Spectral analysis of the alloy. How to choose a metal and alloy analyzer: market overview and reviews of the best models. Equipment used for chemical analysis

Every year, the methods of laboratory analysis in the ANO "Center for Chemical Expertise" are becoming more and more perfect, but the leading positions are undoubtedly occupied by the spectral analysis of metals and products made from them. Moreover, it is based not only on identifying the elements that make up their composition, but also on determining their characteristics and properties that must be taken into account in these systems.

What is spectral analysis of metals

This study allows you to identify the quantitative and qualitative composition of metals. It is carried out directly in the laboratory by experienced professionals. Determination of quantitative indicators consists in calculating the volume, the content of impurities indicated in numbers and percentage.

And the definition quality indicators means identifying the properties and characteristics of a particular component. Analyzes of both types are carried out using modern equipment and special reagents.

Selection of materials for spectral research

Undoubtedly, spectral analysis of metals can be performed in different areas... Most often it is applicable in the field of metallurgy, because institutions must be created there that, using special methods, identify quality production products specific brand.

Spend this analysis it is advisable before purchasing a large batch of metal products in order to make sure of its quality and suitability. For the analysis, the customer will only need to provide the laboratory with a small sample of the metal and fill out the necessary documentation. You can get the research results on hand within the time frame established by a specific institution.

SPECTRAL ANALYSIS (with the help of emission spectra) has application in almost all sectors of the economy. It is widely used in the metal industry for the rapid analysis of iron, steel, cast iron, as well as various special steels and finished metal products, to establish the purity of light, non-ferrous and precious metals... Spectral analysis is widely used in geochemistry in studying the composition of minerals. IN chemical industry and related industries, spectral analysis is used to establish the purity of manufactured and used products, for the analysis of catalysts, various residues, sediments, muds and rinsing waters; in medicine - for the discovery of metals in various organic tissues. A number of special problems, difficult to solve or not at all solvable in any other way, are solved by means of spectral analysis quickly and accurately. This includes, for example, the distribution of metals in alloys, investigation of sulfide and other inclusions in alloys and minerals; this kind of research is sometimes referred to as local analysis.

The choice of one or another type of spectral apparatus from the point of view of the sufficiency of its dispersion is made depending on the goal and objectives of spectral analysis. For the study of platinum metals (Ru, Rh, Pd, Os, Ir, Pt), as well as Fe, Co, Ni, Cr, V, Mo, W, Ti, Mn, Zr, Re, Nb, and Ta, quartz spectrographs with higher dispersion, giving for wavelengths 4000-2200 Ӑ a strip of the spectrum with a length of at least 22 cm. For other elements m. b. Apparatuses are used that give spectra 7-15 cm long. Spectrographs with glass optics are generally of lesser importance. Of these, combined instruments are convenient (for example, from the firms of Gilger and Füss), which, if desired, can be used as a spectroscope and spectrograph. The following energy sources are used to obtain spectra. one) Flame burning mixture - hydrogen and oxygen, a mixture of oxygen and lamp gas, a mixture of oxygen and acetylene, or finally air and acetylene. In the latter case, the temperature of the light source reaches 2500-3000 ° C. The flame is most suitable for obtaining spectra of alkali and alkaline earth metals, as well as for elements such as Cu, Hg and Tl. 2) Voltaic arc. a) Normal, ch. arr. direct current, with a force of 5-20 A. It is used with great success for the qualitative analysis of difficult-to-fuse minerals, which are introduced into the arc in the form of pieces or finely ground powders. For the quantitative analysis of metals, the use of a conventional voltaic arc has a very significant drawback, which is that the surface of the analyzed metals is covered with an oxide film and the arc burning eventually becomes uneven. The temperature of the voltaic arc reaches 5000-6000 ° C. b) Intermittent arc (Abreissbogen) of direct current with a strength of 2-5 A at a voltage of about 80 V. With the help of a special device, the arc is interrupted 4-10 times per second. This method of excitation reduces oxidation of the surface of the analyzed metals. At a higher voltage - up to 220 V and a current of 1-2 A - an intermittent arc can also be used for the analysis of solutions. 3) Spark discharges, obtained using an induction coil or, more often, a DC or (preferably) AC transformer with a power of up to 1 kW, giving in the secondary circuit 10000-30000 V. Three types of discharges are used, a) Spark discharges without capacitance and inductance in the secondary circuit, called sometimes high voltage arc (Hochspannungsbogen). The analysis of liquids and molten salts using such discharges is highly sensitive. b) Spark discharges with capacitance and inductance in the secondary circuit, often also called condensed sparks, represent a more universal source of energy, suitable for exciting the spectra of almost all elements (except for alkali metals), as well as gases. The connection diagram is given in Fig. one,

where R is a rheostat in the primary circuit, Tr is an alternating current transformer, C 1 is a capacitance in the secondary circuit I, S is a switch for changing the inductance L 1, U is a synchronous breaker, LF is a spark arrestor, F is a working spark gap. The secondary circuit II is tuned into resonance to the secondary circuit I using inductance and variable capacitance C 2; a sign of the presence of resonance is the highest current shown by the milliammeter A. The purpose of the secondary circuit II of the synchronous breaker U and the spark arrester LF is to make electrical discharges possibly uniform both in nature and in number for a certain period of time; during normal work, such additional devices are not introduced.

When researching metals in the secondary circuit, a capacity of 6000-15000 cm3 and an inductance of up to 0.05-0.01 N are used. For the analysis of liquids, a water rheostat with a resistance of up to 40,000 Ohm is sometimes introduced into the secondary circuit. Gases are investigated without inductance with small capacitance. c) Discharges of Tesla currents, which are carried out using the circuit shown in Fig. 2,

where V is a voltmeter, A is an ammeter, T is a transformer, C is a capacitance, T-T is a Tesla transformer, F is a spark gap where the analyte is injected. Tesla currents are used to study substances that have a low melting point: various plant and organic preparations, precipitates on filters, etc. In the spectral analysis of metals, in the case of a large number of them, they are usually themselves electrodes, and they are given some form, for example, from those shown in FIG. 3,

where a is an electrode made of the analyzed thick wire, b is of tin, c is a bent thin wire, d is a disk cut from a thick cylindrical rod, e is a shape cut from large pieces of casting. In quantitative analysis, it is necessary to always have the same shape and size of the electrode surface exposed to sparks. With a small amount of the analyzed metal, you can use a frame made of some pure metal, for example, of gold and platinum, in which the analyzed metal is fixed, as shown in FIG. 4.

Quite a number of methods have been proposed for introducing solutions into a light source. When working with a flame, a Lundegaard atomizer is used, which is shown schematically in FIG. 5 together with a special burner.

