The equations for energy, momentum, frequency, wavelength and also Schr?dinger equation of the electromagnetic wave in the atom are derived using the model of atom by analogy with the transmission line. The action constant A0 = (μ0/ε0)1/2s02e2 is a key term in the above mentioned equations. Besides the other well-known quantities, the only one unknown quantity in the last expression is a structural constant s0. Therefore, this article is dedicated to the calculation of the structural constant of the atoms on the basis of the above mentioned model. The structural constant of the atoms s0 = 8.277 56 shows up as a link between macroscopic and atomic world. After calculating this constant we get the theory of atoms based on Maxwell’s and Lorentz equations only. This theory does not require Planck constant h, which once was introduced empirically. Replacement for h is the action constant A0, which is here theoretically derived, while the replacement for fine structure constant α is 1/(2s02). In this way, the structural constant s0 replaces both constants, h and α. This paper also defines the stationary states of atoms and shows that the maximal atomic number is equal to 2s02 = 137.036, i.e., as integer should be Zmax=137. The presented model of the atoms covers three of the four fundamental interactions, namely the electromagnetic, weak and strong interactions.
Although for a long time classical physics and Maxwell’s equations were pushed out of research in the field of atomic phenomena, the article [
According to [
where vector E is the electric field strength, vector H is the magnetic field strength,
The same form of wave equations as (1), but only in one dimension, is also present on the parallel-wire transmission line (Lecher's line), consisting of a pair of ideal conductive nonmagnetic parallel wires of radius ρ, separated by δ, where the ratio
where u is voltage at the entrance of the element dz of Lecher's line, i is electric current at the entrance of the element dz of Lecher’s line,
Systems with the same differential equations behave equally. Hence, in [
According to the model presented in [
where r is the radius of the circular orbit of the electron, q is the charge of the electron (q = –e), e is elementary charge, Q is the charge of the nucleus (Q = Ze), Z is atomic number (which theoretically is not in integral domain), m is the electron rest mass,
The kinetic energy of the electron, [
or
On the other hand, electromagnetic energy is
The single Equation (6) has two unknowns, i.e., parameter C and variable Θ. By using Diophantine equations we get one of the many solutions: C = 4πεr, and
Equation (6), which represents the electromagnetic energy in an atom, can be written like this [
where
is the action of the electromagnetic oscillator, and
is the characteristic impedance of Lecher’s line, while
is the structural coefficient of Lecher’s line.
From Equation (7), in fact from
Therefore in an atom with multiple electrons there are multiple natural frequencies ν. It should be noted that the same expression as (11) came from simultaneously multiplying and dividing the right side of the original expression
Equation (11) shows that the natural frequency ν does not depend on the amount of charge in the atom, but depends on the properties of the space (μ, ε), structural coefficient of Lecher's line σ(χ), (which for its part depends only on the parameters of Lecher's line d and r), and the radius r of the circular orbit. Indeed, electromagnetic oscillations in the atom require a charge, but amount of the charge does not affect the amount of the frequency ν.
If Equation (4) is inserted in Equation (11) we obtain natural frequency of electromagnetic wave in an atom:
In the Equation (12) charges q and Q appear in the form of the product |qQ|. In order to satisfy the condition that the frequency ν is independent of the charge, and to avoid direct or indirect involvement of the charge, the total product
is called the structural constant of the atom. In the article [
From Equations (8), (9) and (13) follows
where
is action constant, which implies that μr = εr. Electromagnetic energy in Equation (7), i.e., Eem = Aν, using Equation (14), we can now write as:
By solving this equation we obtain:
Taking into account that Eem = eV it is as follows:
The extended Duane–Hunt's law we get from Equations (14) and (16) using Eem=eV [
From Equations (5) and (16) follows:
Thanks to the analogy between the electromagnetic wave in the atom and the wave of voltage and current at the Lecher's line, according to Equations (1) and (2), and
i.e., in normalized form [
where
for which the solutions are (
The momentum of the electromagnetic wave in the atom is equal to the momentum of a photon [
In accordance to the law of conservation of momentum, this momentum is equal to the linear momentum of the electron, [
By applying the expressions
= r/[|qQ|/(4pεmc2)], normalized velocity of the electron β = v/c, kinetic energy of the electron *K = K/(mc2), potential energy of the electron *U = U/(mc2), total mechanical energy of the electron *W = *K + *U, electromagnetic energy of the atom *Eem = –*W, momentum of electromagnetic wave in the atom *pem = pem/(mc), wavelength of the electromagnetic wave in the atom *l = l/[A0/(mc2)], normalized phase velocity of the electromagnetic wave in the atom uem/c and uem/v, normalized action of the electromagnetic oscillator A/A0, relative permittivity and relative permeability of the space in the hydrogen atom *εr(H) = *μr(H) = εr(H)/10 = μr(H)/10 and also relative permeability and relative permeability of the space within atom of bismuth *εr(Bi) = *μr(Bi) = εr(Bi)/10 = μr(Bi)/10, product of relative permittivity and normalized phase velocity εr(H)uem/c = μr(H)uem/c in the case of hydrogen, product of relative permittivity and normalized phase velocity εr(Bi)uem/c = μr(Bi)uem/c in the case of bismuth, the ratio of wavelength and radius *l/*r = (l/r)/(4pεcA0/|qQ|) = 2uem/c, the ratio of the frequency f of rotation of the electron orbiting atom and the frequency ν of the electromagnetic wave *(f/ν) = (f/ν)/(4εcA0/|qQ|), all versus normalized frequency of the electromagnetic wave in the atom ν/ν0, where ν0 = A0/mc2; n+1 = 1, 2, 3, … is ordinal number of stationary orbits in the atom, n–1 = 1/1, 1/2, 1/3, … 1/n, for hyperon X0 n–1 = 137.03587.
