Let us consider one more method for the analysis of solids, also related to quantum radiation, but lying in the shorter wavelength part of the spectrum. X-ray diffraction analysis(XRD) is a method for studying the structure of bodies using the phenomenon of X-ray diffraction. This method involves the study of the structure of a substance based on an estimate of the spatial distribution of the scattered X-ray intensity.
Since the wavelength of X-rays is comparable to the size of an atom and the lattice constant of a crystalline body, when a crystal is irradiated with X-rays, a diffraction pattern will be observed, which 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 the order of units of angstroms is used.
Metals, alloys, minerals, inorganic and organic compounds, polymers, amorphous materials, liquids and gases, protein molecules, nucleic acids, etc. are studied by X-ray diffraction analysis. This is the main method for determining the structure of crystals. In their study, RSA provides the most reliable information. In this case, not only regular single-crystal objects, but also less ordered structures, such as liquids, amorphous bodies, liquid crystals, polycrystals, etc., can be analyzed.
On the basis of numerous already deciphered atomic structures, the inverse problem is also solved: according to the X-ray pattern of a polycrystalline substance, for example, alloyed steel, alloy, ore, lunar soil, the crystalline structure of this substance is established, i.e., phase analysis is performed.
In the course of X-ray diffraction, the sample under study is placed on the beams of X-rays and the diffraction pattern resulting from the interaction of the rays with the substance is recorded. The next step is to analyze
Rice. 15.35.
diffraction pattern and by calculation the mutual arrangement of particles in space, which caused the appearance of this pattern, is established. Figure 15.35 shows a photograph of an analytical setup that implements the X-ray diffraction method.
X-ray diffraction analysis of crystalline substances is performed in two stages. The first is the determination of the dimensions 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.
The second stage is the calculation of the electron density inside the unit cell and the determination of the coordinates of the atoms, which are identified with the position of the electron density maxima. Such data are obtained by measuring the intensity of the diffraction peaks.
There are various experimental methods for obtaining and recording a diffraction pattern. In any method, there is an X-ray source, a system for separating a narrow beam of X-rays, a device for fixing and orienting the sample relative to the beam axis, and a receiver of radiation scattered by the sample. The receiver is a photographic film, or ionization or scintillation counters of X-ray quanta, or another device for fixing information. The registration method using counters (diffractometric) provides the highest accuracy in determining the intensity of the registered radiation.
The main methods of X-ray imaging of crystals are:
- Laue method;
- powder method (debyegram method);
- rotation method and its variation - swing method.
When shooting Laue method a beam of non-monochromatic radiation falls on a single-crystal sample (Fig. 15.36, A). Diffract only those rays whose wavelengths satisfy the Wulf-Bragg condition. They form diffraction spots on Lauegram(Fig. 15.36, b) which are located along ellipses, hyperbolas and straight lines, necessarily passing through the spot from the primary beam. 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.
Rice. 15.36. X-ray survey according to the Laue method: A - irradiation scheme: b- a typical Lauegram; / - X-ray beam; 2 - collimator; 3 - sample; 4 - diffracted beams; 5 - flat film
By the nature of the spots on the Laue patterns, one can reveal internal stresses and other defects in the crystal structure. The indexing of individual spots is 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.
Using powder method(Fig. 15.37, A), as well as in the X-ray imaging methods described below, monochromatic radiation is used. The variable parameter is the angle of incidence 0, since crystals of any orientation with respect to the direction of the primary beam are always present in a polycrystalline powder sample.
Rice. 15.37. X-ray powder method: A- scheme of the method; b- typical powder radiographs (debyegrams); 1 - primary beam; 2- powder or polycrystalline sample; 3 - diffraction cones
Rays from all crystals in which planes with some interplanar distance dhkj are in a "reflecting position", i.e., they satisfy the Wulf-Bragg condition, form a cone around the primary beam with a raster angle of 40 °.
To each dukt corresponds to its 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. 15.37, b). Knowing the distances between the symmetrical "arcs", it is possible to calculate the corresponding interplanar distances d in a crystal.
In modern devices, instead of a film rolled over a cylindrical surface, a sensor with a small aperture and an area of the receiving window is used, which is discretely moved along a cylindrical surface, removing diffractogram.
The powder method is the simplest and most convenient in terms of experimental technique, but the only information it provides - the choice of interplanar distances - allows one to decipher only the simplest structures.
IN rotation method the variable parameter is the angle 0. Shooting is done on a cylindrical film. During the entire exposure time, the crystal rotates uniformly around an axis coinciding with some important crystallographic direction and the axis of the cylinder formed by the film. The diffraction rays travel along the generatrices of the cones, which, when intersecting with the film, give lines consisting of spots (layer lines).
The rotation method provides more 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.
This method simplifies the identification of radiographic spots. So, if the crystal rotates around the lattice axis, then all spots on the line passing through the trace of the primary beam have indices (A, To, ABOUT), on adjacent layer lines - respectively (A, k, I) and (A, A, I) etc. However, the rotation method does not provide all possible information, since it is not known at what angle of rotation of the crystal around the rotation axis this or that diffraction spot was formed.
When researching 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 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 each diffraction spot appeared.
Even more complete information is provided by X-ray goniometer methods. X-ray goniometer- This is a device that simultaneously registers the direction of X-rays diffracted on the sample under study and the position of the sample at the time of occurrence of diffraction.
One of these methods is 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 by a cylindrical screen, and the spots of the remaining diffraction cone are “unfolded” over the entire area of the photographic film by its reciprocating axial movement synchronously with the rotation of the crystal. This makes it possible to determine at what orientation of the crystal each spot appeared. weissenbergograms.
There are other survey methods that use the simultaneous simultaneous movement of the sample and film. The most important of them are reciprocal lattice photography method And Burger's precessional method. In this case, photographic registration of the diffraction pattern is used. 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.
X-ray diffraction analysis makes it possible to establish the structure of crystalline substances, including such complex ones as biological objects, 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 also used to study the crystalline state of polymers, amorphous and liquid bodies. X-ray diffraction patterns of such samples contain several blurred diffraction rings, the intensity of which sharply decreases with increasing angle of incidence 0. Based on the width, shape, and intensity of these rings, a conclusion is made about the features of the short-range order in a 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. Radiography includes, along with full or partial RSA, also other methods of using x-rays: x-ray flaw detection(translucence), X-ray spectral analysis, X-ray microscopy and etc.
Determination of the structure of pure metals and many alloys based on XRD ( crystal chemistry of alloys)- one of the leading branches of metallurgy. No state diagram of metal alloys can be considered reliably established if these alloys have not been studied by XRD methods. X-ray diffraction analysis made it possible to study deeply the structural changes that occur in metals and alloys during their plastic and heat treatment.
The X-ray diffraction method also has limitations. To perform a complete X-ray diffraction analysis, it is necessary that the substance crystallizes well with the formation of stable crystals. Sometimes it is necessary to carry out studies at high or low temperatures. This greatly complicates the experiment.
A complete study is very laborious, 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 painstaking work is performed by automatic microdensitometers and PC-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).
In this regard, specialized software packages have been developed and widely used to solve SAR problems, which make it possible to automate the process of measurements and interpretation of their results. However, even with the involvement of computer technology, the determination of the structure remains difficult.
The use of several counters in the diffractometer, which record reflections in parallel, makes it possible to reduce the time of the experiment. Diffractometric measurements surpass photorecording in sensitivity and accuracy, making it possible to determine the structure of molecules and the general nature of the interaction of molecules in a crystal.
An X-ray diffraction study does not always make it possible to judge with the required degree of reliability the differences in the nature of chemical bonds within a molecule, since the accuracy of determining bond lengths and bond angles is often insufficient. A serious limitation of the method is also the difficulty of determining the positions of light atoms, and especially hydrogen atoms.
Ministry of Education and Science of the Russian Federation
Federal State Budgetary Educational Institution of Higher Professional Education
Kuban State University
Physics and Technology Faculty
Department of Physics and Information Technology
Direction 03.03.02 Physics
COURSE WORK
X-ray diffraction analysis of crystals and interpretation of diffraction patterns
Completed by Purunova A.M.
2nd year student
Head Skachedub A.V.
Krasnodar 2015
ABSTRACT
Coursework 33 pages, 11 figures, 16 formulas, 10 sources.
