The world is beautiful

So, let’s completely abstract ourselves and forget any classical definitions. Because with pin is a concept unique to the quantum world. Let's try to figure out what it is.

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Spin and angular momentum

Spin(from English spin– rotate) – the intrinsic angular momentum of an elementary particle.

Now let's remember what angular momentum is in classical mechanics.

Momentum is a physical quantity that characterizes rotational motion, more precisely, the amount of rotational motion.

In classical mechanics, angular momentum is defined as the vector product of a particle’s momentum and its radius vector:

By analogy with classical mechanics spin characterizes the rotation of particles. They are represented in the form of tops rotating around an axis. If a particle has a charge, then, when rotating, it creates a magnetic moment and is a kind of magnet.

However, this rotation cannot be interpreted classically. All particles, in addition to spin, have an external or orbital angular momentum, which characterizes the rotation of the particle relative to some point. For example, when a particle moves along a circular path (an electron around a nucleus).

Spin is its own angular momentum , that is, characterizes the internal rotational state of the particle regardless of the external orbital angular momentum. Wherein spin does not depend on external movements of the particle .

It is impossible to imagine what is rotating inside the particle. However, the fact remains that for charged particles with oppositely directed spins, the trajectories of motion in a magnetic field will be different.

Spin quantum number

To characterize spin in quantum physics, it was introduced spin quantum number.

Spin quantum number is one of the quantum numbers inherent in particles. Often the spin quantum number is simply called spin. However, it should be understood that the spin of a particle (in the sense of its own angular momentum) and the spin quantum number are not the same thing. The spin number is denoted by the letter J and takes a number of discrete values, and the spin value itself is proportional to the reduced Planck constant:

Bosons and fermions

Different particles have different spin numbers. So, the main difference is that some have a whole spin, while others have a half-integer. Particles with integer spin are called bosons, and half-integer ones are called fermions.

Bosons obey Bose-Einstein statistics, and fermions obey Fermi-Dirac statistics. In an ensemble of particles consisting of bosons, any number of them can be in the same state. With fermions, the opposite is true - the presence of two identical fermions in one system of particles is impossible.

Bosons: photon, gluon, Higgs boson. - in a separate article.

Fermions: electron, lepton, quark

Let's try to imagine how particles with different spin numbers differ using examples from the macrocosm. If the spin of an object is zero, then it can be represented as a point. From all sides, no matter how you rotate this object, it will be the same. With a spin of 1, rotating the object 360 degrees returns it to a state identical to its original state.

For example, a pencil sharpened on one side. A spin of 2 can be imagined as a pencil sharpened on both sides - when we rotate such a pencil 180 degrees, we will not notice any changes. But a half-integer spin equal to 1/2 is represented by an object, to return which to its original state you need to make a revolution of 720 degrees. An example would be a point moving along a Mobius strip.

So, spin- a quantum characteristic of elementary particles, which serves to describe their internal rotation, the angular momentum of a particle, independent of its external movements.

We hope that you will master this theory quickly and be able to apply the knowledge in practice if necessary. Well, if a quantum mechanics problem turns out to be too difficult or you can’t do it, don’t forget about the student service, whose specialists are ready to come to the rescue. Considering that Richard Feynman himself said that “no one fully understands quantum physics,” it is quite natural to turn to experienced specialists for help!

A positive number - the so-called spin quantum number , which is usually called simply spin (one of the quantum numbers).

In this regard, they speak of a whole or half-integer spin of a particle.

The existence of spin in a system of identical interacting particles is the cause of a new quantum mechanical phenomenon that has no analogy in classical mechanics: exchange interaction.

The spin vector is the only quantity that characterizes the orientation of a particle in quantum mechanics. From this position it follows that: at zero spin, a particle cannot have any vector or tensor characteristics; vector properties of particles can only be described by axial vectors; particles can have magnetic dipole moments and cannot have electric dipole moments; particles can have an electric quadrupole moment and cannot have a magnetic quadrupole moment; A nonzero quadrupole moment is possible only for particles with a spin not less than unity.

The spin momentum of an electron or other elementary particle, uniquely separated from the orbital momentum, can never be determined through experiments to which the classical concept of particle trajectory is applicable.

