Vortex-ring-fractal Structure of Hydrogen Atom

Pavel Ošmera

Abstract—This paper is an attempt to attain a new andprofound model of the nature’s structure using a vortex-ring-fractal theory (VRFT). Scientists have been trying to explainsome phenomena in Nature that have not been explained sofar. The aim of this paper is the vortex-ring-fractal modelingof elements in the Mendeleev’s periodic table, which is not incontradiction to the known laws of nature. We would like tofind some acceptable structure model of the hydrogen as avortex-fractal-coil structure of the proton and a vortex-fractal-ring structure of the electron. It is known that planetary modelof hydrogen is not right, the classical quantum model is tooabstract. Our imagination is that the hydrogen is a levitationsystem of the proton and the electron.

Index Terms—Model of hydrogen atom, vortex-ring-fractalstructures, covalent bond

I. INTRODUCTIONThe electrical force decreases inversely with the square

of distance between charges. This relationship is calledCoulomb’s law. There are two kinds of “matter”, which wecan call positive and negative. Like kinds repel each other,while unlike kinds attract – unlike gravity, where onlyattraction occurs [8]. When charges are moving theelectrical forces depend also on the motion of charges in acomplicated way [1].

Fractals seem to be very powerful in describing naturalobjects on all scales. Fractal dimension and fractal measureare crucial parameters for such description. Many naturalobjects have self-similarity or partial-self-similarity of thewhole object and its part [3].

Most of our knowledge of the electronic structure ofatoms has been obtained by the study of the light given outby atoms when they are exited. The light that is emitted byatoms of given substance can be refracted or diffracted intoa distinctive pattern of lines of certain frequencies andcreate the line spectrum of the atom.

The careful study of line spectra began about 1880. Theregularity is evident in the spectrum of the hydrogen atom.The interpretation of the spectrum of hydrogen was notachieved until 1913. In that year the Danish physicist NielsBohr successfully applied the quantum theory to thisproblem and created a model of hydrogen. Bohr alsodiscovered a method of calculation of the energy of thestationary states of the hydrogen atom, with use of Planck’sconstant h. Later in 1923 it was recognized that Bohr’sformulation of the theory of the electronic structure ofatoms to be improved and extended. The Bohr theory didnot give correct values for the energy levels of helium atomor the hydrogen molecule-ion H2

+, or of any other atom withmore than one electron or any molecule.

Manuscript received July 14, 2009. This work was supported in part bythe Czech Ministry of Education under Grant No: MSM 00216305529����������� ���� �� � �������� ��� � ��� ����� ����� �� �� ���102/09/1668: Control Algorithm Design by Means of EvolutionaryApproach.

P. Ošmera is with the European Polytechnical Institute Kunovice,Osvobození 699, 686 04 Kunovice, Czech Republic(e-mail: osmera @fme.vutbr.cz).

During the two-year period 1924 to 1926 the Bohrdescription of electron orbits in atoms was replaced by thegreatly improved description of wave mechanics, which isstill in use and seems to be satisfactory. The discovery by deBroglie in 1924 that an electron moving with velocity v hasa wavelength ����mev [2]. The theory of quantummechanics was developed in 1925 with the Germanphysicist Werner Heisenberg. Early in 1926 an equivalenttheory, called wave mechanics, was independentlydeveloped by Austrian physicist Ervin Schroedinger.Important contributions to the theory were also made by theEnglish physicist Paul Adrien Maurice Dirac. The mostprobable distance of the electron from the nucleus is thusjust the Bohr radius rB (ro); the electron is, however, notrestricted to this distance. The electron is not to be thoughtof as going around the nucleus, but rather as going in andout, in varying directions, so as to make the electrondistribution spherically symmetrical [2].

Matter is composed of tiny atoms. All the atoms of anyelements are identical: they have the same mass and thesame chemical properties. They differ from the atoms of allother elements. Twenties-century X-ray work has shownthat the diameters of atoms are of the order 0.2 nm (2x10-10

m). The mass and the positive charge are concentrated in atiny fraction of the atom, called nucleus. The nucleusconsists of protons (p) and neutrons (n). Protons andneutrons are made up of smaller subatomic particles, suchas quarks. Both protons and neutrons have a massapproximately 1840 times greater than an electron (e). Themore energy an electron has, the further it can escape thepull of the positively charged nucleus. Given sufficientenergy, an electron can jump from one shell to higher one.When it falls back to a lower shell, it emits radiation in theform of a photon.

