Top page (correct Bohr model of helium. )

Strange "spin" is NOT a real thing

Rare earth doesn't have "spin".

- "Spin - orbit" coupling is illusion.
- Band theory is completely useless.
- Spin Hall effect = mathematical trick.

*(Fig.1) Spin - orbit coupling doesn't really exist.*

It is said that spin - orbit interaction is related to spin Hall effect (= SHE, see Wiki ) and magnetoresistance.

But unfortunately, these phenomena have **nothing** to do with unrealistic spin.

*(Fig.2) Spin-orbit coupling based on relativistic effect is too weak. ← illusion !*

They insist spin orbit coupling is caused by relativistic effect (= see Wiki ).

But if so, the **magnitude** of spin orbit coupling becomes much **smaller** than the experimental values
( this site p.6 and this site ).

On this site ( p.2, left ), it is written

-----------------------------------

"For example, for GaAs one finds λ = **5.3** Å^{2}.

Thus, the SO (= spin-orbit ) coupling in n-GaAs is by **six** orders of magnitude **stronger** than in vacuum ( λ = **-3.7 × 10 ^{-6}** Å

This enhancement of SO coupling is critical for developing large extrinsic spin currents.

------------------------------------

This wide **discrepancy** in the λ shows the concept of spin-orbit coupling is completely **false**.

In spite of this obvious fact, they **like** to use the word of "spin-orbit coupling" in various (top) journals.

Again, this **wrong** usage is one of **mind-controls**.

*(Fig.3) Electric field (= E ) changes into magnetic field (= B ) under Lorentz transformation ?*

First we explain why spin-orbit coupling means **relativistic** effects.

In Fig.3, an electron with spin is moving in x direction at the speed of "v" under the electric field E_{y} (= S frame ).

In the rest frame of this electron ( v = 0, S' frame )
, this electric field **transforms** into magnetic field B according to the relativistic electromagnetism.

*(Eq.1) Lorentz transformation. *

As shown on this site, Lorentz transformation of electric field becomes like Eq.1.

B' is magnetic field from the viewpoint of moving electron (= S', rest frame ).

Here, the magnetic field in S frame is supposed to be zero ( B = 0 ).

( Only electric field E_{y} exists in S frame. )

So approximately, Eq.1 becomes Eq.2.

*(Eq.2) *

The magnetic energy is magnetic moment (= μ ) × magnetic field (= B ).

The μ of an electron's spin is minus Bohr magneton.

*(Eq.3) *

In the rest frame of an electron (= S' ), the magnetic field (= B_{z} ) is caused in the minus z direction.

If this electron has "spin", the magnetic interactive energies are split into two depending on spin "up" or "down" directions.

*(Eq.4) *

σ is Pauli matrices.

So, s = 1/2 σ.

*(Eq.5) *

The electron's velocity (= v ) can be expressed by wavenumber (= k ) as shown in this Wiki.

And Thomas factor (= 1/2 ) is added.

Substituting Eq.4 and Eq.5 into Eq.3, the magnetic energy (= - μB ) based on relativistic effect becomes

*(Eq.6) *

As a result,

*(Eq.7) *

This spin orbit coupling constant (= λ ) by relativity becomes

*(Eq.8) Relativistic effect ?*

But the experimental values of spin-orbit coupling constant in solids are **much larger** than Eq.8, as follows,

*(Eq.9) Actual solids.*

where

*(Eq.10) *

If you compare Eq.8 (= relativsitic effect ) and Eq.10, you easily find the concept of spin-orbit coupling is **illusion**.

Originally spin itself has **NO** reality at all.

In spite of this fact, why the current physicists continue believing **illusory** spin-orbit coupling ?

There were not alternative theories of quantum mechanics in 1920's due to lack of computers.

This is the reason why strange spin has survied even now, though they try **not** to ask what "spin" really is.

*(Fig.4) Spin-orbit coupling based on relativistic effect is "illusion".*

The serious problem is, if we continue believing this false fact, much more people all over the world would **waste** their lives in **wrong** theories.

And we **cannot** develop the science due to the troublesome "Shut up and calculate!" attitude in the current quantum mechanics.

