several grams. The duration of the analysis is from several minutes to 1 - 2 hours. The greatest difficulties arise in the analysis of elements with small Z and work in the soft region of the spectrum.

The results of the analysis are affected by the overall composition of the sample (absorption), the effects of selective excitation and absorption of radiation by satellite elements, as well as the phase composition and grain size of the samples.

X-ray spectral analysis has proven itself in the determination of Pb and Br in oil and gasoline, sulfur in gasoline, impurities in lubricants and wear products in machines, in the analysis of catalysts, in the implementation of express silicate analyzes, and others.

To excite soft radiation and use it in analysis, bombardment of samples with a-particles (for example, from a polonium source) is successfully used.

An important field of application of X-ray spectral analysis is the determination of the thickness of protective coatings without disturbing the surface of products.

In those cases where high resolution is not required in the separation of the characteristic radiation from the sample and the analyzed elements differ in atomic number by more than two, the crystalless method of X-ray spectral analysis can be successfully applied. It uses direct proportionality between the energy of a quantum and the amplitude of the pulse that it creates in a proportional or scintillation counter. This allows you to select and investigate the pulses corresponding to the spectral line of the element using an amplitude analyzer.

An important method of X-ray spectral analysis is the analysis of microvolumes of a substance.

The basis of the microanalyzer (Fig. 3) is a microfocus X-ray tube combined with an optical metal-microscope.

A special electron–optical system forms a thin electron probe that bombards a small, approximately 1–2 μm, area of the studied thin section, placed on the anode, and excites X-rays, the spectral composition of which is further analyzed using a spectrograph with a curved crystal. Such a device makes it possible to carry out X-ray spectral analysis of a thin section “at a point” for several elements or to investigate the distribution of one of them along a selected direction. In the raster microanalyzers created later, the electronic probe runs around a given surface area of the analyzed sample and makes it possible to observe on the monitor screen a picture of the distribution of chemical elements on the surface of a section that is magnified tenfold. There are both vacuum (for the soft region of the spectrum) and non-vacuum versions of such devices. The absolute sensitivity of the method is 10 -13 -10 -15 grams. With its help, the phase composition of alloyed alloys is successfully analyzed and their degree of homogeneity is studied, the distribution of alloying additives in alloys and their redistribution during aging, deformation or heat treatment are studied, the diffusion process and the structure of diffusion and other intermediate layers are studied, the processes accompanying processing and soldering of heat-resistant alloys, and also explore non-metallic objects in chemistry, mineralogy and geochemistry. In the latter case, a thin layer (50–100Å) of aluminum, beryllium, or carbon is preliminarily deposited on the surface of the thin sections.

Rice. 3. Scheme of the X-ray microanalyzer Castaing and Guinier:

1 - electron gun; 2 - diaphragm; 3 – the first converging electrostatic lens; 4 - aperture diaphragm; 5 - the second collecting electrostatic lens; 6 – test sample; 7 – X-ray spectrometer; 8 - mirror; 9 – objective of a metallographic optical microscope; HV - high voltage.

An independent section of X-ray spectral analysis is the study of the fine structure of X-ray absorption and emission spectra of atoms in chemical compounds and alloys. A detailed study of this phenomenon opens the way for an experimental study of the nature of interatomic interaction in chemical compounds, metals and alloys and the study of the energy structure of the electronic spectrum in them, the determination of effective charges concentrated on various atoms in molecules, and the solution of other problems of chemistry and physics of condensed matter.

X-ray diffraction analysis

X-ray diffraction analysis of crystalline substances is divided into two stages.

Determination of the size of the elementary cell of a crystal, the number of particles (atoms, molecules) in the elementary cell and the symmetry of the arrangement of particles (the so-called space group). These data are obtained by analyzing the geometry of the arrangement of diffraction peaks.

Calculation of the electron density inside the unit cell and determination of the coordinates of atoms, which are identified with the position of the electron density maxima. These data are obtained by analyzing the intensity of the diffraction peaks.

Methods of X-ray shooting of crystals.

It directly follows from the Wulf-Bragg condition that when registering a diffraction pattern, one of the two parameters included in it, ¾ l - wavelength or q - angle of incidence, must be variable.

The main X-ray films of crystals are: the Laue method, the powder method (Debyegram method), the rotation method and its variation - the rocking method and various methods of the X-ray goniometer.

In the Laue method, a beam of non-monochromatic (“white”) beams is incident on a single-crystal sample (Fig. 4a). Diffract only those rays whose wavelengths satisfy the Wulf-Bragg condition. Diffraction spots on the laugram (Fig. 4b) are located along ellipses, hyperbolas and straight lines, which necessarily pass through the spot from the primary beam.

Rice. 4. a - Scheme of the X-ray method according to Laue: 1 - beam of X-rays incident on a single-crystal sample; 2 - collimator; 3 - sample; 4 - diffracted beams; 5 - flat film;

b – typical Lauegram.

In the powder method (Fig. 5.a), as well as in all other X-ray imaging methods described below, monochromatic radiation is used. The variable parameter is the angle q of incidence, since crystals of any orientation with respect to the direction of the primary beam are always present in a polycrystalline powder sample.

Figure 5.a - scheme of X-ray photography by the powder method: 1 - primary beam; 2 - powder or polycrystalline sample; 3 - photographic film rolled around the circumference; 4 - diffraction cones; 5 - "arcs" on the film, arising when its surface intersects with diffraction cones;

b – typical powder X-ray pattern (dibayegram).

The rays from all the crystals, in which the planes with the given interplanar distance d hk1 are in the "reflecting position", that is, they satisfy the Wulf-Bragg condition, form a cone with a raster angle of 4q around the primary ray. Each d hk1 corresponds to its own diffraction cone. The intersection of each cone of diffracted X-rays with a strip of photographic film rolled up in the form of a cylinder, the axis of which passes through the sample, leads to the appearance of traces on it that look like arches located symmetrically with respect to the primary beam (Fig. 5.b). Knowing the distances between the symmetrical "arcs", it is possible to calculate the corresponding interplanar distances d in the crystal.

The powder method is the simplest and most convenient from the point of view of the experimental technique, however, the only information it provides is the choice of interplanar distances, which makes it possible to decipher the very simple structures.

In the rotation method (Fig. 6.a), the variable parameter is the angle q.

Shooting is done on a cylindrical film. During the entire exposure time, the crystal rotates uniformly around its axis, which coincides with some important crystallographic direction and with the axis of the cylinder formed by the bar. The diffraction rays go along the generatrices of the cones, which, when crossing with the film, give lines consisting of spots (the so-called layer lines (Fig. 6.b).

Rice. 6.a - scheme of X-ray survey according to the rotation method: 1 - primary beam;

2 - sample (rotates in the direction of the arrow); 3 – cylindrical film;

b – typical X-ray of rotation.

The method under consideration simplifies the indexing of X-ray spots. So if the crystal rotates around the axis from the lattice, then all spots on the line passing through the trace of the primary beam have indices (h, k, 0), on layer lines adjacent to it - respectively (h, k, 1) and (h, k,1 ¯) and so on. However, the rotation method does not provide all possible information, so it is never known at what angle of rotation of the crystal around the rotation axis this or that diffraction spot was formed.

In the rocking method, which is a variant of the rotation method, the sample does not rotate completely, but "rocks" around the same axis in a small angular interval. This facilitates the indexing of spots, since it makes it possible, as it were, to obtain an x-ray pattern of rotation in parts and to determine, to an accuracy of the rocking interval, at what angle of rotation of the crystal to the primary beam certain diffraction spots appeared.

The methods of X-ray goniometer provide the richest information. X-ray goniometer, a device with which you can simultaneously record the direction of X-rays diffracted on the test sample and the position of the sample at the time of occurrence of diffraction. One of them, the Weissenberg method, is a further development of the rotation method. In contrast to the latter, in the Weissenberg X-ray goniometer (Fig. 7), all diffraction cones, except for one, are covered with a cylindrical screen, and the spots of the remaining diffraction cone (or, what is the same, a layer line) “unroll” over the entire area of the photographic film by means of its return translational axial movement synchronously with the rotation of the crystal. This makes it possible to determine at what orientation of the crystal each spot of the Wassenbergogram appeared.

Rice. Fig. 7. Schematic diagram of the Weissenberg X-ray goniometer: 1 - a fixed screen that passes only one diffraction cone; 2 - a crystal rotating around the X-X axis; 3 – cylindrical photographic film moving forward along the X – X axis synchronously with the rotation of crystal 2; 4 – diffraction cone passed through by a screen; 5 - primary beam.

Application of X-ray diffraction analysis.

The method of X-ray diffraction analysis also has serious limitations. For a complete X-ray diffraction analysis, it is necessary that the substance crystallizes well and gives sufficiently stable crystals. Sometimes it is necessary to conduct research at high or low temperatures. This greatly complicates the experiment. A complete study is very time consuming, time consuming and involves a large amount of computational work.

Bibliography

Zhdanov G.S. Solid State Physics, Moscow, 1962.

Blokhin M.A., Physics of X-rays, 2nd ed., M., 1957.

Blokhin M.A., Methods of X-ray spectral studies, M., 1959.

Vanshtein E.E., X-ray spectra of atoms in molecules of chemical compounds and in alloys, M.-L., 1950.

Bokay G.B., Poray-Koshits M.A., X-ray diffraction analysis, M., 1964.

Shishakov N.A., Basic concepts of structural analysis, M., 1961.