KeywordsKey words: X-ray diffraction analysis, diffractograms, methods of X-ray diffraction analysis, crystals, radiation. Object of study: The study of X-ray diffraction analysis of crystals Subject of study:Crystal Research methods:Theoretically master three methods of X-ray diffraction analysis and interpretation of diffraction patterns The purpose of the course work:Get theoretical knowledge about the methods of researching crystals The objectives of the course work are: To study the scientific literature on the topic of X-ray diffraction analysis of crystals and interpretation of diffraction patterns Conclusions:Mastered the X-ray diffraction analysis of crystals and learned the methods of deciphering diffraction patterns Offers:Three methods of crystal analysis have been proposed. Introduction 1. Historical background Obtaining and properties of x-rays 3. Types of interaction of X-rays with matter Methods of X-ray diffraction analysis X-ray phase analysis 6. Method of photographing the reciprocal lattice 7. Using the results of X-ray diffraction analysis to determine the coordinates of atoms 8. Functional diagram of the device and the principle of crystal formation Conclusion Bibliography INTRODUCTION
X-ray diffraction analysis is one of the diffraction methods for studying the structure of a substance, which is based on the diffraction of x-rays on the analyzed object (three-dimensional crystal lattice). Obtained in the course of research, the diffraction pattern directly depends on the wavelength of X-rays, as well as on the structure of the object. There are different methods of analysis that are used to study metals, inorganic and organic compounds, alloys, polymers, minerals, liquids and gases, crystals, etc. X-ray diffraction analysis (hereinafter referred to as X-ray diffraction analysis) is the main method for determining the atomic structure of a crystal, which includes the space group of the unit cell, its shape and size, and determine the symmetry group of the crystal. Also, due to the fact that a large number of atomic structures have already been deciphered, it is possible to establish the crystalline composition of substances, that is, to perform phase analysis. X-ray radiation is not excited in the studied sample during X-ray diffraction analysis (if the sample emits fluorescent radiation during structural studies, this effect is a side, harmful effect). X-rays emitted by an X-ray tube are diffracted by the crystal lattice of the sample under study. Further, the analysis of the diffraction pattern is carried out and, by means of calculations, the mutual arrangement of particles in space is established, which caused the appearance of this pattern. There are three fundamentally different methods of XRD of crystals: Rotation method (monochromatic radiation is used) Powder method (using monochromatic radiation) Laue method (white X-ray spectrum is used). The rotation method is usually divided into two types: the rotation (rocking) method and the X-ray goniometric method. Learn RSA; Analyze the interpretation of the diffractograms. The structure of the work consists of an introduction, 8 chapters, a conclusion and a list of references. 1. Historical background
In 1912, German physicists M. Laue, W. Friedrich and P. Knipping discovered X-ray diffraction on crystals. They directed a narrow beam of X-rays at the crystal and recorded a diffraction pattern on a photographic plate placed behind the crystal. It consisted of a large number of spots, which were located regularly. Each spot is a trace of a diffraction beam scattered by a crystal. This radiograph is called the Lauegram (Figure 1). Figure 1 - Lauegram of a beryl crystal taken along the 2nd order symmetry axis This theory of X-ray diffraction by crystals made it possible to relate the wavelength radiation, crystal elementary cell parameters a,b,c, incident angles ( α 0,β 0,γ 0) and diffraction ( α 0,β 0,γ 0) rays by the ratios: a (cosα-cosα 0) = hλ(cosβ-cosβ 0) = kλ (1)
With ( cosγ -cosγ 0) = lλ ,
where h, k, l are integers (Miller indices). In order for a diffraction beam to arise, it is necessary that condition (1) be satisfied, i.e., that in parallel beams the path difference between those beams that are scattered by the atom corresponding to neighboring lattice sites should be equal to an integer number of wavelengths. In 1913 W.L. Bragg and G.W. Wulff showed that any of the diffraction rays can be considered as a reflection of the incident beam from one of the systems of crystallographic planes. As a method, PCA was developed by Debye and Scherrer. 2. Obtaining and properties of X-rays
To obtain x-rays, special vacuum devices are used - x-ray tubes. X-ray radiation occurs in the anode of an X-ray tube when it is bombarded with a beam of accelerated electrons, and the accelerating voltage should be 10-70 kilovolts (the accelerating voltage used in structural analysis lies precisely in this interval). High voltage is obtained using high voltage transformers. The circuits of many X-ray machines contain high-voltage capacitors, which make it possible to supply a voltage to the tube that is twice the voltage on the secondary winding of the transformer. Some circuits contain kenotrons (powerful vacuum diodes) that remove the function of rectifying the current from the x-ray tube. Modern installations for x-ray diffraction analysis are also equipped with ferroresonant stabilizers and mains voltage correctors, electronic stabilizers of the X-ray tube filament current. Registration of the radiation scattered by the sample can be carried out both on a film and by ionization methods. Schematic diagrams of some X-ray machines are shown in Figures 2-6. Figure 2 - Schematic diagram of the device 1 Figure 3 - Schematic diagrams of devices 2, 3 Figure 4 - Schematic diagrams of devices 4, 5 The external view of the spectrum of X-rays coming out of the anode of an X-ray tube is quite complex (see Fig. 5) and depends on the anode material, the magnitude of the anode current and the voltage on the tube. At low voltages (limited from above by a value determined for each material), the tube generates only a continuous spectrum of x-rays. This spectrum, also called "white", continuous or bremsstrahlung, has a pronounced intensity maximum and a short-wavelength boundary. Position of intensity maximum I m and shortwave border λ 0 does not change when the current through the tube changes, but moves towards shorter waves with increasing voltage on the tube (Figures 5.2, 5.3). With a further increase in voltage, the spectrum can change radically (Figure 5): intense, sharp lines of the so-called "characteristic" radiation appear against the background of the continuous spectrum. The voltage that must be applied to the tube in order to obtain lines of characteristic radiation is called the excitation potential of this line. 3. Types of interaction of X-rays with matter
One of the first properties of x-rays, discovered in their study, was a high penetrating power. Indeed, the x-ray beam shines through many opaque (for visible light) objects. However, the intensity of the radiation that has passed through the substance is less than the intensity of the initial beam. The mechanisms of attenuation of X-rays by a substance are different for different parts of the X-ray spectrum and different irradiated substances (Figure 6). Figure 6 - Diagram of the mechanisms of attenuation of X-rays by a substance The above diagram shows through which channels the energy of an X-ray beam incident on a substance is spent. The total, or total, attenuation of X-rays is the sum of true absorption and scattering. True absorption corresponds to the conversion of electromagnetic field energy into other types of energy (kinetic energy of photoelectrons) or secondary (fluorescent) radiation. X-rays can be scattered coherently (without changing the wavelength) or incoherently (Compton scattering). the occurrence of fluorescent radiation should not be considered as one of the types of scattering, since in this case, the energy of the primary X-ray radiation is completely spent on the ionization of the internal energy levels of the atoms of the irradiated substance, as a result of which the characteristic radiation of the atoms of the irradiated substance arises. The initial x-ray radiation of the primary beam is completely absorbed by this substance. It can be said that during coherent absorption the primary radiation quantum vanishes, while during scattering it changes its direction. 4. Methods of X-ray diffraction analysis
Three methods are used in X-ray diffraction analysis to overcome such a difficulty as the absence of reflections from an ideal immobile crystal in monochromatic radiation. Lauegram method The Laue method is applied to single crystals. The sample is irradiated with a beam with a continuous spectrum, the mutual orientation of the beam and the crystal does not change. The angular distribution of diffracted radiation has the form of individual diffraction spots (Lauegram). The scheme is shown in Figure 7. Figure 7 - Scheme of the Laue method This method cannot be used to determine the lattice constants. In particular, it is used to orient single crystals and analyze the perfection of single crystals in terms of size and the correct location of points on the X-ray pattern. Rotating or rocking crystal method The rotating or rocking crystal method is shown in Figure 8. Figure 8 - Ewald's construction for the rocking crystal method Black dots - nodes of the reciprocal lattice when the crystal is stationary; bright points - nodes of the reciprocal lattice when they hit the Ewald sphere during the rotation of the crystal In contrast to the Lauegram method, monochromatic radiation is used here (we take into account that the radius of the Ewald sphere is constant), and the points of the reciprocal lattice hit this sphere by rotation (turns) of the direct and reciprocal lattices of the sample. This method is less convenient for orientation. Powder method The powder method (debyegram method) is based on the use of polycrystals or fine powders made from single crystals as samples. It was proposed in 1916 by Dibay and Scherrer. It is widely used to determine the structure of crystals. Figure 9 - Powder method (debyegram method): a - Ewald's construction; 1 - lines of intersection of the Ewald sphere 2 with spheres 3, on which the initial points of the reciprocal lattice vectors B lie; b - scheme of the experiment: 4 - radiation source (X-ray tube), 5 - sample, 6 - photographic film; c - unrolled film In this method, samples of actually used sizes contain more than ~ 108 particles, the orientation of the crystallographic axes, in which more or less uniformly distributed in all directions. We will draw the reciprocal lattice vectors for these particles so that their end points coincide (Figure 8, a). Then their initial points will lie on a sphere of radius B. This sphere will intersect along the circle 3 with the Ewald sphere. And since the relative linewidth (i.e. ∂k/k) of the radiation used in X-ray analysis is ~10 -4, then almost all points of this circle will be the initial points of the wave vectors of the scattered radiation k 2. Other vectors of the reciprocal lattice (one of them is shown in Figure 8a - b ") will give other circles of intersection with the Ewald sphere and other vectors of scattered radiation k 2".
The scheme of the experiment is shown in Figure 8b and the unfolded X-ray film in Figure 8c. Arcs of different radii will appear on this film, they allow you to find the reciprocal lattice vectors and find the lattice constant. . X-ray phase analysis
X-ray phase analysis is also often referred to as substance identification. The purpose of identification is to establish the phase composition of the sample, i.e. the answer to the question: "What crystalline phases are present in this sample?" The fundamental possibility of X-ray phase analysis is based on the fact that each crystalline substance has its own (and only it) interplanar distances and hence a "proper" set of spheres populated by reciprocal lattice sites. According to this principle, the X-ray pattern of each crystal is strictly individual. Polymorphic modifications of the same substance will give different radiographs. The last provision helps to understand the fundamental difference between phase analysis and all other types of analysis (chemical, spectral): X-ray phase analysis registers the presence of one or another type of crystal lattice, and not atoms or ions of a certain kind. Using phase analysis, the chemical composition of the sample can be determined; the reverse is not always possible. X-ray phase analysis is indispensable in the analysis of mixtures of modifications of the same substance, in the study of solid solutions, and in general in the study of state diagrams. The sensitivity of the method is low. Usually the phases present in the mixture in an amount<1%, уже не могут быть обнаружены рентгеновским методом. К тому же чувствительность метода зависит от состава пробы. The detectability of one phase in another depends on many circumstances: on the atomic numbers of the sample components, on the size and symmetry of the unit cell of crystals, on the scattering and absorption abilities of all constituent phases. The higher the scattering power and the lower the absorption coefficient for the atoms that make up the lattice of the phase, the less the amount of this phase can be detected. But the lower the symmetry of the crystal lattice of the substance under study, the more it is needed for detection. The latter is due to the fact that the decrease in symmetry leads to an increase in the number of lines in the X-ray pattern. In this case, the integrated radiation intensity is distributed over a larger number of lines, and the intensity of each of them decreases. In other words, lowering the symmetry leads to a decrease in the repeatability factor for these planes . For example, in a crystal lattice with cubic symmetry , and in the triclinic syngony it is only 2: And .
