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Thematic Section: Theoretical Research. _____________________________________________________ Full Paper
Subsection: Theory of Structure of Matter.
Registration Code of Publication: 15-44-11-1
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(English Preprint)
Contributed: October 20, 2015.
Planetary model of the hydrogen atom and hydrogen-like structures
© Alexey A. Potapov,*+ and Yury V. Mineev
Irkutsk, Russia. Phone: +7 (395) 246-30-09. E-mail: [email protected]
___________________________________
*Supervising author; +Corresponding author
Keywords: hydrogen atom, electron, potential and kinetic energy, resistance, electron capture.
Abstract
The article discusses planetary model of Rutherford-Bohr in the application to the hydrogen atom and
hydrogen-like structures. The basis of the theory has been based on the laws of conservation of energy and
momentum of an amount of movement. The discrete nature of the optical spectrum of the hydrogen atom has
been described. The mechanism of electron capture by the nucleus of an atom has been proposed. The
program for computer modeling has been developed. The examples of computer research presented for the
hydrogen atom and hydrogen-like of cations.
The basis for constructing the whole of physics is the atom of hydrogen. It is the first chemical
element in the periodic table. It has the simplest structure. Moreover, there is a known accurate
solution to the equation of motion for it.
Electron motion equation. The atom of hydrogen is formed from elementary structural units, i.e.
electron and proton. The only way for them to co-exist is the dynamic system represented by the
proton as the center of attraction and the electron rotating around it. The mass of the proton is much
greater than the mass of the electron, and the distance between them is considerably greater than the
size of microparticles. Such parameters make the problem of atom description close to the problem
of the planet motion around the sun, also known as the Kepler problem [1, 2] fundamental for the
natural science. The Kepler problem about the electron motion in a centrally symmetric electric field
of the nucleus is solved on the basis of energy conservation law and the angular momentum . In
polar coordinates, these laws lead to two differential first-order equations with respect to unknown
functions r(t) andφ(t) [2, 3]:
whereeZ – nuclear charge of the atom,
(2)
In the limiting case of circular motion
= 0, equation (1) takes the following form
where
and
kinetic and potential energies, respectively, r – actual distance between the
nucleus and the electron; L– angular momentum equal to
; v – orbital speed of the electron
with mass m; eZ – nuclear charge.
The first summand of energy
in (3) 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 (3), the difference between the exponents of
the first and second summands with a distance equal to ris essential. This leads to the fact that as a
result of superposition of atom functions
and
, theresultant dependency
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):
Based on (4), we can find the bond energy corresponding to the equilibrium state ofaB
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where – bond energy, aB – the Bohr radius. At Z =1, we obtain an expression for the bond energy
of the hydrogen atom
Value
is in complete accord with the experiment,
eV, where IH – the ionization
potential determined by experiment. The Bohr radius calculated on the basis of (5) equals
=
0.529 Å.
An objection is usually raised to the planetary model of the atom. It uses a supposition that the
electron falls on the nucleus due to the losses of atom energy for radiation conditioned by the
electron rotation. At present, it is verified that during the electron motion at a constant orbital speed,
the atom in a steady state does not radiate [2] primarily because the circular motion implies that the
radial speed always remains perpendicular to the direction of the orbital speed of the electron, and
therefore equals zero. As for the centripetal acceleration as a potential reason for the electron
radiation, it is automatically compensated for by the centrifugal acceleration that accompanies the
motion of rotation [2, 4].
A distinguishing feature of electrodynamic description of the hydrogen atom using equation (3)
consists in the fact that it is based on the law of angular momentum conservation L = mvBaB. Within
this theory, there is no need to apply Bohr’s hypothesis about the quantization of angular momentum
of the electron as a value divisible by the Planсk constantnħ. Here, by the Planck constant h = 2 ħ
we mean a world constant introduced by Planck to explain the ability of a black body to radiate. The
numerical value of the Planck constant
was obtained by Planck on the basis of
experimental data [5].
On the other hand, the Planck constant is nothing but a moment of momentum (=angular momentum
= rotational momentum)
representing a universal physical value, for which the
fundamental conservation law is fulfilled. If we follow this assumption, the initial reason for atom
energy quantization observed in the experiment is not the quantization of the angular momentum of
the electron nħ (since it contradicts the law of angular momentum conservation) but the quantization
of the atom radius (see below). It is the law of angular momentum conservation that serves as the
basis for the rationale for the planetary model of the hydrogen atom. The planetary model makes it
possible to explain the nature of atom stability, the discreteness of energy levels, and the mechanism
of atom formation from the same standpoint.