X-ray structural analysis

methods for studying the structure of matter by distribution in space and intensities of X-ray radiation scattered on the analyzed object. R. s. a. along with neutron diffraction (See Neutron diffraction) and electron diffraction (See Electron diffraction) is a diffraction structural method; it is based on the interaction of X-rays with the electrons of matter, which results in X-ray diffraction. The diffraction pattern depends on the wavelength of the X-rays used (See X-rays) and the structure of the object. To study the atomic structure, radiation with a wavelength of X-ray structural analysis of 1 Å, i.e., of the order of the size of atoms, is used. R.'s methods with. a. study metals, alloys, minerals, inorganic and organic compounds, polymers, amorphous materials, liquids and gases, protein molecules, nucleic acids, etc. Most successfully R. with. a. used to establish the atomic structure of crystalline bodies. This is due to the fact that Crystals have a strict periodicity in their structure and represent a diffraction grating for X-rays created by nature itself.

History reference. The diffraction of X-rays by crystals was discovered in 1912 by the German physicists M. Laue, W. Friedrich, and P. Knipping. Directing a narrow beam of X-rays at a stationary crystal, they registered a diffraction pattern on a photographic plate placed behind the crystal, which consisted of a large number of regularly arranged spots. Each spot is a trace of a diffraction beam scattered by the crystal. radiograph , obtained by this method is called the Lauegram (See Lauegram) ( rice. one ).

The theory of X-ray diffraction on crystals developed by Laue made it possible to relate the wavelength λ of radiation, the parameters of the unit cell of the crystal a, b, c(see Crystal lattice) , angles of the incident (α 0 , β 0 , γ 0) and diffraction (α, β, γ) beams by the ratios:

a(cosα - cosα 0) = hλ ,

b(cosβ - cosβ 0) = kλ, (1)

c(cosγ - cosγ 0) = lλ ,

In the 50s. R.'s methods of page began to develop rapidly. a. with the use of computers in the technique of the experiment and in the processing of x-ray diffraction information.

Experimental methods R. with. a. X-ray cameras and X-ray diffractometers are used to create conditions for diffraction and registration of radiation. The scattered X-ray radiation in them is recorded on photographic film or measured by nuclear radiation detectors. Depending on the state of the sample being studied and its properties, as well as on the nature and amount of information that must be obtained, various methods of R. s are used. a. Single crystals selected for the study of the atomic structure must have dimensions X-ray structural analysis 0.1 mm and, if possible, have a perfect structure. The study of defects in relatively large, almost perfect crystals is carried out by X-ray topography, which is sometimes referred to as x-ray topography. a.

The Laue method is the simplest method for obtaining X-ray patterns from single crystals. The crystal in Laue's experiment is stationary, and the X-rays used have a continuous spectrum. Location of diffraction spots on the Laue patterns ( rice. one ) depends on the symmetry of the crystal and its orientation with respect to the incident beam. The Laue method makes it possible to establish whether a crystal under study belongs to one and 11 Laue symmetry groups and to orient it (i.e., determine the direction of the crystallographic axes) with an accuracy of several arc minutes. By the nature of the spots on the Lauegrams, and especially by the appearance of Asterism a, one can reveal internal stresses and some other defects in the crystal structure. The Laue method checks the quality of single crystals when choosing a sample for its more complete structural study.

Sample rocking and rotation methods are used to determine the repeat periods (lattice constant) along the crystallographic direction in a single crystal. They allow, in particular, to set parameters a, b, c unit cell of a crystal. This method uses monochromatic X-ray radiation, the sample is brought into oscillatory or rotational motion around an axis coinciding with the crystallographic direction, along which the repeat period is examined. The spots on the rocking and rotation radiographs obtained in cylindrical cassettes are located on a family of parallel lines. The distances between these lines, the radiation wavelength, and the diameter of the X-ray camera cassette make it possible to calculate the required repetition period in the crystal. The Laue conditions for diffraction rays in this method are satisfied by changing the angles included in relations (1) during rocking or rotation of the sample.

X-ray methods. For a complete study of the structure of a single crystal by X-ray methods. a. it is necessary not only to establish the position, but also to measure the intensities of as many diffraction reflections as possible, which can be obtained from the crystal at a given radiation wavelength and all possible orientations of the sample. To do this, the diffraction pattern is recorded on photographic film in an X-ray goniometer (See X-ray goniometer) and measured using a Microphotometer a the degree of blackening of each spot on the x-ray. In an X-ray diffractometer, one can directly measure the intensity of diffraction reflections using proportional, scintillation, and other X-ray photon counters. To have a complete set of reflections, X-ray goniometers take a series of X-ray patterns. On each of them, diffraction reflections are recorded, the Miller indices of which are subject to certain restrictions (for example, reflections of the type hk 0, hk 1 etc.). Most often, an X-ray goniometric experiment is performed using the Weisenberg methods. Burger ( rice. 2 ) and de Jong-Bowman. The same information can be obtained with the help of rocking radiographs.

To establish an atomic structure of medium complexity (X-ray structural analysis of 50-100 atoms in a unit cell), it is necessary to measure the intensities of several hundreds and even thousands of diffraction reflections. This very time-consuming and painstaking work is performed by automatic microdensitometers and computer-controlled diffractometers, sometimes for several weeks or even months (for example, in the analysis of protein structures, when the number of reflections increases to hundreds of thousands). By using several counters in the diffractometer, which can register reflections in parallel, the time of the experiment can be significantly reduced. Diffractometric measurements are superior to photographic recording in terms of sensitivity and accuracy.

Method for the study of polycrystals (Debye - Scherrer method). Metals, alloys, crystalline powders consist of many small single crystals of a given substance. For their study, monochromatic radiation is used. The X-ray pattern (Debyegram) of polycrystals consists of several concentric rings, each of which merges reflections from a certain system of planes of differently oriented single crystals. Debyegrams of various substances have an individual character and are widely used to identify compounds (including those in mixtures). R.s.a. polycrystals allows you to determine the phase composition of the samples, determine the size and preferred orientation (texturing) of grains in the substance, control the stresses in the sample and solve other technical problems.

Study of amorphous materials and partially ordered objects. A clear X-ray pattern with sharp diffraction maxima can only be obtained with a complete three-dimensional periodicity of the sample. The lower the degree of ordering of the atomic structure of the material, the more blurred, diffuse character is the X-ray radiation scattered by it. The diameter of a diffuse ring in an X-ray diffraction pattern of an amorphous substance can serve as a rough estimate of the average interatomic distances in it. With an increase in the degree of order (see Long-Range Order and Short-Range Order) in the structure of objects, the diffraction pattern becomes more complicated and, consequently, contains more structural information.

The small-angle scattering method makes it possible to study the spatial inhomogeneities of a substance, the dimensions of which exceed the interatomic distances, i.e. range from 5-10 Å to X-ray structural analysis 10,000 Å. The scattered X-ray radiation in this case is concentrated near the primary beam - in the region of small scattering angles. Small-angle scattering is used to study porous and finely dispersed materials, alloys and complex biological objects: viruses, cell membranes, chromosomes. For isolated protein molecules and nucleic acids, the method allows determining their shape, size, molecular weight; in viruses - the nature of the mutual stacking of their components: protein, nucleic acids, lipids; in synthetic polymers - packing of polymer chains; in powders and sorbents - the distribution of particles and pores by size; in alloys - the occurrence and size of phases; in textures (in particular, in liquid crystals) - the form of packing of particles (molecules) into various kinds of supramolecular structures. The X-ray small-angle method is also used in industry to control the processes of manufacturing catalysts, fine coals, etc. Depending on the structure of the object, measurements are made for scattering angles from fractions of a minute to several degrees.

Determination of the atomic structure from X-ray diffraction data. Deciphering the atomic structure of a crystal includes: establishing the size and shape of its elementary cell; determination of whether a crystal belongs to one of the 230 Fedorov (discovered by E. S. Fedorov (see Fedorov)) crystal symmetry groups (see Crystal symmetry); obtaining the coordinates of the basic atoms of the structure. The first and partially second problems can be solved by the Laue methods and rocking or rotation of the crystal. It is possible to finally establish the symmetry group and coordinates of the basic atoms of complex structures only with the help of complex analysis and laborious mathematical processing of the intensity values of all diffraction reflections from a given crystal. The ultimate goal of such processing is to calculate the values of the electron density ρ( x, y, z) at any point of the crystal cell with coordinates x, y, z. The periodicity of the crystal structure allows us to write the electron density in it through the Fourier series :

where V- unit cell volume, Fhkl- Fourier coefficients, which in R. s. a. are called structural amplitudes, i= hkl and is related to the diffraction reflection, which is determined by conditions (1). The purpose of summation (2) is to mathematically assemble the X-ray diffraction reflections to obtain an image of the atomic structure. To produce in this way image synthesis in R. s. a. This is due to the lack of lenses for x-rays in nature (in visible light optics, a converging lens serves for this).

Diffraction reflection is a wave process. It is characterized by an amplitude equal to ∣ Fhkl∣, and phase α hkl(by the phase shift of the reflected wave with respect to the incident), through which the structural amplitude is expressed: Fhkl=∣Fhkl∣(cosα hkl +i sinα hkl). The diffraction experiment makes it possible to measure only reflection intensities proportional to ∣ Fhkl∣ 2 , but not their phases. Phase determination is the main problem in deciphering the crystal structure. The determination of the phases of structural amplitudes is fundamentally the same for both crystals consisting of atoms and for crystals consisting of molecules. Having determined the coordinates of atoms in a molecular crystalline substance, it is possible to isolate its constituent molecules and establish their size and shape.

