Full Paper
Thematic Section: Theoretical Research.
Reference Object Identifier – ROI: jbc-02/16-47-7-112
Subsection: Theory of the Matter Structure.
Publication is available for discussion in the framework of the on-line Internet conference “Butlerov readings”.
http://butlerov.com/readings//
Submitted on September 08, 2016.
Empirical support for the planetary model of the hydrogen atom
© Alexey A. Potapov,*+ and Yury V. Mineev
1
Butlerov Science Foundation. Bondarenko St., 33-44. Kazan, 420066. Russia.
Phone: +7 (843) 231-42-30; +7 (395) 246-30-09. E-mail: [email protected]
2
Training Department. East-Siberian Institute of the Russian Interior Ministry.
Lermontov St., 110. Irkutsk, 664074. Russia. Phone: +7 (3952) 410-146. E-mail: [email protected]
___________________________________
*Supervising author; +Corresponding author
Keywords: planetary model of the hydrogen atom, electron, stability, fundamental constants.
Abstract
The hydrogen atom is a basis for understanding electron structure of atoms that make up the periodic
table. Until recently, concepts of the structure of the hydrogen atom remained controversial. In this paper, the
planetary model of the hydrogen atom is assumed, which is the development of the Rutherford-Bohr atom
model. A theoretical description of the planetary model of the hydrogen atom, based on solving the classical
Kepler’s problem about electron motion in the central field of the nucleus is given. The necessary requirement
for substantiation of any theory is an experiment. The data of the polarizability measurement, radius, the
electric and magnetic moments of atoms, as well as the Stark and Zeeman effects presented as the argument of
the model and its theoretical description. The central point of the work is the implementation of comparing of
the bond energy obtained from the data of wave measurements of the Rydberg constant, and bond energies
obtained by calculation according to the fundamental constants of the charge and mass of the electron and the
application of the law of conservation of movement. The error in matching the calculation and the experiment
data does not exceed the error in determining the physical constants used in the calculations. The error is
estimated at the level of several units of the 6th decimal places. Subjective factor is completely eliminated
here, because while analyzing, the official data on fundamental constants were accepted as the initial data for
comparison.
Dynamic state of an atom explains extremely high structure stability of the hydrogen atom, which is
provided by automatic maintenance of the energy state of the atom at the level given by nature. Autoregulation mechanism is based on the extreme nature of the potential function of the hydrogen atom and the
action of the energy conservation law and the law of conservation of angular momentum. Specially made
computing experiment and the computer model of rotational motion of the electron in the central field of the
nucleus serve here as an illustration of the mechanism of the hydrogen atom stability.
The analysis has to help in confirmation of the planetary model of the hydrogen atom and finish all
discussions about the structure of the hydrogen atom.
Introduction
1. The problem
The modern stage of atomic physics is often called the final stage. According to this, the
development of atomic physics is finished because of fundamental limitations, which appear in the
process of researching atom. These limitations are formulated as the Heisenberg uncertainty
principle. The principle comes down to the fact that we cannot determine the position of the electron
in the atom and new laws, which are different from classical laws for microscopic objects, start
acting at the atomic level [1-3].
This paradigm was a foundation of Copenhagen doctrine, was discovered by N. Bohr,
W. Heisenberg, M. Born, P. Dirac, W. Pauli, and J. von Neumann in the twenties of the XX century.
According to this doctrine, properties of microscopic objects are specifically quantum-mechanical
(non-classical); on the other hand, these properties characterize operations in laboratories (not
particles of matter), which are described only by classical method. There is an obvious contradiction:
quantum-mechanical symbols are related to non-quantum (=classical) facts (Bunge M. Philosophy of
Physics. Moscow. 1973. 132p (russian)).
112 ________ © Butlerov Communications. 2016. Vol.47. No.7. _________ Kazan. The Republic of Tatarstan. Russia.
EMPIRICAL SUPPORT FOR THE PLANETARY MODEL OF THE HYDROGEN ATOM _______________ 112-124
The source of the contradiction is a statement, that physical theory does not give us the right
description of reality; it just describes human experience. As Copenhagen doctrine says:
“Autonomous quantum events do not exist, there are only quantum elements depending on observer;
what exists in a particular quantum state is generated by the observer.” It is this statement is not
consistent with common practice. Everything that surrounds us is the atomic-molecular systems and
there is no need to have an observer to these systems exist and function. At the same time, an
observer is also an atomic-molecular object and his existence does not need for continual observing
for atomic and molecular condition (Bunge M. Philosophy of Physics, M., 1973, 133 p).
Research showed that quantum-mechanical method is absolutely useless as a method of
studying atomic structure and as a method of its description. It is impossible to create a physical
theory which would be based on non-physical supposition such as postulate the impossibility of the
existence of autonomous (independent of the observer) structures. The orthodox interpreting of
quantum mechanics denies the physics, subduing it the psychophysiology human-observer. The
problem of understanding an atomic structure is discussed so far.
Nevertheless, numerous educational and monographic literatures on quantum mechanics have
unsubstantiated references to the fact that its main provisions have the necessary theoretical and
experimental study. Stereotype expressions of atoms often appear in popular scientific literature and
mass media.
On the other hand, there are quite convincing theoretical and logical arguments in favor of
classical approach in atomic physics and planetary model of the atom, which is a development of
Rutherford-Bohr model [5-8].
Results and discussion
2.1 Planetary model of hydrogen atom
Nowadays, the planetary model of hydrogen atom is the most appropriate. The equation of
motion of hydrogen atom has been obtained in this model. According to the model, the atom is the
electron hard linked with the proton. The dynamic system is the only way the proton and the electron
can co-exist. In this case, proton is a center of attraction for spinning electron. The problem of
describing of hydrogen atom is close to the problem of planets moving around the Sun, also known
as a classical Kepler problem. The Kepler problem of the electron motion in a centrally symmetric
electric field is solved on the basis of energy conservation law and the angular momentum 𝐿. In
polar coordinates, these laws lead to two differential first-order equations with respect to unknown
functions r(t) and φ(t) [6, 9]:
𝑚
𝜺(𝑟) = 2 (𝑟̇ 2 + 𝑟 2 𝜑̇ 2 ) −
where eZ – nuclear charge of the atom.
