−

= =

−

(64)

This expression, with Equation (33), |U| = |qQ|/(4πεr), leads to:

em

0

2||

u

A

λ

=U

. (65)

Equation (65) is confirmed by Equations (17) and (57).

4.4. Synchronized (Stationary) States of Atoms

A minimum of two separate oscillating processes are performed simultaneously within an atom, i.e., the circular

motion of electrons around the nucleus and oscillation of electromagnetic wave energy [3]. The time period of

M. Perkovac

246

one circular tour of electrons around the nucleus is Te = 2rπ/v = 1/f, where f is the frequency of circulation of

electrons around the nucleus. The duration of the period of the electromagnetic wave is Tem = 1/ν. Hence, ν/f =

2πνr/v. Using Equation (64), as well as v/c = β and

λ

ν = uem follows, Figure 4:

e

em 0

0

||

,

4

4.

||

TqQ

Tf A

A

f

qQ

ν

ε

εβ

ν

= =

=

v

c

(6 6 )

Long term existence of the rotation of electrons and long term existence of the electromagnetic wave in the

atom (stationary state) is only possible if there is synchronism between them (synchronously stationary state)

[3,4]. Namely, to be coherent with the active power of the electromagnetic wave in an atom, the electron needs

to oscillate (i.e., rotate) with dual frequency of the wave, because the active power of wave oscillates with dual

frequency 2

ω

= 2(2πν), (this will be further discussed in Sub–Heading 4.5). This means that in the synchron-

ously stationary state of the atom, the time period of electron rotation Te is a half period of Tem (or, for reasons of

synchronism, is n±1-multiple of a half period of Tem), i.e., Te = n±1Tem/2, where n+1 = 1, 2, 3,…is ordinal number

of stationary orbits in the atom, when an electron moves away from the nucleus (or n–1 = 1, 2, 3,…, when the

electron approaches the nucleus). Equation (66) gives the speed of electron in a synchronously stationary state [4]

(compare with [15]):

n10

1| |

2

qQ

A

n

ε

±

=v

. (67)

The Equations (31) and (67) give the radius of the electron orbits in the synchronously stationary states:

( )

22

2n

0

1

n

1(/)

||

Ac

rn m qQ

ε

±

−

=π

v

. (68)

From Equations (66), (67) and (68) follows [4]:

( )

2

n323 2

1n

0

1 ||

41(/)

qQ m

fAc

n

ε

±

=−

v

, (69)

( )

2

n223 2

1n

0

1 ||

81( /)

qQ m

Ac

n

νε

±

=−

v

(70)

and, [3],

n

n1

2fn

ν

±

=

. (71)

The total mechanical energy of an electron Wn = –Eem(n) follows from Equations (54) and (70):

( )( )

2

22

2 22

n22

22 222 2

11

nn

00

1 ||1 ||( ).

81( /)81( /)

qQ mqQ m

W mcmc

Ac Ac

nn

εε

±±

=− −++

−−

vv

(7 2)

For energies much smaller than mc2:

( )

2

n222 2

1n

0

1 ||

81( /)

qQ m

WAc

n

ε

±

≈− −v

. (73)

If assume the maximum speed of electron is equal to the speed of light in a given medium, i.e., according to

Equation (28) vmax = uem = F(χ)/(μ*ε*)1/2 (to increase the speed of electron should be n±1 = 1/nmax) from Equa-

tions (51) and (67) we get:

2

0

max 2()

s

nF

Z

χ

=

. (74)

M. Perkovac

247

From Equation (74) follows the greatest possible atomic number Zmax when nmax is minimal and F(χ) is max-

imal, actually when nmax = 1 and F(χ) = 1, i.e.,

22

0

max 0

max

2() 2

s

Z Fs

n

χ

= =

. (75)

4.5. Wave Equations of the Electromagnetic Wave in the Atom

Wave equations of electromagnetic wave in an atom are expressed by Equations (14) and (15). If we insert

phase velocity uem, expressed from Equation (62), i.e.,

()

2

2

2

em 2

1

212

eV/ mc

eV

umeV /mc

−

=−

, (76)

in Equations (14) and (15), we obtain

( )

( )

22

222

2

22

222

2

21 20,

1

21 20.

