The magnetic properties of any body are determined. Magnetic properties of matter (3) - Abstract

In all bodies placed in a magnetic field, a magnetic moment arises. This phenomenon is called magnetizing.

A magnetized body (magnetic) creates an additional magnetic field with induction B′ Which interacts with induction B 0 = μ a H caused by macroscopic currents. Both fields give the resulting field with induction B, which is obtained as a result of vector addition B' and B 0 .

Closed currents circulate in the molecules of the substance; each such current has a magnetic moment; in the absence of an external magnetic field, the molecular currents are oriented chaotically and the average field created by them will be equal to zero. Under the influence magnetic field the magnetic moments of molecules are oriented mainly along the field, as a result of which the substance is magnetized. The measure of the magnetization of a substance (magnet) is the magnetization vector. Vector magnetization I is equal to the vector sum of all magnetic moments p m molecules enclosed in a unit volume of a substance:

The quantity χ is called magnetic susceptibility- dimensionless quantity.

In the SI system:	In the SGSM system:
B′ = μ I	B′ = 4χ I	2)
B = μ 0 H + μ I	B = H+ 4χ I	3)
μ = 1 + χ	μ = 1 + 4π χ	4)

Curve expressing the relationship between H and B or H and I is called magnetization curve.

Substances for which χ> 0 (but insignificantly) are called paramagnetic ( paramagnets); substances for which χ< 0, называются диамагнитными (diamagnets). Substances for which χ is much greater than unity are called ferromagnets.

Ferromagnets differ from paramagnets and diamagnets in a number of properties.

but) The magnetization curve of ferromagnets has a complex character (Fig. 1); for paramagnets it is a straight line with a positive angular
coefficient, for diamagnets - a straight line with a negative slope. The magnetic susceptibility and permeability of ferromagnets depend on the field strength; paramagnets and diamagnets do not have this relationship.

For ferromagnets, the initial magnetic permeability (μinit) is usually indicated - the limiting value of the magnetic permeability when the field strength and induction are close to zero, i.e.

The curve of μ versus H for ferromagnets passes through a maximum. The tables usually indicate the maximum value (μ max).

b) The magnetic susceptibility of ferromagnets increases with increasing temperature. At a certain temperature T k a ferromagnet is transformed into a paramagnet; this temperature is called Curie temperature (Curie point). At temperatures above the Curie point, the substance is paramagnetic. Near the Curie temperature, the magnetic susceptibility of the cutting ferromagnet increases.

The magnetic susceptibility of diamagnets and some paramagnets (for example, in alkali metals) does not depend on temperature. The magnetic susceptibility of paramagnets (with a few exceptions) varies inversely with the absolute temperature.

in) A demagnetized ferromagnet is magnetized by a magnetic field; addiction B(or I) from H during magnetization, the curve 0–1 will be expressed (Fig. 1). This curve is called the initial magnetization curve. The magnetization in weak fields grows rapidly, then the growth slows down and, finally, a saturation state sets in, in which the magnetization practically remains constant with a further increase in the field.

The maximum value of magnetization is called saturation magnetization (I s).

When decreasing H to zero B(and I) will change along the curve 1–2; there is a lag of the change in induction from the change in the field strength. This phenomenon is called magnetic hysteresis.

The magnitude of the induction conserved in the ferromagnet after the field is removed (when H= 0) is called residual induction ( B r). Figure 1 B r is equal to the segment 0–2. To demagnetize a ferromagnet, you need to remove the residual induction. To do this, you need to create a field in the opposite direction. The change in induction in the bottom of the opposite direction will be depicted by the curve 2–3–4.

Field strength H c(segment 0–3 in Fig. 8), at which the induction is zero, is called coercive intensity (force).

Addiction B(or I) from periodically changing magnetic field strength from + H before - H expressed by a closed curve 1–2–3–4–5–6–1. This curve is called hysteresis loop.

For one cycle of changing the field strength from + H before - H energy is consumed proportional to the area of the hysteresis loop.

The properties of ferromagnets are explained by the presence of regions in them, which in the absence of an external magnetic field are spontaneously magnetized to saturation. These areas are called domains. But the location and magnetization of these regions are such that the total magnetization of the entire body is zero even in the absence of a field.