The air blown through the nebulizer ВС captures the test liquid, which is poured in an amount of 3-10 cm 3 into recess C, and in the form of fine dust carries it to the burner A, where it mixes with the gas. For the introduction of solutions into the arc, as well as into the spark, pure carbon or graphite electrodes are used, on one of which a recess is made. It should be noted, however, that it is very difficult to cook the coals completely clean. The methods used for cleaning - alternating boiling in hydrochloric and hydrofluoric acids, as well as calcining in a hydrogen atmosphere up to 2500-3000 ° C - do not give coal free of impurities, Ca, Mg, V, Ti, Al, remain (albeit traces), Fe, Si, B. Coals of satisfactory purity are also obtained by calcining them in air using an electric current: a current of about 400 A is passed through a carbon rod 5 mm in diameter, and the strong heating achieved in this way (up to 3000 ° C) is sufficient for so that most of the coal pollutants evaporate within seconds. There are also such methods of introducing solutions into a spark, where the solution itself is the lower electrode, and the spark slips onto its surface; the other electrode can be any pure metal. An example of such a device is the one shown in FIG. 6 liquid electrode Gerlach.

The recess, where the test solution is poured, is faced with platinum foil or covered with a thick layer of gilding. FIG. 7 shows the Hitchen apparatus, which also serves to introduce solutions into the spark.

From vessel A, the test solution in a weak stream flows through tube B and quartz nozzle C into the sphere of action of spark discharges. The bottom electrode, soldered into a glass tube, is attached to the apparatus by means of a rubber tube E. The attachment C shown in FIG. 7 separately, has a cutout on one side for grouting. D - glass safety vessel in which a round hole is made for the ultraviolet rays to escape. It is more convenient to make this vessel quartz without a hole. The upper electrode F, whether graphite, carbon or metal, is also fitted with a splash plate. For the "high voltage arc", which greatly incandescents the analytes, Gerlach, when working with solutions, uses electrodes with cooling, as is shown schematically in Fig. 8.

On a thick wire (6 mm in diameter), a glass funnel G is fixed with a cork K, where ice pieces are placed. A round iron electrode E with a diameter of 4 cm and a height of 4 cm is fixed at the upper end of the wire, on which a platinum cup P is placed; the latter should be easily removable for cleaning. The top electrode should also be b. thick to avoid melting. When analyzing small amounts of substances - sediments on filters, various powders, etc. - you can use the device shown in Fig. nine.

A lump is made of the test substance and filter paper, moistened for better conductivity with a solution, for example, NaCl, placed on the lower electrode, sometimes consisting of pure cadmium, enclosed in a quartz (worse than glass) tube; the top electrode is also some kind of pure metal. For the same analyzes when working with Tesla currents, a special spark gap design is used, shown in Fig. 10 a and b.

In the round hinge K, an aluminum plate E is fixed in the desired position, on which a glass plate G is applied, and on the latter - preparation P on filter paper F. The preparation is moistened with some acid or salt solution. This whole system is a small capacitor. Closed glass or quartz vessels are used to study gases (Fig. 11).

For the quantitative analysis of gases, it is convenient to use gold or platinum electrodes, the lines of which can be used for comparison. Almost all of the devices mentioned above for introducing substances into the spark and arc are fixed in special tripods during operation. An example would be the Gramont tripod shown in FIG. 12:

using screw D, the electrodes are simultaneously moved apart and moved; screw E serves to move the upper electrode parallel to the optical bench, and screw C for lateral turns of the lower electrode; screw B serves for lateral rotation of the entire upper part of the tripod; finally, using screw A, you can raise or lower the entire upper part of the tripod; H - a stand for burners, glasses, etc. The choice of an energy source for a particular research purpose can be done, guided by the following example table.

Qualitative analysis... In qualitative spectral analysis, the discovery of any element depends on many factors: on the nature of the element being determined, the energy source, the resolution of the spectral apparatus, and also on the sensitivity of photographic plates. The following guidelines can be made regarding the sensitivity of the assay. When working with spark discharges in solutions, it is possible to open 10 -9 -10 -3%, and in metals 10 -2 -10 -4% of the investigated element; when working with a volt arc, the opening limits are about 10 -3%. The absolute amount that m b. open when working with a flame, is 10 -4 -10 -7 g, and with spark discharges 10 -6 -10 -8 g of the investigated element. The greatest sensitivity of the discovery relates to metals and metalloids - B, P, C; less sensitivity for metalloids As, Se and Te; halogens, as well as S, O, N in their compounds are not at all m. b. open and m. b. discovered only in some cases in gas mixtures.

For a qualitative analysis, the “last lines” are of the greatest importance, and in the analysis the task is to determine the most accurate wavelengths of spectral lines. In visual studies, wavelengths are counted along the spectrometer drum; these measurements can be considered only approximate, since the accuracy is usually ± (2-З) Ӑ and in the Kaiser tables about 10 spectral lines belonging to different elements for λ 6000 and 5000 Ӑ and about 20 spectral lines for λ ≈ 4000 Ӑ. The wavelength is determined much more accurately by spectrographic analysis. In this case, on the spectrograms using a measuring microscope, the distance between lines with a known wavelength and a determined one is measured; according to Hartmann's formula, the wavelength of the latter is found. The accuracy of such measurements when working with a device that gives a spectral strip about 20 cm long is ± 0.5 Ӑ for λ ≈ 4000 Ӑ, ± 0.2 Ӑ for λ ≈ 3000 Ӑ, and ± 0.1 Ӑ for λ ≈ 2500 Ӑ. The corresponding element is found by the wavelength in the tables. The distance between the lines during normal work is measured with an accuracy of 0.05-0.01 mm. It is sometimes convenient to combine this technique with recording spectra with so-called Hartmann shutters, two types of which are shown in Fig. 13, a and b; with the help of their spectrograph slit, you can make different heights. FIG. 13c schematically shows the case of a qualitative analysis of a substance X - the establishment of elements A and B in it. The spectra of FIG. 13d show that in substance Y, in addition to element A, the lines of which are indicated by the letter G, there is an impurity, the lines of which are indicated by z. Using this technique, in simple cases, you can perform a qualitative analysis without resorting to measuring the distances between the lines.

Quantitative analysis... For quantitative spectral analysis, the lines with the highest possible concentration sensitivity dI / dK, where I is the intensity of the line, and K is the concentration of the element giving it, are of the greatest importance. The greater the concentration sensitivity, the more accurate the analysis. Over the years, a number of methods for quantitative spectral analysis have been developed. These methods are as follows.

I. Spectroscopic methods (no photographic survey) almost all are photometric methods. These include: 1) Barratt's method. Simultaneously, the spectra of two substances are excited - the test and the standard - visible in the field of view of the spectroscope side by side, one above the other. The ray path is shown in FIG. 14,

where F 1 and F 2 are two spark gaps, the light from which passes through the Nicolas prisms N 1 and N 2, polarizing the rays in mutually perpendicular planes. With the help of the prism D, the rays enter the slit S of the spectroscope. In his telescope is placed the third prism of Nicolas - the analyzer, - rotating which they achieve the same intensity of the two compared lines. Previously, when studying standards, i.e., substances with a known content of elements, a relationship is established between the angle of rotation of the analyzer and the concentration, and a diagram is drawn from these data. When analyzing by the angle of rotation of the analyzer, the desired percentage is found from this diagram. The accuracy of the method is ± 10%. 2). The principle of the method is that the light rays after the spectroscope prism pass through the Wollaston prism, where they diverge into two beams and polarize in mutually perpendicular planes. The ray path is shown in FIG. 15,

