11.1 - Introduction.
In this
chapter, we will give a physical description of the absolute changes that
happen inside an hydrogen atom when it is accelerated to a high velocity.
We have seen that this acceleration produces an increase of the electron
mass and in the Bohr radius. We also notice from chapter three that the
principle of mass-energy conservation is respected inside the hydrogen
atom without having to involve any change of electric charge when the
hydrogen atom is brought to high velocity. We will now show how the
absolute parameters of the hydrogen atom change when it acquires kinetic
energy. We present some considerations to the problem of internal
potentials inside the nucleus of atoms. Finally, we will see how the
nature of the interactions taking place inside nuclei can be predicted
using these considerations.
11.2 - Transformations inside
Fast Moving Atoms.
We have seen in
equation 3.4 that when the velocity of the hydrogen atom increases, the
absolute value of the Bohr radius a increases according to:
| av[rest] = gao[rest] |
11.1 |
where
ao[rest] is the Bohr radius at rest in rest units, and
av[rest] is the Bohr radius at velocity v, also in rest
units. The units in this chapter will always be rest units so that we will
drop the index [rest]. Let us use a numerical example to illustrate some
of the absolute changes taking place inside atoms accelerated to a high
velocity. When an hydrogen atom moves at v = 0.866c, then g = 2. We will consider the hydrogen atom in its ground
state but one can see that the transformations can be applied, in a
similar way for any excited state. From equation 11.1, when v = 0.866c, we
have:
We know from equation
2.23 that the electron mass increases when it moves faster:
Therefore the absolute
electron mass of a hydrogen atom moving at velocity v = 0.866c
becomes:
Figure 11.1
Figure 11.1,
illustrates the simultaneous increase of the Bohr radius and of the
electron mass when g = 2. Let us examine how
these results are compatible with the Bohr model, the de Broglie
wavelength of particles and quantum mechanics.
11.3 - Electric
Potentials.
Let us examine first
the compatibility between the description given above with the laws
regulating the electron and the proton in the hydrogen atom. We recall
that when an atom is accelerated to a high velocity, the electric charges
and the absolute electric field around those charges do not have to change
in order to remain compatible with the principle of mass-energy
conservation. The electric energy Eo of the electron in the
electric field of the proton is:
 |
11.5 |
where k is the Coulomb
constant, e+ and e- are the
electric charges of the proton and the electron and ao
is the average distance between the electron and the proton which
corresponds to the Bohr radius. Putting equation 11.2 in 11.5, we find
that the internal electric energy Ev inside the moving hydrogen
atom is:
 |
11.6 |
Since the compatibility
between observations and mass-energy conservation has been obtained
without modifying the electric charge when a particle is
accelerated, we can write:
| e-o [rest] =
e-v [rest] |
11.7 |
The problem of an
electric charge moving in a variable gravitational potential of the Sun,
has been considered previously (in the case of the Mercury problem) in
chapter one, and will be discussed later in detail in a separate
paper. This will not be discussed here.
In the case
of kinetic energy, in order to be able to establish comparisons, all
parameters are calculated using rest units. According to the Bohr model of
the atom, when the electron of the hydrogen atom moves in an electric
field (i.e. the field of the proton), one must have an equilibrium between
the attracting electric force and the centrifugal force. The
electric and the centrifugal forces are defined as:
 |
11.8 |
In order to
be compatible with Newton physics, these forces must be equal. We
have:
 |
11.9 |
Equations 11.8 and 11.9
give:
 |
11.10 |
For the hydrogen atom
at rest, the distance a is equal to the Bohr radius
ao. We have:
 |
11.11 |
For the hydrogen atom
at velocity v, a = av and we have:
 |
11.12 |
We recall that the
parameters  and  here are the electron velocities with respect to the proton. The
velocity of the hydrogen atom is expressed using g (and v without a subscript). Using equations 11.2 and
11.4 in equations 11.11 and 11.12 gives (for g =
2):
 |
11.13 |
where is the electron velocity with respect to the proton when the
hydrogen atom is at rest and is the electron
velocity with respect to the proton when the hydrogen atom has the
velocity v = 0.866c. From equation 11.13, the electron velocity (with
respect to the nucleus) is reduced by half when the hydrogen atom is
accelerated to a velocity v = 0.866c.
In order to be
compatible with the Bohr equation and quantum mechanics, the length of the
circumference of the orbit of an electron around a proton must be equal to
an integer number of the wavelength of the electron. In the case of the
hydrogen ground state, the electron wavelength must be equal to the length
of one circular orbit. The de Broglie wavelength l is given by:
 |
11.14 |
Putting equations 2.22,
2.23 and 11.13 in 11.14, the wavelength  in the
moving frame is :
 |
11.15 |
Equation 11.15 shows
that the electron wavelength orbiting the moving atom
is twice as long as the electron wavelength of the atom at rest.
Consequently, the radius of the electron orbit is twice as large, when the
atom moves at velocity v=0.866c. This satisfies perfectly the
conditions required above in equation 11.2 when the requirements of
quantum mechanics are applied.
