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Kip Thorne introduces his article on black holes in Scientific American by stating:Or all the conceptions of the human mind from unicorns to gargoyles to the hydrogen bomb perhaps the most fantastic is the black hole: a hole in space with a definite edge over which anything can fall and nothing can escape; a hole with a gravitational field so strong that even light is caught and held in its grip; a hole that curves space and warps time. Like the unicorn and the gargoyle, the black hole seems much more at home in science fiction or in ancient myth than in the real universe. Nevertheless, the laws of modern physics virtually demand that black holes exist. In our galaxy alone there may be millions of them.Thorn is saying that because the ―laws‖ of modern physics require them, black holes must exist. However, it is more rational to conclude that those ―laws‖ which give rise to the gargoyles, unicorns, and black holes of physics are wrong—that, as ordinarily expected, deductions from false premises yield bizarre results. Let us now investigate how such concepts as the black hole arose historically. The theory of the black hole stems from the theories of general relativity, the nuclear atom, and the hydrogen-to-helium conversion process in stars.In the 1930s, Subrahmanyon Chandrasekhar‘s investigation of stellar evolution and structure led him to conclude that, in the process of converting hydrogen to helium, most stars lose energy and contract until internal pressures become great enough to cause collapse of atomic structure. Back in 1924, Sir Arthur Eddington had suggested that the high density of the white dwarf companion of the bright star Sirius was due to ―electron degeneracy,‖ with all electrens stripped from individual atoms. Chandrasekhar seemed to provide an explanation of how this could occur.At this point someone might have pointed to a simpler solution: perhaps the nuclear atom concept was incorrect because of the grave difficulty in explaining the high density of the white dwarfs. Perhaps atoms do not have electrons circling around ihem at relatively large distances. Perhaps the postulated hydrogento-helium conversion process in stars was incorrect...Chandrasekhar believed that a ―non-relativistic gas‖ at the center of a white dwarf could always adjust itself until the gravitational forces compressing the star are countered. liowever, according to the theory of general relativity, with a certain limiting mass, the gravitational forces are net countered fully and so the star does not come into equilibrium. The limiting mass, termed the Chandrasekhar limit, has been calculated to be 1.2 solar masses.²Oppenheimer and Volkoff considered what would happen to stars of mass larger than the Chandrasekhar limit. As the central density increases, inverse beta decay would take place, driving electrons into protons. Thus increasingly rich neutron elements would be formed—giving rise to a ―neutron star.‖ Recently astronomers have concluded that the pulsars are neutron stars.
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RECIPROCAL SYSTEM
DYNAMICS
Ronald W. Satz
RECIPROCAL SYSTEM DYNAMICS
RONALD W. SATZ
Some Myths of Modern Physics
White Lies About Black Holes
A Tall Tale: A Review of Stephen W. Hawking‘s A Brief History of Time
Four Scientific Mysteries Unraveled
Reference Systems
Reference Systems and Speed Limits in the Reciprocal System
The Lorentz Transformation
The Two-Photon Problem
A Note on Scalar Motion
A Note on the Force of the Space-Time Progression
On the Nature of Undisplaced Space-Time
Clock Space, Coordinate Space, Clock Time, Coordinate Time
Particle Physics
Cosmic Rays and Elementary Particles
Identification of Cosmic Particles 3695 MeV/c² and 3105 MeV/c²
A Note on the Cosmic Proton
The Cohesive Energies of Crystals of the Elements
Progress on the Theoretical Calculation of the Cohesive Energy of the Elements
Equation of State of Solid Matter
Further Mathematics of the Reciprocal System
A New Derivation of Planck‘s Constant
Time Region Particle Dynamics
Calculation of the Dissociation Energy of Diatomic Molecules
The Liquid State in the Reciprocal System: The Volume/Pressure Relation, part I
The Liquid State in the Reciprocal System: The Volume/Pressure Relation, part II
Electricity and Magnetism
Permittivity, Permeability and the Speed of Light in the Reciprocal System
The Unit of Magnetic Charge
Photoionization and Photomagnetization
Theory of Electrons and Currents
Astrophysics
Hubble‘s Law and the Reciprocal System
Globular Cluster Mechanics in the Reciprocal System
Stellar Energy Generation in the Reciprocal System
The Gravitational Attraction of the Galaxy
Discussion of Larson‘s Gravitational Equation
The Gravitational Formula at High Velocities
A Crucial Experiment
The Interaction of Alpha Particles and Gold Atoms
Proving Rutherford Wrong
A Proposal for a Crucial Experiment
More Calculations with the R.S. Scattering Equation
Detailed Steps for the Design and Performance of the Proposed Crucial
Experiment More Details for the Proposed Crucial Experiment
WHITE LIES ABOUT BLACK HOLES
Kip Thorne introduces his article on black holes in Scientific American by stating:
Or all the conceptions of the human mind from unicorns to gargoyles to the hydrogen
bomb perhaps the most fantastic is the black hole: a hole in space with a definite edge
over which anything can fall and nothing can escape; a hole with a gravitational field so
strong that even light is caught and held in its grip; a hole that curves space and warps
time. Like the unicorn and the gargoyle, the black hole seems much more at home in
science fiction or in ancient myth than in the real universe. Nevertheless, the laws of
modern physics virtually demand that black holes exist. In our galaxy alone there may be
millions of them.
Thorn is saying that because the ―laws‖ of modern physics require them, black holes
must exist. However, it is more rational to conclude that those ―laws‖ which give rise to
the gargoyles, unicorns, and black holes of physics are wrong—that, as ordinarily
expected, deductions from false premises yield bizarre results. Let us now investigate
how such concepts as the black hole arose historically. The theory of the black hole stems
from the theories of general relativity, the nuclear atom, and the hydrogen-to-helium
conversion process in stars.
In the 1930s, Subrahmanyon Chandrasekhar‘s investigation of stellar evolution and
structure led him to conclude that, in the process of converting hydrogen to helium, most
stars lose energy and contract until internal pressures become great enough to cause
collapse of atomic structure. Back in 1924, Sir Arthur Eddington had suggested that the
high density of the white dwarf companion of the bright star Sirius was due to ―electron
degeneracy,‖ with all electrens stripped from individual atoms. Chandrasekhar seemed to
provide an explanation of how this could occur.
At this point someone might have pointed to a simpler solution: perhaps the nuclear atom
concept was incorrect because of the grave difficulty in explaining the high density of the
white dwarfs. Perhaps atoms do not have electrons circling around ihem at relatively
large distances. Perhaps the postulated hydrogento-helium conversion process in stars
was incorrect...
Chandrasekhar believed that a ―non-relativistic gas‖ at the center of a white dwarf could
always adjust itself until the gravitational forces compressing the star are countered.
liowever, according to the theory of general relativity, with a certain limiting mass, the
gravitational forces are net countered fully and so the star does not come into
equilibrium. The limiting mass, termed the Chandrasekhar limit, has been calculated to
be 1.2 solar masses.²
Oppenheimer and Volkoff considered what would happen to stars of mass larger than the
Chandrasekhar limit. As the central density increases, inverse beta decay would take
place, driving electrons into protons. Thus increasingly rich neutron elements would be
formed—giving rise to a ―neutron star.‖ Recently astronomers have concluded that the
pulsars are neutron stars.
However, it must be pointed out that there is no evidence that pulsars are neutron rich, in
the same way that there is no evidence that white dwarfs are electron degenerate. In order
to obtain such densities with the nuclear atom concept, those deductions might be correct.
But there is an atomic theory, developed by Mr. Larson, that explains such high densities
without the use of the nuclear atom...
According to current theory if the remaining mass exceeds two solar masses, it will
continue to contract to a ―Schwarzchild singularity,‖ a bottomless pit, a black hole. The
properties of a black hole are supposed to be:
1. a gravitational field so strong that not even light can escape, and thus no observer
can see any phenomena occurring within the Schwarzchild radius;
2. a curvature of the space-time whirlpool becoming infinite at the central
singularity;
3. a circumference of 19 kilometers multiplied by the mass of the hole and divided
by the mass of the sun;
4. a mass of between 3 and 50 solar masses;
5. a compositicn of matter compressed to near infinite density, losing in the process
every property of separate identity except mass, electric charge, and angular
momentum.
To say the least, such properties are astounding. It is a relief to know that the Reciprocal
System, developed by Mr. Larson, contains no such theoretical objects: Here is a brief
tabulation of the relevant points of the Reciprocal System:
1. Atoms are not composed of electrons, protons, and neutrons, but are whole units
comprised of various rotational motions; at equilibrium there is equality of inward
and outward forces on groups of atoms; great compression can take place without
―electron degeneracy‖ or ―neutron formation.‖
2. In the sector of the universe in which we live there are two regions. In the time-
space region, gravitation is inward, whereas the space-time pregression is
eutward. In the time region, which lies within unit space, the motions are
reversed: gravitation is outward, the progressicn inward. According to the theory,
during a type I supernova explosion, part of the material is dispersed outward in
space (to form a red giant star) and part dispersed outward in time (to form a
white dwarf). The expansion outward in time is equivalent to a contraction in
space—hence the extreme density of the white dwarfs. Type I supernovae occur
because a thermal limit is reached in the energy conversion process taking place.
Type II supernovae occur because of a stellar age limit. Here, instead of a white
dwarf being formed, a pulsar is formed. The type II process is what ultimately
produces the quasars. All of these high speed explosion products—white dwarfs,
pulsars, and quasars—originate from expansion in time.
3. A different mechanism of energy generation is postulated, which in turn produces
a different pattern of stellar evolution. In the Reciprocal System, stars slowly
increase in mass and temperature until the destructive thermal limit of the iron
group elements is reached. At this point, a type 1 supernova occurs, creating a red
giant and white dwarf. Gravity acts in both directions to bring the white dwarf and
red giant back to the main sequence. There is no stellar death into a black hole.
Simply, a succession of type 1 supernovae occur until the star reaches its upper
age limit and terminates in a type II supernova, producing pulsars which
eventually leave this sector for the space-time region.
What about claims of observation of a black hole? Kip Thorne states that he is 90%
certain of a black hole in Cygnus X-1. It seems that its mass is eight times that of the sun
—meaning that in current theory, a white dwarf and neutron star are ruled out. However,
in the Reciprocal System no such mass limit exists. In fact, it is apparent that the ―black
hole‖ in Cygnus X-l, because of its copious emissions of radio waves and X-rays, is
really a body that will eventually become a pulsar. It is a product of a type II supernova.
At present the periodicity of its radiation is not distinguishable from continuous radiation,
but as the high speed explosion product moves outward it will be. Thus here is a test
between current theory and Reciprocal System—we predict that this so-called black hole
will turn out to be a pulsar, but a pulsar that is more massive than any neutron star could
be.
Currently, it is postulated that black holes account for the great mass discrepancy in giant
elliptical galaxies. Mr. Larson provides the explanation from the Reciprocal System:
A star pressure is building up in the interiors of the older galaxies; that is, an increasing
proportion of the constituent stars are being accelerated to uttra high speeds by the energy
released in the explosion of stars that reach the destructive age limit. The cores of these
galaxies are thus in the same condition as the white dwarf stars and quasars; their densiiy
is abnormally high because the introduction of the time displacement of the ultra high
speeds reduces the equivalent space occupied by the central portion of the galaxy. In
brief, we may say that the reason for the abnormal relation between mass and luminosity
in the giant ellipticals is that these galaxies have white dwarf cores—not white dwarf
stars in the core, but white dwarf cores.
It seems that many individuals are intrigued with the term ―black hole.‖ Perhaps we could
retain this term in the Reciprocal System to denote the location at which mass has left this
sector for the inverse sector, the space-time sector:
References
1. K. S. Thorne, ―The Search for Black Holes,‖ Scientific American, December
1974, p. 32.
2. Martin Harwit, Astrophysical Concepts, (New York: John Wiley and Sons, 1973),
p. 359.
3. J R. Oppenheimer and G. M. Volkoff, ―On Massive Neutron Cores,‖ Phys. Rev.,
56, 455 (1939).
4. Dewey B. Larson, Quasars and Pulsars, (Portland, Oregon: North Pacific
Publishers, 1971), pp. 148-149.
A TALL TALE
A Review of Stephen W. Hawking‘s A Brief History of Time
This paper will critique the conventional physical theory espoused by Stephen Hawking
(hereafter abbreviated to SH) in his best-selling book. From the perspective of the
Reciprocal System (hereafter abbreviated to RS), the book is full of errors on practically
every page. Still I urge members of ISUS to read it-SH does a good job of presenting the
―establishments‖ viewpoint and its worth pondering his thinking. The book,s dust jacket
proclaims SH to be the ―most brilliant theoretical physicist since Einstein‖ yet he begins
his conclusion (p. 171) with the words ―We find ourselves in a bewildering world.‖ This
is a far cry from my Unmysterious Universe, published in 1971.
SH is confused as to whether the universe was created or not. He states that in ―imaginary
time‖ the universe has no beginning or end, no singularities or boundaries—it simply is.
But in ―real time‖ the universe does have a beginning and an end at singularities that
form a boundary to space-time and at which the laws of science break down (p. 139). The
beginning singularity is, of course, called the ―big bang‖, when the universe was
―infinitesimally small and infinitely dense‖(p. 8) and ―infinitely hot‖ (p. 117) and space-
time had infinite curvature [!]. Time, and by implication, space-time, had no meaning
prior to the beginning (p. 8). SH defines an event (p. 23) as ―something that happens at a
particular point in space and at a particular time. So one can specify it by four numbers or
coordinates.‖ This is the conventional four-dimensional space-time, 3 spatial coordinates
and 1 temporal coordinate, which is quite different from the 3-dimensional space-time of
the RS (where each dimension is a dimension of motion, not of space or time
individually). In the RS, space-time has an inbuilt expansion and hence there is no need
for a ―big bang‖ to explain the recession of the distant galaxies. SH states (p. 33) that
―Space and time are now dynamic quantities: when a body moves, or a force acts, it
affects the curvature of space and time—and in turn the structure of space-time affects
the way in which bodies move and forces act.‖ Certainly there is no physical evidence for
this; bare space-time is nonphysical since it cannot be changed into something else—it is
a simply a reference system for motion. SH does not specify a mechanism for producing
this alleged warping of space-time by the ―distribution of mass and energy in it‖ (p. 29).
SH treats space and time as both purely relative, not absolute (p. 21, 33); he says that
"each observer must have his own measure of time, as recorded by a clock carried with
him, and identical clocks carried by different observers would not necessarily agree.― Of
course, this subjectivist belief leads to numerous logical contradictions and is wrong;
space-time is the fundamental component of the universe and must thus be absolute
(there is nothing for it to be ―relative to‖ since it is itself the reference).
An electromagnetic field fills the space-time of SH. There can be (p. 18)―wavelike
disturbances in the combined electromagnetic field‖; (p, 38) ―..visible light consists of
fluctuations or waves, in the electromagnetic field‖. But (p, 54) ―Although light is made
up of waves, Planck‘s quantum hypothesis tells us that in some ways it behaves as if it
were composed of particles: it can be emitted or absorbed only in packets or quanta.‖
This is the old wave-particle duality. Compare this with the definition of photon in the
RS: a linear vibration within a same space-time unit, which itself progresses
perpendicularly, the combination thus generating a wave motion. SH states (p. 31) that
―Light rays too must follow geodesics in space-time‖and (p. 28) ―,nothing can travel
faster than light‖. In the RS, spatial motion is limited to speeds lower than that of light,
but temporal speeds are higher: the speed of light is the midpoint of the speed range of
the universe, not the upper limit. According to SH (p. 117) ―One second after the big
bang, (the temperature] would have fallen to about ten thousand million degrees... At this
time the universe would have contained mostly photons, electrons, and neutrinos... and
their antiparticles, together with some protons and neutrons.‖ SH does not provide a
fundamental definition of electrons or neutrinos (the quark hypothesis does not include
them; quarks are the undefined elementary particles of matter). He states (p. 65) that ―A
proton contains two up quarks and one down quark; a neutron contains two down and one
up.‖ (p. 73) ―One cannot have a single quark on its own... confinement prevents one from
observing an isolated quark or gluon‖ (which carries the ―strong nuclear force‖]. (p, 75)
―,the uncertainty principle means that the energy of the quarks inside the proton cannot
be fixed exactly. The proton would then decay.‖ (p, 73) ―another possibility is a pair
consisting of a quark and an antiquark... such combinations make up the particles known
as mesons, which are unstable because the quark and antiquark can annihilate each other,
producing electrons and other particles.‖ Each subatom has a spin, but (p. 66) ―quantum
mechanics tells us that the particles do not have any well-defined axes‖. (p. 67) ―Particles
of spin 1/2... make up the matter of the universe, and particles of spin 0, 1, and 2 give rise
to forces between the matter particles.‖ (p. 69) ―The electric repulsive force between two
electrons is due to the exchange of virtual photons (spin 1)‖ and (p. 70) ―the force
between two matter particles is pictured as being carried by a particle of spin 2 called the
graviton.‖ This is all in contrast with the RS, in which each subatom is a set of quantized
spins of a photon, with definite axes. The mesons are actually cosmic atoms in the
process of converting to the prevailing structures of our sector. The apparent force
interactions are not due to exchange of particles (virtual or otherwise), but rather
represent an interaction between the particle and the omnipresent space-time progression.
SH continues (p. 117): ―About one hundred seconds after the big bang, the temperature
would have fallen to one thousand million degrees, the temperature inside the hottest
stars. At this temperature protons and neutrons would no longer have sufficient energy to
escape the attraction of the strong nuclear force, and would start to combine together to
produce the nuclei of atoms of deuterium... The deuterium nuclei then would have
combined with more protons and neutrons to make the helium nuclei... and also small
amounts of a couple of heavier elements, lithium and beryllium... Within only a few
hours of the big bang, the production of helium and other elements would have stopped.‖
SH doesn‘t question the validity of the nuclear theory of the atom. No mention is made of
the instability of the neutron: how can an atom be stable if one of its main components is
known to be unstable? Also, why don‘t the alleged protons in the nucleus repel each
other? To explain the ―strong nuclear force‖ as due to the exchange of unobserved
―gluons‖ is mystical. Also, the alleged orbiting electrons are thought not to combine with
protons and neutralize their charges, whereas other pairs of oppositely charged particles
do. Finally, why should the production of new elements cease? It seems more likely that
over the course of thousands, or millions, or billions of years that atoms would continue
to combine to form heavier atoms, and these would join to form ever more complex
molecules. SH states (p. 60) that ―Since the structure of molecules and their reactions
with each other underlie all of chemistry and biology, quantum mechanics allows us in
principle to predict nearly everything we see around us, within the limits set by the
uncertainty principle. (In practice, however, the calculations required for systems
containing more than a few electrons are so complicated that we cannot do them.)‖ Well,
what good is a theory if we can only use it in principle?
