magnetic moment. Kvant

Various media, when considering their magnetic properties, are called magnets .

All substances in one way or another interact with a magnetic field. Some materials retain their magnetic properties even in the absence of an external magnetic field. The magnetization of materials occurs due to the currents circulating inside the atoms - the rotation of electrons and their movement in the atom. Therefore, the magnetization of a substance should be described using real atomic currents, called Ampere currents.

In the absence of an external magnetic field, the magnetic moments of the atoms of a substance are usually randomly oriented, so that the magnetic fields they create cancel each other out. When an external magnetic field is applied, the atoms tend to orient their magnetic moments in the direction of the external magnetic field, and then the compensation of magnetic moments is violated, the body acquires magnetic properties - it becomes magnetized. Most bodies are magnetized very weakly and the magnitude of the magnetic field induction B in such substances differs little from the magnitude of the magnetic field induction in vacuum. If the magnetic field is weakly amplified in a substance, then such a substance is called paramagnetic :

( , , , , , , Li, Na);

if it weakens, then it diamagnetic :

(Bi, Cu, Ag, Au, etc.) .

But there are substances that have strong magnetic properties. Such substances are called ferromagnets :

(Fe, Co, Ni, etc.).

These substances are able to retain magnetic properties even in the absence of an external magnetic field, representing permanent magnets.

All bodies when they are introduced into an external magnetic field are magnetized to one degree or another, i.e. create their own magnetic field, which is superimposed on an external magnetic field.

Magnetic properties of matter are determined by the magnetic properties of electrons and atoms.

Magnetics consist of atoms, which, in turn, consist of positive nuclei and, relatively speaking, electrons revolving around them.

An electron moving in an orbit in an atom is equivalent to a closed circuit with orbital current :

where e is the electron charge, ν is the frequency of its orbital rotation:

The orbital current corresponds to orbital magnetic moment electron

,

(6.1.1)

where S is the area of ​​the orbit, is the unit normal vector to S, is the electron velocity. Figure 6.1 shows the direction of the orbital magnetic moment of an electron.

An electron moving in an orbit has orbital angular momentum , which is directed opposite to and is related to it by the relation

where m is the mass of the electron.

In addition, the electron has own angular momentum, which is called electron spin

,

(6.1.4)

where , is Planck's constant

The spin of an electron corresponds to spin magnetic moment electron directed in the opposite direction:

,

(6.1.5)

The value is called gyromagnetic ratio of spin moments

It can be proved that the torque M acting on a circuit with current I in a uniform field is directly proportional to the area flown by the current, the strength of the current and the induction of the magnetic field B. In addition, the torque M depends on the position of the circuit relative to the field. The maximum torque Miaks is obtained when the plane of the contour is parallel to the lines of magnetic induction (Fig. 22.17), and is expressed by the formula

(Prove this using formula (22.6a) and Fig. 22.17.) If we denote then we get

The value characterizing the magnetic properties of a circuit with current, which determine its behavior in an external magnetic field, is called the magnetic moment of this circuit. The magnetic moment of the circuit is measured by the product of the current strength in it and the area flowed around by the current:

The magnetic moment is a vector, the direction of which is determined by the rule of the right screw: if the screw is turned in the direction of the current in the circuit, then the translational movement of the screw will show the direction of the vector (Fig. 22.18, a). The dependence of the torque M on the orientation of the contour is expressed by the formula

where a is the angle between the vectors and B. From fig. 22.18, b it can be seen that the equilibrium of the circuit in a magnetic field is possible when the vectors B and Rmag are directed along the same straight line. (Think about the case in which this equilibrium will be stable.)

MAGNETIC TORQUE- physical. quantity characterizing the magnetic. charge system properties. particles (or individual particles) and determining, along with other multipole moments (electric dipole moment, quadrupole moment, etc., see Multipoli) the interaction of the system with the external. el-magn. fields and other similar systems.

According to the ideas of the classical electrodynamics, magnet. the field is created by moving electric. charges. Although modern theory does not reject (and even predicts) the existence of particles with magnetic. charge ( magnetic monopoles), such particles have not yet been experimentally observed and are absent in ordinary matter. Therefore, the elementary characteristic of the magnet. properties turns out to be exactly the M. m. A system that has a M. m. (axial vector) creates a magnetic field at large distances from the system. field

(- radius vector of the observation point). A similar view has an electric. dipole field, consisting of two closely spaced electric. charges of opposite sign. However, unlike electrical dipole moment. M. m. is created not by a system of point "magnetic charges", but by electric. currents flowing within the system. If a closed electric density current flows in a limited volume V, then the M. m. created by him is determined by the f-loy

In the simplest case of a closed circular current I, flowing along a flat coil of area s, , and the vector of the M. m. is directed along the right normal to the coil.

