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 Origin of exchange interaction The exchange interaction describes the spin-dependent Coulomb interaction between electrons. The Coulomb repulsion between two electrons is smaller, when their spins are opposite, and is larger, when their spins are parallel. When two electrons of opposite spins approach each other, the breaking of time- inverse symmetry slowly disappears and system of two elementary particles transforms into a system of one particle. As a result, their mutual repulsion decreases and their interaction with surrounding electrons and nuclears is changed The time-inverse symmetry is not broken for an electron state, which occupied by two electrons of opposite spins. It literally means that such state does not have any spin at all. It also means that the state should be considered as one particle without spin with charge -2e instead of two electrons with opposite spins and charge -e and -e. When two electrons of opposite spins approach each other, they are monotonically transformed from the system of elementary particles into a system of only one elementary particle. As a result, the Coulomb repulsion between these two electrons monotonically decreases and additionally the Coulomb interaction between each electron and surrounding electrons and nuclears is changed as well. click on image to enlarge it

#### (3) Features how Quantum Field of Electrons is divided into particles (electrons).

Origins of exchange interaction

1.Antiferromagnetic

very strong at short distance

due to reduction of the Coulomb repulsion force between electrons of opposite spins 2.Ferromagnetic (long distance), antiferromagnetic (short distance)

due to spin dependence of the Coulomb attraction force between electrons and atomic nuclears 3. Antiferromagnetic weak/moderate due to spin-dependence of the Coulomb repulsion force between electrons

#### Why there is no Coulomb repulsion between parts of elementary particles?

 A single elementary particle does not have parts and it can not interact with itself The picture shows what would happen if an electron could interact with itself. Since there is no holding force between parts of an electron, the repelling Coulomb's force would blow up the electron. click on image to enlarge it

#### 3) Even though an elementary particle has a finite size and it can be charged, there is no Coulomb's interaction between sizes of the electron. Otherwise, he repelling Coulomb's force would blow up the electron.

Fig.2. Coulomb repulsion energy between two electrons as function of distance between them For longer distances, the repulsion energy is reverse proportional to distance between electrons E~1/x and it is spin-independent.

For shorter distance, the energy is spin dependent. In the case of the opposite spins (blue line), the repulsion vanishes at shorter distances. In the case of the parallel spins (red line), the repulsion become infinitely large at shorter distances

#### How two elemental particles (two electrons) are transformed into one elementary particle ?

Fig.3. Two electrons of opposite spins, when combined, form an elementary particle without spin.

Each quantum state can be occupied by two electrons of opposite spins.

When a quantum state is occupied by one electron, it is an elementary particle with charge -e and spin=1/2

When a quantum state is occupied by two electrons, it is an elementary particle with charge -2e and spin=0 #### Imaginary case 1: If electron were only a particle, not a wave

Fig.4. Wavefunction of a system of two electrons in the vicinity of two nuclears. Imaginary case of electron as only a particle, not a wave

Each electron is distributed around one nuclear. There is an overlap of electron wavefunction in the middle #### Imaginary case 2: If electron were only a wave, not a particle

Fig.5 Wavefunction of a system of two electrons in the vicinity of two nuclears. Imaginary case of electron as only a wave, not a particle

There is no any difference between electron 1 and electron 2. Their wave functions are exactly the same. #### Spin. Spin symmetry. Symmetric and antisymmetric wave functions.

Fig.6 Probability to find one electron at coordinate x1 and second electron at coordinate x2 of a system of two electrons in the vicinity of two nuclears. Realistic case when electrons have properties of both a wave and a particle

Animation parameter: Distance between two nuclears. Black balls show the positions of nuclears.

# Spin=1. Asymmetric wavefunction  #### Ferromagnetic exchange interaction between localized electrons due to Coulomb attraction to nuclear

Fig.7 (left) Probability to find one electron at coordinate x1 and second electron at coordinate x2=-x1-d, where d is distance between nuclears. (right) Coulomb attraction energy between electrons and nuclears as function of distance between nuclears.

Animation parameter: Distance between two nuclears. Black balls show the positions of nuclears. #### A. No. It is not. The charge of a "full" state is -2e. In the case if two "full" state are placed at same point (if they the same wavefunction), it would infinite repulsion between them.