where S is the slit, P is the spectroscope prism, W is the Wollaston prism. In the field of view, two spectra B 1 and B 2 are obtained, lying side by side, one above the other; L - magnifier, N - analyzer. If you rotate the Wollaston prism, the spectra will move relative to each other, which makes it possible to combine any two of their lines. For example, if an iron containing vanadium is analyzed, then the vanadium line is aligned with any nearby monochromatic iron line; then, turning the analyzer, they achieve the same brightness of these lines. The analyzer's angle of rotation, as in the previous method, is a measure of the concentration of the desired element. The method is especially suitable for the analysis of iron, the spectrum of which has many lines, which makes it possible to always find lines suitable for research. The accuracy of the method is ± (3-7)%. 3) Occhialini's method. If the electrodes (for example, the analyzed metals) are placed horizontally and the image is projected from the light source onto the vertical slit of the spectroscope, then both in spark and in arc discharges, impurity lines can be used. open depending on the concentration at a greater or lesser distance from the electrodes. The light source is projected onto the slit using a special lens equipped with a micrometer screw. During the analysis, this lens moves and the image of the light source moves with it until any impurity line in the spectrum disappears. The measure of impurity concentration is the reading on the lens scale. Currently, this method is also developed for work with the ultraviolet part of the spectrum. It should be noted that Lockyer used the same method for illuminating the slit of the spectral apparatus and he developed a method for quantitative spectral analysis, the so-called. method of "long and short lines". 4) Direct photometry of spectra... The methods described above are called visual methods. Instead of visual studies, Lundegard used a photocell to measure the intensity of spectral lines. The accuracy of determination of alkali metals when working with a flame reached ± 5%. With spark discharges, this method is not applicable, since they are less constant than a flame. There are also methods based on changing the inductance in the secondary circuit, as well as using artificial attenuation of the light entering the spectroscope until the spectral lines under investigation disappear in the field of view.

II. Spectrographic methods... With these methods, photographic images of spectra are studied, and the measure of the intensity of spectral lines is the blackening they give on a photographic plate. The intensity is assessed either by the eye or photometrically.

A. Methods without the use of photometry. 1) Last lines method... When the concentration of any element in the spectrum changes, the number of its lines changes, which makes it possible, under unchanged operating conditions, to judge the concentration of the element being determined. A number of spectra of substances with a known content of the component of interest are photographed, the number of its lines is determined on the spectrograms, and tables are compiled, which indicate which lines are visible at these concentrations. These tables serve further for analytical definitions. When analyzing on the spectrogram, the number of lines of the element of interest is determined and the percentage content is determined from the tables, and the method does not give its unambiguous figure, but the concentration limits, that is, "from-to". It is most reliably possible to distinguish concentrations that differ from each other by a factor of 10, for example, from 0.001 to 0.01%, from 0.01 to 0.1%, etc. Analytical tables are important only for well-defined operating conditions, which may vary greatly from laboratory to laboratory; in addition, careful observance of the constancy of working conditions is required. 2) Comparative Spectra Method... several spectra of the analyte A + x% B are photographed, in which the content x of element B is determined, and in the intervals between them on the same photographic plate - the spectra of standard substances A + a% B, A + b% B, A + c% B , where a, b, c is the known percentage of B. On the spectrograms, the intensity of the lines B determines between which concentrations the value of x is. The criterion for the constancy of operating conditions is the equality of the intensity on all spectrograms of any nearby line A. When analyzing solutions, the same amount of any element is added to them, giving a line close to lines B, and then the constancy of operating conditions is judged by the equality of the intensity of these lines. The smaller the difference between the concentrations a, b, c, ... and the more precisely the equality of the intensities of the lines A is achieved, the more accurate the analysis. A. Rice, for example, used the concentrations a, b, c, ..., related to each other, as 1: 1.5. The method of comparative spectra is adjacent to the method of "selection of concentrations" (Testverfahren) according to Guettig and Thurnwald, which is applicable only to the analysis of solutions. It consists in the fact that if in two solutions containing a% A and x% A (x is greater or less than a), which can now be determined from their spectra, then such an amount of n of element A is added to any of these solutions so that the intensity of its lines in both spectra becomes the same. This will determine the concentration x, which will be equal to (a ± n)%. You can also add some other element B to the solution to be analyzed until the intensity of certain lines A and B is equal and, by the amount of B, estimate the content of A. 3) Homologous pairs method... In the spectrum of substance A + a% B, the lines of elements A and B are not equally intense, and if there are a sufficient number of these lines, you can find two such lines A and B, the intensity of which will be the same. For another composition, A + b% B, the other lines A and B will be the same in intensity, etc. These two identical lines are called homologous pairs. The concentrations B, at which one or another homologous pair is realized, are called fixing points this pair. To work on this method, preliminary compilation of tables of homologous pairs is required using substances of known composition. The more complete the tables, that is, the more they contain homologous pairs with fixing points differing as little as possible from each other, the more accurate the analysis. A fairly large number of these tables have been compiled, and they can be used in any laboratory, since the conditions of the discharges when they are compiled are precisely known, and these conditions can be used. perfectly reproduced. This is achieved using the following simple technique. In the spectrum of substance A + a% B, two lines of element A are selected, the intensity of which varies greatly depending on the magnitude of self-induction in the secondary circuit, namely one arc line (belonging to a neutral atom) and one spark line (belonging to an ion). These two lines are called fixing pair... By selecting the value of self-induction, the lines of this pair are made the same and the compilation is carried out under these conditions, which are always indicated in the tables. Under the same conditions, an analysis is carried out, and the percentage is determined by the implementation of a particular homologous pair. There are several modifications of the homology pair method. Of these, the most important is the method auxiliary spectrum, used when elements A and B do not have a sufficient number of lines. In this case, the lines of the spectrum of element A are associated in a certain way with the lines of another, more suitable element G, and element G begins to play the role of A. The method of homologous pairs was developed by Gerlach and Schweitzer. It is applicable to both alloys and solutions. Its accuracy is on average about ± 10%.

IN. Methods using photometry... 1) Barratt's method. FIG. 16 gives an idea of \u200b\u200bthe method.

F 1 and F 2 are two spark gaps, with the help of which the spectra of the standard and the analyte are simultaneously excited. The light passes through 2 rotating sectors S 1 and S 2 and with the help of the prism D forms spectra, which are located one above the other. By selecting the cuts of the sectors, the lines of the investigated element are obtained with the same intensity; the concentration of the element to be determined is calculated from the ratio of the cut sizes. 2) is similar, but with one spark gap (Fig. 17).

The light from F is divided into two beams and passes through the sectors S 1 and S 2, with the help of the Hüfner rhombus R, two spectral strips are obtained one above the other; Sp is the spectrograph slit. The cutouts of the sectors are changed until the intensity of the impurity line and any nearby line of the main substance is equal, and the percentage of the element being determined is calculated from the ratio of the cutout values. 3) When used as a photometer rotating logarithmic sector the lines are wedge-shaped on the spectrograms. One of such sectors and its position relative to the spectrograph during operation are shown in Fig. 18, a and b.

