The electron has its own mechanical angular momentum L s, called spin. Spin is an integral property of an electron, like its charge and mass. The electron spin corresponds to its own magnetic moment P s, proportional to L s and directed in the opposite direction: P s = g s L s, g s is the gyromagnetic ratio of spin moments. Projection of the own magnetic moment onto the direction of vector B: P sB =eh/2m= B , whereh=h/2, B =Bohr magneton. The total magnetic moment of the atom p a = the vector sum of the magnetic moments of the electron entering the atom: P a =p m +p ms. Experience of Stern and Gerlach. By measuring magnetic moments, they discovered that a narrow beam of hydrogen atoms in a non-uniform magnetic field splits into 2 beams. Although in this state (the atoms were in the S state), the angular momentum of the electron is 0, as well as the magnetic moment of the atom is 0, so the magnetic field does not affect the movement of the hydrogen atom, that is, there should be no splitting. However, further research showed that the spectral lines of hydrogen atoms exhibit such a structure even in the absence of a magnetic field. Subsequently, it was found that this structure of spectral lines is explained by the fact that the electron has its own indestructible mechanical moment, called spin.
21. Orbital, spin and total angular and magnetic moment of the electron.
The electron has its own angular momentum M S, which is called spin. Its value is determined according to the general laws of quantum mechanics: M S = h= h[(1/2)*(3/2)]=(1/2) h3, M l = h – orbital moment. The projection can take on quantum values that differ from each other by h. M Sz =m S h, (m s =S), M lz =m l h. To find the value of the intrinsic magnetic moment, multiply M s by the ratio s to M s, s – intrinsic magnetic moment:
s =-eM s /m e c=-(e h/m e c)=- B 3, B – Bohr Magneton.
The sign (-) because M s and s are directed in different directions. The electron moment is composed of 2: orbital M l and spin M s. This addition is carried out according to the same quantum laws by which the orbital moments of different electrons are added: Мj= h, j is the quantum number of the total angular momentum.
22. An atom in an external magnetic field. Zeeman effect .
The Zeeman effect is the splitting of energy levels when atoms are exposed to a magnetic field. Level splitting leads to the splitting of spectral lines into several components. The splitting of spectral lines when emitting atoms are exposed to a magnetic field is also called the Zeeman effect. Zeeman splitting of levels is explained by the fact that an atom having a magnetic moment j acquires additional energy E=- jB B in a magnetic field, jB is the projection of the magnetic moment onto the direction of the field. jB =- B gm j , E= B gm j , ( j =0, 1,…, J). The energy level is split into sublevels, and the magnitude of the splitting depends on the quantum numbers L, S, J of a given level.
Intrinsic mechanical and magnetic moments (spin)
RATIONALE FOR THE EXISTENCE OF SPIN. The Schrödinger equation allows one to calculate the energy spectrum of hydrogen and more complex atoms. However, experimental determination of atomic energy levels has shown that there is no complete agreement between theory and experiment. Precise measurements revealed the fine structure of the levels. All levels, except the main one, are split into a number of very close sublevels. In particular, the first excited level of the hydrogen atom ( n= 2) split into two sublevels with an energy difference of only 4.5 10 -5 eV. For heavy atoms, the magnitude of fine splitting is much greater than for light atoms.
It was possible to explain this discrepancy between theory and experiment using the assumption (Uhlenbeck, Goudsmit, 1925) that the electron has another internal degree of freedom - spin. According to this assumption, the electron and most other elementary particles, along with the orbital angular momentum, also have their own mechanical angular momentum. This intrinsic moment is called spin.
The presence of spin on a microparticle means that in some respects it is like a small spinning top. However, this analogy is purely formal, since quantum laws significantly change the properties of angular momentum. According to quantum theory, a point microparticle can have its own moment. An important and nontrivial quantum property of spin is that only it can set a preferred orientation in a particle.
The presence of an intrinsic mechanical moment in electrically charged particles leads to the appearance of their own (spin) magnetic moment, directed, depending on the sign of the charge, parallel (positive charge) or antiparallel (negative charge) to the spin vector. A neutral particle, for example, a neutron, can also have its own magnetic moment.
