Top page (correct Bohr model including helium. )

Strange "spin" is NOT a real thing

Singlet and Triplet don't mean "Spin".

*(Fig.1) Hydrogen and potassium fine structure (= D line ).*

On this page, we show sodium D-lines energy splitting is **too wide** to explain by the weak relativistic spin-orbit interactions.

It is quite natural that we think these unrealistically **large** effects in D lines are due to some **Coulomb** interactions with inner electrons.

On this page, we show D-line or fine structure energy splittings in all other alkali metals such as potassium (= K ), rubidium (= Rb ), and cesium (= Cs ) are unrealistically **too big** for the tiny relativistic spin-orbit effect.

On this site, **hydrogen** fine structure (= doublet ) between *2p3/2* and *2p1/2* is about **0.000045 eV** (= 0.4 T ).

The fine structure between *4p3/2* and *4p1/2* of **potassium** is as big as **0.00715 eV**, as shown on this site (p.7, n.67) or nist where 4p3/2-4p1/2 = 57.7 cm-1 equals 0.00715 eV.

*(Fig.2) Spin-orbit interaction ? → Central charge Z becomes "too big".*

*(Fig.3) Energy difference between p3/2 and p1/2*

Relativistic spin-orbit magnetic energy splitting between two energy levels (= same principal number n, same orbital angular momentum l = 1 , different total angular momentum j = 1/2 and j = 3/2 ) is proportional to **Z ^{4}/n^{3}** where Z is the effective central charge containing nuclear charge and all inner electrons except one valence electron ( this p.9-(1.37), this p.9-(5.32), this p.3-left-1st-paragraph, this p.9-lower ).

In hydrogen atomic D-line or fine structure, Z = 1 and n = 2 (= energy splitting between 2p3/2 and 2p1/2).

In potassium atom, n =4.

From these values and experimental D-line energy splitting ( H atom-0.000045 eV vs. K-0.00715 eV, this p.5-Table 1 ), we can determine effective central charge Z of potassium.

*(Fig.4) Z of potassium is too big (= 5.97 ).*

From Fig.3 and Fig.4, the effective core charge (= effective charge combining K nuclear + all inner electrons) of K becomes as large as Z = **5.97**, which is **unrealistic**.

This core charge is the total charge summing up K nucleus and **all** 1n, 2n, 3n electrons.

The charge of K+ ion ( excluding only 4s electron ) is about **+1e**.

So, this core charge "Z" **must** be close to "**1**" also in K.

In conclusion, it is very **impossible** that the fine structure of potassium D lines is caused by spin-orbit interaction.

.

*(Fig.5) Coulomb and large (core) orbit - orbit interactions generate "K" D lines.*

Alkaline large D-line energy splitting is caused by Coulomb electric interaction between inner and outer electrons, Not by paradoxical tiny relativistic spin-orbit magnetic interaction.

*(Fig.6) What is effective core charge Z from ionization energy ?*

As shown in Fig.4, if spin-orbit interaction is real, effective core charge Z of potassium becomes much bigger (= **5.97** ) than "1".

We can know the real effective core charge Z from **ionization** energy of potassium ( 4s ) outer electron.

This ionization energy of potassium (= valence electron ) is **4.34** eV.

*(Fig.7) hydrogen-like atoms*

Fig.7 is energy levels expressed by "n (= principal quantum number )" and "Z (= central charge )" of hydrogen like atoms.

As you know, hydrogen ionization energy of 1s electron is **13.606** eV.

In 1s hydrogen, Z = 1 and n = 1.

*(Fig.8) True effective core charge is Z = 2.259 in potassium.*

Considering the principal quantum number ( n = 4 ), ionization energy (= 4.34 eV ) in "K" and Fig.7, we get the effective central charge Z = **2.259** in K 4s electron.

This **true** effective core charge (= 2.259 ) based on experimental ionization energy is much **smaller** than Z = **5.97** in Fig.4 assuming potassium large D-line splitting may be caused by tiny relativistic spin-orbit magnetic interaction.

As a result, the concept of (relativistic) spin-orbit coupling is too weak (= so need fictional large central charge Z = 5.97 ) to explain large D-line energy splitting between 4p3/2 and 4p1/2 in potassium.

*(Fig.9) Fine structure of Rb.*

As shown on this site (p.5, Table 1), the D-line energy difference between 5p3/2 and 5p1/2 in Rb is **0.02946** eV.