The number of components of the wave function that describes an elementary particle in quantum mechanics increases with the spin of the elementary particle. Elementary particles with spin are described by a one-component wave function (scalar), with spin 1 2 (\displaystyle (\frac (1)(2))) are described by a two-component wave function (spinor), with spin 1 (\displaystyle 1) are described by a four-component wave function (vector), with spin 2 (\displaystyle 2) are described by a six-component wave function (tensor).

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  Although the term spin refers only to the quantum properties of particles, the properties of some cyclically acting macroscopic systems can also be described by a certain number that shows how many parts the rotation cycle of a certain element of the system must be divided into in order for it to return to a state indistinguishable from the initial one.

  The simplest example of a spin is a whole spin equal to 1:

  if you take a vector (for example, put a pen on the table) and rotate it by 360 degrees, then this vector will return to its original state (the handle will again lie the same as before the rotation).

  It's also easy to imagine spin equal to 0:

  this is the point - she looks the same from all sides, no matter how you slice it.

  A little more difficult with the whole spin equal to 2 :

  you will need to come up with an object that behaves the same way as in the previous example with spin 1, but when rotated 180 degrees (that is, half the full rotation) - this is also simple - you need to take a bidirectional vector (a real-life example would be an ordinary pencil , only sharpened on both sides or not sharpened at all - the main thing is that it is without inscriptions and monochromatic, Hawking used an ordinary playing card such as a king or queen as an example ) - and then after turning to 180 degrees it will return to a position indistinguishable from its original one.

  But with a half-integer spin equal 1 / 2 you will have to go into 3 dimensions:

  • If you take a Mobius strip and imagine that an ant is crawling along it, then, having made one revolution (traversing 360 degrees), the ant will end up at the same point, but on the other side of the sheet, and to return to the point from where it started, it will have to go all the way 720 degrees .
  • Another example is the four-stroke internal combustion engine. When the crankshaft is rotated 360 degrees, the piston will return to its original position (for example, top dead center), but the camshaft rotates at 2 times slower and will complete a full rotation when the crankshaft is rotated 720 degrees. That is, when the crankshaft is turned 2 revolutions, the internal combustion engine will return to the same state. In this case, the third measurement will be the position of the camshaft.

  Examples like these can illustrate the addition of spins:

  • Two identical pencils sharpened only on one side (the “spin” of each is 1), fastened to each other so that the sharp end of one is next to the blunt end of the other. Such a system will return to an indistinguishable state from the initial state when rotated only 180 degrees, that is, the “spin” of the system becomes equal to two.
  • Multi-cylinder four-stroke internal combustion engine (the “spin” of each cylinder is 1/2). If all cylinders operate in the same way, then the conditions in which the piston is at the beginning of the power stroke in any of the cylinders will be indistinguishable. Consequently, a two-cylinder engine will return to a state indistinguishable from the original one every 360 degrees (total "spin" - 1), a four-cylinder engine - after 180 degrees ("spin" - 2), an eight-cylinder engine - after 90 degrees ("spin" - 4 ).

  Spin properties

  Any particle can have two types of angular momentum: orbital angular momentum and spin.

  Unlike orbital angular momentum, which is generated by the motion of a particle in space, spin is not associated with motion in space. Spin is an internal, exclusively quantum characteristic that cannot be explained within the framework of relativistic mechanics. If we imagine a particle (for example, an electron) as a rotating ball, and spin as the torque associated with this rotation, then it turns out that the transverse velocity of the particle shell must be higher than the speed of light, which is unacceptable from the position of relativism.

  “In particular, it would be completely meaningless to imagine the intrinsic momentum of an elementary particle as a result of its rotation “around its own axis””

  Being one of the manifestations of angular momentum, spin in quantum mechanics is described by the vector spin operator s → ^ , (\displaystyle (\hat (\vec (s))),) the algebra of whose components completely coincides with the algebra of orbital angular momentum operators ℓ → ^ . (\displaystyle (\hat (\vec (\ell ))).) However, unlike orbital angular momentum, the spin operator is not expressed in terms of classical variables, in other words, it is only a quantum quantity. A consequence of this is the fact that spin (and its projections onto any axis) can take not only integer, but also half-integer values ​​(in units of the Dirac constant ħ ).

