Top page ( correct Bohr model )

Truth of electromagnetic fields.

All true atomic structures.

*(Fig.1) 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.2) 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.3) 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.4) 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.5) 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.6) 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.7) 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.7 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.8) Coulomb force = centrifugal force. *

In Fig.8, the sum of Coulomb force ( from nucleus and another electron ) is equal to the centrifugal force.

*(Fig.9) Total energy E = sum of potential and kinetic energies. *

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

Solving Fig.7-9, we can obtain the equations of atomic radius "r" and total energy "E".

*(Fig.10) 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.11) 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.10, we obtain Fig.12.

*(Fig.12) Beryllium effective central charge Z = 2.26. ← good !*

As seen in Fig.12, 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.13) 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.14) 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.13 !

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

*(Fig.15) 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.15, 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.16) 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.17) Carbon = four valence electrons. *

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

So we assume regular tetrahedron like Fig.17 to get atomic radius.

*(Fig.18) Total potentian energy V. *

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

*(Fig.19) Coulomb force toward center acting on each electron. *

Fig.19 is the Coulomb force toward center acting on each electron.

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

*(Fig.20) n × de Broglie wavelength orbit ! *

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

*(Fig.21) Total energy E and radius r in carbon-like atoms *

From Fig.8-20, we obtain total energy E and radius r, as shown in Fig.52.

*(Fig.22) 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.21, we get radius **0.64** Å and Z = **4.2**, which is very *good*.

*(Fig.23) 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.24) 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.25) 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.26) 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.27) 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.28) 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.28, **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.29) Regular octahedron = Oxygen-like. *

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

*(Fig.30) Total potential energy V in oxygen-like atoms. *

Fig.30 is total potential energy V in oxygen-like atoms.

*(Fig.31) Force acting on each electron *

Fig.31 is Coulomb force acting on each electron in oxygen-like atoms.

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

*(Fig.32) *

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

*(Fig.33) Total energy E and radius of oxygen-like atoms *

Solving Fig.30-32, we get Fig.64.

*(Fig.34) 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.35) *

Using calculation results by de Broglie wavelength in oxygen-like and silicon, we can explain experimental bond length, as shown in Fig.35.

*(Fig.36) 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.37) 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.38) 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.