Top page ( correct Bohr model )

Truth of electromagnetic fields.

All true atomic structures.

*(Fig.1) "Photon" is just abstract math symbol (= a ^{†} ), without physical shape.*

Quantum mechanics claims that the classical Coulomb force is caused by photon exchange.

But **clear** mechanisms of "*attractive*" and "**repulsive**" Coulomb forces remain **unknown**.

In fact, this **photon** causing *Coulomb* force is **virtual** photon, so **NOT** real ( see this, this, this ).

Surprisingly, this **virtual** photon is superluminal tachyon, **violating** Einstein's relativity ( see this, this ).

By the way, what is the physical **shape** and *size* of a single photon ?

Unfortunately, quantum mechanics cannot answer this **basic** question.

In QED, these *photons* must be described using **abstract** Feynman diagram.

This Feynman QED has **NO** ability to show photon's **clear** picture at all.

It just shows **meaningless** math *symbols* ( see this, this ).

For example, a photon is **created** (= a^{†} ), or *annihilated* (= a ). That's all, **NOT** physics.

*(Fig.2) The sum of quark's charge becomes integer (= +e ) !? ← Too good to be true ! *

It is said that the proton is composed of three quarks with **fractional** charges.

But this *fractional* charges ( +2/3e, -1/3e ) can **NEVER** be *isolated*, so quark has **NO** real evidence. See this and this.

They just **jumped** to a conclusion that Jets (= only electrons and lights can be detected ) may be quark or gluon. In standard model, quarks are just **meaningless** math symbols with **NO** physical shapes.

Originally, fractional charges such as +2/3 (= up ) and -1/3 (= down ) quarks are completely **independent** from (= *NO* relation with ) electrons. ( Why 3/4e, 4/7e is **impossible** ? )

But the **sum** of them (= 2 × up + 1 × down quarks ) becomes just **+e** charge *proton*, which number is "*accidentally*" just the Same as **-e** charge **electron** in the universe ! ← **Too good** to be true!

On the other hand, the number of positron, which is said to be the **partner** of electron, is much **smaller** than electrons, and **unstable** antiparticles *disappear* **instantly** (= within 10^{-12} seconds ! ).

And these antiparticles **disobey** *basic* physical law such as total **momenum** conservation.

So standard model based on these quarks **lacks** reality from the beginning.

*(Fig.3) In usual β decay, W boson of "80 × proton" mass cannot be found !*

Standard model argues, in ordinary β decay, a neutron splits into a proton and very **heavy** W boson.

But the **mass** of this W boson is **80 ×** *protons*, **violating** energy conservation. See this, this.

Higgs theory insists only **Higgs** (= *125* GeV ) can give **mass** to W boson, when it is **produced**.

The lifetimes of Higgs and W boson are very **short** ( ~10^{-22} sec ), so they easily **decay**.

When we get some radioactive nuclides, β decay **can** be seen even in **ordinary** laboratory **without** gigantic accelerators (= LHC ).

But it's **impossible** to detect W and Higgs boson decay (= **200 ×** protons energy per neutron ! ) in *ordinary* laboratory, meaning W and Higgs inside LHC have nothing to do with β decay, so **unreal**.

*(Fig.4) Neutron is made of three "fractional" charges ? ← NOT confirmed !*

Though the current particle physics claims that a neutron is made of three " fractional" charge quarks.

But these fractional charges (= 1/3e ) have **never** been observed. Just *speculation*.

It is known that a **neutron** easily decays into a pair of *proton* and **electron** in 15 minutes.

So the total charge is conserved. Neutral neutron = plus proton + minus electron.

From this fact, it is quite **natural** to think a neutron is made of a proton and an electron !

Because quarks of fractional charge can **NEVER** been isolated, so they **lack** experimental proofs, *forever*.

*(Fig.5) An integer times de Broglie wavelength ← Wave ends fit each other, stable. *

As seen in Davisson-Germer experiment, each electron has **wave**-nature, and their destructive interference **affects** the electron's motions.

It is known that Bohr model gives **exact** energy levels, when its orbit satisfies an *integer* times de Broglie wavelength. "Integer" means their wave ends **fit** each other, **avoiding** destructive interference.