1

meV/mc

eV t

eV/ mc

meV/mc

eV t

eV/ mc

−∂

∇− =

∂

−

−∂

∇− =

∂

−

E

E

H

H

(77)

Wave Equations (14), (15) or (77) have a lot of solutions. We will apply the solutions that correspond to the

atom and the transmission line, i.e. , to the LC network. These solutions are standing waves [6,8]:

()

()

x0

0

y

2

(,)sincos,

2

(,)cossin,

/

z

E ztEt

Ez

H ztt

ω

λ

ω

λ

µε

π

=

π

= −

(78)

where E0 is the maximum value, i.e., the amplitude of electric field strength E, Ex(z,t) is the x-component of the

electric field strength dependent on the z-axis and the time t, and Hy(z,t) is the y-component of the magnetic field

strength H dependent on the z-axis and the time t,

ω

= 2πν. All mathematical operations we perform for the

y-component of the magnetic field Hy(z,t) can be performed for the x-component of the electric field Ex(z,t) in

the same way.

In the standing waves (78) the energy oscillates between the electric and magnetic form. The electrical energy

is at a maximum when the magnetic energy is zero, and vice versa. Furthermore, the standing wave transfers no

energy through the space because the average active power of the wave is equal to zero. The current value of the

active power oscillates in both directions, + and – of z axis, with dual frequency 2ω from point to point of z axis

[8]. As already mentioned, this is why (for the maintenance of stationary state of the atom) the electron has to

rotate twice as fast compared to the lower harmonics (n+1), or twice as fast compared to the upper harmonics

(n–1), i.e., f = 2(ν/n±1) in accordance with Equation (71).

If we use the second derivative with respect to z of the y-component Hy(z,t) of the magnetic field strength in

Equation (78), we get: ∂2Hy(z,t)/∂z2 + (2π/

λ

)2Hy(z,t) = 0. After inclusion of the wavelength

λ

from Equation (60)

we obtain:

22 23

yy

2 224

0

(,) 8(12 )(,) 0

(1 )

H ztmeVeV /mcHzt

zAeV/ mc

∂π−

+=

∂−

. (79)

If eV/mc2<<1, then eV”K = W–U, and Equation (79) becomes

22

yy

22

0

(,) 8()(, )0.

H ztmW UHzt

zA

∂π

+− =

∂

(80)

The second derivative of Hy(z,t) with respect to t gives:

M. Perkovac

248

∂2Hy(z,t)/∂t2 +

ω

2Hy(z,t) = 0. After inclusion of the angular frequency

ω

= 2πν from Equation (56) we obtain:

2

22

yy

22

0

(,) 1 /2(,) 0

1/

H zteV mc

eVHzt

A

teV mc

∂

2π −

+=

∂−

. (81)

If eV/mc2 << 1, then eV”K = W – U, and Equation (81) becomes

22

y2y

22

0

(,) ()(,) 0

H ztW UHzt

tA

∂4π

+− =

∂

. (82)

5. Calculation of the Structural Constant s0

Only the structural constant s0 of the atom is unknown in previous expressions. This constant can be determined

in several ways, e.g., by measuring two quantities, the voltage V and frequency ν and calculating the action con-

stant A0 by Duane-Hunt’s law, i.e., using Equations (15) and (19), [17]. However, here we will use a more direct

theoretical calculation, with only one empirical item necessary [18].

Namely, the increase of the nuclear charge in the atom increases atomic number Z. In accordance with Equa-

tion (49), the value of structural coefficient σ(χ) = s02/Z is assigned to each atom. So, greater atomic number

means a lower structural coefficient σ(χ).

On the other hand, there is a critical nuclear charge which ensures stability of the atom [9,19]. In other words,

reducing σ(χ) grows instability of the atom. In general, the higher atomic number means less stability (i.e.,

smaller half-life, or t1/2) of the atom, starting from bismuth 83Bi (Z = 83, t1/2 = 6 × 1026 s, [16]) to ununoctium

118Uuo (Z = 118, t1/2 = 5 ms), [exceptions are atoms of technetium (43Tc, Z = 43, t1/2 = 1.3 × 1014 s) and prome-

thium (63Pm, Z = 63, t1/2 = 5.6 × 108 s )].

For the calculation of structural constant s0 it is enough to find only one associated pair of σ(χ) and Z. The

curve σ(χ) has no extremes, Figure 5. Thus it is not easy to find a mentioned pair of σ(χ) and Z. In that sense, a

better situation is with the phase velocity uem, specifically with the normalized phase velocity uem(ε*μ*)1/2 = F(χ)

of electromagnetic wave in the atom, Figure 5, [9]. Neither of these two curves have extremes, but there is a

sharp knee on F(χ) which can be used to determine the structural constant s0.