It is easy to solve the problem that is the reverse of the structural interpretation: the calculation of the known atomic structure of structural amplitudes, and from them - the intensities of diffraction reflections. The trial and error method, historically the first method of deciphering structures, consists in comparing experimentally obtained ∣ Fhkl∣ exp, with values calculated on the basis of the trial model ∣ Fhkl∣ calc. Depending on the value of the divergence factor

A fundamentally new way to deciphering the atomic structures of single crystals was opened by the use of the so-called. Paterson functions (functions of interatomic vectors). To construct the Paterson function of some structure consisting of N atoms, we move it parallel to itself so that the first atom hits the fixed origin first. Vectors from the origin to all atoms of the structure (including a vector of zero length to the first atom) will indicate the position N maxima of the function of interatomic vectors, the totality of which is called the image of the structure in the atom 1. Let's add more to them N maxima, the position of which will indicate N vectors from the second atom placed at the parallel transfer of the structure to the same origin. After doing this procedure with all N atoms ( rice. 3 ), we'll get N 2 vectors. The function describing their position is the Paterson function.

For the Paterson function R(u, υ, ω) (u, υ, ω - coordinates of points in the space of interatomic vectors), one can obtain the expression:

from which it follows that it is determined by the moduli of structural amplitudes, does not depend on their phases, and, therefore, can be calculated directly from the data of a diffraction experiment. Difficulty in interpreting a function R(u, υ, ω) consists in the need to find the coordinates N atoms from N 2 her maxima, many of which merge due to overlaps that arise when constructing the function of interatomic vectors. The easiest to decrypt R(u, υ, ω) the case when the structure contains one heavy atom and several light ones. The image of such a structure in a heavy atom will differ significantly from other images of it. Among the various methods that make it possible to determine the model of the structure under study by the Paterson function, the most effective were the so-called superposition methods, which made it possible to formalize its analysis and perform it on a computer.

Methods of the Paterson function encounter serious difficulties in studying the structures of crystals consisting of identical or similar atoms in atomic number. In this case, the so-called direct methods for determining the phases of structural amplitudes turned out to be more effective. Taking into account the fact that the value of the electron density in a crystal is always positive (or equal to zero), one can obtain a large number of inequalities to which the Fourier coefficients (structural amplitudes) of the function ρ( x, y, z). Using the methods of inequalities, it is relatively easy to analyze structures containing up to 20–40 atoms in the unit cell of a crystal. For more complex structures, methods based on a probabilistic approach to the problem are used: structural amplitudes and their phases are considered as random variables; distribution functions of these random variables are derived from physical representations, which make it possible to estimate, taking into account the experimental values of the moduli of structural amplitudes, the most probable values of the phases. These methods are also implemented on a computer and make it possible to decipher structures containing 100–200 or more atoms in a unit cell of a crystal.

So, if the phases of the structural amplitudes are established, then the electron density distribution in the crystal can be calculated from (2), the maxima of this distribution correspond to the position of the atoms in the structure ( rice. 4 ). The final refinement of the coordinates of atoms is carried out on a computer Least squares method om and, depending on the quality of the experiment and the complexity of the structure, makes it possible to obtain them with an accuracy of up to thousandths of an Å (with the help of a modern diffraction experiment, one can also calculate the quantitative characteristics of thermal vibrations of atoms in a crystal, taking into account the anisotropy of these vibrations). R. s. a. makes it possible to establish more subtle characteristics of atomic structures, for example, the distribution of valence electrons in a crystal. However, this complex problem has so far been solved only for the simplest structures. For this purpose, a combination of neutron diffraction and X-ray diffraction studies is very promising: neutron diffraction data on the coordinates of atomic nuclei are compared with the spatial distribution of the electron cloud obtained using X-ray diffraction. a. To solve many physical and chemical problems, X-ray diffraction studies and resonance methods are jointly used.

The pinnacle of R.'s achievements. a. - deciphering the three-dimensional structure of proteins, nucleic acids and other macromolecules. Proteins in natural conditions, as a rule, do not form crystals. To achieve a regular arrangement of protein molecules, proteins are crystallized and then their structure is examined. The phases of the structural amplitudes of protein crystals can only be determined as a result of the joint efforts of radiographers and biochemists. To solve this problem, it is necessary to obtain and study crystals of the protein itself, as well as its derivatives with the inclusion of heavy atoms, and the coordinates of the atoms in all these structures must coincide.

About numerous applications of methods of R. of page. a. to study various violations of the structure of solids under the influence of various influences, see Art. Radiography of materials.

Lit.: Belov N.V., Structural crystallography, Moscow, 1951; Zhdanov G. S., Fundamentals of X-ray diffraction analysis, M. - L., 1940; James R., Optical principles of X-ray diffraction, trans. from English, M., 1950; Boky G. B., Poray-Koshits M. A., X-ray analysis, M., 1964; Poray-Koshits M.A., Practical course of X-ray diffraction analysis, M., 1960: Kitaygorodsky A.I., Theory of structural analysis, M., 1957; Lipeon G., Cochran V., Determination of the structure of crystals, trans. from English, M., 1961; Weinshtein B.K., Structural electron diffraction, M., 1956; Bacon, J., Neutron Diffraction, trans. from English, M., 1957; Burger M., Structure of crystals and vector space, transl. from English, M., 1961; Guinier A., X-ray diffraction of crystals, trans. from French, Moscow, 1961; Woolfson M. M., An introduction to X-ray crystallography, Camb., 1970: Ramachandran G. N., Srinivasan R., Fourier methode in crystallography, N. Y., 1970; Crystallographic computing, ed. F. R. Ahmed, Cph., 1970; Stout G. H., Jensen L. H., X-ray structure determination, N. Y. - L., .

V. I. Simonov.

Rice. 9. a. Projection onto the ab plane of the function of interatomic vectors of the mineral baotite O 16 Cl]. The lines are drawn through the same intervals of values of the function of interatomic vectors (lines of equal level). b. The projection of the electron density of baotite onto the ab plane, obtained by deciphering the function of interatomic vectors (a). The electron density maxima (clumps of lines of equal level) correspond to the positions of atoms in the structure. in. Image of a model of the atomic structure of baotite. Each Si atom is located inside a tetrahedron formed by four O atoms; Ti and Nb atoms are in octahedra composed of O atoms. SiO 4 tetrahedra and Ti(Nb)O 6 octahedra in the baotite structure are connected as shown in the figure. Part of the unit cell of the crystal corresponding to Fig. a and b are marked with a dashed line. Dotted lines in fig. a and b determine the zero levels of the values of the corresponding functions.

Physical Encyclopedia - X-RAY STRUCTURAL ANALYSIS, study of the atomic structure of a sample of a substance according to the X-ray diffraction pattern on it. Allows you to establish the distribution of the electron density of a substance, which determines the type of atoms and their ... ... Illustrated Encyclopedic Dictionary

- (X-ray diffraction analysis), a set of methods for studying the atomic structure of a substance using X-ray diffraction. According to the diffraction pattern, the distribution of the electron density of the substance is established, and according to it the type of atoms and their ... ... encyclopedic Dictionary

- (X-ray structural analysis), research method atomic mol. buildings in c, ch. arr. crystals, based on the study of diffraction arising from the interaction. with the test sample of X-ray radiation of a wavelength of approx. 0.1 nm. Use Ch. arr… Chemical Encyclopedia - (see X-RAY STRUCTURAL ANALYSIS, NEUTRONOGRAPHY, ELECTRONOGRAPHY). Physical Encyclopedic Dictionary. Moscow: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1983... Physical Encyclopedia

Determination of the structure in in and materials, i.e., finding out the location in space of their constituent structural units (molecules, ions, atoms). In the narrow sense, S. a. determination of the geometry of molecules and mol. systems, which are usually described by a set of lengths ... ... Chemical Encyclopedia

Brest, 2010

Three methods are mainly used in X-ray diffraction analysis

1. Laue method. In this method, a radiation beam with a continuous spectrum is incident on a stationary single crystal. The diffraction pattern is recorded on a still photographic film.

2. Single crystal rotation method. A beam of monochromatic radiation is incident on a crystal rotating (or oscillating) around a certain crystallographic direction. The diffraction pattern is recorded on a still photographic film. In a number of cases, the film moves synchronously with the rotation of the crystal; this variation of the rotation method is called the layered line sweep method.

3. Method of powders or polycrystals (Debye-Scherrer-Hull method). This method uses a monochromatic beam of rays. The sample consists of a crystalline powder or is a polycrystalline aggregate.

Laue method

The Laue method is used at the first stage of studying the atomic structure of crystals. It is used to determine the syngony of the crystal and the Laue class (the Friedel crystal class up to the center of inversion). According to Friedel's law, it is never possible to detect the absence of a center of symmetry on a Lauegram, and therefore adding a center of symmetry to the 32 crystal classes reduces their number to 11. The Laue method is mainly used to study single crystals or coarse-grained samples. In the Laue method, a stationary single crystal is illuminated by a parallel beam of rays with a continuous spectrum. The sample can be either an isolated crystal or a fairly large grain in a polycrystalline aggregate.

The formation of a diffraction pattern occurs during the scattering of radiation with wavelengths from l min \u003d l 0 \u003d 12.4 / U, where U is the voltage on the X-ray tube, to l m - the wavelength that gives the intensity of the reflection (diffraction maximum) exceeding the background at least by 5 %. lm depends not only on the intensity of the primary beam (atomic number of the anode, voltage and current through the tube), but also on the absorption of X-rays in the sample and the film cassette. The spectrum l min - l m corresponds to a set of Ewald spheres with radii from 1/ l m to 1/l min , which touch the node 000 and OR of the crystal under study (Fig. 1).