Sector clipping obeys the equation

- lg Ɵ \u003d 0.3 + 0.2l

where Ɵ is the length of the arc in parts of the full circle, located at a distance I, measured in mm along the radius from its end. A measure of the intensity of the lines is their length, since with a change in the concentration of an element, the length of its wedge-shaped lines also changes. Preliminarily, a diagram of the dependence of the length of any line on the% content is plotted using samples with a known content; during analysis, the length of the same line is measured on the spectrogram and the percentage is found from the diagram. There are several different modifications of this method. It is necessary to point out the modification of Scheibe, who used the so-called double logarithmic sector. This sector is shown in FIG. 19.

The lines are then examined using a special apparatus. Accuracy achievable using logarithmic sectors, ± (10-15)%; Scheibe's modification gives an accuracy of ± (5-7)%. 4) The photometry of spectral lines with the help of light and thermoelectric spectrophotometers of various designs is quite often used. Convenient are thermoelectric photometers designed specifically for quantitative analysis. For example, in FIG. 20 shows the scheme of the Scheib photometer:

L - constant light source with condenser K, M - photographic plate with the investigated spectrum, Sp - slit, O 1 and O 2 - objectives, V - shutter, Th - thermoelement, which is connected to the galvanometer. A measure of the intensity of the lines is the deflection of the galvanometer needle. Less commonly, self-recording galvanometers are used, which record the line intensity in the form of a curve. The analysis accuracy when using this type of photometry is ± (5-10)%. When combined with other methods of quantitative analysis, the accuracy can be used. increased; for example, three lines method Scheibe and Schnettler, which is a combination of the method of homologous pairs and photometric measurements, in favorable cases can give an accuracy of ± (1-2)%.

Provision of services for the chemical analysis of metal

We can perform the following works:

Chemical composition, chemical analysis of metal:

Determine the chemical composition of steels and alloys

Confirm steel grades

Restore product documentation

Confirm or deny the certificate

Incoming inspection of metals and alloys

Sort ferrous and non-ferrous scrap

Determine the chemical composition of ore rocks

Select an analogue of steels and alloys (using special program - brand of steels Win Steel 8.0 Prof)

Mechanical tests:

Compression and stretching

Determination of hardness

Cooperation options:

Testing at the customer's facility

Sample testing in our laboratory

Traveling to the regions and receiving samples through transport companies

	Efficiency		Visit of a specialist to the customer's site
	Work throughout the Russian Federation		Highly qualified specialists
	Work in accordance with GOST		Selection of analogues of steels and alloys
	Specialist consultation		One-click application (order a service from the site)

"Steel. Method of X-ray fluorescence analysis"		GOST 12353-78, GOST 12344-2003, GOST 12345-2001, GOST 12350-78, GOST 12346-78, GOST 12347-77, GOST 12348-78, GOST 12352-81, GOST 12355-78

Equipment used for chemical analysis

ALL EQUIPMENT HAS A VALID CERTIFICATE OF VERIFICATION.

	X-MET 8000 is a portable X-ray fluorescence energy dispersive spectrometer with the ability to determine light elements Mg, Al, Si, P, S in accordance with GOST 28033-89. The range of the measured elements: from Mg to Bi.
	PMI MASTER UVR is a mobile optical emission metal analyzer that allows high-precision analysis and determination of the grade of any steel and alloys with the ability to analyze carbon, sulfur, phosphorus.
	ARC-MET-8000 portable optical emission analyzer operating in argon mode. With the ability to determine and excellent repeatability of results for carbon, sulfur, phosphorus and boron.
	Rockwell stationary hardness tester METOLAB101 The stationary hardness tester is used to measure the hardness of hard alloys, as well as hardened and unhardened steels, castings, bearing steels, aluminum alloys, thin plates of hard alloys, copper, zinc, chromium and tinned surface coatings, etc. by the Rockwell method. Certificate of type approval of measuring instruments RU.C.28.002.A No. 63563.

Measurement sequence

1		2	X-MET 8000 PMI MASTER UVR	3

Determination of the chemical composition of the sample

Today, chemical analysis of metals - steeloscopy - does not require breaking the integrity of the tested structure or preparing samples. To make a spectral analysis and determine the physicochemical characteristics of metals and alloys, it is also not necessary to contact a laboratory: the modern photoelectric method of spectral analysis allows you to control the quality of finished products even in the field.

Why do you need spectral analysis of metals and alloys?

Spectral analysis of metals using stationary or portable devices using the method of X-ray fluorescence spectral analysis of steel in accordance with GOST 28033–89 is designed to help specialized enterprises in metal sorting.

This solution demonstrates a number of advantages. It won't take long to carry out an examination of the metal. The result will be known in a few minutes. Such a mini-laboratory for the chemical analysis of metal will significantly reduce the costs of the manufacturing enterprise, large retailer and utilities. The price set for the spectral analysis of the metal in specialized organizations and the schedule of their work no longer matters: once having bought a metal analyzer and having completed a training course for specialists who will work with it in the future, your company will be able to organize a spectral analysis of the metal at a convenient time and in a convenient place.

Chemical analysis of the metal is used in the following cases:

Confirmation of the brand, confirmation of certificates.

Sorting of scrap metals and alloys. In this area, falsifications are quite common, but if the inspectors use chemical analysis, the determination of the metal, which gives the most accurate result, is guaranteed to save the company from losses.

Calibration programs of the device.

What substances does the analysis of the chemical composition of metals work with?

X-ray fluorescence analysis of the chemical composition of metals and alloys is carried out in the laboratory using an X-MET 7500 X-ray fluorescence analyzer with the ability to determine the light elements Mg, Al, Si, P, S in accordance with GOST 28033-89. The range of the measured elements: from Mg to Bi. The method is suitable for determining the chemical composition and grade of steel and other metals. In particular, it is allowed:

• chemical analysis of aluminum alloys;
• chemical analysis of titanium alloys;
• analysis of iron alloys, etc.

The universal program for chemical analysis of alloys uses several fundamental parameters for the analysis of metals and alloys, a standard set of 33 elements: Mg, Al, Si, P, S, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, As , Se, Y, Zr, Nb, Mo, Pd, Ag, In, Sn, Sb, Hf, Ta, W, Pt, Ir, Au, Pb, Bi in concentrations from 0 to 100%. Suitable for the analysis of metals on any basis: Pb, W, Au, etc., ferroalloys

How does chemical analysis of metals and alloys work?

In order to make a chemical express analysis of a metal, it is enough to attach one of the devices we are selling to its surface. The X-ray fluorescence method is based on the dependence of the intensity of the characteristic fluorescence lines of an element on its mass fraction in the sample.

Spectral Analysis Instruments

What is an X-ray fluorescence analyzer?

An X-ray fluorescence spectrometer is analytical instrument, which identifies each chemical element present in the test sample.

This device also detects total chemical elements in the sample.

X-MET 7500

X-ray fluorescence analysis of the chemical composition of metals and alloys is carried out using an X-MET 7500 X-ray fluorescence analyzer with the ability to determine the light elements Mg, Al, Si, P, S in accordance with GOST 28033-89. The range of measured elements: from Mg to Bi. The X-ray fluorescence method is based on the dependence of the intensity of the characteristic fluorescence lines of an element on its mass fraction in the sample.