  Spin experiences quantum fluctuations. As a result of quantum fluctuations, only one spin component can have a strictly defined value, for example. In this case, the components J x , J y (\displaystyle J_(x),J_(y)) fluctuate around the average value. Maximum possible component value J z (\displaystyle J_(z)) equals J (\displaystyle J). At the same time the square J 2 (\displaystyle J^(2)) the total spin vector is equal to J (J + 1) (\displaystyle J(J+1)). Thus J x 2 + J y 2 = J 2 − J z 2 ⩾ J (\displaystyle J_(x)^(2)+J_(y)^(2)=J^(2)-J_(z)^(2 )\geqslant J). At J = 1 2 (\displaystyle J=(\frac (1)(2))) the root mean square values ​​of all components due to fluctuations are equal J x 2 ^ = J y 2 ^ = J z 2 ^ = 1 4 (\displaystyle (\widehat (J_(x)^(2)))=(\widehat (J_(y)^(2)))= (\widehat (J_(z)^(2)))=(\frac (1)(4))).

  The spin vector changes its direction during the Lorentz transformation. The axis of this rotation is perpendicular to the momentum of the particle and the relative velocity of the reference systems.

  Examples

  The spins of some microparticles are shown below.

  spin	common name for particles	examples
  0	scalar particles	π-mesons, K-mesons, Higgs boson, 4 He atoms and nuclei, even-even nuclei, parapositronium
  1/2	spinor particles	electron, quarks, muon, tau lepton, neutrino, proton, neutron, 3 He atoms and nuclei
  1	vector particles	photon, gluon, W- and Z-bosons, vector mesons, orthopositronium
  3/2	spin vector particles	Ω-hyperon, Δ-resonances
  2	tensor particles	graviton, tensor mesons

  As of July 2004, the baryon resonance Δ(2950) with spin 15/2 has the maximum spin among known baryons. The spin of stable nuclei cannot exceed 9 2 ℏ (\displaystyle (\frac (9)(2))\hbar ) .

  Story

  Mathematically, the theory of spin turned out to be very transparent, and later, by analogy with it, the theory of isospin was constructed.

  Spin and magnetic moment

  Despite the fact that spin is not associated with the actual rotation of the particle, it nevertheless generates a certain magnetic moment, which means it leads to additional (compared to classical electrodynamics) interaction with the magnetic field. The ratio of the magnitude of the magnetic moment to the magnitude of the spin is called the gyromagnetic ratio, and, unlike the orbital angular momentum, it is not equal to the magneton ( μ 0 (\displaystyle \mu _(0))):

  μ → ^ = g ⋅ μ 0 s → ^ . (\displaystyle (\hat (\vec (\mu )))=g\cdot \mu _(0)(\hat (\vec (s))).)

  The multiplier introduced here g called g-particle factor; the meaning of this g-factors for various elementary particles are actively studied in elementary particle physics.

  Spin and statistics

  Due to the fact that all elementary particles of the same type are identical, the wave function of a system of several identical particles must be either symmetric (that is, does not change) or antisymmetric (multiplied by −1) with respect to the interchange of any two particles. In the first case, the particles are said to obey Bose-Einstein statistics and are called bosons. In the second case, the particles are described by Fermi-Dirac statistics and are called fermions.

  It turns out that it is the value of the particle's spin that tells us what these symmetry properties will be. The theorem relating spin to statistics, formulated by Wolfgang Pauli in 1940, states that particles with integer spin ( s= 0, 1, 2, …) are bosons, and particles with half-integer spin ( s= 1/2, 3/2, …) - fermions.

  Definition 1

  Electron spin(and other microparticles) is a quantum quantity that has no classical analogue. This is an internal property of the electron, which can be likened to charge or mass. The concept of spin was proposed by American physicists D. Uhlenbeck and S. Goudsmit in order to explain the existence of the fine structure of spectral lines. Scientists have suggested that the electron has its own mechanical angular momentum, which is not associated with the movement of electrons in space, which was called spin.

  If we assume that an electron has a spin (its own mechanical angular momentum ($(\overrightarrow(L))_s$)), then it must have its own magnetic moment ($(\overrightarrow(p))_(ms)$). In accordance with the general conclusions of quantum physics, spin is quantized as:

  where $s$ is the spin quantum number. Drawing an analogy with the mechanical angular momentum, the spin projection ($L_(sz)$) is quantized in such a way that the number of orientations of the vector $(\overrightarrow(L))_s$ is equal to $2s+1.$ In the experiments of Stern and Gerlach, scientists observed two orientation, then $2s+1=2$, therefore, $s=\frac(1)(2)$.