Then, for all OR nodes lying between these spheres, the Laue condition will be satisfied (for a certain wavelength in the interval (l m ¸ l min)) and, consequently, a diffraction maximum appears - a reflection on the film. For shooting according to the Laue method, a RKSO camera is used (Fig. 2).

Rice. 2 Chamber RKSO

Here, the primary X-ray beam is cut out by aperture 1 with two holes 0.5–1.0 mm in diameter. The aperture size of the diaphragm is chosen so that the cross section of the primary beam is greater than the cross section of the crystal under study. Crystal 2 is mounted on goniometric head 3, which consists of a system of two mutually perpendicular arcs. The crystal holder on this head can move relative to these arcs, and the goniometric head itself can be rotated through any angle around an axis perpendicular to the primary beam. The goniometric head makes it possible to change the orientation of the crystal with respect to the primary beam and set a certain crystallographic direction of the crystal along this beam. The diffraction pattern is recorded on photographic film 4 placed in a cassette, the plane of which is perpendicular to the primary beam. On the cassette in front of the film is a thin wire stretched parallel to the axis of the goniometric head. The shadow of this wire makes it possible to determine the orientation of the film with respect to the axis of the goniometric head. If sample 2 is placed in front of film 4, then the X-ray patterns obtained in this way are called Laue patterns. The diffraction pattern recorded on a photographic film located in front of the crystal is called an epigram. On Lauegrams, diffraction spots are located along zonal curves (ellipses, parabolas, hyperbolas, straight lines). These curves are plane sections of the diffraction cones and touch the primary spot. On epigrams, diffraction spots are located along hyperbolas that do not pass through the primary beam.

To consider the features of the diffraction pattern in the Laue method, a geometric interpretation is used using a reciprocal lattice. Lauegrams and epigrams are a reflection of the reciprocal lattice of a crystal. The gnomonic projection constructed according to the Lauegram makes it possible to judge the mutual arrangement of the normals to the reflecting planes in space and to get an idea of the symmetry of the crystal reciprocal lattice. The shape of the Lauegram spots is used to judge the degree of perfection of the crystal. A good crystal gives clear spots on the Lauegram. The symmetry of crystals according to the Lauegram is determined by the mutual arrangement of spots (the symmetrical arrangement of atomic planes must correspond to the symmetrical arrangement of reflected rays). (See fig. 3)

Rice. Fig. 3 Scheme of taking X-ray images according to the Laue method (a - in transmission, b - in reflection, F - focus of the X-ray tube, K - aperture, O - sample, Pl - film)

Single crystal rotation method

The rotation method is the main one in determining the atomic structure of crystals. This method determines the size of the unit cell, the number of atoms or molecules per cell. The space group is found from the extinction of the reflections (accurate to the center of inversion). Data from the measurement of the intensity of the diffraction peaks are used in calculations related to the determination of the atomic structure. When taking X-ray images by the rotation method, the crystal rotates or oscillates around a certain crystallographic direction when it is irradiated with monochromatic or characteristic X-rays. The primary beam is cut out by a diaphragm (with two round holes) and enters the crystal. The crystal is mounted on the goniometric head so that one of its important directions (such as , , ) is oriented along the axis of rotation of the goniometric head. The goniometric head is a system of two mutually perpendicular arcs, which allows you to set the crystal at the desired angle with respect to the axis of rotation and to the primary x-ray beam. The goniometric head is driven into slow rotation through a system of gears with the help of a motor. The diffraction pattern is recorded on a photographic film located along the axis of the cylindrical surface of a cassette of a certain diameter (86.6 or 57.3 mm).

In the absence of an external cut, the crystals are oriented by the Laue method. For this purpose, it is possible to install a cassette with a flat film in the rotation chamber. The diffraction maxima on the X-ray pattern of rotation are located along straight lines, called layer lines. The maxima on the radiograph are located symmetrically with respect to the vertical line passing through the primary spot. Rotational X-ray diffraction patterns often show continuous bands passing through diffraction maxima. The appearance of these bands is due to the presence of a continuous spectrum in the X-ray tube radiation along with the characteristic spectrum.

When the crystal rotates around the main crystallographic direction, the reciprocal lattice associated with it rotates. When the nodes of the reciprocal lattice cross the propagation sphere, diffraction rays arise, which are located along the generatrix of the cones, the axes of which coincide with the axis of rotation of the crystal. All nodes of the reciprocal lattice intersected by the propagation sphere during its rotation constitute the effective region, i.e. determine the region of indices of diffraction maxima arising from a given crystal during its rotation. To establish the atomic structure of a substance, it is necessary to indicate the X-ray patterns of rotation. Indexing is usually done graphically using reciprocal lattice representations. The rotation method determines the crystal lattice periods, which, together with the angles determined by the Laue method, make it possible to find the unit cell volume. Using data on the density, chemical composition and volume of the unit cell, the number of atoms in the unit cell is found.

Powder method

In the usual method of studying polycrystalline materials, a thin column of ground powder or other fine-grained material is illuminated with a narrow beam of X-rays with a certain wavelength. The ray diffraction pattern is fixed on a narrow strip of photographic film rolled up in the form of a cylinder, along the axis of which the sample under study is located. Relatively less common is shooting on flat photographic film.

The schematic diagram of the method is given in fig. 4.

Rice. 4 Schematic diagram of powder shooting:

1 - diaphragm; 2 - the place of entry of rays;

3 - sample: 4 - place where the rays exit;

5 - camera body; 6 - (photographic film)

When a beam of monochromatic rays is incident on a sample consisting of many small crystals with various orientations, then the sample will always contain a known number of crystals, which will be located in such a way that some groups of planes will form an angle q with the incident beam, which satisfies the conditions of reflection.

15.1 Physical features of X-ray diffraction analysis

X-ray diffraction analysis is based on the phenomenon of X-ray diffraction, which occurs when X-rays are scattered by crystalline substances. They study the arrangement of atoms in crystalline materials and the processes associated with the rearrangement of atoms in crystals. With the help of X-ray diffraction analysis, state diagrams of alloys are studied, internal stresses, dimensions and orientation of crystallites are determined, the decomposition of supersaturated solid solutions, and many other practically important problems are solved.

X-ray diffraction analysis is widely used in the study of structural imperfections in crystals, the presence of which determines many properties of materials. X-ray diffraction makes it possible to study the mosaic structure of crystals, reveal dislocations, determine the sizes of substructural components, their misorientation, and the type of subgrain boundaries.

X-ray diffraction methods for studying the crystal structure of solids have played an important role in the development of materials science. The method of X-ray diffraction made it possible to determine the atomic-crystal structure of solids and to study the stable and metastable states of metals and alloys, as well as the phenomena that occur during their thermal and mechanical processing, and, thus, to understand the mechanism of structural processes.

A large number of works have been carried out in order to establish a relationship between the atomic-crystal structure and the properties of materials. As a result, atomic-crystal structure data have become a necessary characteristic of materials. Structural characteristics calculated from X-ray diffraction data are widely used in the development of metal processing modes and for the control of technological processes.

The techniques of X-ray diffraction analysis are diverse, which makes it possible to obtain rich information about various details of the structure of materials and its changes during various processing methods.

X-rays are produced when matter is bombarded by fast moving electrons. Diffraction methods use X-rays with a wavelength of about 10 -10 m = 10 -8 cm = 0.1 nm, which is approximately equal to the interatomic distances in a crystalline substance.

For X-ray diffraction, a potential difference of up to 50 kV is used. at the moment the electron reaches the anode, the energy of the electrons will be equal to eU, where e is the charge of the electron, U is the potential difference applied to the electrodes.

When electrons decelerate in the target - the anode mirror, the electron will lose energy E 1 - E 2, where e and E 2 are the electron energies before and after the collision. If braking occurs fast enough, then this energy loss will turn into radiation in accordance with the law:

hν = E 1 – E 2 , (15.1)

where h is Planck's constant; ν is the frequency of the emitted x-rays.

If an electron loses all of its energy in one collision, then
the maximum frequency of the generated radiation is determined by the equation:
hνmax = eU. (15.2)

Since , where c is the speed of light, λ is the wavelength of radiation, it follows from here that the minimum value of the wavelength will be equal to:

At U = 50 kV, the length λ min is approximately equal to 0.025 nm. In most cases, on its way, an electron collides with several atoms, losing part of the energy with each collision, and thus generating several photons, and each of them corresponds to a wave whose length exceeds λ min.

Thus, white radiation is formed - a continuous (continuous) spectrum, which has a sharp boundary in the short-wavelength part and only gradually decreases towards longer wavelengths. Figure 15.1.

In fact, less than 1% of the kinetic energy of electrons is converted into X-rays. The efficiency of this transformation depends on the substance of the anode mirror and increases with an increase in the atomic number Z of its constituent atoms. Combining this effect with that obtained by increasing the voltage U, it can be established that the total X-ray intensity is approximately proportional to ZU 2 .

For tubes with a tungsten anode at U = 20 kV η = 0.12%, at U = 50 kV η = 0.27%. The extremely small η excitations of the continuous spectrum at a relatively low voltage are explained by the fact that most of the electrons (≈99%) gradually waste their energy when interacting with the atoms of the anode material to ionize them and increase the anode temperature.

At a certain accelerating voltage, X-ray characteristic radiation occurs. Figure 15.2.

