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Exchange Interaction

Spin and Charge Transport

The exchange interaction forces the spins of electrons to align either parallel (ferromagnetic exchange interaction) or antiparallel (antiferromagnetic exchange interaction) . The origin of the exchange interaction is the spin-dependent Coulomb interaction.

There are three contributions to exchange interaction between localized electrons: 1) spin-dependent electron-electron Coulomb repulsion at a short distance (antiferromagnetic); (2) spin-dependent electron-nuclear Coulomb attraction (ferromagnetic); 3) spin-dependent electron-electron Coulomb repulsion at a longer distance (ferromagnetic);

 


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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.
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(1). General origin of the exchange interaction

(2) 3 major types to the exchange interaction

(2a) type 1: the strong antiferromagnetic exchange interaction due to spin- dependent the Coulomb's repulsion between electrons at a short distance;

(2b) type 2: the moderate ferromagnetic exchange interaction (in rare case it is antiferromagnetic) due to spin- dependent the Coulomb's attraction between electrons and nuclears;

(2c) type 3: the weak/moderate ferromagnetic exchange interaction due to spin- dependent the Coulomb's repulsion between electrons at a longer distance due the fermion nature of electrons;

(4) Why there is no Coulomb repulsion between parts of elementary particles?

(5) How Quantum Mechanics distinguish between two particles?

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

(7) Magnetic field Hexchange of the exchange interaction

(8) Bethe–Slater curve

(9) Spin Waves & Exchange interaction

(10) 3 types of the magnetic field: (1) conventional magnetic field; (2) Spin-orbit magnetic field; (3) magnetic field of the exchange interaction.

Questions & Answers

(q00) about spin dependence of the strength of the Coulomb repulsion
(q01) why two electrons, which occupy one state, can be considered as one elementary particle
(q02) about Remaining Interactions of the elementary particle, which consists of two electrons
(q03) about the fine structure vs. absence of Spin-Orbit interaction
(q0) about polarity of exchange integral, about the reason why the exchange interaction changes from antiferromagnetic to ferromagnetic.
(q1) about Hund’s rule
(q2) about Majorana fermion
(q3) Protons are fermions. Similar to electrons, can two protons of opposite spins occupy one quantum state?
(q4) The electron might be a point-like particle with infinitely small size. Therefore, the Coulomb interaction inside the electron has no meaning?
(q5) Proton has the charge radius, which could be measured experimentally. Does electron has the charge radius?

() Relation between precession damping and exchange. Spin relaxation: individual for each spin (electron) or collective for all electrons (spins) simultaneously?

() spin wave & spin precession
() spin wave as a source of the spin damping
() strength of the exchange interaction
() spin dumping for an individual electron
() spin of one individual electrons vs. the spin as a component of the total spin
() magnetic domain & spin damping

.........


General origin of the Exchange interaction

(origin of the exchange interaction in short): Spin-dependent Coulomb interaction

(origin of the exchange interaction in in details): The reason, why the Coulomb interaction is spin- dependent, is following: When two electrons of opposite spins occupy one quantum state, they literally become one single new particle with charge of -2e and no spin. When two electrons

The primary object of our Universe are Quantum Fields (the Electromagnetic field, the Quantum Field of Electrons, the Quantum field of Quarks etc), but not particles (an electron, an photon, a quark). A particle is just a stable quantum state of a specific quantum field.

s s

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.


Types of Exchange interaction

Why there are different types of the exchange interaction?

A In general the exchange interaction describes a process of the conversion of two particles into one particle, when distance between two electrons becomes short. It is a conversion of two electrons with opposite spins into one particle without spin. When two electrons of opposite spins are very far from each other, they are nearly identical a simple wave function. However, when distance between electrons becomes short, they are partially converted into one particle. They are still two particles, but different particles as they have been at a longer distance. The conversion is described by the symmetric and antisymmetric parts of their common wavefunction.

The Coulomb interaction of modified electrons during their conversion from two to one particle with themselves and with surrounding environment is described by different types of the exchange interaction.

 

There are 3 major types to the exchange interaction:

(type 1) the strong antiferromagnetic exchange interaction due to spin- dependent the Coulomb's repulsion between electrons at a short distance;

Two electrons of opposite spins may occupy one quantum state. It literally means that in the case when the spins are opposite there is no Coulomb's repulsion between electrons. In contrary, when spin are parallel, the Coulomb's repulsion between electrons is very strong. It means that at very short distance the Coulomb's repulsion between electrons strongly depends on their spin directions. The Coulomb's interaction between electrons at a short distance forces the electron spins to align antiparallel.