SH continues (p. 117): ―The universe as a whole would have continued expanding and
cooling, but in regions that were slightly denser than average, the expansion would have
been slowed down by the extra gravitational attraction. This would eventually stop
expansion in some regions and cause them to start to recollapse... in this way disklike
rotating galaxies were born.‖ SH uses the uncertainty principle to explain the non-
uniform density. He says (p. 140) that ―... there must have have been some uncertainties
or fluctuations in the positions or velocities of the particles. Using the no boundary
condition, we find that the universe must in fact have started off with just the minimum
possible non-uniformity allowed by the uncertainty principle.‖ The RS explanation is
much better: the two main forces of the universe are the space-time progression and
gravitation; where gravitation is stronger, galaxies and stars are formed; where the
progression is stronger, the galaxies move away from each other. Also, the latest
evidence is that the formation of galaxies is not a one-time occurrence—observations
indicate that galaxy building is going on right now, just as the RS predicts.
An average galaxy has a hundred billion stars. SH states (p. 82) that ―A star is formed
when a large amount of gas (mostly hydrogen) starts to collapse in on itself due to
gravitational attraction. As it contracts the atoms of the gas collide with each other more
and more frequently and at greater and greater speeds—the gas heats up. Eventually, the
gas will be so hot that when the hydrogen atoms collide they no longer bounce off each
other, but instead coalesce to form helium. The heat released in this reaction, which is
like a controlled hydrogen bomb explosion, is what makes the star shine. This additional
heat also increases the pressure of the gas until it is sufficient to balance the gravitational
attraction, and the gas stops contracting... Stars will remain stable like this for a long
time... Eventually, however, the star will run out of its hydrogen and other fuels.
Paradoxically, the more fuel a star starts off with, the sooner it runs out.‖ This is
paradoxical indeed, for the larger stars should be the oldest and the smaller stars the
youngest. In fact, the observed evidence indicates that the astronomers have the
evolutionary sequence precisely upside down. SH continues (p. 83, 84, 87): ―When a star
runs out of fuel, it starts to cool off and so to contact... If a star‘s mass is less than the
Chandrasekhar limit, it can eventually stop contracting and settle down to a possible final
state as a white dwarf with a radius of a few thousand miles and a density of hundreds of
tons per cubic inch... [If a stars mass is greater than the Chandrasekhar limit, the star will
eventually collapse to a black hole] in which neither light nor anything else can escape.‖
In the RS, stars slowly accumulate mass, rather than losing mass, and there is no end in a
black hole. Rather at the mass limit, a supernova explosion occurs, and a red giant/white
dwarf pair is formed (or a planetary satellite system) and both stars eventually return to
the main sequence. Its interesting that SH proposes a radical theoretical change in the
black hole construct: he says that actually a black hole would emit particles and radiation
and (p. 115) ―..the black hole, along with any singularity inside it, would evaporate away
and eventually disappear.‖
SH continues (p. 46): ―The present evidence therefore suggests that the universe will
probably expand forever, but all we can really be sure of is that even if the universe is
going to recollapse, it wont do so for at least another ten thousand million years, since it
has already been expanding for at least that long.‖ Compare this with the RS: there are
two sectors, the material sector, and the cosmic sector. Outward spatial expansion in the
material sector is terminated with a galactic explosion which sends the matter over to the
cosmic (inverse) sector, where outward temporal expansion (the inverse of spatial
expansion) occurs. This expansion in turn is terminated with a cosmic galactic explosion
which sends the matter back to the material sector. Thus the main process in the universe
is cyclic, rather than a singular one-time expansion. Also the net total displacement in the
universe is zero, because the number of cosmic displacement units balances the number
of material displacement units (whereas SH states (p. 129) ―... the total energy of the
universe is exactly zero... this negative gravitational energy exactly cancels the positive
energy represented by the matter.‖)
Having carefully studied this book, I think the supporters of the RS have nothing to fear.
If this is the best popular rendition of conventional theoretical physics, the future looks
bright for the spread of the RS, which is a unified, general theoretical system. (Contrast
the RS with the current ―Grand Unified theories‖, which (p. 156) ―are not very
satisfactory because they do not include the force of gravity and because they contain a
number of quantities, like the relative masses of different particles, that cannot be
predicted from the theory but have to be chosen to fit observations‖. Further details of the
RS refutation of the ideas expressed by SH can be found in Dewey Larson‘s book The
Universe of Motion (particularly chapter 29) and his article ―The Mythical Universe of
Modern Astronomy,‖ Reciprocity, Vol. XII, No. 2, Autumn, 1982.
FOUR SCIENTIFIC MYSTERIES UNRAVELLED
I. Quarks
―Quarks are fantastic jive‖ :
James Joyce might have said if alive.
―We started with three
For atomic debris,
And now we find we have five.‖
(F. A. Moen)
(A wake for a fellow named Finnegan
Gave theorists quarks to begin again.
Having swallowed down four,
They keep asking for more
And a theory by which they can win again.)
(Ed., Science News, August 7, 1976)
The present scientific premise is that matter is ultimately composed of a set of matter
particles called quarks. Originally three quarks and three anti-quarks were postulated.
These were called ―up,‖ ―down ― and ―strange. To account for recently observed particle
behavior, theorists hsve postulated a fourth quark a ―charmed‖ quark as it‗s called. To get
around the Fermi-Dirac statistics, an additional quantum number has now been
postulated, ―color.‖ Then a class of intermediate particles called gluons was formulated to
carry the force between quarks. Finally other theorists began to feel that now quantum
numbers beyond charm and color are at work. One theorist said that beyond charm and
color lay truth and beauty. Thus the current hypothesis is a set of as many as seven quarks
and seven antiquarks, which are supposed to combine to form all hadrons. The leptons
(the electron, the muon, and the two kinds of neutrino) are considered different, showing
no intelnal quark structure.
To me such terms as ―strange,‖ ―color,‖ and ―charm‖ are mere names covering a great
deal or ignorance. They remind me of such terms as ―phlogiston‖ and ―levity.‖
Phlogiston, in early chemical theory, was the hypochetical principle of fire, of which
every combustible substance was in part composed.. Levity was a term employed by the
Montgolfier brothers to explain why smoke rose in a chimney. None of these terms really
advanced scientific understanding.
The Reciprocal System developed by Dewey Larson avoids any such mysterious terms.
The basic premise here is that the fundamental component of the universe is motion,
existing in discrete units. Each atom is composed of three types of motion or rotational
spin, rather than threa or more types of quarks. The difference between ―leptons‖ and
―hadrons‖ lies in the number of dimensions of the motion.
The quark theory gives no indication as to how matter can change into radiation and vice
versa. By contrast the Reciprocal System provides a simple answer: both atoms and
radiation are forms of motion and one can simply change one form of motion into
another.
II. Nuclear Atom
According to present theory, all forces between entities result from the exchange of
quanta. It has been theorized that the force between the supposed neutron and proton in
the atom results from an exchange of mesons. These mesons were supposed to be
charged, thus producing a current when in motion. Benson T. Chertok spent some ten
years of work at the Stanford Linear Accelerator . Center to find these meson-exchange
currents. As quoted in the Science News of May l0, 1975, Chertok said his experiment
found no evidence for these mesonexchange currents.‖Ten years of work shot down,‖ he
says.
This is typical of present practice, where theorists keep on postulating entities that
experimenters do not find, and experimenters keep on finding entities that theorists have
not postulated.
In the Reciprocal System, the atom is not compesed of neutrons, protons, and electrons,
and thus no force is needed to explain the attraction between protons and neutrons. The
atom is really one whole unit, composed of three different types of rotational spin.
III. Electrostatics
Present theory assumes that matter is composed of a large number of charged electrons
circling nucleuses of neutrons and protons. It is interesting to derive a consequence of
this hypothesis, showing its implausibility. This derivation comes from the
Encyclopaedia Britannica's article on electricity.
The normal density of solid matter is around five grams par cubic centimeter or 5000
kilograms per cubic meter and a gram molecule of a single compound occupies
something between five and 100 cubic centimeters. Thus the number of atoms per cubic
centimeter is in the region of 1022 to 1023 . The number of electrons pet atam varies from
one to more than 92 in the periodic table, but the number of relatively loosely bound
valence electrons per atom that can be appreciably influenced by electric fields is usually
small, in many cases there is only one per atom. ,As a general rule these are of the order
of 1023 effective electrons per cubic centimeter of 1029 per cubic meter. The total charge
associated with these electrons is thus some 1010 coulombs per cubic meter. The number
of electrons lying within one atomic diameter of the surface of a solid is 1029(2/3) per
square meter (i.e., about l020) a:nd so the available charge per square meter is of the order
of 10 coulombs per square mater. Now a surface charge density of 10 coulombs per
square meter is associated with an electric field normal to the surface of 10/ = 1012 volts
per meter. Even in extreme cases fields rarely exceed 108 volts per meter, and so, in
general the surface charges that appear in elec-troscatics correspond to fractional changes
in the internal charge distribution. Furthermore, a field of 1012 volts per meter
corresponds to a field of 100 volts per angstrom. A field of this strength would be
sufficient to disrupt and destroy the surface atoms. . . . !
It certainly wasn't the purpose of the author of the article to question the validity of
present atomic theory -- but he has surely provided an excellent disproof !
Thankfully, electric charges are not inherently in matter of the Reciprocal System.
IV. Gravitation
Present theories assume that the gravitational effect is propagated at the speed of light.
However, following LaPlace it is possible to show that if gravitational force is
propagated, the velocity must be at least 10°c!
(i) Assume that the gravitational force on the earth due to the sun is ―falling‖ upon the
earth with a velocity of ß.
(ii) Let r be the radius of the earth's orbit and let v be the earth's orbital valocity. Then the
component of solar acceleration in the direction of the earth's motion is
a = GM /r² / v/ß
(iii) By the method of perturbations, this acceleration can be shown to change the radius
of the earth's orbit at the rate
dr/dt = 2 · v²/ß
(iv). The celestial longitude 1(t) of the earth may by expanded in a power series:
1(t) = n t + A(n t)² + . . . (3)
where n is the present mean motion given by
no = (v/r) PRESENT (4)
and t is the time measured from the present. Now
n = v/r = dl/dt (5)
Differentiating eq. (3) twice gives
d²1/dt² = dn/dt = 2An ² (6)
From orbital mechanics,
n²r³ = const (7)
Using eqs. (7) and (2),
dn/dt = -3n² / v/ß (8)
Thus the coefficient of the second term of the celestial longitude equation is
A = -2/3 / v/ß
(v) In Che case of the earth the velocity is
v = 10-4 c (10)
If ß were equal to c, then A would have to be -1.5 x 10-4. In one century, the second
power term in equation (3) would become -60 radians or -1.2‖ x l07. Ant the largest
admissible variation is 2‖ . Therefore, ß must be greater than 6 x l06 c. This same result
occurs if other planets are considered. The assumption that gravitational force travels
with a speed of ß seems always to lead to a relation
ß > 106 c (11)
The relativists try to get around this problem by asserting that it is the gravitational
potential that is propagated. But they don't explain just how that is accomplished.
The Reciprocal System solves the problem by showing that the gravitational effect is not
propagated at all. Gravitational motion is simply an interaction between the inherent
rotational spin of the atoms and the translational progression of space-time.
One reason why Newton's Law of Gravitation is no longer considered fully tenable is tho
observation of the advance of the perihelion of the planet Mercury. The teory of General
Relativity came up with a rather complicated expression to explain the effect. However,
as I showed mathematically in a previous paper, this effect is really in the same class as
those other high velocity effects. And so the mathematical tretatment is of the same
nature as that of Special Relativity.
In the case of a planet orbiting the sun, the gravitational expression becomes
F = G m m2 / r² (1-v²/c²)½ (12)
In the case of an object moving toward another object, the expression is
F = G m m2 r² (1-v²/c²) (13)
These expressions provide an obvious unification of gravitational. and electrical effects,
and I'm amazed that these expressions haven't been stumbled on before The advance of
the perihelion of Mercury is fully explained by eq. 12. Note that eq. 13 shows that force
vanishes at the speed of light—this indicates that the natural datum of the universe is the
speed of light, rather than zero velocity.
Tn sum up some problems of modern physics:
1. Originally 3 quarks and 3 anti-quarks were postulated as the fundamental
components of matter. To account for recent experiments, this has been expanded
into 7 quarks and 7 anti-quarks—and the end is not in sight.
2. The meso-exchange currents supposed to carry the force between nucleons have
not bcen found.
3. The assump:ion of electric charges in matter leads to the conclusion that an
electric field 100 volts per angstrom is developed on surface atoms, an effect
certainly not observed.
4. The assumption a finite velocity the gravitational force leads to the conclusion
that this must be at least 106 c, rather than c.
Reciprocal System is. able solve all ef these problems in elegant fashion. It for such,
reasons I entitled my book Unmysterious Universe.
References
1. Science News, August 7, 1976.
2. Science News, June 26, 1976.
3. Science News, May 10, 1975.
4. ‖Electricity,‖ Encylopaedia Britannica, Fifteenth Edition, Vol. 6, pp. 550-551.
5. H. P. Robertson and Thomas W. Noonan, Relativity and Cosmology
(Philadelphia:W. B. Saunders Company, 1968), 401-403.
6. R.W. Satz, Question Box,‖ Reciprocity, Vol. IV, No. 2, July, 1974.
REFERENCE SYSTEMS AND SPEED LIMITS
IN THE RECIPROCAL SYSTEM: A REVIEW
Current theoretical physics views time as one-dimensional and constituting a kind of
quasispace which joins with the three dimensions of space to form a four-dimensional
space-time framework, within which physical objects move one-dimensionally. This view
has been formulated to help explain some of the new phenomena discovered in the
twentieth century, such as the very small, the very large, and the very fast. These
phenomena exist outside of our normal everyday world, where Newton‘s laws
predominate and where space seems to be totally separate from time. However, even with
this modern framework, most of the phenomena remain mysteries, in whole or in part.
In contrast, the Reciprocal System of theory, originated by D.B. Larson,¹ postulates that
both space and time have three-dimensional aspects and join together to form one entity,
space-time or motion, which itself is. three-dimensional. Space and time are the two
reciprocal aspects of motion and have no properties other than what they have in motion.
Here, space-time or motion is theorized to be the sole component of the universe, not the
framework or the background for particles of matter. Matter in the theory is itself a form
of rotational motion and may move translationally in more than one-dimension
coincidentally. To be sure, these ideas are novel and unfamiliar but they do overcome the
current difficulties in treating phenomena of the very small, the very large, and the very
fast, as will be demonstrated in this paper.
The conventional physical reference system is based on three dimensions of space and
one of time. The space is considered to be stationary and the time is considered to be
ffowing. Within this space, material objects move as a function of time in one dimension
in a specific vectorial direction. This one-dimension of motion may be resolved into three
components, one along each of the three orthogonal axes of the reference system (usually
denoted x, y, and z).
In the Reciprocal System, space and time each have the properties of the other. Time is
three-dimensional, like space, and space progresses, like time. Of course in a
gravitationally-bound material environment, space appears to be stationary and
threedimensional and time appears to be onedimensional and progressing, and so the
conventional reference system works for this situation. In a gravitationally-bound cosmic
(or inverse) environment, where space and time are interchanged, time would appear to
be stationary and three-dimensional, and space would appear to be one-dimensional and
progressing. The inverse of the conventional reference system would work in this
situation. So, in the Reciprocal System two types of reference systems exist: the first with
threedimensions of space and one of time, and the second with three dimensions of time
and one of space. The first is applicable to objects which are aggregated in space (as in
our ordinary material sector) and the second is applicable to objects which are aggregated
in time (as in the cosmic sector). Conventional physical science does recognize anti-
particles and anti-galaxies but does not stipulate an ―anti-reference system.‖
A common mistake of students of the Reciprocal System is to deduce from the above that
there are thus six dimensions of the universe, three of space and three of time. This is not
so, however. All that actually exists are three dimensions of motion, not three dimensions
of space or three dimensions of time individually. Of course, where convenient, we can
mentally fix one component, while allowing the other to move. This has the effect of
concentrating on one aspect of each of the components while ignoring the others. But it is
important not to forget that space and time do not exist separately; they are bound
together in units of motion, which are the actual reality.
Outside of our gravitationally-bound region, what happens? It is an observed fact in
astronomy that distant galaxies are moving away from our galaxy (and all others) at
speeds approaching that of light. The current explanation is that this is a result of a
hypothetical big bang some 16 billion years ago. But this is not, of course, the
explanation of the Reciprocal System. Here, the cause is the space progression, which
manifests itself when gravitation is attenuated. It is an effect brought about by the motion
of the natural reference system relative to our conventional spatial reference system. The
equation for this motion (at the full speed of light) in our sector is
x1² + x2² + x3² = c²t² (1)
(where c, of course, is the speed of light). Please note that because the motion is actually
scalarly outward only, the imputation of a specific vectorial direction for a particular
space unit is arbitrary.
Similarly there is an outward motion of cosmic galaxies relative to a cosmic observer‘s
galaxy, such motion also approaching the speed of light in the limit. Again, this is not due
to some sort of big bang in the cosmic sector; rather it is due to the motion of the natural
reference system relative to the three-dimensional temporal reference system. At the full
speed of light, the equation is:
t1² + t2² + t3² = x²/c² (2)
Again, the assigned vectorial directions are arbitrary, since this is actually a scalar motion
outward in all directions.
The two major sectors of the universe, the material sector and the cosmic sector, each
with their appropriate reference system, are stable. In between these two sectors is an
unstable transition zone, which cannot be represented properly by either reference
system.
In the material sector, the velocities of ordinary phenomena are below that of the speed of
light. In the cosmic sector the inverse velocities of ordinary cosmic phenomena are below
that of the speed of light. Hence the speed of light is the dividing line between the two
sectors, and, in fact, the photons are not actually part of either sector, but at the boundary.
They have no independent motion (other than their vibration) and so remain in the same
spacetime location, which is carried outward by the space-time progression (in a
perpendicular direction) with respect to the conventional space or time reference system.
From the standpoint of a gravitationally-bound cosmic aggregate, the photons appear to
be moving outward in all coordinate time directions. Actually, however, the photons
remain stationary in the natural reference system; the matter particles are moving inward
in space and the cosmic particles are moving inward in time, and so the opposite motions
are imputed to the photons.
Because motion can be three-dimensional, the actual separation between the two major
sectors of the universe is scalar unit speed in all three dimensions, or 3c. Each of the three
dimensions is limited to a maximum of one unit of speed and if motion were limited to
one dimension we would agree with current physics that nothing could travel faster than
the speed of light. However, because the motion is not so limited, the actual limit is 3c.
Notice that we are summing the value of speed of each dimension. These are, after all,
scalar quantities, which can only be added or subtracted. It is not possible to add
vectorially the motions in the different dimensions of multi-dimensional motion! The
sequence of additions of units of speed are: first one unit, in dimension l; then the second
unit, in dimension 2; and finally the third unit, in dimension 3.
In our sector, velocity is measured as s/t. In the cosmic sector it is t/s. To material
observers, photons appear to move through space; to cosmic observers, photons appear to
move through time. Both observers see photons as being the upper limit of speed (or
energy). Many beginning students of the Reciprocal System conclude that when we talk
about motion at speeds faster than that of light, we are referring to rate of change of
position in space. But this is not so. At speeds above the speed of light the rate of change
applies to change of position in time, which moves objects further apart in time, or
(equivalently) moves objects closer together in space.