𝑒2𝑍
𝑟
и 𝐿 = 𝑚𝑟 2 𝜑̇
(1)
The given equation is usually solved by means of a transition from the radius-vector
derivatives with respect to time to the derivatives with respect to the angle
𝑑𝑟 𝑑𝜑
𝐿
𝑑𝑟
𝑟̇ = 𝑑𝜑 𝑑𝑡 = 𝑚𝑟 2 𝑑𝜑.
(2)
The solution to this equation is function [10]
1
𝑚𝑒𝑞
= 𝐴 𝑐𝑜𝑠𝜑 + 𝐿2 ,
where А – an arbitrary constant determined on the basis of the initial conditions.
𝑟
(3)
Equation (3) represents the electron motion trajectory. At the same time, it is the equation of
conic section in polar coordinates, which has the following form [10]
1
1− э𝑐𝑜𝑠𝜑
= эс ,
(4)
where э – eccentricity, с – a parameter of the electron motion trajectory, which corresponds to
4 possible types of functions: 1) hyperbola, at э > 1; 2) ellipse, at 0< э < 1; 3) parabola, at
э = 1; 4) circle, at э = 0.
𝑟
In the limiting case of circular motion
= 0, equation (1) takes the following form
©Бутлеровские сообщения. 2016. Т.47. №7. ___________________ http://butlerov.com/ __________________ 113
Full Paper ____________________________________________ A.A. Potapov, and Yu.V. Mineev
𝐿2
𝑒2𝑍
𝜺(𝑟) = 𝜺𝑘 + 𝜺𝒑 = 2𝑚𝑟 2 − 𝑟
(5)
where 𝜺𝒌 and 𝜺𝒑 − kinetic and potential energies, respectively, r – actual distance between
the nucleus and the electron; L – angular momentum equal to 𝐿 = 𝑚𝑣𝑟; v – orbital speed
of the electron with mass m; eZ – nuclear charge.
The first summand of energy 𝜺(𝑟) in (5) represents the kinetic energy of the electron motion,
while the second summand – the potential energy as the result of the Coulomb interaction between
the nuclear charge +eZ and the electron. For equation (5), the difference between the exponents of
the first and second summands with a distance equal to r is essential. This leads to the fact that as a
result of superposition of atom functions 𝜺К (𝑟) and 𝜺𝒌 (𝑟), the resultant dependency 𝜺(𝑟) acquires a
characteristic minimum of potential energy that corresponds to the equilibrium state of the atom.
This state is determined in a usual way (by finding the extremum):
𝑑𝜺
𝑑𝑟
2𝐿2
= − 2𝑚𝑟 3 +
𝑍𝑒 2
𝑟2
=0
(6)
Based on (6), we can find the bond energy corresponding to the equilibrium state of aB. For
hydrogen atom there is a known accurate solution to the electro dynamic problem of electron motion
ein the central field of the nucleus charge eZ [6].
𝜺=−
𝑍2𝑒2
2𝑎𝐵
𝑒2
и при Z = 1, 𝜺𝐻 = − 2𝑎
𝐵
(7)
Among the atom family only the hydrogen atom satisfies the circular orbit condition э = 0.The
physical meaning of the circular orbit of hydrogen atom consists in the fact that in the absence of
external disturbing factors, the electron motion in the central nuclear field is determined by the
absolute equality of nucleus and electron charges.
An important is the fact that the angular momentum is constant in the planetary model (1)
(according to the conservation law) but it is quantized in the Bohr model.
2.2 The basis of experimental method of atomic research
The difficulty in making a theory of electron atomic structure is connected with the difficulty
in getting the information of internal structure of atom. The fact is that the outer electric shell of
atom behaves like an effective screen for external probing of electric fields. This fact limits
possibilities in experimental methods of researching intra-atomic structure. On the other hand,
applied methods must not be distractive to get more accurate information of the structure of the
atom. That means that in the process of measuring effect on an atom has to satisfy the condition of
smallness of the perturbation of the electron shell of a test atom of ion. Concerns related to the
unavailability of electron structure of atoms became the reason to think that the possibilities of an
empirical method of studing the internal structure of atoms are limited [1].
Analysis of the state of actual problem shows us the following facts. Firstly, nowadays there is
accurate measuring equipment to measure polarizability, which provides the condition of smallness
of the perturbation of an atom state. Secondly, we can apply spectroscopic methods to the hydrogen
atom and to the hydrogen-like structure because of their features of electronic structure. Vast
empirical material has been accumulated with the help of this equipment including data of nonresonance and spectroscopic measurements. Based on these data it managed to develop consistent
dipole-shell model, not only hydrogen atom, but also many-electron atoms.
The only way to get initial information of structure of an atom is the black box method [6, 11,
12]. A theoretical basis of this method is the theory of linear response which is described by the ratio
𝑥 = 𝜒𝑋, whose meaning is to describe the nature of response of x substance and external action X;
where 𝜒 – the generalized susceptibility, which is a response function of x and also it brings us initial
information of structure of matter. In the electrical (optical) experiment the electrical field X = E is
used as the acting field, the dielectric susceptibility 𝜒 = 𝜒е or refractive index n is the measure and
the polarization x = P is used as the response. In general case the measurement 𝜒 is a result of joint
action of all consisting atoms in it, then
𝜒е = 𝛼𝑁
(8)
114 _____________ http://butlerov.com/ _____________ ©Butlerov Communications. 2016. Vol.47. No.7. P.112-124.
EMPIRICAL SUPPORT FOR THE PLANETARY MODEL OF THE HYDROGEN ATOM _______________ 112-124
where N – an atomic density, and a proportionality factor 𝛼 – polarizability – in this case is
used as an elementary property of the atom.
As a ‘property’, the atom polarization is a function of ‘consist’ N, ‘structure’s and ‘bond
energy’ U. It happens if the condition of smallness of interatomic interactions is satisfied (when 𝑈 →
0)). In practice, this condition is satisfied if the measurements of 𝜒 and N were made in the gaseous
𝜒
phase at a low pressure 𝛼 = lim 𝑁 [12].
𝑁→0
The polarizability 𝛼 is classified as an atomic constant and it reflects the fundamental property
of the electron shells which are best observed in the atomic deformation under the influence of
external electric fields [3, 4]. This means the polarizability is a source of the required information of
intra-atomic structure.
The study of the atomic polarizability provided a break through in recognizing in principle the
possibility of studying the structure and allowed to disclose (“decode”) the electronic structure of
atoms [5-9].
