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Basic properties of electron spin

Magnetism of electron gas

This page describes the basic properties of the electron spin. It describes what is the spin direction, spin projection and spin precession and how they should be treated in a description of electron gas.

 

 

Results in short :

Precession of electron spins.

 

 

To note
An electron is a quantum mechanical object and its spin should be treated quantum -mechanically The conventional representation of spin as a 3D spatial vector may tempt to apply the operational rules of a geometrical vector to spin, which is wrong for the most cases.
When working with spin, it is important to understand that spin is not a conventional 3D spatial vector. Spin is described by a spinor and it possesses its own specific properties.

Spinors

 
To read facts about spinors, click here to expand

In contrast to a particle with zero spin, which is described by a single wavefunction, the properties of an electron, which has ½ spin, are described by a linear combination of two wave functions (for example ). The coefficients in this combination depend on the direction of spin. For a given direction of spin, the wavefunction of an electron will be

The spinor S describes the coefficients in (1) as


Examples (conventional basis for )_:

When the spin is parallel and anti parallel to the x-axis, spinors will be

When spin is parallel and anti parallel to the y-axis, spinors will be

When spin is parallel and anti parallel to the z-axis, spinors will be

The spinors are the eigenvectors of the Pauli matrixes

For the spin-up spinors the eigenvalue is +1 and for the spin-down spinors the eigenvalue is -1.


 

Spin-up and Spin-down electrons

Spinors for spin-up and spin-down electrons are orthogonal, therefore their dot product is zero. For example, for spinors

the dot product is

Eq. (6b) means that a spinor, which describes an electron with spin directed in some direction, can be decomposed as a sum of spinors corresponding to opposite spins.

Also, the scalar product of spinors is important (See. Ref. 1), because it is the only bilinear form obtained from two spinors which is invariant to the rotation of the coordinate system.

The scalar product of spinors for spin-up and spin-down electrons is equal to 1:

[1] Landau and Lifshitz "Quantum Mechanics. Non-relativistic Theory" page 208 (file page 222). Djvu-file can be downloaded here or the book can be read online here.

Spinors in Wikipedia

 

 

 


All properties of an electron or a group of electrons are described by a spinor.

What is the spinor?

A. The spinor gives the electron wave function for a given direction of the electron spin. For a single electron, the spinor contains two wave functions. One wavefunction describes the electron wavefunction for one direction of its spin (for example, when the spin direction is along the z axis) and the second wavefunction describes the electron wavefunction for opposite direction of the spin (for example, when the spin direction is opposite to the z axis)

The electron wave function is a scalar or a spinor?

A. The wavefunction of an electron is a scalar, which describes the probability to find the electron at each point in space. The spinor describes, which scalar wave function the electron has for each direction of its spin. Since the spin direction may change in time, it is convenient to describe an electron by a spinor.

The

Can the spin of one electron vary in space and in time?

A. The spin is intristic feature of an electron. One electron has one spin value and one spin direction over its all spacial size. In a system of several electrons there may be a spacial variation of spin value and spin direction. For example, in an atom the nuclear and electrons have different spin directions and different spin value.

The spin direction of an electron may change in time. The rotation of the spin direction in a magnetic field is called the spin precession.

 



Direction of Spin

Note: an electron always has a defined direction of its spin. This basic property of an elementary particle came from the fact that the time inversion symmetry is broken for the electron (See here).

An electron always has a defined spin direction under any circumstances. Along with its wavefunction, it is a basic signiture of any electron.

The spin direction can be found from the spinor of the electron. Click below to see how.

Result:

When the spin direction is in the xy-plane and φ is the angle of spin with respect to the x-axis, the spinor is calculates as

When the spin direction is in the xz-plane and θ is the angle of spin with respect to the x-axis, the spinor is calculates as

For the orbitary spin direction, the spinor is calculated as

Direction of spin. How to find direction of spin from its spinor? How to find the spinor from the direction of the spin? ( click here to expand)

It is possible to find the spin direction of a "spin" state if its spinor is known.

In order to show how to do this, I will consider two cases. The first case is the case when the spin direction is only in the xy-plane. The second case is the case when the spin direction is only in the xz-plane.


case 1: Spin is in the xy-plane

The Puali matrix for an electron, whose spin is in the xy-plane, is

where φ is the angle of spin with respect to the x-axis.

The spinor is an eigen vector of the Pauli matrix. Therefore, the spinor of an electron, whose spin is in the xy-plane, is calculated as

case 2: Spin is in the xz-plane

The Puali matrix for an electron, whose spin is in the xz-plane, is

where θ is the angle of spin with respect to the x-axis.