II. THE SPIN OF THE ELECTRON

It was discovered in 1925 that the electron hasproperties corresponding to its spin S. It can be described asrotating about an axis of a ring structure of the electron (seeFig.1 and Fig.7) [7]. The spin of the electron is defined asangular momentum [8]:

������������

×=(1)

For the spin on axis z:

eee

z vrN

mNS = (2)

where me is the mass of the electron, re is the radius of theelectron and N is number of substructures inside thestructure of the electron. In [8] the formulae for radius re ofthe electron are:

20

2

2 1

8 eee vm

er ⋅=

επ(3)

mmee vvvvππ22 ±=±== (4)

Proceedings of the World Congress on Engineering and Computer Science 2009 Vol IWCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-17012-6-8 WCECS 2009

h

evm

0

2

4ε= (5)

where er is mean radius, v is mean velocity of the electron

[8], ev is mean velocity of the subelectron, vm is maximum

translation velocity of the electron and vme is maximumvelocity of the subelectron -1e (maximum rotational velocityof the electron) if the electron has distance do (see Fig.2 andFig.7) and minimum energy Eqo , see equation (33) or [8].

Mean radius er is:

=⋅= 2

2

02

2

48 mee vm

er

πεπ

(6)The spin Sz on axis z:

=±== emeeeez rvmrvmSπ2

(7)where

�

�±=�

�

(8)

The result in (8) is in coincidence with the generallyequation for the spin, where ms is spin quantum number [8].

We can suppose that a fractal structure of the electron has�������� � � � �o (see Fig.1a):

2222

λλλπ =⋅== N

N

Nr oe (9)

or from Fig.1b

λπ =⋅ er22 πλ4

=er (10)

where N is number of subparts (for example: number ofsubelectrons). Formulae (7) with (10) lead to:

=⋅⋅⋅=⋅== eee

eeez rvN

mN

hrvmS

π22

1

(11)

vm

h

vm

hN

vm

h

eeo

ee

±==⋅== λλ (12)

vve = (13)

It can be an explanation of de Broglie’s equation for awavelength ����mv because we suppose that the electronenergy Et of translation movement is the same as therotational energy Er of the rotating electron [8].

Fig. 1 The electron that is moving with velocity v has awavelength ����mev (de Broglie’s equation):� ������� ������ � ��� �o in the fractal-ring

structure of the electron,b) An inner ring of the electron with spin quantum

number: 1/2 (twice around annuloid to match).It is the 21-multiple “double-helix-line” structure

(here only one “double helix” with markers 1,2).

III. MODEL OF HYDROGEN ATOM

In a new model of the hydrogen atom with a levitatingelectron [7], [8] there is attractive (electric) force F+ and(magnetic) repellent force F- :

−=−= −+ 4

2

2

2 1

4 d

d

d

eFFF o

oπε(14)

The hydrogen atom can have the electron on left side or onright side (see Fig.4a, 4b). The attractive force F+ isCoulomb’s force. The repellent force F- is caused withmagnetic fields of the proton and the electron (see Fig.2). Adistance between the electron and the proton is d in (14).The electron moves as “Yo-Yo” (see Fig. 3) The Bohr radius rB (or ro) has the same size as the

distance mdo111029.5 −⋅≈ [2] in our vortex-fractal-ring

model [8].

22

16

32 2

20

4

220

0

2o

ee

d

em

h

e

h

m

e ==⋅= εεε

�� smh =±=⋅±=

2

1

22

1

π

=⋅⋅±=2

20

0

2

24

2

em

h

h

em

e

e

εεπ

=⋅=2

0

2

4

1

8 me vm

e

ε

πλ

4⋅⋅⋅= e

e vN

mN

Proceedings of the World Congress on Engineering and Computer Science 2009 Vol IWCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-17012-6-8 WCECS 2009

Fig.2 The levitating electron in the field of the proton (thefractal structure model of hydrogen H is simplified[8]).