As shown on this page and this page, Alkali (= **isolated** sodium ) doublets **cannot** be explained by spin-orbit interactions, because spin orbit coupling is **too weak** to explain actual energy splitting.

The important point is that usual textbooks are **hiding** this serious defect about spin -orbit, and insist "spin" has factual grounding.

This is a kind of **frauds**, if they know the above results.

*(Fig.5) Band theory = ancient documents, it's completely useless.*

Band structure consists **only** of wave number (= k ) and energies, **NOT** including space coodinate.

( **Lacking** x,y,z coodinate means this band theory is **NOT** applicable to the actual sicence. )

In spite of this **old** concept, it is used even now in condensed matter physics. This is **disaster**.

To treat many-body complicated wavefunctions, they needed to expand wavefunction in the form of **plane** wave solutions (= e^{ikx} ) in 1920 - 1930s.

Considering abstract periodicity of solids, they paid attention to first ( small ) wave number zone.

This is called Brillouin zone ( see this site ).

*(Fig.6) "Band theory" and "Brillouin zone" are useful ?*

The concept of space coodinate (= x, y, z ) is **indispensable** for applied physics.

We often save different kinds of informations in **different** places inside the computer memory.

In these cases, the position (= x, y, z ) inside memory is very important.

So the band theory based **ONLY** on wavenumber and Brillouin zone is completely **useless**.

These old concepts become like ancient documents and survive **only** for papers in various journals (= **NOT** real world at all ).

*(Fig.7) Time gap without computers produced band theories.*

Solving many-body Schrodinger equations + complicated Slater determinants is **impossible**.

And of course, there were **NO** high efficient computers in 1920s - 1960s.

In spite of these conditions, they needed to treat complicated solid states consisting of **infinite** atoms.

Within **pen** and **paper** theories, all they could do is rely on simple plane wave solutions (= e^{ikr} ).

The problem is, even after high efficient computers appeared, these **dead** concepts such as band theory and Brillouin zone are left like ancient documetns, as shown on this page

As far as we **stick** to these old concepts and continue "Shut up and calculate!" quantum mechanics and spin, the science would **never** advance.

*(Fig.8) Exchange correlation functions of DFT are "free" paremeters.*

Band theory for solids completely **depends** on density functional theory (= DFT ).

Most textbooks and academic papers insist these DFT are first-principle (= ab-initio ).

But unfortunately, this DFT is **NOT** ab-initio at all.

DFT was introduced to treat complicated **many-body** solid states in 1960s.

For this calculation, they thought up **many** kinds of exchange correlation functionals (= E_{xc} ).

The important point is that these exchange-correlation functionals are like **free** parameters we can freely choose.

On this site p.9, "the central problem in DFT is that there is **NO** unique prescription for how to find the exchange and correlation functionals".

So the word of "ab-initio DFT" gives ordinary people **wrong** impressions.

This is **dangerous**.

*(Fig.9) Exchange energy = Pauli exclusion principle ?*

For example, in lithium atom, the third electron **cannot** enter 1s orbitals.

This strong **repulsive** effect needs to be expressed using Slater **determinants**.

As you see Fig.9, these **complicated** determinants are one of the main reasons why the current quantum mechanics **cannot** be applicable to actual larger molecules.

To do with this problem, all they could do was rely on some **approximations**.

Exchange functionals (= E_{x} ) were approximate forms to describe this Pauli exclusion principle.

Various forms of this functionals have been produced until now, and we can **freely** choose one of them.

*(Fig.10) Correlation energy = electron's wavefunctions repel each other ?*

In multi-electrons' atoms, each electron is **avoiding** each other.

When one of their wavefunctions is determined, other wavefunctions need to be **changed** to avoid it.

This means there are **NO** constant probability densities in multi-electron atoms, different from hydrogen.

For example, as shown on this page, they prepared **five** kinds of charge parameters (= Z ) inside one helium atom's wavefunction to describe repulsive motions of two electrons.

In solid states, of course, they had to use some approximations (= correlation functional, E_{c} ).

The form and parameters in these correlation functionals are completely **arbitrary**.