The spinor of an electron, whose spin is in the xz-plane, is calculated as


 

In the case of an arbitrary direction of spin, the spinor can be found by combining (d2) and (d4) and it can be expressed as

 

where teta can be found as follows

If a spinor of a "spin" state is known, it can be simplified to the form (d5) . Therefore, the angles phi and theta can be calculated

 

The spin direction describes the current direction into which the time-inverse symmetry is broken for the electron.

 


 

Spin projection

It is possible to find a projection of spin onto any direction. It is also possible to decompose spin directed in one direction as a sum of spins directed parallel and anti parallel to some other direction (for example, a sum of spin-up and spin-down).

Spin projection (p) (click here to expand)

It should be noted that spinors of two spins, which are directed parallel and antiparallel to some axis, are orthogonal. Therefore, their dot product equals zero

Using the property (p1), a spin can be represented as a sum of two spins of opposite directions as

 

Example 1 Spin directed along the x-axis is decomposed as a sum of opposite spins directed along the z-axis.

It should be noted that this is impossible for a conventional spacial vector. From (2) and (5), we have

 

Example 2 spin directed in the xy-plane is decomposed as a sum of opposite spins directed along the x-axis.

Rewriting (d2) and (3)

where phi is the angle of spin with respect to the x-axis. The spinor of spin directed in the xy-plane is

From Eq.(p1)

or

the probability to be in spin up and spin down states will be

in the case of small phi Eq. (p7) is simplified to

 

 


 

Expected value

If we have some physical quantity, which depends on spin, its expectation value can be calculated as follows

Expected value (e) (click here to expand)

If we have some physical quantity, which depends on spin.

q

q

 

 


 

 

Spin in a magnetic field

 

Below it is shown how to describe the Zeeman effect and the Larmor precession of spin in a magnetic field using spinors.

How it is done?
1. We calculate the spinor for the case electron spin is either parallel or anti paralel to magnetic field.
2. We rotate the spinor out of magnetic field direction and we calculate how it behaves.

 

Step 1. The calculation of the spinor for the case when the spin direction is either parallel or antiparallel to the magnetic field.

In a magnetic field the electrons, whose spin directed along and opposite to the magnetic field, have different energy (Zeeman effect). The difference in energy is given as

where g is the g-factor, is the Bohr magneton.

For example, in the case when a magnetic field H is applied along the z-axis, the electrons with spin directed parallel and anti parallel to the z-axis have energy

where E0 is the electron energy in the absence of a magnetic field.

From Eq.(21), the wavefunction of spin-down and spin-up electrons can be expressed as

or

Eqs. (23) descibe electron wavenction for two directions of the spin by taking the projection on the z axis of folowing spinor:

 

Step 2. The calculation of the spinor for the case when the spin direction is either parallel or antiparallel to the magnetic field.

The Eqs. (23) describe the case when both the spin direction and the direction of the magnetic field are along the z-axis. When the spin direction turns away from the z-axis through angle theta, the electron wavefunction should still be described by Eq.(23) but using the spinor corresponding to the turned spin. Substituting Eq. (d.4) for the spinor turned by angle theta away from the z-axis into Eq(23) gives

Eq. (24) can be simplified to

where S(t) is the time-dependent spinor

which describes an electron, whose spin direction has time-independent angle theta with respect to the z-axis and the time –dependent angle phi in the xy-plane. Comparing Eqs. (d5) and (26), the angle of spin in the xy-plane with respect to the x-axis is

Therefore, Eq (25) described a precession of spin around the magnetic field with the Larmor frequency

Fig 3. Animated figure. Precession of spins of a conduction sp-electron (blue arrows) due to the exchange interaction with the local d-electrons. The precession axis is along the spin direction of the local d-electrons (red arrow). The angle between spin directions of sp- and d-electrons is 70 deg.

Eq.(24) and Eq (25) are equivalent. In Eq (24) the spinor is time-independent and it describes two electron states of different energy. In contrast, in Eq. (25) the spinor is time-dependent and it describes electron states of the same energy, but with spin precession.

Therefore, there are two fully equivalent representations of spin in a magnetic field. In the first representation the spin state is described as a superposition of two spin states with different energy and opposite time-independent spin-directions. In the second representation the spin state is described as a state with a constant energy, but with time-dependent spinor. It corresponds to the time-dependent spin direction or the spin precession.

 

The spin precession is not specific for the interaction of spin with magnetic field, but is a general property of spin.