Fig.3 Displacement velocity v and rotation velocity ve ofthe electron on Fig.2

Fig.4 Distances do = ro between proton and electron [8]a) left side orientation of hydrogenb) right side orientation of hydrogenc) the hydrogen molecule-ion H2

+

d) the hydrogen molecule H2 with covalent bond

To calculate quantum model of hydrogen we use radius re

of the electron, which was derived in [8]:

2

2

2

2

4 e

o

e

oe v

v

m

er

πµ= (15)

for 2

cvo =

ooo

cv

µε2

1

2

22 == (16)

Fig.5 Vortex-fractal structure of the neutron [8]

Fig.6 Vortex-fractal structure of the proton [8]

=⋅=⋅=2

2

2

2

2

2

2

2

244 ee

o

e

o

e

oe v

c

m

e

v

v

m

er

πµ

πµ

20

2

2 1

8 ee vm

e ⋅=επ

(17)

IV. STRUCTURE OF ELECTROMAGNETIC FIELD

Electric lines or rays with different substructures(positron and electron substructures) repel each other (seeFig.8). The same types of substructures (lines) attract eachother and create braids. The same behavior has magneticfield - magnetic lines or rays with electron and positronsubstructures repel each other (see Fig.9). Electric lines are

do d

v , ve +vm

+vme

-vm

-vme

v= ve=0

Proceedings of the World Congress on Engineering and Computer Science 2009 Vol IWCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-17012-6-8 WCECS 2009

formed from electron subparts (-4e) or/and positron subparts(-4��� ������ ���� � � �� ��� � �� ���� �� ����� �� �

-3e)or/and positron subparts (-3

��� ���� ���� � ��� � �

perpendicular to magnetic lines or rays. The ray is a “braid”of lines (see Fig.8).

Fig.7 Vortex-fractal ring structure of the electron [8], [10]

Fig.8 Structure of the electric field

Fig.9 Structure of the magnetic field

Energy Eo of the quite and free electron 0e [8], which hasvelocity v=0 and quite mass meo, can be calculated fromkinetic energy of their subparts: subelectrons -1e withvelocity v-! (ve), subsubelectrons -2e with velocity v-2 (vo),subsubsubelectrons -3e with velocity v-3,subsubsubsubelectrons -4e with velocity v-4 , andsubsubsubsubsubelectrons -5e with velocity v-5 (see Fig.11):

If velocities of substructures are [7], [8]:

�==���

2543

cvvvvo ≈=== −−− (18)

then their inner kinetic energy is:

++≈ −32

3322

2 2

1

2

1Nv

N

mNv

N

mE eo

oeo

o

(19)2cmE eoo ≈ (20)

This result is in coincidence with the well-knownEinstein’s law. Mass is a measure of the amount of matter inan object. The object’s inertia is proportional to its mass,and Einstein showed that mass is actually a very compactform of energy. Figure 10 explains a particle structure of the photon and awave behavior of the light ray, which consists from morephotons arranged in the series (sequence, string). A vortexpair is created from “bath” vortex VB an a “tornado’ vortexVT with flow of energy E [8].

Fig.10 Structure of light as a ring particle or a wave energystructure

VI. QUANTUM MODEL OF HYDROGEN

To calculate a quantum model of hydrogen we use radius re

of the electron, which was derived in [8]:

2

2

2

2

4 e

o

e

oe v

v

m

er ⋅=

πµ

(22)

2

2

2

2

422

e

o

e

oe v

v

m

er ⋅=

πµππ (23)

Let us assume that [18]:

ooo

cv

µε2

1

2

22 == (24)

On a circumference of a circle with re (see Fig.1) have to ben �� � ��������������� ���� ��� ����ev (n is quantumnumber) [8]:

==== 2

2

2

22

2

2

2

2 1

42422

eeoee

o

e

o

e

oe vm

e

v

c

m

e

v

v

m

er

πεπµ

πµππ

cmNvN

mNv

N

meo

eoeo =++ −−52

5542

44 2

1

2

1

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ISBN:978-988-17012-6-8 WCECS 2009

(25)

nhv

e

eo

=1

2

2

πε(26)

nhv

e

mo

=22

2 ππε

(27)

where vm is maximum velocity of the electron if the electronhas distance do and minimum energy Eqo [8]:

h

e

nv

omn ε4

1 2

= (28)

Energy Et of translation movement of the electron is:

22

4

22

16

1

2

1

h

em

nvmE

o

emet ε

== (29)

Energy Er of rotation of the electron is due to a maximalsymmetry in the multiple double-helix structure of theelectric field has the same size as energy Et of translationmovement. Total energy Eq of rotating and moving electronis:

=+= rtq EEE

(30)

For quantum number n=1

���

���

�

�

������

��

�

≈=ε

(31)

eVd

e

d

eEE

oooooqo 6.13

1

84

31

64

3 22

≈===πεπε

(32)

ooo

eqo d

e

h

emE

1

88

2

22

4

πεε== (33)

mem

hd

e

oo

112

2

1029.5 −⋅≈= ε(34)

It is the same result as Bohr obtained [2] but with quitedifferent hydrogen model.