This type of control is used in the following cases:

• Determination of the chemical composition of steels and alloys.
• Restoration of product documentation.
• Confirmation of the brand, confirmation of certificates.
• Incoming inspection of metals and alloys.
• Sorting of scrap metals and alloys.
• Selection of analogs of steels and alloys (using a special program - the brand of steels Win Steel 7.0 Prof).

What parameters can be determined by chemical analysis of a metal?

A set of 8 specialized empirical programs is available to the user: "low alloy steels and cast irons", "stainless steels", "tool steels", "aluminum alloys", "copper alloys", "cobalt alloys", "titanium alloys", "nickel alloys" ... The choice of the program with which it is planned to determine the chemical composition of the metal is carried out automatically.

• Spectrum identification software (yes / no).
• Program for the analysis of carbon, low alloy steels and cast irons.
• Program for the analysis of stainless steels.
• Tool for the analysis of tool steels.
• Program for the analysis of copper alloys.
• Program for the analysis of nickel alloys.
• Program for the analysis of titanium alloys.
• Program for the analysis of cobalt alloys.
• Program for the analysis of aluminum alloys.
• Identification programs (yes / no).
• The function of automatically determining the type of material and selecting the required program for analysis.
• Automatic concentration correction for small and complex samples.
• One point recalibration function.
• Built-in grades of metals and alloys, the ability to correct and add grades.
• The ability to average the results over at least 50 measurements to obtain reliable results when analyzing heterogeneous samples.
• The ability to create reports in a protected from correction PDF format according to a user's template with the ability to place a company logo, measurement results, measurement error, measurement time and duration, operator name and other information at the user's choice

Rostov-on-Don 2014

Compiled by: Yu.V. Dolgachev, V.N. Pustovoit Optical emission spectral analysis of metals. Methodical instructions for laboratory practice / Rostov-on-Don. DSTU Publishing Center. 2014 .-- 8 p.

The methodological instructions are developed for use by students when performing a laboratory workshop on the disciplines "Nondestructive Methods of Control of Materials", "Physicochemistry of Nanomaterials", "Nanotechnologies and Nanomaterials" and are intended for the practical development of theoretical ideas about the structure and properties of materials, obtaining skills in analyzing the chemical composition of metals and alloys,.

Published by the decision of the methodological commission

faculty of Mechanical Engineering Technologies and Equipment

Scientific editor, Doctor of Technical Sciences, Professor Pustovoit V.N. (DSTU)

Reviewer Doctor of Technical Sciences, Professor Kuzharov A.S. (DSTU)

 Publishing Center DSTU, 2014

Optical emission spectral analysis of metals

PURPOSE OF THE WORK: to get acquainted with the purpose, capabilities, principle of operation of the Magellan Q8 spectral analyzer and to perform a chemical analysis of a metal sample.

1. Basic theoretical concepts

1.1. Purpose of optical emission spectral analysis

Today, the analysis of the chemical composition has found wide application in many sectors of the national economy. The quality, reliability, durability of the product largely depend on the composition of the alloy used. The slightest deviation from the specified chemical composition can lead to a negative change in properties. A special danger lies in the fact that this deviation can be visually invisible and, as a result, undetectable without special devices. Human senses do not make it possible to analyze such parameters of the metal as its composition or the grade of the alloy used. One of the devices that allows one to obtain the necessary information on the chemical composition of the alloy is an optical emission spectrometer.

An optical emission spectrometer is used to measure the mass fraction of chemical elements in metals and alloys and is used in analytical laboratories of industrial enterprises, in workshops for quick sorting and identification of metals and alloys, as well as for analyzing large structures without violating their integrity.

1.2 Principle of operation of an optical emission analyzer

The principle of operation of the spectrometer is based on measuring the radiation intensity at a certain wavelength of the emission spectrum of the atoms of the analyzed elements. Radiation is excited by a spark discharge between the auxiliary electrode and the analyzed metal sample. During the analysis, argon flows around the object under study, making it more visible for study. An emission spectrometer records the radiation intensity and, based on the data obtained, analyzes the composition of the metal. The content of elements in the sample is determined by the calibration dependences between the intensity of emission radiation and the content of the element in the sample.

The spectrometer consists of a spectrum excitation source, an optical system, and an automated control and recording system based on an IBM-compatible computer.

The spark excitation source of the spectrum is designed to excite the emission light flux from the spark between the sample and the electrode. The spectral composition of light is determined by the chemical composition of the test sample.

Currently, the most optimal arrangement of the optical system is considered to be the Paschen-Runge scheme (Fig. 1).

Fig. 1 Optical system according to the Paschen-Runge scheme

When the atoms excited by the glow discharge move to a lower orbit, they emit light. Each emitted wavelength is characteristic of each atom that emitted it. The light is focused on the entrance slit of the spectrometer and split into a concave holographic grating in accordance with the wavelengths. After that, through the precise setting of the exit slit, the light enters the photomultiplier corresponding to the element.

To ensure good transparency, the optical chamber must be evacuated. In addition, the system must be independent of external conditions (temperature and air pressure). At present, stationary optical spectrometers are thermally stabilized with an accuracy of tenths of a degree.

The process of measuring and processing the output information is controlled from the built-in IBM-compatible computer using a special software package. The program is used to configure the device, build calibration curves based on the analysis of standard samples, optimize its parameters, control spectrometer modes, process, save and print measurement results.

1.3 Installing Magellan Q8

Qantron Magellan (Magellan Q8) is an optical emission analyzer with vacuum optics from Bruker (Fig. 2). Allows you to determine the chemical composition of alloys based on iron (steel and cast iron), copper (bronze, brass, etc.), aluminum (duralumin, etc.). The plant is equipped with sensors that determine the percentage of elements such as carbon, nitrogen, phosphorus, sulfur, vanadium, tungsten, silicon, manganese, chromium, molybdenum, nickel, aluminum, cobalt, copper, niobium, titanium, tin, boron, iron, zinc , tin, beryllium, magnesium, lead.

The installation is calibrated using calibration samples of various steels, cast irons, bronze, and aluminum alloys. The accuracy of determining the chemical composition of alloys is up to hundredths of a percent.

Figure: 2. Installing Magellan Q8

Spectral analysis I Spectral analysis

a physical method for the qualitative and quantitative determination of the atomic and molecular composition of a substance, based on the study of its spectra. Physical basis of S. a.- Spectroscopy of atoms and molecules, it is classified according to the purposes of analysis and types of spectra (see. Optical spectra). Atomic S. and. (ACA) determines the elemental composition of the sample from the atomic (ionic) emission and absorption spectra, molecular S. a. (MSA) - the molecular composition of substances by molecular absorption spectra, luminescence (see Luminescence) and Raman scattering of light (See Raman scattering of light).

Emission S. and. produced by the emission spectra of atoms, ions and molecules, excited by various sources of electromagnetic radiation in the range from γ-radiation to microwave. Absorptive S. and. carried out according to the absorption spectra of electromagnetic radiation by the analyzed objects (atoms, molecules, ions of a substance in various states of aggregation).

History reference. IN ASA is based on the individuality of the emission and absorption spectra of chemical elements, first established by G.R.Kirkhhoff and R. Bunsen (1859-61). In 1861 Kirchhoff proved on the basis of this discovery the presence of a number of elements in the solar chromosphere, laying the foundation for astrophysics. In 1861-1923, 25 elements were discovered with the help of ASA. In 1932, deuterium was discovered by the spectral method.