This satisfies the wave
condition of the constructive interference of the electron wave after each
translation since the radius of the orbit (therefore its circumference) of
the moving atom is twice as large as the radius for the atom at rest as
illustrated on figure 11.1.
Furthermore, when all
these fundamental conditions are perfectly satisfied, the frequency of
emission of light between electronic transitions is reduced by two, since
the energy between the states is reduced by two when the atom is moving,
exactly as observed experimentally from the red shift of spectral lines
and from the slowdown of moving atomic clocks. It is this absolute
reduction of frequency of a moving clock located in a moving frame that
has been erroneously interpreted by Einstein as time dilation.
We must then
conclude that the predicted absolute change of parameters inside a moving
frame, resulting from mass-energy conservation is coherent inside moving
atoms. We must also note that all the transformations given above are in
perfect agreement with constant absolute electric charges in all frames
when kinetic energy is added to atoms.
(see equations 11.7 and 11.8). We recall that this absolute
electric field is similar to the absolute gravitational field shown in
chapters four and five. This agreement proves the invariability of the
electric forces as well as the quadratic decrease of the electric field
around charges in all frames. This result agrees perfectly with the
well observed experiment showing that electric charges moving at high
velocity in a magnetic field travel along a larger radius of curvature
corresponding to a different value of e/m. This smaller ratio of electric
charge over electron mass is due to an increase of mass (due to
kinetic energy) while there is no
change of electric charge. Therefore, experimental
results show that the ratio of the electric charge over the mass of
electrons (e/m) is different in fast moving particles.
11.4 - Sommerfeld Fine
Structure.
The prediction of the
advance of the perihelion of Mercury seen in chapter five is not the sole
example of the success of the principle of mass-energy conservation and
classical mechanics. There is also a well documented example in atomic and
molecular physics in which it is clearly observed that the principle of
mass-energy conservation influences the electronic structure inside atoms.
There are many similarities between Mercury moving inside the
gravitational field of the Sun and the electrons of atoms orbiting inside
the electric potential of the proton. However, an important difference is
that the electron mass is not concentrated into a relatively small
location with respect to the size of the atom contrary to the case of
Mercury and the Sun.
Since electrons exist
as waves, the electric potential between the electron cloud and the proton
can be calculated using the wave distribution given in
quantum mechanics. This leads to the same average energy and
distance ao that we would find if all the electron was
concentrated at a distance equal to the Bohr radius from the proton.
Consequently, one can calculate the potential of that electron cloud using
quantum mechanics, as if it were located at a distance from the proton,
equal to the Bohr radius. That electron cloud can either oscillate through
the proton if the angular momentum is zero or around it, if the angular
momentum is not zero.
When the electron cloud
is trapped into the electric field of a proton, an hydrogen atom is
formed. During its formation, energy is given up as emitted radiation.
This is similar to the energy that Mercury must release when it is trapped
into the Sun's gravitational potential. The electron cloud can be
distributed according to many configurations having different energies
corresponding to different quantum states. Consequently, during the
formation of each of those states, the electron loses mass the same way
Mercury does when it is trapped in the Sun's gravitational
potential.
Let us use the Bohr
model in which an electron moves on an orbit around the nucleus. We know
that the Rydberg states of hydrogen correspond to electrons traveling on
an orbit whose circumference is exactly equal to an integer number of the
wavelength of the electron. Then, there is a constructive interference of
the electron wave when moving to the next orbit around the nucleus. The
number of wavelengths forming the orbit is equal to the principal quantum
number. This model is compatible with the energies calculated by quantum
mechanics.
Experimentally, after
the Rydberg states were measured, it was noticed that the transitions
between these states are not as simple as originally expected. It was
discovered that the transitions between each pair of states are generally
made of several very close spectral lines. Sommerfeld carried out
calculations using general relativity and he discovered that instead of
simple transitions between quantum states, there should be multiple
transitions due to the fine structure.
Due to the change of
electron mass as a function of its distance from the proton, the
wavelength of the electron changes. Consequently, the radius of the orbit
changes because it is necessary to have an integer number of wavelengths
in a circumference. Due to that change of the distance from the proton,
the electrostatic potential changes so that the electron energy becomes
different. Consequently, the force (not the field) between
the electron and the proton does not follow exactly a quadratic function.
Therefore the electron orbit around the proton precesses as in the case of
Mercury around the Sun as given in equation 5.52. Due to this precession,
the transitions between different quantum states have slightly different
energies depending on the relative direction of the velocity of the
electron around the nucleus involved in the quantum transition.
Experimentally, the fine structure is well known. The Sommerfeld fine
structure constant is equal to:
 |
11.16 |
where h is the Planck
parameter.
This fine structure
term is observed between all quantum states as long as transitions are
allowed by the selection rules. Sommerfeld's fine structure is explained
in many textbooks [1].
It is often illustrated by precessing ellipses forming rosettes identical
to the path of Mercury on figure 6.2.