VI. CONCLUSIONS

It seems that gravitation lines (from graviton fibers) arein the same axes as levitating electrons [8]. Gravitation linesrepel each other and due to the two bodies are attracted. Thegravitation lines are braids, which are created from gravitonfibers. A quantum-foam-space that is full of graviton fibersand ring structures we would like to call gravitonum (or

shortly gravum) to distinguish from the terms: vacuum,ether etc. The gravum is a space where are highly organizedand lower organized substructures. It means they are in adifferent level of self-organizational state (see Fig.11).Smaller substructures can create dark matter and darkenergy. The name osmeron (see Fig.11) was derived fromthe name “Osmera” of Egyptian deity with 4 pairs of godsas primary creative forces (from a chaos beginning).Osmerons are very small. It explains why they haveimmeasurable size and mass. From the basic osmeronsubstructure the electron, proton and neutron can be created[8].

The exact analysis of real physical problems is usuallyquite complicated, and any particular physical situation maybe too complicated to be analyzed directly by solving thedifferential equations. Ideas as the field lines (magnetic andelectric lines) are very useful for such purposes. We thinkthey are created from self-organized subparts of gravum(see Fig.11, Fig.8 and Fig.9). A physical understanding is acompletely nonmathematical, imprecise, and inexact, but itis absolutely necessary for a physicist [1]. It is necessary tocombine an imagination with a calculation in the iterationprocess [4]. Our approach is given by developing graduallythe physical ideas – by starting with simple situations andgoing on towards the more and more complicated situations.But the subject of physics has been developed over the past200 years by some very ingenious people, and it is not easyto add something new that is not in discrepancy with them.The vortex model (see Fig.7) of the electron was inspired byvortex structure in the PET-bottle experiment with one holeconnector ($3 souvenir toy, Portland, Oregon 2004) [7], ourconnector with 2 or 3 holes [7], [8] and levitating magnet“levitron” (physical toy). The “ring theory” is supported byexperiments in [9] and [11] too. Now we realize that thephenomena of chemical interaction and, ultimately, of lifeitself are to be understood in terms of electromagnetism,which can be explain by vortex-ring-fractal structure indifferent state of self-organization inside gravum.

ACKNOWLEDGMENT:This work has been supported by Czech Ministry of

Education No: MSM 00216305529 Intelligent Systems in���������� �� � �� �� ������������ �������

Algorithm Design by Means of Evolutionary Approach.

REFERENCES

[1] Feynman R.P., Leighton R.B., Sands M.: The FeynmanLectures on Physics, volume I-III, Addison-Wesleypublishing company, 1977

[2] Pauling L.: General Chemistry, Dover publication, Inc, NewYork, 1988

[3] Ian Stewart: Does God play Dice, The New Mathematics ofChaos, Penguin, 1997

[4] Li Z., Halang W. A., Chen G.: Integration of Fuzzy Logic andChaos Theory; paragraph: Osmera P.: Evolution ofComplexity, Springer, 2006 (ISBN: 3-540-26899-5)

[5] Huggett S.A., Jordan D.: A Topological Aperitif, Springer-Verlag, 2001

[6] Thorne, K.S.: Black Holes & Time Warps, W.W.Norton &Company, Inc., 1994

[7] Ošmera, P.: The Vortex-ring Structures and Mendeleev’sTable, Proceedings of WCECS2007, October 24-26, 2007,San Francisco, USA, 152-157

22

4

2 8

12

h

em

nE

o

et ε

==

vm

hn

vm

hnn

eee 2

1

2

1

2=== λ

Proceedings of the World Congress on Engineering and Computer Science 2009 Vol IWCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-17012-6-8 WCECS 2009

[8] Ošmera, P.: Vortex-fractal-ring Structure of Hydrogen,Proceedings of MENDEL 2009, Brno, Czech Republic (2009)81-92.

[9] Mauritsson J.: online.itp.ucsb.edu/online/atto06/mauritsson/

[10] �apková T.: Modeling Fractal Objects in Blender, bachelorwork, UTB, Zlín, 2009

[11] Lim, T.T.: serve.me.nus.edu.sg/limtt/

Fig.11 Fractal-ring structure of Universe (ring theory) [5], [6], [8]

Proceedings of the World Congress on Engineering and Computer Science 2009 Vol IWCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-17012-6-8 WCECS 2009