The existence of a spin in an electron was indicated by the experiments of Stern and Gerlach (1922) by observing the splitting of a narrow beam of silver atoms under the influence of an inhomogeneous magnetic field (in a homogeneous field the moment only changes orientation; only in an inhomogeneous field does it move translationally either along the field or against it). depending on the direction relative to the field). Unexcited silver atoms are in a spherically symmetric s-state, that is, with an orbital momentum equal to zero. The magnetic moment of the system, associated with the orbital motion of the electron (as in the classical theory), is directly proportional to the mechanical moment. If the latter is zero, then the magnetic moment must also be zero. This means that the external magnetic field should not affect the movement of silver atoms in the ground state. Experience shows that such an influence exists.
In the experiment, a beam of silver, alkali metal and hydrogen atoms was split, but Always only observed two bundles, equally deflected in opposite directions and located symmetrically relative to the beam in the absence of a magnetic field. This can only be explained by the fact that the magnetic moment of the valence electron in the presence of a field can take on two values, identical in magnitude and opposite in sign.
The experimental results lead to the conclusion that that the splitting in a magnetic field of a beam of atoms of the first group of the Periodic Table, which are obviously in the s-state, into two components is explained by two possible states of the spin magnetic moment of the valence electron. The magnitude of the projection of the magnetic moment onto the direction of the magnetic field (it is this that determines the deflection effect), found from the experiments of Stern and Gerlach, turned out to be equal to the so-called Bohr magneton
The fine structure of the energy levels of atoms that have one valence electron is explained by the presence of spin in the electron as follows. In atoms (excluding s-state) due to orbital motion, there are electric currents, the magnetic field of which affects the spin magnetic moment (the so-called spin-orbit interaction). The magnetic moment of an electron can be oriented either along the field or against the field. States with different spin orientations differ slightly in energy, which leads to the splitting of each level into two. Atoms with several electrons in the outer shell will have a more complex fine structure. Thus, in helium, which has two electrons, there are single lines (singlets) in the case of antiparallel electron spins (the total spin is zero - parahelium) and triple lines (triplets) in the case of parallel spins (the total spin is h- orthohelium), which correspond to three possible projections onto the direction of the magnetic field of orbital currents of the total spin of two electrons (+h, 0, -h).
Thus, a number of facts led to the need to attribute a new internal degree of freedom to electrons. For a complete description of the state, along with three coordinates or any other triple of quantities that make up the quantum mechanical set, it is also necessary to specify the value of the spin projection onto the chosen direction (the spin modulus does not need to be specified, because as experience shows, it does not change for any particle under what circumstances).
The spin projection, like the orbital momentum projection, can change by a multiple of h. Since only two electron spin orientations were observed, Uhlenbeck and Goudsmit assumed that the electron spin projection S z for any direction can take two values: S z = ±h/2.
In 1928, Dirac obtained a relativistic quantum equation for the electron, from which the existence and spin of the electron follows h/2 without any special hypotheses.
The proton and neutron have the same spin 1/2 as the electron. The photon's spin is equal to 1. But since the photon's mass is zero, then two, not three, of its projections +1 and -1 are possible. These two projections in Maxwell's electrodynamics correspond to two possible circular polarizations of an electromagnetic wave, clockwise and counterclockwise relative to the direction of propagation.
PROPERTIES OF TOTAL MOMENTUM IMPULSE. Both the orbital momentum M and the spin momentum S are quantities that take only quantum discrete values. Let us now consider the total angular momentum, which is the vector sum of the mentioned moments.
We define the operator of the total angular momentum as the sum of the operators and
The operators and commute, since the operator acts on the coordinates, but the operator does not act on them. It can be shown that
that is, the projections of the total angular momentum do not commute with each other in the same way as the projections of the orbital momentum. The operator commutes with any projection, from which it follows that the operator and the operator of any (but one) projection correspond to physical quantities and are among those that are simultaneously measurable. The operator also commutes with operators and.