In order to study the capabilities of the planetary model of the hydrogen atom and hydrogen-like
structures, we developed a computer program (see Appendix, including Fig. 2÷4).
Hydrogen atom stability. The equation of electron motion along a circular orbit is described by
potential function
according to (3). The minimum of function
corresponds to the
equilibrium state of the atom. In this case, the energy balance is achieved, when the centripetal
energy of the Coulomb attraction is balanced by the centrifugal repulsion energy. The phenomenon
of atom stability is based on the law of angular momentum conservationL = const.
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PLANETARY MODEL OF THE HYDROGEN ATOM AND HYDROGEN-LIKE STRUCTURES ________________ 1-15
To explain the mechanism of the hydrogen atom stability, Fig. 1 presents
repulsive
and
attractive
branches of the potential function (shown with a dotted line) and the potential function
itself
(shown with a solid line). Value
corresponds to the minimum of the potential curve
, which represents the result of superposition of the attractive and repulsive branches of the given
function. The resultant energy minimum determinesthe atom stability.
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 also possible due to the balance between the forces of the
Coulomb attraction of the electron to the nucleus and the forces of centrifugal repulsion that appear
when the electron is rotating in the central field of the nucleus.
As long as the energy of external forces does not exceed the energy of the first excited state of the
atom, the atom stays in its initial state owing to the maintenance of the constant value of the angular
momentum
. If the atom radius is reduced by
with respect to the equilibrium state
as a result of a disturbance (for instance, as a result of atom-atom collisions), the orbital speed of
the electron should increase by
by virtue of the constancy of the angular momentum, so that
product
remains unchanged. Therefore, any disturbance in the atom state causes a response
aimed at restoring the initial state. In fact, the atom has an automatic control system, where the
control actions are performed at a feedback signal that appears when the initial state of the atom is
disturbed. The process of automatic system control is based on the laws of energy and angular
momentum conservation according to (1) and (2).
The planetary electrodynamic model of the hydrogen atom satisfies the Lyapunov stability criterion,
according to which derivative
of the given function F in the neighborhood of the studied point (a
= aB) should have an opposite sign of the initial function F. In this study, such a function is
represented by function
as in (3).
To study the stability of the hydrogen atom and hydrogen-like structures, a computer program was
developed (see Appendix, including Fig. 5).
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Full Paper __________________________________________________________ A.A. Potapov, and Yu.V. Mineev
The nature of discreteness of the hydrogen atom energy spectrum.According to one of Bohr’s
postulates, the electron can jump from one stationary orbit nj to another ni, emitting or absorbing a
certain portion of energy equal to a change in energy from to
at a definite frequency fi, so that
h
,
(6)
whereh – the Planck constant, which is introduced using the current hypothesis about the corpuscular
character of radiation [6]. In this case, it is implied that the discreteness of optical spectrum of the
hydrogen atom is defined by the discreteness of the angular momentum nħ. Energy levels of the
optical spectrum are calculated on the basis of the ratio
, where
– bond energy of the
hydrogen atom in an undisturbed state, п– the main quantum number
( n  1, 2, )[6].
The discreteness of the optical spectrum, and therefore the discreteness of the atom energy levels in
the proposed planetary atom model equation (3) do not result from the postulate on the quantization
of the angular momentum. They result from the periodical character of the electron rotation.
In fact, the parameter of rotational motion of the electron is represented by the period of rotation ТВ
of the electron along the orbit with a radius of atom
frequency of electron rotation
nucleus charge
period of electron gyration Тп =
radius
and the orbital speed
angular momentum conservation
discrete values of speeds
divisible values of orbit radii
and orbital speed
or by the
which in turn are determined by the central field of the
. At the given strength of the electric field nucleus Е, the
in the п-thorbit (where п = 1,2,...) is strictly specified by the
of the electron in the given n-th orbit. By virtue of the law of
, discrete values of radii
should correspond to the
, so that
or
. It follows that the
correspond to the normal frequencies of the hydrogen
atom
which in fact determine the discreteness of energy atom levels according to (6).