When a ferromagnet is in a magnetic field, the boundaries between domains are displaced (in weak fields) and the domain magnetization vectors are rotated in the direction of the magnetizing field (in stronger fields), as a result of which the ferromagnet is magnetized.

A ferromagnet placed in a magnetic field changes its linear dimensions, that is, it is deformed. This phenomenon is called magnetostriction. Elongation depends on the nature of the ferromagnet and the strength of the magnetic field.

The magnitude of the magnetostrictive effect does not depend on the direction of the field; in some substances, shortening (nickel) is observed, while in others, elongation (iron in weak fields) along zero. This phenomenon is used to generate ultrasonic vibrations with frequencies up to 100 kHz.

Numerous experiments indicate that all substances placed in a magnetic field are magnetized and create their own magnetic field, the action of which is added to the action of an external magnetic field:

where is the magnetic induction of the field in the substance; - magnetic induction of the field in vacuum, - magnetic induction of the field, which has arisen due to the magnetization of the substance.

In this case, the substance can either strengthen or weaken the magnetic field. The effect of a substance on an external magnetic field is characterized by a quantity called the magnetic permeability of the substance.

Magnetic permeability is a physical scalar quantity that shows how many times the induction of a magnetic field in a given substance differs from the induction of a magnetic field in a vacuum.

Substances that weaken the external magnetic field are called diamagnets(bismuth, nitrogen, helium, carbon dioxide, water, silver, gold, zinc, cadmium, etc.).

Substances that enhance the external magnetic field - paramagnets(aluminum, oxygen, platinum, copper, calcium, chromium, manganese, cobalt salts, etc.).

For diamagnets> 1. But in both cases, the difference from 1 is small (several ten-thousandths or hundred-thousandths of a unit). So, for example, bismuth = 0.9998 = 1,000.

Some substances (iron, cobalt, nickel, gadolinium and various alloys) cause a very large increase in the external field. They are called ferromagnets... For them = 10 3 -10 5.

For the first time, the explanation of the reasons due to which bodies have magnetic properties was given by Ampere. According to his hypothesis, elementary elements circulate inside molecules and atoms. electric currents, which determine the magnetic properties of any substance.

It has now been established that all atoms and elementary particles do have magnetic properties. The magnetic properties of atoms are mainly determined by their constituent electrons.

According to the semiclassical model of the atom, proposed by E. Rutherford and N. Bohr, electrons in atoms move around the nucleus in closed orbits (in the first approximation, we can assume that they are circular). The motion of an electron can be represented as an elementary circular current, where e is the charge of the electron, v is the frequency of rotation of the electron in its orbit. This current forms a magnetic field, which is characterized by a magnetic moment, its modulus is determined by the formula, where S is the orbital area.

The magnetic moment of an electron due to its motion around the nucleus is called orbital magnetic moment... The orbital magnetic moment is a vector quantity and the direction is determined by the right-hand screw rule. If the electron moves clockwise (Fig. 1), then the currents are directed counterclockwise (in the direction of movement of the positive charge), and the vector is perpendicular to the orbital plane.

Since the planes of the orbits of different electrons do not coincide in an atom, their magnetic moments are directed at different angles to each other. The resulting orbital magnetic moment of a many-electron atom is equal to the vector sum of the orbital magnetic moments of individual electrons.

Uncompensated orbital magnetic moment is possessed by atoms with partially filled electron shells. In atoms with filled electron shells, it is 0.

In addition to the orbital magnetic moment, the electron also has intrinsic (spin) magnetic moment, which was first established by O. Stern and V. Gerlach in 1922. The existence of a magnetic field for the electron was explained by its rotation around its own axis, although one should not literally liken the electron to a rotating charged ball (top).

It has been reliably established that the magnetic field of an electron is the same inherent property as its mass and charge. An electron, in a very rough approximation, can be thought of as a very small ball surrounded by electric and magnetic fields (Fig. 2). The magnetic fields of all electrons are the same, just as their masses and charges are the same. Spin magnetic moment is a vector directed along the axis of rotation. He can orient himself in only two ways: either along ... or against ... If there is an external magnetic field in the place where the electron is located, then either along the field or against the field. As shown in quantum physics, in the same energy state there can be only two electrons, the spin magnetic moments of which are opposite (Pauli's principle).