The polarizability can be measured both at low frequency and at optical frequency range. The
optical polarizability of the atom is the electrical polarizability. It was separated as a different
subclass because it has a feature which appears in information experiment in the form of optical and
X-ray spectra. Generally, in the optical range we measure not the dielectric susceptibility 𝜒е but the
absorption intensity 𝜒𝑒" (or the radiation intensity), which correspond to the resonance frequency this
spectral line [11, 12]. In this case, the experiment describes as before, then
𝑃" = 𝜒𝑒" 𝐸
where 𝑃" – the parameter, which appears the absorption intensity of the incident
electromagnetic wave 𝐸 = 𝐸0 exp(𝑖𝜔𝑡) where 𝐸0 and 𝜔 – the amplitude and
the frequency this electromagnetic wave.
(9)
𝜒"
Value 𝜒𝑒" (equation (9)) determines that 𝛼𝑒" = 𝑁𝑒 is the reduced value, which describes
individual properties of atoms. In other words, it is like an ability of atoms to resonant interaction
with the electromagnetic field. The data 𝛼𝑒" bring us useful information of discrete (quantum)
character of atomic energy level.
In this case values 𝜒𝑒" and 𝛼𝑒" are imaginary components of complex susceptibility 𝜒 ∗ and
polarizaility 𝛼 ∗ and they determine as 𝜒 ∗ = 𝜒 ′ + 𝑖𝜒 " and 𝛼 ∗ = 𝛼′ + 𝑖𝛼 " , where 𝜒 ′ and 𝛼 ′ are real
components of susceptibility and polarizability, respectively. Such a complex representation of
measured values is widespread in the physics of dielectrics to describe the behavior of substance in
the frequency domain [11, 12]. In this case the behavior 𝜒′ reflects polarization properties and the
behavior 𝜒" reflects matter properties, which are connected with the energy absorption of the
analyzing field E.
We can apply the same description in a form of dispersion equation to the resonance
phenomena. But there is a question, how accurate information in the spectroscopic experiment is, if
we use the method of response? In general, the condition of smallness of the perturbation ∆𝜺 of
initial energy state of 𝜺 object is a criterion of applicability the theory of perturbation, which is a
basis of the method of response, so ∆𝜺 ≪ 𝜺. But the atom is not a usual object. Its energy levels are
correlated and subject to the certain law. As for the hydrogen atom and the hydrogen-like structure,
this condition ∆𝜺 ≪ 𝜺 comes down to the condition of smallness of the bond energy of the electron
to the energy of the nucleus, which is always true. Apparently, it is necessary to talk not about the
perturbation of the energy state of an atom, but about typical state of hydrogen atoms and hydrogenlike structures, which reflect their own nature and electron state.
The problem is getting more difficult when we consider many-electron atoms, because when
one of the electrons transits to excited state, then it leads to changing of energy state of the atomic
nucleus. In fact, it is equivalent to the perturbation of initial atomic state and the researchers problem
is to establish the magnitude of the value this perturbation and its influence to the accuracy of
measuring of the relevant energy level ε . From this point of view, the measurement problems of the
optical spectra of atoms and the ionization potentials can be regarded as parameters, which bring us
©Бутлеровские сообщения. 2016. Т.47. №7. ___________________ http://butlerov.com/ __________________ 115
Full Paper ____________________________________________ A.A. Potapov, and Yu.V. Mineev
the most direct information of the structure of atoms. It is necessary and sufficient to determine
systematic error of the measurement of bond energy or ionization potential I atom.
So, information about the atomic structure can be obtained by resonant or non-resonant
experiments. A quantitative measure of this information is the dielectric susceptibility and/or
refractive index, which are caused by the individual properties of atoms. Along with it, research
methods of atoms which are based on the observation of the response of atoms to ‘force’ the electric
and/or magnetic field are widespread, such as methods of Stark and Zeeman [12].
2.3 The measured values
Polarizability. Researches of the atomic polarizability largely predicted a break through in the
recognition of the fundamental possibility of understanding the electronic structure of atoms [6]. The
important fact is the polarizability α is measured with high accuracy (up to 0.02%) and in the
condition of smallness of the perturbation of the initial atom state. This fact is the necessary and
sufficient condition for obtaining reliable information of intra-atomic structure and by that the
problem of ‘uncertainty’ of electronic atom configuration disappears.
The presence of high-precision data on polarizabilities of atoms allows us to consider the
problem of the atomic structure of a qualitatively new level. The starting point for such consideration
is a conceptual position, according to which, the phenomenon of deformation polarization acts as a
source of initial information of electronic structure of atoms.
In equation (8) the polarizability 𝛼 is an initial value, due to the electronic structure of the
atom. The atom polarizability is a function of ‘composition’, ‘structure’ and the bond energy. The
term ‘composition’ in this case is the total number of electrons and a nucleus, which has a charge
+eZ, where Z – serial number of the periodic table element. For each element, the ‘composition’ of
the atom is uniquely determined and it can be regarded as given. As for ‘structure’ of the atom, this
term in this case is about its electronic configuration (the relative position of the electrons in the
atom).This ‘structure’ of the atom directly linked to the choice of an acceptable model of the atom as
a basis for constructing a theory of its electronic structure. It is clear, that the ‘structure’ of the atom
is directly related to the energy of sub-atomic interactions, it means that in general the polarizability
of the atom is a function of the structures and energy 𝜺 at a given ‘composition’ N atom, this is
𝛼 = 𝛼(𝑠, 𝜺)|𝑁=𝑐𝑜𝑛𝑠𝑡 . Values s and 𝜺 are interrelated and interdependent, that allows the possibility
of building a consistent model of the electronic structure of atoms.
The author's studies let us to clarify the relationship of the polarizability with a radius of the
hydrogen atom [6, 13]. The special feature of the hydrogen atom structure and hydrogen-like cations
is their dipole structure. The presence of the dipole moment is shown in an extremely high chemical
activity of the hydrogen atom and hydrogen-like cations.
Their dipole moment p is based on rigidly interconnected the nucleus and the electron, р =
𝑒𝑎𝐵 , where аВ – atomic radius. The physical meaning of the dipole moment is that in the hydrogen
atom the electron and the proton are rigidly interconnected by Coulomb forces and they are one thing
in form of a stable atomic structure.