Thus, it is not the angular momentum that changes in the transition from one energy level to another,
as the Bohr theory requires, but the radius of the atom and the orbital speed of the electron. The
electron transition to another "allowed" orbit means a transition to the orbit with the "allowed"
radius . In this case, the sought equation (3) takes the following form
Here, the discreteness of radii of the excited atom is taken into account and r=ап=п is substituted;
the second summand considers that an n-fold increase in radius
leads to an n-fold decrease in the
energy of the nucleus bond with the electron.
Equation (7) corresponds to the well-known equation in atomic physics, which gives a proper
description of the optical spectrum of the hydrogen atom for (6). However, the energy absorption in
the optical experiment is defined by the resonance condition, i.e. by the equality between the normal
atom frequency
and the frequency of the external source of oscillations
i.e.
Thus,
according to the law
, the observed discreteness of the hydrogen atom energy is the
consequence of the resonance mechanism of interaction between the atom and the external field at
normal frequencies of the hydrogen atom . This is where the physical meaning of quantization of
the bond energy of an electron with a hydrogen atom nucleus lies.
Within the framework of the planetary electrodynamic model, it is possible to explain the mechanism
and the nature of optical spectra. The bond energy
and radius
of the hydrogen atom are
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PLANETARY MODEL OF THE HYDROGEN ATOM AND HYDROGEN-LIKE STRUCTURES ________________ 1-15
interrelated and interdependent. Both of these values characterize the stability of the atom in its
dynamic state. As we have already mentioned, the maintenance of the atom stability is possible due
to the balance of forces of the Coulomb attraction between the nucleus and the electron and to the
forces of centrifugal repulsion. As long as the energy of external forces W does not exceed the
energy
of the first excited state of the atom at n=2, the atom stays in its initial state at a constant
level owing to the law of angular momentum conservation
. After the
achievement of energy W of the external field of value
, which corresponds to the energy of the
first excited state of the atom
and the distance between the electron and the nucleus equal
to
, the resonance condition
is fulfilled and energy W ofthe external source is
resonantly absorbed, which is revealed on the basis of the observed first absorption line of the optical
spectrum. State
is quasi-stable; it is maintained on conditionthat
, where
is
the energy of the second excited state at n = 3. Owing to the extreme characteristic of the potential
function
, within this interval of energies the electron stays in a quasi-equilibrium state
by the law of angular momentum conservation according to the mechanism described
above. When limit W =
is achieved, another resonance
takes place, which is
accompanied by the absorption of energy W of the external radiation source and registered as the
second absorption line in the optical spectrum of the hydrogen atom. The observed phenomenon
appears each time, when the condition of resonance
corresponding to the discrete
(quantum) energy levels
and fixed distances
is satisfied.
Polyvalent single-electron cations that are hydrogen-like by definition can also serve as an
illustraton. They can be obtained by removing all but one electron from the external atom shells. At
Z , equation (3) gives a description of the electron motion in the field of positive charge +eZ. In
terms of structure, polyvalent cations are different from the hydrogen atom only in the size of the
radius
. For them, the expression for energy bond is satisfied within the whole range Z
[2]. High accuracy of calculations is confirmed by experimental data on ionization potentials.
It seems important that there are certain ratios between the parameters of cations and the parameters
of the hydrogen atom in an unperturbed state: for the bond energy of cations
, where
– the energy of a bond between the electron and the hydrogen atom nucleus, and for
the radius of circular orbit of the cation
, where
orbit of the hydrogen atom nucleus, Z - index number of the element.
This result is obtained when the radius of cation
in the initial equation (3), so that
- radius of the circular
and the charge of cation +eZare substituted
The quadratic dependency of the cation bond energy
is determined by the fact that with a Zfold decrease in cation radius in a completely bound system nucleus-electron, the bond energy
increases to Z times, so that
.
Here, the hydrogen atom acts as a structure-forming element of hydrogen cations in the
sequence
according to the atomic number Z of the element in the
periodic table. The discreteness of this sequence is determined by the discreteness of the nucleus
charge +eZ. The bond energy connected to the radius of cations also forms a numerical sequence
.
