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Magnetic moment

the main quantity characterizing the magnetic properties of a substance. The source of magnetism, according to the classical theory of electromagnetic phenomena, are electric macro- and microcurrents. An elementary source of magnetism is considered a closed current. From experience and classical theory electro magnetic field it follows that the magnetic actions of a closed current (current-carrying circuit) are determined if the product ( M) current strength i to the contour area σ ( M = iσ /c in the CGS system of units (See CGS system of units), With - speed of light). Vector M and is, by definition, M. m. It can also be written in a different form: M = m l, where m- the equivalent magnetic charge of the circuit, and l- the distance between the "charges" of opposite signs (+ and - ).

M. m. have elementary particles, atomic nuclei, electron shells of atoms and molecules. The mechanical mass of elementary particles (electrons, protons, neutrons, and others), as shown by quantum mechanics, is due to the existence of their own mechanical moment - Spin a. The nuclear masses are composed of the intrinsic (spin) masses of the protons and neutrons that form these nuclei, as well as the masses associated with their orbital motion within the nucleus. The molecular masses of the electron shells of atoms and molecules are made up of spin and orbital molecular masses of electrons. The spin magnetic moment of an electron m cn can have two equal and oppositely directed projections on the direction of the external magnetic field N. The absolute value of the projection

where μ in \u003d (9.274096 ± 0.000065) 10 -21 erg/gs - Boron magneton, h - Planck constant , e and m e - the charge and mass of the electron, With- the speed of light; S H- projection of the spin mechanical moment on the direction of the field H. The absolute value of spin M. m.

where s= 1 / 2 - spin quantum number (See quantum numbers). The ratio of the spin M. m. to the mechanical moment (back)

since spin

Studies of atomic spectra have shown that m H cn is actually not equal to m in, but m in (1 + 0.0116). This is due to the action on the electron of the so-called zero-point oscillations of the electromagnetic field (see Quantum electrodynamics, Radiative corrections).

The orbital M. m. of an electron m orb is related to the mechanical orbital moment orb by the relation g opb = |m orb | / | orb | = | e|/2m e c, that is, the magnetomechanical ratio g opb is two times less than g cn. Quantum mechanics allows only a discrete series of possible projections m orb onto the direction of the external field (the so-called spatial quantization): m H orb = m l m in , where m l - magnetic quantum number taking 2 l+ 1 values ​​(0, ±1, ±2,..., ± l, where l- orbital quantum number). In multielectron atoms, the orbital and spin magnetisms are determined by the quantum numbers L and S total orbital and spin moments. The addition of these moments is carried out according to the rules of spatial quantization. Due to the inequality of magnetomechanical relations for the electron spin and its orbital motion ( g cn ¹ g opb) the resulting M. m. of the atomic shell will not be parallel or anti-parallel to its resulting mechanical moment J. Therefore, one often considers the component of the total M. m. in the direction of the vector J equal to

where g J is the magnetomechanical ratio of the electron shell, J is the total angular quantum number.

M. m. of a proton whose spin is

where Mp is the mass of the proton, which is 1836.5 times greater m e , m poison - nuclear magneton equal to 1/1836.5m c. The neutron, on the other hand, should have no MM, since it is devoid of charge. However, experience has shown that the MM of the proton m p = 2.7927m is poison, and that of the neutron m n = -1.91315m is poison. This is due to the presence of meson fields near nucleons, which determine their specific nuclear interactions (see Nuclear forces, Mesons) and affect their electromagnetic properties. The total M. m. of complex atomic nuclei are not multiples of m poison or m p and m n. Thus, M. m. nuclei of potassium