This fine structure of Rb is much bigger than hydrogen (= 0.000045 eV ), too.

*(Fig.10) Effective central charge Z = 10.25 is too large.*

Like potassium case, we can compute effective central charge based on spin-orbit and ionization energies.

Central charge based on the spin-orbit energy splitting equation becomes too large ( Z = **10.05** ) in comparison to true effective charge ( Z = **2.77** ) based on experimental ionization energy (= 4.177 eV ).

This result shows spin-orbit interaction of Rubidium is **wrong**, too.

*(Fig.11) Fine structure of Cs.*

As shown on this site (= p.5, Table 1 ), the energy difference between 6p3/2 and 6p1/2 in Cs is **0.06869** eV (or 0.06875eV ).

This fine structure of Cs is much **bigger** than hydrogen (= 0.000045 eV ), too.

*(Fig.12) Effective central charge Z = 14.248 is too large.*

Like potassium case, we can compute effective central charge based on spin-orbit and ionization energies.

Central charge assuming the tiny relativistic spin-orbit interaction may cause D-line energy splitting becomes too large ( Z = **14.248** ) in comparison to true effective charge ( Z = **3.209** ) based on experimental ionization energy (= 3.893 eV ).

This result shows spin-orbit interaction of cesium needing the unrealistically-large central charge Z =14.248 is **wrong**, too.

*(Fig.13) Magnetization of metals really proves spin-orbit (= LS ) coupling ?*

One of the main experimental results of strange Spin is magnetization of 4f rare-earth elements.

Magnetic property under some magnetic field should **obey** Lande g-factor ( see this site ) and LS coupling rules, according to quantum mechanics.

In fact, it is known that "**3d**" metals such as Ti, V, Cr, Mn, Co, Ni and Cu do **NOT** obey Lande g factor.

Their experimental grounding is **Only 4f** rare-earth **ions**.

All other elements show more complicated magnetic property, which **cannot** be described by simple Lande g factor.

( Even if you search for "effective Bohr magneton", **only** rare-earth ions can be found on the internet. )

Even in case of rare-earth ions, they introduced very **artificial** rules to fit them to experimental results.

So Spin-orbit interactions in various metals are also very **doubtful** and unrealistic.

*(Fig.14) Angular momentums (= L ) of "3d" orbitals are all zero ? ← Wrong !*

According to the standard quantum mechanics, angular momentum of "3d" orbitals is **L = 2**, which is **NOT** zero.

But in "3d" metallic ions, the calculated resuts of spin-orbit coupling (+ Hund rule ) are completely **different** from the experimental values, if they have angular momentum.

To **fit** them to experimental data, they suddenly **delete** all angular mometum of 3d orbitals !

( See also this site (p.426) or this site p.59. )

This is clearly **artificial** rule, and shows spin orbit couplings ( Hund rule, Lande g factor ) **break** down.

They insist transition-metallic ions are NOT independent from **neighboring** ions, and all of their orbital angular momentums are **cancelled** out. ( → L = 0 ).

But this interpretation is very **forcible** and convenient.

**Only** orbital magnetic moment becomes **zero** ? Though spin magnetic moment is completely **intact**. This is strange.

*(Fig.15) "4f" orbitals are very far away from "4d" orbitals ! ← Aritificial tricks, again.*

Different from "3d" metals, it is known that magnetization of "**4f**" rare-earth ions is **close** to experimental results.

( See this site (p.425) or this site p.59. )

But in fact, they adopt some very artificial **tricks** in rare-earth elements.

One of the tricks is the **positions** of 4f orbitals in the periodic table.

As shown in Fig.15, higher energy levels such as 5s, 5p, 5d, and 6s appear **before** 4f orbitals !

In palladium (= Pd ), all of 4d orbitals are already **occupied**, but we have to **wait** for "4f" orbital to appear until "Ce" !

*(Fig.16) Ionization : Nd → Nd3+. 5s and 5p electrons are skipped ??*

Surprisingly, rare earh ions obey very **unreasonable** rules, as shown in Fig.16.

In the ionization from Nd to Nd3+, three electrons in 6s and 4f electrons are removed, though 5s and 5p electrons are completely **intact**.