An important factor determining the sensitivity of the method is the size of the crystals of the substance under study: the smaller the crystals (at L 10-6cm), the greater the blurring of the interference lines, and with a small amount of phase, the blurred lines can merge with the background. The sensitivity of the method increases significantly with X-ray photography in monochromatic radiation, because monochromatization leads to a sharp decrease in the background level. It should be remembered that the use of monochromators entails a significant increase in exposure. You can increase the sensitivity by skillful selection of radiation, shooting modes, using modern equipment (diffractometers). However, under any shooting conditions, the sensitivity limit is set, first of all, by the sample itself: its composition and structural state. Phase analysis methods are based on the fact that each substance gives a certain set of interference lines, which is independent of other substances present in the sample. The intensity ratio of the lines of a given phase does not change, although the intensity of each line is proportional to the content of the phase in the substance (if absorption is neglected). Quantitative phase analysis is currently carried out mainly with the help of diffractometers, but in some cases the photographic method is also used. All the methods of quantitative phase analysis developed so far 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. Let us briefly consider some of the methods of quantitative phase analysis. The method of homologous pairs was developed by V.V. Nechvolodov, is used for photographic registration of scattered radiation. It does not require the use of a reference sample and can be used to study two-phase systems if the absorption coefficient of the determined phase does not noticeably differ from the absorption coefficient of the mixture. The table of homologous pairs of lines is calculated theoretically or compiled on the basis of experimental data. Homologous pairs are found on x-ray diffraction patterns of mixtures to find pairs of lines that have different blackening densities and belong to different phases. Knowing the indices of these lines, the content of the analyzed phase is found from the table of homologous pairs. The internal standard method (mixing method) is used in the quantitative analysis of two- and multi-phase mixtures. A certain amount (10-20%) of the reference substance is mixed into the powder of the test substance, with the interference lines of which the lines of the phase under study are compared. This method can be used for both photographic and ionization registration of a diffraction pattern. The reference substance must meet the following conditions: a) the lines of the standard should not coincide with the strong lines of the phase under study; b) the mass absorption coefficient of the reference substance should be close to the absorption coefficient of the analyzed sample; c) the size of the crystals should be 5-25 microns. The external standard (independent standard) method is used when the test sample cannot be powdered. It is also often used to standardize shooting conditions. In the photographic recording method, a reference substance in the form of a thin foil is glued onto a cylindrical or flat surface of the sample. When using a diffractometer, an external standard is set on a cuvette containing the test sample, or a reference substance is periodically taken. The analysis is carried out using a graduated graph built on reference mixtures (2)
The error of the independent standard method is small, in the most favorable cases it is 1.0-0.5%. It is expedient to use the external standard method where serial studies with high expressivity are required and where the analyzed samples have a qualitatively homogeneous and relatively constant quantitative composition. For serial analysis of complex mixtures, it is advisable to use specialized multichannel X-ray diffractometers. The sensitivity of the method in this case reaches 0.05%. x-ray diffraction pattern atom crystal 6. Method of photographing the reciprocal lattice
X-ray patterns can be viewed as distorted projections of reciprocal lattice planes. It is much more interesting to obtain an undistorted projection of the reciprocal lattice. Consider how undistorted projections are obtained. Imagine that a flat film is on one of the reciprocal lattice planes perpendicular to the axis of rotation. As the reciprocal lattice rotates, the film will intersect the reflection sphere along with the corresponding plane. The diffraction rays that appear at the moments of intersection of the nodes of the reciprocal lattice with the reflection sphere will fall only at those points of the film under which these nodes are located. Then you get something like a contact photograph of a reciprocal lattice grid. Due to the fact that the reciprocal grating and the sphere of reflection are artificial constructions and can be depicted at any scale, the photographic film is placed not on the most photographed reciprocal grating plane, but at some distance from it. In this case, the film rotation axis will not coincide with the crystal rotation axis. Bundle of primary rays S 0 falls on crystal A at some angle to its axis of rotation. By changing this angle, one can change the opening of the interference cones. An annular screen makes it possible to cut out a certain interference cone, the opening of which is chosen so that this cone passes through the annular opening of the screen. The setting data (the distance between the axis of rotation and the axis of rotation of the crystal) depend on one quantity - from the distance of the investigated nth plane of the reciprocal lattice to the zero nd x = n/I (3) Angle µ n is determined from the ratio , (4)
because "= OO" + OB (5) . (6)
, (7)
, (8)
, (9)
, (10)
That's why (11)
On the radiograph, the projection of the reciprocal lattice is enlarged. The magnification scale is determined from the ratio , (12)
Those. the scale K is the same for all planes of the reciprocal lattice (for a given screen and a certain wavelength). Indexing of radiographs obtained in the KFOR chamber is not difficult. It consists in establishing directions on the grid corresponding to the two most characteristic nodal lines - the axes of the reciprocal lattice a x and b x . It is necessary to start indexing not from the zero grid, but from any nth one, because extinctions on the X-ray pattern of the zero grid can lead to an erroneous judgment about the directions with the smallest translations. It is advisable to superimpose a kphorogram, for example, from the I-th plane on a kphorogram from the zero plane. Then the resulting grid will allow you to easily determine two reflection indices; the third index is determined by the layer line number. The X-ray pattern obtained in the chamber for photographing the reciprocal lattice during the rotation of the crystal around a certain axis makes it possible to determine the lattice periods along the other two axes, as well as the angle between these axes. 7. Using the results of X-ray diffraction analysis to determine the coordinates of atoms The first and partially the second problems can be solved by the Laue methods and rocking or rotation of crystals. It is possible to finally establish the symmetry group and coordinates of the basic atomic complex structures only with the help of complex analysis and laborious mathematical processing of the intensities of all diffraction reflections from a given crystal. The ultimate goal of such processing is to calculate, from experimental data, the values of the electron density ρ( x, y, z). The periodicity of the crystal structure allows us to write the electron density in it through the Fourier series: , (13)
where V is the unit cell volume, Fhkl - Fourier coefficients, which are called structural amplitudes in X-ray diffraction analysis, .
Each structural amplitude is characterized by three integers. Diffraction reflection is a wave process. It has an amplitude equal to , and phase αhkl (by the phase shift of the reflected wave with respect to the incident), through which the structural amplitude is expressed: . (14)
The diffraction experiment makes it possible to measure only reflection intensities proportional to , but not their phases. Phase determination is the main problem in deciphering the crystal structure. The definition of 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, the reverse of the structural interpretation: the calculation of the known atomic structure of the structural amplitudes, and on them - the intensities of diffraction reflections. The trial and error method, historically the first method for 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, the trial model is accepted or rejected. , (15)
In the 30s. more formal methods have been developed for crystalline structures, but more formal methods for non-crystalline structures, but for non-crystalline objects, trial and error is still practically the only means of interpreting the diffraction pattern. 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. The vectors from the origin to all atoms of the structure (including the vector of zero length up to the first atom) will indicate the positions of N maxima of the function of interatomic vectors, the totality of which is called the image of the structure in atom 1. Let us add N more maxima to them, the position of which will indicate N vectors from the second atom, placed at the parallel transfer of the structure to the same origin. Having done this procedure with all N atoms (Figure 10), we will get N2 vectors. The function describing their position is the Patterson function. Figure 10 - Scheme for constructing the Patterson function for a structure consisting of 3 atoms For the Patterson function P(u ω ) (u ω - coordinates of points in the space of interatomic vectors), you can get the expression: , (16)
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 the function P (u ω ) consists in the need to find the coordinates of N atoms from N2 of its maxima, many of which merge due to overlaps that arise when constructing the function of interatomic vectors. The easiest to decipher P (u 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 from the Patterson 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 Patterson function face serious difficulties in studying the structures of crystals consisting of atoms of the same or similar 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 that obey the Fourier coefficients (structural amplitudes) of the function ρ( x, y, z). Using the methods of inequalities, one can relatively simply 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 distribution of the electron density of atoms in the structure can be calculated (Fig. 10). The final refinement of the atomic coordinates is carried out on a computer using the least squares method and, depending on the quality of the experiment and the complexity of the structure, makes it possible to obtain them with an accuracy of thousandths of A (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). X-ray diffraction analysis also 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. To solve many physical and chemical problems, X-ray diffraction studies and resonance methods are used together. The pinnacle of achievements in X-ray diffraction analysis is the deciphering of 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. 8. Functional diagram of the device and the principle of signal formation
X-ray devices must meet some basic requirements, which are dictated by the Wulf-Bragg law and R-ray optics: the possibility of obtaining poly- and monochromatic radiation; beam focusing; ensuring automatic enforcement of the law; averaging the reflection over the sample surface; proportionality of the radiation detector to the number of x-ray quanta; automatic marking of the diffraction angle. Figure 11 shows the functional structure of the DRON-3M device. Figure 11 - Structure of the DRON-3M device: 1 - X-ray tube; 2 - sample; 3 - radiation detector; 4 - goniometer mechanism with automatic diffraction angle; 5 - power supply system; 6 - cooling system; 7 - detector signal processing system; 8 - recorder Monochromaticity is achieved by using a metal foil that transmits one wavelength and, if possible, absorbs one wavelength. This property is possessed by nickel foil, which absorbs 97% of the radiation from the copper anticathode and transmits with very low absorption. λ = 1.54Å. Lenses capable of focusing R-rays are not yet available. Therefore, in the design of the device, special devices are used to select and measure θ - goniometers. The goniometer automatically maintains focusing of radiation at any diffraction angle. It automatically fulfills the Wulf-Bragg law due to the fact that the angular velocity of rotation of the sample on which the radiation falls is at any time 2 times less than the velocity of the radiation detector (receiver). Due to this, at any time the detector is located at an angle of 2 θ to the incident radiation, and the sample at an angle θ.
The averaging of all arrangements of the reflection planes in the sample occurs due to its rotation around an axis perpendicular to the reflection plane. As a signal detector, a scintillation counter with a photomultiplier is used, which has a good proportionality to the number of X-ray quanta supplied to it. On the recorder tape, to facilitate the interpretation of X-ray patterns, the diffraction angle is automatically marked by a special electronic circuit associated with the mechanism of movement of the sample located in the goniometer. CONCLUSION
At present, it is difficult to find an area of human activity where X-rays would not be used. 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. 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 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. 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. 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 record reflections in parallel, can reduce the time of the experiment. Diffractometric measurements are superior to photographic recording in terms of sensitivity and accuracy. Allowing one 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. As a result of the course work, I have mastered the following general cultural and professional competencies: ) (OK-12) the ability to master the basic methods, methods and means of obtaining, storing, processing information, to have the skills to work with a computer as a means of managing information. The work used sources taken from the worldwide network "Internet". In the course of writing this term paper, I studied a number of different books and publications on the Internet. With their help, this work is full of various facts that were unknown to me up to this point. ) (GPC-7) The ability to obtain and use in their activities knowledge of a foreign language. Thanks to the knowledge of a foreign language, literature in English was used when writing the work. While writing this work, I found material in a foreign language. In order to use the information found, it was necessary to translate the articles into Russian, which she managed by including the translated text in her work. ) (PC-1) The ability to use specialized knowledge in the field of physics for the development of specialized physical disciplines. The information I have studied on this topic will help me not only in writing this term paper, but will also be useful in the future in an in-depth study of crystals, X-ray diffraction analysis, and also for preparing for exams. LIST OF CITATIONS
1.Gurevich, A.G. Physics of Solids. - Proc. allowance for universities / FTI im. A.F. Ioffe RAS.- St. Petersburg: Nevsky Dialect; BVH-Petersburg, 2004.-320 p.: ill. 2.Zhdanov, G.S. Fundamentals of X-ray diffraction analysis.- Moscow.- Gostekhizdat.-1940.-76 p.: ill. .Pokoev, A.V. X-ray diffraction analysis. - Moscow. - ed. 2, - 1981. - 127 p. .Rakhimova, N.T. Coursework on the topic "X-ray diffraction analysis".- Ufa.-2012.-30 p. .Belov, N.V. Structural crystallography. - St. Petersburg. - ed. 4, 1951.-97 p. ."Wikipedia". - Internet Encyclopedia .James, R. Optical principles of X-ray diffraction - Moscow. - Gostekhizdat. - ed. 1, 1950.-146 p. 8.Johnston W.D., Jr. Nonlinear optical coefficients and the Raman scattering efficiency of LO and TO phonons in acentric insulating crystals // Phys. Rev. B. - 1970. - V.1, No. 8. - P.3494-3503.Similar works to - X-ray diffraction analysis of crystals and interpretation of diffraction patterns
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FEDERAL EDUCATIONAL AGENCY STATE EDUCATIONAL INSTITUTION OF HIGHER PROFESSIONAL EDUCATION BASHKIR STATE UNIVERSITY CHEMICAL AND TECHNOLOGICAL FACULTY DEPARTMENT OF TECHNICAL CHEMISTRY
Course work
"X-ray diffraction analysis"
Lecturer: Doctor of Physical and Mathematical Sciences, Prof. Chuvyrov A.N.
Student: Rakhimova N.T.