(type 2) the moderate ferromagnetic exchange interaction (in rare case it is antiferromagnetic) due to spin- dependent the Coulomb's attraction between electrons and nuclears;

Due to fermion nature of an electron, the wave function of two electrons is symmetrical, when spin are antiparallel, and the wave function is antisymmetric, when spin are parallel. When two localized electrons are located at two neighbor atoms, antisymmetric wave functions means that probability to find electrons is smaller at point, which is between atoms, than at point in the vicinity of nuclear. In contrast, symmetrical wave functions means that probability to find electrons is nearly the same between atom and in the vicinity of nuclear. Therefore, when spins are parallel, the probability for an electron be at vicinity of nuclear is larger and the absolute value of the negative nuclear-electron interaction energy becomes larger comparing to the case of the antiparallel spins. It forces the spins to align ferromagnetically.

(type 3) the weak/moderate ferromagnetic exchange interaction due to spin- dependent the Coulomb's repulsion between electrons at a longer distance due the fermion nature of electrons;

Due to fermion nature of an electron, the wave function of two electrons is symmetrical, when spin are antiparallel, and the wave function is antisymmetric, when spin are parallel. The Coulomb's repulsion between electrons is smaller when a distance between electrons is longer. In the case of antisymmetric wave function (parallel spins) the average distance between two electrons is a little bit longer than for the case of symmetric wave function (antiparallel spins) . Therefore, the Coulomb's repulsion energy is smaller for parallel spins. It forces the spins to align ferromagnetically.

Additionally there is an asymmetric exchange interaction, which is called the Dzyaloshinskii-Moria Interaction


Note: The force of the exchange interaction is not some "magical" quantum-mechanical force. It is just the Coulomb interaction, but spin-dependent. The exchange interaction acts on the electron spins in order to align the spins such that the Coulomb interaction energy is minimized. There are several reasons why the Coulomb interaction becomes spin-dependent, which are:

(1) Switching-off of the Coulomb's repulsion between two electrons of opposite spins, when they occupy one quantum state.

(2) Dual particle-wave nature of an electron.

(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

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2.Ferromagnetic (long distance), antiferromagnetic (short distance)

due to spin dependence of the Coulomb attraction force between electrons and atomic nuclears

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3. Antiferromagnetic weak/moderate

due to spin-dependence of the Coulomb repulsion force between electrons

 

 

 

 

 

 


Origin of First Type/Contribution for the exchange interaction:

The origin of strong antiferromagnetic exchange interaction

between electrons at a short distance

the following describes the first reason why the coulomb's repulsion between electrons depends on their spin directions

There is a strong antiferromagnetic exchange interaction due to spin- dependent the Coulomb's repulsion between electrons at a short distance.

Its origin is explained as followed: Two electrons of opposite spins may occupy one quantum state. It literally means that in the case when the spins are opposite there is no Coulomb's repulsion between electrons. In contrary, when spin are parallel, the Coulomb's repulsion between electrons is very strong. It means that at very short distance the Coulomb's repulsion between electrons strongly depends on their spin directions. The Coulomb's interaction between electrons at a short distance forces the electron spins to align antiparallel.

 

 

Q. A quantum state can be occupied by two electrons of opposite spins. This literally means that two negatively- charged particles being at the same point do not repel each other. How is it possible? Should the Coulomb repulsion be infinitely large?

Q. The Coulomb repulsion force between particles of the same charge is very strong, when distance between particles is short. For example, in order to reach a distance within a grasp of the strong force, protons short be at temperature more than 10 million degrees C , which is the temperature at center of a star (for example, temperature of the core of Sun is 15.7 million degrees). In contrast, two electrons of of opposite spins at room temperature can be at exactly same point. How the huge Coulomb repulsion force between them is compensated??

When two electrons of opposite spins occupy one quantum state, they become one elementary particle and there is no any the Coulomb repulsion inside of this elementary particle. Therefore, there is no any Coulomb repulsion between two electrons of opposite spins, when they occupies the same quantum state and they are at exactly same point in space.