The presence of the atom dipole moment assumes its interaction with an external electric field
E both orientation mechanism and deformation mechanism [12-14]. This means, that the
polarizability 𝛼𝐻 which is observed in the experiment has two components 𝛼𝐻 = 𝛼𝑜𝑟 + 𝛼𝑑 .
Orientation polarizability 𝛼𝑜𝑟 determines the ability of atoms to orient their dipole moment p along
the acting field E. The limiting factor that prevents the free orientation of the dipole is ‘strength’ of
the rotational motion of an electron in a circular orbit. The kinetic energy 𝜺𝑘 , which is equal the half
of the potential energy 𝜺𝒑 (𝜺𝑘 =13.6 eV) corresponds to this force FK.
Earlier this contribution to the calculations was not accurate. Entering the correction 𝛼𝑜𝑟 to the
polarizability 𝛼𝐻 provides its connection with the radius of the formula 𝛼𝑑 = 3𝑎𝐵3 . Consistency of
calculating radius 𝑎𝐵 with the experiment is an additional confirmation of the planetary model of the
hydrogen atom. It is an important result because until recently it was believed that the impossibility
of the classical description of the polarizability of the hydrogen atom is the rationale for the
transition to the quantum-mechanical description [12].
116 _____________ http://butlerov.com/ _____________ ©Butlerov Communications. 2016. Vol.47. No.7. P.112-124.
EMPIRICAL SUPPORT FOR THE PLANETARY MODEL OF THE HYDROGEN ATOM _______________ 112-124
The magnetic dipole moment is directly connected to the electric moment. There is an equation
𝑒𝐿
′
for Bohr magnet on 𝜇𝑚
= 2𝑚𝑐 . Taking into account the equality 𝐿 ≡ ħ this formula can be
𝑒𝐿
𝑒𝑚𝑎
𝑣
1
transformed to 2𝑚𝑐 = 2𝑚𝐵 · 𝑐 = 2 𝑒𝑎𝐵 𝛼. The coefficient ½ is here as a result of an erroneous
interpretation of the magnetic moment of the hydrogen atom [13, 15]. In fact, the magnetic moment
of the hydrogen is entirely determined by the orbital magnetic moment of the electron, so that the
′
corrected value of the magnetic moment should be equal 𝜇𝑚 = 2𝜇𝑚
, and as a final result 𝜇𝑚 =
𝑒𝑎𝐵 𝛼 = 𝑝𝑒 𝛼.
The origin of the magnetic moment is connected to the dynamic effect which appears in the
𝑣
depending electron charge quantity on the speed of its movement, 𝑒 ∗ = 𝑒 𝑐 , in particular the Lorentz
effect [15, 16] so that 𝜇𝑚 = 𝑒 ∗ 𝑎𝐵 . In fact, the magnetic dipole moment is orthogonal component of
the electric moment which occurs due to the rotation of the electron.“Magnetic” dipole moment is a
derived quantity of electric dipole moment.
The physical meaning of the constant 𝛼, which is included to the magnetic momentum
𝑒2
expression, revealed after its conversion 𝛼 = 𝑚𝑣𝑎
𝐵𝑐
𝑒2
= 2𝑎 ∙
𝐵
2
𝑚𝑣 2
𝑣
𝑣
∙ 𝑐 = 𝑐 = 𝑎𝐵
𝜔
с
using the interconnection
between the Planck constant and the angular momentum, 𝐿 ≡ ħ, where 𝐿 = 𝑚𝑣𝑎𝐵 – orbital speed of
the electron, m – electron mass, e – electron charge, c – speed of light, 𝜔 – angular frequency of
electron rotation. Here
𝑒2
2𝑎𝐵
and
atom, respectively, so that
𝑚𝑣 2
2
are equal to each other, total and kinetic energy of the hydrogen
𝑣
𝛼 = 𝑐 = 𝑎𝐵
𝜔
с
[1, 2]. This ratio sets the direct connection between
constant of the fine structure and the radius of the hydrogen atom 𝑎𝐵 . The physical meaning of this
connection is the perturbation of the electron static state (quiescent state) by the Lorentz force that
occurs when an electron rotates in a circular orbit in a central field of the nucleus [15, 16]. A
quantitative measure of the perturbation is an orbital speed of v electron. The fine structure constant
exists due to the dynamic behavior of the electron which characterizes the hydrogen atom, 𝜺 =
2
𝑚𝜔 2 𝑎𝐵
2
= 𝑚𝑐 2 𝛼 .
The equality above 𝐿 ≡ħ directly follows from the known ratio 𝜺 = ℎ𝑓, where the Planck
constant h is a coefficient connects the radiant energy with the radiant frequency f [16]. From this
𝜺
ratio it follows that ℎ = 𝑓 =
𝑚𝑣 2
𝑓
=
2𝜋𝑚𝑣𝑎𝐵 𝑓
𝑓
= 2𝜋𝐿. Value 𝜺 is determined here as an electrostatic
potential of the nuclear charge. In its field is happening accelerated electron motion in the radiation
𝑚𝜔 2 𝑎2
ℎ
process, so that = 2 𝐵 = 𝑚𝑐 2 𝛼. Consequently 𝐿 ≡ħ, where ħ = 2𝜋 – reduced Planck constant, L
– angular momentum, 𝐿 = 𝑚𝑣𝑎𝐵 , where m – electron mass, v – orbital speed of the electron. On the
other hand, radius
is directly connected to the reduced Planck constant ħ according to the
definition if angular momentum as a product of impulse р = mv with radius of circular orbit 𝑎𝐵
so 𝐿 = 𝑚𝑣𝑎𝐵 . This ratio reveals the physical meaning of the Planck constant; it is equal to
numerical value of angular momentum, which is an atomic constant, according to the law of
conservation of momentum. In fact, it means that the Planck constant appeared because of the
hydrogen atom.
Bond energy (ionization potential). Bond energy is an absolute characteristic of atomic state.
It is a part of original equation of electron motion (1). Ionization potential is the energy required to
remove an electron from atom or ion. Ionization potential is a measure of the atomic stability and it
acts as a passport characteristic of an atom. In general, we cannot equate the value of the ionization
potential with bond energy of an electron 𝜺 in an atom. There is a difference between value I and
real bond energy, because of the uncertainty of the atomic energy state in the process of removing
the electron during its ionization. In this respect, the hydrogen atom and hydrogen-like systems
occupy a special position. For them, the ionization potential with a high accuracy is equal to the bond
energy.