High sensitivity and the ability to determine many elements in low-mass samples have made ASA an effective method for the qualitative analysis of the elemental composition of objects. In 1926 it. physicist V. Gerlach laid the foundation for quantitative S. and. For S.'s development and. G.S. Landsberg, S.L. Mandel'shtam, A.K. Rusanov (Moscow), A.N. Filippov, V.K.Prokofiev (Leningrad), and others played an important role in its implementation at industrial enterprises of the USSR.

Atomic Spectral Analysis (ASA)

Emission ASA consists of the following main processes:

1) selection of a representative sample reflecting the average composition of the analyzed material or the local distribution of the determined elements in the material;

2) introduction of the sample into the radiation source, in which the evaporation of solid and liquid samples, the dissociation of compounds and the excitation of atoms and ions occur;

3) converting their luminescence into a spectrum and registering it (or visual observation) using a spectral device (see Spectral devices) ;

4) interpretation of the obtained spectra using tables and atlases of spectral lines of elements.

At this stage, quality ASA ends. The most effective is the use of sensitive (so-called "last") lines that remain in the spectrum at the minimum concentration of the element being determined. Spectrograms are viewed on measuring microscopes, comparators, spectroprojectors. For a qualitative analysis, it is sufficient to establish the presence or absence of analytical lines of the determined elements. By the brightness of the lines during visual inspection, one can give a rough estimate of the content of certain elements in the sample.

Quantitative ASA is carried out by comparing the intensities of two spectral lines in the spectrum of the sample, one of which belongs to the element being determined, and the other (comparison line) to the main element of the sample, the concentration of which is known, or to an element specially introduced at a known concentration ("internal standard").

The quantitative ASA is based on the relationship between the concentration from determined element with the ratio of the intensities of the determined impurity line ( I 1) and comparison lines ( I 2):

I 1 / I 2 \u003d ac b

(permanent a and b determined empirically), or

lg ( I 1 / I 2) = b lg from + lg a.

Using standard samples (at least 3), you can plot the dependence of lg ( I 1 / I 2.) from lg s (calibration graph, fig. one ) and determine from it a and b. The values I 1 and I 2 can be obtained directly by photo-electric registration or by photometry (measuring the density of blackening) of the determined impurity line and the comparison line during photo registration. Photometry is performed on a microphotometer.

To excite the spectrum in ASA, different light sources are used and, accordingly, different methods of introducing samples into them. The choice of the source depends on the specific conditions of the analysis of certain objects. The type of source and the method of sample introduction are the main content of the particular ACA methods.

The first artificial light source in the ACA was the flame of a gas burner - a very convenient source for quickly and accurately identifying many elements. The temperature of the flames of combustible gases is not high (from 2100 K for a hydrogen-air mixture to 4500 K for a rarely used oxygen-cyanogen mixture). With the help of flame photometry, about 70 elements are determined from their analytical lines, as well as from the molecular bands of compounds formed in flames.

Electric light sources are widely used in emission ASA. In a direct current electric arc between specially cleaned carbon electrodes of various shapes, in the channels of which the test substance is placed in a crushed state, it is possible to simultaneously determine dozens of elements. It provides a relatively high temperature for heating the electrodes and favorable conditions for the excitation of sample atoms in the arc plasma; however, the accuracy of this method is low due to the instability of the discharge. Raising the voltage to 300-400 in or going to high-voltage arc (3000-4000 in), the accuracy of the analysis can be increased.

More stable excitation conditions are created by the AC arc. In modern alternating current arc generators (see, for example, fig. 2 ), various excitation modes can be obtained: low-voltage spark, high-frequency spark, alternating current arc, pulsed discharge, etc. Such light sources with different modes are used in the determination of metals and hardly excitable elements (carbon, halogens, gases contained in metals, etc. .d.). High voltage condensed spark ( fig. 3 ) serves primarily as a light source for metal analysis. The stability of the spark discharge makes it possible to obtain a high reproducibility of the analysis; however, complex processes occurring on the surfaces of the analyzed electrodes lead to changes in the composition of the discharge plasma. To eliminate this phenomenon, it is necessary to perform preliminary firing of samples and normalize the shape and size of samples and standard samples.

In ASA, it is promising to use stabilized forms of electric discharge such as plasmatrons of various designs, high-frequency induction discharge, microwave discharge generated by magnetron generators, and high-frequency torch discharge. With the help of various methods of introducing analytes into the plasma of these types of discharge (blowing of powders, spraying solutions, etc.), the relative accuracy of the analysis is significantly increased (up to 0.5-3%), including the components of complex samples, the content of which is tens of%. In some important cases of the analysis of pure substances, the use of these types of discharge reduces the limits of determination of impurities by 1–2 orders of magnitude (up to 10 -5 -10 -6%).

Atomic absorption S. and. (AAA) and atomic fluorescent S. a. (AFA).In these methods, the sample is converted to steam in an atomizer (flame, graphite tube, plasma of a stabilized RF or microwave discharge). In AAA, light from a source of discrete radiation, passing through this pair, is attenuated and the degree of attenuation of the intensities of the lines of the element being determined is judged on its concentration in the sample. AAA is carried out on a special spectrophotometer ah. The AAA technique is much simpler in comparison with other methods; it is characterized by a high accuracy in determining not only small, but also large concentrations of elements in samples. AAA successfully replaces labor-consuming and time-consuming chemical methods of analysis, without yielding to them in accuracy.

In AFA, atomic vapors of the sample are irradiated with the light of a resonant radiation source and the fluorescence of the element being determined is recorded. For some elements (Zn, Cd, Hg, etc.), the relative limits of their detection by this method are very small (Spectral analysis 10 -5 -10 6%).

ASA allows measurements of the isotopic composition. Some elements have spectral lines with a well-resolved structure (for example, H, He, U). The isotopic composition of these elements can be measured on conventional spectral instruments using light sources that produce thin spectral lines (hollow cathode, electrodeless HF and microwave lamps). To carry out isotopic spectral analysis of most elements, high-resolution instruments are required (for example, the Fabry-Perot etalon). Isotope spectral analysis can also be carried out on the electronic-vibrational spectra of molecules, measuring isotopic shifts of the bands, which in some cases reach significant values.

Express methods of ASA are widely used in industry, agriculture, geology, and many other areas of the national economy and science. ASA plays a significant role in nuclear engineering, the production of pure semiconductor materials, superconductors, etc. More than three-fourths of all analyzes in metallurgy are performed using ASA methods. Operational (within 2-3 min) control during melting in open-hearth and converter production. In geology and geological exploration, about 8 million analyzes are carried out per year to evaluate deposits. ASA is used for environmental protection and soil analysis, forensic science and medicine, seabed geology and the study of the composition of the upper atmosphere, in isotope separation and determination of the age and composition of geological and archaeological sites, etc.

