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8.
THEORY OF SPECTRA OF THE ATOMS AND THE IONS
The development
of the theory of spectroscopy was initiated by Balmer-Rydberg and Niels Bohr
when they were able to calculate the spectrum of the hydrogen atom [26], [53].
Schoedinger’s equation was a climax of the theory of spectroscopy [111], [122].
It gave the possibility to calculate the spectra of all hydrogen-like atoms
(the atoms with one electron). But the possibilities of Schroeginger’s equation
were very limited. The spectra of all further electrons (beginning from the
atomic nucleus) are calculated with the help of Schroedinger’s equation no
more. As it has been noted in the fundamental work [9], in these cases the
approximate methods are implemented in order to calculate the spectra of the
atoms and the ions.
The main drawback of Schroedinger’s equation is in the fact that it does
not give the opportunity to determine the exact position of the electron in the
atom, it shows only the density of its presence in this or that field.
Empirical character of the approximate methods excludes the possibility of the exact
determination of the position of the electron in the atom and makes the
formation of the notions concerning its interaction with the atomic nucleus
difficult. Due to it, in modern quantum physics the electrons are distributed
along the shells, levels and sublevels.
One should note a large work carried out by the spectroscopists -
experimenters [5], [25], [60], [61], [62]. They have measured the
characteristics of several hundred thousands spectral lines, but there is no theory,
which could allow to calculate these spectra. But the models of the photon and
the electron, the main participants of the process of formation of the spectra
of the atoms and ions found by us, give the possibility to begin the
elaboration of such theory.
As both the photon and the electron have spins equal to Planck’s
constant, which is vector quantity, the rules of addition and subtraction of
energies of the photons and the electrons should be based on the rules of vector algebra [43].
8.1. Spin of the Photon and the Electron
In quantum physics, the notion of spin in characterizes the rotation of the particles. According to Maxwell
theory and the electromagnetic wave model originating from it, the spin of the
photon is considered to be equal to a unit and is directed along the trajectory
of its motion. The values ±1/2 are
ascribed to spin of the electron, and the direction is perpendicular to its
orbital plane in the atom [148]. Actually, it is an orbital spin of the
electron.
In classical mechanics, the concept of moment of momentum (angular
momentum) of the body in relation to an axis, which cross its center of mass,
corresponds to the concept of spin. We have already shown that energy of the
photon 
and energy
of the
free electron is determined according to the identical formulas:
(198)
(199)
As during absorption or emission of the photons by the electrons their
energies are added or subtracted and as they are vector values, this addition
and subtraction should be carried out according to the rules of vector algebra.
From the physical point of view when
the photon is absorbed by the electron, their axis of rotation should be
parallel. It should originate from the mathematical models of the formation of
the spectra as well.
Let us analyse only one energy transition of the electron in the
hydrogen atom. Binding energy 
of the electron while
its staying at the first energy level of this atom is equal to ionization
energy 
of the hydrogen atom,
i.e. 
electron volt (eV).
When the electron absorbs the photon with energy of 10.20 eV and goes to the
second energy level, energy of its connection with the nucleus is reduced and becomes
equal to 3.40 eV. Naturally, when the photon is absorbed by the electron, their
energies are added, and we should write
(200)
But this result conflicts with the experiment, which points out to the
fact that binding energy of the electron with the nucleus after the photon’s
absorption is not increased, it s decreased, and it becomes equal to 3.40 eV,
not 23,80 eV. That’s why the previous ratio should be written in the following
way [109]
(201)
In order to liquidate a contradiction in the formula (201), it has been
agreed that energy of the electron in the atom should be considered as negative
one, and the formula (201) should be written in the following way
(202)
But it is very difficult to agree. The thing is that the electron in the
atom has potential component and kinetic component of its full energy. The
above-mentioned agreement is acceptable for potential energy, but it cannot be
spread to kinetic energy. That’s why it is necessary to try to find more
convincing proof of reasonableness of the agreement being reflected in the
formula (202).
First of all, there is no complete energy 
of the electron in
the formula (202). The value 13.60 eV is equal to ionization energy 
of the hydrogen atom.
The sense of this energy is in the fact that if the electron absorbs the photon
with energy of 13.60 eV, it will lose the connection with the nucleus after it
and will become free. It means that the value of 13.60 eV corresponds to
binding energy of the electron with the nucleus of the hydrogen atom at the moment
when it is on the first energy level. Energy 
is energy of the
absorbed photon, which provides a transition of the electron to the second
energy level, that’s why energy 
, which is equal to difference 13.60-10.20 = 3.40 eV, corresponds
to binding energy of the electron with the nucleus of the atom at the moment of
its stay at the second energy level [109].
Let us introduce full energy 
of the free electron
into the equation (202) [109].