  In this case, the projection of the spin onto the direction of the external magnetic field is determined by the formula:

  where $m_s=\pm \frac(1)(2)$ is the magnetic spin quantum number.

  It turned out that the experimental data led to the need to introduce an additional internal degree of freedom. To fully describe the state of an electron in an atom, the following are needed: principal, orbital, magnetic and spin quantum numbers.

  Dirac later showed that the presence of spin follows from the relativistic wave equation he derived.

  Atoms of the first valence group of the periodic system have a valence electron located in the state with $l=0$. In this case, the angular momentum of the entire atom is equal to the spin of the valence electron. Therefore, when they discovered for such atoms, spatial quantization of the angular momentum of an atom in a magnetic field, this became evidence of the existence of spin in only two orientations in an external field.

  The spin quantum number, different from other quantum numbers, is fractional. The quantitative value of the electron spin can be found in accordance with formula (1):

  For the electron we have:

  It is sometimes said that the spin of an electron is oriented towards or against the direction of the magnetic field strength. This statement is inaccurate. Since this actually means the direction of its component $L_(sz).$

  where $(\mu )_B$ is the Bohr magneton.

  Let us find the ratio of the projections $L_(sz)$ and $p_(ms_z)$, using formulas (4) and (5), we have:

  Expression (6) is called the spin gyromagnetic ratio. It is twice the orbital gyromagnetic ratio. In vector notation, the gyromagnetic ratio is written as:

  Experiments by Einstein and de Haas determined the spin gyromagnetic ratio for ferromagnets. This made it possible to determine the spin nature of the magnetic properties of ferromagnets and obtain the theory of ferromagnetism.

  Example 1

  Exercise: Find the numerical values ​​of: 1) the electron’s own mechanical angular momentum (spin), 2) the projection of the electron’s spin onto the direction of the external magnetic field.

  Solution:

  As a basis for solving the problem, we use the expression:

  where $s=\frac(1)(2)$. Knowing the value $\hbar =1.05\cdot (10)^(-34)J\cdot s$, let’s carry out the calculations:

  As a basis for solving the problem, we use the formula:

  where $m_s=\pm \frac(1)(2)$ is the magnetic spin quantum number. Therefore, the calculations can be made:

  Answer:$L_s=9.09\cdot (10)^(-35)(\rm J)\cdot (\rm s),\ L_(sz)=\pm 5.25\cdot (10)^(-35) J\cdot s.$

  Example 2

  Exercise: What is the spin magnetic moment of the electron ($p_(ms)$) and its projection ($p_(ms_z)$) to the direction of the external field?

  Solution:

  The spin magnetic moment of an electron can be determined from the gyromagnetic relation as:

  The electron's own mechanical angular momentum (spin) can be found as:

  where $s=\frac(1)(2)$.

  Substituting the expression for the electron spin into formula (2.1), we have:

  We use the quantities known for the electron:

  Let's calculate the magnetic moment:

  From the experiments of Stern and Gerlach it was found that $p_(ms_z)$ (projection of the electron’s own magnetic moment) is equal to:

  Let's calculate $p_(ms_z)$ for the electron:

  Answer:$p_(ms)=1.6\cdot (10)^(-23)A\cdot m^2,\ p_(ms_z)=9.27\cdot (10)^(-24)A\cdot m^ 2.$

  In this regard, they speak of a whole or half-integer spin of a particle.

  The existence of spin in a system of identical interacting particles is the cause of a new quantum mechanical phenomenon that has no analogy in classical mechanics, exchange interaction.

  The spin vector is the only quantity that characterizes the orientation of a particle in quantum mechanics. From this position it follows that: at zero spin, a particle cannot have any vector or tensor characteristics; vector properties of particles can only be described by axial vectors; particles can have magnetic dipole moments and cannot have electric dipole moments; particles can have an electric quadrupole moment and cannot have a magnetic quadrupole moment; A nonzero quadrupole moment is possible only for particles with a spin not less than unity.