Lit .: Zaidel A. N., Fundamentals of spectral analysis, M., 1965; Spectral analysis methods, Moscow, 1962; Emission spectral analysis of atomic materials, L. - M., 1960; Rusanov A.K., Fundamentals of quantitative spectral analysis of ores and minerals. M., 1971; Spectral Analysis of Pure Substances, ed. X. I. Zilberstein, [L.], 1971; Lvov B. V., Atomic absorption spectral analysis, M., 1966; Petrov AA, Spectral isotopic method of materials research, L., 1974; Tarasevich N.I. Semenenko K.A., Khlystova A.D., Methods of spectral and chemical-spectral analysis, M., 1973: Prokofiev V.K., Photographic methods of quantitative spectral analysis of metals and alloys, part 1- 2, M. - L., 1951; Menke G., Menke L., Introduction to Laser Emission Microspectral Analysis, trans. from it., M., 1968; Korolev NV, Ryukhin VV, Gorbunov SA, Emission spectral microanalysis, L., 1971; Tables of spectral lines, 3rd ed., M., 1969; Striganov A.P., Sventitsky N.S., Tables of spectral lines of neutral and ionized atoms, Moscow, 1966.

L. V. Lipis.

Molecular Spectral Analysis (MSA)

The MSL is based on a qualitative and quantitative comparison of the measured spectrum of the test sample with the spectra of individual substances. Accordingly, qualitative and quantitative ISA are distinguished. ISA uses different kinds molecular spectra (See Molecular Spectra) , rotational [spectra in the microwave and long-wave infrared (IR) regions], vibrational and vibrational-rotational [absorption and emission spectra in the mid-IR region, Raman spectra, IR fluorescence spectra], electronic, electronic vibrational and electronic-vibrational-rotational [absorption and transmission spectra in the visible and ultraviolet (UV) regions, fluorescence spectra]. ISA allows the analysis of small quantities (in some cases, fractions mcg and less) substances in various states of aggregation.

The main factors that determine the capabilities of ISA methods:

1) information content of the method. It is conventionally expressed by the number of spectrally resolvable lines or bands in a certain interval of wavelengths or frequencies of the studied range (for the microwave range it is Spectral analysis 10 5 , for the mid-IR region in the spectra of solid and liquid substances Spectral analysis 10 3);

2) the number of measured spectra of individual compounds;

3) the existence of general laws between the spectrum of a substance and its molecular structure;

4) sensitivity and selectivity of the method;

5) the universality of the method;

6) simplicity and availability of spectra measurements.

Quality ISA establishes the molecular composition of the sample under study. The spectrum of a molecule is its unambiguous characteristic. The most specific spectra of substances in a gaseous state with an allowed rotational structure, which are investigated using spectral devices of high resolution. The most widely used are the IR absorption and Raman spectra of substances in liquid and solid states, as well as absorption spectra in the visible and UV regions. The widespread introduction of the Raman method was facilitated by the use of laser radiation for their excitation.

To increase the efficiency of MSA, in some cases, the measurement of spectra is combined with other methods of identification of substances. Thus, the combination of chromatographic separation of mixtures of substances with the measurement of the IR absorption spectra of the isolated components is becoming more widespread.

Quality ISA also includes the so-called. structural molecular analysis. It has been established that molecules with the same structural elements exhibit common features in the absorption and emission spectra. This is most clearly manifested in vibrational spectra. Thus, the presence of a sulfhydryl group (-SH) in the structure of a molecule entails the appearance of a band in the spectrum in the range 2565-2575 cm -1, nitrile group (-CN) is characterized by a band 2200-2300 cm -1 etc. The presence of such characteristic bands in the vibrational spectra of substances with common structural elements is explained by the characteristic frequency and shape of many molecular vibrations. Such features of vibrational (and to a lesser extent electronic) spectra in many cases allow one to determine the structural type of a substance.

Qualitative analysis greatly simplifies and accelerates the use of computers. In principle, it can be fully automated by entering the readings of spectral instruments directly into a computer. Its memory should contain spectral characteristic features of many substances, on the basis of which the machine will analyze the substance under study.

Quantitative ISA the absorption spectra is based on the Bouguer-Lambert-Beer law e , establishing a connection between the intensities of the incident and transmitted through the substance I light from the thickness of the absorbing layer I and substance concentration from:

I(l)\u003d I 0 e - χ cl

The χ coefficient is a characteristic of the absorption capacity of the component to be determined for a given radiation frequency. An important condition conducting quantitative MSA - independence of χ on the concentration of the substance and constancy of χ in the measured frequency range determined by the width of the spectrophotometer slit. MSA based on absorption spectra is carried out mainly for liquids and solutions; for gases it becomes much more complicated.

In practical ISA, the so-called. optical density:

D \u003d In (/ o //) \u003d χ cl.

If the mixture consists of n substances that do not react with each other, then the optical density of the mixture at the frequency ν is additive: m points of the mixture spectrum ( m ≥ n) and solving the resulting system of equations:

For quantitative MSA, spectrophotometers are usually used, which make it possible to measure f (v) in a relatively wide range of v. If the absorption band of the test substance is sufficiently isolated and free from the overlap of the bands of other components of the mixture, the investigated spectral region can be distinguished, for example, using an interference filter a . On its basis, specialized analyzers widely used in industry are designed.

In quantitative MSA based on Raman spectra, most often the line intensity of the determined component of the mixture is compared with the intensity of a certain line of a standard substance measured under the same conditions (the “external standard” method). In other cases, a standard substance is added to the test substance in a certain amount (“internal standard” method).

Among other methods of qualitative and quantitative MSA, fluorescence analysis has the highest sensitivity, but under ordinary conditions it is inferior to vibrational spectroscopy methods in universality and selectivity. Quantitative MSA based on fluorescence spectra is based on the comparison of the luminescence of a solution of the test sample with the luminescence of a number of standard solutions of similar concentration.

Of particular importance is MSA using the technique of frozen solutions in special solvents, such as paraffins (see Shpolsky effect). The spectra of substances in such solutions (Shpolsky spectra) have a pronounced individuality, they are sharply different for structurally similar and even isomeric molecules. This makes it possible to identify substances that cannot be determined from their fluorescence spectra under normal conditions. For example, the Shpolsky method makes it possible to carry out a qualitative and quantitative analysis of complex mixtures containing aromatic hydrocarbons. Qualitative analysis in this case is carried out according to the luminescence and absorption spectra, quantitative - according to the luminescence spectra by the methods of "internal" and "external" standards. Due to the extremely small width of the spectral lines in the Shpolsky spectra, this method manages to achieve the threshold detection sensitivity of some polyatomic aromatic compounds (Spectral analysis 10 Spectral analysis 11 g / cm 3).

Lit .: Chulanovsky VM, Introduction to molecular spectral analysis, M. - L., 1951; Bellamy L., Infrared spectra of complex molecules, trans. from English, M., 1963; Application of spectroscopy in chemistry, trans. from English., M., 1959; Determination of the individual hydrocarbon composition of straight-run gasolines by the combined method, M., 1959; Judenfrend S., Fluorescent analysis in biology and medicine, trans. from English, M., 1965.

V. T. Aleksanyan.

Fig. 2. Schematic diagram of an alternating current arc of double power supply: A - ammeter; R 1 and R 2 are rheostats; Tr - step-up transformer: K - inductance coil; AP - analytical interval; P - auxiliary gap; C 1 and C 2 are capacitors.