 

Spin direction vs. spin projection?

Along any direction (for example, the direction of the magnetic field, spin can have only two projections ½ and -½). In contrast, spin can have any direction in the 3D space. This looks like as two controversial statements and sometimes it confuses the understanding of spin properties.

In fact, this is just two identical representations of a "spin" state in a magnetic field. In the first representation a "spin" state is represented as a sum of two projections with different energies. The energies are different for the projection with spin parallel or anti parallel to the magnetic field. There is a finite probability for a “spin” state to be in a state with either higher energy or lower energy. In the second representation, the “spin” state has a defined energy, but there is spin precession around the magnetic field. It should be noted that the precession frequency (the Larmor frequency) does not depend on the angle between the spin direction and the magnetic field and there is a spin precession even in the case when the spin parallel to the magnetic field.

 


 

 

Fig 1. Three possibilities of occupation of an electron state. Only a state, which is filled by one electron, has a defined spin direction.

Quantum states and Spin.

1. The conduction electrons can be in "spin", " full" and "empty" states. Only "spin" states have spin directions. The spin of "full" and "empty" states is zero and these states do not have a spin direction.

2. There are two electrons of opposite directions in a "full" state, but the information about the spin direction of electrons before been scattered into a "full" state is not memorized in the "full" state. Therefore, the two electrons of opposite spin directions can have spin directed in any direction with equal probability.

3. "spin" states have a spin direction. "full" and "empty" states do not have a spin direction

4. There are two identical representations of a "spin" state in a magnetic field. In the first representation a "spin" states is represented as a sum of two projections with different energies. The energies are different for the projections with spin parallel are antiparallel to the magnetic field. In the second representation, there is a spin precession around the magnetic field. The energy of the "spin" state is constant.

 

States filled by one or two electrons or not filled

An electron quantum state can either be filled by no electrons or one electron or two electrons of opposite spins. We define these three states as “full”, “empty” and “spin” states (See Fig.1).

Only the "spin" state can have a spin direction. The spin of "full" and "empty" states is zero and these states do not have a spin direction. This can be easily understood by comparing an electron with a photon. The spin direction can be thought of as analogous to the polarization direction of a photon. The analog of a "spin" state is linearly polarized light. It always has a defined direction of polarization (it could be s- or p- polarized or somewhere in between). The analog of a "full" state is unpolarized light (light of equal p- and s-polarizations). The polarization direction in this case can not be defined (The intensity of light is equal for any polarization direction).

1. "spin" state

"spin" state is an electron state in which one of the two possible places is occupied by an electron. The wavefunction for a "spin" state is

The time inverse symmetry is broken for a "spin" state. When the time inverse symmetry is broken, there is always a direction along which it is broken. Of course, it is the spin direction.

The "spin" state always has a spin direction along which the time inverse symmetry is broken.

A change of the spin direction can be described by a corresponding change of the coefficients a1 and a2 in the spinor (7).

 

2. "full" state

"full" state is a electron state in which both of two possible places for an electron are occupied. The "full" state has spin "0" and it is described by a scalar wavefunction, which remains unchanged when the coordinate system is rotated.

Such a wavefunction can be expressed as the scalar product of spinors for spin-up and spin-down electrons.

It is important to emphasize that the wave functions (8) is the same for electrons with any spin direction. This means that the wavefunction (8) is the same for any combination of the spinors of opposite spins.

The time inverse symmetry is not broken for a "full" state. A wavefunction of a "full" state is invariant under axis rotation.

 

 

Fig 2."full" state. The "full" state does not have spin direction. Therefore, all these representation of "full " state are identical and indistinguishable. A wavefunction of a "full" state is invariant under axis rotation.

 

 

Important note

The " full" state can not have any direction!!! For example, three "full" states shown in Fig. 2 are indistinguishable.

Note: A wavefunction of a particle with zero spin is invariant under axis rotation. Therefore, all properties of the particles should also be invariants under axis rotation

3. "empty" state

"empty" is a electron state in which both of the two possible places for an electron are unoccupied.

Similar to the "full" state, the " empty" state has no spin direction and the time inverse symmetry is not broken for the "empty" state.

 


 

Only "spin" states may contribute to magnetic properties of conduction electrons. The "full" and "empty" states have zero spin and they do not contribute to the magnetic properties.

This could be understood by a comparison with EPR, where only unpaired electrons contribute to the EPR signal (Details, see here)

 

 

 

 

 

 

 


 

The same content can be found in V. Zayets JMMM 356 (2014)52–67 (click here to download pdf)

 

 

 

 

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