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LandauLifshitz Equations
Spin and Charge TransportThe LandauLifshitz Equation describes the precession of the magnetic moment around a magnetic field.
It should be noted that due the relativistic nature of the electromagnetic field when an electron moves in a static electrical field it experiences an effective magnetic field. (See here) in a static magnetic field it experiences an effective electrical field. (See here)
LandauLifshitz Equations:where γ is is the electron gyromagnetic ratio, M is magnetization and H_{eff} is the total magnetic field, which includes the external magnetic field, demagnetization field and effective magnetic field of spinorbit interaction. The Landau–Lifshitz–Gilbert equation is similar, but it describes differently the damping term:
Q. Is the LandauLifshitz Equation the equation of the classic mechanic or the Quantum mechanic??A. Both. The LandauLifshitz Equation describes the Larmor precession , which is the classical effect. Also, The LandauLifshitz Equation describes oscillations between two wavefunction of the spinor, which is a QuantumMechanical effect. note: LandauLifshitz Equation describes two very different processes: (1) the first term describes spin precession. It is a basic quantum mechanical properties of the spin (the timeinverse symmetry) (2)
Spin precessionThe spin precesses around a magnetic field with the Larmor frequency: The spin precesses counterclockwise about the direction of the magnetic field. The spin precession is a quantummechanical effect. It describes an electron state when electron energy is between the spinup and spindown energy. The wave function of electron during the precession is intermixture of the wave function spindown and spinup state. The spindown and spinup states are the electron states when the electron spin is either parallel or anti parallel to the magnetic field.
note: The spin precession does not minimize the energy of an electron in the magnetic field
During spin precession the spin direction changes. Does it violate the spin conservation law?A. No, the spin precession does not violate the spin conservation law. During the spin precession the electron spin does not change. The electron spin can be either parallel to the applied magnetic field or antiparallel or between these direction. The electron wavefunction for the case when the spin is between parallel and antiparallel directions is a combination of the wavefunction of parallel and antiparallel directions. Such combination describes a state of the spin precession. The states, when electron spin is parallel, antiparallel or at angle to magnetic field and precess, are absolutely equal and describe eigen state of an electron. Even more, it is correct to say that there is a spin precession even for case spin is parallel and antiparallel to the magnetic field, but the precession radius is zero.
Damping of the spin precessionThe spin damping describes the process of alignment of electron spin along a magnetic field. During the spin damping process the direction of electron spin is changing. During the spin damping, the spin is not conserved!! Another particle with the spin should interact with the electron in order to conserve the spin during spin damping. For example, it could be a photon (spin=1) or magnon or nuclears with nonzero spin. Note:The spin damping is a long process. It takes many spinprecession periods during the spin damping until the spin is aligned along the magnetic field. The spin damping is the long process because the interaction with another particle with
Note: The mechanisms of spin damping are different for localized delectrons and conduction electrons.The reason: The different size. The localized delectrons have a size about the size of atomic orbital ~ 1 nm. The conduction electrons have a size of ~31000 nm.
In case of conductive electrons, the spin damping is the collective process when the different contributions of many conduction electrons causes the spin damping. Many conduction electrons experience the spin damping together at the same. In case of localized electrons, the spin damping is the individual process when each localized electron experiences the spin damping individually and independently from other localized electrons.
Mechanisms of the spin damping:Localized delectrons (1) emitting of photon; (See here) (2) interaction with magnons (spin waves) (3) Conduction spelectrons (1) emitting of photon (See here); (2) dephasing of precession;
(3)
Q. Why conduction electrons do not support magnons and spin waves?
Spindepended force, which electron experiences in a gradient of magnetic field
According to the Laws of Mechanics, a force acts on an object in the direction, in which the total energy of the is minimized. The electron energy in a magnetic field is S*H/2. In a gradient of magnetic field, a force acts on an electron. The direction of this force depends on the electron spin. When spin is parallel to the magnetic field, the force acts so that electron moves in the direction from a smaller to a larger magnetic field. When spin is antiparallel to the magnetic field, the force acts so that electron moves in the direction from a larger to smaller magnetic field.
Note: This force causes the repelling or attraction between two permanent magnets, which we may experience in everyday life.
The quantummechanical limitation on possible precession angles:
Transversal symmetry and spin precession
Spin damping mechanisms:  emission of a photon interaction with a photon
Spin damping due to emission of a photon
Note: It could be a pumping of the spin precession due to absorption of a photon. The electron magnetic resonance (EMR) and nuclear magnetic resonance (NMR) the nu are based on this effect. Q. Only circular polarized wave has spin. Is in EMR and NMR, circular polarized microwave radiation is used. A. No. The electromagnetic wave, which are used in the EMR and NMR, is not polarized. The spin absorbed the required polarization. The wave of other polarization remains unabsorbed.
size dependence:
Magnetic moment induced by the orbital momentIn an atom in a gas, both the spin and electron orbital moment contribute to the atom magnetic moment. In crystal the orbital moment usually is ignored. It is only partially true. There can be a large orbital moment for both the localized and delocalized electrons in a crystal, but interaction of the orbital moment with magnetic field is different in the crystal than in a gas. 1) Orbital moment in a crystal does not precess around a magnetic field 2) There is a difference in energies for orbital moment directed along and in opposite to magnetic field (Zeeman effect). 3) Because of the orbital moment, a magnetic field breaks the timeinverse symmetry for the orbitals. The distribution of the orbitals with the orbital moment along and opposite to the magnetic field are different. Because of the breaking of the timeinverse symmetry, there could be a significant spinorbit interaction.
Difference between the spin and the orbital moment
The spin and the orbital moment interact differently with a magnetic field If the spin interacts only directly with the magnetic field, the orbital moment additionally interacts with relativistic electrical field (Lorentz electric field) induced by the magnetic field.
This field is different for the electrons, which rotate in clockwise and anticlockwise directions. Therefore, the orbital distribution becomes different for two electrons, which rotates in the opposite directions. The timeinverse symmetry is broken !!! This breaking of the time inverse symmetry may cause a significant enhancement of the magnetic field due to the spin orbit interaction. Note: For simplicity of understanding, the electron orbit is shown as a 2D circle. The 2D circle can represent a 3D spherical orbit deformed in one direction. This effect exists for any realistic orbit. Note: Even though the figure shows the classical view of the electron orbit, the quantum mechanical treatment gives exactly the same result. All electrons, including the innerorbit electrons and the electrons of an inert gas, experiences. This effect contributes substantially to diamagnetic properties of gases and solids.
In a crystal the electron orbital can not be rotated, even though the electron may experience some orbital torque (See below). Only the electron spin can precess around a magnetic field This effect also can break the timeinverse symmetry of the orbit.
Precession of orbital moment in a magnetic field
To see how the symmetry of the electron orbital is related to the orbital moment , click here
Direction and value of orbital moment is directly related to the shape electron orbital. The orbital is asymmetrical in the direction of the orbital moment.
The precession of the orbital moment literally means the precession of electron orbit as well.
The electron
Why there can not be a precession of the orbital moment in a crystal?
The orbital moment of electrons in a solid can not precess around a magnetic field!!!

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