Again, DFT is **NOT** ab-initio at all.

*(Eq.11) Hamiltonian of multi-electron atoms.*

Eq.11 is Hamiltonian of multi-electron atoms.

H_{0} is the sum of each kinetic energy and potential energy between nucleus and electron.

The second term of right side in Eq.11 represents repulsive forces of electrons.

*(Eq.12) Slater determinants = Pauli exclusion principle ?*

Generally, determinants become **zero**, when two lines ( or columns ) of them are equal.

They take advantage of this property in determinants to express Pauli exclusion principle.

Because according to Pauli exclusion, two electrons **cannot** be in the same states (= lines ).

As I said many times, the current quantum mechanics NEVER try to investigate the origin of this Pauli exclusion principle.

As shown on this page, electron spin is **too weak** to explain this strong repulsive force.

*(Eq.13) *

Using Eq.11 and Eq.12, the total energy becomes like Eq.13.

J_{ij} is the repulsive Coulomb energy between i and j electrons.

K_{ij} represents **exchange** energy based on Slater determinants, as follows.

*(Eq.14) *

The sum of them is called exchange correlation energy (= E

*(Eq.15) Density functional theory is NOT ab-initio.*

As I said, this exchange and correlation functionals can be **freely** changed as we like.

So DFT is **NOT** first principle at all, though they **always** call it "ab-initio DFT" in solid physics.

This misuse is very dangerous.

*(Eq.16) Exchange energy in LDA.*

On this site (p.4), "the problem is that functional E_{xc} is **NOT** known.

The simplist approximation is local density approximation (LDA) using **uniform** electron gas.

Of course, electron's distributions in the actual solids are **far** from uniform.

So LDA **fail** in various materials.

Eq.16 is exchange term of LDA.

*(Eq.17) LDA - VWN correlation function.*

On this site p.4, there are several expressions for LDA correlation energy which have been obtained by **fitting** to the results of accurate QMC calculations.

Most popular choice in LDA correlation functions is VWN-LDA as shown in Eq.17.

ε of Eq.17 is on this site (p.9).

This LDA correlation function contains as many as **12** fitting parameters.

So it is NOT first-principle at all.

*(Eq.18) LDA - PW correlation functional*

One of other LDA correlation functionals is PW-LDA like Eq.18.

This functional contains **6** fitting parameters.

Of course, there are other correlation functionals in LDA.

So, LDA is NOT a single pattern.

We can **freely** change correlation functionals depending on circumstances.

Quantum Monte Carlo simulation (= QMC ) can give random and **wrong** results.

The results of this QMC **changes** depending on electron's density (= ρ ).

**NO** convenient LDA functionals have been found to satisfy all conditions.

*(Eq.19) Hybrid functional.*

This B3LYP functionals have been used most often in qunatum chemistry.

The important point is that this B3LYP contains many

So, again, this B3LYP is **NOT** ab-initio at all.

*(Eq.20) *

B3LYP contains B88 exchange function.

B88 is one of GGA functional, which adds some gradient of electron's density to uniform LDA.

ζ is semi-**empirical** parameter.

There are **many** kinds of GGA functionals.

Depending on the situation, one of these functionals is **chosen**.

*(Eq.21) *

Eq.19 contains LYP exchange functional, which is one of GGA.

There are five semi-**empirical** parameters in LYP.

In the current DFT, they are spending much time in finding better exchange correlation functionals.

Because the form of these functionals are completely **free**.

If DFT is NOT first principle, these efforts are just **waste** of time.

*(Fig.11) Spin Hall effect ?*

In spin Hall effect, even under NO external magnetic field, "up" ( down ) spin electron turns toward left ( right ), when some electric field (= E ) is applied ( see this site or this site ).

We have **NOT** yet arrived at the definite conclusion about the true mechanism of this spin Hall effect.

In extrinsic mechanism, spin-polarized electrons are **scattered** by some impurities through **spin-orbit** interaction.

( This is called "skew scattering" or "side jump". )

In intrinsic mehacanism, they insist some band structure ( Berry phase ) or spin precession cause spin Hall effect.

But these intrinsic theoirs are very **artificial** and have some defects, as I say later.