Group: HFMM-3
1. Introduction
2. Historical background
3.1 Nature of SAR signals
7. Literature
1. Introduction
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 ~1E is used, i.e. about the size of an atom.
Metals, alloys, minerals, inorganic and organic compounds, polymers, amorphous materials, liquids and gases, protein molecules, nucleic acids, etc. are studied by 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. 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 diffraction pattern of a polycrystalline substance, for example, alloyed steel, alloy, ore, lunar soil, that is, 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.
There are three fundamentally different methods of X-ray imaging of crystals, two of which, the rotation method and the powder method, use monochromatic radiation, and the third, the Laue method, uses the white spectrum of X-rays. A variation of the rotation method is the crystal rocking method. In addition, the rotation method can be divided into two types: in one case, the rotation or rocking of the crystal occurs when the film is stationary (the usual method of rotation or rocking), and in the other, the film moves simultaneously with the rotation of the crystal (methods of scanning layered lines or, as they are often called, x-ray methods).
The diffractometric method also adjoins the X-ray goniometry methods. Its main difference is that X-rays are recorded not by photographic film, but by an ionization device or a scintillation counter.
2. Historical background
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 a crystal. An X-ray pattern obtained by this method is called a Laue pattern (Fig. 1).
diffraction x-ray atomic
Rice. 1. Laue diagram of a NaCl single crystal. Each spot is a trace of X-ray diffraction reflection. Diffuse radial spots in the center are caused by the scattering of X-rays by thermal vibrations of the crystal lattice.
The theory of X-ray diffraction on crystals developed by Laue made it possible to relate the wavelength l of radiation, the parameters of the unit cell of a crystal a, b, c (see Crystal lattice), the angles of the incident (a0, b0, g0) and diffraction (a, b, g) rays by the relations:
a (cosa--cosa0) = hl,
b (cosb -- cosb0) = kl, (1)
c(cosg -- cosg0)=ll,
where h, k, l are integers (Miller indices). For the occurrence of a diffraction beam, it is necessary to fulfill the above Laue conditions [equations (1)], which require that in parallel beams the path difference between the beams scattered by atoms corresponding to neighboring lattice sites be equal to an integer number of wavelengths.
In 1913, W. L. Bragg and, simultaneously with him, G. W. Wulff, proposed a more illustrative interpretation of the appearance of diffraction rays in a crystal. They showed that any of the diffraction beams can be considered as a reflection of the incident beam from one of the systems of crystallographic planes. In the same year, W. G. and W. L. Braggy were the first to study the atomic structures of the simplest crystals using X-ray diffraction methods. In 1916, P. Debye and the German physicist P. Scherrer proposed the use of X-ray diffraction to study the structure of polycrystalline materials. In 1938 the French crystallographer A. Guinier developed the method of small-angle X-ray scattering for studying the shape and dimensions of inhomogeneities in matter.
The applicability of X-ray diffraction analysis to the study of a wide class of substances, the industrial need for these studies stimulated the development of methods for deciphering structures. In 1934, the American physicist A. Paterson proposed to study the structure of substances using the function of interatomic vectors (the Paterson function). The American scientists D. Harker, J. Kasper (1948), W. Zachariasen, D. Sayre, and the English scientist W. Cochran (1952) laid the foundations for the so-called direct methods for determining crystal structures. N. V. Belov, G. S. Zhdanov, A. I. Kitaigorodskii, B. K. Weinshtein, M. Porai-Koshits (USSR), L. Pauling, P. Ewald, M. Burger, J. Carle, G. Hauptman (USA), M. Wulfson (Great Britain) and others made a great contribution to the development of Paterson and direct methods of X-ray diffraction analysis. Bernal (30s) and successfully continued by J. Kendrew, M. Perutz, D. Crowfoot-Hodgkin and others, played an exceptionally important role in the development of molecular biology. In 1953, J. Watson and F. Crick proposed a model of the deoxyribonucleic acid (DNA) molecule, which agreed well with the results of X-ray studies of DNA obtained by M. Wilkins.
3. Experimental methods of X-ray diffraction analysis
3.1 Nature of SAR signals
To obtain information about the spatial structure of a substance, including polymers, X-ray radiation is used, the wavelength of which is from 0.1 to 100 E. In practice, to obtain signals (reflexes) from polymers, a copper anti-cathode and a nickel filter are most often used, with which a K-line with a wavelength = 1.54 E is separated from the continuous spectrum of R-radiation. and supramolecular structure (small-angle R-ray diffraction).
The concepts of "diffraction" and "interference" of rays are known to you from the course of physics.
SAR is based on the phenomena of reflection, scattering, diffraction and interference of R-rays. Diffraction is inherent in all types of radiation: if there are several slits (defects) in the screen, each of them turns out to be a source of circular or spherical waves (Fig. 1). These waves interfere (interact) with each other, mutually annihilating in some places and amplifying in others (Fig. 2).
In 1913, father and son Bragg (British) and the Russian scientist Wulff discovered that a beam of R-rays emerging from a crystal behaves as if it were reflected from a mirror, a plane.
Let us consider several atomic layers located at a distance from each other, which is commensurate with or slightly greater than radiation. If a beam of R-rays is directed at such an object, then the point to which electromagnetic radiation reaches can be a source of reflected radiation. Let us assume that the angle of incidence of a plane wave, then the angle of reflection will be equal to 2.
Rice. 1. Diffraction by a single slit
Rice. 2. Wave interference
Due to reflection from different atomic layers, a path difference appears equal to, where is a positive integer, is the wavelength of the incident and reflected R-radiation. From simple geometric considerations, we find that the path difference is equal to
Equation (1) is commonly referred to as the Wulff-Bragg law for X-ray diffraction by crystals. A diagram illustrating this law is shown in fig. 3.
Rice. 3. Scheme of the course of incident and reflected rays in a crystal
For a three-dimensional lattice with an identity period in each direction (i.e. for a bulk crystal lattice), three diffraction conditions must be met that determine the values of three angles - 1, 2, 3.
where n, m, k are integers.
However, three angles in space cannot be chosen arbitrarily, since the angles between an arbitrary straight line and three mutually perpendicular coordinate axes are connected by a geometric condition
Equations (2) and (3) have solutions, i.e. make it possible to calculate the angles 1, 2, 3 for a grating with given parameters, not at any wavelength, but only those that ensure the compatibility of equations (2) and (3). All other waves dissipate without giving maxima.
Processing of the results is reduced to calculating the sizes of identity periods d (interplanar distance) at a known and experimentally determined angle for the maximum of reflected radiation. The structural ordering of the arrangement of macromolecules and their parts determines the existence of several levels of periodicity, characterized by their period value, each of which corresponds to its own reflection maximum angle.
3.2 Forms of presentation of the results of x-ray diffraction analysis
Methodically, X-ray diffraction analysis is performed according to one of three options, which differ in the way the sample is prepared and the form in which the results are presented.
The Laue method is used to study single crystals of rather large sizes (more than 1 mm in the perimeter). The sample is placed in the path of continuous (polychromatic) X-ray radiation, in which there is always a wavelength that satisfies equations (2) and (3). The X-ray pattern is a system of spots (point reflections) from different orders. For polymers, this method is practically not used because of the difficulties in obtaining single crystals of such sizes.
The method of rotation or rocking (the Bragg method) is based on the use of a single crystal and monochromatic radiation. During rotation or rocking, the crystal can turn in such a plane for which the Wulf-Bragg law is satisfied. Whenever this happens, a corresponding reflex occurs, fixed by a photographic film placed on the inside of the cylinder, in the center of which the sample rotates or oscillates.
The powder method is most suitable for polymers. To obtain an X-ray pattern, a monochromatic beam of R-rays is directed to a polycrystalline sample (powder). When the beam meets the crystal, the orientation of which with respect to the incident radiation satisfies the Wulf-Bragg law, diffraction occurs from each system of equally oriented planes. The X-ray pattern is obtained in the form of concentric circles (rings) fixed by a photographic film located perpendicular to the incident beam behind the sample.
The X-ray diffraction pattern can be written as a dependence of the integral intensity of any diffraction reflection on angle 2. Figure 4 shows conditional diffraction patterns for polymers with a high degree of crystallinity (a), mixed structure (b), and amorphous structure (c).
Rice. 4. Typical diffractograms of polymers: shaded area - amorphous halo; 01, 02, 03 - reflexes
3.3 Using the results of X-ray diffraction analysis to solve problems in polymer technology
X-ray diffraction analysis gives an idea of the structure of a polymer material and its change under the influence of various factors associated with processing conditions: temperature, load, orientation, etc. Control of the polymer structure in the technology of its production makes it possible to choose the optimal conditions for the synthesis of polymers with desired properties. In the course of exposure to the polymer, one can immediately obtain information about phase transitions and conformations of macromolecules.
Diffraction of R-beams at small angles makes it possible to judge the structural ordering in the arrangement of macromolecules and their parts in the region of short-range and long-range order, the density of amorphous interlayers, and the imperfection of crystalline structures. All this is important for predicting the behavior of a polymer under thermomechanical effects under processing conditions.
The advantage of X-ray diffraction analysis in comparison with electron microscopy, which makes it possible to obtain such information about the supramolecular structure, is the simplicity of sample preparation in the powder method, a large amount of information with less time spent on analysis.
3.4 Functional diagram of the device and the principle of signal formation
X-ray devices must meet some basic requirements, which are dictated by the Wulf-Bragg law and R-ray optics:
The possibility of obtaining poly- and monochromatic radiation;
Beam focusing;
Ensuring automatic enforcement of the law;
Reflection averaging over the sample surface;
Proportionality of the radiation detector to the number of X-ray quanta;
Automatic marking of the diffraction angle.
On fig. 5 shows the functional structure of the DRON-3M device.
Rice. 5. Block diagram of the DRON-3M device:
1 - x-ray tube; 2 - sample; 3 - radiation detector; 4 - goniometer mechanism with automatic diffraction angle; 5 - power supply system; 6 - cooling system; 7 - detector signal processing system; 8 - recorder
Monochromaticity is achieved by using a metal foil that transmits one wavelength and, if possible, absorbs other wavelengths. This property is possessed by nickel foil, which absorbs 97% of the radiation from the copper anticathode and transmits with very low absorption = 1.54 E.
Lenses capable of focusing R-rays are not yet available. Therefore, in the design of the device, special devices are used to select and measure angles - goniometers. The goniometer automatically maintains focusing of radiation at any diffraction angle. It automatically fulfills the Wulf-Bragg law due to the fact that the angular velocity of rotation of the sample on which the radiation falls is at any time 2 times less than the velocity of the radiation detector (receiver). Due to this, at any time the detector is located at an angle of 2 to the incident radiation, and the sample at an angle.
The averaging of all arrangements of the reflection planes in the sample occurs due to its rotation around an axis perpendicular to the reflection plane.