Comparison between different representation of an electron

Probability of one electron to be at point x1 and second electron at point x2

##### black balls show position of nuclears

as a wave

as a particle

real electron

between a wave and a particle   ## 3.Origin of weak moderate ferromagnetic exchange interaction

#### Bethe–Slater curve

 Bethe–Slater curve It is empirical curve, which represents the measured exchange interaction as distance between localized electrons. from Wikipedia #  Spin waves are waves of magnetization direction of localized electrons. They may propagate a long with only a weak absorption. The red arrows show spins of localized d-electrons. Click on image to enlarge it

## Spin Waves

### Effective magnetic field of exchange interaction Hexchange The effective magnetic field Hexchange of the exchange interaction, which an electron experiences from all neighbor electrons vs. material Curie temperature.

(fact) At Tc the thermal fluctuations breaks exchange alignment of neighbor electrons. Therefore, the electron thermal energy becomes larger than the energy of the exchange interaction.

See more detailed explanation here
click on image to enlarge it

## 3 types of magnetic field

Type 1: Conventional magnetic field Type 2: Magnetic field of Spin-orbit interaction Type 3: effective magnetic field of the exchange interaction   This is the conventional magnetic filed. This magnetic field "fills" all the space and all electrons experience equally this magnetic field.

The magnetic field HSO of the spin- orbit interaction is the component of the electromagnetic field similar to the conventional magnetic field. However, each electron experiences an individual HSO of different magnitude and direction. The HSO of each electron does not influence the spin any neighbor electron. Even electrons, which rotate around the same nuclear, may have different HSO when their orbital symmetry is different.

In the exchange field the electron spin is aligned either parallel or anti parallel to the spin direction of its neighbor electrons. The exchange field is not a component of the electromagnetic field. However, in a solid the spin properties of the electron in a exchange field are exactly the same as in a conventional magnetic field. Therefore, the exchange field can be assigned as an effective magnetic field.
The HSO is originated from electric field of a nuclear and depends on the orbital symmetry of the electron. That is why the HSO is individual for each electron. Details about the spin-orbit interaction are here. Details about the exchange interaction are here
The spin properties of electrons are exactly the same for each type of the magnetic field. In an equilibrium the electron spin is aligned along the total magnetic field, which is a vector sum of all three types of the magnetic field. There is a spin precession before the alignment.
click on image to enlarge it

#### Questions & Answers

about spin dependence of the strength of the Coulomb repulsion () Q. Now I have a special interest on coulomb repulsion energy between two electrons as function between them in your contents. Do you have a paper on this subject? If you have it, please let me know about your publication information.

The dependence of the strength of the Coulomb repulsion between two electrons on their mutual spin directions is the origin of the exchange interaction. More specifically, the dependence of the Coulomb repulsion on the degree of the breaking of the time-inverse symmetry in the system of two electrons. When electrons have opposite spins and distance between them is reduced, the degree of the broken time inverse symmetry for two electrons decreases. As a result, the strength of the coulomb repulsion decreases as well. When two electrons occupy one quantum state, the time-inverse symmetry is not broken and there is no Coulomb repulsion between two electrons. The Coulomb repulsion becomes zero.

The reason for that is that two electrons become one particle with charge -2e and no spin, when two electrons occupy one quantum state. There is nothing that could distinguish two separate particles of one quantum state in this case. There is no single property, which could be associated and make a difference between two different particles (electrons) of one state. Two electrons, which occupy one quantum state, have zero total spin and are described by a scalar wave function (not a spinor). In case if such two electrons were two particles with a zero total spin, it would be possible to distinguish whether their spin directions are up and down or left and right or front and back. However, it could not be distinguished. When two electrons occupy one quantum state, the time inverse symmetry is not broken. Therefore, there is no spin inside. It is neither up and down nor left and right nor front and back. It is one particle with zero spin, for which the time-inverse symmetry is not broken.