According to the Reciprocal System, motion exists only in discrete units, so the question
arises, how can we have fractional units? Obviously the velocities on Earth are only a
tiny fraction of the speed of light, so how can this be? Recall that the theory requires that
we start with one unit, not from zero units. This one unit of motion is equal to one unit of
energy, because of the reciprocal relation between space and time. To achieve effective
translational speeds below unity we simply subtract the appropriate number of energy
units from one. The equation in natural units is
v = 1-1/x (3)
where x is the number of one-dimensional energy units (with dimensions t/s). Note that as
x is increased the speed is increased, and in the limit reaches 1 (or c). In the time region,
the region inside unit space, the numerical value of the energy term must be squared, for
reasons given by Larson.² So the equation actually is
v = 1-(1/x)² (4)
Suppose x has the value n. Motion at this speed often appears in combination with a
motion 1-1/m² that has the opposite vectorial direction. The net result is
vH = 1/n²-1/m)² (5)
This is clearly recognized as the Rydberg relation (in natural units) that defines the
spectral frequencies of atomic hydrogen—the possible speeds of the hydrogen atom.
Other elements have similar relations for their thermal motion. The important point is that
translational motion is quantized; it is not a continuum. To this extent, we can agree with
current theory.
Because of the ability of adding or subtracting energy units to the three basic speed
ranges, we can have speeds of 1 - 1/x, 2 - 1/x, and 3 - 1/x. Larson denotes the speed range
1 - 1/x ―low speed‖; the speed 2 - 1/x, ―intermediate speed‖; and the speed 3 - 1/x, ―high
speed.‖ Because of the one-dimensional nature of energy, it is not possible to go from
one speed range to the next by simply adding more energy. The only way to accomplish
this is by direct addition of units of speed. And the only way that can be accomplished is
by huge stellar or galactic explosions.
In a Type I supernova explosion (caused when the temperature limit of the iron group of
elements is reached) part of the material moves outward in space at speeds less than that
of light and part moves outward in time (or inward in space) at speeds in the intermediate
range. This results in a very dense compact star, a white dwarf, stationary in space,
surrounded by a cloud of dust and gas moving spatially outward. Eventually the speeds of
the particles of the white dwarf fall below that of unit speed (or c) and the white dwarf
begins to expand in space, eventually becoming a normal star on the main sequence. The
dust cloud recondenses into a red giant star, which also eventually returns to the main
sequence.
A Type II supernova is even more powerful. Here the explosion results in high speed
motion of the compact star away from the scene, together with the usual cloud of dust
and gas. This compact start, or pulsar, is similar to the white dwarf, except that it has an
additional translational motion in the high speed range (or third dimension of motion).
Once the effect of gravitation is attenuated (and the net speed goes above two units) the
pulsar will leave our sector and move into the cosmic sector, where the processes of that
sector will disintegrate the spatial aggregate and recondense it in time by means of
cosmic gravitation. It will thus disappear from our view.
In the central regions of the largest galaxies, the spheroidal galaxies, consisting of 1012 to
1013 stars, the matter is at the upper limit of age. Instead of isolated Type II supernova
explosions, a whole chain reaction of such explosions occurs, resulting in galactic
fragments (between 7 x 107 to 2 x 109 stars) being ejected. The fragments ejected at
upper-range speeds are the quasars; those at low or intermediate speeds are the radio
galaxies. Note that although the quasar itself is moving at high speed, the particles of
which it is composed are moving at intermediate speed, hence the compact structure.
The Reciprocal System of theory explains many of the puzzling characteristics of
quasars. One such characteristic is the observed double image of some quasars. Larson
explains this as follows:
Scalar motion does not distinguish between the direction AB and the direction BA. The
lateral recession outward from point X is therefore divided equally between a direction
XA and the opposite direction XB by the operation of probability. Matter moving
translationally at upper range speeds thus appears in the reference system in two locations
equidistant from the line of motion in the coincident dimension (the optical line of sight,
in most cases).³
Hence there is no need for such a hypothesis as a ―gravitational lens.‖
Initially the explosion speed of the quasar is applied to overcoming the effect of
gravitation, and thus there is a rapid change of position in the reference system. As
gravitation is gradually overcome, the net speed increases, but the rate of change of
position decreases, because the speed in the explosion dimension is not visible in the
reference system. However, the speed in the explosion (high speed) dimension can be
detected by the shift in frequency toward the red of the radiation coming from the quasar
and received on Earth. This Doppler shift is a measure of the scalar sum of the outward
motions of the quasar, both that due to the recession and that due to the explosion speed.
It is a direct speed measurement, and the relative adjustment factors do not apply to it.
Hence values greater than 1 actually do mean speeds greater than c, which is what the
Reciprocal System requires. Thus the quasars are not nearly as distant as the current
cosmological explanation of the quasars‘redshifts suggests.
Another unusual characteristic of the quasars is their seemingly impossible great energy
generation. But the conventional assumption is that this energy is carried away in all
three dimensions by radiation. In the Reciprocal System, the radiation that comes from an
object at the upper range of speed is distributed two-dimensionally. As Larson states:
If we find that we are receiving the same amount of radiation from a quasar as from a
nearby star, and the quasar is a billion (109) times as far away as the star, then if the
quasar radiation is distributed over three dimensions, as currently assumed, the quasar
must be emitting a billion billion (1018) times as much energy as the star. But on the basis
of the two-dimensional distribution that takes place in equivalent space, according to the
theory of the universe of motion, the quasar is only radiating a billion (109) times as
much energy as the star,... which is equivalent to no more than a rather small galaxy.4
Larson calculates5 that the explosion redshift is a function of recession redshift and
normally takes the value 3.5 z½. The quasar will begin converting to the cosmic status
when this speed reaches a value of 2.0. The corresponding redshift is then 0.3265, and the
total quasar redshift (the sum of the recession redshift and the explosion redshift) is
2.3265. According to the observers, there is a sudden cutoff in the distribution of quasars
above a Doppler shift of 2.2, which is consistent with the theory. (By probability there
will be a few quasars that linger on for higher redshifts).
The boundary between the two major sectors of the universe is thus quite unstable, as it is
filled with material quasars and pulsars and cosmic quasars and pulsars, all in the process
of moving to the opposite sector. It is also filled with material white dwarfs and cosmic
white dwarfs which will eventually return to being normal stars in their respective
sectors. The boundary here is a speed or energy boundary. There is another boundary
within each sector at unit distance or unit time. In the material sector, the region inside
unit distance represents and important subdivision; in the cosmic sector, the region inside
unit time also represents an important subdivision. Reversals of motion occur at these
unit distance or time boundaries just as they do at the unit velocity or energy boundaries.
Inside unit space, only motion in time is possible, because fractional space units do not
exist. But the motion in time may be expressed in equivalent space units by means of the
reciprocal relation. Similarly, inside unit time, only motion in space is possible, because
fractional time units do not exist. This motion in space may also be expressed in terms of
equivalent time units by means of the reciprocal relation.
Inside unit distance or time, the space-time progression and gravitation reverse directions.
The progression is always away from unity and gravitation always toward it. So outside
unit space or time, the progression moves the very large spheroidal galaxies apart; inside
unit space or time, the progression moves the atoms of matter or cosmic matter close
together, in opposition to gravitation, which in this region is a force of repulsion. Atoms
of matter or cosmic matter can thus reach equilibrium positions in the solid state at the
locations where the two forces reach equality. There is thus no need for the ad hoc
electrical forces of conventional theory, which, in any case, supply only one of the two
necessary forces for equilibrium.
I have thus shown, in brief, that the Reciprocal System of theory, based on the novel
concept of three dimensions of motion, with space and time being reciprocally related,
can handle some of the current problems in physics: those phenomena involving the very
small, the very large, and the very fast. The very small, the atoms and the subatoms, are
subject to the relations that apply inside unit space or unit time, where the roles of the
progression and gravitation are reversed from what they are in the time-space (material)
or space-time (cosmic) regions. The very large, the spheroidal galaxies, are moving away
from each other at speeds approaching that of light because of the space-time progression
and the attenuation of gravitation at great distances. After many billions of years of
aggregation, these galaxies are now in the process of discombobulating—emitting jets of
high speed gases, quasars, and radio galaxies—because their matter has reached the upper
limit of age. The very fast, the quasars, have extraordinarily high redshifts because they
are moving at speeds faster than light and they will ultimately disappear into the cosmic
sector.
References
1. D. Larson, Nothing But Motion, Portland, Oregon: North Pacific Publishers, 1979,
p. 30.
2. D. Larson, The Structure of the Physical Universe, Portland, Oregon: North
Pacific Publishers, 1959, p. 19.
3. D. Larson, The Universe of Motion, Portland, Oregon: North Pacific Publishers,
1984), p. 301.
4. D. Larson, Ibid., pp. 287-288.
5. D. Larson, Ibid., p. 210.
THE LORENTZ TRANSFORMATION
Question: Please provide a detailed rationale of how the RS theory produces the correct
answer to this ―Lorentz transform‖ problem:
M
L <———|———> R
When L and R travel at the speed of light relative to M, Larson says the speed of R
relative to L is 2 units of space divided by two units of time; thus, the velocity of R
relative to L is 2/2 = l. Now suppose that L and R both travel at C/2 relative to M. If we
seemingly follow the same procedure as above, it appears that the total distance involved
is (½+ ½) and the total time involved is (1 + 1), so that the velocity of R relative to L
should be distance/time - 1/2. Obviously somethirg is wrong. What?
Answer: Assume that the partIcles traveling with the speed C/2 are atoms. Then the rate
of motion of the atom toward R relative to the motion of the atom toward L is 0.8C. The
RS theory thus offers the same answer to your question as does the Lorentz
transformation equation. The mode of motion of a photon (vibration) is different from
that of an atom (a combination of vibration and rotation). Photons remain in the space-
time locations in which they originate; atoms do not. The space-time locations of photons
move at the unit rate C. Atoms do not remain in the space-time locations in which they
originate. Therefore, the procedure for calculating the rate of motion of two photons
going apart from each other is not applicable to calculate the motion rate of two atorns
moving apart from eacil other, each at v = C/2 with respect to M. The relative motion rate
of the two atorns in this case is
0.5C + 0.5C 1.0C
u = ————— = ——— = 0.8C (1)
1 +[(0.5C) (0.5C)/C²]
1.25
The procedure, equation (1), is called the Lorentz transformation equation. How does the
RS theory arrive at the Lorentz equation? How does the RS theory deduce this equation?
This question amounts to asking how does RS theory imply the transformation equation:
vdb + wba
uda = —————
1 +[(vdb) (wba)/C²]
This Lorentz equation or law about the composition of velocities follows from the RS
theory, because the latter assumes that a light photon remains in the space-time location
in which it originates and further assumes that the location progresses at unit speed or at
the uniform rate of C = 3 x 105 km/sec., independent of the motion of source or detector
of the photon. These assumptions are incompatible with the Newtonian-Galilean
transformation equation, the Newtonian law of the composition of velocities, uda = vdb +
wba ; uda = - uad.
The fact that the velocity of light is independent of the velocity of the source of the light
implies that any finite velocity of the source, when added to the velocity of light, yields a
resultant for the light whose magnitude equals that of the speed of light.
Now in Newtonian physics, when three particles A, B, D are moving in a straight line,
and if U is the velocity of A relative to D, V is the velocity of D relative to B and W is
the velocity of B relative to A, then uad + vdb + wba = 0.
However, the just stated fact and RS principle asserts that when v = C, then u = - C,
whatever value w may have. This implies that the equation u+v+w = 0 is not true when
velocities commensurable with that of light are involved: it works satisfactorily only
when all the velocities are small compared with C.
How then to deduce the correct form of tkre law ef composition of velocities for
velocities of any magnitude is now the task. Specify then that the exact relation between
the three velocities is F (u,v,w) = 0. Agree that wba = - wab etc.
By permuting the three particles A, B and D note that the function F has to be a
symmetric function of u, v and w.
Further, the function F has to be a linear function so that may yield a one-valued solution
when solved with respect to u, v or w. Consequently, the equation assumes the form
g + h (u+v+w) + k(vw + wv) + 1 (uvw) = 0
Since when w = 0, u = - v, then g - ku² = 0 for all values of u and so g and k are zero.
Thus, the equation takes the form h(u+v+w)+1(uvw)=D. Also, u= -C when v=C, no
matter what the value of w. Hence hw-1C²=0 and h=1C². Therefore,
1C²(u+v+w) +1(uvw)=0
or
u+v+w + (uvw/C²) = 0
This is the exact relation which replaces the Newtonian relation u+v+w=0. This exact
relation implies that
-u-uvw/C²=v+w
and
-u[1+(vw/C²)]=v+w
-u=v+w/[1+(vw/C²)]
-uad = uda
Therefore
vdb + wba
uda = —————
1 +[(vdb) (wba)/C²]
Q.E.D.
Thus the Lorentz Law of the composition of velocities is simly the mathematically
equivalent expressin of every physical theory which assumes that the speed of radiation
in vacuo is independent of the mtion of the radiation source.
—Reciprocity IV.1 (April 1974)
THE TWO-PHOTON PROBLEM
Question: How can two photons from separate sources meet if their space-time locations
are moving away from each other with the space-time progression?
Answer :
1. The essential point here is that if an object is in motion relative to a stationary
reference system, and acquires an additional motion, this new motion does not
replace the previously existing motions; it adds to them.
2. A completely free object is moving outward from all other such objects by reason
of the space-time progression (Motion I). Two such objects having no other
motions therefore cannot collide.
3. Gravitationally bound objects without independent motions are likewise moving
outward from all other similar objects (Motion I), but coincidentally are moving
inward toward all of these objects at the same rate of speed, by reason of
gravitation (Motion II). Two such objects maintain the same separation, and
therefore cannot collide.
4. An object A in a gravitationally bound system may acquire an independent
motion in any direction (Motion III). The sum of all three of the motions of this
object (equal to its independent motion) may then carry it to a point where it will
collide with a similar object B.
5. A photon released from object A participates in all three of the motions of that
object, and inasmuch as it is not under any restraint in the dimensions
perpendicular to the direction of Motions I and II, it is also moved outward at unit
speed in one of these dimensions by the space-time progression. (This motion can
be in any direction relative to the reference system, as the gravitational motion is
random) . The second progression is Motion IV; that is, it is an addition to all of
the other three motions. The net resultant of all four motions is a combination of
Motion III and Motion IV. If object A maintains the same speed and direction, the
motion of the photon, as seen in the context of a stationary reference system, is
directly outward from object A. The emitted photon may therefore collide with
any object B in the gravitational system, or with a photon emitted from object E.
—Reciprocity V. 3 (October 1975)
A Note on Scalar Motion
Beginning students of the Reciprocal System often have difficulty understanding scalar
motion, confusing it with vectorial motion. I will attempt here to clarify matters.
Assume as a thought experiment, a spherical light source in gravitational equilibrium
with us, the observers. In our ordinary 3-D spatial reference system, the source is
stationary, and photons are streaming away from it at the speed of light in all directions.
Without the knowledge of the Reciprocal System, you might conclude that the photons
have independent motion and are moving vectorially through coordinate space away from
the source. Iiowever, the theory says that this is not the case from the standpoint of the
true, natural reference system. The photons actually have no independent motion and thus
are stuck in the same space-time units in which they originate. It is the aioms of the
source that have independent motion and are moving against the space-time progression.
This resulting motion is termed gravitation and is always inward. Most importantly, this
motion is scalar: it is inward in all directions, with no one direction favored. It is because
the source is moving inward in all directions that makes it appear that the photons are
moving outward in all directions. Since the inward gravitational motion is taking place in
space, the motion imputed to the photons is outward in space.
Likewise for the cosmic sector: the cosmic atoms are moving inward in time, and so
cosmic observers would conclude that the photons from their light are moving outward in
coordinate time. In actuality, of course, the photons remain in the same absolute
spacetime locations and are not moving either in coordinate space or coordinate time!
Now suppose, as in the Einstein-Podolsky Rosen experiment, that two photons originate
at the same event and move in opposite directions. In the material sector, the motion
appears to be outward in space; in the cosmic sector, the motion appears to be outward in
time. In actuality, we are moving inward in space away from the photons, and cosmic
observers are moving inward in time away from photons. We have no independent
motion in coordinate time (at low vectorial speeds), and since the photons do not either,
we are able to effect a change in both photons by means of a change in one of them.
Likewise, the cosmic observers have no independent motion in coordinate space (at low
vectorial inverse speeds), and since the photons do not either, the cosmic observers are
able to effect a change in both photons by means of a change in one of them. (Existents
which are contiguous in either space or time may both be affected by application of a
suitable single force).
Because photons are stationary in the natural reference system, they are nat ―lost‖ from
either sector and are not ―disappearing over the time or space horizon‖; the universe is
not ―running down‖ toward a slow ―heat death‖.
A NOTE ON THE FORCE OF THE
SPACE-TIME PROGRESSION
In a previous paper of mine¹ I discussed Hubble‘s Law and the Reciprocal System. I
integrated Hubble‘s equation to obtain the following equation representing the distance of
a galaxy from our galaxy as a function of time:
r = ro + (vi - ro)e}Ht
(1)
where ro is the gravitational limit, H is Hubble‘s constant, and ri is the initial distance.
Now if we go back to Hubble‘s original equation,
V = Hr (2)
we can differentiate this, instead of integrating it as before.
dv/dt = H dr/dt = a = Hv = H²r (3)
Hence, the force of the space-time progression must be
Fp = mG² H²r + FG¹-G² (4)
where mG² is the mass of the galaxy moving away and FG¹-G² is the gravitational attraction.
This equation should have many applications in the Reciprocal System.
References
1. R. W. Satz, ―Hubble‘s Law and the Reciprocal System,‖ Reciprocity, Vol. VII, No. 3
ON THE NATURE OF
UNDISPLACED SPACE-TIME
Question: With respect to what is space-time moving? If there is not something
more fundamental than space-time with respect to which space-time, itself, is
moving, then space-time cannot properly be said to move (or progress) at all.
Answer: The term ―space-time,‖ as used in the Reciprocal System of theory, is
equivalent to, and interchangeable with, the term ―motion,‖ in the broadest sense of
the latter term, and the general nature of the answer to the foregoing question can
readily be seen if the equivalent term is substituted for ―space-time‖ in the wording
of the question. No one appears to have any difficulty in recognizing that the end of
a unit of time is later—more advanced—than the beginning of that unit; that is,
there is a progression from the beginning to the end. Furthermore, it is commonly
understood that this is simply a progression, not a progression relative to something
else, and hence a unit of time, a section of the progression, is a self-contained entity.
As the published expositions of the Reciprocal System have demonstrated, the
concept of a universe of motion requires that space be defined in exactly the same
terms as time, except that it is the inverse quantity. Thus the end of a unit of space
is also more advanced—that is, more distant (the spatial equivalent of later)—than
the beginning; not more distant from something but simply more advanced. Space,
too, is a progression, and since both of its components progress, notion (space-time)
is likewise a self-contained progression; it is not a ―something‖ that progresses
relative to something else.
Of course, a certain amount of mental effort is required in order to lift our thinking
out of the grooves in which it has been running so long, but obviously if it is
possible to conceive of time as a progression, independently of any hypothetical
background—a mental feat that seems to present no particular difficulty—then it is
also possible to conceive of an inverse quantity of exactly the same general nature.