Figure 15.1. Continuous spectrum obtained from

tungsten target

Figure 15.2. K spectra of Mo and Cu at 35 kV,

The α-line is a doublet.

The intensity of these lines can be hundreds of times greater than the intensity of any other continuous spectrum line in the same wavelength range. Characteristic radiation occurs when an incident electron has a sufficiently high energy to knock out an electron from one of the inner electron shells of an anode mirror atom, and the resulting vacancy is occupied by an electron from a higher energy level, the excess energy is realized in the form of radiation. The wavelength of the emitted wave is determined by the energy difference between these two levels, and thus, an increase in voltage, although it contributes to an increase in intensity, does not change the wavelength of the characteristic radiation of the anode.

The spectra of characteristic waves are quite simple and are classified in ascending order of wavelengths K, L, M - series in accordance with the level from which the electron was knocked out. K-series lines are obtained if an electron is knocked out from the deepest K-level, and the vacancy thus formed is filled by an electron from a higher level, for example L or M. If an electron is knocked out from the next L level and replaced by an electron from the M or N level , L-series lines appear. Figure 15.3.

Figure 15.3. Transitions between energy

levels that form X-ray spectra

Each series occurs only when the accelerating voltage exceeds a certain critical value U 0 , which is called the excitation potential.

The value of the excitation potential U 0 is associated with the smallest wavelength of this series λ min:

The excitation potentials of the series are arranged in the following order: U N< U M < U L < U K . Например, для вольфрама U N = 2,81 кВ; U L = 12,1 кВ и U K = 69,3 кВ. Потенциал возбуждения данной серии растёт с увеличением атомного номера материала анода. Спектры характеристического излучения различных элементов одинаковы по своему строению.

In the practice of X-ray diffraction analysis, the K-series is most often used, which consists of four lines: α 1 , α 2 , β 1, β 2 . The wavelengths of these lines are arranged in the sequence λ α 1 > λ α > λ β 1 > λ β . The ratio of the intensities of these lines for all elements is approximately the same and approximately equal to I α 1: I α 2: I β 1: I β 2 .

With an increase in the atomic number of an element, the spectra of characteristic radiation shift towards short wavelengths (Moseley's Law).

where σ is the screening constant; ; n and m are integers for the K-series n = 1, for the L-series n = 2.

15.2 X-ray characteristic radiation sources

An X-ray tube is a source of X-rays arising in it as a result of the interaction of fast-flying electrons with

atoms of an anode placed in the path of electrons.

To excite x-rays in x-ray tubes, the following must be ensured: obtaining free electrons; communication of large kinetic energy to free electrons, from
several thousand to 1-2 million electron volts; interaction of fast-flying electrons with anode atoms.

X-ray tubes are classified according to certain criteria. According to the method of obtaining free electrons. A distinction is made between ionic and electron tubes. In ion tubes, free electrons are created as a result of the bombardment of a cold cathode by positive ions, which occur in a rarefied up to 10 -3 - 10 -4 mm Hg. in a gas when a high voltage is applied to them. In electron tubes, free electrons are formed due to thermionic emission of a cathode heated by a current.

According to the method of creating and maintaining a vacuum. Soldered and collapsible tubes are used. In sealed tubes, a high vacuum is created during manufacture and maintained throughout the entire period of operation. Violation of the vacuum causes the failure of the tube. In collapsible tubes, vacuum is created and maintained by a vacuum pump during operation.

By appointment, the tubes are used for transillumination of materials - X-ray flaw detection. For structural analysis - X-ray diffraction method. For medical purposes - diagnostic and therapeutic.

The main type of tubes used in X-ray diffraction analysis are sealed electron tubes. Figure 15.4.

They are a glass container into which two electrodes are introduced - a cathode in the form of an incandescent tungsten wire spiral and an anode in the form of a massive copper tube. A high vacuum of 10 -5 - 10 -7 mm Hg is created in the balloon, which ensures the free movement of electrons from the cathode to the anode, thermal and chemical insulation of the cathode, and prevents the occurrence of a gas discharge between the electrodes.

When a tungsten coil, heated by a filament current to 2100 - 2200 ° C, emits electrons, they, being in the field of high voltage applied to the poles of the tube, rush at high speed to the anode. Hitting the platform at the end of the anode (anode mirror), the electrons are sharply decelerated. Approximately 1% of their kinetic energy is then converted into the energy of electromagnetic oscillations - X-ray characteristic radiation, the rest of the energy is transformed into heat released at the anode.

Figure 15.4. Scheme of soldered electronic

X-ray tube BSV-2 for structural

analysis: 1- cathode; 2 - anode; 3 - windows for release

x-rays; 4 - protective cylinder;

5 - focusing cap

The relatively soft beams typically emitted by structural analysis tubes with wavelengths of 0.1 nm or more are very strongly absorbed by the glass. Therefore, to release X-rays, special windows are soldered into the cylinders of these tubes, made either from an alloy of hetane containing light elements (beryllium, lithium, boron), or from metallic beryllium.

The focus of the tube is called the area on the anode, on which electrons fall and from which X-rays are emitted. Modern x-ray tubes have a round or bar focus. Accordingly, the cathode is made either in the form of a spiral placed inside the focusing cup, or in the form of a helical line located inside the half-cylinder.

An X-ray tube anode for structural analysis is a massive hollow cylinder made of a material with high thermal conductivity, most often copper. A plate is pressed into the end wall of the anode - the anticathode (anode mirror), which slows down the electrons emitted from the cathode. In tubes for structural analysis, the anode mirror is made of the metal whose characteristic radiation is used to obtain a diffraction pattern in solving specific problems of X-ray diffraction analysis.

The most common tubes with anodes made of chromium, iron, vanadium, cobalt, nickel, copper, molybdenum, tungsten, tubes with silver and manganese anodes are used. The end face of the anode in tubes for structural analysis is cut off at an angle of 90° to the anode axis.

The most important characteristic of the tube is the power limit:

P = U I W (15.6)

where U is the high voltage value, V; I - tube current, A.

In some problems of X-ray diffraction analysis, especially those requiring high-resolution X-rays, the imaging efficiency depends on the size of the focus and, therefore, is determined by the specific power of the tube - the power emitted per unit area of the anticathode. For such conditions, sharp-focus tubes are designed, for example, BSV-7, BSV-8, BSV-9 and microfocus tube BSV-5.

15.3 Methods for recording the characteristic

x-ray radiation

To register X-rays, ionization, photographic, electrophotographic and luminescent methods are used.

The ionization method makes it possible to measure the intensity of x-rays with great accuracy over a relatively small area limited by measuring slits. The method is widely used in X-ray diffraction analysis, when it is necessary to know the exact relationship between the intensities and the profile of diffraction peaks.

The photographic method of recording diffraction maxima has become widespread. Possesses documentation and high sensitivity. The disadvantages of the method include the need to use photographic material, which complicates the registration of X-rays.

The electrophotographic method (xeroradiography) is a relatively simple method, the advantage of which lies in the possibility of successively obtaining a large number of images on one plate.

The method of observing an image on a luminous screen is highly productive and does not require expenditure on photographic materials. One of the disadvantages of the method is its low sensitivity in detecting defects (lack of documentation.

ionization method.

X-rays passing through a gas ionize its molecules. As a result, the same number of ions of different sign is formed. In the presence of an electric field, the emerging ions begin to move towards the corresponding electrodes. The ions that have reached the electrodes are neutralized, and a current appears in the external circuit, which is recorded. Figure 15.5.

Figure 15.5. Dependence of ionization current i

on the voltage on the electrodes U: I - saturation region;

II - area of full proportionality; III - region

incomplete proportionality; IV - region of equal impulses

A further increase in voltage up to U = U 2 does not cause an increase in the ionization current, only the ion velocity increases. At U ≥ U 2, the ion velocity becomes sufficient for ionization of gas molecules through collision - impact ionization, and the current begins to increase with increasing voltage due to gas amplification. The gas amplification factor up to U ≤ U 3 depends linearly on the applied voltage - the region of full proportionality, and can reach 10 2 - 10 4 .

At U ≥ U 3 there is a violation of the linearity of the gas amplification - an area of incomplete proportionality. At U ≥ U 4, in the case of a photon passing between the electrodes with an energy sufficient to form at least one pair of ions, an avalanche discharge occurs - a region of equal pulses, in which the passage of ionizing particles of different energies corresponds to the appearance of identical current pulses. A further increase in voltage leads to the appearance of a self-discharge.

The ionizing effect of X-rays is used to register them. Applied devices operating in various areas of gas discharge:

Ionization chambers - in the saturation area;

Proportional meters - in full proportional mode;

Gas-discharge counters - in the field of equal impulses.

ionization chambers.

Operate in saturation mode. The saturation voltage depends on the shape of the electrodes and the distance between them. For absolute measurements of the dose of x-rays, normal chambers are used, which can be cylindrical or flat. The chamber has three electrodes isolated from the body, made in the form of rods or tubes with a diameter of several millimeters: one measuring "A" and two protective "B".

proportional counters.

With an increase in the electric field strength in the ionization chamber, the electrons formed under the action of X-rays can acquire energy sufficient for impact ionization of neutral gas molecules. The electrons generated during secondary ionization can create further ionization. Gas amplification factor 10 4 - 10 6 .

Chambers operating under conditions of gas amplification are called proportional counters, since when a quantum of ionizing radiation enters them, an impulse arises on the electrodes that is proportional to the energy of this quantum. Proportional counters are especially widely used for recording long-wavelength X-rays.

Geiger counters.