and using Equation (19) it becomes
Phase velocity of the electromagnetic wave in an atom is obtained by multiplying Equations (19) and (27):
From (25) and (27) follows
The ratio of the wavelength of the electromagnetic wave in the atom and the atom radius are obtained from Equations (4) and (27) using Eem=eV and Equation (29),
This expression, with
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 [
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) [10,11]. 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 2w = 2(2pν), (this will be further discussed in Sub–Heading 2.9). This means that in the synchronously 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, 2, 3, … is ordinal number of stationary orbits in the atom (or n–1 = 1/1, 1/2, 1/3, … 1/n). Equation (33) gives the speed of electron in a synchronously stationary state ([
The Equations (3) and (34) give the radius of the electron orbits in the synchronously stationary states:
From Equations (33), (34) and (35) follows [
and, [
The total mechanical energy of an electron Wn= –Eem(n) follows from Equations (17) and (37):
For energies much smaller than mc2:
If assume the maximum speed of electron is equal to the speed of light in a given medium, i.e., according to Equation (22) vmax
From Equation (41) follows the greatest possible atomic number Zmax when nmax is minimal and
Wave equations of electromagnetic wave in an atom are expressed by Equation (1). If we insert phase velocity uem in these equation from Equation (29), i.e.,
we obtain
Wave equations (1) and (44) have a lot of solutions. We will apply the solutions that correspond to the transmission line, i.e., to the LC network. These solutions are standing waves [12,13]:
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, w = 2pν. 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 (45) the energy oscillates between the electric and magnetic form. The electrical energy is 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 2w from point to point of z axis [
If we use the second derivative with respect to z of the y-component Hy(z,t) of the magnetic field strength in Equation (45), we get: ¶2Hy(z,t)/¶z2 + (2p/l)2Hy(z,t) = 0. After inclusion of the wavelength l from Equation (27) we obtain:
If eV/mc2 << 1, then eV » K = W – U, and Equation (46) becomes
The second derivative of Hy(z,t) with respect to t gives:
¶2Hy(z,t)/¶t2 + w2Hy(z,t) = 0. After inclusion of the angular frequency w = 2pν from Equation (19) we obtain:
If eV/mc2 << 1, then eV » K = W – U, and equation (48) becomes
Structural constant of the atom s0 can be determined in several ways, e.g., by measuring two quantities, the voltage V and frequency ν and calculating the action constant A0 by Duane–Hunt’s law, i.e., using Equations (15) and (19), [
Namely, the increase of the nuclear charge in the atom increases atomic number Z. In accordance with Equation (13), 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 [2,14]. In other words, to reduce σ(χ) means to grow instability of the atom. In general, the higher atomic number means the less stability (i.e., the less half–life, or t1/2) of the atom, starting from bismuth 83Bi (Z = 83, t1/2 = 6 × 1026 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,
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 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 increase the atomic number (starting from the first radioactive element bismuth, 83Bi, 7.848 MeV, to the unoctium, 118Uuo, 7.074 MeV, about 0.31% decrease for each 35 atoms in that area [
Before calculating, we observe the first derivative F'(χ) of the curves of normalized phase velocity F(χ) of electromagnetic waves in an atom (
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 (
The border crossing from the transition zone to the stable zone (i.e., χ0 = 2.382 788), in accordance with the experiments, [
In classical physics, the use of Maxwell’s equations and Lorentz theory of electrons gives us equations that describe the behavior of atom. For this description, usual classical physical constants are sufficient. The only new necessary physical constant is the structural constant of the atom s0. This new physical constant was calculated in the framework of a unified theory based on the model of the atom by analogy with the transmission line. To
calculate it, we need only one empirical item, i.e., the atomic number of the first unstable atom in periodic table of elements. It is known, that it is bismuth, with atomic number Z = 83. From this data, the structural constant s0 is calculated. Thus, we get s0 = 8.277 ± 0.43%. Comparing with the fine structure constant we get s0 = 8.27756, which is consistent with the calculation performed here (the relative difference is less than 0.0068%). This constant, thanks to the expressions (μ0/ε0)1/2s02e2 and (2s02)–1, can replace two existing constants, Planck’s constant h and the fine structure constant α. We get also the stationary states of the atoms and the maximal atomic number Z = 2s02 = 137.036, i.e., as integer should be Zmax = 137. The difference between 2s02 and Zmax we call the slipping. This indicates that, in addition to integer synchronously stationary states, non-integer (asynchronously) stationary states in atoms are also possible. Now, just a mention, as it seems to asynchronously stationary states can produce a continuous spectrum of atoms. This, however, requires more detailed research. This model of the atoms covers the energies of the electromagnetic, weak and strong fundamental interactions [
Wolfram Research, Inc. Mathematica software is used by courtesy of Systemcom Ltd., Zagreb, Croatia. The author thanks Ms. Erica Vesic for editing this article in English, Mr. Damir Vuk for the useful discussions and Prvomajska TZR Ltd., Zagreb, Croatia, www.prvomajska-tzr.hr, for the financial support.
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