To characterize the magnetic state of macroscopic bodies, the average value of the resulting magnetic force of all microparticles forming the body is calculated. Referred to a unit volume of a body, the magnetic field is called magnetization. For macrobodies, especially in the case of bodies with atomic magnetic ordering (ferro-, ferri-, and antiferromagnets), the concept of average atomic M. m. is introduced as the average value of M. m. per one atom (ion) - the carrier of M. m. in body. In substances with a magnetic order, these average atomic molecular masses are obtained as the quotient of the division of the spontaneous magnetization of ferromagnetic bodies or magnetic sublattices in ferri- and antiferromagnets (at absolute zero temperature) by the number of atoms that carry the molecular mass per unit volume. Usually these average atomic molecular weights differ from the molecular weights of isolated atoms; their values ​​in Bohr magnetons m in turn out to be fractional (for example, in the transition d-metals Fe, Co and Ni, respectively, 2.218 m in, 1.715 m in and 0.604 m in) This difference is due to a change in the motion of d-electrons (carriers of M. m.) in a crystal compared to motion in isolated atoms. In the case of rare-earth metals (lanthanides), as well as non-metallic ferro- or ferrimagnetic compounds (for example, ferrites), the unfinished d- or f-layers of the electron shell (the main atomic carriers of M. m.) of neighboring ions in the crystal overlap weakly, therefore, a noticeable collectivization of these there are no layers (as in d-metals), and the molecular masses of such bodies change little in comparison with isolated atoms. The direct experimental determination of MM on atoms in a crystal became possible as a result of the use of magnetic neutron diffraction, radio spectroscopy (NMR, EPR, FMR, etc.) and the Mössbauer effect. For paramagnets, it is also possible to introduce the concept of average atomic magnetism, which is determined through the Curie constant found experimentally, which is included in the expression for the Curie law a or the Curie-Weiss law a (see Paramagnetism).

Lit.: Tamm I. E., Fundamentals of the theory of electricity, 8th ed., M., 1966; Landau L. D. and Lifshitz E. M., Electrodynamics of continuous media, Moscow, 1959; Dorfman Ya. G., Magnetic properties and structure of matter, Moscow, 1955; Vonsovsky S.V., Magnetism of microparticles, M., 1973.

S.V. Vonsovsky.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what the "Magnetic moment" is in other dictionaries:

Dimension L2I SI units A⋅m2 ... Wikipedia

The main quantity characterizing the magn. properties in wa. The source of magnetism (M. m.), according to the classic. email theory. magn. phenomena, yavl. macro and micro (atomic) electric. currents. Elem. a closed current is considered a source of magnetism. From experience and classic. ... ... Physical Encyclopedia

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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 magnetic moment is very important in explaining various physical phenomena occurring 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.

Any substances. The source of the formation of magnetism, according to the classical electromagnetic theory, are microcurrents arising from the movement of an electron in orbit. The magnetic moment is an indispensable property of all nuclei, atomic electron shells and molecules without exception.

Magnetism, which is inherent in all elementary particles, is due to the presence of a mechanical moment in them, called spin (its own mechanical momentum of quantum nature). Magnetic properties atomic nucleus are made up of the spin momenta of the constituent parts of the nucleus - protons and neutrons. Electronic shells (intraatomic orbits) also have a magnetic moment, which is the sum of the magnetic moments of the electrons located on it.

In other words, the magnetic moments of elementary particles are due to the intra-atomic quantum mechanical effect, known as the spin momentum. This effect is similar to the angular momentum of rotation about its own central axis. Spin momentum is measured in Planck's constant, the fundamental constant of quantum theory.

All neutrons, electrons and protons, of which, in fact, the atom consists, according to Planck, have a spin equal to ½. In the structure of an atom, electrons, rotating around the nucleus, in addition to the spin momentum, also have an orbital angular momentum. The nucleus, although it occupies a static position, also has an angular momentum, which is created by the effect of nuclear spin.

The magnetic field that generates an atomic magnetic moment is determined by various forms this angular momentum. The most noticeable contribution to the creation is made by the spin effect. According to the Pauli principle, according to which two identical electrons cannot be simultaneously in the same quantum state, bound electrons merge, while their spin momenta acquire diametrically opposite projections. In this case, the magnetic moment of the electron is reduced, which reduces the magnetic properties of the entire structure. In some elements that have an even number of electrons, this moment decreases to zero, and the substances cease to have magnetic properties. Thus, the magnetic moment of individual elementary particles has a direct impact on the magnetic properties of the entire nuclear-atomic system.