In many-electron atoms, the spin magnetic moments of individual electrons, like the orbital moments, add up as vectors. In this case, the resulting spin magnetic moment of the atom for atoms with filled electron shells is 0.

The total magnetic moment of an atom (molecule) is equal to the vector sum of the magnetic moments (orbital and spin) of the electrons entering the atom (molecule):

Diamagnets consist of atoms that, in the absence of an external magnetic field, do not have their own magnetic moments, since all spin and all orbital magnetic moments are compensated for.

The external magnetic field does not act on the entire atom of the diamagnet, but acts on individual electrons of the atom, the magnetic moments of which are nonzero. Let at a given moment the speed of the electron makes a certain angle (Fig. 3) with the magnetic induction of the external field.

Thanks to the component, the Lorentz force (directed towards us in Fig. 3) will act on the electron, which will cause additional (apart from other motions, in which the electron participates in the absence of a field), circular motion. But this movement is an additional circular current that will create a magnetic field characterized by a magnetic moment (induced), directed towards the right-hand screw. As a result, diamagnets weaken the external magnetic field.

Paramagnets are made up of atoms that have the net magnetic moment of the atom. In the absence of an external field, these moments are oriented chaotically and the substance as a whole does not create a magnetic field around itself. When paramagnets are placed in a magnetic field, predominant orientation of vectors along the field (this is prevented by the thermal motion of particles). Thus, the paramagnet is magnetized, creating its own magnetic field, which coincides in direction with the external field and amplifies it. This effect is called paramagnetic. When the external magnetic field is weakened to zero, the orientation of the magnetic moments due to thermal motion is violated and the paramagnet is demagnetized. In paramagnets, a diamagnetic effect is also observed, but it is much weaker than the paramagnetic one.

There are microscopic circular currents ( molecular currents). Later, after the discovery of the electron and the structure of the atom, this idea was confirmed: these currents are created by the movement of electrons around the nucleus and, since they are oriented in the same way, in total form a field inside and around the magnet.

On the picture but the planes in which the elementary electric currents are located are oriented randomly due to the chaotic thermal motion of atoms, and the substance does not exhibit magnetic properties. In a magnetized state (under the action, for example, of an external magnetic field) (figure b) these planes are oriented in the same way, and their actions are summed up.

Magnetic permeability.

The reaction of the medium to the action of an external magnetic field with induction B0 (field in vacuum) is determined by the magnetic susceptibility μ :

where IN Is the induction of the magnetic field in the substance. The magnetic permeability is the same as the dielectric constant ɛ .

According to their magnetic properties, substances are divided into diamagnetics, paramagnets and ferromagnetics... Diamagnets have a coefficient μ , which characterizes the magnetic properties of the medium, is less than unity (for example, in bismuth μ = 0.999824); in paramagnets μ > 1 (for platinum μ - 1,00036); in ferromagnets μ ≫ 1 (iron, nickel, cobalt).

Diamagnets are repelled by a magnet, paramagnets are attracted to it. By these signs, they can be distinguished from each other. For many substances, the magnetic permeability hardly differs from unity, but for ferromagnets it greatly exceeds it, reaching several tens of thousands of units.

Ferromagnets.

Ferromagnets exhibit the strongest magnetic properties. The magnetic fields that are created by ferromagnets are much stronger than the external magnetizing field. True, the magnetic fields of ferromagnets are not created due to the circulation of electrons around nuclei - orbital magnetic moment, and due to the proper rotation of the electron - the intrinsic magnetic moment, called spin.

Curie temperature ( Twith) Is the temperature above which ferromagnetic materials lose their magnetic properties. It has its own for each ferromagnet. For example, for iron T with= 753 ° С, for nickel T with= 365 ° С, for cobalt T with= 1000 ° C. There are ferromagnetic alloys in which T with < 100 °С.

The first detailed studies of the magnetic properties of ferromagnets were carried out by the outstanding Russian physicist A.G. Stoletov (1839-1896).

Ferromagnets are used quite widely: as permanent magnets (in electrical measuring instruments, loudspeakers, telephones, etc.), steel cores in transformers, generators, electric motors (to enhance the magnetic field and save electricity). On magnetic tapes, which are made of ferromagnets, sound and images are recorded for tape and video recorders. On thin magnetic films, information is recorded for storage devices in electronic computers.