**Tedious** explanations "Bohr model's accelerating electron radiates energy. " are based on completely wrong assumptions. Because there are **NO** energies around only a **single** electron !

On this page, we can show the **true** mechanism of emission of **electromagnetic** waves and fields.

*(Fig.6) Why does a neutron decay into proton and electron in 15 minutes ? *

Considering a neutron easily decays into a proton and an electron, we naturally think a neutron is some transient form of a pair of *proton* and **electron**.

As I said, each electron's motion is **stable**, when its orbit satisfies an **integer** times de Broglie wavelength, which is the reason neutron easily **decays** into a *hydrogen* atom ( n → e+, e- ).

But **short** distance between a proton and an electron inside a neutron generates strong *Coulomb* **attraction**, which is thought to **postpone** this neutron decay. It takes as much as **15** minutes.

*(Fig.7) Neutron → proton + e-, Proton → neutron +e+ ? Perpetual machine ?*

**Neutron** is slightly heavier than a proton, so it can decay into a *proton* and an electron.

Surprisingly, quantum mechanics claims a *proton* can also decay into
a **neutron** and a positron !

A proton is **lighter** than a neutron, so this *inverse* process is **unreasonable**.

If so, resultant neutron decays into proton, again, and that proton decays into a neutron, again ...

This is like a "fantasy" **perpetual** machine.

So the beta plus decay emitting positron **violates** energy conservation.

Electron capture is what actually happens in positron emission.
Unstable positron itself cannot be confirmed directly. It is estimated **only** from the detection of high energy light.

*(Fig.8) Strong nuclear force is due to short-distance Coulomb attraction ! *

Standard model claims that strong nuclear force binding positive protons inside nuclei is caused by quarks and gluon. But **neither** quarks nor gluons can be isolated, so these are only **hypothetical**.

It is known that all nuclei are composed of protons and neutrons.

The point is, neutrons (= proton + **electron** ) are *indispensable* for making each nucleus.

A **electron** inside neutron is *binding* protons by a very "**short**-distance" **Coulomb** force.

When an electron is **between** two protons, total Coulomb potential energy becomes *negative*, and **stable**.

Inside tiny nucleus, these **distances** (= r ) are extremely **short**, so *Coulomb* attraction is close to **infinity** ( theoretically ), which can explain strong nuclear force, **NOT** depending on "fantasy" quark !

*(Fig.9) Nuclear structures in Deuterium and Helium-4.*

Deuterium has one proton and one **neutron**.

So it is thought that an **electron** ( inside *neutron* ) **attracts** two protons between them.

In the same way, helium-4 having two protons and two *neutrons* uses two **electrons** as a "binder". Nuclear space is very **small**, so each electron is **quivering** between protons.

On the other hand, in neutron, an electron tends to **rotate** around only **one** proton.

As I said, this orbital motion needs an **integer** times de Broglie wavelength, which breaks neutron.

**More** than one protons are indispensable for electron's *oscillations* **among** protons.

Like "**spring**" motion, this oscillation conserves energy and can be **stable**

*(Fig.10) Heavy and slower protons cannot bind many electrons stably !*

Then why is each *nucleus* always **positive**, and it can **never** be negative ?

Eelectron is very **light** and moves much **faster** than heavy proton.

So each electron can **capture** protons, before slow protons go away from other protons.

But very *heavy* protons **cannot** play a role of nuclear "binder".

When there are more electrons than protons, each electron **repelling** each other, easily **scatters**.

*Heavy* proton **cannot** go ahead of electrons to prevent them from scattering.

This is the reason why each nucleus is always **positive** including *neutrons* (= electrons ).

*(Fig.11) Graviton can cause "time dilation" ? What mechanism ? *

The present physics is "**disaster**". There is only 10 dimensional string theory as unified theory with *gravity*.
So unless you believe extradimensions and 10^{500} many worlds, you will be **persecuted**.

These quantum gravity theories just show **abstract** "graviton" operators ( see this p.3 ), **without** ability to describe concrete physical mechanism.

They insist gravity causes time dilation. But how can each "graviton" **handle** the clock time, like *magic* ? These are just **unrealitic** and strange. Graviton by *extra*dimensions are just "**imaginary**" concepts !