We determined the state of the electron in the field of the central force by three quantum numbers: n, l, m. Quantum levels E n were generally determined by two quantum numbers n, l. In this case, the electron spin was not taken into account. If we also take spin into account, then each state turns out to be essentially double, since two spin orientations are possible S z = hm s ; m s = ±1/2. Thus, a fourth is added to the three quantum numbers m s, that is, the wave function taking into account spin should be denoted.
For each term E n,l we have (2 l+ 1) states differing in the orientation of the orbital momentum (number m), each of which in turn decomposes into two states that differ in spin. Thus, there is 2(2 l+ 1) -fold degeneracy.
If we now take into account the weak interaction of the spin with the magnetic field of orbital currents, then the energy of the state will also depend on the orientation of the spin relative to the orbital momentum. The energy change during such an interaction is small compared to the energy difference between levels with different n,l and therefore the new lines that arise are close to each other.
Thus, the difference in the orientations of the spin moment with respect to the internal magnetic field of the atom can explain the origin of the multiplicity of spectral lines. From the above it follows that for atoms with one optical electron, only doublets (double lines) are possible due to two orientations of the electron spin. This conclusion is confirmed by experimental data. Let us now turn to the numbering of atomic levels taking into account the multiplet structure. When taking into account the spin-orbit interaction, neither the orbital momentum nor the spin momentum have a specific value in a state with a specific energy (the operators do not commute with the operator). According to classical mechanics, we would have the precession of the vectors and around the total torque vector, as shown in Fig. 20. The total moment remains constant. A similar situation occurs in quantum mechanics. When taking into account the spin interaction, only the total moment has a certain value in a state with a given energy (the operator commutes with the operator). Therefore, when taking into account the spin-orbit interaction, the state should be classified according to the value of the total moment. The total moment is quantized according to the same rules as the orbital moment. Namely, if we introduce the quantum number j, which sets the moment J, That
And the projection to some direction is 0 z has the meaning J z = hm j, wherein j= l + l s (l s= S), if the spin is parallel to the orbital moment, and j= | l - l s| if they are antiparallel. In a similar way m j = m + m s (m s= ±1/2). Since l,m are integers, and l s , l m- halves, then
j = 1/2, 3/2, 5/2, … ; m j= ±1/2, ±3/2, … , ± j.
Depending on the orientation of the spin, the energy of the term will be different, namely it will be for j = l+ ½ and j = |l- S|. Therefore, in this case, the energy levels should be characterized by the numbers n,l and the number j, which determines the total moment, that is, E = E nlj.
The wave functions will depend on the spin variable S z and will be different for different j: .
Quantum levels at a given l, differing in meaning j, are close to each other (they differ in the spin-orbit interaction energy). Four of numbers n, l, j, m j can take the following values:
n= 1, 2, 3,…; l= 0, 1, 2,…, n- 1; j = l + l s or | l - l s |; l s= ±1/2;
-j ? m j ? j.
The value of the orbital moment l is denoted in spectroscopy by the letters s, p, d, f, etc. The main quantum number is placed in front of the letter. The number is indicated at the bottom right j. Therefore, for example, the level (therm) with n= 3, l = 1, j= 3/2 is designated as 3 R 3/2. Figure 21 shows a diagram of the levels of a hydrogen-like atom taking into account the multiplet structure. Lines 5890? and 5896? form
famous sodium doublet: yellow lines D2 and D1. 2 s-term is far away from 2 R-terms, as it should be in hydrogen-like atoms ( l-degeneracy removed).
Each of the levels considered E nl belongs to (2 j+ 1) states differing in number m j, that is, the orientation of the total moment J in space. Only when an external field is applied can these merging levels separate. In the absence of such a field we have (2 j+ 1)-fold degeneracy. So term 2 s 1/2 has degeneracy 2: two states that differ in spin orientation. Term 2 R 3/2 has fourfold degeneracy according to the orientations of the moment J, m j= ±1/2, ±3/2.
ZEEMAN EFFECT. P. Zeeman, studying the emission spectrum of sodium vapor placed in an external magnetic field, discovered the splitting of spectral lines into several components. Subsequently, on the basis of quantum mechanical concepts, this phenomenon was explained by the splitting of atomic energy levels in a magnetic field.