It means that from Quantum mechanical point of view , the one quantum state, which is occupied by two electrons, is one elementary particle without parts, but it is not a set of two interacting elementary particles. As any features (electron spin or electron special position), which could distinguish between two electrons, are fully dissolved after two electrons occupies one state, two elementary particles becomes one elementary particle (at least as Quantum mechanic sees or defines an elementary particle).

Since an elementary particle does not have internal parts, there is no repulsion or attraction inside of the elementary particle.There is nothing inside of an elementary particle, which could repel each other. That is the reason why there is no Coulomb repulsion between two electrons which occupy one quantum state. Please note that this case is very different from the case when two electrons of opposite spins occupy two different quantum states.

Even though their total spin may be zero, they are always two distinguished particles. When distance between two electrons of opposite spins is reduced, the strength of the Coulomb repulsion is reduced from its "normal" value to zero. In case of parallel spins, the strength of the Coulomb repulsion behaves normally: it increases for a shorter distance. The dependence of the Coulomb repulsion on the spin direction and distance between electrons is called the exchange interaction. Please note that the same story can be explained based on a symmetrical and antisymmetric wave function.

I have two papers on this subject:

V. Zayets "Spin rotation after a spin-independent scattering. Spin properties of an electron gas in a solid", Journal of Magnetism and Magnetic Materials 356 (2014)52–67

V. Zayets, "Spin transport of electrons and holes in a metal and in a semiconductor", Journal of Magnetism and Magnetic Materials 445, pp 53–65 (2018) .

The papers are about the spin statistics, but not the exchange interaction. However, both effects are based on the same feature of the time- inverse symmetry, so they could be helpful to understand it.

(why two electrons, which occupy one state, can be considered as one elementary particle) Q. In the case of two electrons in an atom, they exist in the same quantum energy state (in the same orbital), and in pairs according to the Pauli principle. They do not become one particle. Why don't they become one particle forever? What is the difference ? only distance? And how long should the distance between the two electrons with opposite spin be within approximately in order for them to become one? is it predictable? And I heard that there is an electron-electron interaction between electrons with opposite spin. So when two electrons become one particle as you say, is the electron-electron correlation small enough to ignore? .

Two electrons, which occupy one quantum state, become one elementary particle. Otherwise, the Coulomb repulsion between them would be infinite. This is the origin of the Pauli principle and the reason why two particles can occupy one quantum state despite the infinite repulsion between them. This fact can be understood from the Quantum Mechanic.

Whether the two particles can be called one particle is a matter of definition. However, the definition of an elementary particle is rather fixed in the Quantum Mechanic.

It is important that the elementary subject of the Quantum Mechanic is not the elementary particle, but the symmetry or, to be more precise, the broken symmetry, which is called the Quantum Fields in the Quantum Mechanic. An elementary particle is just a stable state of several broken-symmetries. The degree, of how much the specific symmetry is broken, is fixed for an elementary particle. It is the basic principle of the Quantum Mechanic, which is called the Noether principle. This important principle was well recognized by all founders of the Quantum Mechanic

From this Quantum mechanical point of view, the two electrons, which occupy one quantum state, become one elementary particle, because the quantum state, which they occupy, has one set of several broken symmetries and it is a stable state of a fix number of broken symmetries, which means by definition it is one particle.

(about Remaining Interactions of the elementary particle, which consists of two electrons)

An elementary particle does not have any internal parts and there is no interaction inside of an elementary particle. As a consequence:

----(consequence 1: ) Two electrons of one state are fully undistinguished from each other.

---- (consequence 2: ) Coulomb repulsion: There is none between two electrons.

------(consequence 3: ) Exchange interaction: There is none.

Two electrons, which occupy one quantum state, do not experience any Exchange interaction between themselves or with a neighbor electron. The inner-shell electron does experience any exchange interaction.

------(consequence 4: ) Spin-orbit interaction: There is none.

Neither of the electrons experiences the Spin-orbit interaction. The inner-shell electron does experience any Spin-orbit interaction.

(the fine structure vs. absence of Spin-Orbit interaction)

An exception is an optical transition. The spin-orbit interaction leads to the fine structure.

At the optical transition, one electron transits into the upper energy level and another electron remains in the ground level. Therefore, two electrons are different,the two electrons are not one elementary particle anymore and each electron experiences the spin-orbit interaction individually, which leads to the fine structure in the absorption spectrum of an atom.