*(Fig.12) Field "imbalance" and their oscillation cause macroscopic gravity. *

It is known that gravity is extremely **weaker** (= about 10^{-39} times Coulomb force ! ) than other fundamental forces. So it is natural to think gravity is **NOT** fundamental force but some **macroscopic** effect of other forces.

On this page, we can explain the **Coulomb** mechanism using "field *density*" **imbalance**.

Basically, "field" tends to be **uniform** in pressure, so *high* and *low* pressure regions **attract** each other.

We can think gravity as **large**-scale *imbalance* in field's **density** (= *medium* filling all space ). If there are higher (= denser ) pressure areas, they tend to **approach** lower pressure areas.

When a large number of the same kind of field attraction **synchronize**, it generates "gravity", as macroscopic effect. This idea is more **natural** than other "fantasy" 10 dimensions and graviton.

*(Fig.13) Hydrogen atom. Proton (= nucleus) is 1840 times heavier than electron. *

Proton is about 1840 times **heavier** than an electron.

So proton (= nucleus ) is almost **at rest** in *hydrogen* atom.

But of course, from the fact of reduced mass, proton is thought to be **moving** by Coulomb force inside atoms. Due to total momentum conservation, proton's *momentum* is always just the **opposite** of electron ( MV = mv ).

According to de Broglie theory ( wavelength = h/momentum ), both electron and proton's de Broglie wavelength should be the same. But they are not.

As seen in Fig.13, proton is **confined** to about *1840* times **smaller** region than the electron's orbit. So proton is almost **at rest**.

*(Fig.14) While electron is orbiting, proton is almost "stationary". *

Their same (= opposite ) momentums should generate the same de Broglie wavelength.

But proton's moving area is 1840 times **shorter** than electron's orbit.

So the proton is considered to be **oscillating** in almost the **same** place, instead of orbiting, like a electron inside neutron.

In fact, proton's magnetic moment (= g-factor × angular momentum ) is NOT proton's Bohr magneton, which is so **weak** that proton's **actual** "spinning" can influence its magnetism.

On the other hand, large magnetic moment of electron (= 658 times proton ) indicates it is due to electron's **orbital** motion, NOT by unrealistic tiny spin.

As **light** electron has to move to the **opposite** side of nucleus in which electron's *uniform* oscillation **cannot** be kept. So an electron must *rotate* around with de Broglie waves, **not** oscillate.

*(Fig.15) After static electron, positron are created, Coulomb energy is -infinity ! *

It is known that antiparticles such as positrons do **NOT** obey total momentum conservation. In fact, these antiparticles **disobey** energy conservation law, too !

In pair production, they say, when a *photon* has **more** energy than two rest mass energies ( mc^{2} of electron ), it can generate a pair of an electron and a positron.

But an electron and a positron has the oppoosite **±** charges. So the instant both these "*static*" particles are generated, they start to **attract** each other by **Coulomb** force.

This is strange. The first photon (= light ) energy is just *2mc ^{2}*. But just after an electron and proton (= at

*(Fig.16) Total energy is NOT conserved in pair production ! *

This strong **Coulomb** attraction between *stationary* electron and positron show the total energy is **NOT** conserved in pair production. So positron **cannot** prove Einstein mass relation ( E = mc^{2} ) at all !

In fact, antiparticles such as positrons **cannot** be isolated from other abundant electrons and protons. So they just **mistake** electrons ( or protons ) **scattered** ( by Coulomb ) as antiparticles.

Because inside *infinite* different particles, we **cannot** know every single Coulomb **scattering** correctly. It is **impossible** to determine vague antiparticle only from Lorentz *magnetic* force.

In this p.6,10, they say even after several different detectors, they may **mistake** proton for positron. Of course these "mistake" probability itself is just **speculation**. No body knows the exact effects of multi-particle **scattering**.

*(Fig.17) Spin energy (= 1 K ) vs. actual ferromagnetic energy (= 1043 K ). *

Though ferromagnetism ( ex. permanent magnet of iron ) is said to be caused by spin ( see this ), this is **impossible**. Because spin-spin *magnetic* energy is **too weak** to explain this ferromagnet.

As shown in this p.6 , this p.1, this (= intro ), **spin**-spin dipole *magnetic* energy is only **0.0001** eV (= 1 K in temperature expression ), which is much lower than iron ferromagnetic temperature (= 1043 K ! ).