Although there is no theory about the connection between the phase velocity of electromagnetic waves in the

atom and the stability of the atoms, it is still possible to use this mathematical benefit of sharp knee for those

2.2

2.4

2.6

0.2

0.4

0.6

0.8

1

2.0

83Bi

0.825

402

()

σχ

2.129

531 7

114

Fl

Stable

Unsta b le

118

Uuo

Transition

0.329

18

0.738

105

2

0

2=137.017 s

()

F'

χ

2

([ln(/2/4-1)] / (ln+1/4) Fχ=χ+χχ)

00

×=()=0.825 40283 = 8.277sσχ Ζ

χ

0

=2.382

788

Z

th

=83±1/2

−

()

F''

χ

82Pb

22

00 00

=/A se

µε

∗∗

()

em

=Fu

ε

χµ

2

( )[ln( /2/4-1)](ln+1/4)σχ=χ+χχ

0

δ

/

ρ

=

χ

Figure 5. Structural coefficient of Lecher’s line σ(χ), norma-

lized phase velocity of the electromagnetic wave in the atom

F(χ) = uem(ε*μ*)1/2, the first derivative of the normalized phase

velocity F'(χ), inverted second derivative of the normalized

phase velocity F'' (χ), all versus ratio δ/ρ = χ of the transmis-

sion Lecher’s line, consisting of a pair of ideal conducting

nonmagnetic parallel wires of radius ρ separated by δ, which

represents a model of an atom.

M. Perkovac

249

atoms, in which there is the lower phase velocity of electromagnetic waves that exhibit greater instability. Use of

this result will be discussed just a little bit later.

The nuclear binding energy per nucleon slightly decreases with increasing atomic number (starting from the

first radioactive element bismuth, 83Bi, 7.848 MeV, to the ununoctium, 118Uuo, 7.074 MeV, about 0.31% de-

crease for each of the 35 atoms in that area [20]). Physically this means that the boundary between stable and

unstable areas is not emphasized. Mathematically it allows that between the two areas there exists so-called

transition area, Figure 5. This is, at the moment, the most accurate way to determine the boundary between sta-

ble atoms and the others. Indeed, the first unstable atom must be located on that border. This is a key fact to de-

termine the structural constant s0 of the atom.

Before calculating, we observe the first derivative F ‘(χ) of the curves of normalized phase velocity F(χ) of

electromagnetic waves in an atom (Figure 5). When this derivative is greater than 1, it means that the phase ve-

locity rapidly declines, that is a zone of unstable atoms (2 < χ < 2.1295317). It should be noted that the situation

χ ≤ 2 is theoretically impossible because then there is no Lecher’s line.

When the second derivative F’’(χ) of the normalized phase velocity F(χ) is greater than 1, it means that the

phase velocity starts to rapidly decline (Figure 5), this is a transition zone (2.129 531 7 < χ < 2.382 788).

The border crossing from the transition zone to the stable zone (i.e., χ0 = 2.382 788), in accordance with the

experiments, [20], is closest to the bismuth atom. Bismuth atom (83Bi) is the first unstable atom, in the entire

chain of stable atoms, which ends with lead (82Pb). The corresponding value of the structural coefficient in that

place is σ(χ0) = 0.825402, Figure 5. Bismuth is a chemical element with atomic number Z = 83, with half–life

more than a billion times the estimated age of the universe. Even though charges in reality take discrete values

(e, 2e, 3e, Ze), theoretical value of Z in Equation (49) can be within the range Zth = 83 ± 1/2. Thus, according to

Equation (49) we get the structural constant of the atom [0.825402 × (83 ± 1/2)]1/2, i.e., 8.252 < s0 < 8.302, with

a mean value 8.277 and with sample standard deviation ±0.035355 or as a percentage s0 = 8.277 ± 0.43%. Com-

paring with the fine structure constant we get s0 = 8.277 56, which is consistent with the calculation performed

here (the relative difference is less than 0.0068%).

6. System of the Elements

After calculating the structural constants s0 the structural coefficient

σ

of each element can be determined. For

this purpose we use Equations (46) and (49), Figure 6, i.e.,

2

20

1

( )ln1ln

24 4

s

Z

χχ

σχ χ

=+−+=

. (83)

It can be seen that there is a maximal atomic number, 137, just as it is defined in Equation (75). Recently dis-

covered element (Z = 118) is ununoctium [Joint Institute for Nuclear Research (JINR) by Yuri Oganessian and

his group in Dubna, Russia, 2002].

A century well-known behavior of hydrogen atoms describing Equations (70) and (72) if we put n±1 = n+1 (in

the case of n±1 = 1 we get Lyman series, in case of n±1 = 2 Balmer series, n±1 = 3 Paschen series, n±1 = 4 Brackett

series, n±1 = 5 Pfund series).