If the current is created by the stationary movement of point electric. charges with masses having velocities , then the resulting M. m., as follows from f-ly (1), has the form

where is meant microscopic averaging. values ​​over time. Since the vector product on the right side is proportional to the momentum vector of the particle's momentum (it is assumed that the speeds ), then the contributions of the dep. particles in M. m. and at the moment of the number of movements are proportional:

Proportionality factor e/2ts called gyromagnetic ratio; this value characterizes the universal connection between the magnetic. and mechanical charge properties. particles in the classical electrodynamics. However, the movement of elementary charge carriers in matter (electrons) obeys the laws of quantum mechanics, which makes adjustments to the classical. picture. In addition to the orbital mechanical moment of motion L The electron has an internal mechanical moment - back. The total magnetic field of an electron is equal to the sum of the orbital magnetic field (2) and the spin magnetic field.

As can be seen from this formula (following from the relativistic Dirac equations for an electron), gyromagnet. the ratio for the spin turns out to be exactly twice that for the orbital momentum. A feature of the quantum concept of magnet. and mechanical moments is also the fact that the vectors cannot have a definite direction in space due to the non-commutativity of the projection operators of these vectors on the coordinate axes.

Spin M. m. charge. particles defined f-loy (3), called. normal, for an electron it is magneton Bora. Experience shows, however, that the M. m. of an electron differs from (3) by an order of magnitude ( is the fine structure constant). A similar supplement called abnormal magnetic moment, arises due to the interaction of an electron with photons, it is described in the framework of quantum electrodynamics. Other elementary particles also have anomalous magnetic properties; they are especially large for hadrons, to-rye, according to modern. representations, have vnutr. structure. Thus, the anomalous M. m. of the proton is 2.79 times greater than the "normal" one - the nuclear magneton, ( M- the mass of the proton), and the M. m. of the neutron is equal to -1.91, i.e., it is significantly different from zero, although the neutron does not have electric power. charge. Such large anomalous M. m. hadrons due to internal. the movement of their constituent charges. quarks.

Lit .: Landau L. D., Lifshits E. M., Field Theory, 7th ed., M., 1988; Huang K., Quarks, leptons and gauge fields, transl. from English, M., 1985. D. V. Giltsov.

It is known that the magnetic field has an orienting effect on the loop with current, and the loop rotates around its axis. This happens because in a magnetic field a moment of forces acts on the frame, equal to:

Here B is the magnetic field induction vector, is the current in the frame, S is its area and a is the angle between the lines of force and the perpendicular to the frame plane. This expression includes the product , which is called the magnetic dipole moment or simply the magnetic moment of the frame. It turns out that the magnitude of the magnetic moment completely characterizes the interaction of the frame with a magnetic field. Two frames, one of which has a large current and a small area, and the other has a large area and a small current, will behave in a magnetic field in the same way if their magnetic moments are equal. If the frame is small, then its interaction with the magnetic field does not depend on its shape.

It is convenient to consider the magnetic moment as a vector, which is located on a line perpendicular to the plane of the frame. The direction of the vector (up or down along this line) is determined by the "rule of the gimlet": the gimlet must be placed perpendicular to the frame plane and rotated in the direction of the frame current - the direction of movement of the gimlet will indicate the direction of the magnetic moment vector.

Thus, the magnetic moment is a vector perpendicular to the plane of the frame.

Now let's visualize the behavior of the frame in a magnetic field. She will strive to turn around like that. so that its magnetic moment is directed along the magnetic field vector B. A small loop with current can be used as the simplest "measuring device" to determine the magnetic field vector.

Magnetic moment is an important concept in physics. Atoms are made up of nuclei around which electrons revolve. Each electron moving around the nucleus as a charged particle creates a current, forming, as it were, a microscopic frame with current. Let us calculate the magnetic moment of one electron moving in a circular orbit of radius r.

Electric current, i.e., the amount of charge that is transferred by an electron in orbit in 1 s, is equal to the charge of the electron e, multiplied by the number of revolutions it makes:

Therefore, the magnitude of the magnetic moment of the electron is:

It can be expressed in terms of the magnitude of the angular momentum of the electron. Then the value of the magnetic moment of the electron associated with its orbital motion, or, as they say, the value of the orbital magnetic moment, is equal to:

An atom is an object that cannot be described using classical physics: for such small objects, completely different laws apply - the laws of quantum mechanics. Nevertheless, the result obtained for the orbital magnetic moment of the electron turns out to be the same as in quantum mechanics.

Otherwise, the situation is with the electron's own magnetic moment - the spin, which is associated with its rotation around its axis. For the spin of an electron, quantum mechanics gives the value of the magnetic moment, which is 2 times greater than classical physics:

and this difference between orbital and spin magnetic moments cannot be explained classically. The total magnetic moment of an atom is made up of the orbital and spin magnetic moments of all electrons, and since they differ by a factor of 2, a factor appears in the expression for the magnetic moment of the atom characterizing the state of the atom:

Thus, an atom, like an ordinary loop with current, has a magnetic moment, and in many respects their behavior is similar. In particular, as in the case of a classical frame, the behavior of an atom in a magnetic field is completely determined by the magnitude of its magnetic moment. In this regard, the concept of a magnetic moment is very important in explaining various physical phenomena that occur with matter in a magnetic field.