©Бутлеровские сообщения. 2016. Т.47. №7. ___________________ http://butlerov.com/ __________________ 117
Full Paper ____________________________________________ A.A. Potapov, and Yu.V. Mineev
Bond energy of hydrogen atom is very important for our research. It is directly connected with
the Rydberg fundamental constant, so that [13, 17].
𝜺 = сℎ𝑅𝑦.
(10)
Constant Ry, appearing in (10), formed because the energy of spectral term used to be
presented in wave numbers 1/𝜆 in spectroscopy. If the spectral terms expressed in energy units, the
Rydberg Ry should be multiplied by Planck constant and the speed of light c, so that 𝑅𝑦 ∙ ℎс =
𝑚𝑒 4 ℎс
4𝜋ħ3 𝑐
=
𝑚𝑒 4
2ħ2
𝑒4
𝑒2
= 2𝑚𝑣3 𝑎2 = 𝑚𝑣3 𝑎
𝐵
𝐵
𝑒2
2𝑎𝐵
𝑒2
= 2𝑎 = 𝜺𝐻 . In this case, Rydberg Ry takes the meaning of the
𝐵
1
bond energy 𝜺𝐻 of the hydrogen atom. An important is the fact that the Rydberg constant 𝑅𝑦 = 𝜆 is
determined experimentally on the basis of measurements of the wavelength  absorption with an
accuracy of 7 decimal places, 𝑅𝑦 = 10973731.77 m-1 [17]. Constants appearing in formula (10) the
speed of light с = 2.99792458 m/s and the Planck constant h = 6.626176·10-34J·s are also determined
with an accuracy of 6 decimal places. Inputting these numerical values into the formula (10) we
obtain a numerical value for bond energy 𝜺 = 2.17990707·10-11 erg = 13.605803 eV (here is the
equivalent1 eV = 1.6021892·10-12 erg).
On the other hand, we can express the bond energy of the hydrogen atom in terms of physical
constants as the theory parameters, so that [13, 17].
𝜺=
𝑚𝑒 4
2ℏ2
.
(11)
Substituting the known values in this formula е = 1.6021892·10-19 C = 4.803242·10-10 SGS,
ℏ =1.0545887·10-34 J·s and 𝑚 = 9.109534·10-31 kg [17] we get 𝜺 = 2.17990622·10-11 erg =
13.605797 eV.
Comparison of the numerical values for (10) and (11) gives ∆𝜺 = 6·10-6 eV, so the relative
∆ε
error in calculation of the bond energy is equal to ε ≈ 4·10-7, which corresponds to the order of
magnitude of error in the formulas (10) and (11) of physical constants. It was obtained a truly
fantastic result. It is difficult to assume that the resulting agreement between theory and experiment
in the sixth decimal place is random. Subjective contribution is completely eliminated, since official
data on fundamental constants [17] were used as input. Only this single result has to convince any
skeptic or an apologist of quantum mechanics in the legitimacy of the planetary model of the
hydrogen atom.
Stark effect is a phenomenon when there is the imposition of the electric field on atoms then
their energy levels are shifted and split into sublevels [18, 19]. There are linear and quadratic Stark
effects, depending on the nature of the bond energy of the atom ε by an external electric field E.
An important consideration for the Stark and Zeeman effects seems the presence of the
hydrogen atom dipole moment p, which involves its interaction with the external electric field E for
strain [12] and orientation mechanism [14]. It means that the experimentally observed polarizability
𝛼𝐻 , as noted above, has two components 𝛼𝐻 = 𝛼𝑜𝑟 + 𝛼𝑑 . Feature of orientation polarizability 𝛼𝑜𝑟 of
hydrogen atom is that it is elastic due to opposition from the centrifugal ‘force’ of the rotational
motion of the electron.
This force 𝐹𝐾 corresponds to the kinetic energy 𝜺𝐾 . In view of smallness of the interaction
energy 𝑝𝐸 compared with the energy 𝜺𝐾 orientation-elastic elastic polarizability defined as 𝛼𝑜𝑟 =
〈𝑝𝑜𝑟 〉 =
2𝑝2
3𝜺𝐾
[12, 13].
Deformation polarizability 𝛼𝑑 is an atomic constant that represents a fundamental property of
atoms, which manifests itself in an elastic deformation of the electron shells under the influence of
an electric field.
Deformation polarizability of the hydrogen atom is defined by its radius by the formula [12,
13] 𝛼𝑑 = 3𝑎𝐵3 , the calculation of which is consistent with the value obtained earlie 𝛼𝑑 = 0.467 Å3 .
Deformation polarizability of an atom can also be calculated in terms of the H. Lorentz theory and
his oscillation model [13].
118 _____________ http://butlerov.com/ _____________ ©Butlerov Communications. 2016. Vol.47. No.7. P.112-124.
EMPIRICAL SUPPORT FOR THE PLANETARY MODEL OF THE HYDROGEN ATOM _______________ 112-124
The physical meaning of Stark effect more fully disclosed in the dipole-shell model [12],
according to which a hydrogen atom is represented as a rotating dipole 𝑝 = 𝑒𝑎𝐵 , formed rigidly
interconnected nucleus and the electron. In the external electric field E dipole has the torque 𝑁 =
𝑝 × 𝐸, in result we obtain orientation dipole moment рор. Simultaneously, the electric field E induces
a dipole moment, defined as 𝑝𝑖 = 𝛼𝑑 Е. Actually, exactly orientation and deformation polarization
are the perturbation factor of the initial states of the atoms, and are the basis of the mechanism of
formation of the Stark splitting. In the first case, the interaction of an atomic system with the electric
field E occurs by direct electrostatic mechanism Δε = pE, manifested in a linear effect, while in the
second case (in the case of a non-linear effect) occurs by the induction mechanism ∆𝜺 = 𝑝𝐸.
According to this, we can consider that the Stark effect is universal and inherent in all atoms.
Because of the circular rotation of the dipole moment of the hydrogen atom, the energy
increment ±∆𝜺 due to the electric field E, symmetric with respect to the initial state of 𝜺 atom. This
explains the symmetrical nature of the Stark splitting relatively unperturbed line of the optical
spectrum.