If the voltage at the anode of the proportional counter is high enough, then the output pulses will not be proportional to the primary ionization and their amplitude, at a certain voltage, reaches a constant value, independent of the type of ionizing particles. This mode of operation of the counter is called the region of equal impulses or the Geiger region.

In the region of equal momenta, when a radiation quantum enters the counter, an electron avalanche arises, which, when moving towards the anode, excites the atoms of the noble gas that fills the counter. Excited atoms emit quanta of ultraviolet radiation, which contributes to the further propagation of the discharge along the anode filament. Counters with an organic additive have a limited service life due to the decomposition of the quenching additive 10 8 - 10 9 counts. Halogen counters can count up to 10 12 - 10 13 pulses.

Counters are characterized by parameters: efficiency, dead time and stability.

The time interval during which the counter is not able to register newly arriving radiation quanta is called dead time, which is determined by the time of movement of positive ions to the cathode, in Geiger counters it is 150-300 μs.

For X-ray diffraction analysis, counters of the MSTR-3 type are produced for the long-wave region of the spectrum, λ = 0.15 - 0.55 nm, MSTR-5 for the short-wave region of the spectrum, λ = 0.05 - 0.2 nm and the MSTR-4 counter.

scintillation counters.

Scintillation counters are among the most advanced instruments for measuring the intensity of X-ray radiation. The counters consist of a transparent luminescent crystal - a scintillator and a photomultiplier tube (PMT). NaI or KI crystals activated with a small amount of thallium are used as scintillators. Symbol - NaI (TI) or KI (TI).

A feature of scintillation counters is the proportional relationship between the ionizing ability of the particle and, consequently, the energy and amplitude of the voltage pulse at the output of the photomultiplier. The presence of such a relationship allows using amplitude analyzers to isolate pulses corresponding to quanta of a certain energy - to measure the intensity of radiation corresponding to a certain wavelength. The dead time of the counters is 1-3 μs, which allows you to increase the counting rate to 5·10 4 without any noticeable miscalculation.

semiconductor counters.

Semiconductor (germanium and silicon) counters have been used to register X-rays. The counter is a semiconductor diode with a p-n junction, to which a bias voltage is applied in the non-conducting direction. The bias voltage expands the carrier-depleted layer, creating a sufficiently sensitive effective volume for detecting ionizing particles.

Photographic method of registration.

A special X-ray film is used for photographic recording of X-rays. The photographic effect of X-rays is produced only by that fraction of them that is absorbed in the photographic emulsion. This proportion depends on the wavelength of the X-rays and decreases with decreasing wavelength. The emulsion layer of X-ray film absorbs ~30% of X-ray energy at 0.11 nm and only 1% at 0.04 nm. An increase in the sensitivity of the film to short-wavelength radiation can be achieved by using intensifying screens.

Xeroradiographic method (xerography).

This method retains the main advantages of the photographic method, but is more economical. The method uses special aluminum plates, on which a layer of amorphous selenium 100 μm thick is applied by vacuum deposition. Before x-ray shooting, the plate is placed in a special charger.

luminescent method.

Some substances emit visible light when exposed to x-rays. The energy yield of such a glow is small and amounts to a few percent of the absorbed X-ray energy.

Of particular interest are phosphors - substances that give the highest output of visible luminescence. The best phosphor with a yellow-green glow is a mixture of Zs + CdS. This mixture at different ratios between the components makes it possible to obtain a glow with a different spectral composition.

15.4 X-ray diffraction

With respect to X-ray diffraction, the crystal

considered as a three-dimensional diffraction grating. A plane monochromatic wave is incident on a linear diffraction grating. Figure 15.6.

Figure 15.6. Diffraction from a flat grating

Each hole in the grating becomes a source of radiation of the same wavelength λ. As a result of the interference of waves emitted by all holes in the grating, diffraction spectral lines of various orders are formed: zero, first, ... n-th. If the difference in the path of rays coming from adjacent holes in any direction is one wavelength, then a spectral line of the 1st order appears in this direction. The spectral line of the 2nd order arises at a path difference of 2λ, the spectrum of the nth order - at a path difference of nλ. For the occurrence of a diffraction maximum, the path difference must be equal to nА, where n is an integer, the relation must be satisfied: a(сosα ± сosλ 0) = nλ

In a crystal, a, b, c are the lengths of the axes of the crystal lattice, α 0, β 0, γ 0, α, β, γ are the angles formed with the axes by the primary and diffracted beams.

The occurrence of a diffraction maximum from a three-dimensional crystal lattice is determined by the system of Laue equations:

where h, k, l are integers, called reflection indices or Laue indices.

The Bragg equation defines the condition of X-ray diffraction that occurs when X-rays pass through a crystal, and have such a direction that they can be considered as the result of reflection of the incident beam from one of the systems of grating planes. Reflection occurs when the condition is satisfied:

2d sinθ = nλ, (15.8)

where θ is the angle of incidence of the primary X-ray beam on the crystallographic plane, d is the interplanar distance, n is an integer. Figure 15.7.

Figure 15.7. Diagram of Bragg's law derivation

In accordance with the Laue equations, each reflection is characterized by indices (hkl), Miller indices () determine the system of crystallographic planes in the lattice. Miller indices do not have a common factor. There are relations between Laue indices (hkl) and Miller indices (h’k’l’): h = nh’, k = nk", l = n1"

The system of Laue indices with a common factor n means that there is an n-th order reflection from the lattice planes with Miller indices (h’ k’ l’).

For example, reflections with Laue indices (231), (462), (693) are reflections of the 1st, 2nd and 3rd orders from lattice planes with Miller indices (231).

In the case of a cubic system, the interplanar distance d and the unit cell parameter "a" are related by the relation:

where (h'k'l') are Miller's xes.

Thus, for a cubic crystal, the Bragg equation can be written as:

When using Laue indices, equation (15.10) will look more simple:

The values of the Laue and Miller indices for crystals of different crystal groups (syngonies) are given in various reference literature on X-ray diffraction analysis.

15.5 Methods for indicating diffraction spectra

The interplanar distances d i corresponding to the individual values of the reflection angles in θ i are interconnected by the following equation:

In equation (15.12) a, b, c, α, β, γ denote the unit cell periods and axial angles, hkl are the indices of the considered plane of the crystal lattice.

Knowing the periods of the elementary cell of any substance, it is possible for each plane characterized by certain values of the indices (hkl) to calculate from equation (15.12) the corresponding interplanar distances d hkl.

In practice, the periods of an elementary cell are determined based on the known values of d i . The problem would be relatively simple if three integers (indices) corresponding to the individual values of d i were known. Then it would be possible to use six values of d hkl from the system of equations (15.12) and calculate the unknown constants: a, b, c, α, β, γ.

Equation (15.12) is greatly simplified for crystalline substances with high symmetry. Therefore, one should start with indexing the X-ray diffraction pattern of a material with a cubic structure.

Indexing of materials with a cubic structure

For a cubic lattice a = b = c, α = β = γ = 90°. After substitution into equation (15.12) and after calculating the determinants, the equation is converted to the form:

From the Wulf-Braggs equation it follows:

Hence:

As a result of measurements of the radiograph, after recalculating the arcs into angles, we obtain a series of values θ i , and sin θ i ;. These quantities can be denoted by ordinal "i", in ascending order, but it is impossible to apply the indices hkl characteristic of them. The experimentally known values are sin 2 θ i , not sin 2 θ hkl .

The problem of deciphering X-ray diffraction patterns of materials with a cubic structure is reduced to the selection of values for a series of integer values. This problem cannot be solved unambiguously without additional conditions.

Therefore, various methods for indicating the obtained radiographs are used: the method of differences, fan charts, various nomograms, and many other special methods.

15.6 Qualitative X-ray phase analysis

Phase analysis is the establishment of the number of phases in a given system and their identification. The X-ray method of phase analysis is based on the fact that each crystalline substance gives a specific interference pattern with a certain number, arrangement and intensity of interference lines, which are determined by the nature and arrangement of atoms in this substance.

Each phase has its own crystal lattice. The families of atomic planes that form this lattice have their own set of values of interplanar distances d hkl that is characteristic only for this lattice. Knowing the interplanar distances of an object makes it possible to characterize its crystal lattice and, in many cases, to establish a substance or phase. Data on interplanar distances for various phases are given in the reference literature.

Determination of the phase composition of polycrystalline substances by their interplanar distances is one of the most common and relatively easy to solve problems of X-ray diffraction analysis.

This problem can be solved for any polycrystalline substance, regardless of the type of its crystal lattice.

From the Wulf-Bragg formula (nλ = 2dsinθ) follows:

λ is the wavelength of the characteristic radiation in which the X-ray pattern was obtained, the value is known, then the problem of determining the interplanar distances is reduced to determining the diffraction angles θ.

There are practically no two crystalline substances that would have the same crystal structure in all respects, therefore, X-ray patterns almost unambiguously characterize this substance and no other. In a mixture of several substances, each of them gives its own X-ray diffraction pattern independently of the others. The resulting x-ray pattern of the mixture is the sum of a series of x-ray patterns that would have been obtained if each substance were taken one by one.

X-ray diffraction analysis is the only direct way to identify phases that even the same substance can have. For example, analysis of six modifications of SiO 2 , modifications of iron oxides, crystal structures of steels and other metals and alloys.

X-ray phase analysis is widely used in metallurgical production to study source materials: ores, flux enrichment products, agglomerates; smelting products in the production of steels; for the analysis of alloys during their thermal and mechanical processing; for the analysis of various coatings made of metals and their compounds; for the analysis of oxidation products and in many other industries.