So they argue this ferromagnetism is caused by "**quantum** exchange" interaction (= Pauli ).

But quantum mechanics **NEVER** tries to *clarify* what this exchange force **really** is.

To express this strange exchange force, all they can do is rely on *abstract* math symbols.

In Hubbard spin ( see this p.5 ), it just shows **meaningless** electron operators, **without** ability to describe **concrete** atomic states, **useless**, forever.

*(Fig.18) Spin is too weak ! Coulomb interaction causes ferromagnet !*

It is known that iron (= Fe ), Ni, Co can be ferromagnet even at room temperature.

As I said, spin magnetic moment (= Bohr magneton ) is **too weak** (= 1 K ) to cause ferromagnet at 1043 K in Fe.

So we have to consider **other** strong interactions such as **Coulomb** forces.

In Fig.18, iron atomic raius (= 0.93 Å ) is **suited** for *synchonizing* with neighboring atoms.

Like in this section, if we designate de Broglie wavelength ( Fe = 4 ), atomic structure ( Fe = regular hexahedron ), and total 1-8th ionization energies, we can know the atomic radius ( Fe = 0.93 Å ). See this.

We use lattice constant of iron, which direction is **most** involved in ferromagnet.

*(Fig.19) Titanium electron is too influenced by other neuclei. *

In atoms such as titanium, which electron's number is smaller than iron, their atomic *radius* is **longer** due to **weak** central charge Ze+.

In this case, each valence electron is too much **restricted** by *neighboring* nuclei, which **prevents** them from responding to external magnetic fields. As a result, ferromagnet **cannot** be seen in this titanium.

*(Fig.20) Copper has 11 valence electrons. ← Too shrinked ! *

On the other hand, copper has as many as **11** valence electrons.

So its atomic radius **shrinks** too much to sinchronize with other nuclei under external magnetic field.

In Copper, each electron is **more** influenced by other valence electrons belonging to the **same** atom, which **prevents** them from synchronizing with external magnetic fields.

As a result, Fe, Ni, Co have atomic *radius*, which length is just **suited** to cause stable ferromagnet.

*(Fig.21) Magnetic flux (= Φ ) inside hole is quantized = n × h/2e. *

It is known that magnetic flux (= Φ = B × area ) passing through (= trapped by ) the hole inside superconductor become quantized ( Φ = n × h/2e ).

Quantum mechanics tries to explain this quantized magnetic flux using Cooper with **fictitious** phonons and monopole.

But we can explain this phenomenon using **real** de Broglie waves. See this p.23, this.

*(Fig.22) Lorentz force = centrifugal force. *

When the applied magnetic field is **B**, Lorentz
magnetic force becomes **evB**, which is perpendicular to electron's motion. And the centrifugal force is *mv ^{2}/r*, where v and r are velocity and radius.

We assume the orbital length is just an **integer** (= n ) times de Broglie wavelength ( = h/mv ). "h" and "m" are Planck constant and electron's mass.

*(Fig.23) *

Solving two equations of Eq.22, we have Eq.23 and Eq.24.

*(Fig.24) Quantized magnetic flux (= Φ = n × h/2e ). *

The area of circle is "πr^{2}", so magnetic flux (= Φ ) becomes B (= magnetic field ) × πr^{2}, which result gives just an **integer** times magnetic flux quantum (= h/2e ) !

*(Fig.25) *

So we can prove magnetic flux quntum using **real** de Broglie waves, **NOT** depending on "*fictitious*" quasiparticle phonon.

*(Fig.26) Quantum mechanics cannot solve many-body atoms → quasiparticle ! *

According to phys.org, even *recent* journals (= Phys.Rev., PNAS ) deal with **unreal** anyon with **fractional** charge, though fractional charge ( ex. 1/3e ) cannot be isolated.

As shown in this, this, it's **impossible** to *isolate* and observe **quarks** with fractional charge in actual experiments. So these anyons with fractional charge are just **fictitious** quasiparticles.

Quantum mechanics **cannot** solve and predict **multi**-electron atoms, and **lacks** reality in wavefunctions ( see many-worlds ), causing "camouflage" *target*, **quasiparticle**.