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Full Paper __________________________________________________________ A.A. Potapov, and Yu.V. Mineev
It has to be mentioned that reliable data on ionization potentials
of single-electron cations
appeared only in the 1970s. This fact placed considerable constraints on researchers' possibilities for
the interpretation of the hydrogen atom structure. The appearance of the data on ionization potentials
of polyvalent cationsallowed the application of the planetary model to single-electron cations with
Z
as well, up to Z =100(!), i.e. to the single-electron cations of practically all elements of the
periodic table [2].
Today it is well-established that the hydrogen atom and hydrogen-like atoms have a planetary
structure, according to which a single-point electron rotates along the circular orbit in the central
field of the nucleus charge. All the structures derived from the hydrogen atom are circles, whose
radius corresponds to the sequence
,
, ,
,
,
,
,
quantitative description of single-electron structures is given using radius
, ,
,
.A
and bond energy
.
Thus, all theoretical and experimental data available at present confirm the applicability of
the planetary model of thehydrogen atom and hydrogen-like structures. The theoretical description of
the given planetary model is based on the fundamental classical laws of mechanics, electrostatics,
and electrodynamics. Within the framework of the planetary model, the essence of the atom as an
object of scientific knowledge is revealed.
The equations given above served as the basis for constructing the algorithm of the computer
program (see Appendix, Fig. 3).
Mechanism of the electron capture by the atomic nucleus.The formation of stable atomic structures
is connected to the phenomenon of electron capture by the nucleus or a positive ion. The idea of
electron capture was introduced by N. Bohrin the course of his studies on thedevelopment of a shell
model for multi-electron atoms [7].The real problem is directly connected to the aforementioned
dynamic theory of electron motion in the central field of attraction of the nucleus[1 ÷ 3].
The phenomenon of electron capture by the nucleus is observed in the evolutionary process of
transition from the plasma state of matter (as a sum of positive and negative charges) to the bound
atomic state. One of the main plasma parameters is the so-called Debye screening length , which
characterizes the screening efficiency of the charge of the selected ion.This means that the electrons
near the ion are attracted to it, and therefore, screen the electric field created by the selected ions. As
a result, the electric field of the ion weakens and becomes insignificant at a distanceof
. In
equilibrium, the space charges of electrons and nuclei (or ions )compensate for each other so
that the resulting field in plasma turns out to be equal to zero. This is a quasi-neutral plasma state.
A transition of plasma charges to the bound atomic state takes place due to the collisions between the
charged particles caused by a decrease in temperature Т. The mechanism for binding the charges
consists in the fact that the condition of plasma quasi-neutrality is not satisfied at some critical values
of temperature Тand concentration of ions N, and the electrons get in the sphere of the Coulomb
nuclear attraction. Under certain favorable conditions (“impact” parameter and an appropriate speed
of electron motion), there appears a possibility of atom formation as a system of completely bound
and mutually connected electron and nucleus.
The following mechanism of electron capture can be suggested. In the general case, the kinetic
energy of free electron motion in the central nuclear field (core) is formed by the radial and
azimuthal components, which represent the result of speed vector resolution into radial and
azimuthal components, so that [2, 8]
where
translational speed of an electron,
angular momentum of the electron-ion pair
expressed using the relative radius vector, r,
variables of distance
and angle, which determine the position of an electron with respect to the nucleus (or the atomic
core).
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PLANETARY MODEL OF THE HYDROGEN ATOM AND HYDROGEN-LIKE STRUCTURES ________________ 1-15
The total energy of a pair of particles that are not connected to each other consists of kinetic and
potential components[2, 9, 10]
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
The solution to this equation is function[3]
whereА – an arbitrary constant determined on the basis of the initial conditions.
Equation(11) represents the electron motion trajectory. At the same time, it is the equation of
conic section in polar coordinates, which has the following form [3]
where э – eccentricity, с – a parameter of the electron motion trajectory, which corresponds to 4
possible types of functions: 1) hyperbola,at э ; 2) ellipse,at 0
; 3) parabola,at
; 4)
circle,at
.
The azimuthal contribution of kinetic energy to (10) represents the centrifugal energy. It plays
the role of a repulsive barrier to the attraction forces from the nucleus (ion). The smaller the distance
between the particles, the “higher” the barrier. When centrifugal barrier
reaches its maximum,
radial member
in (10) becomes zero. At this turning point the electron starts moving either along
the parabola (at
or along a closed elliptical orbit (at
. Therefore, the nucleus (atomic
core) captures the electron on condition that0
andε . In the general case, the electron
capture results in the formation of a stable stationary elliptical orbit, whose equation (according to
(12)) is as follows
whereа – semi-major axis of the ellipse.