Fig. 3. Diagram of a condensed spark generator with a control gap: AP - an adjustable analytical gap formed by vanadium electrodes; R 1 - rheostat; Tr - supply transformer; C - capacitor; L - inductor; P - control gap; R 2 - blocking resistance.

II Spectral analysis

linear operators, a generalization of the theory of eigenvalues \u200b\u200b(see Eigenvalues) and eigenvectors (see Eigenvectors) of matrices (i.e. linear transformations in a finite-dimensional space), which grew out of problems in mechanics, to the infinite-dimensional case (see Linear operator, Operator theory). In the theory of oscillations, the motion of a system with n degrees of freedom in the vicinity of a stable equilibrium position, which is described by a system of linear differential equations of the form x is n-dimensional vector of deviations of the generalized coordinates of the system from their equilibrium values, and A - symmetric positive definite matrix. Such movement can be represented as an overlay n harmonic vibrations (so-called normal vibrations) with circular frequencies equal to the square roots of all possible eigenvalues \u200b\u200bλ k matrices A. Finding the normal vibrations of the system here is reduced to finding all the eigenvalues \u200b\u200bλ k; and eigenvectors x k matrices A.The set of all eigenvalues \u200b\u200bof a matrix is \u200b\u200bcalled its spectrum. If the matrix A - symmetric, then its spectrum consists of n real numbers λ 1, ..., λ n (some of them may coincide with each other), and the matrix itself, using a transition to a new coordinate system, can be reduced to a diagonal form, i.e., the corresponding linear transformation A in n-dimensional space (the so-called self-adjoint transformation) admits a special representation - the so-called. Spectral decomposition of the form

where E 1 , ..., E n -operators of projection to mutually perpendicular directions of eigenvectors x 1 , ......, x n. Asymmetric matrix A (which corresponds to a non-self-adjoint linear transformation) has, generally speaking, a spectrum consisting of complex numbers λ 1, ..., λ 1, and can only be transformed to a more complex than the diagonal Jordan form [cf. Normal (Jordan) form of matrices (See Normal form of matrices)] corresponding to the representation of the linear transformation A, more complex than the usual spectral decomposition described above.

When studying oscillations around the equilibrium state of systems with an infinite number of degrees of freedom (for example, a homogeneous or inhomogeneous string), the problem of finding the eigenvalues \u200b\u200band eigenvectors of a linear transformation in a finite-dimensional space has to be extended to a certain class of linear transformations (i.e., linear operators) in an infinite-dimensional linear space. In many cases (including, in particular, the case of string vibrations), the corresponding operator can be written in the form of acting in the function space f(x) integral operator A, so here

where TO(x, y) - given on a square a≤ x, y≤ b continuous function of two variables satisfying the symmetry condition TO(x, y) \u003d K(y, x). In these cases, the operator A always has a complete system of pairwise orthogonal eigenfunctions (See Eigenfunctions) φ k, to which there corresponds a countable sequence of real eigenvalues \u200b\u200bλ k, which together make up the spectrum of the operator A.If we consider the functions on which the operator acts A, as vectors of Hilbert space, then the action Awill, as in the case of a finite-dimensional self-adjoint transformation, reduce to stretching the space along the system of mutually orthogonal axes φ k with stretch coefficients λ k (for λ k 0 such stretching has the meaning of stretching with the coefficient | λ k| combined with mirroring), and the operator itself A here again will have a spectral expansion of the form

where E k - directional projection operators φ k.

S. a., Originally developed for integral operators with symmetric kernel TO(x, y), defined and continuous in some bounded domain, was then extended within the framework of the general theory of operators to many other types of linear operators (for example, to integral operators with a kernel that has a singularity or given in an unbounded domain, differential operators in spaces of functions of one or several variables, etc. etc.), as well as on abstractly defined linear operators in infinite-dimensional linear spaces. It turned out, however, that such an extension is associated with a significant complication of the semiconductor arithmetic, since for many linear operators the eigenvalues \u200b\u200band eigenfunctions, understood in the usual sense, do not exist at all. Therefore, in the general case, the spectrum has to be defined not as a set of eigenvalues \u200b\u200bof the operator A, but as a collection of those values \u200b\u200bfor which the operator ( A- λ E) -1, where E - the identical (unit) operator does not exist, or is defined only on a non-dense set, or is an unbounded operator. All eigenvalues \u200b\u200bof the operator belong to its spectrum and together form its discrete spectrum; the rest of the spectrum is often called the continuous spectrum of the operator [sometimes, the continuous spectrum is only called the collection of those λ for which the operator ( A -λ E) -1 is defined on a dense set of elements of space, but is unlimited, and all points of the spectrum that are not included in either the discrete or the continuous spectrum are called the residual spectrum].

Most developed by S. and. self-adjoint linear operators in a Hilbert space (generalizing symmetric matrices) and unitary linear operators in the same space (generalizing unitary matrices). Self-adjoint operator A in a Hilbert space always has a purely real spectrum (discrete, continuous, or mixed) and admits a spectral decomposition of the form

where E(λ) - t. n. expansion of unity (corresponding to the operator A), i.e., a family of projection operators (See Projection operator) , satisfying special conditions... The points of the spectrum in this case are the points of growth of the operator function E(λ) ; in the case of a purely discrete spectrum, they are all jumps E(λ) , so here

and the spectral expansion (*) is reduced to the expansion

A unitary operator in a Hilbert space has a spectrum located on the circle | λ | = 1, and admits a spectral decomposition of a related (*) form, but with the replacement of integration from -∞ to ∞ by integration over this circle. A special class of normal operators in a Hilbert space, representable in a similar representation (*), is also studied, but where the integration on the right-hand side is extended to a more general set of points λ of the complex plane, which is the spectrum A. As for S. and. of non-self-adjoint and non-normal linear operators generalizing arbitrary nonsymmetric matrices, then numerous papers by J. Birkhoff (USA), T. Carleman (Sweden), M. V. Keldysh, M. G. Kerin (USSR), B. Sökefalvi-Nagy (Hungary), N. Dunford (USA), and many other scientists, but nevertheless the corresponding theory is still far from complete.

S. a. linear operators has a number of important applications in classical mechanics (especially the theory of oscillations), electrodynamics, quantum mechanics, theory of random processes, differential and integral equations, and other areas of mathematics and mathematical physics.

Lit .: Courant P., Hilbert D., Methods of Mathematical Physics, trans. from it., 3rd ed., t. 1, M. - L., 1951; Akhiezer N.I., Glazman I.M., Theory of linear operators in Hilbert space, 2nd ed., Moscow, 1966; Plesner A.I., Spectral theory of linear operators, Moscow, 1965; Rise F., Szekefalvi Nagy B., Lectures on functional analysis, trans. from French., M., 1954; Szekefalvi-Nagy B., Foias Ch., Harmonic analysis of operators in Hilbert space, trans. with French., M., 1970; Dunford N., Schwartz J.T., Linear Operators, trans. from English, part 2-3, M., 1966-74; Keldysh M.V., Lidskii V.B., Problems of the spectral theory of non-self-adjoint operators, in: Tr. 4th All-Union Mathematical Congress, vol. 1, Leningrad, 1963, p. 101-20.

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