Deformation polarizability of the hydrogen atom is relatively small and its linear effect in
comparison with other atoms is most pronounced. The linear contribution of other dipole atoms (for
example, alkali metal atoms and haloids) represented less, but it is not completely eliminated,
according to the atomic physics.
So, the Stark effect, which is based on the interaction of the dipole moment of the atom with an
external electric field, virtually confirms the validity of the dipole-shell model of atoms and,
respectively, the planetary model of the hydrogen atom.
Zeeman effect appears in splitting of atomic spectral lines under the influence of a magnetic
field. Zeeman effect finds its qualitative explanation in the framework of classical ideas on the
assumption about the presence of atomic magnetic moment. The interaction of the magnetic dipole
moment 𝜇𝐻 with the field H is represented as a small perturbation, so that ∆𝜺𝐻 = −𝜇𝐻 𝐻.
As said above, the magnetic dipole moment 𝜇𝑚 is an orthogonal component of the electric
𝑣
moment 𝑝𝑒 = 𝑒𝑎𝐵 consequently 𝜇𝑚 = 𝑝𝑒 𝑐 which occurs due to the orbital rotation of the electron
[13, 15]. This conclusion is a consequence of more common statement that magnetic field is a result
of moving electric charges [20, 19]. There is no principal difference between electric and magnetic
field. But there is a difference in the way they come from. In one case, it is created by the static
electric charges on the plates of the capacitor, and in other case – by electric currents in constructions
like solenoids. So, there is also no principal difference between Stark and Zeeman effects.
The meaning of the quantitative difference between Stark and Zeeman effects becomes clear.
The meaning is in the difference between electrostatic field Еst and electrodynamic (‘magnetic’) field
𝑣
Еdyn(≡Н),the connection between them is formed by the ratio Еdyn = 𝑐 Еst, where v – translational
speed of the movement of electrons in a conductor, c – speed of light [10]. So, Zeeman effect has an
electrical nature and electrostatic mechanism of its forming as a result of interaction of the
orthogonal component of the dipole moment of atoms (or molecules) to the electric field created by
strong electric currents. According to the physical meaning, Stark effect does not differ from the
Zeeman effect.
Stark and Zeeman effects may be the direct experimental proof of the planetary model of the
hydrogen atom of Rutherford-Bohr. The most obvious is a linear Stark effect, which appears only if
an atom has a permanent dipole moment.
2.4 Atom stability
The problem of atom stability is the key for the development of atomic physics. Discussion on
different aspects of this problem continues so far. There are some theses of this discussion in [12,
21]. Objection against the classical description of the hydrogen atom by (1), in fact, is reduced to the
problem of its stability. If an atom is a rotating dipole, it seems, it must inevitably radiate energy and
that is why the electron must fall into the nucleus. In many textbooks on atomic physics and radiation
theory this task is as an example of the non-applicability of the classical theory to describe atomic
©Бутлеровские сообщения. 2016. Т.47. №7. ___________________ http://butlerov.com/ __________________ 119
Full Paper ____________________________________________ A.A. Potapov, and Yu.V. Mineev
objects. At the same time, there is no rigorous solution, but there are only evaluative results 10−10 −
10−8 с and even 10−15 с. How right is this statement?
The answer to this question helps to give specially organized computing experiment: in the
simplest of the planetary model of the hydrogen atom the electron rotates around a fixed nucleus.
The main parameters of the model in polar coordinates are: module of radius vector – the distance
from the electron to the nucleus r, the angle of rotation φ and the time interval dt, when we can make
one calculation. Intermediate parameters are: azimuth speed, which helps to determine the speed of
𝑣 𝑑𝑡
rotation, i.e., the increment of the rotation angle 𝑑𝜑 = 𝑎𝑟𝑐𝑡𝑔 ( 𝑎𝑟 ) and the radial speed 𝑣𝑟 , i.e. the
encroaching speed of electron to the nucleus, which determines the increment of module of radius
vector 𝑑𝑟 = 𝑣𝑟 𝑑𝑡. Full speed is determined as 𝑣 = √𝑣𝑎2 + 𝑣𝑟2 .
Equation of motion is given by the balance of forces of the Coulomb attraction and centrifugal
𝑒2
𝑣𝑎2
𝑒
𝑟
force: 𝑎 = 𝑚
−
𝑟2
. Acceleration of radiation deceleration is calculated by the known formula for
2
2 𝑒 𝑑𝑎⃗⃗
finite motion 𝐹⃗𝑟𝑎𝑑 = 3 𝑐 3 𝑑𝑡 . Taking into account that the module
2
𝑒2
𝑣3
,
3 𝑚𝑒 𝑐 3 𝑟 2
𝑑𝑎⃗⃗
𝑑𝑡
is equal to
𝑣3
𝑟2
, we have 𝑎𝑟𝑎𝑑 =
which is the reason for reducing the azimuth speed. On the basis of these theses can be
formed infinite algorithm. The result of its fulfillment will be the result of a computing experiment.
The trajectory of the approach of the electron to the nucleus is quite interesting (see Appendix, fig. 2).
According to the planetary model, equilibrium state of an atom corresponds to the minimum of
function 𝜺(𝑟) (5) and the electron is in a potential well formed by the electron attraction energy of
the electron to the nucleus and by the energy of the centrifugal repulsion. So that any deviations from
the equilibrium position of the atom through a feedback mechanism based on the law of conservation
of momentum, tends to return the system to its original state. This provides stability of the hydrogen
atom.
Fig. 1. The potential function of the hydrogen atom (solid line)
as a sum of attractive and repulsive branches (dotted lines)
The maintenance of the atom stability is possible due to the balance between the forces of the
Coulomb attraction of the electron to the nucleus and the forces of centrifugal repulsion [16]. As
(2)
long as the energy of external forces W does not exceed the energy 𝜺𝐻 of the first excited state of
the atom (n = 2), the atom stays in its initial state owing to the maintenance of the constant value of
the angular momentum L = mvBaB = const. Indeed, expected fall of the electron to the nucleus
reduces the radius of а atom. But due to the angular momentum conservation law, reducing the
radius of the atom must be accompanied by a simultaneous increase in the speed of electron motion
v, so that the product va remains unchanged and equal vBaB (see Appendix, fig. 3). Similarly, the
120 _____________ http://butlerov.com/ _____________ ©Butlerov Communications. 2016. Vol.47. No.7. P.112-124.