 

Q. Why two protons of opposite spin direction can not occupy one quantum state and can not be at same point at same time, but electrons can?

A. It is because an electron is an elementary particle, which does not contain any part. In contrast, proton is not elementary particles. It consists of two up-quarks and one down-quark. Only an elementary particles have such property that two charged elementary particles can be at same point in the space.


 

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.

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Simple answer is:

An elementary particle does not have parts, therefore there is nothing inside an elementary which could repel each other.

 

If we could assume that there is something inside electrons, which could repel each other, it should be another force, which should be directed opposite to the Coulomb's force and which should hold the electron together. Otherwise, the electron would be turn apart. For example, in an atomic nuclear repelling Coulomb force between protons is compensated by attractive strong force of the strong interaction. Similar inside a proton, Coulomb force is compensated by the force of the strong interaction between quarks. For electron there is no "other" force, which could hold electron against blowing up. That literally means that an electron does not have any parts, which could repel each other.

Q. Is it possible that an electron interact with itself as whole object without parts?

A. No.

If it were the case, the electron energy would depend on the electron size. For the longer electron length, the smaller interaction would be between the edges of the electron and the smaller would be the total energy of the electron. Since the length of delocalized electrons equals to the mean-free path, in a semiconductor the band gap energy would substantially depend on the crystal quality of the semiconductor. From a variety of experiments, it is very clear that it is not the case.

 

1) A single elementary particle can not interact with itself

2) An elementary particle does not have parts.

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

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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

Figure 2 shows the Coulomb repulsion energy between two electrons. The Coulomb interaction between two identical elementary particles depends on their mutual spin directions.

When electron spins are parallel (blue line), the Coulomb interaction between is nearly the same as the Coulomb interaction between any two charged objects. The repulsion energy sharply increases for shorter distances as 1/x, where x is the distance between electrons.

 

When electron spins are opposite, for longer distances the repulsion energy increases following the 1/x law, but at shorter distances the repulsion energy diverges from the 1/x law and decreases to zero. At zero distance, there is no any repulsion between electrons, because they becomes one particle.

There is an energy barrier with energy of Eb between cases when the electrons are far away and when the electrons of opposite spins are the same point.

It requires an energy larger than Eb in order to scatter an electron from a "full" state.

 

 

 

 

Q. Protons are fermions. Similar to electrons, can two protons of opposite spins occupy one quantum state?

A. No. Proton is not elementary particle. It has "parts" (quarks). Only two elementary particles can occupy one quantum state.

If two protons of opposite spins could be able to occupy one quantum state, the thermonuclear reaction inside core of the Sun would be much faster and the Sun would be burnt out already a long time ago.

 

Q. The electron might be a point-like particle with infinitely small size. Therefore, the Coulomb interaction inside the electron has no meaning?

A. The size of electron is not as small. The mean-free path is the effective length of delocalized conduction electrons. The radius of orbit is the effective radius (size) of the localized electrons. In a high-crystal-quality semiconductor the electron length (the mean-free path) can be as long as a few micrometers.

Besides, the size of electron can be too small, because of the Heisenberg's uncertainty principle.

Q. Proton has the charge radius, which could be measured experimentally. Does electron has the charge radius.

A. The charge radius has a little meaning for an electron. If in the case of the proton the charge is curried by quarks. Since the quarks are bounded by the strong interaction and the strong interaction sharply increase for a longer distance between quarks, there is a rather sharp boundary for charge distribution in the proton and the charge radius has clear meaning. In contrast, the electron is an elementary particle and its charge distribution is fully described by its wave function. It could be either wide (delocalized electrons) or narrow (localized electrons). The parameters like the orbital moment or the mean-free path are better describes the charge distribution of the electron.

 

How Quantum Mechanics distinguish between two particles?

In Quantum mechanics a system of many electrons describes by a single wavefunction. There is Coulomb repulsion between each two electrons. Therefore, there is an interaction between parts of this function. In contrast, the an elementary particle (one electron) is described by a wavefunction as well, but this wavefunction does not describes any internal interaction. How it is possible?

A single elementary particle is described by a single wavefunction with spatial coordinates (x1,y2,z2). The wavefunction describes the interaction of the particle only with external objects, but not any intrinsic interaction.