At
, equation (13) represents a description of the circular orbit.
The stationary electron motion trajectory in the central field of ionqis described using equation [2, 9,
10]
where kinetic energy includes radial
and azimuthal
components in the general case;
value
represents the potential energy of electron attraction to the ion. The difference between
equations (10) and (14) consists in the fact that the first one refers to the free (unbound) electron and
the second one – to the bound electron in the orbit.
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Full Paper __________________________________________________________ A.A. Potapov, and Yu.V. Mineev
The stability of the bound state of charges in an atom is achieved owing to the law of angular
momentum conservation (see above).The potential function
of such a system represents the
dependency of the total energy on distance between the charges (Fig.1). It has a characteristic
minimum, which ensures the stability of the atom as a whole.
Based on the previous analysis, we can explain the phenomenon of electron capture. It
consists in the competition between the Coulomb forces of electron attraction to the nucleus, on the
one hand, and the forces of centrifugal repulsion, on the other hand. The electron capture by the
nucleus (atomic core) becomes possible when the total energy εfor (14) is greater than the kinetic
energy εКof translational motion of the electron.
As it has already been mentioned, condition
is true for the circular orbit of the
electron. This condition is satisfied when the kinetic component of energy
equals the total energy
of the atom in absolute value. In this case, the radial component of energy in (14) becomes equal to
zero. Among all the elements of the periodic table, only the hydrogen atom satisfies this condition.
The physical meaning of the circular orbit of the hydrogen atom consists in the fact that in the
absence of external disturbing factors, the electron motion in the central nuclear field is determined
bythe absolute equality of nucleus and electron charges. With respect to other atoms and ions, the
hydrogen atom represents a standard that reveals the mechanism of structure formation.
The phenomenon of electron capture by the nucleus was studied using computer
modeling(see Appendix, including Fig. 7, 8, 9).
Conclusions
1. In the planetary model, the hydrogen atom single-electron polyvalentcations represent a
bound system that consists of a nucleus (the center of attraction) and a single point
electron that rotates around it in a circular orbit.
2. The stability of the atom and hydrogen-like cations is ensured owing to the balance
between the Coulomb forces of electron attraction from the nucleus and the forces of
centrifugal repulsion conditioned by the rotational motion of the electron. While in the
steady state, the electron is in a potential well formed due to the joint action of the
Coulomb energy of electron attraction to the nucleus and the energy of centrifugal
repulsion, which have different power law dependencies on the inverse distance between
the nucleus and the electron: linear and square ones, respectively.
3. The discreteness of energy levels of the hydrogen atom and single-electron cationsis
conditioned by the periodicity of the electron rotation in circular orbits, which are usually
given by the orbit radius and the orbital speed.
4. Hydrogen atom and single-electron cations are formed as a result of the capture of a free
electron by the nucleus to the circular orbit, which is possible due to the forces of
attraction from the nucleus charge on the one hand, and, on the other hand, to the forces
of centrifugal repulsion that appear in the central field of the nucleus due to the electron
speed resolution into radial and azimuthal components.
References
[1] V.V. Multanovsky. Classical mechanics. M.: Drofa. 2008. 384p.
[2] A.A. Potapov. Renaissance of the classical atom. М.: Publishing house “Nauka”. LAP LAMBERT
Academic publishing. 2011. 444p.
[3] Ch. Kittel, W. Knight, M. Ruderman. Mechanics. М: Nauka. 1983. 448p.
[4] I.E. Pekhotin. Axiom of circular motion. M.: Publishing house «Sputnik». 2010. 64p.
[5] O.P. Spiridonov. Fundamental physical constants. M.: «Vysshayashkola». 1991. 340p.
[6] P. Atkins. Quanta. M.: Mir. 1977. 496p.
[7] N. Bohr. Selected studies. M.: Nauka. 1970. 584p.
[8] W. Flygare. Molecular structure and dynamics. M.: Mir. 1982. Vol.1 and 2. 872p.
[9] A.A. Potapov. Science of substance: state of research. Butlerov communications. 2011. Vol.24. No.1. P.1-15.