« Physics - grade 11 "

The magnetic field is created by electric currents and permanent magnets.
All substances placed in a magnetic field create their own magnetic field.

Magnetization of the substance.

All substances placed in a magnetic field are magnetized, that is, they themselves become sources of a magnetic field.
As a result, the vector of magnetic induction in the presence of matter differs from the vector of magnetic induction in vacuum.

Ampere's hypothesis

The reason why bodies have magnetic properties was established by the French physicist Ampere: the magnetic properties of a body can be explained by currents circulating inside it.

Inside molecules and atoms, there are elementary electric currents that are formed as a result of the movement of electrons in atoms.
If the planes in which these currents circulate are randomly located in relation to each other due to the thermal motion of the molecules, then their actions are mutually compensated, and the body does not exhibit any magnetic properties.

In the magnetized state, the elementary currents in the body are oriented so that their actions add up.

The strongest magnetic fields are created by substances called ferromagnets.
Permanent magnets are made of them, since the ferromagnetic field does not disappear after the magnetizing field is turned off.

Magnetic fields are created by ferromagnets not only due to the circulation of electrons around nuclei, but also due to their own rotation. In ferromagnets, there are regions called domains about 0.5 microns in size.

If the ferromagnet is not magnetized, then the orientation of the domains is chaotic, and the total magnetic field created by the domains is zero.
When an external magnetic field is switched on, the domains are oriented along the lines of the magnetic induction of this field, and the induction of the magnetic field in ferromagnets increases, becoming thousands and even millions of times greater than the induction of the external field.

Curie temperature.

At temperatures higher than a certain one for a given ferromagnet, its ferromagnetic properties disappear.
This temperature is called Curie temperature by the name of the French scientist who discovered this phenomenon.
When heated, magnetized bodies lose their magnetic properties.
For example, the Curie temperature for iron is 753 ° C.
There are ferromagnetic alloys in which the Curie temperature is less than 100 ° C.

The use of ferromagnets

There are not so many ferromagnetic bodies in nature, but they have found wide application.
For example, a core placed in a coil amplifies the magnetic field it generates without increasing the current in the coil.
The cores of transformers, generators, electric motors, etc. are made of ferromagnets.

When the external magnetic field is turned off, the ferromagnet remains magnetized, that is, it creates a magnetic field in the surrounding space.
Because of this, permanent magnets exist.

Ferrites are widely used - ferromagnetic materials that do not conduct electric current, this chemical compounds iron oxides with oxides of other substances.
One of the known ferromagnetic materials - magnetic iron ore - is ferrite.

Ferromagnets are used for magnetic recording of information.
Magnetic tapes and magnetic tapes are made from ferromagnets, which are used for sound recording in tape recorders and for video recording in video tape recorders.

Sound is recorded on a tape using an electromagnet, the magnetic field of which changes in time with sound vibrations.
When the tape moves near the magnetic head, various sections of the film are magnetized.

Magnetic induction head diagram

where
1 - electromagnet core;
2 - magnetic tape;
3 - working gap;
4 winding of the electromagnet.

The development of magnetic recording technology has led to the emergence of magnetic micro-heads, which are used in computers, allowing to create a high density of magnetic recording, so on a ferromagnetic hard disk with a diameter of several centimeters up to several terabytes (10 12 bytes) of information is stored. Reading and writing of information on such a disk is carried out using a micro-head. The disc rotates at a tremendous speed, and the head floats above it in a stream of air, which prevents the possibility of mechanical damage to the disc.

\ (~ \ vec B = \ vec B_0 + \ vec B_1, \)

where \ (~ \ vec B \) is the magnetic induction of the field in the substance; \ (~ \ vec B_0 \) is the magnetic induction of the field in vacuum, \ (~ \ vec B_1 \) is the magnetic induction of the field arising due to the magnetization of the substance. In this case, the substance can either strengthen or weaken the magnetic field. The effect of a substance on an external magnetic field is characterized by the value μ, which is called the magnetic permeability of the substance

\ (~ \ mu = \ dfrac B (B_0). \)

Magnetic permeability is a physical scalar quantity that shows how many times the induction of a magnetic field in a given substance differs from the induction of a magnetic field in a vacuum.