Ferromagnetic elements with an odd number of electrons will always have non-zero magnetism due to the unpaired electron. In such elements, neighboring orbitals overlap, and all spin moments of unpaired electrons take the same orientation in space, which leads to the achievement of the lowest energy state. This process is called exchange interaction.

With this alignment of the magnetic moments of ferromagnetic atoms, a magnetic field arises. And paramagnetic elements, consisting of atoms with disoriented magnetic moments, do not have their own magnetic field. But if you act on them with an external source of magnetism, then the magnetic moments of the atoms will even out, and these elements will also acquire magnetic properties.

Experiments by Stern and Gerlach

In $1921$, O. Stern put forward the idea of ​​an experiment in measuring the magnetic moment of an atom. He carried out this experiment in co-authorship with W. Gerlach in $1922$. The method of Stern and Gerlach uses the fact that a beam of atoms (molecules) is able to deviate in an inhomogeneous magnetic field. An atom that has a magnetic moment can be represented as an elementary magnet with small but finite dimensions. If such a magnet is placed in a uniform magnetic field, then it does not experience force. The field will act on the north and south poles of such a magnet with forces that are equal in magnitude and opposite in direction. As a result, the center of inertia of the atom will either be at rest or move in a straight line. (In this case, the axis of the magnet can oscillate or precess). That is, in a uniform magnetic field there are no forces that act on an atom and impart acceleration to it. A uniform magnetic field does not change the angle between the directions of the magnetic field induction and the magnetic moment of the atom.

The situation is different if the external field is inhomogeneous. In this case, the forces that act on the north and south poles of the magnet are not equal. The resulting force acting on the magnet is non-zero, and it imparts an acceleration to the atom, along the field or against it. As a result, when moving in an inhomogeneous field, the magnet under consideration will deviate from the original direction of movement. In this case, the size of the deviation depends on the degree of field inhomogeneity. In order to obtain significant deviations, the field must change sharply already within the length of the magnet (the linear dimensions of the atom are $\approx (10)^(-8)cm$). Experimenters achieved such heterogeneity with the help of the design of a magnet that created a field. One magnet in the experiment looked like a blade, the other was flat or had a notch. The magnetic lines thickened at the "blade", so that the intensity in this area was significantly greater than at the flat pole. A thin beam of atoms flew between these magnets. Individual atoms were deflected in the generated field. Traces of individual particles were observed on the screen.

According to the concepts of classical physics, magnetic moments in an atomic beam have different directions with respect to some axis $Z$. What does it mean: the projection of the magnetic moment ($p_(mz)$) on this axis takes all the values ​​of the interval from $\left|p_m\right|$ to -$\left|p_m\right|$ (where $\left|p_( mz)\right|-$ magnetic moment modulus). On the screen, the beam should appear expanded. However, in quantum physics, if quantization is taken into account, then not all orientations of the magnetic moment become possible, but only a finite number of them. Thus, on the screen, the trace of a beam of atoms was split into a certain number of individual traces.

The experiments performed showed that, for example, a beam of lithium atoms split into $24$ beams. This is justified, since the main term $Li - 2S$ is a term (one valence electron with spin $\frac(1)(2)\ $ in the s-orbit, $l=0).$ it is possible to draw a conclusion about the magnitude of the magnetic moment. This is how Gerlach proved that the spin magnetic moment is equal to the Bohr magneton. Studies of various elements showed complete agreement with theory.

Stern and Rabi measured the magnetic moments of nuclei using this approach.