When placed in an external field, a substance can react to this field and itself become a source of a magnetic field (be magnetized). Such substances are called magnets(compare with the behavior of dielectrics in an electric field). According to their magnetic properties, magnets are divided into three main groups: diamagnets, paramagnets, and ferromagnets.

Different substances are magnetized in different ways. The magnetic properties of matter are determined by the magnetic properties of electrons and atoms. Most of the substances are weakly magnetized - these are diamagnets and paramagnets. Some substances under normal conditions (at moderate temperatures) are capable of being magnetized very strongly - these are ferromagnets.

Many atoms have a net magnetic moment equal to zero. Substances made up of such atoms are diamagetics. These include, for example, nitrogen, water, copper, silver, common salt NaCl, silicon dioxide Si0 2 . Substances, in which the resulting magnetic moment of the atom is different from zero, belong to paramagnets. Examples of paramagnets are: oxygen, aluminum, platinum.

In what follows, when speaking of magnetic properties, we will have in mind mainly diamagnets and paramagnets, and the properties of a small group of ferromagnets will sometimes be specially discussed.

Let us first consider the behavior of matter electrons in a magnetic field. Let us assume for simplicity that the electron rotates in the atom around the nucleus with a speed v along an orbit of radius r. Such a motion, which is characterized by an orbital angular momentum, is essentially a circular current, which is characterized, respectively, by an orbital magnetic moment.

volume r orb. Based on the period of revolution around the circumference T= - we have that

an arbitrary point of the orbit the electron per unit time crosses -

once. Therefore, the circular current, equal to the charge passing through the point per unit time, is given by the expression

Respectively, orbital magnetic moment of an electron according to the formula (22.3) is equal to

In addition to the orbital angular momentum, the electron also has its own angular momentum, called back. Spin is described by the laws of quantum physics and is an inherent property of an electron - like mass and charge (see more details in the quantum physics section). The intrinsic angular momentum corresponds to the intrinsic (spin) magnetic moment of the electron r sp.

The nuclei of atoms also have a magnetic moment, but these moments are thousands of times smaller than the moments of electrons, and they can usually be neglected. As a result, the total magnetic moment of the magnet R t is equal to the vector sum of the orbital and spin magnetic moments of the electrons of the magnet:

An external magnetic field acts on the orientation of particles of a substance that have magnetic moments (and microcurrents), as a result of which the substance is magnetized. The characteristic of this process is magnetization vector J, equal to the ratio of the total magnetic moment of the particles of the magnet to the volume of the magnet AV:

Magnetization is measured in A/m.

If a magnet is placed in an external magnetic field В 0, then as a result

magnetization, an internal field of microcurrents B will arise, so that the resulting field will be equal to

Consider a magnet in the form of a cylinder with a base area S and height /, placed in a uniform external magnetic field with induction At 0 . Such a field can be created, for example, using a solenoid. The orientation of microcurrents in the outer field becomes ordered. In this case, the field of microcurrents of diamagnets is directed opposite to the external field, and the field of microcurrents of paramagnets coincides in direction with the external field.

In any section of the cylinder, the orderliness of microcurrents leads to the following effect (Fig. 23.1). Ordered microcurrents inside the magnet are compensated by neighboring microcurrents, and uncompensated surface microcurrents flow along the lateral surface.

The direction of these uncompensated microcurrents is parallel (or anti-parallel) to the current flowing in the solenoid creating an external zero. In general, they Rice. 23.1 give the total internal current This surface current creates an internal microcurrent field B v moreover, the connection between the current and the field can be described by the formula (22.21) for the zero of the solenoid:

Here, the magnetic permeability is taken equal to unity, since the role of the medium is taken into account by introducing the surface current; the density of winding turns of the solenoid corresponds to one for the entire length of the solenoid /: n = one //. In this case, the magnetic moment of the surface current is determined by the magnetization of the entire magnet:

From the last two formulas, taking into account the definition of magnetization (23.4), it follows

or in vector form

Then from formula (23.5) we have

The experience of studying the dependence of the magnetization on the strength of the external field shows that the field can usually be considered weak, and in the expansion in a Taylor series, it is sufficient to confine ourselves to a linear term:

where the dimensionless coefficient of proportionality x - magnetic susceptibility substances. With this in mind, we have

Comparing the last formula for magnetic induction with the well-known formula (22.1), we obtain the relationship between magnetic permeability and magnetic susceptibility:

We note that the values ​​of the magnetic susceptibility for diamagnets and paramagnets are small and are usually modulo 10 "-10 4 (for diamagnets) and 10 -8 - 10 3 (for paramagnets). In this case, for diamagnets X x > 0 and p > 1.