*(Fig.12) Spin Hall effect = Spin ? illusion ?*

In spin Hall effect experiments, they did **NOT** detect "spin" itself ( see this site ).

They just measure the slight change of reflected **light** polarization.

Signal is very weak. This experiment is very difficult, so they don't know the directions of "up" or "down" spins.

Only asymmetry of **light** polarization can be detected.

They use Kerr effect to know "spin".

But this Kerr effect is purely **classical** mechanical ones like Faraday effect, as shown in Wiki.

Basically it is natural each electron inside semiconductor is moving, **rotating** around some positive nucleus.

So we can say spin Hall effect is caused by some **circular** orbit (= classical rotation ) of electrons.

*(Fig.13) Spin Hall = Magnus effect ? Faster-than-light spinning ?*

Some physicists say spin Hall effect is due to Magnus effect ( see Wiki Magnus and spin Hall ).

In Magnus effect, **spinning** ball curves away from its principal path.

Because the directions of spin and air flow are parallel or antiparallel in each side of the spinning balls.

The problem is spinning speed of point-like electron must reach more than **100 times** speed of light ( > 100c ), if "spin" is actual spinning, as shown on this page.

So electron spin **cannot** use this Magnus effect as the cause of spin Hall effect.

On the other hand, if we consider spin Hall effect is caused by **circular** orbit of electron, Magnus effect can be applied naturally.
(= each **de Broglie** wave flow's effects )

*(Fig.14) Directions of electric field (= E ) become opposite → "spin" Hall effect ?*

Besides Magnus effect, we can consider the influence of external electric field.

In Fig.11 and Fig.14, one side of this electron's rotation is parallel to the external electric field.

Another side is antiparallel.

This **difference** in rotational direction may be the cause of "spin" Hall effect, though it does NOT depend on strange "spin".

On the other hand, spinning electron is **point**-like particle.

So it causes **NO** difference with respect to directions of external electric field on the surface of **tiny** electrons.

*(Fig.15) Spin Hall effect = Rotation.*

In spin Hall experiments, they detected the directions of polarized **lights** ( NOT spin ! ).

In **normal** Zeeman effect, the change of **orbital** angular momentum generates ( left or right ) circular **polarized** lights depending on the change of ± m ( see this site ).

And Kerr and Faraday effects are originally **classical** effects.

Considering spinning speed of electron becomes much faster than light ( see also this site ), it is quite natural that we think spin Hall effect is due to **circular** orbit of electrons.

*(Fig.16) Point-like electron → Only random scattering is left.*

As I said in Fig.13 and Fig.14, point-like electrons **cannot** rely on classical Magnus effect and the difference in the external electric field direction as the cause of spin Hall.

So there is ONLY **random** scattering left to explain spin Hall effect, if you believe "spin".

But as I said in Fig.2, spin-orbit interaction by realtivity is **too weak** to explain this scattering.

And **random** scattering means electrons turn toward left or right randomly.

So it generates **NO** spin currents, because the average is almost zero.

This is strange.

First, we explain intrinsic effect based on Rashba model, as shown in this site and this paper.

As far as I read this paper, they seem to use some mathematical **tricks** to get universal values.

And it is known this intrinsic effect **vanishes** due to some impurities as shown in this site and this paper (p.6, 7).

From here, we explain about this paper.

This mechanism is very complicated, and based on some **tricks**.

*(Fig.17) Rashba intrinsic electric field.*

Some semiconductors originally have intrinsic electric field (= E_{rash} ) in the z direction.

In Fig.17, an electron is moving in **y** direction ( k_{y} is wavenumber ).

According to relativity, magnetic field (= B_{ras} is in - x direction ) is generated in the rest frame of this electron.

As an electron has **negative** charge, its spin turns in x direction.

**Δ _{1}** is

The

*(Eq.22) Hamiltonian.*

Considering spin-orbit coupling energy, Hamiltonian becomes like Eq.22.

λ is the Rashba coupling constant ( λ > 0 ).

σ means Pauli matrices ( s = 1/2σ ).

*(Eq.23) Larmor precession.*

Eq.23 is Larmor precession ( see Wiki ).