As a signal detector, a scintillation counter with a photomultiplier is used, which has a good proportionality to the number of X-ray quanta supplied to it. On the recorder tape, to facilitate the interpretation of X-ray patterns, the diffraction angle is automatically marked by a special electronic circuit associated with the mechanism of movement of the sample located in the goniometer.
4. Interpretation of diffractograms and processing of analysis results
4.1 Determining the size of structural elements
When studying X-ray patterns or diffraction patterns obtained from samples of different polymers or one polymer, but obtained under different conditions, it was noticed that the same X-ray reflections have different widths. This is explained by the small sizes of crystallites and their defects. If we do not take into account the contribution of defectiveness to the signal expansion, then it is possible to determine the sizes of crystallites from the expansion of the reflection, since the contribution of defectiveness is an order of magnitude lower.
The size of a crystallite (L) is its effective size, i.e. some value characterizing the order of size of the crystallite. The value of L can be calculated using the Scherer formula
where is the crystallite size, angstroms; - wavelength, angstroms; - line extension, radians; - Bragg angle, degree; k is a coefficient depending on the shape of the crystallite.
The value is determined at the level of half the height of the line maximum after subtracting the background and the amorphous halo, if it is under the peaks of crystallinity. The coefficient k = 0.9 if the shape of the crystallite is known, and k = 1 if the crystal is spherical. In the latter case, L = 0.75D, where D is the diameter of the sphere. For a powder consisting of homogeneous grains of volume V, with an error less than 20%, the crystal volume is L3 with an error less than 50%.
To obtain the correct value, a standard is used, most often NaCl, with the most intense reflection at 2 = 31–34, or a well-crystallized reference sample of the polymer under study with sufficiently large grains. For him
where B is the line width of the studied polymer; - line width of the standard.
The standard and the test sample are examined with the same slit width and a decrease in the intensity of the primary beam for the standard (the correction should be sufficiently small). On a diffraction curve recorded on a chart tape, the line width is measured in millimeters. In order to apply formulas (4) and (5), it is necessary to perform a recalculation. For example, let one angular degree on the tape correspond to a distance of 27.3 mm. In turn, it is known that one radian corresponds to approximately 57.3 degrees. Then for L in angstroms we get
At 2 \u003d 20º, \u003d 1.54 E, \u003d 2.2 mm. L = 1000 E, and at
= 220 mm and the same values of other parameters L = 10 E. With
= 220 mm, a line of very wide intensity, practically poorly observed, and at = 2.2 mm, this is the maximum measurable line.
Therefore, the limits of application of the method are effective crystallite sizes from 10 to 1000 E. Most industrial polymer samples have crystallite sizes of 50–500 E, i.e. within the limits of applicability of the XRD method. The measurement error is 10-20%.
4.2 Determination of the degree of crystallinity of polymers
X-ray diffraction analysis makes it possible to carry out phase analysis of polymers. A particular case of X-ray phase analysis is the determination of the so-called X-ray degree of crystallinity of polymers. There is a relationship between this characteristic and some properties of polymers (density, hardness, melt yield strength, etc.). However, a change in the degree of crystallinity alone cannot explain the behavior of polymers under different conditions. Additional information about the change in the supramolecular structure obtained by other methods is also required. The X-ray degree of crystallinity does not always coincide with the same characteristic determined by other methods: IR, NMR spectroscopy, dilatometry, thermal methods, etc.
The degree of crystallinity () characterizes the proportion of regularly packed molecules in relation to completely disordered molecules, i.e. the ratio of the crystalline and amorphous phases in the polymer (relative degree of crystallinity), %, is calculated by the formula
The total degree of crystallinity of the polymer, %, is calculated by the formula
where is the area of the crystalline part (above the halo); is the area of the amorphous part (under the halo).
Rice. 6. Dividing the area under the diffraction curve:
background line; - halo line; 1 - isotactic polystyrene; 2 - poly-4-methylpentene-1; 3 - polytetrafluoroethylene; 4 - polypropylene oxide
Practically on the diffraction pattern, the areas under the crystalline peaks and the amorphous holo are measured in a certain limited range of Bragg angles, taking into account the correction for the background, and the ratio of these areas is found. The areas are measured with a planimeter, by cells of millimeter paper or by the weight method: the cut out areas and 1 cm2 of the same paper on which they are applied are weighed, and the areas of each figure are found from the proportion. Examples of area division are shown in fig. 6.
The division of the area under the diffraction curve into crystalline and amorphous parts causes certain difficulties and errors, which depend on the shape of the curve. When carrying out such a procedure, one can use the Hermans empirical criterion, according to which there is always a point between two peaks that does not belong to any of them, if the reflection maxima are separated by at least 2 = 3º from each other. The intensities of the crystalline peaks and the amorphous halo should be measured over as wide a range of the scattering angle as possible.
5. 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 the belonging of a crystal to one of the 230 Fedorov (discovered by E. S. Fedorov) crystal symmetry groups; 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, from experimental data, the values of the electron density r(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:
c(x, y, z) = 1/V? Fhkl exp [-2pi (hx + ky + lz)], (2)
where V is the volume of an elementary cell, Fhkl are the Fourier coefficients, which in R. s. A. are called structural amplitudes, i = v-1. Each structural amplitude is characterized by three integers hkl and is associated with 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 SFhklS and a phase ahkl (phase shift of the reflected wave with respect to the incident), through which the structural amplitude is expressed: Fhkl =SFhkl--S(cosahkl + isinahkl). The diffraction experiment makes it possible to measure only reflection intensities proportional to SFhklS2, but not their phases. Phase determination is the main problem in deciphering the crystal structure. The definition of 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 the structural amplitudes, and from them the intensities of diffraction reflections. The trial and error method, historically the first method for deciphering structures, consists in comparing the experimentally obtained SFhklSexp with the values of SFhklScalc calculated on the basis of a trial model. Depending on the value of the divergence factor
the trial model is accepted or rejected. In the 30s. more formal methods have been developed for crystalline structures, but for non-crystalline objects, trial and error is still practically the only means of interpreting the diffraction pattern.
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. The vectors from the origin to all atoms of the structure (including the vector of zero length up to the first atom) will indicate the position of N maxima of the function of interatomic vectors, the totality of which is called the image of the structure in atom 1. Let us add N more maxima to them, the position of which will indicate N vectors from the second atom, placed at the parallel transfer of the structure to the same origin. Having done this procedure with all N atoms (Fig. 3), we will get N2 vectors. The function describing their position is the Paterson function.
Rice. 3. Scheme for constructing the Paterson function for a structure consisting of 3 atoms.
For the Paterson function P(u, u, w) (u, u, w are the coordinates of points in the space of interatomic vectors), one can obtain the expression:
P (u, x, u) \u003d 2 / V? |Fhkl|2 cos 2р (hu + kх + lш), (4)
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. The difficulty in interpreting the function P (u, u, w) lies in the need to find the coordinates of N atoms from N2 of its maxima, many of which merge due to overlaps that arise when constructing the function of interatomic vectors. The simplest case for deciphering P (u, u, w) is 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 r(x, y, z) obey. 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 (Fig. 3). The final refinement of the coordinates of atoms is carried out on a computer using the least-squares method and, depending on the quality of the experiment and the complexity of the structure, makes it possible to obtain them with an accuracy of thousandths of E (using 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 achievements in X-ray diffraction analysis is the deciphering of 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.
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 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.
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 record reflections in parallel, can reduce the time of the experiment. Diffractometric measurements are superior to photographic recording in terms of sensitivity and accuracy.
Allowing one 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.
7. Literature
1) N. V. Belov, Structural Crystallography, Moscow, 1951;
2) Zhdanov G. S., Fundamentals of X-ray diffraction analysis, M. - L., 1940;
3) James R., Optical principles of X-ray diffraction, Moscow, 1950;
4) Boky G. B., Poray-Koshits M. A., X-ray diffraction analysis. M., 1964;
5) Igolinskaya N.M., X-ray diffraction analysis of polymers, Kemerovo., 2008;
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X-RAY STRUCTURAL ANALYSIS(X-ray diffraction analysis) - methods for studying the atomic structure of matter by distribution in space and intensities of X-ray scattered on the analyzed object. . R. s. A. crystalline materials allows you to set the coordinates of atoms with an accuracy of 0.1-0.01 nm, determine the characteristics of these thermal atoms, including anisotropy and deviations from harmonics. law, receive on eksperim. . to data of distribution in space of density of valence electrons on chemical. bonds in crystals and molecules. These methods are used to study metals and alloys, minerals, inorganic. and organic compounds, proteins, nucleic acids, viruses. Specialist. R.'s methods with. A. allow to study polymers, amorphous materials, liquids, gases.
Among the diffraction methods for studying the atomic structure of matter R. s. A. is the naib. widespread and developed. Its capabilities are complemented by methods neutronography And electronography.Diffraction the picture depends on the atomic structure of the object under study, the nature and wavelength of the x-rays. radiation. To establish the atomic structure of matter Naib. efficient use of x-rays. radiation with a wavelength of ~ 10 nm or less, i.e., of the order of the size of atoms. Especially successfully and with high accuracy R.'s methods of page. A. explore the atomic structure of the crystal. objects, the structure of which has a strict periodicity, and they, thus, are natural. three-dimensional diffraction. grating for x-ray radiation.
Historical reference
At the heart of R. s. A. crystalline substances lies the doctrine of . In 1890 Russian. crystallographer E. S. Fedorov and German. mathematician A. Schonflis (A. Schonflis) completed the derivation of 230 space groups characterizing all possible ways of arranging atoms in crystals. X-ray diffraction. rays on crystals, which is experimental. R.'s foundation with. a., was discovered in 1912 by M. Laue (M. Laue) and his collaborators W. Friedrich and P. Knipping. The theory of X-ray diffraction developed by Laue. rays on crystals made it possible to relate the radiation wavelength, the linear dimensions of the unit cell of the crystal a, b, c, the angles of the incident and diffraction rays by the relations 
Where h, k,l- whole numbers ( crystallographic indices). Relations (1) are called the Laue equations, their fulfillment is necessary for the occurrence of X-ray diffraction. rays on a crystal. The meaning of equations (1) is that between parallel beams, scattered atoms, corresponding to neighboring lattice sites, must be integer multiples.
In 1913, W. L. Bragg and G. V. Wulff showed that diffraction. x-ray the beam can be considered as a reflection of the incident beam from a certain crystallographic system. planes with interplanar spacing d: 
where is the angle between the reflecting plane and the diffraction. beam (Bragg angle). In 1913-14, W. G. and W. L. Braggi were the first to use X-ray diffraction. rays for experiments. verification of the atomic structure of crystals NaCl, Cu, diamond, etc., previously predicted by W. Barlow. In 1916, P. Debye and P. Scherrer proposed and developed diffraction. methods of X-ray diffraction studies of polycrystalline. materials ( Debye - Scherrera method).
as a source of x-rays. radiation were used (and are still used) soldered x-rays. tubes with anodes from dec. metals and, therefore, with different corresponding characteristics. radiation - Fe (= 19.4 nm), Cu (= 15.4 nm), Mo (= 7.1 nm), Ag (= 5.6 nm). Later, an order of magnitude more powerful tubes with a rotating anode appeared; powerful, having a white (continuous) radiation spectrum source - X-ray. synchrotron radiation. With the help of a system of monochromators, it is possible to continuously change the synchretron X-ray used in the study. radiation, which is of fundamental importance when used in R. s. A. effects of anomalous scattering. As a radiation detector in R. s. A. serves as an x-ray. photographic film, to-ruyu displace scintillation and semiconductor detectors. Efficiency will be measured. systems has increased dramatically with the use of coordinate one-dimensional and two-dimensional detectors.