Additional complication is that the one-particle state is slightly influenced by the two-separate- electron state . It is a general feature of the Quantum Mechanic. For example in the case of an atom, even though there are no electrons in the excited state, still the excited state slightly influences the ground state. Similarly, even though two electrons in one state is one elementary particle, there is a small influence of its two particle virtual state. (q1) In the case of two electrons in an atom, they exist in the same quantum energy state (in the same orbital), and in pairs according to the Pauli principle. They do not become one particle.

(a1) Two electrons, which occupy one quantum state, become one particle. It is the origin of the Pauli principle. In this case, all symmetry breaking satisfies the definition of a single particle. (q2) Why don't they become one particle forever?

(a2) It is a fully- normal quantum state, one or two electrons can be excited to another quantum state. For example, in the electron gas of conduction electrons in a metal, the electron scatterings between the electron states, which occupied by two electrons and which occupied by one electron, occur after 10-100 picosecond after an electron scattered in (for electron energy is near the Fermi energy). For these electrons, the lifetime of the two-electron state is about 10-100 ps. (q3) What is the difference ? only distance?

(a4) Everything. The wavefunction of two electrons should be absolutely identical. The position, width, energy, wave vector, all should be identical. (q4) And how long should the distance between the two electrons with opposite spin be within approximately in order for them to become one? is it predictable?

(a4) The distance should be zero in order for two particles to become one elementary particle. Additionally, all other parameters should be absolutely identical (width, energy, wave vector) The Coulomb repulsion between two electrons is reduced when the distance between electrons is reduced. The reduction can be calculated using the classical method of the symmetrical and asymmetrical wavefunctions. It is specific for each quantum state. (q5) And I heard that there is an electron-electron interaction between electrons with opposite spin. So when two electrons become one particle as you say, is the electron-electron correlation small enough to ignore?

(a5) The electron-electron correlation is for electrons of different quantum states. There is no interaction or correlation inside of an elementary particle, because the elementary particle does not have parts. There might be electron-electron correlation between electrons of different quantum states.

about polarity of exchange integral, about the reason why theexchange interaction changes from antiferromagnetic to ferromagnetic. (from SAROJ KUMAR MISHRA) Q. In the exchange interaction energy equation there is an exchange integral term Jex. so the question is that, if Jex is positive then why all the spins will be aligned parallel, and if Jex is negative then why all the spins will be aligned antiparallel

The exchange integral is only a mathematical trick to somehow describe the exchange interaction. The reason, for which it is introduced, is so that minimizing the total energy gives either ferromagnetic (parallel) or antiferromagnetic (antiparallel) spin alignment. Only for this reason, the exchange integral is either positive or negative. In fact, the physics of the exchange is more rich, complex and interesting. In order to understand it and, therefore, the polarity of the exchange interaction, let me explain it for a set of electrons aligned in a 1D line. The Coulomb interaction between two neighbor electrons depends on the mutual spin directions. The repulsion between electrons is largest when their spins are parallel and is smallest when their spins are anti parallel. Since the energy of the repelling is positive, the minimum of the energy corresponds to antiparallel alignment of spins, negative exchange integral and the antiferromagnetic exchange interaction. I would like to emphasize that the exchange interaction between two electrons is always antiferromagnetic and therefore the exchange integral is always negative. If in the previous example there were only electrons, in the next example, additionally there are positively-charged nuclei at the position of each electron. There is no exchange interaction between nuclei and the electrons. There are inner-shell electrons and the electron under consideration occupies the external shell, which is relatively far from the nucleus. The energy of the attractive Coulomb interaction between electron and nucleus is negative and its absolute value is larger when the distance between electron and nucleus is shorter. The repelling Coulomb force from the left and the right neighbor electrons forces to shrink the electron orbital pushing the electron closer to the nucleus and, therefore, makes smaller the energy of the Coulomb interaction between the electron and the nucleus .Therefore, the repulsion between each two neighbor electrons has two opposite contributions to the total energy. The energy increases due to an increase of the repulsion Coulomb energy between two electrons and the energy decreases due to a decrease of the attraction Coulomb energy between the electron and the nucleus. When the latter prevails, the increase of the repulsion between neighbor electrons causes a decrease of the total energy. Since the repulsion between neighboring electrons is larger when their spins are parallel, the total energy is smaller for the parallel spins, the exchange is ferromagnetic and the exchange integral (the addition to the total energy) is positive. Note: the identical result can be obtained considering the symmetrical and anti symmetrical wavefunctions.