Quasiparticle are **fictitious** (= *unreal* ) elementary particles ( see this, this ).

But all the current condensed matter physics can do is **invent** these **unreal** particles.

Of course, we **cannot** directly observe fractional charges themselves.

They just **artificially** defined fractional charge only from *fractional* quantum Hall effect.

From the fact that electrical conductance can be a **fraction** × e^{2}/h, they **imagine** *fictitious* anyons with **fractional** charge, which is just abstract math symbol with **NO** shape. See this (1.1).

*(Fig.27) Electric current J becomes "fraction" ×e ^{2}h E by de Broglie waves ! *

In Fig.27, the electric field (= E_{y} ) is applied in y direction, and the
magnetic field (= B ) is applied in z direction. Each electron is rotating by Lorentz force.

Furthermore, this **whole** circular orbit is moving at velocity "v" in x direction.

Because applied electric force (= eE_{y} ) and magnetic force (= evB ) are **equal** to each other.

*(Eq.28)*

From Eq.27, we find the **balanced** velocity in x direcction, as shown in Eq.28.

We assume each circular orbit satisfies an **integer** times de Broglie wavelength under mgnetic field B.

*(Fig.29) Each circular orbit in Fig.28 satisfies n × de Broglie wavelength !*

Solving Eq.29, the **area** (= πr^{2} ) of each circle becomes

*(Eq.30)*

The electric current (= J ) is equal to charge e × electron density (= N ) × velocity v.

Electron density N is equal to "k" (= number of electron included in one circle ) / πr^{2}.

*(Fig.31) *

From Eq.28, Eq.30, Eq.31, we obtain the value of **fractional** (= 2k/n ) quantum Hall effect.

"**n**" is **de Broglie** wavelength of each circular orbit. "**k**" is electron's **number** in one circle.

*(Fig.32) 2 × de Broglie wavelength orbit. *

Fig.32 shows **2** × de Broglie wavelength orbit.

This includes **two** pairs of the opposite charges.

We assume each orbit contains only **one** electron ( so **k = 1**, and another is "*hole*" )

*(Fig.33) Overlapping with the same phases in de Broglie waves makes them stable. *

As shown in Fig.33, an electron of each circular orbit **always** meets the **same** phase of other de Broglie waves.

The same-phase *overlapping* between each electron and hole makes each electron's circular motion **stable**. In Fig.33, n = 2 (= de Broglie ) and k = 1 (= electron's number ), so an **integer** Hall effect.

*(Fig.34) Each electron always meets the same-phase hole ! *

As shown in Fig.33 and Fig.34, each electron **always** meets the **same**-phase *hole* of other de Broglie waves, **avoiding** destructive interference.

As a result, this electron's orbit and an integer ( 2k/n = **1** ) Hall conductance are **stable** with respect to de Broglie wave's interference.

*(Fig.35) 3 layers of electric currents gives "3" integer Hall effect ! *

Each layer of Eq.33 gives an integer "1" as the amount of electric current J in quantum Hall effect.

So when there are **three** layes, this integer becomes "**3**".

As a result, we can explain all integer quantum Hall effect !

*(Fig.36) A half of orbits are "empty" → fractional (= 1/2 ) quantum Hall effect. *

When we **reduce** the whole amount of electric current, a part of circular orbits become **empty** with NO electrons.

Due to electric *uniformity*, the interval between empty orbits becomes **equal**.

In Fig.36, a half of circular orbits contains NO electrons.

In this case, "k" = 1/2, so it gives fractional (= **1/2** ) quantum Hall effect !.

If we think various different patterns using Fig.35 and Fig.36, we can explain all kinds of *fractional* (+ integer ) quantum Hall effect, **NOT** relying on "fantasy" anyon.

*(Fig.37) Maximum orbits = midpoint lines + 2 (= two perpendicular orbits )*

On this page, we show all atomic structures and the number of *midpoint* lines ( related to **de Broglie** wavelength ) influences the number of **maximum** orbital number in the periodic table.

In this section, we prove atomic structure based on *de Broglie* wavelength can explain various molecular **bond** length.

*(Fig.38) Rough estimtion of helium-like structure. *

As shown on this page, when two electrons in the same orbit, it gives a little different energy values.