Quantity n quality of information obtained with the help of R. s. a., depend on the accuracy of measurements and processing of experiments. data. Diffraction processing algorithms. data are determined by the used approximation of the theory of interaction x-rays. radiation with matter. In the 1950s the use of computers in the technique of X-ray diffraction experiments and for processing experiments began. data. Created fully automated systems for the study of crystalline. materials, to-rye conduct an experiment, processing experiments. data, main procedures for constructing and refining the atomic model of the structure and, finally, graphic. presentation of research results. However, with the help of these systems, it is not yet possible to study automatically. mode crystals with pseudosymmetry, twin samples and crystals with other structural features.
Experimental Methods x-ray structural analysis
To implement the diffraction conditions (1) and register the position in space and the intensities of the diffracted X-ray. radiation serve as x-rays. cameras and x-rays. diffractometers with registration of radiation respectively photogr. methods or radiation detectors. The nature of the sample (single crystal or polycrystal, a sample with a partially ordered structure or an amorphous body, liquid or gas), its size, and the problem to be solved determine the required exposure and the accuracy of the scattered x-ray recording. radiation and, consequently, a certain method of R. s. A. To study single crystals when used as a source of x-rays. sealed-off X-ray radiation. tube sufficient sample volume ~10 -3 mm 3 . To obtain high-quality diffraction picture, the sample must have the most perfect structure, and its blockiness does not interfere with structural studies. The real structure of large, almost perfect single crystals is investigated by x-ray topography, to-ruyu is sometimes also referred to as R. s. A.
Laue method- the simplest method for obtaining x-ray patterns of single crystals. The crystal in Laue's experiment is motionless, and the x-ray used. radiation has a continuous spectrum. The location of the diffraction spots on Lauegrams depends on the size of the unit cell and crystal symmetry, as well as on the orientation of the sample with respect to the incident x-ray. beam. The Laue method makes it possible to attribute a single crystal to one of the 11 Laue symmetry groups and to establish the orientation of its crystallographic. axes to within angular. minutes (see Laue method). By the nature of diffraction spots on the Lauegrams, and especially by the appearance of asterism (blurring of spots), it is possible to identify the internal. stresses and certain other structural features of the sample. The Laue method checks the quality of single crystals and selects the most. perfect samples for a more complete structural study (X-ray goniometric methods; see below).
Methods of rocking and rotation of the sample determine the periods of repetition (broadcast) along the given crystallographic. directions, check the symmetry of the crystal, and measure the intensity of the diffraction. reflections. The sample is vibrated during the experiment. or rotate. movement about an axis coinciding with one of the crystallographic. axes of the sample, to-ruyu previously oriented perpendicular to the incident X-ray. beam. Diffraction picture created by monochromatic. radiation, is registered on X-ray. film in a cylindrical cassette, the axis of which coincides with the axis of oscillation of the sample. Diffraction Spots with such shooting geometry on a developed film turn out to be located on a family of parallel lines (Fig. 1). Return period T along the crystallographic direction is:
Where D- the diameter of the cassette, - the distance between the corresponding straight lines on the radiograph. Since it is constant, the Laue conditions (1) are satisfied by changing the angles during rocking or rotation of the sample. Usually on the x-ray patterns of rocking and rotation of the sample, diffraction. spots overlap. To avoid this unwanted effect, you can reduce the angle. sample oscillation amplitude. This technique is used, for example, in R. s. A. proteins, where rocking radiographs are used to measure diffraction intensities. reflections. 
Rice. Fig. 1. X-ray diffraction pattern of the rocking of the seidoserite mineral Na 4 MnTi(Zr,Ti) 2 0 2 (F,OH) 2 2.
X-ray methods. For a complete structural study of a single crystal by X-ray methods. A. it is necessary to determine the position in space and measure the integral intensities of all diffraction. reflections arising from the use of radiation with a given. To do this, during the experiment, the sample must, with an accuracy of the order of arc. minutes to take orientations, for which the conditions (1) are satisfied consistently for all families of crystallographic. sample planes; while many are registered. hundreds and even thousands of diff. reflexes. When registering diffraction x-ray pictures. photographic film, the intensities of reflections are determined by a microdensitometer by the degree of blackening and the size of the diffraction. spots. In decomp. types of goniometers are implemented diff. geom. diffraction registration schemes. paintings. A complete set of diffraction intensities. reflections are obtained on a series of radiographs, reflections are recorded on each radiograph, on crystallographic. indexes to-rykh superimposed def. restrictions. For example, reflections of the type hk0, hk1(rice. 2)
. To establish the atomic structure of a crystal whose unit cell contains ~100 atoms, it is necessary to measure several. thousand diff. reflections. In the case of protein single crystals, the volume of the experiment increases to 10 4 -10 6 reflections. 
Rice. Fig. 2. X-ray diffraction pattern of the mineral seidoserite, obtained in the Weisenberg X-ray goniometer. The registered diffraction reflections have indices. Reflections located on the same curve are characterized by a constant index k.
When replacing film with X-ray counters. quanta increase the sensitivity and accuracy of measuring diffraction intensities. reflections. In modern automatic diffractometers are provided with 4 axes of rotation (3 for the sample and 1 for the detector), which makes it possible to implement diffraction registration methods of various geometries. reflections. Such a device is universal, it is controlled by a computer and specially developed algorithms and programs. The presence of a computer allows you to introduce feedback, optimization of measurements of each diffraction. reflections and, therefore, natures. way to plan the entire diffraction. experiment. Measurements of intensities are made with the statistic required for the structural problem to be solved. accuracy. However, an increase in the intensity measurement accuracy by an order of magnitude requires an increase in the measurement time by two orders of magnitude. The quality of the test sample imposes a limitation on the accuracy of measurements. For protein crystals (see below), the experiment time is reduced by using two-dimensional detectors, in which many measurements are carried out in parallel. tens of diffraction reflections. In this case, the possibility of optimizing measurements at the level of department is lost. reflex.
Method for the study of polycrystals (Debye-Scherrer method). For R. s. A. crystalline powders, ceramic materials, etc. polycrystalline. objects consisting of a large number of small, randomly oriented relative to each other single crystals, monochromatic is used. x-ray radiation. X-ray from polycrystalline. sample (de-bayogram) is a collection of concentric-rich. rings, each of which consists of diffraction. reflections from diff. crystallographic systems oriented in different grains. planes with a certain interplanar distance d. Kit d and their corresponding diffraction intensities. reflections are individual for each crystal. substances. The Debye-Scherrer method is used in the identification of compounds and the analysis of mixtures of polycrystalline. substances by quality. and quantities. the composition of the components of the mixture of phases. An analysis of the distribution of intensities in Debye rings makes it possible to estimate the grain sizes, the presence of stresses, and preferential orientations (texturing) in the arrangement of grains (see Fig. Radiography of materials, Debye - Scherrera method).
In the 1980s - 90s. in R. s. A. began to apply the method of clarifying the atomic structure of the crystal. substances by diffraction. data from polycrystalline. materials proposed by X. M. Rietveld (N. M. Rietveld) for neutron diffraction. research. The Rptveld method (full-profile analysis method) is used when the approximate structural model of the studied compound is known; in terms of the accuracy of the results, it can compete with X-ray diffraction methods for studying single crystals.
Study of amorphous materials and partially ordered objects. The lower the degree of ordering of the atomic structure of the analyte, the more blurred, diffuse character is the X-ray scattered by it. radiation. However, the diffraction studies of even amorphous objects make it possible to obtain information about their structure. Thus, the diameter of the diffuse ring in the X-ray pattern from an amorphous substance (Fig. 3) allows us to estimate the avg. interatomic distances in it. With an increase in the degree of order in the structure of objects, diffraction. the picture becomes more complex (Fig. 4) and therefore contains more structural information. 
Rice. 3. X-ray pattern of an amorphous substance - cellulose acetate.
Rice. 4. Radiographs of biological objects: a - hair; b - sodium salt of DNA in a wet state; c - DNA sodium salt textures.
Small angle scattering method. In the case when the dimensions of the inhomogeneities in the object of study exceed the interatomic distances and range from 0.5-1 to 10 3 nm, i.e., many times greater than the wavelength of the radiation used, the scattered X-ray. radiation is concentrated near the primary beam - in the region of small scattering angles. The intensity distribution in this region reflects the structural features of the object under study. Depending on the structure of the object and the size of the inhomogeneities, the X-ray intensity. scattering is measured in angles from fractions of a minute to several. degrees.
low angle scattering is used to study porous and fine materials, alloys and biol. objects. For protein molecules and nucleic acids in solutions, the method allows, with a low resolution, to determine the shape and size of an individual molecule, they say. mass, in viruses - the nature of the mutual stacking of their constituent components (protein, nucleic acids, lipids), in synthetic. polymers - the packing of polymer chains, in powders and sorbents - the distribution of particles and pores by size, in alloys - to fix the appearance of new phases and determine the size of these inclusions, in textures (in particular, in liquid crystals) - the packing of particles (molecules) into various kinds of supramolecular structures. The low-angle method proved to be effective. scattering and for studying the structure of Langmuir films. It is also used in the industry to control the manufacturing processes of catalysts, fine coals, etc.
Analysis of the atomic structure of crystals
Determination of the atomic structure of crystals includes: establishing the shape and dimensions of the unit cell, the symmetry of the crystal (its belonging to one of the 230 Fedorov groups) and the coordinates of the basic atoms of the structure. Precision structural studies allow, in addition, to obtain quantities. characteristics of thermal motions of atoms in a crystal and the spatial distribution of valence electrons in it. Methods of Laue and rocking the sample determine the metric of the crystal. gratings. For further analysis, it is necessary to measure the intensities of all possible diffraction. reflections from the test sample for a given l. Primary processing of experiments. data takes into account the geometry of the diffraction. experiment, absorption of radiation in the sample, and other more subtle effects of the interaction of radiation with the sample.
The three-dimensional periodicity of a crystal makes it possible to expand the distribution of its electron in space in a Fourier series:
Where V- the volume of the unit cell of the crystal, Fhkl- Fourier coefficients, which are in R. s. A. called structural amplitudes. Each structural amplitude is characterized by integers h, k, l- crystallographic. indices in accordance with (1) and uniquely corresponds to one diffraction. reflection. Expansion (2) is physically realized in diffraction. experiment.