If there is any difficulty in so doing, it does not arise from the nature of the concept
itself, but from an unwillingness, or inability, to let go of ideas that are derived from
premises that have no relevance in a universe of motion. When space and time are
viewed in terms of these concepts there should be no obstacle to recognizing that
motion (or space-time) is a similar self-contained progression, According to the
fundamental concept on which the new theoretical system is based, the unit of this
progression—the unit of motion—is the basic entity of the universe, that from
which all else is constructed. It cannot be related to anything ―more fundamental.‖
The idea of a background to which motion must be related belongs to some concept
such as that of a universe of matter; it has no place in a universe of motion, where
motion itself is the ultimate reality.
CLOCK SPACE, COORDINATE SPACE;
CLOCK TIME, COORDINATE TIME:
What is the difference? At last year‗s ISUS convention, a number of individuals expressed difficulty in
comprehending the difference between clock space and coordinate space and the
difference between clock time and coordinate time. This note will review these concepts
to aid the understanding of these individuals.
Larson states [1]: ―We begin with one-dimensional space s and one-dimensional time t....
Dividing space by time we obtain velocity s/t....‖ Space and time do not exist separately;
they exist only as aspects of motion. Motion in the most general sense is thus a relation of
space to time, and in the Reciprocal System space and time have no properties other than
what they have as aspects of motion. In defining motion, we can start with units of space
and time, as Larson did in the quotation, or with units of motion and define space and
time as the two aspects of that motion; the definitions are equivalent.
The basic space-time unit is thus one-dimensional and is a progression. We reject the
Relativity doctrine that space and time are joined in four-dimensional continuum and that
space and time magnitudes are purely relative. From the postulates of the Reciprocal
System we compute the absolute natural unit of space to be 4.558816x10-6 cm and the
absolute natural unit of time to be 1.520655x10-16 sec. Their ration is 2.997930x1010
cm/sec, the speed of light. The progression originates everywhere and is thus
omnipresent. Larson stats [2]: ―Unit velocity is a ... true physical datum with a finite
magnitude.‖ Thus we begin with clock space-time, rather than with coordinate space or
coordinate time (or a combination of coordinate space with clock time). Conventional
physicists (and individuals new to the Reciprocal System) keep trying to start with some
type of 3-D or higher metric; we reject this approach entirely.
Coordinate space and coordinate time result from clock space and clock time. Larson
explicitly states [3]: ―There is a general framework of the universe, an extension space,
generated by translational motion....‖ ; likewise, there is an ―extension‖ time, generated
by translational motion, the progression. This motion is scalarly outward in all directions
and thus the overall distribution of the 1-dimensional progressing units is 3-dimensional.
In The Unmysterious Uniuerse [4], I wrote ―...with respect to any particular progressing
unit, coordinate space and coordinate time include all other progressing units.... The
progression of a single unit of space is one-dimensional, but the progression of all space
units is distributed in three-dimensions.‖ Stationary coordinate space (or stationary
coordinate time) can arise only in the context of a gravitationally-bound material system
(or cosmic system), in which the atoms of matter (or c-matter) have neutralized the space
progression (or time progression).
When Larson states [5] that ―undisplaced space-time is the physical equivalent of
nothing‖ he means that a unit of space-time is not a photon, a subatom, an atom, or an
electric or magnetic charge. These other entities are interchangeable, either directly or
indirectly, but a unit of spacetime is not. It cannot be changed into something else, and it
cannot be used as an energy source. But this does not mean that space-time is ―nothing‖;
it is unit motion, not zero motion, and is every bit as much an existent as anything
physical. Larson says [6] ―In terms of [a] building analogy, [space and time] correspond
to the bricks of which the building [i.e., the universe] is to be constructed.‖
REFERENCES:
1. D. Larson, New Light on Space and Time (Portland Oregon: North Pacific
Publishers, 1965), p. 226.
2. Ibid, p. 83.
3. Ibid, p. 242.
4. R. Satz, The Unmysterious Uniuerse (Troy, NY: Troy Printers, 1971), p.24, 27.
5. D. Larson, op. cit, p. 242.
6. D. Larson, op. cit, p. 240. 5
COSMIC RAYS AND ELEMENTARY PARTICLES
A View of the Reciprocal System
Introduction
Recently the existence of two new particles has been discovered at the Brookhaven
National Laboratory and the SLAC-Lawrence Berkeley Laboratory. They have been
named the psi [3695) and psi (3100) resonances (although their exact masses, in MeV/c²,
are still somewhat in dispute). A whole bumper crop of theories has sprung up to explain
these heavy resonances. In this endeavor physicists are postulating quantities such as
―charm, color, paracharge, gentleness and chimerity.‖ These terms are added to the
original term, ―strangeness.‖ In this paper an explanation of these particles, others like
them, and their natural origins, will be offered without the use of any quantity such as
―charm.‖
I. Critique of Present Theory
Around the turn of the present century physicists had just two particles to work with: the
electron and the proton. The theory then was that the atom was composed only of these
particles. In 1932 the neutron was discovered and added to the list of particles contained
within the atom. At the same time the positron was discovered, but not added to the list of
particles contained within the atom. Then during the 1940s the first two mesons, the
muon and the pion, were discovered. The physicists found that they could use the pi
meson to ―explain‖ the force of attraction between protons and neutrons within the
postulated atomic nucleus. But they knew of no function for the mu meson.
With the increase of energy in the linear accelerators and storage rings in the last two
decades, scores of new particles were found. Certainly Physicists could not assume that
all played a role in atomic structure. They decided that some particles are more
elementary than others: the present theory is that all the particles of the universe are made
up of three ―quarks‖ and three ―anti-quarks.‖. These somehow combine to make up every
other particle. (The original theory postulated an eight-fold way, but that hypothesis has
broken down).
There are two immediate criticisms to the quark hypothesis:
1. In all attempts to find quarks, no one has succeeded—it seems that every time an
attempt is made, new particles are discovered, but unfortunately, they aren‘t
quarks.
2. The quark theory gives no indication as to how matter can change into radiation
and vice versa.
Another aspect of present theory concerns the forces between particles. In essence the
theory states that all forces arise from the exchange of quanta, as follows:
1. Gravity results from the exchange of ―gravitons.‖
2. Electromagnetic forces result from the exchange of ―virtual photons.‖
3. The weak beta decay of atoms results from the exchange of ―the weak boson W.‖
4. The strong nuclear forces result from the ―exchange of mesons between
nucleons.‖
5. Chemical attraction between atoms results from the exchange of electrons.
All of these hypotheses suffer from one and the same defect: none of the exchange
particles has ever been observed in action doing what they are supposed to be doing
Despite major efforts no one has observed gravitons, weak bosons, exchange electrons or
exchange photons. (The last category is, by definition, not observable; hence there is no
way to test the latter theory.) Nor has the actual mechanism of coupling ever been
specified. It may be true that an exchange of gifts in the human realm promotes bonds of
friendship and it is true that our rate of effecting closer bonds is limited to the speed of
light. But the atomic realm is different. There the communication of forces is evidently
instantaneous; there is no time factor in Newton‘s law of gravitational force or
Coulomb‘s law of electrostatic force. To postulate an exchange of force and to limit the
propagation speed to that of light is, in a sense, anthropomorphic.
II. The Theory of the Reciprocal System
In the Reciprocal System as advanced by D. B. Larson in his books, beginning with the
Structure of the Physical Universe, no particle is posited as being actually elementary.
The fundamental component of the universe is not a set of matter particles (quarks).
Rather, the fundamental component is motion, existing in discrete units. Quantitatively,
the motion may be above or below unit value (one unit of space per unit of time). In the
sector we live in, the motion is ordinarily below unity. According to the theory, another
sector exists with motion above unity.
In our sector exists a series of chemical elements and subatomic particles, each a specific
quantity of motion. Likewise the other sector has a series of chemical elements and
subatomic particles: In our sector there exist stupendous galactic explosions resulting in
quasars, whose matter, the theory postulates, leaves our sector and enters the inverse
sector. Likewise, in the other sector stupendous galactic explosions occur, in which
inverse matter is dispersed into our sector. This dispersed, very energetic inverse matter
may be identified as cosmic rays.
In Mr. Larson‘s theory an atom, either in this sector or the other, is not composed of
protons, neutrons and electrons (or their ―antiparticles‖). Rather each atom and each
cosmic atom has a specific quantity of rotational motion about three perpendicular axes.
All of the particles, whether atoms, mesons, resonances or baryons, are units of motion,
some having more than others. The rotational motion making up the particles can, under
certain conditions, convert to linear motion or radiation. Forces arise from the interaction
of the particles‘ motion with that of the space-time progression. The latter is the general
translational outward motion of the universe, which arises from the equivalence of single
units of space and time. Since the particles and the progression are always interacting,
there is no Propagation of gravitational or electrostatic force and thus NO speed limit.
In our sector the most common element is hydrogen. According to theory, H² (deuteron)
is the natural atom, and H¹ is H² with a magnetic charge. Likewise in the other sector the
most common element is designated cosmic hydrogen, co-H² (co-deuteron). This co-H² is
the most common natural cosmic atom, and co-H¹ is co-H² with a cosmic magnetic
charge.
In the cosmic ray stream cosmic hydrogen should be much more abundant than other
cosmic elements. As elements constructed of motion above unity, these cosmic elements
should have properties the inverse of those ordinarily associated with corresponding
elements in our sector. this means that their masses must be the reciprocals of the masses
of their namesakes in this sector. Mathematically, the mass of a cosmic element is
m-co-element = 1/n natural mass units
= 2/n atomic mass units
= 364.66/n electron mass units
= 1862.95/n MeV/c²
where n is the cosmic atomic number. In the case of cosmic isotopes,
m-co-elements =1862.95/ (n +½G) MeV/c²,
where G is the number of cosmic gravitational charges. Once in our sector the co-
elements are subject to the same forces that produce material isotopes. Let I be the
number of material isotopic charges. Then the complete mass equation (ignoring
secondary mass effects) is
m-co-elements = (1862.95/ [n+½G) + I [931 .478) MeV/c²,
where I and G must have opposite signs for compatible motions.
Using the first equation above for cosmic deuteron, I find that it has the same mass as
deuteron in our sector. Once in our sector, with its lower translational velocity and its
change in space-time reference point, the cosmic deuteron transforms to the deuteron of
our sector. This is a true CPT transformation: an inversion of space-time coordinates
followed by an interchange of a particle and "antiparticle". Momentum and energy are
conserved in this transformation.
III. Critiquing the Reciprocal Theory
Old readers of Mr. Larson's books will note that here I am disagreeing with the original
presentation of the theory. There it was said that cosmic helium transformed to cosmic
krypton, emitting neutrons along the way, until finally the cosmic krypton converted into
a neutron (or equivalent). Although the present author agrees with Mr. Larson on the
framework of the theory, here we differ on details. Since the New Science Advocates is
not a religious sect, but a scientific body, members have the freedom, indeed the
obligation, to question the details of the theory. Cosmic krypton cannot decay to a
neutron because to do so Would violate the conservation law of momentum and energy.
One body decays, except in the case of hydrogen, are simply impossible. Furthermore, a
cosmic element in the middle of the cosmic periodic table of elements cannot emit a
neutron to move to the last column in the table, anymore than an element in the middle of
our periodic table could accept a neutron to move to the last column of the table.
I thus conclude that cosmic hydrogen transforms to our hydrogen, rather than that cosmic
krypton transforms to a neutron.
Once transformed, the new deuteron is unstable in our atmosphere and soon decays to a
proton, electron and neutrino. Thus, the cosmic ray stream is composed mostly of high
energy protons, precisely as observed.
Other cosmic elements and their isotopes exist in the cosmic ray stream and are produced
in laboratories on earth. Because of the variability in G and I, two different particles can
sometimes have the same mass. Thus identification of particles cannot be based solely on
mass. Some of the better known heavy mesons (hyperons) together with their probable
identifications are here listed:
SYMBOL OBSERVED MASS N G I IDENTIFICATION
3695 1 -1 0 co-H¹
1 0 2 co-H²+2
½ 0 0 co-n
751 2 1 0 co-He5
1672.5 3 -1 1 co-Li5+1
547 3 1 0 co-Li7
1304 5 0 1 co-B10+1
1236 6 0 1 co-C12+1
3100 6 0 3 co-C12+3
1198 7 0 1 co-N14+1
1118 10 0 1 co-Ne20+1
I think that the recently found psi [3695) particle is either cosmic hydrogen, isotope 1 or
cosmic deuteron with two material isotopic charges or cosmic neutron. A firm decision
will have to await experimental results.
On the basis of preliminary calculation I tentatively submit that the other recently found
particle psi (3100), is cosmic carbon with three material isotopic charges.
In addition to the heavy mesons two light mesons are commonly observed in the cosmic
ray streams and produced in laboratories on earth: the muon and the pion. I agree with D.
B. Larson that the muon is co-argon and the pion is co-silicon. On earth these mesons are
created from kinetic energy; an energetic proton strikes another proton, producing a third
particle, the pion. This pion decays to a muon and neutrino. I do not agree with Mr.
Larson that the muon then decays to co-cobalt and then to co-krypton. My reading of the
evidence indicates that the muon simply decays to positrons and neutrinos. The rotational
kinetic energy is converted to linear kinetic energy of simple rotational units, positrons or
electrons or neutrinos. The light mesons are created from kinetic energy and to kinetic
energy they return.
Before it decays, the pion is ―strongly interacting,‖ because it has both space and time
displacements. The muon is ―weakly interacting,‖ because it has only space
displacements. This is also the reason why muons are the ―hard‖ component of cosmic
rays—they can penetrate many meters into the ground. In the decay processes certain
conservation laws seem to hold true. Physicists are currently proposing one new
conservation law after another (strangeness number, hypercharge, paracharge, charm ,
baryon number, lepton number, etc.). In the Reciprocal System there is but one
conservation law: space-time displacements can be neither created nor destroyed. Energy,
t/s, and thus momentum, (t/s) , are conserved. In some reactions certain groups of space-
time units are conserved, for example, electric charge.
In many cases the heavy mesons decay to the lambda meson, which then ejects a neutron.
The remaining pion, if neutral, decays to two gamma rays. If charged, the pion decays to
a muon, which then transforms to a positron (or electron) and neutrino. The somewhat
lighter mesons, the eta, the rho, and the (small letter) omega, decay to two or more pions.
The theoretical explanation of decay process involves probability: smaller quantities are
more probable than larger ones. In the steps from rotational kinetic energy to translational
kinetic energy, the fewest number of particles are utilized. The decay pattern of the new
particles appears to be along these lines. The psi (3695) decays to the psi (3100), emitting
two pions, and then the psi (3100) decays to two muons. Many questions, however, still
remain to be answered.
CONCLUSION
The following are the essentials of the new theory:
1. A sector, the inverse of ours, is hypothesized, which provides a natural source for
scores of new particles, such as the mesons.
2. The cosmic rays are a stream of such particles coming from the inverse sector.
3. The masses of these particles are the inverses of their namesakes in our sector.
4. Inverse deuteron is equivalent to and converts to our deuteron. The most abundant
element in the cosmic sector converts to the most abundant element in our sector.
5. Cosmic atoms may have cosmic isotopic charges and/or material isotopic charges.
6. The cosmic elements in the cosmic ray stream, other than hydrogen, decay
eventually to the kinetic energy of simple rotational units, the electron, the
positron, the neutrino. In our laboratories mesons are created out of kinetic
energy. In our atmosphere the natural mesons return to kinetic energy.
7. All conservation laws relate to one: space-time displacements are neither created
nor destroyed.
Reciprocity Vol. V, No. 2 (May, 1975)
LETTER TO EDITOR
Dear Prof. Meyer,
In my opinion, the development of the consequences of the postulates of the Reciprocal
System of theory, and the correlation of these consequences with the results of
observation, have now been carried far enough to make it evident that the theoretical
system is basically correct. There are, however, many questions still remaining with
reference to the details, even in the areas that have already been studied, and, of course,
there are a great many other areas yet to be examined. I believe it is very desirable to
encourage free and open discussion of the theory and its application, so that we can have
the benefit of as many points of view as possible in extending and clarifying the
theoretical structure, and I want to avoid saying or doing anything that might give the
impression that I am trying to discourage dissenting opinions. For that reason I would
prefer not to comment at this time on Ronald Satz‘ article discussing the newly
discovered heavy ―resonances,‖ except to say that I agree with his conclusions l, 2, 3 and
7, and in part, with conclusion 5. I hope that readers of RECIPROCITY will give this
article careful consideration, and will not hesitate to express their opinions, pro and con,
in ―letters to the editor.‖
D. B. Larson
IDENTIFICATION OF COSMIC PARTICLES
3695 MeV/C² AND MeV/C²
In November, 1974, two teams, one at the Brookhaven National Laboratory and the other
at the Lawrence Livermore Laboratory, announced the discovery of a new particle with a
mass equivalent to 3,105 MeV/c² of energy. The lifetime of this particle is about 10-20
second, considered by some to be a remarkably long lifetime for a particle of this heavy
mass. This particle is named with the Greek letter, psi, and is referred to as a psi
resonance.
Shortly afterward, the two teams discovered a second psi resonance with a mass
equivalent to 3,695 MeV/c² of energy and lifetime of about 10-20 second. Cosmic decay of
the 3,695 MeV/c² particle apparently results in production of 3,105 MeV/c² particle.
Discovery of these two new physical entities is exciting news from the frontiers of
physics. How the psi resonances fit into the physical scheme of things has remained a
mystery until now. The discovery of the mere existence of these high-energy particles has
been deemed so important that the leaders of the two teams, Drs. Samuel Ting and
Burton Richter, were awarded the 1976 Nobel Prize in physics for this discovery.
In the Reciprocal System psi resonances and other related cosmic particles are identified
as specific isotopes of cosmic chemical elements.
The identification procedure depends on the convergence of several lines of approach,
including theoretical computation ot the mass and lifetime of each particle and also
examination whether and how ic can fit into the regular cosmic decay sequence after the
particle enters the material sector.
Cosmic element mass once the cosmic element enters the material sector is generally
made up ot its rotational mass,the inverse of the material element mass (Figures 1 and 2),
and of its material gravitational charges (Figure 3) acquired with entry into the material
sector (Larson, 1979).
Figure 1
COMPUTATION OF COSMIC ELEMENT MASS
1 atomic mass unit = 1.66 × 10-27 kg.
c = 2.99 x 108 m/ s ; c² = 8.94 × 1016 m²/ s²
Equivalent energy of 1 a.m.u. = mc²
1 a.m.u. = (1.66 x 10-27 kg) (8.94 × 1016 m ²/ s² = 14.9 x 10-11 J
1 electron volt = 1 ev = 1.6 x 10-19 J
Energy equivalent of 1 a.m.u. = 14.9 × 10-11 J / 1.6 X 10-19J
1 atomic mass unit = 931.15 MeV/c²
Mass of a material atom of atomic number Z:
m = 1862.30Z MeV/c² (1862.3 = 2 (931.15))
Mass of a cosmic atom is INVERSE mass
We observe cosmic mass as 1862.30/Z MeV/c²
Figure 2
COMPUTATION OF COSMIC ELEMENT MASS
Let Z = atomic number of cosmic element
cosmic mass = 1862.30/Z MeV/c²
Alternative Procedure
Instead of atomic number units (Z),
use atomic mass (or weight) units to express osmic mass.
Atomic weight units are half the size of units of atomic number.
