Top page (correct Bohr model )

Strange "spin" is NOT a real thing

Fine-structure by Bohr-Sommerfeld model.

- Spin-orbit coupling in quantum mechanics is illusion.
- Computing fine structure using JAVA program.
- Calculating approximate form of fine structure.

*(Fig.1) Spin-orbit interaction is completely false.*

As shown on this site and this site, it is said that sodium D lines are due to spin-orbit interaction , which is one of relativistic effects.

But on this page, we can prove spin-orbit coupling is too weak to explain alkali metals such as Na, K, ...

So spin-orbit coupling is completely an **illusory** concept.

*(Fig.2) Dirac's hydrogen = Bohr-Sommerfeld model.*

As shown on this site and this site (p.12) , Bohr-Sommerfeld model gives completely the **same** fine structure as Dirac hydrogen.

Compare this page (= Dirac hydrogen ) and this page (= Sommerfeld model ).

This accidental coincidence is very **artificial**.

Of course, Sommerfeld model appeared **earlier** than Dirac equation, so it is obvious that the latter solution **imitates** the former one.

About the Dirac hydrogen solution, see this site (p.9) and this site (last eq.).

*(Fig.3) The same approximate fine structure formula.*

On this site (p.9) and this site (p.13), we can get the simple form of energy difference (= fine structure ), which is proportional to Z^{4}/n^{3}.

"Z" is effective central charge, which includes **all** other charges of nucleus and **inner** electrons.

And "n" is the principal quantum number, which expresses major energy levels.

In this section, we have derived this approximate formula from the original Dirac (= Sommerfeld ) hydrogens.

*(Fig.4) *

For example, first we get the energy difference between **2p3/2** and **2p1/2** of hydrogen atom.

The principal quantum number "n" is "2" in both states.

Total angular momentum is j = 3/2 in 2p3/2, and j = 1/2 in 2p1/2.

In Bohr-Sommerfeld model, radial number n_{r} = 0 (1) and angular number n_{φ} = 2 (1) corresponds to 2p3/2 (2p1/2).

*(Fig.5) ↓ This formula gives the same results.*

Approximately, we can use Fig.5 instead of the results of Fig.4 as fine structure.

When substituting **l = 1** ( angular momentum in Dirac form ) into Fig.5, this formula gives the fine structure between 2p3/2 and 2p1/2.

Of course, Bohr-Sommerfeld model gives completely the **same** form of Fig.5.

Though Bohr model does NOT have spin, these two models are based on the same mechanism ?

This is a very strange **coincidence** !

*(Fig.6) Hydrogen fine structure.*

As shown on this site, **hydrogen** fine structure (= doublet ) between *2p3/2* and *2p1/2* is about **0.000045 eV**.

This is a very small value, from which we find spin magnetic moment (= Bohr magneton, again coincidence ! ) is **too weak** to explain important physical phenomena.

*(Fig.7) Energy unit conversion.*

When you see hydrogen energy levels of Nist, energy unit of **cm-1** is used instead of eV.

So using this site, we convert the unit of cm-1 into eV (= electronvolt ).

*(Fig.8) Nist hydrogen energy data.*

From the hydrogen energy levels of 2p3/2 and 2p1/2 in Nist, we can get the values of Fig.8 in eV.

So the energy difference between these two states becomes

*(Fig.9) *

This energy difference is called fine structure, which is the same as Fig.6.

As shown in this section, this energy difference (= ΔE ) can be expressed as the simple form of Fig.10.

*(Fig.10) Fine structure*

On this page, we investigate whether the equation of Fig.10 is always valid in various atoms and ions.

*(Fig.11) Fine structures of H and He+*

If you see this site, you find the energy difference between 2p3/2 and 2p1/2 in singly ionized helium (= He+ ) becomes **0.00072 eV**.

*(Fig.12) Central charge Z*

As shown in Fig.12, the central positive charge Z in H and He+ are Z = 1 and Z = 2, respectively.

*(Fig.13) Computing each fine structure.*

Using the formula of Fig.10, we find this formula is **true** in both hydrogen and helium ion.

Because the energy difference of He+ (= 0.00072 eV ) is about **16** times larger than hydrogen (= 0.000045 eV ).

This value of **16** is **equal** to Z^{4}, when substituting **Z = 2** into this energy formula.

In this section, we can confirm this formula is correct using JAVA program.

*(Fig.14) Fine structure of Li and Be+.*

From this site and this site , we can know the energy difference between 2p3/2 and 2p1/2 in lithium and Beryllium ion.

*(Fig.15) Central charge of Li and Be+ are close to H and He+.*

Of course, the central core charge of lithium atom is **Z = 1**.