"S" is electron's spin direction. "B" is magnetic field.

g_{e} is spin g factor (= **2** ).

*(Eq.24) *

An electron has negative charge, so negative sign is added in Eq.23 and Eq.25.

*(Eq.25) *

"n" and "S" (= spin ) are in the same direction.

Δ(t) is also in the spin direction, which is opposite to magnetic field (= B ).

( Unit of this Δ(t) is the same as energy (= J ). )

Using Eq.25, Eq.23 becomes

*(Eq.26) *

Δ_{1} is the direction in which spin **first** turns.

So in Fig.17, Δ_{1} is in **x** direction.

The direction of Δ_{2} is defined like Eq.27.

( If Δ_{1} is in x direction, Δ_{2} is in y direction ).

*(Eq.27) *

From Eq.26, we obtain

*(Eq.28) *

First spin direction is +x, so Δ_{3} (= z component ) is zero in Eq.28.

And,

*(Eq.29) *

Δ_{2} component (= perpendicular to Δ_{1} ) of spin is very small (= spin precession ).

Spin component n_{2} is proportional to Δ_{2}, so

*(Eq.30) *

From Eq.28 and Eq.30, we have

*(Eq.31) *

From Eq.22 and Eq.25, Δ (= - B direction ) can be expressed as

*(Eq.32) *

The wavenumber k is p/ħ.

First, this k is supposed to turn in y direction ( k = k_{y} ).

( And hat z is - z direction ).

So Eq.32 becomes

*(Eq.33) *

Here the external electric field E_{x} is applied like Fig.18.

After that, this electron drifts toward - x direction.

And due to relativistic effect, new effective magnetic field (= B_{new} ) is generated in the z direction.

As a result, this spin turns in -z direction (= down ) due to its negative charge.

But some **tricks** need to be used in this calculation.

*(Fig.18) Electric field E _{x} is applied. → spin down ?*

From Eq.27, Δ_{2} is in the k_{y} (= k ) direction in this case.

The time derivative of this "k" component (= Δ_{2} ) of Eq.32 becomes

*(Eq.34) *

In Eq.34, due to new electric field E_{x}, this electron is accelerated in - x direction.

So we use

*(Eq.35) *

Momentum p is equal to ħk ( k is wavenumber ).

The time derivative of momentum is electric force (= -eE ) in Eq.35.

But Eq.34 is a little strange.

Negative sign (= -e ) in inner product should **not** be left, I think.

Substituting Eq.33 and Eq.34 into Eq.31, and add 1/2 (= spin ), we have

*(Eq.36) *

As you see Eq.36, depending on **k _{y}** (= moving ) direction, the z component of spin becomes

So they insist this is spin Hall effect.

But it is

*(Eq.37) *

Electron's velocity is expressed as momentum (= ħk ) / m.

Spin current ( j ) is equal to spin angular moementum (= ħS ) × velocity.

*(Eq.38) *

So the integral of wave number becomes like Eq.38.

k_{F,+} and k_{F,-} are maximum and minimum wave numbers in this system.

These maximum and minimum wave numbers are **arbitrary** ( ← **Trick** ).

*(Eq.39) *

Substituting Eq.36 and Eq.38 into Eq.37, and integrating it, we get Eq.39.

Here they suddenly define like Eq.40.

*(Eq.40) *

*(Eq.41) *

So they insist universal electric conductivity (= -e/8π ) can be gotten.

But as you see Eq.40, this definition can be **changed** as we like.

The choice of Eq.40 value leads to the result of Eq.41

*(Eq.42) *

Comparing Eq.8 and Eq.40, the difference between k_{F,+} and k_{F,-} become like Eq.42.

I **cannot** find proper reason why the condition of Eq.42 is chosen.

*(Fig.19) Total (= average ) spin is always zero in this theory.*

According to this theory, when electron is moving in y direction, its spin becomes "down".

When electron is moving in - y direction, spin becomes "up".

The external electric field is only in x direction.

So the **average** velocity in y direction is always **zero**.

This means the average spin is always **zero** in **any** parts of this system.
This result is **inconsistent** with the experiment of Fig.12 (= spin accumulation in each sides ).