Then cosmic mass = 3724.61/m MeV/c²
This is the mass of cosmic atom (isotope)
in the condition in which it enters material sector.
m here represents atomic weight units
Figure 3
COMPUTATION OF COSMIC ELEMENT MASS
after element enters material sector.
Mass of cosmic element in atomic weight units when it enters material
sector:
Cosmic mass = 3724.61/m MeV/c2, m here represents atomic weight units.
Superscripts for isotope symbols are atomic weight units.
After entering material sector cosmic atoms
may acquire gravitational charges of material type.
Mass of each gravitational charge is one atomic weight unit = 931.15 MeV.
The psi resonance with a mass equivalent to 3695 MeV/C² has been identified as the
isotope of cosmic hydrogen, c-H², cosmic deuteron with two material gravitational
charges (Figure 4). This is a deduction from the Reciprocal System theory and the
achievement of Ronald W. Satz (1975) and Larson (1979).
Figure 4
IDENTIFICATION OF 3695 MeV/c² PARTICLE
Identified by R. W. Satz as ―cosmic deuteron with two material isotope
charges‖ (c-H²).
Rotational mass of a material hydrogen (H²) atom is 1.007405 units of
atomic number scale.
Mass of a cosmic H² atom is the reciprocal of this number = 0.99265 units.
For hydrogen Z = 1, first portion of
Cosmic mass of c-H² = 1862.31 (0.99265/Z:
Rotational cosmic mass of c-H² = 1848 MeV/c2
After entry to material sector c-H² acquires two material gravitational
charges
2(931.15 MeV/c²) = 1862.3 MeV/c²
Total cosmic mass of c-H² =
1848 MeV/c² + 1862 MeV/c² = 3710 MeV/c²
Observed mass of c-H² reported as 3695 MeV/c²
The psi resonance with a mass equivalent of 3105 MeV/c² has bean identified as an
isotope of cosmic helium, c-He³ with two material gravitational charges (Figure 5). This
is an achievement of D.B. Larson (1979).
Figure 5
IDENTIFICATION OF 3105 MeV/c² PARTICLE
Identified by D. B. Larson as cosmic helium with two material gravitational
charges (c-He³).
The material He³ isotope is a He atom (mass = 4 atomic weight units) with a
one-unit negative gravitational charge (one negative atomic weight unit).
The mass of the isotope is then 3 atomic weight units.
The cosmic He³ isotope is a similar but inverse structure, with a net mass of
3 cosmic atomic weight units.
Since the c-He3 isotope has a mass of 3 cosmic atomic weight units, its
rotational mass as observed in the material sector is 3724.61/3 = 1242
MeV/c².
After entry to material sector the c-He³ isotope adds two material
gravitational charges mass 931.15 each making total mass 3104 MeV/c².
Observed mass reported as 3105 MeV/c² .
Some 20 years ago Larson (1959) already identified as isotopes of other cosmic chemical
elements the muon, the pion, the lambda, sigma, xi and omega particles (Table 1).
TABLE 1
SOME COSMIC ELEMENT ISOTOPES IDENTIFIED
Isotope
Cosmic
Mass
3724.61/ m
MeV/ c²
Gravitational
Number
Charges
mass
MeV/c²
Total
Mass
MeV/c²
Observed
Mass
MeV/c²
Name
c-H² 1848 2 1862 3710 3695 psi
c-He³ 1242 2 1862 3104 3105 psi
c-Li5 745 1 931 1676 1673 omega
c-B10 373 1 931 1304 1321 xi
c-N14 266 1 931 1197 1197 sigma
c-Ne20 185 1 931 1116 1116 lambda
c-Si27 138 0 0 138 140 pion
c-Ar35 106 0 0 106 106 muon
References
Dewey B. Larson, The Structure of The Physical Universe, North Pacific Publishers,
1959.
Dewey B. Larson, Nothing But Motion, Volume I of a Revised and Enlarged Edition of
The Structure of The Physical Universe, 1979. North Pacific Publishers.
Ronald W. Satz, Cosmic Rays and Elementary Particles: A View of the Reciprocal
System Reciprocity Vol. V, no. 2 (May 1975)
A NOTE ON THE COSMIC PROTON
The rotational displacements of the material proton (a single rotating system of a 2R
photon) are 2-1-(1); the rotational displacements of the cosmic proton (a single rotating
system of a ½ R photon) are (2)-(1)-1. Both protons can take a positive or negative
charge (because both have the necessary space and time displacements). In the material
sector, rotational time displacements are predominant, hence the opposing rotational
vibrational space displacements (positive charges) are more common; in the cosmic
sector, rotational space displacements are predominant, hence the opposing rotational
vibrational time displacements (negative charges) are more common.
So we would expect the observed material proton to carry a positive charge (usually) and
the observed cosmic or ―anti‖ proton to carry a negative charge. The first observed ―anti‖
proton (in 1955) was created by the bombardment of one proton on another at rest. Either
this proton and the subsequent ones created in this manner are actually material protons
with negative charges or they are cosmic protons. Certainly the ones that are used in the
annihilation experiments are cosmic protons; two material protons would combine to
form deuterium, not terminate rotational displacements. Because material deuterium and
cosmic deuterium have theoretically identical masses, each of the individual rotating
systems has identical mass.
Thus the mass of the material proton and the cosmic proton should be identical! And both
should be perfectly stable. The only way to tell them apart would be by their subsequent
reaetions. Two cosmic protons would combine to form cosmic deuterium; with two
gravitational charges, the mass of this particle comes to 3710.91 Mev, which is elose to
the observed mass of the psi particle. This particle then follows the decay sequence
described in Larson‗s book Nothing But Motion, chapter 15. So the forthcoming
experiment should see what happens when two ―anti‖ protons combine; our predictions
are clear.
—Cf. D. B. Larson’s comments in a letter to R. W. Satz, dated Sept. 22, 1988
THE COHESIVE ENERGY OF THE ELEMENTS AT
ZERO TEMPERATURE AND ZERO EXTERNAL
PRESSURE
The equation for the internal energy of a substance is
u = h - pv (1)
where h is enthalpy, p is pressure, and v is volwne. At zero absolute temperature, the
enthalpy is zero.
uo= -pv (2)
For a gas at zero temperature governed by the ideal gas law, the internal energy must also
be zero. This is not so with a solid. Larson has shown that the equivalent of an external
pressure exists which provides the cohesion of the solid state. This pressure arises from
the force of the space-time progression, which is inward directed within the time region.
With zero external pressure and zero temperature, the internal energy must equal the
cohesive energy. Letting * be the internal pressure in kN/m² and vo be the volume in m³
/mole, and dropping the sign convention, we obtain the cohesive energy in kJ/mole:
uo = po vo (3)
However, as shown in reference one, motion in the time region (whether inward or
outward) is effective only hatf the time. This reduces the cohesive energy given by
equation (3) by a factor of two.
uo = ½po vo (4)
This equation is directly applicable to the "rare gas" elements.
The equation for molar volume is
vo = GNso ³ (5)
where so is the nearest neighbor distance N is Avogadro's number, and G is a geometric
factor. For face-centered-cubic crystals,
Gfcc= .707 (6)
For body-centered-cubic crystals,
Gbcc = .770 (7)
For other crystals,
(GMW/density) x 10-6
G = ———————— (8)
so³N
where GMW is the gram molecular weight, density is in grams per cubic centimeter, and
so is in meters (10 Angstroms).
In Chapter 25 of reference one, Larson derives the equation for the internal pressure in
natural units:
aZR
Po = ———————— (9)
312.89(so/sut)³
where a is the effective displacement in the active dimension, Z is either the electric
displacement or the second magnetic disptacement (depending on the orientation of the
atom), and R is the number of rotational units. sut' is the time region natural unit of space,
given by sut = (1/156.44) su (10)
In kN/m² the value for po becomes
Po = 4.177 x 10-17 aZr kN
—— —— (11)
So³ m²
Then,
uo = 12.57GaZR kJ/mole (12)
The parameters a, Z, and R have been deduced by Larson for most of the elements, but
not yet for the rare gas elements. Pending this, the value of the internal pressure can be
determined as the reciprocal of twice the initial compressibility (equation 25-14,
reference one):
po = 1/2kT (13)
Table I gives the values for po , vo, and uo for the rare gas elements. Overall the values
compare within 8% of the experimental values.
Elements other than the rare gas elements have electric displacement and this must
obviously have an effect on cohesive energy. The additional energy is given by this
expression:
ut' = INEu (1/156.44)4 (14)
where I is an integer or half integer value, N is Avogadro's number, and Eu is the natural
unit of energy. Alternatively from the cohesive energy standpoint, the effective volume,
v, may be altered. The factor is the interregional ratio (applicabte to energy, as well as
force). I is one for most of the displacement one elements, one and one-half or two for
displacement two elements, three or more for displacement three elements, and from 3½
to 5½ for displacement four etements. I can be zero or negative for the electronegative
etements. An exact equation for I cannot as yet be given.
The final reduced equation for cohesive energy is
kJ
uo = 12.57 GaZR + 50.31 —— (15)
mole
Table II gives the values of G, a, Z R, I, and uo for most of the remaining elements,
together with the experimental values from reference two. Usually agreement is within a
few percent.
Present atomic theory has nothing comparable to equation (15). The so-called Lennard-
Jones potential commonly used is empirically based and has not been deduced from first
principles--and even then it has usually been applied only to the noble elements and a few
other elements of 1ow atomic weight. Thus we have here a definite advantage of the
Reciprocal System over current theory.
References
1. Dewey B. Larson, Nothing But Motion, Vol. 1 of the revised Structure of the
Physical Universe, presently in manuscript form.
2. C. Kittel, Introduction to Solid State Phisics, Fifth Edition (New York: John
Wiley & Sons, Inc., 1976 , p. 74.
Table 1
—————————————————————————–———————
kN m³ kJ kJ
Element — — — —
po m²
vo mole
uo mole
uexp mole
—————————————————————————–———————
Helium 8.56x104 1.950x10-5 .835 --
Neon 5.00x105 1.395x10-5 3.488 1.92
Argon 5.33x105 2.227x10-5 5.935 7.74
Krypton 8.93x105 2.806x10-5 12.53 11.2
Xenon 9.52x105* 3.528x10-5 16.79 15.9
Radon 12.30x105* 3.584x10-5* 22.04 19.5
*Estimated values based on trend line analysis or assumed specific rotational values.
Table II
—————————————————————————–—————
——
kJ
kJ
Element Form G a Z R I
——
——
uo mole
uexp mole
—————————————————————————–—————
——
Li bcc .770 4 1 1 2½ 164.5 158
Be hcp .752 4 4 1 3½ 327.3 320
C dia 1.554 4 6 1 5 720.3 711
Na bcc .770 4 1 1 1 89.0 107
Mg hcp .780 4 3 1 1 157.1 145
Al fcc .707 4 5 1 3 328.6 327
Si dia 1.543 4 5 2 -6½ 448.9 446
K bcc .770 4 1 1 1 89.0 90.1
Ca fcc .707 4 3 1 1½ 182.1 178
Ti hcp .731 4 8 1 3½ 470.1 468
V bcc .770 4 8 1 4 510.9 512
Cr bcc .770 4 8 1 2 410.3 395
Mn cu.com. 1.087 4 8 1 -3 286.3 282
Fe bcc .770 4 8 1 2 410.3 413
Co hcp .696 4 8 1 3 430.9 424
Ni fcc .707 4 8 1 3 435.3 428
Cu fcc .707 4 8 1 1½ 359.9 336
Zn hcp .809 4 4 1 -1 112.4 130
Ge dia 1.541 4 4 1 1 360.2 372
Rb bcc .770 4 1 1 1 89.0 82.2
Sr fcc .707 4 3 1 1 156.9 166
Zr hcp .731 4 6 1½ 5½ 607.5 603
Nb bcc .770 4 8 1½ 5 716.1 730
Mo bcc .770 4 8 2 1 669.7 658
Ru hcp .730 4 8 2 1½ 662.7 650
Rh fcc .707 4 8 2 0 568.8 554
Pd fcc .707 4 8 1½ -1 376.3 376
Ag fcc .707 4 8 1 0 284.4 284
Cd hcp .816 4 4 1 -1 113.8 112
In tet .762 4 4 1 2 253.9 243
Sn dia 1.543 4 4 1 0 310.3 303
Sb rho 1.227 4 4 1 0 246.8 265
Cs bcc .770 4 1 1 1 89.0 77.6
Ba bcc .770 4 2 1 2 178.0 183
La hex .721 4 4 1 5½ 421.7 431
Ce fcc .707 4 4 1 5½ 418.9 417
Pr hex .722 4 4 1 4 346.4 357
Nd hex .698 4 4 1 4 341.6 328
Sm com .716 4 4 1 1 194.3 206
Gd hcp .722 4 4 1 5 396.7 400
Dy hcp .732 4 4 1 3 298.1 294
Ho hcp .732 4 4 1 3 298.1 302
Er hcp .736 4 4 1 3½ 324.1 317
Tm hcp .679 4 4 1 2 237.2 233
Yb fcc .707 4 2 1 1½ 146.5 154
Lu hcp .732 4 4 1 5½ 423.9 428
Ta bcc .770 4 8 2 3 770.3 782
W bcc .770 4 8 3 -1½ 853.8 859
Ir fcc .770 4 8 3 -3½ 677.2 670
Pt fcc .770 4 8 2 0 568.8 564
Au fcc .770 4 8 1½ -3 275.7 284
Tl hcp .690 4 4 1 1 189.1 182
Pb fcc .770 4 4 1 2½ 268.0 265
Bi rho 1.224 4 3 1 1 234.9 210
Th fcc .770 4 8 1 6 586.2 598
U com .998 4 8 1 3 552.3 536
—————————————————————————–—————
——
THE EQUATION OF STATE OF SOLID MATTER
For many years scientists and engineers have had available an excellent equation of state
for gaseous matter. Now, at last, the Reciprocal System of Dewey B. Larson is able to
give us an exact equation of state for solid matter. This paper will present a unified
treatment of the subject, with Reference 1 as the starting point.
I. Volume of Solid as a Function of Temperature with Pressure Constant
From the material presented in Chapter 8 of Ref. 1, I have drawn a generalized plot of
thermal expansion coefficient versus temperature, Figure l. The symbols are defined as
follows:
ß = thermal expansion coefficient
T = temperature
TM = temperature of solid end point (at or close to melting point)
VM = volume at solid end point
T1 = first transition temperature
ß0 = initial value of thercnal expansion coefficient at absolute zero temperature
V0 = initial volume at absolute zero temperature
V1 = volume at tranSition temperature
ß0‘ = initial vatue of thermal expansion coefficient based on second segment of curve
V0‘ = initial volume based on second segment of curve
With the initial votume of the first segment of the curve included, eq. (8-4) of Ref. 1
becomes
V = V0 + K/n³ T² (1)
where K is a constant and n is the number of rotational units that are themally
vibrating.This equation can be put into a more usable form involving T, T1, ß1, end ß0 —
all of which can be determined from theory. The thermal coefficient of expansion at
temperature T is
ß = 1/V dV/dT + ß0
= 2KT/n³ [1/V0 + KT²/n³ + ß0 (2)
= 2KT/n³V0 + KT² + ß0
At T1,
ß1 = 2KT1/n³V0 + KT1² + ß0
Then,
ß1ß0/2T1 = K/n³V0 + T1²
So,
2T1/ß1-ß0 = n³V0/K + t1²
K/n³ = V0/2T1/ß1-ß0 - T1² (3)
Therefore,
V = V0 + V0T²/2T1/ß1-ß0 - T1² (4)
This equation holds from T = 0 to T = T1. Larson has deduced the following values of ß1,
ß0, and T1:
ß0 = 5.17 * 10-6/°K for one unit
= 10.3 * 10-6/°K for two units
= 15.5 * 10-6/°K for three units
= 20.7 * 10-6/°K for four units units
= units = [3576/TM]
ß0 = -2/7 * ß1 for electropositive elements
= -1/7 * ß1 for some 2lectronegative alements
Tl = 8.98 (a + z + y)°K
a, z, y are from Table 22 of Ref. 1
Thus given the volume of the solid at zero temperature, che rotational factcrs of the
element, and the solid and point temperature, the volume V at any other temperature, (up
to T1) can be easily determined.
The equation for the volume for temperatures above T1 has the same form as eq. (4):
V0 T²
V = V01 + ———
2T1 (5)
——— - T²
ß1-ß0 For this equation to be of use,V0‘ and 0‘ must be expressed in terms of known quantities
such as VM and TM. Now,
V0‘ TM²
VM = V0‘ + ———–
2T1
———– - T1²
ß1-ß0‘
V0‘ T1²
V1 = V0‘ + ———
–
2T1
———
– - T1²
ß1-ß0‘
ln the equation for V1, solve for V0‘ and put in equation for VM:
V1
(
TM²
)
VM = —————— 1 + —————
1 + T1² 2T1
——––—– ——– - T1²
2T1 ß1ß0‘
———– - T1²
ß1-ß0‘
Or,
TM²
1 + —————–
2T1
——–– - T1²
ß1-ß0‘
VM/V1 = ———————
T1²
1 + —————–
2T1
——–– - T1²
ß1-ß0‘
Let,
1
C2 = —————
2T1 (6)
——– - T1²
ß1-ß0
Then,
VM 1 + C2TM²
—– = ————–
V1 1 - C2T1²
Solve for C2:
VM - V1
C2 = ——————— (7)
V1TM² - VMT1²
From eqs. (6) and (7), ß0‘, can be found:
2T1
ß0‘ = ß1 -
V1TM² - VMT1² (8)
+ T1²
VM - V1
This value of ß0‘ can then be substituted into the equation for V0‘:
V1
VO‘ = ———————–
T1²
1 + —————— (9)
2T1
———– - T1²
ß1 - ßo‘
With ß0‘ and V0‘ known, eq. (5) is ready for use. Larson has deduced the following
values of TM and VM:
TM = 1.80 * T1 for one rot. unit vibrating
= 4.56 * T1 for two rot. units
= 9.32 * T1 for three rot. units
= 17.87 * T1 for four rot. units
GM
VM = 1.0625 V0 —–
G0
where G0 is the initial crystal geometric constant and GM is the final one (some solids
cnange crystalline structure as they expand).
The ratio VM/V1 can be generalized to any pair of final to initial volumes:
Vf 1 + C2Tf²
— = ———— (10)
Vi 1 + C2Ti²
Compare this with the equation for a gas:
Vf Tf
— = — (11)
Vi Ti
II. Volume of Solid as a Function of Pressure with Temperature Constant
The comoression of a solid by hydrostatic pressure is discontinuous at certain aressures
nere denoted as P1, P2, P3, P4 etc. At these pressures the internal pressure P0 can change to
P01, P02, P03, P04, etc., thus altering the slope of the compression curve. Larson has shown
that between the transitions, the volume naries as the inverse square root of the pressure.