**6s** orbitals are **lower** than **4f** orbitals with respect to energy !?

( 6s orbitals appear **before** 4f orbitals in Fig.15. )

This **unrealistic** jump to "6s" orbital is indispensable for describing Cs D lines.

They insist 5s and 5p orbitals exist **outside** of 4f orbital.

If so, Nd3+ ion should remove 5p electrons instead of 4f.

This is clearly **artificial** trick.

*(Fig.17) J = L -S or J = L + S ?? ← Trick.*

As I explain later, they apply **different** rules to small and large 4f metals.

The upper case of Fig.17 uses the equation of J = L - S.

The lower case of Fig.17 uses the equation of J = L + S.

Though directions of angular momentum and spin are the **same** in them, they intentionally **change** the sign of them.

As far as I check various other atoms and ions, this **ad-hoc** rule was introduced **ONLY** for rare-earth ions.

( Because pure effective Bohr magneton is valid **ONLY** in 4f rare-earth ions. )

This site (table 4) show, in other orbitals such as 3d, 4d, 5d, 5f **excluing** ONLY "4f", the influences by **ligand-field** become too **strong** to be neglected.

This means **all** these orbitals **disobey** simple Lande g factor.

*(Eq.1)*

Eq.1 is magnetic moment of some atom (ion).

μ_{B} is Bohr magneton, which magnitude is as same as Bohr's ground state orbit.

"J" is total angular momentum combining L and S.

*(Eq.2) g _{J} = Lande g factor*

As shown on this page, Lande g factor (= g_{J} ) is based on very **unnatural** precessions.

In multi-electron atoms, we need to sum up each spin like Eq.2, they insist.

The magnetic moments themselves are the **same** in spin and orbital angular momentums.

But in LS coupling, they consider "S" and "L" have completely different properties. This is strange.

( So, they clearly **separate** spin and orbital like S = S1 + S2 + S3 .. , L = L1 + L2 + L3 .. )

*(Eq.3) E = magnetic energy.*

Under some external magnetic field H, the magnetic energy E is given by magnetic moment (= μ_{J} ) × H.

M_{J} is the component of J in the direction of H.

*(Eq.4)*

The probability of some energy state is given by Boltzmann factor.

So the average magnetic moment M becomes like Eq.4.

B(x) is called "Brillouin function".

*(Eq.5) *

To get effective Bohr magneton, we consider the magnetic moment in the limit that magnetic field H becomes **zero** ( H → 0 ).

Using Eq.5, Eq.4 becomes

*(Eq.6) Effective Bohr magneton in the limit of H → 0.*

In the limit of H → 0, we can get the effective Bohr magneton, which includes a **square root** of J(J+1).

( You need to remember this J(J+1) is gotten based purely on **classical** concepts such as Boltzmann factor, here. )

The important point is that it is extremely **difficult** to measure magnetization in the limit of H → 0.

Because various **noises** and thermal fluctuation **cannot** be neglected under very weak H.

So this effective Bohr magneton needs to be obtained by **extrapolation** based on data of H > 0.

*(Eq.7) True magnetic moment.*

When the external magnetic field H is strong enough, magnetization M of metals becomes like Eq.7.

Eq.7 **doesn't** contain a square root of J(J+1).

This value of Eq.7 under **enough** H is more **reliable** than Eq.6 of very weak H.

See also this site ( p.4-6 ).

*(Fig.18) Effective Bohr magneton in the limit of H → 0 can be believed ?*

As shown on this site (p.422), magetizations of various metals ( ions ) tend to be an **integer** times Bohr magneton (= μ_{B} ) under some **enough** H.

This result indicates some Bohr's **quantization** rule is valid also in these metals.

On the other hand, effective Bohr magneton including a square root of J(J+1) is obtained from the slope of magnetization curve in the limit of **H → 0**.

So these results are very **susceptible** to various other effects, and we need to evaluate these results very **carefully**.

*(Fig.19) Lande g factor is invalid → angular momentum L = 0 !?*

As I said, all "3d" transition metallic ions do **NOT** satisfy Lande g factor and LS coupling rule.

They suddenly make every angular momentum zero ( L = 0 ) to **fit** them to experimental results.

This is very unreasonable.

**Only** magnetic moments caused by **spin** ( NOT orbital ) are left.

How can the law of nature **distinguish** magnetic moments caused by orbital mortion and spin ?

It's **impossible**.