Another elementary particle is is described by a different wavefunction with different spatial coordinates (x2,y2,z2). The Coulomb interaction between particle one and particle 2 is described by integral:

 

 

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

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Since all known elementary particles are waves, it is very common that during an interaction between them, the number of particles changes. For example, the result of interaction of an electron and a positron is only a photon.

All matter in the universe was created literally from nothing 13.8 billion years ago during the inflation period.

Creation or annihilation of particles waves is very common process. How two particles (waves) can be created from nothing could be understood as follows: When there are two absolutely identical waves, but phase shifted 180 degrees relatively each other, their sum gives zero or nothing. Therefore, two particles (for example, an electron and a positron) can combine that result will be nothing. Similarly, if some force changes the phase shift between particles from 180 degrees to any other, from nothing two particles can be created.

 

Mathematically the process when two electrons of opposite spins combine and create one elementary particle without spin ("full" state) can be understood as follows.

Two electrons with opposite spins, which occupy different states, are described by two spinors

They are described by different sets of coordinates (x1,y1,z1) and (x2,y2,z2). It means they are two elementary particles, which interact with other.

When these two electrons of opposite spins occupy one state, they are described by a scalar wave function, which is product of spinors (1.1) and (1.2)

It is important that the scalar wavefunction (1.3) is described by one set of coordinate. It means (1.3) describes one elementary particle.

More about spin basic properties see here

 

 

 

 

 



Origin of Second Contribution for the exchange interaction:

2. Origin of moderate ferromagnetic exchange interaction

due to spin- dependent the Coulomb's attraction between electrons and nuclears

the following describes the second reason why the coulomb's repulsion between electrons depends on their spin directions

There is a moderate exchange interaction between electrons due spin -dependent the Coulomb's attraction between electrons and nuclears. Usually it is ferromagnetic, but it could be antiferromagnetic at a short distance between nuclears in a crystal lattice.

Its origin is explained as follows: Due to fermion nature of an electron, the wave function of two electrons is symmetrical, when spin are antiparallel, and the wave function is antisymmetric, when spin are parallel. When two localized electrons are located at two neighbor atoms, antisymmetric wave functions means that probability to find electrons is smaller at point, which is between atoms, than at point in the vicinity of nuclear. In contrast, symmetrical wave functions means that probability to find electrons is nearly the same between atom and in the vicinity of nuclear. Therefore, when spins are parallel, the probability for an electron be at vicinity of nuclear is larger and the absolute value of the negative nuclear-electron interaction energy becomes larger comparing to the case of the antiparallel spins. It forces the spins to align ferromagnetically.

Bellow it is explained why the wave function of two electrons is either symmetric or antisymmetric depending on mutual spin directions. This property of wave function is originated from the dual nature of electrons and the fermion nature of an electron.

Dual nature of an electron: Wave or Particle?

How to construct a wavefunction of several electrons?

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

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Possibility 1:

Each electron is an individual elementary particle, which is clear distinguished from any other electrons. In this case the electron should have an individual clear- distinguished wave function. The wavefunction of a system of electrons is a vector, each element of which is the individual wavefunction of each electrons.

Possibility 2:

Probability of each electron to be at a spatial point can be independently defined

Then, the probability that 1st electron will be at point r1, 2d electron will be at point r2 and 3d electron will be at point r3 will be

 

Merits of this representation:

1) The electrons are well-separated from each other. Therefore, there is no infinite Coulomb repulsion between electrons.

2) When the nuclears are sufficiently separated from each other, this representation describes correctly two atoms, where around each nuclear only one electron is circling.

Demerits of this representation:

1) Two electrons are well- distinguished from each other.

It can be justified when the nuclears are far away from each other and one electron is near one nuclear and another is near the second nuclear. However, when two nuclears are moved into one point, there is no feature which can distinguish one electron from the second electron.

3) It contradicts the basic assumption of Quantum Mechanics that the electron is a wave. Two electrons should form a common field.

 

 

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.

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In this case the electrons form one join field of "all electrons". The field is only one and each electron is a part of this field. Therefore, each electron mimics the distribution of the common field and the spacial distributions of each electrons are the same.

It is similar to the electrical or magnetic field. The "all electrons" fields induced by different sources sum up each other. For example, the electrical field between somewhere between two electrons is the sum of the electrical field induced by 1st electron plus the electrical field induced by the second electron. At each point it can be distinguished from each source the electrical field is induced. Only the sum of the electrical field by different source are measurable.