The most general way to express this is with the following aquation:
V P0 + Pref
—— = ———— (12)
Vref P0 + P
For the given value of P, the vatues of Pref, Vref, and P0 must be detennined by theory (or
empiricnlly if necessary) before V can be found. For a four transition solid we have the
following:
0 < P < P1 Vref = V0 Pref = 0 P0 = P0
P1 < P < P2 Vref = V1 Pref = P1 P0 P01
P2 < P < P3 Vref = V2 Pref = P2 P0 = P02
P3 < P < P4 Vref = V3 Pref = P3 P0 = P03
Now, in the MKS system,
aZy KN
P0 = 4.177 * 10-23 —– —– (13)
S0³ m²
where a, Z, y are the rotational compression values (simitar to the thermat values) and so
is the base interatomic spacing. At each transition a, Z, y can change (and possible S0),
thus causing P0 to change.
Before continuing the discussion of the equation of state I will discuss some subsidiary
properties of matter: the bulk modulus, the modulus of elasticity, and Poisson‘s ratio.
Larson has derived the equation for compressibility; the solid bulk modulus is the inverse
of this:
B = 2 * P0 (14)
(at zero external pressure and zero temperature for a pure substance). I witl not derive the
equation for the modulus of elasticity, E. In eq. (13) let the constants of the equation be
written as J and generalize s0 (for the moment) to s. Then the initial internal stress is
= -P = -Js-3 (15)
By definition,
d
E = —–
d
And,
d d ds
— = — —
d ds d
where is the strain:
s - s0
= ——–
s0
So,
s = s0 + s0
thus,
ds
— = s0
d
Since
d
— = 3Js-4
ds
then
d
— = 3Js-4 s0
d
and when s = s0 and T = 0 °K for a pure substance,
d
— s0,T = 0 = 3Js0-3
= -3P0
d
E = -3P0 (16)
(stress and pressure are in opposite directions)
Poisson‘s ratio can be determined from the well-known equation
3P0
= .5 - ——– (17)
6 2P0
Thus at zero temperature and pressure for a pure substance,
3P0
n = .5 - ——– = .25 (18)
6 2P0
This is in the ―ball park‖ for most solids; however, most substances used in construction
are impure and at other than zero temperature dnd, in addition, may contain a proportion
of tiquid molecules — thus drastically changing the values of Poisson‘s ratio and the
modulus of elasticity. These considerations will be left to another paper.
Going back to eq. (12) we can generalize to the ratio of final to initial volume within a
segment:
Vf P0 + Pi
— = ———– (19)
Vi P0 + Pf
This compares with the equation for a gas:
Vf Pi
— = — (20)
Vi Pf
III.Volume of Solid as a Function of Both Temperature and Pressure
The solid can be considered to undergo a aressure change at zero temperature and then a
temperature change from the new volume. Let P < P1 Then
P0
V0NEW = V0 ——––
P0 + P
Let T < T1. Then,
V0NEW T²
V = V0NEW + ————————
2T1NEW (21)
———— - T1NEW2
ß1 - ß0
The value of T1 is not the same as before. To get to the original value of V1 the new value
of T1 must be higher
2
T1NEW = ———————
ß1 - ß0
——— + ß1 - ß0 (22)
V1
—————–
V0 (P0/ P0 + P)½
where V1 is calculated from the original T1. I am assuming here that. is as before.
If P < P1 and T > T1, then the value of V0‘ has to be modified, since T1 and TM are
different. I assume that ß0‘ is the same. Then the second term on the right in eq. (8) is the
same and the new value of TM can be found:
( {
T1NEW
[
V1TM2 - VMT12
]
}
VM-V1 +VM
) ½
TMNEW = ———– ———————
+ T12 - T1NEW² — T1NEW
T1 VM - V1 V1
(23)
Eq. (5) becomes
V0NEWT²
V = V0NEW + ———–
2T1NEW (24)
———– - T1NEW2
ß1 - ß0
Equations (21) and (24j (combined) represent the complete equation of the solid state.
IV. Exampte Calculations
As an example, consider one volume unit of silver at zero degrets K and zero external
pressure. Whnt is the volume at temperature T and pressure P?
First the thermal rotational factors, a-Z-y, from Tabte 22 of Ref, 1 are found; they are 4-
3-l. With these, the temperature of the first transition point, T1, can be calcutated:
T1 = 8.98 (a+Z+y) = 8.98 (8) = 71.84oK
Silver has a maximum of four magnetit rotational units vibrating, so the solid end point is
TM = 17.87 * T1 = 17.87 * 71.84 = 1283.78oK
In this case the endgoint appears to be somewhat higher than the empirical melting point,
1234 oK Thus it would seem that the thermal factors at the end point are towered by one
to 3-3-1, so that
TM = 17.87 * 8.98 (3+3+1) = 1123.31oK
Now the number of units to use in selecting ß1 is
[3576 / 1123.31] = 3
and therefore
ß1 = 15.5 * 10-6 / oK
Since siiver is etectronegative,
s0 = - 1/7 ß1 = 1/7 * 15.5 * 10-6 = -2.214 * 10-6
Then from eq. (4),
V T²
— = 1 + ————————
V0 2 * 71.84
———————– - 71.84²
(15.5 + 2.214) * 10-6
V
— = 1 + 1.234 * 10-7 T² T < T1, P = 0
V0
This equation holds good up to T = T1 at which point
V1
— = 1 + 1.234 * 10-7 T1² = 1.0006369
V0
For temperatures above T1, the values of V0‘ and ß0‘ are needed. To calculate ß0‘ I am
going to use the empirical value of TM pending theoretical clarification.
From eqs. (8) and (9),
2 * 71.84
ß0‘ = 15.5 * 10-6 - ————————————
1.0006369 * 1234² - 1.0625 * 71.84²
——————————————— + 7184²
1.0625 - 1.0006369
= 9.647 * 10-6
1.0006369
V0‘ = —————–
1 + 71.84²
———— = 1.0004265
2 * 71.84²
———————– - 71.84²
(15.5 - 9.647) * 10-6
Thus from eq. (5),
1.0004265 T²
V = 1.0004265 + ——————
2 * 7184
———————— - 71.84²
(15.5 - 9.647) * 10-6
V = 1.0004265 + 4.07623 * 10-8 T² T > T1, P = 0, V0 = 1
(Note: no crystal change from FCC is assumed here).
Now we‘11 go on to look at the pressure relations. Assume that P is less than the first
transition pressure P1 (which is approx. 107 KN/m² ) so that the initial compressibility
factors from Table 14 of Ref. 1 can be used: a-Z-y = 4-8-1. From Table 4 of Ref. 1,
s0 = 2.87 x 10-10 m. Then from eq. (13),
P0 = 4.177 * 10-23 (4 * 8 * 1) / (2.87 * 10-10)³
= 5.654 * 107 KN/m²
Since P c p , P = 0, dnd V = V . Then eq. (12) is
V
(
5.554 * 107
)
— = ——————– ½ P < 1.0 * 107 KN/m²
V0 5.554 * 107 + P T = 0
If P = .001 P ,
V/V0 = (P0/1.001 P0)½ = .999500
The bulk modulus B, modulus of elasticity E, and Poisson‘s ratio can now be calculated
for a pure sample of silver at zero temperature:
B = 2 * P0 = 2 * 5.654 * 107 = 1.1305 * 108 KN/m²
E = 13 * P01 = 3 * 5.654 * 107 = 1.1692 * 108 KN/m²
V = .25
For the combined pressure and temperature loading, eq. (21) yields
V0 5.654 * 107
——————– T²
V = V0 5.654 * 107 5.654 * 107 + P
——————– + ————————
5.654 * 107 + P
2 T1
——————— - T12
(15.5 + 2.214) * 10-6
The value of T1 to be used here comes from eq. (22):
T1 = 2
————————————————————
15.5 * 10-6 + 2.214 * 10-6 + (15.5 + 2.214) * 10-6
———————————
1.0006369
————–
If P = .001 P0, then
5.654 * 10T - 1
——————–
5.65 * 107 1P
T1 = 128.24 °K
Putting this value of T1 into the above gives:
V
— = .99950 + 6.91095 * 10-8 T² T < T1
V0 P = .001 P0
The new value of T1 gives the new value of V0‘:
V0‘ = 1.0006369
————–
1 + 128.224
—————————— = 1.0002614
2 * 128.24
—————— - 128.242
(15.5 - 9.647) * 10-6
Thus, for temperatures above T1
V = 1.0002614 + 1,0002614 T²
——————––
2 * 128.24
————–— - 128.24²
(15.5 - 9.647) * 10-6
V = 1.0002614 + 2.28350 * 10-8 T² T > T1, P = .001P0
V0 = 1
Finally, from eq. (23) I find that the new melting temperature is:
TMNEW = 1650.88 °K
(I have assumed, however, that this does not affect the original value of ß1) .
Basically the same procedure could be used with other elements, atloys, and compounds.
Corresponding equations do not exist in quantum mechanics. A solution in ―principle
only‖ is not a true solution. A true solution is based on principle and works in practice.
**********************
Reference
1. Dewey B. Larson, Solid Matter, prepublication version of second volume of the
revised edition of The Structure of the Physical Universe (Portland, Oregon: North
Pacific Publishers, 1980).
FURTHER MATHEMATICS OF THE RECIPROCAL
SYSTEM
This paper will present in the most concrete, explicit manner the mathematics of space–
time, radiation, and matter of the Reciprocal System. Readers without special knowledge
of the Reciprocal System are first urged to study Larson‘s books, especially Nothing But
Motion¹ before undertaking the study of this paper.
I. Mathematics of Space–Time
A. Rectangular Coordinates
Starting from any reference point x0, y0, z0, t0 in the 0–system, the space–time
progression is a spherical expansion. In rectangular coordinates the equation
is
(x–x0)² + (y–y0)
² + (z–z0)
² = c
² t
² (1)
where c is the speed of light. If we choose the reference point to be x0 = 0, y0 = 0, z0 = 0,
t0 = 0, then the equation is simply
x² + y
² + z
² = c
² t
²
Now consider a second system, the 0´–system, moving translationally with respect to the
0´–system in the x–direction. What is the equation for the progression in the 0´–system?
From the inverse Lortentz transformations,
[x´+vt´]² 1
x²= ————— = ———— (x´²+v²t´²+2v x´t´) (2)
[ v²/c²)]² 1-(v²/c²)
[t´+(v/c²)x´]² 1
t²= ————— = ———— (v²/c4)x´²+t´²+(2v/c²)x´t´) (3)
[ v²/c²)]² 1-(v²/c²)
y² = y´² (4)
z² = z´² (5)
Upon substitution, we obtain
x² + y
² + z
² – c
² t² = x´
² + y´² + z´
² – c
²t´² (6)
But since the left side of the equation equals zero, so must the right side:
x´² + y´
² + z´
² = c
²t´
² (7)
Thus the progression as determined by 0' is also spherical. And so the equation for the
progression is invariant under a Lorentz transformation.
B. Polar Coordinates
In polar coordinates the equation is simply
r – r0 = c(t – t0) (8)
Or, letting r0 = 0, t0 = 0, (9)
r/t = c
In the Reciprocal System the speed of light is the natural unit of velocity and so r and t
must take equal natural values. The space–time progression is thus ¹/1,
²/2,
3/3, etc. Thus
one unit of space is equivalent to one unit of time. If there are an infinite number of
space units, there must be an infinite number of time units; if there are a finite number of
space units, there must be a finite number of time units.
II. Mathematics of Radiation
In the Reciprocal System radiation is the combined motion of a simple harmonic
oscillation in one dimension and a uniform translation in a perpendicular direction.
A. Simple Harmonic Oscillation
The equation for a simple harmonic oscillation in one dimension (say the y direction) is
y = A*SIN(–2 fost) (10)
where A is the amplitude and fos is the frequency. Since the oscillation takes place over
one natural space unit, the amplitude must be one–half a natural space unit and thus is
A = .5*Snat = .5 x 4.558816 x 10–8
m = 2.279408 x 10–8
m (11)
for all photons. In observation from the time–space region this value is reduced by the
interregional ratio142.222 to 1.6027 x 10–10
m = 1.6027 Å
The other variable to be determined in eq.(10) is the frequency, fos. In one cycle the
oscillation travels one space unit up and one space unit down, for a total of two units.
The average velocity of the oscillation is th
vos = (2*snat/cycle) * fos (cycles/sec) (12)
The natural unit of frequency must occur when the average velocity is
c.
vos = c = 2*snat * fos nat (13)
But c = snat/tnat´
so snat/tnat = 2*snat * fos nat
Solving for fos nat we have
fos nat = 1/(2*tnat) = 1/(2*1.520655 * 10–16
sec) = 3.2880575 * 1015
cycles/sec (14)
the Rydberg frequency. (Actually, Larson derived the natural unit of time from the
Rydberg frequency, but I think it was instructive to do the reverse, and this method will
be used to calculate rotational and rotational vibration frequencies as well. Of course,
this method assumes that the natural unit of time can be found by some other means.)
Because of the discrete nature of the Reciprocal System, it is only possible to have
integer multiples or reciprocal integer multiples of the Rydberg frequency.
Putting the values of A and fos in eq.(10) we have
y = 1.6027*SIN(–2 n * 3.2880575 * 1015
* t) Å (15)
where
n = 1, 2, 3, ...or (16)
n = ½, ¹/3,
¼, ....
B. Perpendicular Translation
Perpendicular to the oscillation is a translation at unit velocity (the speed of light). Let x
be perpendicular to y.
Then
x = c * t (17)
C. Combined Motion
From eq.(17) t can be found in terms of x and c and put in eq.(15). The result
is
y = 1.6027 * SIN(–2 n * 3.2880575 * 1015
* x/2.997930 * 108) (18)
or y = 1.6027 * SIN (–6.8912465 * 107 * nx)Å
if x is given in meters. This is the equation for a monochromatic wave of radiation in the
Reciprocal System.
III. Mathematics of Matter
Particles of matter consist of rotating photons. Subatoms have one rotating photon;
atoms have two rotating photons (both photons rotate about the same central point). The
rotational motion has a translational effect, which will be discussed after the mathematics
of the rotation has been worked out.
A. Rotation
1. single systems–particles
A photon can rotate around either of two horizontal axes passing through its midpoint,
and also around itself. In the Reciprocal System the true physical zero is motion at unit
speed. Anything physical must have a motion either greater or less than unit speed. This
deviation is called a speed displacement by Larson. The first particle has 1 speed
displacement around one horizontal axis of the photon and is called the rotational base.
Actually there are two rotational bases: one with one speed displacement above unity,
the other with one speed displacement below unity. As will be discussed later, the one
displacement unit neutralizes the translational motion of the photon in the original
dimension, but the progression now continues in the remaining dimension, so the
effective displacement is zero. In the ground state condition, the photon that is being
rotated is one vibrational displacement away from unity (either 2R or (1/2)R). Here is a
table giving the photon frequency, the rotational displacement, the effective rotational
displacement, and rotational speed of the cosmic rotational base and the material
rotational base:
Photon
frequency
Rotational
displacement
Effective
rotational
displacement
Rotational
speed
———— ————— ———————— ——————
Cosmic rotational
base ½R (1)-0-0 0-0-0 2-1-1
Material rotational
base 2R 1-0-0 0-0-0 ½-1-1
In the above table the speeds are calculated from the displacements as follows. For
displace
ments of np, ns, and nE, the speeds are (np+1), (ns + 1), and (nE + 1) for a cosmic particle,
and 1/(np + 1), 1/(ns + 1), and 1/(nE + 1) for a material particle. (Of course material
particles could have high speed electric displacement, and cosmic particles could have
low speed electric displacement).
These speeds can be converted to conventional units, such as revolutions per second, as
follows. In one rotation of a photon about a horizontal axis the tip of the photon coverse
a distance of * snat´ a circumference. The speed of the rotation is
then
Vrot = [( * snat) /rev] * frot (rev/sec) (19)
The natural frequency of rotation must occur when the speed is c.
c = ¹ * snat * frot nat
But c = snat/tnat´
so solving for frot nat, we get
frot nat = 1/(¹tnat)
But tnat = 1/(2R)
so frot nat = 2R
/¹ (20)
where R is the Rydberg frequency, as before. In these terms, then, the cosmic rotational
base is a photon that has a vibrational oscillation of 1.6440288 x 1015
cycles/sec and is
rotating at 4.1864848 x 1015
revolutions/sec around one axis, and 2.0932424 x 1015
revolutions/sec around the other two axes. Likewise, the material rotational base is a
photon that has a vibrational oscillation of 6.576115 x 1015
cycles/sec and is rotating at
1.0466212 x 10 15
revolutions/sec around one axis, and 2.093242 x 1015
revolutions/sec
around the other two axes.
All the other particles have photon vibrational frequencies, rotational displacements,
effective rotational displacements, rotational speeds, and rotational frequencies. Here is a
complete tabulation:
Photon
frequency
Rotational
displacement
Effective
rot.
displacement
Rotational
speed
Rotational
frequency
———— ————— ————— ———————— ——————
M–
positron 2R 1–0–1 0–0–1
½-1–
½ R
/2R
/R/
C–positron ½R (1)–0–(1) 0–0–(1) 2–1–2 4R/
2R/
4R/
M–
electron 2R 1–0–(1) 0–0–(1)
½-1–2 R
/-2R
/4R
/
C–electron ½R (1)–0–1 0–0–1 2–1–½ 4R
/2R
/R/
Photon
frequency
Rotational
displacement
Effective rot.
displacement
Rotational
speed
Rotational
frequency
———— ————— ————— ———————— ————————
M-
massless
neutron
2R 1–1–0 ½ –½–0 ½–
½-1 R
/R/
²R/
C-
massless
neutron
½R (1)–(1)–0 (½)–(½)–
0 2–2–1 4R
/4R
/²R
/
M-
neutrino 2R 1–1–(1) ½–½–(1)
½-½–2 R
/-R
/4R
/
C–neutrino ½R (1)–(1)–1 (½)–(½)–1 2–2–½ 4R
/4R
/R/
Many more permutations appear to be possible, but the probability principles keep
eccentricity to a minimum. Since none of the above particles has an effective
displacement of 1 or more, they are all massless (aside from the mass contribution of an
electric charge). The diameter of all the particles is one natural space unit, reduced by the
(one–photon) interregional ratio, or 3.2054 Å. However, because these particles do not
exert any force in the uncharged state, a particle–measuring probe would not be able to
detect any size of these particles at all.
2. intermediate systems
Intermediate particles have two rotating photons, but one of the two sets has no
effective displacement and thus contributes no primary mass. The two intermediate
particles are the neutron and the mass one hydrogen isotope (and their cosmic analogs).
There are only two kinds of rotations that can combine to form this kind of particle, the
proton type and the neutrino type We identify the combination of the material proton
rotation and the material neutrino rotation as the mass one hydrogen atom; the
combination of the material proton rotation and the cosmic neutrino rotation as the
neutron; the combination of the cosmic proton rotation and the cosmic neutrino rotation
as the mass one atom of cosmic hydrogen; and the combination of the cosmic proton
rotation and the material neutrino rotation as the cosmic neutron. The proton is a single
system with displacements 2–1–(1), effective displacements 1–1–(1), speeds ½-½
-2, and
rotational frequencies 2R/3 –R/ –4R/ . Then we would have the following table for the
neutron and mass one hydrogen.