Because this central charge Z includes **all** charges of nucleus and inner electrons.

In the same way, the central charge of Be+ should be close to **Z = 2** like He+

*(Fig.16) Fine structure formula is effective also in Li and Be+.*

Using the formula of Fig.10 (= Fig.3 ) or computing values by program, we find this fine structure equation is **valid** also in lithium and beryllium ion.

*(Fig.17) Sodium fine structure is too big to be explained by a single electron's spin.*

As shown on this site, **hydrogen** fine structure (= doublet ) between *2p3/2* and *2p1/2* is about **0.000045 eV**.

On the other hand, the fine structure between *3p3/2* and *3p1/2* of **sodium** is as big as **0.0021 eV**, as shown on this site.

Approximately, we can consider the outer 3p ( or 3s ) electron of sodium is moving around **Z = +1** central **core** charge.

( "Core" is the total charge of Na nucleus and **all** electrons contained in n = 1 and 2 orbits. )

From the viewpoint of this outer electron, the core charge is moving around in the opposite direction, which causes magnetic field at the point of the electron having "spin".

As a result, spin-orbit interaction is produced, they insist.

*(Fig.18) Central charge Z = 3.54, which is much bigger than Z = 1 in Na !?*

Both in hydrogen and sodium's outer (= 3p ) electron, the effective central (= core ) charges they feel are about Z = **+1e**.

Because this central charge Z is the **sum** of nucleus and all inner electrons.

But the discrepancy between these H and Na spin-orbital interactions are **too wide**.

This means the effective central charge in sodium is much **bigger** than Z = 1 (= about Z = **3.54** in Na ), which is very **unreasonable** and unrealistic.

*(Fig.19) Effective "Z" becomes too big (= 3.5 ), if spin-orbit coupling is true.*

The central charge which 3p valence electron can feed should be close to Z = 1 also in sodium (= Na ).

But when Z = 1, the calculated spin-orbit energy becomes **too small**, which **cannot** reach 0.0021 eV.

When Z = 3.54, the fine structure formula gives this 0.0021 eV.

This result **proves** the spin-orbit coupling does **NOT** happen in actual fine structure.

*(Fig.20) True effective charge Z = 1.84 from ionization energy.*

As shown on this site, the ionization energy of Na is **5.14 eV** ( n = 3 ).

Comparing this value with hydrogen (= 13.606 eV in n = 1 ), we find the true effective charge of Na is **Z = 1.84**.

This Z is a little bigger than Z = 1 due to the **wide** gap of n = 2 inner electrons' shell.

But of course, it **cannot** reach Z = 3.54 at all.

*(Fig.21) Wide discrepancy in Na and Mg+ fine structure.*

On this site (= Na ) and this site (= Mg+ ) , we can know the energy difference between 3p1/2 and 3p3/2 in Na and Mg+ (= singly ionized magnesium ion ).

*(Fig.22) Effective central charge Z become too big in Na and Mg+.*

From Fig.21 and the fine structure formula, we find the effective central charge of Na and Mg+ become **unrealistically** too big.

Z of Na and Mg+ are 3.54 and 5.39, which is **much bigger** than 1 (= H ) and 2 (= He+ ).

*(Fig.23) Unrealistic effective central charge Z.*

As I said, this effective central charge Z includes all charges of nucleus and **inner electrons**.

So Z of Na and Mg+ must be close to "1" and "2".

These results show the concept of spin-orbit coupling **breaks down** in various metals.

As shown on this page, in larger atoms such as K, Cs .., this result becomes much **worse**, which means Z becomes much more **unrealistic** values.

*(Fig.24) Wider gap in n=2 electron's shell than n = 1.*

The average orbital radius of n = 2 is much longer than n = 1.

( Atomic radius is proportional to a **square** of n ).

So the valence electron of n = 3 in Na can feel much **stronger** Coulomb force through **wider gap** in n = 2 inner electron's shell than hydrogen and lithium.

This wider gap in inner electron's shell is the main reason why fine structure in metals become **large**.

We have to consider **Coulomb** effect instead of (imaginary) spin-orbit interaction.

*(Fig.25) The difference in orbital shapes (= elliptic and circular ) causes large fine structure.*

As shown on this page, the fine structure of Bohr-Sommerfeld model is cuased by the different orbital shapes in **elliptical** and **circular** orbits.

The average radius of them are the same in hydrogen.

But the **perihelion** of elliptical orbits is much **closer** to nucleus than that of circular orbit, which causes the different **Coulomb** interaction.