Therefore, if we assume that there is "all electrons" field and each electron as a wave particle is a part of this field, the wavefunction of three electrons will be

 

 

Merits of this representation:

1) It is following the basic assumption of Quantum Mechanics that the electron is a wave. There are a huge number of experiments, which prove that the electron is a wave (RHEED, LHEED, electron transport in a solid).

2) In the case when all nuclears are moved into a one point, one electron field correctly describes all electron features. The different electrons occupy different quantum state of different energy levels or/and different orbital moment.

Demerits of this representation:

1) The identical wave functions for two or more electrons should cause infinite Coulomb repulsion between electrons.

2) When nuclears are separated far away from each other, still each electron is circulating around both nuclears. It is unrealistic.

 

Case 3: the real electron: between a wave and a particle

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=0. Symmetric wavefunction.

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Spin=1. Asymmetric wavefunction

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When two electrons approach each other, they form a common quantum field, which has a defined spin. The spin of a system of two electrons can be either 0 or 1. An object with a spin has specific properties symmetry for of its wavefunction. It can be shown (see Landau & Lifshitz "Quantum Mechanic" Vol.3, Chap. 62), that a wavefunction of a system of two identical particles is symmetric when the spin of the system is zero and it is asymmetrical when spin is 1.

Therefore, the system of two electrons is described by a spinor (which is similar to Eq.3.1, but it also includes the spin-symmetry properties). In the case of two electrons, the spinor of rank two has two wave functions: symmetrical and asymmetrical.

The symmetric wave function, which corresponds to the case of antiparallel spins of the electrons and total spin equals zero, is

The antisymmetric wave function, which corresponds to the case of parallel spins of the electrons and total spin equals to one, is

In the case of Figs. 4 and 5, the probability to find electron at coordinate x1 does not depend on position of the second electron. In contrast, in Fig.6 the probability of 1st electron to be at coordinate x1 depends on position of the 2d electron (coordinate x2).

 

Similar to Fig.5, Fig. 6 is symmetrical for exchange of x1 and x2. Therefore, as wave all electrons are fully identical and they can not be marked or distinguished from each other.

The probability for the 1st electron to be in the vicinity of the 1st nuclear is highest when the probability of the 2d electron is highest to be in the vicinity of the 2d nuclear. Two peaks can be well- distinguished for both figures of Fig.16. This means that as charge particles the electrons are not the same place. One electrons is near one nuclear and the second electron is

Merits of this representation:

1) It is following the basic assumption of Quantum Mechanics that the electron is a wave.

2) The electrons are well-separated from each other. Therefore, there is no infinite Coulomb repulsion between electrons.

3) When the nuclears in a crystal lattice are sufficiently separated from each other, this representation describes correctly two atoms, where around each nuclear only one electron is circling.

 

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.

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The energy of the attractive Coulomb interaction between electrons and nuclears is negative.

When the probability of electrons to be in the vicinity of nuclears is higher, the interaction between electrons and nuclears is stronger and the interaction energy is lower.

The exchange interaction forces the electron spins into the state of a lower energy.

 

1. Case of ferromagnetic exchange interaction. A longer distance between nuclears.

In this case maximus of the asymmetric wave function of the ferromagnetic state is at the positions of nuclears. At the middle between nuclears the asymmetric wave function becomes zero.

The the symmetric wave function of the antiferromagnetic state has two peaks and the valley between them. However, the probability for an electron between nuclears is higher for the antiferromagnetic state and the probability to be in the vicinity of the nuclears is higher for ferromagnetic state.

Therefore, the interaction is stronger and the energy is lower for the ferromagnetic state.

 

2. Case of antiferromagnetic exchange interaction. A shorter distance between nuclears.

For the symmetric wave function of the antiferromagnetic state, the valley disappeared and there is only one peak. Both nuclears are near maximum of this peak. The probability of electrons be in the vicinity of nuclears is high.

The asymmetric wave function of the ferromagnetic state still has two peaks with a valley between them. The nuclears are near the minimum of the valley. The probability of electrons be in the vicinity of nuclears is low.

Therefore, the interaction is stronger and the energy is lower for the antiferromagnetic state.

 

 

Q. The "full" state (a state, which is occupied by two electrons of opposite spins) has spin equal zero. Is the "full" state is a boson?