  The spin momentum of an electron or other elementary particle, uniquely separated from the orbital momentum, can never be determined through experiments to which the classical concept of particle trajectory is applicable.

  The number of components of the wave function that describes an elementary particle in quantum mechanics increases with the spin of the elementary particle. Elementary particles with spin are described by a one-component wave function (scalar), with spin 1 2 (\displaystyle (\frac (1)(2))) are described by a two-component wave function (spinor), with spin 1 (\displaystyle 1) are described by a four-component wave function (vector), with spin 2 (\displaystyle 2) are described by a six-component wave function (tensor).

  What is spin - with examples

  Although the term “spin” refers only to the quantum properties of particles, the properties of some cyclically acting macroscopic systems can also be described by a certain number that shows how many parts the rotation cycle of a certain element of the system must be divided into for it to return to a state indistinguishable from the initial one.

  It's easy to imagine spin equal to 0: this is the point - she looks the same from all sides, no matter how you slice it.

  Example spin equal to 1, most ordinary objects can serve without any symmetry: if such an object is rotated 360 degrees, then this item will return to its original state. For example, you can put a pen on the table, and after turning it 360°, the pen will again lie the same way as before the rotation.

  As an example spin equal to 2 you can take any object with one axis of central symmetry: if you rotate it 180 degrees, it will be indistinguishable from the original position, and in one full rotation it becomes indistinguishable from the original position 2 times. An example from life would be an ordinary pencil, only sharpened on both sides or not sharpened at all - the main thing is that it is without inscriptions and monochromatic - and then after turning 180° it will return to a position indistinguishable from the original one. Hawking used an ordinary playing card such as a king or queen as an example.

  But with half a whole spin equal 1 / 2 a little more complicated: it turns out that the system returns to its original position after 2 full revolutions, that is, after a rotation of 720 degrees. Examples:

  • If you take a Möbius strip and imagine that an ant is crawling along it, then, having made one turn (traversing 360 degrees), the ant will end up at the same point, but on the other side of the sheet, and to return to the point where it started, it will have to go all the way 720 degrees.
  • four-stroke internal combustion engine. When the crankshaft is rotated 360 degrees, the piston will return to its original position (for example, top dead center), but the camshaft rotates 2 times slower and will make a full revolution when the crankshaft is rotated 720 degrees. That is, when the crankshaft is turned 2 revolutions, the internal combustion engine will return to the same state. In this case, the third measurement will be the position of the camshaft.

  Examples like these can illustrate the addition of spins:

  • Two identical pencils sharpened only on one side (the “spin” of each is 1), fastened with their sides so that the sharp end of one is next to the blunt end of the other (↓). Such a system will return to an indistinguishable state from the initial state when rotated only 180 degrees, that is, the “spin” of the system becomes equal to two.
  • Multi-cylinder four-stroke internal combustion engine (“spin” of each cylinder is equal to 1/2). If all cylinders operate in the same way, then the conditions in which the piston is at the beginning of the power stroke in any of the cylinders will be indistinguishable. Consequently, a two-cylinder engine will return to a state indistinguishable from the original one every 360 degrees (total "spin" - 1), a four-cylinder engine - after 180 degrees ("spin" - 2), an eight-cylinder engine - after 90 degrees ("spin" - 4 ).

  Spin properties

  Any particle can have two types of angular momentum: orbital angular momentum and spin.

  Unlike orbital angular momentum, which is generated by the motion of a particle in space, spin is not associated with motion in space. Spin is an internal, exclusively quantum characteristic that cannot be explained within the framework of relativistic mechanics. If we imagine a particle (for example, an electron) as a rotating ball, and spin as the torque associated with this rotation, then it turns out that the transverse velocity of the particle shell must be higher than the speed of light, which is unacceptable from the position of relativism.

  Being one of the manifestations of angular momentum, spin in quantum mechanics is described by the vector spin operator s → ^ , (\displaystyle (\hat (\vec (s))),) the algebra of whose components completely coincides with the algebra of orbital angular momentum operators ℓ → ^ . (\displaystyle (\hat (\vec (\ell ))).) However, unlike orbital angular momentum, the spin operator is not expressed in terms of classical variables, in other words, it is only a quantum quantity. A consequence of this is the fact that spin (and its projections onto any axis) can take not only integer, but also half-integer values ​​(in units of the Dirac constant ħ ).