Interestingly, the same Equations (70) and (72) describe behavior of neutron, hyperon Λ0 and hyperon Ξ0 if

we put n±1 = n–1 (i.e., n±1 = 1/125.886339 for neutron n0, n±1 = 1/137.03543 for hyperon Λ0, n±1 = 1/137.03587

for hyperon Ξ0). This is the reason that all three of these particles are in the same place as hydrogen (Figure 6),

[21].

To the similar results regarding the elementary particles otherwise on original way came Giuseppe Bellotti,

[22,23]. He was also given new periodic classification of the elements on the basis of standing waves in the

atom, using wave equation of potentials of the electron and positron, similar to the Equation (77) of electric and

magnetic fields in this paper.

7. Conclusions

Maxwell’s equations, together with the Lorentz equations, proved to suffice for the construction of a model of

the atom. This model is made by analogy with the transmission line (Lecher’s line). Using this model laws

M. Perkovac

250

10

10

6

10

12

10

18

10

24

10

30

0.5

1.0

5.0

10

50

2He

3Li

4Be

5B

6C

7N

8O

9F

10Ne

20Ca

30Cn

40Zr

50Sn

60Nd

σ

(

1

H

) =

σ

( 5.043 729 881 41×10

29

) = 68.517 999 540 = s

02

70Yb

80Hg

90Th

100Fm

137Xxs

130Xxl

118Uuo

110Ds

119Xxa

χ

5.04372988141×10

29

σ

Ξ0

Λ0

n0

1H

min.

1H

2He

3Li

max.

0

2

Figure 6. System of the elements, i.e., structural coefficient of Lecher’s line

σ

vs. parameter χ

of Lecher’s line, specifying all known elements, as well as 19 till now undiscovered elements,

starting with atomic number 119, up to and including atomic number 137 (log-log scale). Re-

cently discovered element, in the second year of two thousand, is ununoctium, Z = 118, Dub-

na, Russia (Results from the first 249Cf + 48Ca experiment). Each element has more statio-

nary states which are determined with different amount of n±1. So with n±1 = 1 we obtain Ly-

man series of radiation of the hydrogen atom, with n±1 = 2 Balmer series, with n±1 = 3 Paschen

series, with n±1 = 4 Brackett series, with n±1 = 5 Pfund series. If n±1 = 1/(125.886339) hydro-

gen atom takes on the properties of neutron n0, if n±1 = 1/(137.03543) hydrogen atom takes on

the properties of hyperon Λ0, and if n±1 = 1/(137.03587) hydrogen atom takes on the proper-

ties of hyperon Ξ0. This is why n0, Λ0 and Ξ0 are in the same place where there is hydrogen 1H.

A minimal structural coefficient

σ

min is obtained when the velocity of the electron is equal to

the phase velocity of the electromagnetic wave in the atom, i. e., according to Equations (75)

and (83)

σ

min = s02/Zmax = s02/(2s02) = 1/2. The maximal amount of

σ

arises, according to Equ-

ation (83), when the atomic number is the minimal (Z=1), i.e.

σ

max = s02/Z = s02 = 68.517 999

540. All other elements are within this area.

which apply in quantum mechanics are derived, and are performed and Schrödinger’s equation, with the clear

meaning of the wave function. The wave function represents the electric or magnetic field strength of the elec-

tromagnetic wave in the atom.

Using the synchronization of two phenomena within the atom, the electromagnetic wave and the circular mo-

tion of electrons, stationary states of atoms are derived. It has been shown that two directions relative to the base

state (n = 1) are possible. One is shift of the electron out of the center to the outside (n±1 = n+1; n = 1,2,3…), and

the other is a shift in direction to the center of the atom (n±1 = n–1; n = 1,2,3…). The first is a classic, well-known

for a hundred years. The latter is a novelty, and it makes possible the formation of neutrons and hyperons using

protons and electrons, as is the case by hydrogen.

Structural constant of the atoms was determined with the aid of Maxwell’s and Lorentz’s equations. The

amount of structural constant 8.27756 is determined by the rapid decline of the phase velocity of the electro-

magnetic wave in the atom. It happens to the bismuth atom.

Finally, it was found that the atomic number cannot exceed 137, meaning that it is still theoretically possible

to detect another 19 so far undiscovered elements.

Acknowledgements

Wolfram Research, Inc. Mathematica software is used by courtesy of Systemcom, Ltd., Zagreb, Croatia,

www.systemcom.hr. The author thanks Ms. Erica Vesic for editing this article in English, Mr. Damir Vuk and

Mr. Branko Balon for the useful discussions, Prvomajska TZR, Ltd., Zagreb, Croatia, www.prvomajsk a -tz r.hr

and Drives-Control, Ltd., Zagreb, Croatia, www.drivesc.com, for the financial support.

M. Perkovac

251

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