EMPIRICAL SUPPORT FOR THE PLANETARY MODEL OF THE HYDROGEN ATOM _______________ 112-124
system returns the electron to its initial state, if the orbital radius is briefly increased as a result of
external interaction; in this case, recovery of its initial state is achieved due to reducing orbital speed
of electron (see Appendix, fig. 4). In the atom there is an automatic adjustment of the given energy
state, which happens due to the potential function (5) with typical minimum (fig. 1). So the atom
stability is provided by its own auto regulation system.
Analysis shows that the planetary model of the atom can be also applied to hydrogen-like
cations. The applicability of this model is explained by the structure likeness of two-particle
(electron-nuclear) configuration of one-electron cations to the hydrogen atom. The equation (5) is
applicable to them. Their radii and the bond energy are equal аZ = 𝑎𝐵 /Z and 𝜺𝑍 = 𝜺𝐻 𝑍 2 [5],
respectively. In fact, the hydrogen atom is a structure-forming element here.
The observed discreteness of radii and bond energy of one-electron multiply charged cations is
a consequence of discreteness of nucleus charge + eZ. The bond energy 𝜺𝑍 of cations is determined
by the Coulomb interaction between charges of nucleus +𝑒𝑍 and electron (−𝑒), separated by
𝑎
distance, which is equal 𝑍𝐵 . Radius 𝑎𝐵 of the hydrogen atom serves here as the initial measure of
𝑎
𝑎
𝑎
𝑎
𝐵
sequence of cations radii 𝑍𝐵 , 𝑍−1
, … , 3𝐵 , 2𝐵 , 𝑎𝐵 , which correspond an atomic number of Z element
in the periodic table [6]. Discreteness of this numerical sequence is defined by the discreteness of
nucleus charge +𝑒𝑍 . The bond energy, conjugated to the cations radius, also forms the numerical
sequence {𝜺H 𝑍 2 }.
The hydrogen atom also determines the equidistant law of forming radii ап of the hydrogen
𝜺
atom in the excited state ап = п𝑎𝐵 and energy levels corresponding to them 𝜺𝑛 = 𝑛𝐻𝟐 [6, 16]. Actually,
this is the difference between the planetary model and the Bohr model, where the radius of the atom
is not quantized, but the angular momentum is.
Appendix
1. Infinite cyclic algorithm of calculating model for solving the problem of the electron falling on the
nucleus:
2 𝑒 2 𝑣3
1) 𝑎𝑟𝑎𝑑 = 3 𝑚
𝑒𝑐
3
𝑟2
;
2) 𝑣𝑎′ = 𝑣𝑎 − 𝑎𝑟𝑎𝑑 𝑑𝑡;
𝑒2
3) 𝑎 = 𝑚
𝑒
−
𝑟2
𝑣𝑎′
2
𝑟
;
4) 𝑣 ′ = 𝑣 + 𝑎𝑑𝑡;
5) 𝑣𝑟 = √𝑣 ′ 2 − 𝑣𝑎′ 2 ;
6) 𝑟 ′ = 𝑟 − 𝑣𝑟 𝑑𝑡;
𝑣 𝑑𝑡
7) 𝑑𝜑 = 𝑎𝑟𝑐𝑡𝑔 ( 𝑎𝑟 ) ;
8) 𝜑 ′ = 𝜑 + 𝑑𝜑.
2. Results of computing experiment of the electron falling on the nucleus without feedback
mechanism.
Initial conditions of computing experiment correspond to the Bohr model of the hydrogen atom 𝑎𝐵 =
ℏ2
𝑚𝑒 𝑒 2
.
The end of the experiment is the moment of falling on the nucleus. It is a moment when
azimuthal speed is close to zero and the rotation stops, but electrons move to the nucleus in a straight
line.
Results of computing experiment when 𝑑𝑡 = 10−30 с:
 fall time – 5.1 ∙ 10−17 с;
 distance to the nucleus – 3.7 ∙ 10−4 Å, about 15 times larger than the classical radius of an
electron;
 speed – 1. 1 ∙ 108 m/c.
©Бутлеровские сообщения. 2016. Т.47. №7. ___________________ http://butlerov.com/ __________________ 121
Full Paper ____________________________________________ A.A. Potapov, and Yu.V. Mineev
Fig. 2. The electron falls on the nucleus
Results of computing experiment to determine atom stability to external effects, when there is the
feedback mechanism.
To provide a feedback mechanism in the algorithm above, the first and second rows are
ℏ
replaced by 𝑣𝑎 = 𝑚 𝑟.
𝑒
When the constancy condition is satisfied, the atom can reach a stable state under initial
experimental conditions, which are different from parameters of Bohr hydrogen atom.
Fig. 3. Initial electron position 𝑟 = 1.1𝑎𝐵
Fig. 4. Initial electron position 𝑟 = 0.9𝑎𝐵
Conclusions
1. The currently available experimental data on the polarizability, radius, dipole electric and
magnetic moments of the hydrogen atom, as well as Stark and Zeeman effects can serve as a
reliable rationale for the applicability of the planetary model of the hydrogen atom.
2. Another convincing argument in a favor of the planetary model is an agreement of the bond
energy data obtained on the basis of wave measurement (involving Rydberg fundamental
constant) and the bond energy obtained by calculation according to the fundamental constants of
charge and mass of the electron, taking into account the conservation law of angular momentum.
The matching error does not exceed the error in determining the physical constants used in the
calculations and is estimated with an accuracy of 6 decimal places.
3. Extremely high atomic stability is provided by automatically keeping the energy state of the atom
at the level specified by the nature. Auto regulation mechanism is based on the extreme nature of
the potential function of the hydrogen atom and the action of the conservation law of angular
momentum.
References
[1] W. Heisenberg. Physics and philosophy. Moscow: Nauka. 1989. 400p. (russian)
[2] R. Penrose. The Road tо Reality: A Complete Guide to the Laws of the Universe. Moscow-Izhevsk: RHD.
2007. 912p. (russian)
[3] I. Lakatos. Falsification and the methodology of scientific research programs. In the book: Kuhn T. The
Structure of Scientific Revolutions. Moscow: OOO “AST”. 2001. 608p. (russian)
122 _____________ http://butlerov.com/ _____________ ©Butlerov Communications. 2016. Vol.47. No.7. P.112-124.