  Spin experiences quantum fluctuations. As a result of quantum fluctuations, only one spin component can have a strictly defined value, for example. In this case, the components J x , J y (\displaystyle J_(x),J_(y)) fluctuate around the average value. Maximum possible component value J z (\displaystyle J_(z)) equals J (\displaystyle J). At the same time the square J 2 (\displaystyle J^(2)) the total spin vector is equal to J (J + 1) (\displaystyle J(J+1)). Thus J x 2 + J y 2 = J 2 − J z 2 ⩾ J (\displaystyle J_(x)^(2)+J_(y)^(2)=J^(2)-J_(z)^(2 )\geqslant J). At J = 1 2 (\displaystyle J=(\frac (1)(2))) the root mean square values ​​of all components due to fluctuations are equal J x 2 ^ = J y 2 ^ = J z 2 ^ = 1 4 (\displaystyle (\widehat (J_(x)^(2)))=(\widehat (J_(y)^(2)))= (\widehat (J_(z)^(2)))=(\frac (1)(4))).

  The spin vector changes its direction during the Lorentz transformation. The axis of this rotation is perpendicular to the momentum of the particle and the relative velocity of the reference systems.

  Examples

  The spins of some microparticles are shown below.

  spin	common name for particles	examples
  0	scalar particles	π mesons, K mesons, Higgs boson, 4 He atoms and nuclei, even-even nuclei, parapositronium
  1/2	spinor particles	electron, quarks, muon, tau lepton, neutrino, proton, neutron, 3 He atoms and nuclei
  1	vector particles	photon, gluon, W and Z bosons, vector mesons, orthopositronium
  3/2	spin vector particles	Ω-hyperon, Δ-resonances
  2	tensor particles	graviton, tensor mesons

  As of July 2004, the baryon resonance Δ(2950) with a spin of 15/2 has the maximum spin among the known baryons. The spin of stable nuclei cannot exceed 9 2 ℏ (\displaystyle (\frac (9)(2))\hbar ) .

  Story

  The term “spin” itself was introduced into science by S. Goudsmit and D. Uhlenbeck in 1925.

  Mathematically, the theory of spin turned out to be very transparent, and later, by analogy with it, the theory of isospin was constructed.

  Spin and magnetic moment

  Despite the fact that spin is not associated with the actual rotation of the particle, it nevertheless generates a certain magnetic moment, which means it leads to additional (compared to classical electrodynamics) interaction with the magnetic field. The ratio of the magnitude of the magnetic moment to the magnitude of the spin is called the gyromagnetic ratio, and, unlike orbital angular momentum, it is not equal to the magneton ( μ 0 (\displaystyle \mu _(0))):

  μ → ^ = g ⋅ μ 0 s → ^ . (\displaystyle (\hat (\vec (\mu )))=g\cdot \mu _(0)(\hat (\vec (s))).)

  The multiplier introduced here g called g-particle factor; the meaning of this g-factors for various elementary particles are actively studied in particle physics.

  Spin and statistics

  Due to the fact that all elementary particles of the same type are identical, the wave function of a system of several identical particles must be either symmetric (that is, does not change) or antisymmetric (multiplied by −1) with respect to the interchange of any two particles. In the first case, the particles are said to obey Bose–Einstein statistics and are called bosons. In the second case, the particles are described by Fermi-Dirac statistics and are called fermions.

  It turns out that it is the value of the particle's spin that tells us what these symmetry properties will be. The spin-statistics theorem formulated by Wolfgang Pauli in 1940 states that particles with integer spin ( s= 0, 1, 2, …) are bosons, and particles with half-integer spin ( s= 1/2, 3/2, …) - fermions.

  Generalization of spin

  The introduction of spin was a successful application of a new physical idea: the postulation that there is a space of states that are in no way related to the movement of a particle in the ordinary

  1/2, for a photon 1, for p- and K-mesons 0.
  

  Spin is called also own moment of quantity of movement, they say. systems; in this case, the spin of the system is defined as the vector sum of the spins of individual particles: S s = S. Thus, the spin of the nucleus is equal to an integer or half-integer number (usually denoted by I) depending on whether the nucleus includes an even or odd number and . For example, for 1 H I = 1/2, for 10 V I = 3, for 11 V I = 3/2, for 17 O I = 5/2, for 16 O I = 0. For Not in the ground stateIn the first, the total electron spin is S = 0, in the first S = 1. In modern times. theoretical physics, ch. arr. in theory, spin is often called the total angular momentum of a particle, equal to the sum of the orbital and proper. moments.
  