[10] A.A. Potapov. Theory of matter: perspectives for development of the predictive theory. Butlerov
communications. 2011. Vol.24. No.1. P.31-45.
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PLANETARY MODEL OF THE HYDROGEN ATOM AND HYDROGEN-LIKE STRUCTURES ________________ 1-15
APPENDIX
The computer program basedon the principles of the atomic planetary model is aimed at completing
three main dynamic tasks:
a) electron motion in the hydrogen atom and hydrogen-like cations with a possibility of changing the
charge of the atom;
b) stability of the atom and the process of its excitation;
c) atom formation process (electron capture).
The look of the interface of the program is shown in Fig. 2.
The main algorithm of the program is a linear algorithm in an infinite loop. It can be represented by a
simple flow chart (Fig. 3).
The program code corresponding to the flow chart is written in the following order:
1)
Calculation of the azimuthal speed:
(here and elsewhere is the
assignment operator, as is common for computer programs), where – the reduced Planck constant,
– mass of the electron, – distance between the electron and the nucleus.
2)
Determination of the acceleration:
, here the first summand is
the acceleration due the Coulomb interaction, the second summand is the acceleration determined by
the centrifugal force, the third summand is the acceleration created by the radiation reaction force,
which can be ignored due to its small value.
3)
Based on the acceleration value and the previous value of speed, we calculate the
instantaneous speed as
, where
– a time interval arbitrarily specified by the user (in
the considered program is given in the interval from
to
).
4)
The radial speed is determined on the basis of a suggestion that there exist non-central
action forces best observed in the atomic processes, where they transform the translational motion
into the rotational one by changing the direction of motion and not the speed value.
Therefore,
.
5)
The instantaneous radial speed determines an increase in the distance between the
interacting charges
.
6)
The angle of rotation is determined as
.
7)
The conversion of polar coordinates into Cartesian coordinates is performed
conventionally:
,
, where and
– coordinates of the electron with
respect to the nucleus.
8)
The state of the atom is determined by its energy. Potential energy is calculated as the
,kinetic energy –classically, using formula
Coulomb interaction energy
Or without explanation:
.
;
;
;
;
;
;
;
,
;
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Full Paper __________________________________________________________ A.A. Potapov, and Yu.V. Mineev
,
.
The program makes it possible to change the value of the nuclear charge at user’s request and
monitor the state of any hydrogen-like cation(Fig. 4).
The program also allows the demonstration of the atomic system stability in thecourse of energy
transmission, which does not correspond to the transition from one steady state to another (Fig. 5).
In fact, the excitation process is close to the process explaining the atom stability. It is performed by
a similar means taking into account the requirement for the angular momentum conservation and
orbit quantization (Fig. 6).
The process of electron capture by the proton, i.e. the process of forming the hydrogen atom from
free particles is presented in Fig. 7 and 8. Figure 7 shows that at the beginning, the stationary
electron is at a distance of
.The figure demonstrates the process of electron attraction and
acceleration and a change in the character of its motion from the translational to the rotational one.
Figure 8 describes the process of atom formation when the electron has a considerable initial speed
(20000 m/s) and is at a considerable initial distance (1µm).
If we double the initial speed of the electron (Fig. 9), the capture will not take place. The electron
will pass the nucleus and the electric forces will increase its speed and change its trajectory.
Fig. 2.The look of the program interface
The upper line of the interface has five pull-down menus with commands that allow us to control the object
(electron), speed up and slow down the motion (by choosing a certain time interval), change the scale (one
twip (pixel) corresponds to
; in the given case, the scale is increased fourfold), and monitor the state
of the system (the distance between the nucleus and the electron; relative speed of the electron; potential,
kinetic, and total energy). The main part of the screen is on the left in the middle. It is intended for the
demonstration of the electron motion trajectory. The right part with the graph of the potential function in the
background serves to assess the changes in the energy state of the atom (see Fig. 1).
Fig. 3.Flow chart of the main part of the program
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Fig. 4.A solution to the problem of «Cation nuclear charge selection»
Here you can see the planetary model of the lithium cation (in the same scale as the hydrogen atom model,
Z=3). The values of the orbit radius, speed, kinetic, total, and potential energy are displayed on the screen.