EMPIRICAL SUPPORT FOR THE PLANETARY MODEL OF THE HYDROGEN ATOM _______________ 112-124
[4] L. Accardi. Dialogues about quantum mechanics: Heisenberg, Feynman, Akademus, Candido and a
chameleon on a branch. Moscow-Izhevsk: Institute of Computer Research; SRC “Regular and chaotic
dynamics”. 2004. 448p.
[5] A.A. Potapov. Fundamental basis of the matter structure. Butlerov Communications. 2015. Vol.41. No.2.
P.1-29. ROI: jbc-02/15-41-2-1
[6] A.A. Potapov. Renaissance of the classical atom. Moscow: Nauka, LAPLAMBERT Academic publishing.
2011. 444p. (russian)
[7] A.A. Potapov. Nature and mechanism of binding of atoms. Moscow: RIOR: INFRA-M. 2013. 295p.
[8] A.A. Potapov. Science of substance: the way out of crisis. In coll. “Actual problems of biology, chemistry,
physics”: materials of the international scientific-practical conference. Novosibirsk: Publishing house
“EKOR-kniga”. 2011. P.136-148. (russian)
[9] A.A. Potapov, Y.V. Mineev. Planetary model of a hydrogen atom and hydrogen-like structures. Butlerov
Communications. 2015. Vol.44. No.11. P.1-15. ROI: jbc-02/15-44-11-1
[10] Ch. Kittel, W. Knight, M. Ruderman. Mechanics. Moscow: Nauka. 1983. 448p.
[11] A.A. Potapov. Dielectric method of research substance. Irkutsk: Publishing house of Irkutsk University.
1990. 256p. (russian)
[12] A.A. Potapov. Deformation polarization: the search of optimal models. Novosibirsk: Nauka. 2004.
511p. (russian)
[13] A.A. Potapov. Radius of a hydrogen atom: fundamental constant. Science, Technics and Education.
2015. No.10. P.7-16. (russian)
[14] A.A. Potapov. Orientation polarization: the search of optimal models. Novosibirsk: Nauka. 2000. 336p.
(russian)
[15] A.A. Potapov. On the issue of electron spin. The Way of Science. 2015. No.11. P.19-28. (russian)
[16] A.A. Potapov. Optical spectrum of hydrogen atom: nature and formation mechanism. Nauka and Mir.
2015. No.11. Vol.1. P.13-28. (russian)
[17] Physical Encyclopedic Dictionary. Moscow: Sov. Encyclopedia. 1983. 928p.
[18] A. Sommerfeld. Atomic structure and spectra. Moscow: Technic Theoretical Literature. 1956. Vol.1
592p. (russian)
[19] A.A. Potapov. Nature and mechanism of formation of Stark and Zeeman effects of hedrogen atom.
Nauka and Mir. 2016. Vol.1. No.1. P.32-41. (russian)
[20] E. Pursell. Electricity and Magnetism. Moscow: Nauka. 1975. 440p. (russian)
[21] A.A. Potapov. High technology: quality control algorithm. Moscow: Nauka, LAPLAMBERT Academic
publishing. 2016. 122p.
©Бутлеровские сообщения. 2016. Т.47. №7. ___________________ http://butlerov.com/ __________________ 123
Full Paper ____________________________________________ A.A. Potapov, and Yu.V. Mineev
In the English version of this article, the Reference Object Identifier – ROI: jbc-02/16-47-7-112
Эмпирическое обоснование планетарной
модели атома водорода
© Потапов Алексей Алексеевич1*+ и Минеев Юрий Вячеславович2
Научный фонд имени А.М. Бутлерова. ул. Бондаренко, 33-44. г. Казань, 420066. Россия.
Тел.: +7 (843) 231-42-30; +7 (395) 246-30-09. Е-mail: [email protected]
2
Кафедра профессиональной подготовки. Восточно-Сибирский институт МВД России.
ул. Лермонтова, 110. г. Иркутск, 664074. Россия. Тел.: +7 (3952) 410-146. E-mail: [email protected]
1
_______________________________________________
*Ведущий направление; +Поддерживающий переписку
Ключевые слова: планетарная модель атома водорода, электрон, устойчивость,
фундаментальные константы.
Аннотация
Атом водорода выступает основой понимания электронного строения атомов, составляющих
таблицу Менделеева. До самого последнего времени представления о строении атома водорода
оставались дискуссионными. В настоящей работе принята планетарная модель атома водорода,
являющаяся развитием модели атома Резерфорда-Бора. Дано теоретическое описание планетарной
модели атома водорода, основанное на решении классической задачи Кеплера о движении электрона в
центральном поле ядра. Необходимым требованием для обоснования любой теории является
эксперимент. В качестве аргумента правомерности данной модели и ее теоретического описания
приведены данные измерений поляризуемости, радиуса, электрического и магнитного моментов
атома, а также эффектов Штарка и Зеемана. Центральным пунктом статьи является сравнение энергии
связи электрона с ядром, полученной на основании данных волновых измерений постоянной Ридберга,
с одной стороны, и энергии связи, полученной расчетным путем по данным фундаментальных
констант заряда и массы электрона и применением закона сохранения количества движения, с другой
стороны. Погрешность согласования данных расчета и эксперимента не превышает погрешности
определения физических констант, используемых при расчете, и оценивается на уровне нескольких
единиц 6-го знака после запятой. Субъективный фактор здесь исключен полностью, поскольку при
анализе в качестве исходных данных для сравнения были приняты утвержденные фундаментальные
константы.
Динамическое состояние атома объясняет предельно высокую структурную устойчивость атома
водорода, которая обеспечивается благодаря автоматическому поддержанию энергетического
состояния атома на заданном природой уровне. Механизм авторегулирования основан на
экстремальном характере потенциальной функции атома водорода и действии закона сохранения
энергии и закона сохранения момента количества движения. Иллюстрацией механизма устойчивости
атома водорода стали специально поставленный вычислительный эксперимент и компьютерная модель
вращательного движения электрона в центральном поле ядра.
Данный анализ должен помочь в утверждении планетарной модели атома водородаи положить
конец дискуссии о строении атома водорода.
124 _____________ http://butlerov.com/ _____________ ©Butlerov Communications. 2016. Vol.47. No.7. P.112-124.