Photon
frequency
Rotational
displacement
Effective rot.
displacement
Rotational
speed
Rotational
frequency
———— ————— ————— ———————— ————————
Neutron {
2R 2–1–(1) 1–1–(1) ¹/² –½-2
2R/³
R/
R/
½R (1)–(1)–1 (½)–(½)–
1 2–2–½ 4R
/4R
/R/
H¹ {
2R 2–1
> (1) 1–1
> (1) ¹/³-½
> 2
2R/³
-R/
>4R/ 2R 1–1 ½–½ ½ –½ R
/R/
The new notation makes clear the two photons involved and the five rotations (to be
discussed next).
3. Atomic cycles
Atoms have two rotating photons, but here both systems have effective displacements
and both systems ordinarily have the same velocities. Let the first photon be called A
and the second be called B. A and B are mutually perpendicular. We have the following
five rotations:
(i) the rotation of A about B produces disk a;
(ii) the rotation of B about A produces disk b;
(iii) then disk a can be rotated about A;
(iv) then disk b can be rotated about B;
(v) finally the whole structure can be rotated in the electric dimension.
This last rotation is in the scalar direction opposite to that of the previous rotations.
Cosmic atoms have speeds above unity for the first four types of rotations, whereas
material atoms have speeds below unity for the first four types. The electric rotation may
be above or below unity for both cosmic and material atoms.
The first particle with two effective rotating systems is deuterium, the second is helium,
etc. A table similar to that for the intermediate particles can be made.
Photon
frequency
Rotational
displacement
Rotational
speed
Rotational
frequency
———— ————— ——————— ————————
Deuterium {
2R 2–1
> (1) ¹/³ –
½
> 2
2R/³
R/
>4R/ 2R 2–1 2³–½ 2R
/³R/
Helium { 2R 2–1 >
0 ¹/³-½ > 1 2R
/³-R
/ >2R/
2R 2–1 ½³ –½ R
/³R/
All other atoms can be given appropriate values in the same manner. In the solid state,
however, the values that govern the physical properties are not the actual rotations, but
the relative rotations, and the different values there are not due to inherent differences in
the rotational speeds, but to differences in the orientations of the interacting atoms, and
this will be discussed further later.
4. Electric charges and magnetic charges
According to the Reciprocal System an electric charge is a rotational vibration about the
electric axis, and the magnetic charge is a rotational vibration about one of the magnetic
axes. Both charges have the same natural frequency, calculated as follows. In one cycle
the motion covers a distance of ¹ * snat one way and * snat back, for a total of 2 * snat.
So we have
vch = (2 * snat/cycle) * fch (cycles/sec) (21)
At the unit level, vch = c = snat/tnat, so
snat/tnat = (2 * snat/cycle) * fch nat (22)
Solving for fch nat and recalling that tnat = 1/(2R),
fch nat = R/ (23)
This frequency is one–half that of a full rotation and can thus be considered to be
effective in one direction only half the time. One negative electric charge is a rotational
vibration of R/2 = 5.233106 x 1014
cycles/sec. One positive electric charge is a
rotational vibration of 2R/ = 2.093242 x 1015
cycles/sec. Similarly one unit of magnetic
charge is a rotational vibration of 2R/ = 5.233106 x 1014
cycles/sec, whereas one unit of
isotopic charge is a rotational vibration of R/2 = 5.233106 * 1014
cycles/sec. The
isotopes of atoms result from the addition of isotopic charges.
B. Translation
The rotational motion of particles has a translational effect. The maximum inward
translation is two full units, giving one net inward unit. In terms of rotation we can have
2³ = 8 one–dimensional rotational electric displacements or 4 two–dimensional rotational
magnetic displacements. Note that since 1³ = 1, the first magnetic rotational
displacement, which is ½ unit rotational speed, produces one unit of inward translation
and thus neutralizes the original translational motion of the photon, but the progression
still continues in the third dimension. Thus the rotational base and all the single system
massless particles previously discussed move at the speed of light. Additional magnetic
and electric displacements produce a net inward motion, and the inward motion of a
group of atoms is termed gravitation.
For atoms with magnetic displacements of less than 4 and electric displacements of less
than 8, the frequency of the rotating photons is normally one displacement above unity,
or 2R (the frequency of photons in cosmic atoms is (½)R). When the magnetic
displacement reaches 4 or the electric displacement reaches 8, the rotation must be
extended to a second vibrational displacement unit–which means that the frequencies of
the photons are now 3R (or (¹/3)R for cosmic atoms). As Larson points out, though, it is
possible to have these higher frequency photons even when the rotational displacements
are less than 4 or 8, in which case we can say that the atom is ―excited‖.
After the change to vibration two, two units of vibrational displacement exist to be
rotated, and so each added unit of rotational displacement corresponds to only one–half
unit of added specific speed. Thus the speeds corresponding to magnetic displacements
can be listed as follows:
Magnetic Displacement Magnetic Speeds
1 ½ ½ ½
2 ¹/3 ²/5 ¹/3
3 ¹/4 ¹/3 ¹/4
4 ¹/9 ²/7 ¹/5
5 ²/5 ¹/4 in one
displacement
axis only
And the speeds corresponding to electric displacements can be listed as follows:
Electric Displacement Electric Speeds Electric Displacement Electric Speeds
1 ½ ½ 9 ¹/9 ¹/6
2 ¹/3 ²/5 10 ²/19 ²/13
3 ¹/4 ¹/3 11 ¹/10 ¹/7
4 ¹/5 ²/7 12 ²/21 ²/15
5 ¹/6 ¹/4 13 ¹/11 ¹/8
6 ¹/7 ²/9 14 ²/23 ²/17
7 ¹/8 ¹/5 15 ¹/12 ¹/9
8 ²/17 ²/11 16 ²/25 ²/19
In the solid state, the values for electric rotation can be further altered. Larson states
that a combination of one atom of electric displacement x with another atom of electric
displacement 8–x results in a neutral bond. This bond gives rise to an electric speed of ¹/10 for vibration one, and
¹/5 for vibration two. Also there can be a combination of two 8–
x atoms, which Larson calls a secondary positive bond. In this case the rotational speed
comes to ¹/(18–2x).
One final set of complications involves the lower group elements. Here there is just one
subordinate magnetic displacement unit and thus these elements have less rotational force
and thus are closer together in the solid state. The force is proportional to ln t, where t is
the inverse of the magnetic speed, and since ln 2 is less than 1, atoms that have magnetic
speed greater than ¹/3 in any dimension have no effective force in that dimension. The
number of ―active‖ dimensions is given in the following tabulation.
Lower Group Atomic Table in Solid Stat
Atom.
No Elem.
Oscillation
Frequency
Rotational
Displ. Bond
Rotational
Speed
Rotational
Frequency
Active
Dim.
{ 2R 2-1 >
(1) ¹/³–½ >
¹/10 2R
/³R/ >
R/
1 H
Neutral
1
2R 2-1 ¹/³–
½ 2R
/³R/
{
2R 2-1
> 0
¹/³–½
> 1
2R/³
R/
> 2R/
2 He
Zero
1
2R 2-1 ¹/³–
½ 2R
/³R/
{
3R 2-1
> 1
²/5–²/5
> ½
4R/5
R/5
> R/
3 Li
Positive
2
3R 2-1 ²/5–
²/5
4R/5
R/5
{
3R 2-1
> 2
¹/³–½
> ²/5
2R/³
R/
> 4R/5
4 Be
Positive
2
3R 2-1 ¹/³–
½ 2R
/³R/
{
3R 2-1
> 3
¹/³–½
> ¹/5
2R/³
R/
> 2R/5
5 B
Neutral
1
3R 2-1 ¹/³–
½ 2R
/³R/
{
2R 2-2
> (5)
¹/³–¹/³
> ¹/10
2R/³
2R/³
> R/5
or B
Neutral
1
2R 2-2 ¹/³–¹/³
2R/³
2R/³
{
2R 2-1
> 4
¹/³–½
> ¹/10
2R/³
R/
> R/5
6 C
Neutral
2
2R 2-1 ¹/³–
½ 2R
/³R/
{
2R 2-2
>
¹/³–¹/³
> 1
2R/³
2R/³
> 2R/
or C
Zero
3
2R 2-2 ¹/³–¹/³
2R/³
2R/³
{
2R 2-2
> (3)
¹/³–¹/³
> ¹/10
2R/³
2R/³
> R/5
7 N
Neutral
1½
2R 2-2 ¹/³–¹/³
2R/³
2R/³
{
2R 2-2
> (3)
¹/³–¹/³
> 1
2R/³
2R/³
> 2R/
or N
Zero
3
2R 2-2 ¹/³–¹/³
2R/³
2R/³
{
2R 2-2
> (2)
¹/³–¹/³
> ¹/10
2R/³
2R/³
> R/5
8 O
Neutral
1½
2R 2-2 ¹/³–¹/³
2R/³
2R/³
{
2R 2-2
> (2)
¹/³–¹/³
> 1
2R/³
2R/³
> 2R/5
or O
Zero
3
2R 2-2 ¹/³–¹/³
2R/³
2R/³
{
2R 2-2
> (1)
¹/³–¹/³
> ¹/10
2R/³
2R/³
> R/5
9 F
Neutral
2
2R 2-2 ¹/³–¹/³
2R/³
2R/³
The reader can continue the table all the way to element 118. Again, one must first
determine the kind of bond involved before the electric rotational speed can be
determined.
Since different atoms have different rotational speeds and thus different rotational
forces, a particle probe of equal energy shot at atoms of different elements would
―penetrate‖ to different depths. Thus experimenters have concluded that ―nuclear‖ size is
proportional to atomic weight. Actually what they are measuring is atomic size, and
according to the Reciprocal System this is constant (2.914 Å diameter)–but the force is
proportional to the atomic weight of the atom. Also, where interatomic distances are less
than 2.914 Å, the atoms are partially merged; where distances are greater than 2.914Å,
the atoms are separate.
Reference
1. Dewey B. Larson, Nothing But Motion (North Pacific Publishers: Portland, Oregon,
1979)
A NEW DERIVATION OF PLANCK‘S CONSTANT
To present-day physical science the numerical value of Planck‘s constant is a mystery:
quantum mechanica does not have a theoretical method for its calculation. By contrast the
Reciprocal System of theory derives the value of all physical constants, including
Planck‘s constant, from its fundamental postulates. However, because of errors in the
previous derivations, this paper presents a new, dimensionally sound method for the
calculation.
Larson¹ was the first to attempt to derive Planck‘s constant from the Reciprocal System.
Because of the change in the calculated natural values of mass and energy in the second
edition of his work², the original derivation has been invalidated. The factor of three that
was used is dimensionally incorrect since the photon is a one-dimensional vibration. And
the use of the cgs gravitational constant in such an equation is wrong since the result
cannot be converted to a different system of units such as the Sl (mks) system. The
remainder of Larson‘s original equation (including the use of the interregional ratio and
the square of the natural unit of time) will be shown to be correct.
Nehru³ made the second attempt to calculate the constant. However, he started by setting
the dimensions of energy to be space divided by time, which is, of course, the reverse of
what they are. The rest of the derivation was very tortuous, although he ended up with a
good numerical result (with faulty dimensions).
Let us now proceed with the new derivation. First, consider conceptually the linear
vibration of the photon. The oscillation takes place over one space unitwhich,
simultaneously, is also one time unit. In the material sector of the universe, we define
frequency to be cycles/sec, because here it is time that appears to have a uniform
progression; in the cosmic sector of the universe, hypothetical cosmic observers would
define frequency to be cycles/cm (or some such length unit), because there it is space that
would appear to have a uniform progression. Actually, the photon exists at the boundary
between the two sectors, where both space and time progress uniformly. Here the correct,
nAtural definition of frequency must be cycles/(cm-sec) (or equivalent units). To put it
another way, frequency in the natural sense is the number of cycles per space-time unit.
Photons of all frequencies can be observed in both sectors, and the only way that this
could be possible is if the denominator of the natural definition contains both a space unit
and a time unit. This then causes Planck's constant to have the actual dimensions of erg-
cm-sec. However, if the dimensions of frequency are assumed to be cycles/sec, rather
than cycles/(cm-sec), then the dimensions of Planck‘s constant are erg-sec. Let E be
photon energy, h be Planck‘s constant, and be photon frequency. Then, as usual, we
have
E=h* (1)
In space-time terms, equation (1) is
[
t²
]
t ——– 1
— =
( t/s
) ——
(2)
s ——– s * t
t/s
In the cgs system of units, equation (1) is
[
sec²
]
——– 1
erg =
( sec/cm
) ——–—
(3)
——– cm * sec
erg
Observe, in both cases, the dimensional consistency. Since the oscillation of the photon
takes place with in a unit of space-time, the interregional ratio must be contained within
Planck‘s constant. With this factor and the dimensional information from above, Planck‘s
constant is
1 t²0
h= ———— * ———–
156.4444
( sec/cm
)
———–
erg
where t is the natural unit of time (1.620666 * 10'18 sec).
Ref. 3 states that the ratio of (sec/cm)/erg is 2.236066 x 10-8
. This figure is deduced as
follows. Dimensionally unit mass is t³/s³ , or 3.711381 * 10-32
sec³/cm³. Avogadro‘s
constant is the number of atoms per gram atomic weight. 6.02486 * 10-23
. The reciprocal
of this number, 1.66979 * 10-24
, in grams, is therefore the mass equivalent of unit atomic
weightz². Thus to convert from the unit sec³/cm³ to grams we must divide by 2.236066 *
10-8
. From the euprsssion E = mc² we see that the sa.me conversion factor must apply to
energy (in ergs) to keep the equation balanced. (Nehru³ modi ied his equation to include
secondary mass; however, his rasulting equation is dimensionally incorrsct. Furthermore,
secondary mass varies between the subatoms and atoms and so cannot be a part of the
conversion factor). Thus the numerical value of Planck‘s constant is
h = 6.6102662 g 10-27
erg-sec (5)
(when frequency is assumed to have the dimensions cycles/sec).
This is 99.77% of the egperimental value of 6.6266 * 10-27
erg-sec. Given the
uncertainties involved in the determination of Avogadro‘s constant and the natural unit of
time, the result is satisfaetory. Any improvement in the accuracy of these values would be
reflected in an improvement in the accuracy of the calculation of Planck‘s constant.
References
1. Dewey B. Laraon, The Structure of the Physical Universe (Portland, Oregon:
North Pacific Pub- lishers, 1969), pp. 117-118.
2. Dewey B. Larson, Nothing But Motion (Port- land, Oregon: North Pacific
Publishers,1979), pp. 157-168.
3. K.V.K. Nehru, ―Theoretical Evaluation of Planck’s Constant,‖ Reciprocity, Vol.
XII, No. 3.
TIME REGION PARTICLE DYNAMICS
Mr. Larson has worked. out the static relations between particles in the time region;
specifically, he has calculated the equilibrium interatomic distances for all the elements
and many compounds (see pages 27-49 of The Structure of the Physlcal Universe). This
paper will explore he dynamic relations between particles in the time region.
Consider a particle (say an alpha particle) moving directly towards a stationary atom (say
a gold atom fixed in thin foil). Initially the particle has a velocity vo. Once it enters the
tlme region, that is, when its distance is less than one natural unit of space, two forces are
encountered: the progression and gravitation. In the time region, the progression acts to
bring particles closer together, whereas gravitation acts to repel particles — the reverse of
gravitation in the time-space region. The progression is stronger until the equilibrium
distance is reached, then the gravitational force becomes stronger. I believe that the
equation of motion is
Fp - KG / (xu- x)4 = m d²x / dt² (1)
where
Fp = unit force of the progression
KG = magnitude of the rotatianal motion of the partlcles
xu = natural unit of space
m = mass of the moving particle
x = distance-measured from start of time region
Dividing by m gives
Fp / m - KG / m (xu - x4 = d²x / dt²
The right hand side reduces to
d²x/dt² = dv/dt = dv/dx dx/dt = v dv/dx (2)
Thus
Fp/m - KG/m (xu - x)4 = v dv/dx
Separating variables and integrating, we have
xf
xf
vf
Fp/m dx - KG dx/m (xu -x)4 = vdv
o o vo
or
Fp/m xf - KG /3m [(xu - xf)-3 - (xu)
-3] = ½ (vf² - vo²) (3)
There are two cases of interest with this equation.
Case 1: Suppose we want to know the initial velocity required to bring the particles to a
certain distance apart from each other. Equation (3) is solved for vo, letting vf be zero.
Vo = (2[KG/3m{(xu - xf)-3 - (xu)
-3}-Fp/m xf] )½ (4)
Case 2: Suppose we want to know the final separation between two particles, given vo.
Let
xsep = xu -x
Equation (3) becomes
Fp/m (xu - xsep) - KG/3m xsep-3 - xu
-3] = ½ (vf² - vo²)
With vf = 0, and putting the terms irevolving xsep on one side af the equation, we have
Fp/m xu + KG xu-3/3m + 1/2 vo² = Fp/m xsep + KG/3m xsep
-3
Define the following coefficients:
C2 = Fp/m xu + KG xu-3/3m + ½ vo²
C2 = Fp/m
C3 = KG/3m
C4 = -C1/C2
C5 = C3/C2
With these coefficients, the result is a quartic equation:
xsep4 + C4 xsep
3 + C5 = 0 (5)
This equation can then be solved by the usual means.
Now, going back to equation (3) we can solve for v as a function of x:
v = dx/dt = {2[ Fp/m x - KG/3m[(xu - x)-3 -xu
-3]] + vo²}½
Separating variables and integrating we have
x
t = dx/{2[Fp/m x -KG/3m((xu - x)-3 -xu
-3)] + vo²}½
o
The integral can be evaluated numerically by Romberg‘s method.
Example
Consider an alpha particle moving directly towards a gold atom in a foil, at an initial
velocity of 2.06x107 meters/sec. What is the distance of closest approach? How long does
it take to get there? What happens afterward.?
Here we have
vo = 2.06 x 107 m/sec
m = 6.64 x 10-27 kg
xu = .455884 x 10-7 m
Fp = 1.09699 x 10-3 n
Now,
KG = 1.09699 * 10-3 * (.455884 * 10-7 )4 /(156.44) * ln² tA ln² tB
For gold, tA =4.5; for helium, tB = 3. But helium has only one active dimension so the
force is multiplied by 1/3. Thus
KG = 1.09699 * 10-3 * (.455884 * 10-7 )4 /(156.44)4 * ln² (4.5) ln² (3)
* 1/3 = 7.20006 * 10-42 N-m4
(This assumes that since helium is inert, the electric displacements of gold have no
bearing on the motion). The coefficients are next calculated:
C2 = 2.7438094 * 1015
C2 = 1.6520934 * 1023
C3 = 3.614488 * 10-16
C4 = 4.6872709 * 10-8
C5 = 2.197923 * 10-39
The quartic equation is
xsep4 - 4.6972709 * 10-8 xsep
3 + 2.187823 * 10-39 = 0
The only physical solution is
xsep- = 3.6014328 * 10-11 m .36 Å 3.6226287 * -11 m with revised Fp]
Note that his is cons:lderably greater than that predicted by use of classical atomic theory
and Coulomb‘s law: 2.581 x 1014 m.
Using equation (6) and Romberg‘s method I find that
t = 6.3041312 * 10-16 sec
The average velocity of the particle to the point of closest approach is
4.552396 * 10-8 /6.3041312 * 10-16 = 7.2212742 * 107 m/sec
The initial velocity having been dissipated, the particle goes back to the equilibrium
point. Of course, at room temperature, helium is a gas, and so the particle would not
remain in the time region!