*(Fig.26) Spin-orbit model cannot cause large Coulomb difference.*

On the other hand, if this fine structure is generated by spin-orbit coupling, these two states must have the **same** orbital shape ( ex. 3p ).

The same orbital shape means there are **NO** difference in Coulomb interaction.

So spin-orbit coupling model **cannot** explain large fine structure in metals.

As a result, the fine structure of Bohr-Sommerfeld model proves to be **true** and real.

*(Fig.27) Fine structure by original equation (= Dirac and Sommerfeld ).*

In this section, we compute two equations of Fig.27 and Fig.28 using this JAVA program.

Compile this program as the name of "finest.java".

*(Fig.28) Fine structure by Z^4/n^3 form.*

First, we try to compute fine structure of hydrogen (= energy difference between 2p3/2 and 2p1/2 ).

After running this program, input "1" (= Z ), press enter key.

This Z means the central charge.

Next, we input "2" (= n ) as the principal quantum number, and press enter key.

Lastly, input "1", which means we want to compute the energy difference between j=3/2 (= p3/2 ) and j=1/2 (= p1/2 ).

*(Fig.29) Input each value. Hydrogen 2p3/2-2p1/2 case.*

The computing results become like Fig.30.

The value of "finestructure by original equation" is gotten using Fig.27.

The value of "finestructure by Z^4/n^3 form" is gotten using Fig.28.

*(Fig.30) Computing results.*

As you see Fig.30, the formula of Fig.27 and Fig.28 give the same results.

This means the approximate fine structure form of Z^{4}/n^{3} is **right**.

*(Fig.31) He+ ion case.*

In He+ ion, input Z = 2 instead of Z = 1 in the first question.

Again the formula of Fig.27 and Fig.28 can give almost the same values.

*(Fig.32) *

*(Eq.1) Dirac's hydrogen = Bohr-Sommerfeld model.*

As shown on this site and this site (p.12) , Bohr-Sommerfeld gives completely the **same** fine structure as Dirac hydrogen.

Compare this page and this page .

This accidental coincidence is very **artificial**.

About the Dirac hydrogen solution, see this site (p.9) and this site (last eq.).

*(Eq.2) The same approximate fine structure formula.*

On this site (p.9) and this site (p.13), we can get the simple form of energy difference (= fine structure ), which is proportional to Z^{4}/n^{3}.

"Z" is effective central charge, which includes **all** inner charges of nucleus and **inner** electrons.

And "n" is the principal quantum number, which expresses major energy levels.

*(Eq.3) *

We replace a part of the denominator of Eq.2 by "A".

Of course, Sommerfeld and Dirac equations are the same.

*(Eq.4) Conversion from Sommerfeld to Dirac forms. *

In Bohr-Sommerfeld model, the principal quantum number "n" is the sum of radial (= n_{r} ) and tangential (= n_{φ} ) quantum numbers.

In Dirac equation, they use the **common** variable "**j**" as total angular momentum.

This "j" is the sum of orbital (= "l") and spin (= s ) angular momentums.

So even if two states have **different** orbital angular momentum, their energy levels become the **same**, when their "j" are the **common**.

*(Eq.5) Total energy.*

Using "A" of Eq.3, the energy level of Eq.2 can be expresses as Eq.5.

Here we apply Taylor expansion of Eq.6 to this equation.

*(Eq.6) Taylor expansion.*

where

*(Eq.7) *

From Eq.7, total energy E of Eq.5 becomes

*(Eq.8) *

Using Eq.3 and approximation ( fine structure constant α is small ), we obtain.

*(Eq.9) *

where we use

*(Eq.10) *

From Eq.9, we have

*(Eq.11) *

Furthermore, considering fine structure constant α is very small (= 1/137 ), we can use the following approximation.

*(Eq.12) *

Substituting Eq.11 and Eq.12 into Eq.8, the total energy E becomes.

*(Eq.13) *

When the principal quantum number "n" is the common value, only angular number (= n_{φ} ) influences the energy values.

So we pick up the term (= third term of Eq.13 ) including this n_{φ}.

*(Eq.14) *

As you see Eq.14, this value is proportional to Z^{4}/n^{3}.

We want to calculate the energy difference between two states with angular momentum n_{φ} and n_{φ}+1 in the same "n".

*(Eq.15) *

As shown in Eq.4, these two states correspond to two different "j" in Dirac form.

"l" is the orbital angular momentum.

*(Eq.16) Energy difference (= fine structure ) between two states in the same "n".*

From Eq.14 and Eq.16, we find the energy difference between two states are proportional to **Z ^{4}/n^{3}**.

These two states exist inside the same "n", and their "j" ( or n

2014/8/15 updated. Feel free to link to this site.