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

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as a particle

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real electron

between a wave and a particle

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Q. The system of two electrons have even spin: one or zero. Is it a boson?

A. No. It is not boson. The system of two electrons is just a system of two fermions. Nothing more.

The system of two electrons has a charge -2e and a charged object can not be a boson.

Two bosons can occupy one quantum state. Four electrons can not occupy one quantum state.

 



Origin of Third Contribution for the exchange interaction

3.Origin of weak moderate ferromagnetic exchange interaction

due to spin-dependent Coulomb's repulsion between electrons

the following describes the third reason why the coulomb's repulsion between electrons depends on their spin directions

There is a moderate/small ferromagnetic exchange interaction between electrons due spin- dependent the Coulomb's repulsion between electrons at a longer distance

Its origin is explained as follows: Due to fermion nature of an electron, the wave function of two electrons is symmetrical, when spin are antiparallel, and the wave function is antisymmetric, when spin are parallel. The Coulomb's repulsion between electrons is smaller when a distance between electrons is longer. In the case of antisymmetric wave function (parallel spins) the average distance between two electrons is a little bit longer (red line of Fig. 7 ) than for the case of symmetric wave function (antiparallel spins) (blue line of Fig. 7 ) . Therefore, the Coulomb's repulsion energy is smaller parallel spins. It forces the spins to align ferromagnetically.

 

Each electron should not be considered as an individual object. Only the Quantum Field of All Electrons is an object, which can divided into a number of electrons. There are many possibilities how the Quantum Field of Electrons can be divided into particles (electrons).

Similar, the electromagnetic field could also be divided into particles (phonons) and there are many possibilities how to divide the field into the photons.

In contrast to photons, which do not interact with each other, the electrons repels each other. Therefore, the energy of the Quantum Field of Electrons depends how the field is divided into the particles (electrons). The lowest energy correspond to the ground state.

Under different conditions (different spin direction or different external field), the division the Quantum Field of Electrons into particles (electrons) corresponding to a lowest energy might be different.

Therefore, a different total spin of a system of several electrons corresponds to a different division the Quantum Field of Electrons into particles (electrons) and a different distribution of wave functions of these particles (electrons). Since for the different spin the electron distributions become different, the Coulomb's interaction between the electrons become different as well. It makes the Coulomb's interaction to be spin-dependent.


 


 

Bethe–Slater curve

Bethe–Slater curve

It is empirical curve, which represents the measured exchange interaction as distance between localized electrons.

from Wikipedia

wiki page is here

This empirical curve shows that for a shorter distance between the localized electrons of metals are ferromagnetic and at a shorter distance it is antiferromagnetic.

 

 

 

 

 

 

 

 

 

 

 

 



All above description is valid only for localized electrons !



 

localized and conduction (delocalized) electrons

. The electrons can be either localized or delocalized. The localized electrons usually (but not necessarily) have d- or f- spatial symmetry. Their length is short and it is about the interatomic length. In contrast, the length of the delocalized or conductive electrons is longer. In average, it equals to the electron mean-free path. The spatial symmetry of delocalized electrons is s- or p-. The transport mechanisms for localized and delocalized electrons are different. If the localized electrons move only because of scattering from a state to  a state, the transport of delocalized (conduction) electrons is due their movement between scatterings.

Additionally, the delocalized (conduction) electrons can be divided into two groups. The electrons of one group can freely move in bulk of a metal. The electrons of the second group are bound to a defect or an interface. Since the bounded electrons cannot move, they contribute only to the transport due to the scatterings.

Different types of the exchange interaction

(1) Localized -Localized -----> neighbor to neighbor exchange interaction

d-electron to d-electron

The spin direction of each localized electron is fixed in time.

The exchange interaction can be either ferromagnetic or antiferromagnetic.

Each neighbor electron may have individual magnetization, which is fixed and which does not change in time.

For example, in a ferromagnetic material each delocalized neighbor electron has parallel spin direction.

in a antiferromagnetic material each delocalized neighbor electron has parallel spin direction.

(2) Delocalized -delocalized ----->scatterings

conduction electron to conduction electron

The spin direction of each delocalized (conduction) electron is not fixed in time. The spin direction of delocalized (conduction) electrons very frequently changes in time with about picosecond time scale..

 


 


Exchange interaction between localized and conduction electrons. The sp-d Exchange interaction

(description is here)

 


Spin Waves

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

 

Due to the exchange interaction between localized electrons, the waves of magnetization direction can propagate.