*(Fig.20) Vanadium ion ( V3+ ) : L = 3 → L = 0 !?*

For example, vanadium ion ( V3+ ) has **two** outer electrons in 3d orbitals.

According to Hund rule ( see this site ), the orbital angular momentum (= L ) of V3+ becomes 2+1 = **3**.

And total spin angular momentum is S = **1**.

Though "3d" orbitals clearly **possess** orbital motion, they suddenly **delete** them ( L = 3 → **L = 0** ).

This trick is clearly **unacceptable**.

How can the surrounding enviroment **distinguish** spin and orbital magnetic moments, **conveniently** ?

This mechanism is **too good** to be true. **Impossible**.

*(Fig.21) Magnetic moment of V3+ ion becomes just 2.0 × Bohr magneton (= μ _{B} ).*

If angular momentum (= L ) is zero, and only spin is left, Lande g factor becomes extremely simple ( g_{J} = **2** ).

And total angular momentum J is equal to S ( J = S ).

In V3+, saturation magnetization (= g_{J}J, plateau line of Fig.18 ) becomes just **2.0** × Bohr magneton, which means Bohr-Sommerfeld **quantization** rule is valid in all 3d metals.

*(Fig.22) Bohr quantization rule is valid in all 3d metallic ions*

A square root of S(S+1) is **fictitious** precession state, which **cannot** be directly confirmed.

The expeimental values (= g_{J}J ) which we can confirm are all a **integer** times Bohr magneton (= μ_{B} ), as shown in Fig.22.

So, Bohr-Sommerfeld quantization rule is valid in these ions.

*(Fig.18) Saturation magnetization (= g _{J}J ) under enough H.*

**Neutral** "3d" metals show much more **complicated** magnetic propertoes influenced by other **surrounding** atoms.

These neutral metals do **NOT** satisfy simple Lande g factor, and LS coupling.

It is known that these localization theories **don't** hold in neutral transition metals.

They need completely **different** theories.

*(Fig.23) Rare-earth ions really obey LS coupling ?*

Different from 3d transition metallic ions, "4f" rare earth ions are known to almost agree with LS coupling rule.

"g_{J}" is Lande g factor, which can be gotten from Eq.2.

But as I said, they depend on various **ad-hoc** rules to fit them to experimental results ( see Fig.15, 16, 17 ).

In spite of these ad-hoc rules, **three** rare-earth ions still do **NOT** obey Lande g factor, as shown in Fig.23.

These three elements **expose** the **limit** of these **artificial** tricks.

*(Fig.24) "g _{J}J" becomes an integer times Bohr magneton.*

Almost all textbooks tend to show data as effective Bohr magneton number including a square root of J(J+1).

But these states are **fictitious** precession states, which **cannot** be confirmed **directly**.

Saturation magnetization (= **g _{J}J**, plateau line of Fig.18 ) represents

Suprisingly, almost all rare-earth ions obey Bohr's quantization rule with respect to g_{J}J, as shown in Fig.24.

These results indicate **quantization** of de Broglie waves play a important role in all atoms and ions.

*(Fig.25) Positions of "4f" rare-eath are very far away from 4d atoms !*

As you notice, the positions of "4f" rare-earth elements were very **artificially** determined to fit them to experimental results.

In Pd atom, all of 4d orbitals have already been filled (= 4d × 10 ).

Various **higher** energy levels such as 5s, 5p, 6s, 5d appear **before** "4f" orbitals.

*(Fig.26) Ionization : Nd → Nd3+. 5s and 5p electrons are skipped ??*

Strange to say, the present quantum mechanics insists the energy "**6s**" orbital is **lower** than that of "**4f**" orbital.

( "**4d**" < 5s < 5p < **6s** < 5d < "**4f**". ← Trick ! )

This unrealistic positions of "6s" orbitals are indispensable for explaining Cs D lines.

The **skipping** n= 4, 5 orbitals (= **4f**, 5d, 5f, 5g ) and **jumping** to 6s are very unreasonable and **unacceptable**.

*(Fig.27) Which is true, J = L - S or J = L + S ?*

Pr3+ ( praseodymium ion ) has **two** electrons in 4f orbitals.

According to basic Hund rule, orbital (= L ) and spin (= S ) angular momentums become "5" and "1", respectively, as shown in Fig.27.

The important point is that almost half of "4f" metals should satisfy **J = L - S** instead of J = L + S.