So, if the projection $p_(mz)$ is quantized, the average force that acts on the atom from the magnetic field is quantized along with it. The experiments of Stern and Gerlach proved the quantization of the projection of the magnetic quantum number onto the $Z$ axis. It turned out that the magnetic moments of the atoms are directed parallel to the $Z$ axis, they cannot be directed at an angle to this axis, so we had to accept that the orientation of the magnetic moments relative to the magnetic field changes discretely. This phenomenon has been called spatial quantization. The discreteness of not only the states of atoms, but also the orientations of the magnetic moments of an atom in an external field is a fundamentally new property of the movement of atoms.

The experiments were fully explained after the discovery of the electron spin, when it was found that the magnetic moment of the atom is caused not by the orbital moment of the electron, but by the internal magnetic moment of the particle, which is associated with its internal mechanical moment (spin).

Calculation of the motion of the magnetic moment in an inhomogeneous field

Let an atom move in an inhomogeneous magnetic field, its magnetic moment is equal to $(\overrightarrow(p))_m$. The force acting on it is:

In general, an atom is an electrically neutral particle, so other forces do not act on it in a magnetic field. By studying the motion of an atom in an inhomogeneous field, one can measure its magnetic moment. Let us assume that the atom moves along the $X$ axis, the field inhomogeneity is created in the direction of the $Z$ axis (Fig. 1):

Picture 1.

\frac()()\frac()()

Using conditions (2), we transform expression (1) into the form:

The magnetic field is symmetrical with respect to the y=0 plane. It can be assumed that the atom moves in this plane, which means that $B_x=0.$ The equality $B_y=0$ is violated only in small areas near the edges of the magnet (we neglect this violation). From the above it follows that:

In this case, expressions (3) have the form:

The precession of atoms in a magnetic field does not affect $p_(mz)$. We write the equation of motion of an atom in the space between the magnets in the form:

where $m$ is the mass of the atom. If an atom passes the path $a$ between the magnets, then it deviates from the X axis by a distance equal to:

where $v$ is the speed of the atom along the $X$ axis. Leaving the space between the magnets, the atom continues to move at a constant angle with respect to the $X$ axis in a straight line. In formula (7) the quantities $\frac(\partial B_z)(\partial z)$, $a$, $v\ and\ m$ are known, by measuring z one can calculate $p_(mz)$.

Example 1

Exercise: How many components, when conducting an experiment similar to the experiment of Stern and Gerlach, will the beam of atoms split if they are in the state $()^3(D_1)$?

Solution:

A term splits into $N=2J+1$ sublevels if the Lande multiplier is $g\ne 0$, where

To find the number of components into which the beam of atoms will split, we should determine the total internal quantum number $(J)$, the multiplicity $(S)$, the orbital quantum number, compare the Lande multiplier with zero, and if it is nonzero, then calculate the number sublevels.

1) To do this, consider the structure of the symbolic record of the state of the atom ($3D_1$). Our term is deciphered as follows: the symbol $D$ corresponds to the orbital quantum number $l=2$, $J=1$, the multiplicity of $(S)$ is equal to $2S+1=3\to S=1$.

We calculate $g,$ by applying formula (1.1):

The number of components into which the beam of atoms is split is equal to:

Answer:$N=3.$

Example 2

Exercise: Why was a beam of hydrogen atoms, which were in the $1s$ state, used in the experiment of Stern and Gerlach to detect the spin of an electron?

Solution:

In the $s-$ state, the angular momentum of the electron $(L)$ is equal to zero, since $l=0$:

The magnetic moment of an atom, which is associated with the movement of an electron in orbit, is proportional to the mechanical moment:

\[(\overrightarrow(p))_m=-\frac(q_e)(2m)\overrightarrow(L)(2.2)\]

hence it is equal to zero. This means that the magnetic field should not affect the movement of hydrogen atoms in the ground state, that is, split the flow of particles. But when using spectral instruments, it was shown that the lines of the hydrogen spectrum show the presence of a fine structure (doublets) even if there is no magnetic field. In order to explain the presence of a fine structure, the idea of ​​an intrinsic mechanical angular momentum of an electron in space (spin) was put forward.