The advantages of X-ray phase analysis include: high reliability and rapidity of the method. The direct method is not based on an indirect comparison with any standards or changes in properties, but directly provides information about the crystal structure of a substance, characterizes each phase. It does not require a large amount of substance, the analysis can be carried out without destroying the sample or part, the method allows the estimation of the number of phases in the mixture.

The use of diffractometers with ionization registration of interference lines, for example, URS-50IM, DRON-1, DRON-2.0 and other devices, leads to an increase in the sensitivity of phase analysis. This is due to the fact that when focusing according to the Bragg - Brentano method, the scattered rays are not focused, and therefore the background level here is much lower than with the photographic registration method.

15.7. Quantitative X-ray phase analysis

All developed methods of quantitative phase analysis are based on the elimination or taking into account the causes that cause a deviation from proportionality between the phase concentration and the intensity of the interference line, by which the phase content is determined.

15.7.1 Homology pair method.

The method is used for photographic recording of X-ray patterns and does not require the use of a reference sample and can be used for two-phase systems, provided that the absorption coefficient of the phase being determined does not differ markedly from the absorption coefficient of the mixture.

This condition can be met in some alloys, for example in two-phase (α+β)-brass, in hardened steel containing retained austenite and martensite. The method can also be applied to the analysis of a three-phase mixture if the content of the third phase is not higher than 5%.

The principle underlying the method is that the absorption coefficient of the analyzed phase does not differ from the absorption coefficient of the mixture, and the blackening density of the interference line D on the film is in the linear part of the characteristic curve of the photographic emulsion:

D 1 = k 1 x 1 Q 1 , (15.17)

where k 1 - coefficient of proportionality, depending on photo processing and conditions for obtaining radiographs; x 1 - mass fraction of the phase; Q 1 - reflectivity of the crystal plane (h 1 k 1 l 1).

If a pair of close lines from the phases has the same density of blackening, then, since both lines are on the same X-ray pattern, we can assume k 1 \u003d k 2 and therefore x 1 Q 1 \u003d x 2 Q 2, where x 1 and x 2 are the contents of the phases included into the composition of the material, Q 1 and Q 2 - the reflectivity of the respective planes. Considering that x 1 + x 2 = 1 we get:

The error of quantitative phase analysis when using homologous pairs is ~ 20%. The use of special methods for estimating the intensity of lines reduces the relative error of the analysis to 5%.

15.7.2 Internal standard method (mixing method).

Quantitative phase analysis of two- and multi-phase mixtures can be carried out by mixing a certain amount x s of a reference substance (10–20%) into a powdered sample, with the interference lines of which the lines of the phase being determined are compared. Use the method for both photographic and ionization registration of a diffraction pattern.

It is necessary that the reference substance satisfies the following conditions: the lines of the reference must not coincide with the strong lines of the phase being determined; the mass absorption coefficient for the reference substance μ a should be close to the absorption coefficient c.a of the analyzed sample; the size of the crystallites should be 5 - 25 microns.

The principle of the method - on the X-ray pattern obtained after mixing the reference substance, the intensity of the interference line of the analyzed phase is calculated by the equation:

The ratio I a /I s is a linear function of x a . Having determined the ratio for a number of mixtures with a known content of the analyzed phase, a calibration graph is built. To compare the intensities, a certain pair of lines with indices (h 1 k 1 l 1) of the determined phase and (h 2 k 2 l 2) of the reference substance are selected.

15.7.3 Phase analysis when superimposing lines of determined phases.

In some cases, it is impossible to obtain the lines of the determined phase without superimposing other lines, in particular the lines of the standard substance. The total intensity of the superimposed line I i is measured and the intensities of the well resolved line of the standard substance I 1 are compared. The calculation is carried out according to the formula:

where x a is the mass fraction of the analyzed phase.

For analysis, a straight line graph is built that does not pass through the origin. To build it, three reference mixtures are needed.

15.7.4 Method for measuring the intensity ratios of analytical lines.

The method is applicable to the analysis of multiphase mixtures when all components are crystalline phases. On the diffractometer measure the intensity of the analytical (reference) lines I 1 , I 2 ...1 n one for each phase. Make up a system (n - 1) of equations:

where x 1 x 2, ... x n - mass fractions of phases.

This method is used to carry out a quantitative phase analysis of materials with complex compositions with a relative error of 1 - 3%.

15.7.5 Mass absorption coefficient measurement method.

For the pure phase for the mixture , for the ratio

intensities:

where μ is the absorption coefficient of the sample; μ 1 - absorption coefficient of the 1st phase.

By measuring the absorption coefficient of the sample μ and the intensity of the lines I 1 of the 1st phase, it is possible to determine the mass fraction of the phase x i . The values (I i) 0 and μ i are found from a single measurement on a reference sample from the pure phase. The error in determining u by this method is 2 - 3%.

15.7.6 Method of "external standard" (independent standard).

The method is used in cases where the sample cannot be reduced to powder, and is also often used to standardize imaging conditions.

The ratio of the shooting time of the standard τ s and the sample τ a is determined by the ratio of the arcs occupied by the standard I s and the sample I a on the circumference of the cylinder with a radius equal to the radius of the sample.

Thus, by changing I s it is possible to change the ratio of the lines of the standard and the sample. A calibration graph is built for a certain ratio I s /I a and a certain pair of interference lines. To do this, shoot mixtures with a known phase content and measure the intensities of the lines of the sample (I h 1 k 1 l 1) and the standard (I h 2 k 2 l 2) s. The unknown content of the phase is determined from the calibration graph from the intensity ratio.

When using a diffractometer, a periodic survey of the reference substance is carried out. The analysis is carried out with the help of a calibration graph built on reference mixtures.

It is expedient to use the external standard method where serial phase analysis with high speed is required, and where the analyzed samples have a qualitatively homogeneous and relatively constant quantitative composition.

15.7.7 Overlay method.

The overlay method was developed for a two-phase substance and is based on a visual comparison of X-ray patterns of the studied and reference substances. The overlay X-ray pattern is obtained by alternately exposing pure alloy components to one X-ray pattern, one of which is exposed for time τ 1 and the other for time τ 2 .

To obtain X-ray overlay patterns, you can use a sample in the form of a thin section consisting of two cylindrical sectors, one of which is a pure phase 1, the other is phase 2. The thin section is oriented at an angle ψ with respect to the primary beam s 0 and rotates around the axis AA, perpendicular to to the surface of the cut. Figure 15.8.

Figure 15.8. Overlay survey scheme

As the section rotates, phases 1 and 2 alternately fall under the primary beam. The exposure time of each phase is determined by the opening angle of the corresponding sector:

By changing the angle α, one can obtain X-ray patterns corresponding to different concentrations of phases 1 and 2.

When taking radiographs of the overlay using the thin section method, the intensity of the line I 1 ’ of the structural component of the alloy is determined by the formula:

where Q 1 - the reflectivity of the plane with indices (h 1 k 1 l 1); μ 1 - linear absorption coefficient of phase 1; k 1 - coefficient depending on the Bragg angle θ and shooting conditions; ν 1 = сsecψ + сsec(2ν 1 – ψ); ψ - the angle between the primary beam and the plane of the section.

Similarly for phase 2. The absolute error of the overlay method Δc ~ 5% in the concentration range 10 - 90%. The advantage of the method is its rapidity.

15.8. Methods for practical calculation of unit cell parameters

To determine the periods of the crystal lattice, it is necessary to calculate the interplanar distances of the selected diffraction reflections, to determine their interference indices - the indication of reflections. After indexing X-ray maxima according to the recorded diffraction pattern, the period of a cubic crystal is determined by the formula:

The period of the crystal lattice of the main phase component of the alloy is calculated from several reflections with sufficiently large diffraction angles θ > 60°. The error in the calculation of periods is determined for the reflections used by the formula:

Δa = a ctgθΔθ (15.25)

Δa depends on the angle θ, so the period values obtained from different diffraction maxima cannot be averaged. For the final value of the period of the crystal, the values for reflections with the maximum diffraction angle, or the average of the values of reflections at an angle greater than 70°, are taken. The most accurate value of the period is obtained by graphical extrapolation with plotting the dependence a = f(θ) and extrapolating the period value to the angle θ = 90°. Various extrapolation dependencies are used.

For cubic crystals, the Nelson-Riley extrapolation function gives the best results. Figure 15.9.

Figure 15.9. Extrapolation when determining a period

cubic syngonies: a - aluminum; b - copper

With the right choice of extrapolation functions, the experimental points deviate from the straight line, the magnitude of these deviations is determined by the random error of the experiment. The form of the extrapolation line characterizes the systematic error.

Since the error in determining the unit cell period depends significantly on the diffraction angle, therefore, in order to accurately determine the grating periods, it is necessary to select suitable characteristic radiation (X-ray tube anodes). Diffraction angles in the precision region for cubic crystals with periods of 0.3 - 0.5 nm, depending on the wavelength of the applied radiation, are given in the reference literature.

For crystals of all syngonies, except for the cubic one, the interplanar distances generally depend on all linear lattice parameters. To determine the periods, it is necessary to use as many lines as there are different linear parameters in the lattice of a given syngony.