As far as I check various sites, this **ad-hoc** rule was introduced **ONLY** for explaining the experimental results of these **rare-earth** ions. See also this site.

*(Fig.28) Dysprosium ion ( DY3+ ) , J = L + S !?*

On the other hand, in Dy3+ ion, they use the **different** equation of J = L + S.

This distinction is very **unreasonable**.

As shown in Fig.27 and Fig.28, the **directions** of each orbital and spin do **NOT** change.

So they should use the common formula J = L + S ( **or** L - S ).

*(Fig.29) Directions are unchanged → J = L ± S is artificial trick.*

As shown in Fig.29', all spin directions are suddenly **reversed**, the instant one electron is added to 4f orbitals from Gd3+ to Tb3+.

How clever "spin" can judge and follow these ad-hoc rules ?

This rule is **too good** to be true.

*(Fig.29') All spin directions are reversed from Gd3+ to Tb3+ ??*

This rule is valid **ONLY** in 4f rare-earth ions, which means it was introduced ONLY for 4f metals.

For example, "3d" transition metals do **NOT** obey LS couplicg rule, as I said in Fig.19.

This site (table 4) show, in other orbitals such as 3d, 4d, 5d, 5f **excluding** ONLY "4f", the influences by **ligand-field** become too **strong** to be neglected.

This means almost **all** orbitals **disobey** simple Lande g factor and Hund rule.

On page 358, Physics 1971-1980 by Stig Lundqvist

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In 1925 Hund wrote a paper on the magnetic susceptibilities of **rare earth** compounds which was the crowning achievement of the empiricism ...

He utilized Lande's g factor and Hund rule .....

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And in 1932, Van Vleck extended his theory to Eu and Sm.

*(Fig.30) Saturation magnetization = g _{J} J.*

As I said, effective Bohr magneton including a square root of J(J+1) is fictitious precession states, which cannot be confirmed directly.

( **Only** in the limit that external magnetic field H is **zero**, this relation appears . )

*(Fig.31) Effective Bohr magneton in the limit of H → 0 can be believed ?*

Of course, when the magnetic field H is very weak ( H → 0 ), various noises cause **errors**.

If we consider saturation magnetization (= g_{J}J = plateau line of Fig.31 ), you will find rare earth magnetization is **also** close to an **integer** multiple of Bohr magneton (= μ_{B} ).

*(Fig.32) Cerium ion ( Ce3+ )*

*(Fig.33) Pr3+ and Nd3+ ions.*

As shown in Fig.32 and Fig.33, saturation magnetizations (= g_{J}J ) of Ce3+, Pr3+ and Nd3+ are all **close** to an **integer** multiple of Bohr magneton.

*(Fig.34) These ions are just an integer multiple of Bohr magneton.*

The cases of Tb3+, Dy3+, Ho3+ and Er3+ are very **remarkable**.

They are **just** an integer times Bohr magneton.

So you will find Bohr's quantization rule is almost **valid** even in these multi-electron rare-earth metals.

*(Fig.35) *

As shown in Fig.23, Lande g factor and LS coupling rule are completely **violated** in Eu3+ and Sm3+.

( They artificially try to introduce **other** effects to correct this discrepancy. But they **aren't** natural results at all. )

If we suppose total angular momentum of Eu3+ outer electrons are just "**3**", g_{J} becomes just "**1**".

As a result, g_{J}J becomes just **3.0** μ_{B}.

*(Fig.36)*

In the same way, if we suppose total angular momentum of Sm3+ outer electrons are just "**1.0**", g_{J} becomes just "**1.0**".

As a result, g_{J}J becomes just **1.0** μ_{B}.

*(Fig.37) "g _{J}J" becomes an integer times Bohr magneton also in rare-earth ions.*

In Pm3+ ion, simple LS coupling rule breaks down.

They insist the data of Pm3+ magnetization is **too difficult** to be gotten.

But as you see Fig.37, the magnetization of Pm3+ propably becomes almost **zero**.

This is the reason why they **cannot** measure Pm3+ magnetization.

Even though they use various ad-hoc assumptions such as "4f" **unnatural** positions of periodic table and J = L - S, three rare-earth ions such as Pm3+, Sm3+, Eu3+ still do **NOT** satisfy LS coupling rule.

This result **exposes** the **limit** of these artificial tricks.

2014/1/24 updated. Feel free to link to this site.