Electrons in an atom can only be in certain discrete states, when changing which a quantum of light is emitted or absorbed. The energy of the atomic level depends on the total orbital momentum, characterized by the orbital quantum number L, and the total spin of its electrons, characterized by the spin quantum number S. Number L can only accept integers, and a number S- integers and half-integers (in units h). In direction they can take accordingly (2 L+ 1) and (2 S+ 1) positions in space. Therefore, the data level L And S degenerate: it consists of (2 L+ 1)(2S +1) sublevels, the energies of which (if the spin-orbit interaction is not taken into account) coincide.
The spin-orbit interaction leads, however, to the fact that the energy of the levels depends not only on the quantities L And S, but also on the relative position of the orbital momentum and spin vectors. Therefore, the energy turns out to depend on the total torque M = M L + M S, determined by the quantum number J, and the level with the given L And S splits into several sublevels (forming a multiplet) with different J. This splitting is called fine level structure. Thanks to the fine structure, the spectral lines are also split. For example, D-sodium line corresponds to the transition from the level L = 1 , S= ½ per level c L = 0, S= S. The first of them (levels) is a doublet corresponding to possible values J= 3/2 and J= Ѕ ( J =L + S; S= ±1/2), and the second does not have a fine structure. That's why D-line consists of two very close lines with wavelengths of 5896? and 5890?.
Each level of the multiplet still remains degenerate due to the possibility of orientation of the total mechanical moment in space along (2 j+ 1) directions. In a magnetic field this degeneracy is removed. The magnetic moment of an atom interacts with the field, and the energy of such interaction depends on the direction. Therefore, depending on the direction, the atom acquires different additional energy in the magnetic field, and Zeeman splitting of the level into (2 j+ 1) sublevels.
Distinguish the normal (simple) Zeeman effect when each line is split into three components and the anomalous (complex) effect when each line is split into more than three components.
To understand the general principles of the Zeeman effect, let us consider the simplest atom - the hydrogen atom. If a hydrogen atom is placed in an external uniform magnetic field with induction IN, then due to the interaction of the magnetic moment R m with an external field, the atom will acquire an additional value depending on the modules and mutual orientation IN And pm energy
UB= -pmB = -pmBB,
Where pmB- projection of the magnetic moment of the electron onto the direction of the field.
Considering that R mB = - ehm l /(2m)(magnetic quantum number m l= 0, ±1, ±2, …, ±l), we obtain
Bohr magneton.
Total energy of a hydrogen atom in a magnetic field
where the first term is the energy of the Coulomb interaction between an electron and a proton.
From the last formula it follows that in the absence of a magnetic field (B = 0), the energy level is determined only by the first term. When is B? 0, different permissible values of m l must be taken into account. Since for given n And l the number m l can take 2 l+ 1 possible values, then the initial level will split into 2 l+ 1 sublevels.
In Fig. 22a shows possible transitions in the hydrogen atom between states R(l= 1) and s (l= 0). In a magnetic field, the p-state splits into three sublevels (at l = 1 m = 0, ±1), from each of which transitions to the s level can occur, and each transition is characterized by its own frequency: Consequently, a triplet appears in the spectrum (normal effect Zeeman). Note that during transitions the rules for selecting quantum numbers are observed:
In Fig. Figure 22b shows the splitting of energy levels and spectral lines for the transition between states d(l= 2) and p(l= 1). State d in a magnetic field
is split into five sublevels, state p into three. When taking into account the transition rules, only the transitions indicated in the figure are possible. As can be seen, a triplet appears in the spectrum (normal Zeeman effect).
The normal Zeeman effect is observed if the original lines do not have a fine structure (they are singlets). If the initial levels have a fine structure, then a larger number of components appear in the spectrum and an anomalous Zeeman effect is observed.
MECHANICAL AND MAGNETIC MOMENTS OF ELECTRON
Orbital magnetic moment of an electron
Each current, as is known, generates a magnetic field. Therefore, an electron whose orbital mechanical moment differs from zero must also have a magnetic moment.
From classical concepts, the angular momentum has the form
where is the speed and is the radius of curvature of the trajectory.