For the tetragonal syngony, the calculation of parameters is carried out according to the formulas:

For the hexagonal syngony, the calculation of periods is carried out according to the formulas:

Error in calculation of elementary cell parameters:

The graphical method for accurately determining the dimensions of the unit cell of cubic and uniaxial crystals gives results of sufficiently high accuracy, but for crystals with lower symmetry, it is rational to use the analytical method (Cohen's method). For crystals - rhombic, monoclinic or triclinic, the Cohen method may also be inapplicable, since the presence of a large number of lines makes it impossible to unambiguously indicate higher-order reflections. This difficulty can be minimized by using long-wavelength radiation, then an increase in the angle - the distance between the lines, leads to a decrease in their total number and, consequently, to an increase in the probability of unambiguous indication.

The Cohen method is the processing of experimental data using the least squares algorithm, which allows minimizing random errors, while systematic errors are eliminated by applying an appropriate extrapolation function. The method does not take into account the increasing accuracy of experimental data as the Bragg angle θ approaches 90°.

Thus, various methods for precision calculation of unit cell parameters have been developed and used, which have great practical application in studying the formation of solid solutions of metal alloys, phase and structural transformations during various heat treatment methods, and in many other technically important cases in materials science, physics, and solid state .

The positions of the X-ray diffraction lines from the sample when working on a diffractometer with a counter are set according to the intensity distribution in the diffracted radiation.

The maximum can be taken as the point of intersection with the diffraction profile of the line connecting the midpoints of the horizontal chords drawn at different heights. If the diffraction profile of the line is asymmetric, then all these methods will give different values for the diffraction angle.

Using the center of gravity of the diffraction peak is the most accurate method, since the calculation of the maximum of the diffraction line does not depend on the symmetry of the line. For a correct reading, you need to have a complete diffraction profile of the line.

To find the position of the intensity maxima, the position of the middle of the segments (chords) connecting the points of the line profile lying on opposite sides of the maximum and having equal intensities is determined. The line intensity is defined as the difference between the measured intensity and the background intensity, the change of which within the line is considered to be linear. The resulting points are connected by a curve that is extrapolated to the line profile. Figure 15.10.

Figure 15.10. Determination of the maximum intensity

x-ray reflection method of chords

Figure 15.11. Scheme for determining the center of gravity

diffraction maximum

Determining the center of gravity of the diffraction maximum is a more time-consuming operation. Figure 15.11.

The position of the center of gravity is determined in x units, then converted to 2θ units using the formula:

where θ 1 and θ 2 - the value of the angles (in degrees) corresponding to the beginning and end

measurement area.

Determining the center of gravity consists of the following operations: dividing the interval of angles in which the intensity of the line is nonzero into n segments; measurement of intensity at each point x i calculation of the position of the center of gravity according to the formula (15.30).

15.9 Methods for calculating structural parameters

crystalline materials

15.9.1 Features of the calculation of structural parameters

Internal stresses differ in the volumes in which they are balanced:

Macrostresses, which are balanced in the volume of the entire sample or product, in the presence of macrostresses, the removal of any part of the part leads to an imbalance between the remaining parts, which causes deformation (warping and cracking) of the product;

Microstresses are balanced within individual crystals and can be both non-oriented and oriented in the direction of the force that caused the plastic deformation;

Static distortions of the crystal lattice, which are balanced within small groups of atoms. In deformed metals, static distortions are balanced in groups of atoms lying near grain boundaries, slip planes, and other types of boundaries. Such distortions can be associated with dislocations.

Displacements of atoms from ideal positions (lattice nodes) can occur in solid solutions due to differences in the size of atoms and chemical interactions between like and unlike atoms that form a solid solution.

Stresses of different types lead to different changes in X-ray patterns and diffractograms, which makes it possible to study internal stresses by X-ray diffraction.

The results obtained by X-ray diffraction analysis are widely used in the development of new alloys, in the appointment of processing parameters, and in the control of technological processes. The study of the structure of materials makes it possible to reveal the influence of structural characteristics on the physical and mechanical properties of materials. The methods of X-ray diffraction analysis are diverse, which makes it possible to obtain valuable information about the structure of metals and alloys, which cannot be obtained by other methods.

15.9.2 Methods for determining the magnitude of microstresses

and crystalline blocks by the approximation method

Microdistortions of crystallites lead to broadening of interference lines on X-ray patterns, which can be characterized by the value Δd/d, where Δd is the maximum deviation of the interplanar distance for a given interference line from its average value d. Figure 15.12.

Figure 15.12. Location of the family of atomic planes:

a - the absence of microstresses; b - in the presence of microstresses

In the presence of microstresses, each system of atomic planes with the same interference indices (hkl) has, instead of a strictly defined interplanar distance d hkl, an interplanar distance d + Δd. The magnitude of microstresses is estimated from the magnitude of the relative deformation of the crystal lattice of metals: . For cubic crystals: .

The effect of line broadening in the diffraction pattern is also caused by the dispersity of crystalline blocks (CSR). The line width is affected by the divergence of the primary X-ray characteristic radiation, the absorption of the sample material, the location and size of the illuminating and analytical diaphragms - the geometric factor, the overlap or incomplete separation of α 1 - α 2 doublets.

If the physical state of the sample is known, from which it can be concluded that the physical broadening of the line β with interference indices (hkl) is caused only by the presence of microstresses or only by the dispersion of coherent scattering blocks D hkl is less than 0.1 μm, then the magnitude of the grating distortion in the direction perpendicular to the reflection plane ( hkl) and the size of crystalline blocks are calculated by the formulas:

where λ is the wavelength of the X-ray characteristic radiation.

In most cases, in the studied metal alloys, the broadening of diffraction reflections is caused, in addition to geometric factors, by the presence of microstresses and the dispersion of crystalline blocks. In this case, calculation according to formulas (15.31) is possible only after selecting the factors m - the dispersion of crystalline blocks and n - the presence of microstresses in the physical broadening β of each selected diffraction maximum.

An analysis of the intensity distribution in X-ray reflection makes it possible to establish that the value B - the true broadening of the line, free from the superposition of the doublet α 1 - α 2 is associated with the physical broadening of the line and b - the true geometric broadening of the standard, free from the superposition of the doublet, are determined by the expression:

The functions g(x) and f(x) determine the angular distribution of the diffraction reflection intensity due to the simultaneous effect of the survey geometry, the presence of microstresses, and the dispersion of the coherent scattering regions. These functions are approximated by various expressions that describe the intensity distribution in X-ray reflections with varying degrees of accuracy. For metals with cubic Bravais lattices, the results of sufficiently high accuracy are obtained by approximation by the expression:

With a known approximating function, the true physical broadening β is determined when shooting on a diffractometer or by a photomethod of two maxima from the sample under study and the standard. One of the lines has a small reflection angle with a small sum of squares of interference indices, the second maximum is recorded with the maximum possible reflection angle with a large sum of squares of Miller indices, similar maxima are recorded from a reference sample.

Having determined the half-width of diffraction reflections, the experimental broadening of both the studied sample "B" and the standard "b" is obtained.

The experimental total broadenings B and b, obtained when shooting in characteristic X-rays, are a superposition of the α 1 – α 2 doublet. Therefore, it is necessary to introduce a correction for duplicity, which is calculated according to the equation:

Schematically, the method of extracting the α 1 component from the experimental width of the X-ray maximum is shown in Figure 15.13 (Reschinger's method).

The extrapalation function is chosen depending on the shape of the profile of the diffraction maxima. From the maxima corrected for duplicity, the physical broadening β is found:

Figure 15.13. Correction scheme for

diffraction reflection duplicity

After separating the physical factor of the broadening of X-ray maxima, it is necessary to evaluate the share of the influence of the dispersity of crystalline blocks and the presence of microstresses.

If the crystal blocks are larger than 0.1 µm, then the physical broadening is caused only by microstresses:

from which it follows that the broadening is proportional to tgθ.

If there are no microstresses in the sample, but the crystalline blocks are smaller than 0.1 μm, then the physical broadening is caused only by the dispersity of the blocks:

The broadening is inversely proportional to cosθ.

In most cases, in metal alloys, the broadening of X-ray maxima is caused by both factors: microstresses and dispersion of crystalline blocks. In this case, from the physical factor of broadening β, it is necessary to single out m - broadening caused by the smallness of the blocks and n - broadening caused by the presence of microstresses:

where N(x) is a function of the presence of microstresses; M(x) is a function that determines the dispersion of crystalline blocks.

Equation (15.38) with two unknowns is unsolvable, therefore it is necessary to use two lines of a diffractogram or X-ray pattern, for which the physical broadening factors will be equal:

Let us divide the physical broadening curve into elements with base dу and height f(y). Each such element is affected by the geometric broadening function g(x), which leads to its smearing into a curve similar to g(x). The area of this element is still f(y)dy. The experimental curve h(x), obtained from the sample, is a superposition of many such blurry elements:

Equation (15.41) is a convolution of the functions f (x) and g (x), from the symmetry of the equation it follows:

The functions h(x), g(x) and f(x) can be expressed in terms of the Fourier integrals:

In equations (15.43), the coefficients h(x), g(x) and f(x) are Fourier transforms and can be expressed by the equations:

Equation (15.45) can be represented as:

Considering that lgA BL depends on L, therefore, if we obtain graphs from several lines of the diffraction pattern in the coordinates lgA BL for different diffraction reflections, then we can determine lgA BL and lgA MK.

The number of the Fourier coefficient n is related to the distance in the crystal lattice L by the equation:

where Δ(2θ) is the value of the interval of expansion of the experimental maximum in radians for the selected lines of the diffraction pattern.

Thus, by plotting A n = f(L n) and drawing a tangent (or secant) for different values of L n , the value is determined

Portal for the student. Self-training

Brest, 2010

Laue method

Single crystal rotation method

Powder method

RELATED ARTICLES