Dia- and para-magnets

All substances have certain magnetic properties, that is, they are magnets... For most substances, the magnetic permeability μ is close to unity and does not depend on the magnitude of the magnetic field. Substances for which the magnetic permeability is slightly less than unity (μ< 1), называются diamagnets, slightly more than one (μ> 1) - paramagnets... Substances whose magnetic permeability depends on the magnitude of the external field and can significantly exceed unity (μ »1) are called ferromagnets.

Examples of diamagnets are lead, zinc, bismuth (μ = 0.9998); paramagnets - sodium, oxygen, aluminum (μ = 1,00023); ferromagnets - cobalt, nickel, iron (μ reaches a value of 8⋅10 3).

For the first time, an explanation of the reasons due to which bodies have magnetic properties was given by Henri Ampere (1820). According to his hypothesis, elementary electric currents circulate inside molecules and atoms, which determine the magnetic properties of any substance.

Let's take some solid... Its magnetization is related to the magnetic properties of the particles (molecules and atoms) of which it is composed. Let's consider what circuits with current are possible at the micro level. The magnetism of atoms is due to two main reasons:

1) the movement of electrons around the nucleus in closed orbits ( orbital magnetic moment) (fig. 1);

2) proper rotation (spin) of electrons ( spin magnetic moment) (Fig. 2).

For the curious... The magnetic moment of the circuit is equal to the product of the current in the circuit and the area covered by the circuit. Its direction coincides with the direction of the magnetic induction vector in the middle of the current loop.

Since the planes of the orbits of different electrons in an atom do not coincide, the vectors of the magnetic field inductions created by them (orbital and spin magnetic moments) are directed at different angles to each other. The resulting induction vector of a many-electron atom is equal to the vector sum of the vectors of the field inductions created by individual electrons. Uncompensated fields are possessed by atoms with partially filled electron shells. In atoms with filled electron shells, the resulting induction vector is 0.

In all cases, the change in the magnetic field is due to the appearance of magnetization currents (the phenomenon electromagnetic induction). In other words, the principle of superposition for the magnetic field remains valid: the field inside the magnet is a superposition of the external field \ (~ \ vec B_0 \) and the field \ (~ \ vec B "\) of the magnetizing currents i ′ that arise under the action of an external field. If the field of the magnetization currents is directed in the same way as the external field, then the induction of the total field will be greater than the external field (Fig. 3, a) - in this case we say that the substance enhances the field; if the field of the magnetization currents is directed opposite to the external field, then the total field will be less than the external field (Fig. 3, b) - it is in this sense that we say that matter weakens the magnetic field.

Rice. 3

IN diamagnets molecules do not have their own magnetic field. Under the action of an external magnetic field in atoms and molecules, the field of magnetization currents is directed opposite to the external field, therefore, the modulus of the magnetic induction vector \ (~ \ vec B \) of the resulting field will be less than the modulus of the magnetic induction vector \ (~ \ vec B_0 \) of the external field.

IN paramagnets molecules have their own magnetic field. In the absence of an external magnetic field, due to the thermal motion, the vectors of inductions of magnetic fields of atoms and molecules are oriented randomly, therefore their average magnetization is zero (Fig. 4, a). When an external magnetic field is applied to atoms and molecules, a moment of forces begins to act, tending to rotate them so that their fields are oriented parallel to the external field. The orientation of the paramagnetic molecules leads to the fact that the substance is magnetized (Fig. 4, b).

Rice. 4

The complete orientation of molecules in a magnetic field is impeded by their thermal motion, therefore the magnetic permeability of paramagnets depends on temperature. It is obvious that the magnetic permeability of paramagnets decreases with increasing temperature.

Ferromagnets

The very name of this class of magnetic materials comes from the Latin name for iron - Ferrum. main feature of these substances lies in the ability to maintain magnetization in the absence of an external magnetic field, all permanent magnets belong to the class of ferromagnets. In addition to iron, its "neighbors" according to the periodic table, cobalt and nickel, have ferromagnetic properties. Ferromagnets find a wide practical use in science and technology, therefore, a significant number of alloys with different ferromagnetic properties have been developed.

All these examples of ferromagnets refer to transition metals, the electron shell of which contains several unpaired electrons, which leads to the fact that these atoms have a significant intrinsic magnetic field. In the crystalline state, due to the interaction between atoms in crystals, regions of spontaneous (spontaneous) magnetization arise - domains. The sizes of these domains are tenths and hundredths of a millimeter (10 -4 - 10 -5 m), which significantly exceeds the size of an individual atom (10 -9 m). Within one domain, the magnetic fields of atoms are oriented strictly parallel, the orientation of the magnetic fields of other domains in the absence of an external magnetic field changes arbitrarily (Fig. 5).