The spin waves is a feature of localized electrons. The localized d-electrons can support the spin waves very well.

The conduction electrons may support a spin waves, but only with a longer wavelength. Also, a spin wave of spins of conduction electrons has a substantial loss.

The reasons why the spins of conduction electrons are not so good for the propagation of a spin wave are following:

(1) substantially larger size of a conduction electron;
(2) the conduction electrons experience very-frequent scatterings
(3) a conduction electron moves in space with a speed, which may be faster than the speed of the spin wave.
(4) Conduction electrons significantly overlap each other. Each conduction electron are fully overlapped with millions and more other conduction electrons. It limits a free movement of its spin.

 


Magnetic field Hexchange of the exchange interaction

(representation of the exchange interaction by a magnetic field) Since the exchange interaction directly depends on the electron spin, the exchange interaction of electron with all surrounding electrons can be represented as the effective magnetic field Hexchange, which interacts with the electron spin.

 


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

(note) The exchange interaction is poorly described by the magnetic field Hexchange.

 

(note) the magnetic field Hexchange is just only a representation, but Hexchange is not a real magnetic field

in contrast to exchange magnetic field, which is not a real magnetic field, the effective magnetic field of the spin- orbit interaction is absolutely real and 100%- true magnetic field (see here)

 

(fact) The Hexchange can be evaluated from the Curie temperature of a ferromagnetic material.

The exchange interaction aligns spins of neighbor atoms in a ferromagnetic material. In contrast, thermal fluctuations disaligns the spins. The Curie temperature Tc is the temperature until which there is neighbor - neighbor alignment in ferromagnetic material. At there is a balance between the energy of the exchange interaction and the thermal energy. From this balance condition the exchange energy and the effective magnetic field Hexchange of the exchange interaction are calculated.

 

 

(note) The representation of the exchange interaction by the magnetic field Hexchange can be both helpful and confusing

 

(magnetic field Hexchange is huge!!!, but it has a little physical meaning) The Hexchange is about 1000-2000 T. It is much bigger than the field, which can be achieved by any known magnet. For example, the largest magnetic field achieved by an electromagnet is about 1 T and by a superconductive magnet is about 30 T. Since is just a representation of the exchange interaction, but is not a real magnetic field. The huge value of Hexchange has a little physical meaning.

 

(note ) The exchange interaction only fixes the alignment between spins. When such alignment is considered as one object (a magnetic domain) with a single spin, the Hexchange can be ignored.

(note) Even though Hexchange looks so large, it is very deceiving. Many mechanisms break the alignment due to the exchange interaction. For example, spin wave (see below), magneto- static interaction (see magnetic domains) etc.


3 types of the magnetic field: (1) conventional magnetic field; (2) Spin-orbit magnetic field; (3) magnetic field of the exchange interaction.

 

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.


 

About Hund’s rule

Hund's rule in Wikipedia

Q. , after some time of pondering I remembered about Hund’s rule and stuck in confusion why it would work if the electrons with antiparallel spins don’t repel each other on short distances. At first sight, electrons would always “want" to occupy as little orbitals as possible and make total spin go to zero. So my idea is that it maybe Hund’s rule work because of the energy barrier imposed by electron interaction, because even if the spins are antiparallel they repel until some point, and then the repelling diminishes. I couldn't find approval of that, in fact all the articles about “real” explanation of Hund’s rule available involve some complicated QM which, to my bad, I am not yet acquainted with. Also, in case my intuition is close to the truth, it may be then possible to somehow “force” an additional electron to pair up instead of occupying new orbital. For example, if we send the electron with just the right speed to, say, C+ particle, it may be stuck at the paired up stated and be stable since in this case it would be attached to the nucleus more tightly. I haven’t done any calculations and understand that there may be no possibility for that in case there is no such theoretical range of energies-electron speed, but maybe it would be possible if we could do calculations with different atoms.

 

A. Dependency of orbital occupancy on the electron spin direction is determined by two effects: the exchange interaction and the spin-orbit interaction.

The exchange interactions is forcing electron spins to align anti parallel each other. The the spin-orbit interaction is forcing electron spins to align parallel each other. Reason why:

(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.


 

Q. Is it possible that an elementary particle without charge be a fermion?

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.

 

 

 

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