The magnetic moment of a closed current with area creates a magnetic moment
is the unit normal to the plane, and are the charge and mass of the electron.
Comparing (3.1) and (3.2), we obtain
The magnetic moment is related to the mechanical moment by a multiplier
which is called the magnetomechanical (gyromagnetic) ratio for the electron.
For moment projections we have the same connection
The transition to quantum mechanics is carried out by replacing numerical equations with operator equations
Formulas (3.5) and (3.6) are valid not only for an electron in an atom, but also for any charged particles that have a mechanical moment.
The eigenvalue of the operator is equal to
where is the magnetic quantum number (see Section 2.1)
The constant is called the Bohr magneton
In SI units it is J/T.
In the same way, you can obtain the eigenvalues of the magnetic moment
where is the orbital quantum number.
Recording is often used
Where . The minus sign is sometimes omitted.
Intrinsic mechanical and magnetic moments of an electron (spin)
The electron has a fourth degree of freedom, which is associated with the electron’s own mechanical (and, therefore, magnetic) moment - spin. The presence of spin follows from the relativistic Dirac equation
where is a vector matrix, and are four-row matrices.
Since the quantities are four-row matrices, the wave function must have four components, which can be conveniently written as a column. We will not carry out solutions (3.12), but will postulate the presence of spin (intrinsic moment) of the electron as some empirical requirement, without trying to explain its origin.
Let us briefly dwell on those experimental facts from which the existence of electron spin follows. One such direct evidence is the results of the experience of the German physicists Stern and Gerlach (1922) on spatial quantization. In these experiments, beams of neutral atoms were passed through a region in which a non-uniform magnetic field was created (Fig. 3.1). In such a field, a particle with a magnetic moment acquires energy and a force will act on it
which can split the beam into individual components.
The first experiments examined beams of silver atoms. The beam was passed along the axis, and splitting along the axis was observed. The main component of the force is equal to
If the silver atoms are not excited and are at the lower level, that is, in the () state, then the beam should not split at all, since the orbital magnetic moment of such atoms is zero. For excited atoms (), the beam would have to split into an odd number of components in accordance with the number of possible values of the magnetic quantum number ().
In fact, the beam split into two components was observed. This means that the magnetic moment that causes the splitting has two projections on the direction of the magnetic field, and the corresponding quantum number takes on two values. The results of the experiment prompted the Dutch physicists Uhlenbeek and Goudsmit (1925) to put forward a hypothesis about the electron has its own mechanical and associated magnetic moments.
By analogy with the orbital number, we introduce the quantum number, which characterizes the electron’s own mechanical momentum. Let's determine by the number of splittings. Hence,
The quantum number is called the spin quantum number, and it characterizes the intrinsic or spin angular momentum (or simply “spin”). Magnetic quantum number, which determines the projections of the spin mechanical moment and the spin magnetic moment of the spin, has two meanings. Since , a , then no other values exist, and, therefore,
Term spin comes from the English word spin, which means to spin.
The spin angular momentum of the electron and its projection are quantized according to the usual rules:
As always, when measuring a quantity, one of two possible values is obtained. Before measurement, any superposition of them is possible.
The existence of spin cannot be explained by the rotation of the electron around its own axis. The maximum value of the mechanical torque can be obtained if the mass of the electron is distributed along the equator. Then, to obtain the magnitude of the moment of order, the linear velocity of the equatorial points must be m/s (m is the classical radius of the electron), that is, significantly greater than the speed of light. Thus, a nonrelativistic treatment of spin is impossible.
Let's return to the experiments of Stern and Gerlach. Knowing the magnitude of the splitting (by the magnitude), we can calculate the magnitude of the projection of the spin magnetic moment onto the direction of the magnetic field. It constitutes one Bohr magneton.
We get the connection between and:
Magnitude
is called the spin magnetomechanical ratio and it is twice the orbital magnetomechanical ratio.
The same connection exists between the spin magnetic and mechanical moments:
Let us now find the value:
However, it is customary to say that the spin magnetic moment of an electron is equal to one Bohr magneton. This terminology has developed historically and is due to the fact that when measuring a magnetic moment, we usually measure its projection, and it is precisely equal to 1.