( Two perpendicular orbits give the exact helium ground state energy. ).

But **approximately**, Fig.38 simple helium-like model can be used in estimating atomic **radius**.

One orbital length is supposed to be an **integer** times de Broglie wavelength in Fig.38.

Nuclear central charge is +Ze, atomic radius is "r". "v" is electron's velocity

*(Fig.39) Coulomb force = centrifugal force. *

In Fig.39, the sum of Coulomb force ( from nucleus and another electron ) is equal to the centrifugal force.

*(Fig.40) Total energy E = sum of potential and kinetic energies. *

Total energy E is the sum of all potential and kinetic energies.

Solving Fig.38-40, we can obtain the equations of atomic radius "r" and total energy "E".

*(Fig.41) Atomic radius "r" and total energy E depend on de Broglie wavelength "n".*

Just for calculation, we use Bohr radius and ground state energy of hydrogen atom.

*(Fig.42) Hydrogen Bohr radius and ground state energy. *

Beryllium has **two** valence electrons in n = **2** orbits.

We suppose the sum of 1st and 2nd ionization energies of Be is total energy E in n = 2 orbits.

Inserting E = **27.5** eV (= sum of 1st, 2nd ionization energies ) and n = **2** (= de Broglie wavelength ) into the result of Fig.41, we obtain Fig.43.

*(Fig.43) Beryllium effective central charge Z = 2.26. ← good !*

As seen in Fig.43, when we input n = 2 as de Broglie wavelength, it gives **reasonable** effective central charge Z = **2.26**.

This central charge is the sum of +4e Be nucleus and two 1s electrons. ( +4e -2e = **+2e** )

So the sum of them should be close to "**2**". Due to electron 1s shell **gap**, it's a little *bigger* than 2.

*(Fig.44) Atomic radius when 2 × de Broglie wavelength in Be. *

Furthermore, when Be valence electrons are in **2** × de Broglie wavelength orbits, their orbital radius becomes **1.05 Å**, which can explain **experimental** bond length properly.

*(Fig.45) Experimental Be-Be bond length = 2.22 Å *

In this site, the experimental value of Be-Be bond length is **2.22 Å** ( 1 Å = 1.0 × 10^{-10} m ).

This value is almost **2.10 Å** = 2 × **1.05 Å** of calculation value in
Fig.44 !

So the concept of de Broglie wavelength is **valid** in Beryllium **molecular** bond !

*(Fig.46) n = 1 → Central charge Z is too small, and radius is too short !*

Next we suppose Beryllium valence electrons are in **1** × de Broglie wavelength orbits and input **n = 1** instead of n = 2. The same ionization energy (= 27.5 eV ) is used as total energy E.

As shown in Fig.46, in this case, the effective central charge becomes **too small** ( Z = *1.25* ).

Central charge **must** be **bigger** than 2, so this result is **inconsistent** with the actual Beryllium !

Orbital radius is also **too short** (= 0.53 Å ).

2 × 0.53 Å = *1.06 Å* is much **shorter** than the actual bond length ( 2.22 Å ).

*(Fig.47) Central charge Z is too big, and radius r is too short in n = 3 ! *

Next, if we input n = **3** instead of n = 2 into these equations.

This case also gives **unreasonable** results.

The effective central charge Z = **3.26** is too **big**, with respective to "2".

And atomic radius **1.58** Å is too **long** to explain actual Be-Be bond length (= 2.22 Å ).

As a result, we can prove two valence elecrons of Beryllium in **2** × de Broglie wavelength orbit !

*(Fig.48) Carbon = four valence electrons. *

Carbos and silicons have **four** valence electrons.

So we assume regular tetrahedron like Fig.48 to get atomic radius.

*(Fig.49) Total potentian energy V. *

Total potential energy is the sum of energies among Ze+ center and four electrons.

*(Fig.50) Coulomb force toward center acting on each electron. *

Fig.50 is the Coulomb force toward center acting on each electron.

We suppose this force is equal to **centrifugal** force.

*(Fig.51) n × de Broglie wavelength orbit ! *

Again we suppose each orbit satisfies an **integer** (= n ) times de Broglie wavelength.

*(Fig.52) Total energy E and radius r in carbon-like atoms *

From Fig.49-51, we obtain total energy E and radius r, as shown in Fig.52.

