>
= =
(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 SubHeading 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 eVK = WU, 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 eVK = W U, and Equation (81) becomes
22
y2y
22
0
(,) ()(,) 0
H ztW UHzt
tA
+− =
. (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-Hunts 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 Lechers 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 halflife
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
Maxwells 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 (Lechers 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 Schrödingers 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 Maxwells and Lorentzs 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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