Main the complexity of the structural study lies in the fact that the usual diffraction. experiment makes it possible to measure the intensity of diffraction. bundles I hkl but does not allow fixing their phases. For a mosaic crystal in a kinematic approach 
. Analysis of experiments. The array, taking into account the regular extinctions of reflections, makes it possible to unambiguously establish its belonging to one of the 122 roentgens. symmetry groups. In the absence of anomalous scattering, diffraction the picture is always centrosymmetric. To determine the Fedorov symmetry group, it is necessary to find out independently whether the crystal has a center of symmetry. This problem can be solved on the basis of an analysis of the anomalous component of X-ray scattering. rays. In the absence of the latter, statistical curves are constructed. distributions over their values, these distributions are different for centrosymmetric and acentric crystals. The absence of a center of symmetry can be unambiguously established by physical. properties of the crystal (pyroelectric, ferroelectric, etc.).
The Fourier transform of relation (2) makes it possible to obtain calculated f-ly for calculating the quantities Fhkl(in the general case - complex): 
where - at. X-ray scattering factor radiation by an atom jj, x j , y j , z j- its coordinates; summation goes over all N elementary cell atoms.
The problem, inverse to the structural study, is solved as follows: if the atomic model of the structure is known, then the moduli and phases of the structural amplitudes and, consequently, the diffraction intensity are calculated using (3). reflections. Diffraction experiment makes it possible to measure many hundreds of amplitudes unrelated by symmetry, each of which is determined by (3) by a set of coordinates of the basic (symmetry-independent) atoms of the structure. There are significantly fewer such structural parameters than modules; therefore, there must be connections between the latter. The theory of structural analysis has established connections of various types: inequalities, linear inequalities, structural products, and determinants of the connection of structural amplitudes.
On the basis of naib, effective statistic. ties developed [J. Carle (J. Karle) and X. A. Hauptman (H. A. Hauptman), Nobel Prize, 1985] so-called. direct methods for determining the phases of structural amplitudes. If we take a trio of structural amplitudes large in moduli, the indices of which are related by simple relations h 1 + h 2 + h 3 = 0, k 1 + k 2 + k 3 = 0, l 1 + l 2 + l 3 = 0, then naib. the probable sum of the phases of these amplitudes will be equal to zero:
The probability of fulfilling equality is higher, the greater the product of the special. in the manner of normalized structural amplitudes included in this relation. As the number of atoms increases N in the elementary cell of the crystal, the reliability of the ratio decreases. In practice, much more complex statistics are used. relations and rather rigorous estimates of the probabilities of fulfillment of these relations. Calculations on these ratios are very cumbersome, the algorithms are complex and are implemented only on powerful modern computers. COMPUTER. Direct methods give the first approximate values of the phases and only the max. strong in normalized modules of structural amplitudes.
For the practice of structural studies, automatic procedures are important. refinement of phases of structural amplitudes. Based on an approximate set of phases of the strongest structural amplitudes and according to the corresponding experiments. modules, according to (2), the first approximate distribution of the electron density in the crystal is calculated. Then it is modified on the basis of physical. and crystallochem. information about the properties of this distribution. For example, at all points in space 
, according to the modifications. distribution by Fourier inversion, the refined phases are calculated and, together with the experiment. values are used to construct the next approximation, and so on. After obtaining sufficiently accurate values, according to (2), a three-dimensional distribution of the electron density in the crystal is constructed. It is essentially an image of the structure under study, and all the difficulty in obtaining it is caused by the lack of converging lenses for x-rays. radiation.
The correctness of the obtained atomic model is checked by comparing experiments. and modules of structural amplitudes calculated by (3). Quantity. the characteristic of such a comparison is the divergence factor
This factor makes it possible to obtain the optimum by trial and error. results. For non-crystalline objects it is practically unities. diffraction interpretation method. paintings.
Determining the phases of structural amplitudes by direct methods becomes more complicated with an increase in the number of atoms in the unit cell of the crystal. Pseudosymmetry and certain other features of its structure also limit the possibilities of direct methods.
Another approach to determining the atomic structure of crystals from x-rays. diffraction data was proposed by A. L. Paterson. The atomic model of the structure is based on the analysis of the function of interatomic vectors P(u,v,w)(f-tion of Paterson), which is calculated from the experiment. values . The meaning of this function can be explained using the scheme of its geom. construction. An atomic structure containing in a unit cell N atoms, we place it parallel to itself so that the first atom is at the origin. If we multiply the atomic weights of all atoms of the structure by the value of the atomic weight of the first atom, then we get the weights of the first N peaks f-tsii interatomic vectors. This is the so-called. image of the structure in the first atom. Then, at the origin of coordinates, we place the image of the structure constructed in the same way in the second atom, then in the third, etc. Having done this procedure with all N atoms of the structure, we get N 2 peaks of the Paterson function (Fig. 5). Since atoms are not points, the resulting function P(u,v,w) contains fairly blurred and overlapping peaks: 
Rice. 5. Scheme for constructing the function of interatomic vectors for a structure consisting of three atoms.
[ - volume element in the vicinity of the point ( x,y,z)]. The function of interatomic vectors is constructed from the squares of the modules of experiments. structural amplitudes and is a convolution of the electron density distribution with itself, but after inversion at the origin. 
Rice. 6. Mineral baotite Ba 4 Ti 4 (Ti, Nb) 4 O 16 Cl; a - function of interatomic vectors, projection onto the plane ab, lines of equal level of function values are drawn at equal arbitrary intervals; b - projection of the distribution of electron density on the plane ab, obtained by interpreting the function of interatomic vectors and refining the atomic model, the thickening of lines of equal level correspond to the positions of atoms in the structure; c - projection of the atomic model of the structure onto the ab plane in Pauling polyhedra. Si atoms are located inside tetrahedra of oxygen atoms, Ti and Nb atoms are located in octahedrons of oxygen atoms. The tetrahedra and octahedrons in the structure of baotite are connected as shown in the figure. Ba and C1 atoms are shown by black and light circles. Part of the elementary cell of the crystal, shown in figures a and b, corresponds to the figure in the square marked with dashed lines.
Difficulties in interpretation P(u,v,w) are related to the fact that among N 2 peaks of this function, it is necessary to recognize the peaks of one image of the structure. The maxima of the Paterson function overlap significantly, which further complicates its analysis. Naib. easy to analyze the case when the structure under study consists of one heavy atom and several. much lighter atoms. In this case, the image of the structure in a heavy atom stands out in relief against the background of the remaining peaks P(u,v,w). A number of systematic methods have been developed. analysis of functions of interatomic vectors. Naib. effective ones are superpositions. methods when two or more copies P(u,v,w) in parallel position are superimposed on each other with corresponding offsets. At the same time, peaks that naturally coincide on all copies highlight one or more of the N the original image of the structure. As a rule, for unities. images of the structure have to use add. copies P(u,v,w). The problem comes down to finding the necessary mutual offsets of these copies. After localization on the superposition. synthesis of the approximate distribution of atoms in the structure, this synthesis can be subjected to Fourier inversion, and so on. it allows one to obtain the phases of the structural amplitudes. The latter together with the experiment. values are used for construction. All superposition procedures. methods algorithmized and implemented in automatic. mode on the computer. On fig. 6 shows the atomic structure of the crystal, established by superposition methods according to the Paterson function.
Experiments are being developed. methods for determining the phases of structural amplitudes. Phys. The basis of these methods is the Renninger effect - multibeam X-ray. diffraction. If available at the same time x-ray diffraction reflections, there is a transfer of energy between them, which depends on the phase relationships between the data of the diffraction. bundles. The entire pattern of intensity changes is limited by the arc. seconds and for mass structural studies, this technique is practical. has not yet acquired value.
In independent. section R. s. A. allocate precision structural studies of crystals, allowing to obtain diffraction. given not only the model of the atomic structure of the compounds under study, but also the quantities. characteristics of thermal vibrations of atoms, including the anisotropy of these vibrations (Fig. 7) and their deviations from harmonics. law, as well as the spatial distribution of valence electrons in crystals. The latter is important for studying the relationship between atomic structure and physical. properties of crystals. For precision research, special experimental methods. measurements and processing diffraction. data. In this case, accounting is required at the same time. reflections, deviations from the kinematics of diffraction, taking into account dynamic. corrections of the theory of diffraction, and other subtle effects of the interaction of radiation with matter. When specifying the structural parameters, the name method is used. squares, and the correlation between the refined parameters is of paramount importance. 
Rice. Fig. 7. Ellipsoids of anisotropic thermal vibrations of atoms of the stable nitrogen strong radical C 13 H 17 N 2 O 2.
R. s. A. used to establish a connection between the atomic structure and the physical. properties, superionic conductors, laser and nonlinear optical. materials, high-temperature superconductors, and others. A. obtained unique results in the study of the mechanisms of phase transitions in solids and biol. activity of macromolecules. Thus, the anisotropy of acoustic absorption. waves in paratellurite single crystals is related to the anharmonicity of thermal vibrations of Te atoms (Fig. 8)
. The elastic properties of lithium tetraborate Li 2 B 4 O 7 , opening up prospects for its use as an acoustic detector. waves, due to the nature of the chemical. links in this connection. With the help of R. s. A. study the distribution in the crystal of valence electrons that realize interatomic bonds in it. These relationships can be explored using the strain distribution. electron density, which is the difference
where is the distribution of electron density in the crystal, is the sum of spherically symmetrical densities of free (not chemically bonded) atoms of a given structure, which are located respectively at points with coordinates x i , y i , z i. When established by X-ray. diffraction deformation data. electron density max. it is difficult to take into account the thermal vibrations of atoms, beings. image correlating with the nature and directions of chemical. connections. So, deformation. density reflects the redistribution in space of that part of the electron density of atoms, which is directly involved in the formation of chemical. connections (Fig. 9). 
Rice. Fig. 8. The nearest environment of tellurium by O atoms in the structure (a) and the anharmonic component of the probability density distribution of the Te atom at a given point in space during thermal vibrations (b). Positive (solid) and negative (dashed) lines of equal level are drawn through 0.02 -3.
Rice. Fig. 9. Cross section of the synthesis of the deformation electron density of a Li 2 B 4 O 7 crystal by a plane passing through the O atoms of the triangular group BO 3, in the center of which there is an atom B. The maxima on the segments B - O indicate the covalent nature of the bonds between these atoms. Dashed lines indicate the regions from which the electron density has shifted to chemical bonds. Lines of equal level are drawn through 0.2 .
Rice. 10. Ordered arrangement of Sr atoms over lanthanum positions in the structure Cu atoms
Structural studies of high-temperature superconductors made it possible to establish their atomic structure and its relationship with their physical. properties. It was shown that in single crystals the transition temperature to the superconducting state T s depends not only on the number of Sr, but also on the way it is statistical. accommodation. The uniform distribution of Sr atoms in the structure is optimal for superconducting properties. Sr concentration in def. layers of the structure (Fig. 10) leads to the loss of part of the oxygen in these layers and to a decrease T s. For crystals 
R.'s methods of page. A. ordering in the arrangement of O atoms was established. Within the limits of one crystal, the presence of regions of local composition rhombic in symmetry was established 
With T s ~90 K and regions are in [СuО 6 ]-octahedrons. Oxygen deficiency is shown by the absence of one oxygen vertex in one of the Cu polyhedra. Positions completely occupied by La atoms are shown by black circles. The open circles are the positions of lanthanum, in which all Sr atoms are concentrated and statistically located.