*(Fig.53) True atomic radius r and effective central charge Z in Carbon. *

In this table, we suppose the sum of 1-4th ioniztion energies as total energy of **n = 2** carbon valence electrons.

Inputting E = **148** eV and n = **2** into Fig.52, we get radius **0.64** Å and Z = **4.2**, which is very *good*.

*(Fig.54) Experimental values of C-H, C-C bond length. *

As shown on this and this, **experimental** values of H-C and C-C bond length is **1.09** Å and **1.54** Å

Carbon atomic radius (= 0.64 Å ) **fits** these experimental values very well.

Hydrogen atomic radius (= 0.45 Å ) shrinks, and is a little *shorter* than Bohr radius (= 0.53 Å ) due to attraction of carbon's nucleus.

*(Fig.55) If carbon valence electrons are in 1 or 3 de Broglie wavelength, wrong radius ! *

If we suppose carbon's valence electrons are in *1* or *3* de Broglie wavelength orbits, it gives "too **short**" or "too **long**" atomic radius, compared with the experimental bond lenghs.

Effective central charge (= Z ) are also "too **small**" (= 2.5 ) or "too **large**" (= 5.8 ) compared with the proper value of "**4**". So we find four valence electrons of carbon are in **2** × de Broglie wavelength orbits.

*(Fig.56) Silicon radius and central charge, when 3 × de Broglie wavelength. *

When we suppose four valence electrons of silicon are in **3** × de Broglie wavelength orbits, it gives **good** results. Effective central charge (= **5.05** ) is a little bigger than carbon, due to larger gap.

*(Fig.57) When silicon valence electrons are in 2 or 4 de Broglie wavelength orbits .. *

If we suppose silicon valence electrons are in *2* or *4* de Broglie wavelength orbits, their effective charges become "too **small**" ( Z = 3.6 ) or "too **big**" ( Z = 6.4 ), compared with **4**.

*(Fig.58) Experimental value of Si-Si bond length. *

As shown in this, the exeprimental value of Si-Si bond length is **2.33 Å**.

Atomic radius in 3 × de Broglie wavelength is just **consistent** with this experimental bond length.

1.15 × 2 = **2.30** Å = **2.33** Å, when **3** × de Broglie wavelength !

On the other hand, radii in 2 or 4 de Broglie wavelength are **inconsistent** with the experimental values !

*(Fig.59) Why silicon's effective central charge (= 5.05 ) is bigger than carbon (= 4.2 ) ? *

Though both carbon and silicon have the same four valence electrons, their effective central charges (= Z ) are **different**. Because **larger** *gaps* are open in silicon inner shell.

As shown in Fig.59, **gap** between n = 1 electrons are much **smaller** than that of n = 2 electron shell, which difference gives **larger** effective central charge Z in silicon.

*(Fig.60) Regular octahedron = Oxygen-like. *

Oxygen and sulfur have **six** valence electrons, so we can suppose they have "**octahedral**" electron distribution.

*(Fig.61) Total potential energy V in oxygen-like atoms. *

Fig.61 is total potential energy V in oxygen-like atoms.

*(Fig.62) Force acting on each electron *

Fig.62 is Coulomb force acting on each electron in oxygen-like atoms.

We suppose this is equal to **centrifugal** force.

*(Fig.63) *

Again we suppose each orbit is n × de Broglie wavelength.

*(Fig.64) Total energy E and radius of oxygen-like atoms *

Solving Fig.61-63, we get Fig.64.

*(Fig.65) Oxygen ( Sulfur ) is 2 (3) × de Broglie wavelength. *

When we suppose oxygen valence electrons are **2** × de Broglie wavelength orbits, it gives *good* central charge ( Z = **6.27** ), which is close to "**6**".

Due to larger gap in inner shell, the effective central charge Z of sulfur becomes a little **larger** (= 7.18 ) than oxygen.

*(Fig.66) *

Using calculation results by de Broglie wavelength in oxygen-like and silicon, we can explain experimental bond length, as shown in Fig.66.

*(Fig.67) When two atoms approach too close, repulsion among "electrons" get bigger. *

What determines each molecular bond length ?