*(Fig.20) GaAs, heavy- (HH) and light- (LH) hole bands*

Another main intrinsic mechanism of spin Hall effect is based on band structure. ( Science 301 1348 2003, see this paper ).

According to Luttinger Hamiltonian, the band energies are split into heavy-hole (= HH ) and light hole (= LH ) bands.

HH has total angular momentum j = ± 3/2, LH has j = ± 1/2.

*(Fig.21) Spin Hall effect ?*

According to this theory, particles having +3/2 or +1/2 turn in the direction opposite to particles having -3/2 or -1/2.

Each particle is supposed to be moving inside **either** of HH or LH bands.

But this explanation is **inconsistent** with experimental results, I think.

*(Fig.22) Spin accumulation becomes zero in each band ?*

As I said, each band contains ± angular momentum, because these directions are **random**.

If so, like Fig.19, total spin in each band becomes **always** zero.

Because particles with plus and minus angular momentum are **circulating**, turning in the opposite directions.

Again, this theory is **inconsistent** with experimental results which showed spin accumulation.

*(Eq.43) Luttinger Hamiltonian with external field.*

In this paper, they use the approximation of Luttinger Hamiltonian.

γ_{1} and γ_{2} are **semiempirical** parameters, which are **different** in each material.

"k" is particle's wave number, and "S" denotes total angular momentum.

"-eEx" is electric energy.

*(Eq.44) *

Eq.44 is wave number in the spherical coordinate.

Here they use the following unitary transformation.

*(Eq.45) *

"kS" term of Eq.43 transforms like Eq.46.

Under the unitary transformation of Eq.45, spin S turns toward z direction.

After this manipulation, each partilce is moving with its spin S in the z direction, they insist.

( Of course, this manipulation is very **unnatural**. )

*(Eq.46) *

*(Eq.47) *

Using Eq.47, Hamiltonian of Eq.43 changes into Eq.48 under transformation of Eq.45.

*(Eq.48) *

The last term of Eq.48 is added to the new x variable.

This is clearly

*(Eq.49) Mathematical Trick.*

Magnetic field (= B ) can be expressed as ∇ × A.

( ∇ is **space** ( NOT wave number ! ) derivative, and A is vector potential ).

Wave number (= k ) is clearly different from space variable "x".

But they use the same form of true magnetic field in **pseudo** magnetic monopole like Eq.50.

*(Eq.50) "Fake" magnetic monopole ?*

λ of Eq.50 is total angular momentum.

*(Eq.51) *

*(Eq.52) *

Using Eq.50, Eq.52 becomes

*(Eq.53) *

In Eq.53, depending on the sign of λ, the direction of velocity y changes.

But the new force in y direction in Eq.53 is strange.

There are **NO** forces perpendicular to **both** spin magnetic moment and electric field in the actual world.

( This strange force is **neither** relativistic effect nor Lorentz force. )

And each band contains ±3/2 ( or ±1/2 ) magnetic moments.

So magnetic components of Eq.53 are **cancelled** out. ( +3/2 - 3/2 = 0. )

*(Fig.23) Spin Hall effect ?*

But as I said, this explanation is **inconsistent** with experimental results.

*(Fig.24) Spin accumulation becomes zero in each band ?*

The opposite sign of angular momentum ( ex. ±1/2 ) inside each band **cancel** each other.

Because these particles are **circulating** in y direction.

And k_{z} direction of Eq.53 is **random**.

As a result, total spin accumulation in each site always becomes **zero**.

And the definition of antisymmetric tensor (= F_{ij} ) is NOT true magnetic tensor.

I **couldn't** find reasonable reason of the replacement of Eq.49.

*(Fig.25) Quanum Hall effect = Four space dimensions ? = String theory ?*

Surprisingly, according to top journal ( Science 294 823 2001, see arxiv ), they insist four **space** ( NOT time ) dimensions exist. See also this news

As you know, another dimension is completely fantasy.

You may be surprized at the fact that even top journals believe extra dimensions such as string theory.

2014/2/14 updated. Feel free to link to this site.