(203)
We should remind that 13.60 eV is energy of ionization of the hydrogen
atom. It corresponds to binding energy of the electron with the proton at the
time when the electron is on the first energy level, and 3.40 eV is binding
energy of the electron with the proton, which corresponds to the second energy
of the electron; 10.20 eV is energy of the absorbed photon. We can remove value
from the equation
(203); due to it the equation is not changed, and it will assume the similitude
of the formula (202). It is clear that energy of the electron in the atom is a
positive value, and the equation (202) reflects the change of binding energies of the electron only during
its energy transitions, and minuses before the values 13.60 and 3.40 mean not
negativity of energy, but the process of subtraction of energy being spent for
the connection of the electron with the proton [109].
It is clear that at the time
of the stay of the electron on the first energy level in the hydrogen atom its
full energy 
is reduced by energy
value 13.60 eV of its binding with the nucleus. When the photon with energy 
is absorbed, binding
energy of the electron with the nucleus is reduced up to 13.60-10.20=3.40 eV.
As it is clear, the law of conservation of energy is strictly observed in
relation (203). Let us write similar relations for transition of the electron
from the first to the third energy level and to the fourth one.
,
(204)
(205)
It is easy to note that as the electron is pulled out from the atomic
nucleus, its binding energy 
with the nucleus is
changed according to dependence
,
(206)
where n=1, 2, 3 ….. is a number of the energy
level of the electron in the atom, the main quantum number.
This is a mathematical model of the law of the change of binding energy
with the nucleus of the hydrogen atom and the hydrogen-type atoms. Let us pay
attention to the fact that in this case ionization energy 
is equal to binding
energy 
of the electron with
the nucleus corresponding to the first energy level.
The law of formation of the absorption spectra of the atom of hydrogen
and hydrogen-type atoms originates from the relations (202), (203), (204),
(205) and (206).
(207)
As the spectral lines of absorption coincide with the spectral lines of
emission, the mathematical model of the law of radiation should be the same as
the one of the law of absorption. It is natural that it does not radiate at the
time when the electron is on the first energy level, because this level is the
ultimate one for it. But if it is on the second energy level, it can emit the
photon with energy 
. In this case the equation of the radiation process will be
written in the following way
.
(208)
When the electron is on the third (n=3) energy level and the fourth
(n=4) one, it has binding energies with the nucleus 
and 
. When the electron goes from the third energy level and the
fourth one to the first level, it will emit the photons with energies: 
and 
, and the equations of these processes will be written in the
same way:
,
(209)
.
(210)
In
general, these ratios will be written in the following way
. (211)
If we reduce by 
and convert, we’ll
find [109]
, (212)
It
coincides completely with the equation (207). Thus, the same mathematical model
of the laws of radiation and absorption of the photons by the electrons during
their energy transitions in the atoms originates from the equations of
absorption (203), (204) and (205) and radiation (208), (209) and (210).
Now let us consider the physical sense of energies being included in the
law (207) of formation of the spectra of the atoms and the ions. 
is energy of the
absorbed or emitted photon. 
is ionization energy,
which is equal to energy of the photon after it being absorbed the electron
loses the connection with the nucleus and becomes free. It means that
ionization energy is determined according to the same relations as energy of
the photon 
. Binding energies of the electron with the atomic nucleus
(213)
are equal to the energies of the photons. For
example, in the hydrogen atom the binding energy 
of the electron with
the atomic nucleus, which corresponds to the first energy level, is equal to
energy of its ionization 
. That’s why 
. If it is taken into consideration, the mathematical model
of the law of emission and absorption (207) of the photons by the electrons
during their energy transitions in the atoms can be written in the following
way [109]
. (214)
Or
(215)
We have got the mathematical model of the law of formation of the spectra
of the atoms and the ions, which includes only frequencies of the photons being
absorbed or emitted, i.e. frequencies of rotation of the photons in relation to
their axes. And where is frequency of rotation of the electron round the
nucleus of the atom? None. The energy model of this law (207) has no energy,
which corresponds to the orbital motion of the electron. This is an amazing
fact. For nearly one hundred years we have supposed that the electron in the
atom rotates round the nucleus like a planet round the Sun. The law of
formation of the spectra of the atoms and ions (207), (212), (215), which
describes energy transitions of the electron in the hydrogen atom, denies the
orbital motion of the electron. This law contains no energy, which corresponds
to the orbital motion of the electron; this means that it has no such motion.
This amazing conclusion makes us think about many things, first of all, about
haste in reception of the results of interpretation of many experiments.
The
Foundations of Physchemistry of Microworld
Copyright Ó2003 Kanarev Ph.
M.
Internet Version - http://book.physchemistry.innoplaza.net
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