The Sommerfeld fine
structure constant can be explained more accurately using the principle of
mass-energy conservation as done in the case of the orbit of Mercury.
However, this is beyond the scope of this book. We will limit our
explanations to this qualitative description. We understand now that the
fine structure inside atoms is due to the principle of mass-energy
conservation. Of course, Sommerfeld's calculations do not lead to a
complete agreement in the case of an electron, because one must consider
the electron spin. However, this last correction is irrelevant in the case
of Mercury.
11.5 - Atomic Structure inside
Free Falling Atoms.
Let us study a hydrogen
atom falling freely in a gravitational field. We can assume that the atom
was initially located in outer space before it slowly started to drift and
accelerate gradually toward the Sun. Then, gradually, the hydrogen atom
acquires a high velocity. An observer accompanying the falling mass would
not feel any internal acceleration. We will now calculate
the absolute rate of the falling clock.
Let us examine this
problem separating mathematically the two components of energy acting on
the falling mass. With respect to a rest frame in outer space, the
speeding hydrogen atom is now at a location where there exists a
gravitational potential. We have seen that to calculate the exact mass of
the particle, this potential must be taken into account. Furthermore, the
falling hydrogen has acquired a velocity which must also be taken into
account.
We have seen in
equation 1.22 that the gravitational potential, where the atom is now
located, is such that the mass of the particle has decreased and is now
different from its mass in outer space. We also know that the kinetic
energy increases the mass of the particle by an amount, which must be
equal to the mass lost due to the potential energy.
This can be easily
calculated and we see that the decrease of mass due to the gravitational
potential compensates exactly the increase of mass due to kinetic energy.
Consequently, the absolute mass of the particle (proton and electron) does
not change while it is falling.
11.6 - High Potentials and Higher
Order Terms.
Contrary to Einstein,
in this book we have not arbitrarily postulated that physical quantities
are invariant in all frames. We have used only the principle of
mass-energy conservation. However, we have found that when we consider the
zero and first orders of v/c (or the gravitational potential), the
physical laws appear (almost) invariant in all frames as arbitrarily
assumed by Einstein. In that particular case, the physical consequences
are almost all identical to what Einstein found with his arbitrary
postulate. However, our results are obtained using solely the principle of
mass-energy conservation. The physical laws derived from the use of the
first order of v/c are invariant, up to the point we reach higher order
terms in (v/c)2 and other still higher
terms (however small) that have been neglected.
One could repeat all
the above calculations without neglecting the higher order terms. Then,
one could have an exact answer to the problem of extreme energies. We can
foresee that if we dealt with physical phenomena in which the higher terms
were not negligible (correction due to velocity), the physical laws
observed might be slightly different. Within those physical limiting
conditions, at high energy, the behavior of matter would not correspond to
the description we are used to see in a rest frame and in a frame in which
the ratio v/c is not too high.
We have to realize that
the experimental conditions that correspond to such high energies are
quite common in physics. It is clear that when the nucleus of an atom
emits particles having energies of millions of electron volts, the second
and third order terms of the potential involved are not negligible.
Consequently, we expect that the internal phenomena taking place in the
nucleus of atoms at such high potential taking into account the higher
order terms lead to physics we are not accustomed to see. It is for that
reason that nuclear forces are not familiar to us and to classical
mechanics. We believe that the principle of mass-energy conservation is
one of the ultimate principles in physics that possesses the wonderful
power of informing us, in a logical way, on the correct physical nature of
the forces involved in nuclear and particle physics. Mass-energy
conservation is relevant everywhere in physics and can be
applied everywhere in nature, especially when enormous potentials are
involved as in the nucleus of atoms and at the center of the stars.
A general
study of physics in which the principle of mass-energy conservation is
fully applied is beyond the scope of this book. However, we are convinced
that a physical and realistic description of our physical world can be
logically achieved without having to involve the non-realistic, the
non-conservation of mass-energy and the contradictory hypotheses used in
modern physics [2].
11.7 - References.
[1] H. Semat, Introduction
to Atomic and Nuclear Physics, Holt, Rinehart and Winston, Forth
edition, P. 245, 1962.
[2]
P. Marmet, Absurdities
in Modern Physics: A Solution, Les Éditions du
Nordir, c/o R. Yergeau, 165 Waller Street, Simard Hall, Ottawa, On.
Canada K1N 6N5, 144p. 1993.
11.8 - Symbols and
Variables.
| ao |
Bohr radius of the atom at rest |
| av |
Bohr radius of the moving atom |
| Eo |
Electric energy of the atom at rest |
| Ev |
Electric energy of the moving atom |
| lo |
de Broglie wavelength of the atom at
rest |
| lv |
de Broglie wavelength of the moving
atom |
| mo |
mass of the atom at rest |
| mv |
mass of the moving atom |
| vo |
velocity of the electron relative to the proton of
the atom at rest |
| vv |
velocity of the electron relative to the proton of
the moving atom |
|