(to conclude):

The exchange interaction between two electrons is always negative. It is because of the fundamental property of the broken time inverse symmetry. The exchange interaction occurs because the degree of the broken time- inverse symmetry for a system of two electrons decreases when the distance between them decreases. As a result, the strength of the Coulomb interaction becomes spin-dependent.

When additionally the electron interacts with a nucleus or other electrons, the exchange may become ferromagnetic. It is because the spin- dependent reduction of the repulsion between two electrons may reduce the attraction between the nucleus and the electrons or may increase repulsion between the electron and other electrons. Each process leads to the lower total energy for parallel spin alignment and the ferromagnetic exchange interaction.

The spins are aligned into the directions when the total energy of their interaction is smallest. Otherwise, there is a precession of the spin and, therefore, there is a precession of the magnetic moment, which causes an emission of a circularly- polarized photons and the reduction of the total energy until the energy minimum. See details here.

#### (exchange interaction)      The exchange interaction is originated from the feature of our Nature that the state of a higher symmetry has a lower energy. The Higgs field may be only one exception. The reason why the energy is lower is that the interaction between two particle disappears when two particle join each other to create a single particle of a higher symmetry.

An example is two electrons of opposite spins. For each electron, the time-inverse symmetry is broken and the wavefunction is a spinor. When these two electrons of opposite spins occupy one quantum state, they become one single new particle with charge of -2e and no spin. The time-inverse symmetry for this new particle is not broken and its wavefunction is a scalar.

This new particle is not a simple sum of two electrons of opposite spins. For simple sum of two electrons of opposite spins, the time- inverse symmetry is not broken!. For example, two opposite spins can be directed along the x- axis or the y-axis or the z-axis. Therefore, two electrons of opposite spins occupying one state is really a new particle. It is absolutely not a the sum of two individual particles, which are sitting in one place.

An elementary particle does not have parts. Therefore, there can not be any interaction between nonexistent parts of elementary particle. That is why the Coulomb interaction between charges of two electrons of opposite spins is switched off when two electrons approach each other.

The symmetrical and asymmetrical wavefunction, which are used to describe the exchange interaction, just describe the process how two individual particles are monotonically transformed into one particle as two electrons approach each other.

#### (spin-orbit interaction)      The spin-orbit interaction is just the magnetic field of a relativistic origin, which is forcing all electron spins to align along its own direction. The spin-orbit magnetic field is induced by an electrical field of the nuclear due the finite speed of the electron orbital movement.

When an electron moves in electrical field, it experience a magnetic field. It is a relativistic feature of the electromagnetic field. The similar relativistic effect is Lorentz force: .When an electron moves in a magnetic field, it experience an electrical field, which forces the electron to turn from a straight movement.

#### Majorana fermion is a fermion particle, which does not have antiparticle. The Majorana fermions should be uncharged.

A. No. Only charged particles can be fermions. The Majorana fermion can not exist.

Only one fermion (or two fermion with opposite spins) can occupy one quantum state. In order for an elementary particle to be a fermion, there should be a force, which prevents two or more fermions to occupy one quantum state. In case of a charge particle, the Coulomb repulsion prevents two or more identical particles be at the same place at the same time (occupy the same quantum state).

There should be some force, which repels fermions from each other to prevent occupation of a quantum state by two fermions. The force should be one from known forces of the nature. There is no any "special" "quantum-mechanical" force to do this. Relation between precession damping and exchange. Spin relaxation: individual or collective? ( from Sky) Q. I have some confusion about the pressesion damping for the localized electrons. There are two statements in this subject: (1)"All localized electrons are aligned to each other due to the strong exchange interaction. The spins of these electrons are spatially localized to the size of about one atom. As a result, the spins of neighbor electrons swing with respect to each other (similarly as balls connected by springs)." and (2) "In case of localized electrons, the spin damping is the individual process when each electron experiences the spin damping individually and independently from other localized electrons." In my opinion, the statement (1) means that the pressesion damping of localized electrons is strongly connected with each other, which seems contradict with statement (2).