  The concept of spin was introduced in 1925 by J. Uhlenbeck and S. Goudsmit, who used it to interpret experiments. data on beam splitting in magnetic fields. the field was suggested that it could be considered as a top rotating around its axis with a projection onto the direction of the field equal to In the same year, W. Pauli introduced the concept of spin into mathematics. the apparatus is non-relativistic and formulated the principle of prohibition, which states that the two identities. particles with half-integer spin cannot simultaneously be in the same system (see). According to W. Pauli's approach, there are s 2 and s z, which have their own. values ​​ђ 2 s(s + 1) and ђs z respectively. and act nat. called the spin parts of the wave function a and b (spin functions) in the same way as the orbital angular momentum of the quantity of motion I 2 and I z act on spaces. part of the wave function Y (r), where r is the radius vector of the particle. s 2 and s z are subject to the same commutation rules as I 2 and I z.
  

  Spin. The Breit-Pauli N VR includes two terms that linearly depend on the components of the vector potential A, which determines the external mag. field:
  

  
  

  For a uniform field A = 1/2 IN x r, the x sign means the cross product, and
  

  
  

  Where -magneton. Vector quantitycalled mag. the moment of a particle with charge e and mass m (in this case, an electron), while the vector quantityreceived the name spin magnet moment. Odds ratio before s And l called g-factor ohm of the particle. For 1 H (spin I = 1/2) the g-factor is equal to 5.5854, for the 13 C nucleus with the same spin I = 1/2 the g-factor is equal to 1.4042; possible and negative. g-factors, for example: for the 29 Si nucleus, the g-factor is - 1.1094 (spin is 1/2). The experimentally determined value of the g-factor is 2.002319.
  

  Both for one and for a system or other particles, the spin S is oriented relative to the direction of the uniform field. The projection of the spin S z onto the direction of the field takes 2S + 1 value: - S, - S + 1, ... , S. Number of decomposition. spin projections are called systems with spin S.
  

  Magn. field acting on or nucleus in , m.b. not only external, it can be created, etc., or arise during the rotation of a system of charged particles as a whole. Yes, interaction. mag. field created by i with kernel v leads to the appearance in the Hamiltonian of a term of the form:
  

  where n v is the unit charge and mass of the nucleus in the direction of the radius vector of the nucleus Rv, Z v and M v. Members of the form I v ·I i answer, members of the form I v ·s i - . For atomic and mol. systems, along with those indicated, terms proportional to (s i · s j), (I v · I m), etc. arise. These terms determine the splitting of degenerate energies. levels, and also lead to differences. level shifts, which determines the fine structure and hyperfine structure (see,).
  

  Experimental manifestations of spin. The presence of a non-zero spin of the electronic subsystem leads to the fact that in a homogeneous magnetic field. field, a splitting of energy levels is observed, and the magnitude of this splitting is influenced by the chemical. (cm. ). The presence of non-zero spins also leads to splitting of levels, and this splitting depends on the screening of the external. fields by the environment closest to a given core (see). Spin-orbit interaction leads to strong splitting of the levels of electronic states, reaching values ​​of the order of several. tenths of eV and even several. units eV. It manifests itself especially strongly in heavy elements, when it becomes impossible to talk about this or that spin or, and one can only talk about the total angular momentum of the system. Weaker, but nevertheless clearly detectable when studying the spectra, are the spin-rotation and .
  

  For condenser environments, the presence of particle spins is manifested in magnetic. holy of these environments. At a certain temperature, an ordered state of particle spins ( , ) may occur, located, for example, in crystalline nodes. lattice, and therefore associated with magnetic spins. moments, which leads to the appearance of strong paramagnetism (ferromagnetism, antiferromagnetism) in the system. Violation of the order of particle spins manifests itself in the form of spin waves (see). Interaction own mag. moments with elastic vibrations of the medium are called. spin-phonon interaction (cm. ); it determines the spin-lattice and spin-phonon absorption of sound.