The value of the total energy corresponds to the experimental data on the total ionization potential.
The main screen field shows the trajectory of the electron motion in the orbit, in the center of which there is a
positive nucleus. The graph of the potential function is on the right. For the sake of space-saving,the
maximum value in the graph remains the same for all the cation charges. However, the function minimum is
considerably shifted to the origin of the coordinates.
Fig. 5. Atom stability
The program fulfills the task that consists in explaining the mechanism of atom stability. The hydrogen atom
receives the energy of 7 eV instead of 10.2 eV necessary for a transition from the main level to the first
excitation level.
The menu of excitation is displayed on the upper right of the screen. The main part of the screen shows the
electron trajectory (clockwise motion) that can be divided into four sections: the ground state (a circle with a
smaller radius); effect on the atom (an arc that goes from the left edge of the smaller circle to the upper right
of the larger circle); a short period of motion in the non-stationary orbit (one and a half revolutions around the
circle with a bigger radius); a transition to the stationary orbit (an arc that goes from the lower part of the
bigger circle to the upper part of the smaller circle).
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On the right, the graph of the potential function (a dotted line) shows the energy processes (solid lines): left
vertical line (1→2) represents the process of an instantaneous loss of potential energy by the electron due to
the external influence; upper curve (2→3) represents the process of transition to the forbidden energy level;
the right vertical line(3→4) and the lower curve (4→1) demonstrate an inverse process, i.e. a return to the
previous state.
Fig. 6.Atom excitation
Here, we consider the dynamics of atom excitation using the hydrogen atom as an example. Three episodes
are presented: a transition from the stationary state to the first excited state; a transition from the first excited
state to the second excited state; de-excitation.
The figure modeling the electron trajectory shows three circles that correspond to different energy states and
the arcs that connect these circles indicating the trajectory of transition (the motion is clockwise).
In the graph of the potential function, the processes are duplicated: line1→2→3 – a transition from the main
energy level to the second one (transition energy equals 10.2 eV); line 3→4→5 – a transition from the second
energy level to the third one (transition energy equals 1.9 eV); line5→6→7 – de-excitation and return of the
atom to its ground state (transition energy equals 12.1 eV). The changes in the energy state have a step
structure.
Fig. 7.Electron capture at small (atomic) distances
The figure presents the process of electron capture by the proton, i.e. the process of the simplest atom
formation (hydrogen atom formation) from previouslyunbound particles. At the beginning of motion, the
stationary electron is at a distance of
from the proton (see the data on the upper right of the screen).
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Full Paper __________________________________________________________ A.A. Potapov, and Yu.V. Mineev
The process of acceleration and transformationfrom translational motion to the rotational one (the initial
position of the electron is marked with a point) is vividly seen in the main part of the screen where the
electron motion trajectory is demonstrated.
The final characteristics of the electron (its speed and distance to the proton) are given on the upper left of the
screen. They correspond to the design data for the planetary model.
The graph of the potential function reflects the changes in the energy state of the electron as it approaches the
proton.
Fig. 8.Atom formation process when the electron with some speed is at a considerable initial distance
The figure demonstrates the atom formation process when the electron has a considerable initial speed (20000
m/s) and is at a considerable initial distance (1 µm). These conditions can be considered limiting, since any
further increase in the speed or distance will not lead to the atom formation.
The right part of the screen displays the initial parameters of the electron speed and distance from the proton.
The main screen field shows the electron trajectory, where the electron changes its direction and starts
approaching. The scale is significantly reduced (1:2048). The beginning of motion is marked with a point at
the bottom of the screen. The values of distance, speed, and total energy of the electron are given for the
moment when the program ends. The total energy has a negative value, which indicates the beginning of the
capture. If the motion continues, the capture is inevitable.
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Fig. 9. Electron passing the nucleus
As compared to the previous example, the initial speed of the electron is doubled – 40000 m/s. The initial
distance is the same - 1 µm. For the sake of clarity, the scale is increased fourfold.
The high initial speed of the electron does not allow the proton to capture it. The electron passes the nucleus.
Moreover, the electric forces increase the electron speed and change its trajectory. The electron is not captured
and the atom is not formed.
It has to be noted that the value of the total energy is positive and the value of speed is greater than the initial
value despite the fact that the electron is moving away from the proton. The total energy value is greater than
zero.
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