With T c ~ 60 K. In crystals with an amount of oxygen less than 6.5 atoms per unit cell, along with rhombic regions. symmetries of the local composition, regions of tetragonal symmetry of the local composition appear, which do not pass into the superconducting state.
Rice. 11. Atomic model of the molecule of guanyl-specific ribonuclease C 2, built on the basis of X-ray diffraction study of single crystals of this protein with a resolution of 1.55
To solve many problems of solid state physics, chemistry, molecular biology, etc. The combined use of X-ray diffraction analysis and resonance methods (EPR, NMR, etc.) is very effective. At a research of an atomic structure of proteins, nucleinic to - t, viruses, etc. objects of molecular biology there are specific. difficulties. macromolecules or. larger biol. objects must first be obtained in monocrystalline. form, after which it is possible to apply all methods of R. to their research. a., developed for the study of crystalline. substances. The problem of phases of structural amplitudes for protein crystals is solved by the method of isomorphic substitutions. Along with single crystals of the studied native protein, single crystals of its derivatives with heavy atomic additives isomorphic to the crystals of the studied protein are obtained. Difference Paterson functions for derivatives and native protein make it possible to localize the positions of heavy atoms in the unit cell of a crystal. The coordinates of these atoms and the sets of modules of the structural amplitudes of the protein and its heavy atomic derivatives are used in special. algorithms for estimating the phases of structural amplitudes. In protein crystallography, step-by-step methods for establishing the atomic structure of macromolecules are used with succession. transition from low to higher resolution (Fig. 11). Developed and special methods for refining the atomic structure of macromolecules by X-ray. diffraction data. The volume of calculations is so large that they can be effectively implemented only on the most powerful computers.
R.'s questions with. a., associated with the study of the real structure of a solid body by diffraction. data discussed in Art. Radiography of materials.
Lit.: Belov N.V., Structural crystallography, Moscow, 1951; B about to and y G. B., Poray-Koshits M. A., X-ray structural analysis, 2nd ed., Vol. 1, M., 1964; Lipson G., Kokren V., Determination of the structure of crystals, trans. from English, M., 1956; Burger M., Structure of crystals and vector space, transl. from English, M., 1961; Gin'e A., Radiography of crystals. Theory and practice, trans. from French, Moscow, 1961; Stout G, H., J e n s e n L. H., X-ray structure determination, N. Y.-L., 1968; X e and er D. M., X-ray diffractometry of single crystals, L., 1973; Blundel T., Johnson L., Protein crystallography, trans. from English, M., 1979; Vainshtein BK, Symmetry of crystals. Methods of structural crystallography, M., 1979; Electron and magnetization densities in molecules and crystals, ed. by P. Becker, N. Y.-L., 1980; Crystallography and crystal chemistry, M., 1986; Structure and physical properties of crystals, Barcelona, 1991. V. I. Simonov.
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.
The Kossel method is also used - a stationary single crystal is removed in a widely divergent beam of monochromatic characteristic radiation.
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 = l 0 = 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 by at least 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).
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 the sample 2 is located in front of the 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).
Fig.2 
Fig.3 
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 scheme of the camera for shooting by the method of rotation is shown in Fig. 1.
The primary beam is cut out by diaphragm 2 (with two round holes) and falls on crystal 1. The crystal is mounted on goniometric head 3 so that one of its important directions (such as , [ 010], ) 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 using a motor 4. The diffraction pattern is recorded on photographic film 5 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, a cassette with a flat film is provided 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 (dotted line in Figure 2). 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 a crystal rotates around the main (or important) 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.
Fig.1 
Fig.2 
Method of powders (polycrystals).
The powder method is used to obtain a diffraction pattern from polycrystalline substances in the form of a powder or a massive sample (polycrystal) with a flat microsection surface. When samples are illuminated with monochromatic or characteristic X-ray radiation, a distinct interference effect appears in the form of a system of coaxial Debye cones, the axis of which is the primary beam (Fig. 1).
The diffraction conditions are satisfied for those crystals in which the (hkl) planes form an angle q with the incident radiation. The lines of intersection of Debye cones with the film are called Debye rings. To register an interference pattern in the powder method, several methods are used to position the film in relation to the sample and the primary x-ray beam: shooting on flat, cylindrical, and cone film. Registration can also be done using counters. For this purpose, a diffractometer is used.
With the photographic method of registering an interference pattern, several types of surveys are used:
1.
Flat film. There are two ways to position the film: front and rear (reverse) shooting. In front shooting, the sample is placed in front of the film with respect to the direction of the primary beam of rays. A number of concentric circles are recorded on the film, which correspond to the intersection with the plane of the film of interference cones with an opening angle q< 3 0 0 . Измерив диаметр колец, зарегистрированных на пленке, можно определить угол q для соответствующих интерференционных конусов. Недостатком такого способа съемки является то, что на фотопленке регистрируется только небольшое число дифракционных колец. Поэтому переднюю съемку на плоскую пленку применяют в основном для исследования текстур, при котором необходимо определить распределение интенсивности по полному дифракционному кольцу. При задней съемке образец располагается по отношению к пучку рентгеновских лучей сзади пленки. На пленке регистрируются максимумы, отвечающие углу q >3 0 0 . Reverse shooting is used for accurate determinations of periods and for measuring internal stresses.
2. Cylindrical film.
The axis of the cylinder along which the film is located is perpendicular to the primary beam (Fig. 2).
The angle q is calculated from the measurement of the distances between the lines 2 l, corresponding to the same interference cone, according to the relations:
2l = 4qR; q = (l/ 2R) (180 0 / p),
where R is the radius of the cylindrical cassette along which the film was placed. In a cylindrical camera, the film can be placed in several ways - symmetrical and asymmetric ways of loading the film. With the symmetrical charging method, the ends of the film are located near the diaphragm, through which the beam of primary rays enters the chamber. To exit this beam from the chamber, a hole is made in the film. The disadvantage of this method of charging is that during photo processing the film is reduced in length, as a result, when calculating the X-ray pattern, one should use not the value of the radius R along which the film was located during the shooting, but a certain value R eff. R eff. is determined by shooting a reference substance with known lattice periods. According to the known period of the grating of the standard, theoretically the reflection angles q calc are determined. , from the values of which, in combination with the distances between the symmetrical lines measured from the X-ray diffraction pattern, determine the value of R eff.
With the asymmetric method of loading the film, the ends of the film are placed at an angle of 90 0 with respect to the primary beam (two holes are made in the film for the entry and exit of the primary beam beam). In this way, R eff. determined without taking the standard. To do this, measure the distances A and B between the symmetrical lines on the radiograph (Fig. 3):
R eff. \u003d (A + B) / 2p;
A general view of the Debye camera for shooting debyegrams is shown in Figure 4.
The cylindrical body of the camera is mounted on a stand 3, equipped with three set screws. The axis of the cylinder is horizontal. The sample (thin column) is placed in holder 1, which is fixed in the chamber with a magnet. The centering of the sample when installing it in the holder is carried out in the field of view of a special mounting microscope with low magnification. The photographic film is placed on the inner surface of the housing, pressed with special spacer rings fixed on the inner side of the chamber cover 4. The X-ray beam washing the sample enters the chamber through the collimator 2. Since the primary beam, falling directly on the film behind the sample, veils the X-ray pattern, it is intercepted on the way to the film by a trap. To eliminate the dottedness of the rings on the X-ray diffraction pattern of a coarse-grained sample, it is rotated during shooting. The collimator in some cameras is made in such a way that by inserting lead or brass circles (screens) with holes into special grooves in front and behind it, you can cut out a beam of rays of a round or rectangular cross section (round and slit diaphragms). The dimensions of the apertures of the diaphragm should be chosen so that the beam of rays washes the sample. Typically, cameras are made so that the diameter of the film in it is a multiple of 57.3 mm (ie 57.3; 86.0; 114.6 mm). Then the calculation formula for determining the angle q, deg, is simplified. For example, for a standard Debye chamber with a diameter of 57.3 mm, q i = 2l/2. Before proceeding to the determination of interplanar distances using the Wulf-Bragg formula:
2 d sin q = n l ,
It should be taken into account that the position of the lines on the X-ray diffraction pattern from the column slightly changes depending on the radius of the sample. The fact is that due to the absorption of X-rays, a thin surface layer of the sample, and not its center, participates in the formation of the diffraction pattern. This leads to a shift of the symmetrical pair of lines by:
D r = r cos 2 q , where r is the sample radius.
Then: 2 l i = 2 l meas. ± D 2l - D r.
Correction D 2l associated with a change in the distance between a pair of lines due to film shrinkage during photo processing is tabulated in reference books and textbooks on X-ray diffraction analysis. According to the formula q i \u003d 57.3 (l / 2 R eff.). After determining q i, sinq i is found and the interplanar distance is determined from them for lines obtained in K a - radiation:
(d/n) i = l K a / 2 sin q i K a .
To separate the lines obtained by diffraction from the same radiation planes l K b , filtered characteristic radiation is used or a calculation is carried out in this way. Because:
d / n \u003d l K a / 2 sin q a \u003d l K b / 2 sin q b;
sin q a / sin q b \u003d l K a / l K b " 1.09, whence sinq a \u003d 1.09 sinq b.
In the sinq series, find the values corresponding to the most intense reflections. Next, there is a line for which sinq is equal to the calculated value, and its intensity is 5-7 times less. This means that these two lines arose due to the reflection of rays Ka and Kb, respectively, from planes with the same distance d/n.
Determining the periods of crystal lattices is associated with some errors, which are associated with inaccurate measurements of the Wolf-Bragg angle q. High accuracy in determining the periods (error 0.01-0.001%) can be achieved by using special methods of shooting and processing the results of measuring radiographs, the so-called precision methods. Achieving maximum accuracy in determining the lattice periods is possible by the following methods:
1. using the values of interplanar distances determined from the angles in the precision region;
2. a decrease in error as a result of the use of precise experimental techniques;
3. using methods of graphical or analytical extrapolation.
The minimum error D d/d is obtained when measuring at angles q = 80¸ 83 0 . Unfortunately, not all substances give lines at such large angles on the x-ray. In this case, a line at the largest possible angle q should be used for measurements. An increase in the accuracy of determining cell parameters is also associated with a decrease in random errors, which can only be taken into account by averaging, and taking into account systematic errors, which can be taken into account if the causes of their occurrence are known. Accounting for systematic errors in determining the lattice parameters is reduced to finding the dependence of systematic errors on the Bragg angle q , which allows extrapolation to angles q = 90 0 , at which the error in determining interplanar distances becomes small. Random errors are.