On this page, we prove Coulomb **repulsion** among electrons becomes stronger, when the distance between two atoms is **shorter** than the experimental value in H2, using Virial theorem.

As two nuclei are **closer** to each other, electrons in both atoms are **attracted** toward the *center* of two atoms. This repulsion among electrons gets **stronger** in shorter bond length.

*(Fig.68) H - F bond is very tight and stable. Why ? *

It is known that *hydrogen* tends to be **attracted** and bound to *fluorine* tightly and stably.

Fluorine has **seven** valence electrons and large effective central charge, so its radius is very short.

So there are only very **small** gaps among seven valence electrons in F.

To avoid electrons and pass through this small gap, another atom must have a **small** number of valence electrons.

This is the reason why **hydrogen** atom with only **one** electron tends to be bound to fluorine *tightly*.

*(Fig.69) F-F bond is very unstable due to strong repulsion among electrons. *

When two fluorines are bound to each other, this F-F bond tends to be **unstable**.

Because both these two fluorines have very **small** electrons' gaps.

To **cancel** strong internuclear (= 7e × 7e ) *repulsions*, both-side electrons have to **approach** another nucleus. But very **small** gaps among valence electrons in F **prevent** them.

This is the **true** mechanism of molecular bond length.

If you run this java program, you can know radius (= nuc ) and central charge (= Z ).

Atom | electrons | de Broglie | Radius | charge Z |
---|---|---|---|---|

H | 1 | 1 | 0.45 Å | Z = 1.0 |

Be | 2 | 2 | 1.05 Å | Z = 2.2 |

C | 4 | 2 | 0.64 Å | Z = 4.2 |

N | 5 | 2 | 0.53 Å | Z = 5.2 |

O | 6 | 2 | 0.46 Å | Z = 6.3 |

F | 7 | 2 | 0.40 Å | Z = 7.3 |

Al | 3 | 3 | 1.39 Å | Z = 4.0 |

Si | 4 | 3 | 1.15 Å | Z = 5.0 |

P | 5 | 3 | 0.98 Å | Z = 6.1 |

S | 6 | 3 | 0.86 Å | Z = 7.2 |

Cl | 7 | 3 | 0.76 Å | Z = 8.2 |

Atom | electrons | de Broglie | Radius | charge Z |
---|---|---|---|---|

Na | 1 | 3 | 2.59 Å | Z = 1.8 |

Mg | 2 | 3 | 1.74 Å | Z = 2.9 |

K | 1 | 4 | 3.74 Å | Z = 2.2 |

Ca | 2 | 4 | 2.60 Å | Z = 3.5 |

Ti | 4 | 4 | 1.63 Å | Z = 6.1 |

Cr | 6 | 4 | 1.18 Å | Z = 8.8 |

Mn | 7 | 4 | 1.04 Å | Z = 10.2 |

Fe | 8 | 4 | 0.93 Å | Z = 11.5 |

Co | 9 | 4 | 0.85 Å | Z = 12.8 |

Ni | 10 | 4 | 0.77 Å | Z = 14.2 |

Cu | 11 | 4 | 0.71 Å | Z = 15.5 |

Zn | 12 | 4 | 0.66 Å | Z = 16.8 |

Atom | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | Total |
---|---|---|---|---|---|---|---|---|

H | 13.606 | - | - | - | - | - | - | 13.606 |

Be | 9.323 | 18.211 | - | - | - | - | - | 27.534 |

C | 11.260 | 24.383 | 47.888 | 64.494 | - | - | - | 148.025 |

O | 13.618 | 35.121 | 54.936 | 77.414 | 113.899 | 138.120 | - | 433.108 |

F | 17.423 | 34.971 | 62.708 | 87.140 | 114.243 | 157.165 | 185.186 | 658.836 |

Si | 8.152 | 16.346 | 33.49 | 45.142 | - | - | - | 103.13 |

P | 10.487 | 19.77 | 30.202 | 51.444 | 65.025 | - | - | 176.928 |

S | 10.36 | 23.338 | 34.79 | 47.222 | 72.595 | 88.053 | - | 276.358 |

Cl | 12.968 | 23.814 | 39.61 | 53.465 | 67.8 | 97.03 | 114.196 | 408.883 |

2015/6/7 updated. Feel free to link to this site.