Thus, even in the non-magnetized state, strong magnetic fields exist inside the ferromagnet, the orientation of which changes in a random chaotic manner when passing from one domain to another. If the dimensions of a body significantly exceed the dimensions of individual domains, then the average magnetic field created by the domains of this body is practically absent.

If you place a ferromagnet in an external magnetic field IN 0, then the magnetic moments of the domains begin to rearrange. However, there is no mechanical spatial rotation of the material sections. The process of magnetization reversal is associated with a change in the motion of electrons, but not with a change in the position of atoms in the nodes crystal lattice... Domains with the most favorable orientation relative to the direction of the field increase their size due to neighboring "incorrectly oriented" domains, absorbing them. In this case, the field in the substance increases very significantly.

Properties of ferromagnets

1) the ferromagnetic properties of a substance appear only when the corresponding substance is in crystalline state;

2) the magnetic properties of ferromagnets strongly depend on temperature, since the orientation of the magnetic fields of the domains is impeded by thermal motion. For each ferromagnet there is a certain temperature at which the domain structure is completely destroyed, and the ferromagnet turns into a paramagnet. This temperature value is called Curie point... So for pure iron, the Curie temperature is approximately 900 ° C;

3) ferromagnets are magnetized until saturation in weak magnetic fields. Figure 6 shows how the modulus of magnetic field induction changes. B in steel with a change in the external field B 0 ;

4) the magnetic permeability of a ferromagnet depends on the external magnetic field (Fig. 7).

This is due to the fact that at first with an increase B 0 magnetic induction B grows stronger, and, consequently, μ will increase. Then, at the value of the magnetic induction B´ 0 saturation occurs (μ at this moment is maximum) and with a further increase in B 0 magnetic induction B 1 in the substance ceases to change, and the magnetic permeability decreases (tends to 1):

\ (~ \ mu = \ dfrac B (B_0) = \ dfrac (B_0 + B_1) (B_0) = 1 + \ dfrac (B_1) (B_0); \)

5) residual magnetization is observed in ferromagnets. If, for example, a ferromagnetic rod is placed in a solenoid through which a current flows, and magnetized to saturation (point BUT) (Fig. 8), and then reduce the current in the solenoid, and with it and B 0, then it can be seen that the field induction in the rod during its demagnetization remains all the time greater than in the process of magnetization. When B 0 = 0 (the current in the solenoid is off), the induction will be B r(residual induction). The rod can be removed from the solenoid and used as a permanent magnet. To finally demagnetize the rod, you need to pass a current in the opposite direction through the solenoid, i.e. apply an external magnetic field with the opposite direction of the induction vector. Now increasing the modulus of the induction of this field to B oc, demagnetize the rod ( B = 0).).

Thus, during magnetization and demagnetization of a ferromagnet, the induction B lags behind B 0. This lag is called hysteresis phenomenon... The curve shown in Figure 8 is called hysteresis loop.

Hysteresis(Greek ὑστέρησις - "lagging behind") is a property of systems that do not immediately follow the applied forces.

The shape of the magnetization curve (hysteresis loop) differs significantly for various ferromagnetic materials, which are widely used in scientific and technical applications. Some magnetic materials have a wide loop with high values residual magnetization and coercive force, they are called magnetically hard and are used to make permanent magnets. For other ferromagnetic alloys, small values of the coercive force are characteristic; such materials are easily magnetized and remagnetized even in weak fields. Such materials are called magnetically soft and are used in various electrical devices - relays, transformers, magnetic circuits, etc.

Literature

Aksenovich L.A. Physics in high school: Theory. Tasks. Tests: Textbook. allowance for institutions providing the receipt of general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Minsk: Adukatsya i vyhavanne, 2004 .-- C.330- 335.
Zhilko, V.V. Physics: textbook. allowance for the 11th grade. general education. shk. from rus. lang. training / V.V. Zhilko, A.V. Lavrinenko, L.G. Markovich. - Mn .: Nar. Asveta, 2002 .-- S. 291-297.