The spin precession and the precession damping are a collective effect, when the directions of all spins are parallel all the time. It is because the exchange interaction between spins is very strong. Exceptions are the spin waves and domain walls. The exchange interaction between localized neighbor electrons is very strong, but is not infinitely strong. As a result, a slight misalignment between neighboring spins are possible (spin waves). Also, when some strength is accumulated over many spins (over millions or billions of spins) the parallel alignment between neighboring spins can be broken (a domain wall).

The spin damping is a collective process of the total spin. There is no individual spin damping. The spin-down to spin-up quantum transition of one electron means only change of one component of the total spin and is not related to individual spin of one localized electron.

(spin wave & spin precession)

Since the exchange interaction is not infinitely strong, a slight misalignment between two localized neighbor electrons is possible. Due to such a tiny misalignment, a spin wave exists in a ferromagnetic material. A spin wave is a mixture of an electromagnetic wave and spin precession. The magnetic component of an electromagnetic wave is slightly different at a position of each localized electron. As a result, the spin precession is slightly different between neighboring localized electrons. As you said, the spins of neighboring electrons slightly swing with respect to each other. Even though the spin misalignment between neighboring electrons is very small and the spins of neighboring electrons are still nearly parallel, the misalignment is accumulated with a distance and can be substantial for the electrons separated by a long distance.

(spin wave as a source of the spin damping)

The spin wave is a particle with a non-zero spin. It interacts with the total spin of the nanomagnet causing an electron transition from the higher- energy spin-down energy level to the lower- energy spin-up energy level. This process is called the spin damping and this quantum transition is fully equivalent to the classical precession damping. It is important that the spin wave interacts with the total spin of the whole nanomagnet, but not with individual spin of localized electrons. The interaction is the most efficient when the size of the nanomagnet or size of a magnetic domain matches the wavelength of the spin wave.

(strength of the exchange interaction)

The strength of the effective exchange magnetic field is possible to estimate from Curie temperature (see above). The magnetic field of the exchange interaction is rather high. It is about 1900 Tesla for Co and 900 Tesla for Ni. For example, a large superconducting magnet produces a magnetic field of about 20-40 Tesla. Because of the high strength of the exchange interaction, it is nearly impossible that the spin of one individual localized electron is reversed with respect to the spin direction of all neighboring electrons. Only many electrons can reverse their spins simultaneously and coherently ( a magnetic domain)

(spin dumping for an individual electron)

All individual localized electrons are so strongly glued to each other by the exchange interaction, they behave as one quantum object. Spin of a localized electron is aligned strongly to be parallel to the spins of its neighboring localized electrons. The total spin behaves as one quantuum object. It precesses as one object or tilts its direction as one object and interacts with spin waves (magnons) and circularly- polarized photons as one object.

(spin of one individual electrons vs. the spin as a component of the total spin)

Even when there is a quantum transition of an electron from the spin-down to spin-up energy level (spin damping), it does not mean that one localized electron becomes spin-up in the surroundings of neighboring spin- down electrons. The spins of all neighboring electrons remain parallel (nearly) all the time. The meaning of the transition of one electron from the spin-down to spin-up energy level means that one component of the total spin is changed and, as a result, the precession angle of the total spin becomes larger. All the time the spin of all localized electrons are glued to each other. All spins precess coherently and are always parallel to each other.

(magnetic domain & spin damping)

The strong exchange interaction can be broken at a boundary between magnetic domains. Some effects can accumulate for a larger number of localized electrons. When the number of localized electrons reaches some critical number, a domain wall is formed. The behavior of two neighbor domains may be rather independent. E.g., the magnetic dipole interaction makes magnetization of neighbor domains to be antiparallel. Similarly, the spin precession of the neighbor domains can be at slightly different frequency and the precession angle.