Situations in which the particle is not moving directly towards the atom will be treated in
a future paper.
CALCULATION OF THE DISSOCIATION ENERGY
OF DIATOMIC MOLECULES
This paper presents the first rational calculation of the dissociation energy of diatomic
molecules. Quantum mechanics does not have such a calculation, even in principle. The
importance of this calculation is that it provides additional quantitative verification of the
molecular force and energy concepts of the Reciprocal System.
Dissociation energy is the change in energy (usually expressed in kcal per mole) at
absolute zero temperature in the ideal gas state for the reaction
A-B —> A + B (1)
the products (atoms A and B) being in their ground states and the reactant (molecule A-
B) in the zeroth vibrational level. Note that dissociation energy is slightly different from
bond energy, which is defined as the standard enthalpy change at 25º C for the ideal gas
reaction given above. Calculating dissociation energy rather than bond energy frees us
from having to consider molecular thermal energy.
Now let us proceed to the derivation of the expression for bond dissociation energy from
the principles of the Reciprocal System. A diatomic molecule, as a unit, exists in the
time-space region. However, the two individual atoms of the molecule, relative to one
another, exist in the time region because the interatomic distance is less than one space
unit; hence, time region expressions apply to the attributes of the bond. To quote Larson,
The motion in time which can take place inside the space unit is equivalent to a motion in
space because of the inverse relation between space and time. An increase in the time
aspect of a motion in this inside region (the time region, where space remains constant at
unity) from 1 to t is equivalent to a decrease in the space aspect from 1 to 1/t. Where the
time is t, the speed in this region is equivalent space 1/t divided by time t, or 1/t²[Ref. 1].
Thus,
v = 1/t² (2)
In the Reciprocal System, energy is the reciprocal of speed. Hence, in the time region,
E = t² (3)
This energy equation gives the proper dimensional form of the expression for dissociation
energy. It can be generalized to
E = ta * tz (4)
In application to the problem at hand, ta and tz refer to the rotational time
displacements of the atoms of the molecule, where ta is the primary magnetic
displacement or the secondary magnetic displacement and tz is the second magnetic
displacement or the electric displacement. To justify this intepretation, let us recall
that the two atoms of the molecule are in translational equilibrium; in the Reciprocal
system this means that the scalar translational repulsion effect of the rotational force of
the atoms is equal and opposite to the cohesive translational force of the space-time
progression; the magnitude of the force is thus equal to the translational equivalent force
of the rotational force of the atoms and so the required dissociation energy must equal the
rotational energy. Because of the discrete unit postulate, less than this amount of energy
would be ineffective.
As it stands, equation (4) expresses the energy in natural units of the time region. We
have to convert the equation to an equivalent expression for the time-space region so that
we can compare calculated to observed results. First of all we must apply the fourth
power of the interregional ratio, 1/156.44, to the equation, just as is done in the atomic
force equation.
E = (1/156.44)4 * ta * tz (5)
This is the energy in natural units as would be observed in the time-space region. To
convert this to conventional units of measurement we multiply by the value of the natural
unit of energy expressed in conventional units, Eu.
E = (1/156.44)4 * ta * tz * Eu (6)
The experimental values are expressed as kcal/mole so we must multiply the right side of
the equation (6) by a conversion factor, k, and by Avogadro‘s number, N.
E = (1/156.44)4 * ta * tz * Eu * k * N (7)
Next we must append a factor of ½ to the expression to account for the inherent
vibrational nature of the time region motions and a factor of 1/3 to the expression to
reduce the energy to one dimension. So now we have
E = (1/156.44)4 * ta * tz * Eu * k * N * (1/6) (8)
From Ref. 1, Eu is 1.49175 x 10-3
ergs and N is 6.02486 x 1023
. k is 2.389 x 10-11
kcal/erg. The final working equation is
E = 5.9747 * ta * tz kcal/mole (9)
Displacement ta can range from 1 to 4 and displacement tz can range from 1 to 8. Table I
lists the possible values of E for the various combinations of ta and tz.
I have applied equation (9) to 18 diatomic molecules of the elements. The theoretical and
experimental results are given in table II. Let t1 symbolize the primary magnetic
displacement of an element, t2 the secondary magnetic displacement, and t3 the electric
displacement. It is clear from the table that
ta = t1, or ta = t1 + 1, or ta = t1 - 1, or ta = t2, or ta = t2 + 1, or ta = t2 - 1 (10)
And
tz = t2, or tz = t2 + 1, or tz = t2 - 1, or tz = t3 (11)
For electronegative elements, the 8-t3 rule applies:
tz = 8 - t3, or tz = 8 - t3 + 1 (12)
Generally, only one (if any) of the two displacements has to be incremented or
decremented by 1 to obtain a good fit with the experimental data; the other displacement
equals the rotational displacement (or 8 minus the rotational space displacement) as the
theory requires. Elements that require an increment of displacement usually have low
atomic number; elements that require a decrement of displacement usually have high
atomic number.
The values of ta and tz thus fit the normal variations in the elements that have appeared
in other Reciprocal System calculations. This, together with allowance for experimental
error, allows us to conclude that we have good agreement between theory and reality.
A future paper will apply equation (9) to diatomic molecules of unlike atoms.
References
1. Dewey B. Larson, Nothing But Motion (Portland, Oregon: North Pacific
Publishers, 1979), p. 155.
2. John A. Dean, ed., Lange’s Handbood of Chemistry, Eleventh Edition (New
York: McGraw-Hill Book Company, 1973), pp. 3-123 to 3-127.
Table I: Allowed Values of Dissociation Energy
E kcal/mole ta tz ta tz ta tz
5.9747 1 1
11.9494 1 2 2 1
17.9241 1 3 3 1
23.8988 1 4 2 2 4 1
29.8735 1 5
35.8482 1 6 2 3 3 2
41.8229 1 7
47.7976 1 8 2 4 4 2
53.7723 3 3
59.7470 2 5
71.6964 2 6 3 4 4 3
83.6458 2 7
89.6205 3 5
95.5952 2 8 4 4
107.5446 3 6
119.4940 4 5
125.4687 3 7
143.3928 3 8 4 6
167.2916 4 7
191.1904 4 8
Table II: Calculated and Observed Values of Dissociation Energy
Molecule Displacement Method ta tz Ecalc. Eobs.
As-As 3-3-(3) t2 8-t3 3 5 89.62 91
Cs-Cs 4-3-1 t2-1 t3 2 1 11.95 10.4
Cl-Cl 3-2-(1) t1 t2+1 3 3 53.77 57.07
Cu-Cu 3-3-(7) t1 t2 3 3 53.77 48
F-F 3-2-8 t1 t2 3 2 35.85 36
Ga-Ga 3-3-(5) t1 t2-1 3 2 35.85 35
Au-Au 4-4-(7) t2 8-t3+1 4 2 47.80 52
D-D 2-1-(1) t1 8-t3+1 2 8 95.60 105
I-I 4-3-(1) t1-1 t2-1 3 2 35.85 35.55
Li-Li 2-1-1 t1 t2+1 2 2 23.90 25
P-P 3-2-(3) t1+1 8-t3 4 5 119.49 116.0
K-K 3-2-1 t2 t3 2 1 11.95 11.8
Se-Se 3-3-(2) t2-1 8-t3 2 6 71.70 65
Ag-Ag 4-3-(7) t2 8-t3+1 3 2 35.85 39
Na-Na 2-2-1 t2+1 t3 3 1 17.92 17.3
S-S 3-2-(2) t2 8-t3+1 2 7 83.65 83
Te-Te 4-3-(2) t1-1 t2 3 3 53.77 53
Sn-Sn 4-3-(4) t2-1 8-t3 2 4 47.80 46
Note: the observed values, Eobs., come from Reference 2.
THE LIQUID STATE IN THE RECIPROCAL SYSTEM:
THE VOLUME/TEMPERATURE RELATION,
A CONTEMPORARY MATHEMATICAL TREATMENT
This paper provides a step-by-step procedure for the calculation of liquid specific volume
as a function of composition and temperature, based on the Reciprocal System of D. B.
Larson1. In this theory, each individual molecule may be in the solid, liquid, or gaseous
(or vapor) state, regardless of the state of the majority of molecules of the substance.
First let's define some terms:
= overall specific volume of liquid (cm3/g) (total volume/total mass)
= specific volume increment at 0 oK and that due to the solid molecules in
solution of the liquid (solid volume/total mass)
= specific volume increment due to the liquid molecules of the substance,
temperature above 0 oK (liquid volume/total mass)
= specific volume increment due to the critical (gaseous or vapor) molecules in
solution of the liquid (gaseous volume/total mass)
Then,
(1)
The initial values of these three components are designated . These differ only
by a geometric factor (designated ) applied to a base initial value, ,
determined as follows.
Just as the volume of a gas is determined by the number of molecules, so the
volume of a liquid is determined by the number of volumetric groups which it contains.
In an organic compound, for instance, each of the common interior groups, such as CH2,
CH, or CO, constitutes one volumetric group. The CH3 groups in the end positions of the
aliphatic chains occupy two units each. So hexane, represented as
CH3CH2CH2CH2CH2CH3, has 8 volumetric groups. Let be the number of volumetric
groups and recall that the factor .707 expresses the geometric reduction obtained by the
close-packed arrangement of the liquid groups because of their flexibility of movement.
Then, in natural units, the base initial volume is directly proportional to the number of
volumetric units, reduced by close-packing:
(2)
Let be the molecular weight (non-dimensional) of the molecule of the substance,
be the value of the natural unit of atomic mass in g, and be the value of the natural
unit of liquid volume expressed in cm3. Then, in conventional units, the basic initial value
is
cm3/g (3)
is not the cube of the natural unit of space in the time-space region, which is
applicable only to the gaseous state. Rather, is the cube of the natural unit of space in
the time region, which is 1/156.45 (the inter-regional ratio) of that in time-space region,
or 2.9139 x 10-8
cm. Cubing this we get
cm3
The natural unit of mass is 1 atomic mass unit, so is 1.65979x10-24
g. Putting these
values in eq. 3, we get
cm3/g (4)
For hexane, is 8 and the molecular weight is 86.18. Therefore,
cm3/g
For the critical (gaseous or vapor) specific volume increment, the geometric factor is
always 1.00. For the solid specific volume increment, the geometric factor is .891
(the cube root of .707) where close-packing in the solid state can be achieved. Where
such packing cannot be achieved, the geometric factor is 1.000. The same applies to
the geometric factor for the liquid specific volume increment, . Therefore, the initial
values of the three volume components may be expressed as
(6)
(7)
(8)
In a multi-group molecule, the value of the geometric factors and represent
averages, since some groups may be at .891 while others at 1.000. Let = the number
of close-packed groups per molecule in the solid state, and let = the number of close-
packed groups per molecule in the liquid state. Then
(9)
(10)
For hexane, for instance, is .9864 (with 1 group at .891 and 7 groups at 1.0000, the
average is 7.891/8 or .9864) and is .9728 (with 2 groups at .891 and 6 groups at
1.0000). Therefore, for hexane, the initial values of the specific volume increments are
cm3/g
cm3/g
cm3/g
From eq. 10 it's clear that ordinarily . However, for lower group
elements, hyrdrogen through fluorine, closer packing than normal can be achieved
because of inactive dimensions of the gravitational repulsion force. This means that, in
effect, for lower group elements the geometric factors can be less than .891. We can still
use eq. 10, though, if we allow the value of the number of solid groups to exceed the
number of volumetric units.
Now that we have the initial values as a function of composition, we can determine the
values of the three components as a function of temperature. The solid specific volume
increment not only includes the initial volume at 0 oK but also a factor proportional to the
number of solid molecules in the substance at any temperature, , which can be
determined by probability considerations.
(11)
To use the normal probability function or table we need to know the value of the normal
random variable, zs, applicable. It should be proportional to the difference between the
liquid temperature and the melting point , in degrees K, divided by the melting
point. The coefficient and the intercept have unfortunately not been worked out
theoretically, but are given empirically by Larson (Ref. 1) as follows:
(12)
We want the right tail of the distribution, so we subtract the value of the normal function,
denoted by erf(zs), from 1 and then multiply by the average difference in specific volume
between solid and liquid molecules, denoted by :
(13)
Larson uses an average value of of .080 for paraffin hydrocarbons (C14 and below)
and .084 for paraffins above C14 (rather than computing the individual values). For
hexane, = 178 K (-95 oC). At = -50
oC, zs=1.41 and from the normal probability
table, erf(zs) .9207. Subtracting this from 1.0000, we get .0793, which means that 7.93
% of the molecules in the liquid hexane aggregate at -50 oC are in the solid state.
Multiplying this figure by the approximate difference in specific volume between solid
and liquid molecules, .080, we get .0063 cm3/g for the value of .
The thermal motion beyond the initial point of the liquid (considered as starting at
0 oK) is the one-dimensional equivalent of the thermal motion of a gas, and thus the
volume generated is directly proportional to the temperature, . Let be the natural
unit of temperature in the time region (for the condensed states of matter) and be the
temperature factor. Then
(14)
In Ref. 2, Larson derived the value of to be 510.8 K. For simple substances, is 1.
More complex or more electropositive substances have values of of 2 up to 16.
Hexane has a value of 1; water, 2; silver, 16. Compounds of electropositive and
electronegative elements have intermediate values (some with half-integral values, which
are averages), as would be expected.
The gaseous or vapor increment of specific volume depends on the proportion of
critical molecules existing in the aggregate at each temperature, which can be computed
from probability considerations. Larson uses two random variables for this computation,
both a function of the critical temperature, :
(15)
(16)
Then the specific volume increment due to critical molecules in the substance is
(17)
For hexane, = 508 K. At = 210 oC, = .2947 and =.8106. The corresponding
values of the normal probability function are .6144 and .8109. Then, from eq. 17,
cm3/g
The .5747 factor means that 57.47% of the molecules at this temperature are in the
critical state.
Having determined we can now calculate from eq. 1.
To automate the task of comparing the theoretical values with those observed, I've
prepared a computer program and run it on most of the same liquids Larson used in the
original series of papers: hexane, hexadecane, benzene, acetic acid, ethyl acetate, ethyl
chloride, ethanethiol, fluorine, hydrochloric acid, sulfur dioxide, carbon tetrachloride, and
water. Printouts from the program for all of these liquids follow. The observed values
come from the same sources Larson used: Timmermans' Physico-chemical Constants of
Pure Organic Compounds, the American Petroleum Institute, and the International
Critical Tables.
Most of the computer results are in harmony with Larson's manual calculations.
The two seeming exceptions are for acetic acid and water. For acetic acid, Larson used a
value of initial liquid specific volume of .5469, which is .7795 that of his base initial
volume, .7016; but .891 is supposedly the smallest allowed fraction. For water, Larson
used a value of .7640 for both the initial solid and liquid specific volumes, but this is only
.8713 that of his base initial volume, .8769, not .891. Actually. these differences are due
to "hydrogen bonding", which can allow closer packing than normal. In a second
calculation for water, I input 1.78 for and so as to get the initial volumes to be
.7640. The theoretical results computed came out to be much closer to the experimental
ones than the previous run.
To compute the specific volume for any liquid of your choice, follow these steps:
1. Determine the formula of the compound and its molecular weight.
2. From the formula, determine the number of volumetric units and number of
temperature units.
3. Use equation 4 to obtain the base initial volume.
4. Use equations 9 and 10 to compute the geometric factors; some iteration here may be
required to the get the right values.
5. Compute the initial volumes with equations 6, 7, and 8.
6. Using equations 12 and 13, compute the solid specific volume increment, equation 11.
7. Use equation 14 to compute the liquid specific volume increment.
8. Using equations 15 and 16, compute the critical specific volume increment, equation
17.
9. Sum the results to get the final value, from equation 1.
References:
1. D. Larson, The Liquid State , privately circulated series of papers on the liquid state,
circa. 1960-1964. Note: I made use of the papers numbered I, II, II-supplement, and III.
I've reorganized all of the equations and changed some of the symbols for the sake of
clarity. I've also used the latest values of the conversion constants. The computer
program is entirely original.
2. D. Larson, Basic Properties of Matter (Salt Lake City, UT: International Society of
Unified Science, 1959-1988), pp. 59-60.
Appendix: The Computer Program
The following pages show the input screens of the program. The data base
language is filePro Plus and the computation language is TrueBasic. This is the first of
what will be a comprehensive series of programs for the calculation of all properties of
matter based on the Reciprocal System of theory. Eventually the programs will be made
available for purchase.
THE LIQUID STATE IN THE RECIPROCAL SYSTEM:
THE VOLUME/PRESSURE RELATION,
A CONTEMPORARY MATHEMATICAL TREATMENT, PART II
From thermodynamics,¹ the general equation of state of a pure substance is
(1)
where
volume expansivity (2)
and
isothermal compressibility (3)
(Of course, V = volume, P = pressure, T = temperature.)
From my previous paper² (and Larson‘s original work8),
(4)
where
VL = overall specific volume of liquid (cm³/g) (total volume/total mass)
V1 = specific volume increment at 0ºK and that due to the solid molecules in
solution of the liquid (solid volume/total mass)
V2 = specific volume increment due to the liquid molecules of the substance,
temperature above 0ºK (liquid volume/total mass)
V3 = specific volume increment due to the critical (gaseous or vapor) molecules in
solution of the liquid (gaseous volume/total mass)
In this paper we will consider the effect of pressure on a liquid at temperatures below the
liquid natural temperature unit, 510.8ºK. At low temperature, . Pressure has a
different effect on V3 than it has on V2. Also, pressure has a different effect on a liquid at
a temperature above, rather than below, 510.8ºK. These differences will be handled in
another paper.
For a solid under pressure³, the volume is multiplied by , where Po is the internal
pressure and P is the external pressure. For a liquid under pressure, the volume is
multipled by the square of the solid factor, or simply . So,
(5)
It follows that isothermal compressibility is
(6)
It's often easier to work with the bulk modulus, B, which is the inverse of .
(7)
From my previous paper,
cm3/g (8)
since is negligible for most liquids above the melting point.
cm3/g (9)
cm3/g (10)
where nv is the number of volumetric units.
The internal pressure of a liquid is obviously different from that of a solid. The natural
unit of pressure in the Reciprocal System is4
15,538,642 atm. To calculate the internal
pressure of a solid we divide this number by the interregional ratio, 156.45. For a liquid,
we divide by the square of the interregional ratio. Because liquid cohesion is two-
dimensional rather than three-dimensional we must also multiply the expression by 2/3.
Therefore,
atm (11)
This expression is then multiplied by the number of pressure units, np, and divided by the
ratio of the base volume to 1, raised to the 2/3 power. (The solid expression just uses
volume, or so3.) Therefore,
atm (12)
Substituting eq. 10 in eq. 12, we get
atm (13)
np is the number of atoms effectively acting against the external pressure. It is
sometimes, but not usually, equal to the number of volumetric units, nv. Using eq. 8, 9,
and 10, B can be expressed as
atm (14)
Now let's turn to calculating the volume expansivity.
K-1 (15)
where is the value of the expansivity at the end point of the solid.
One could plug (or 1/B) and into eq. 1 and integrate, but the resulting equation is more
complex than eq. 5 and thus not useful.
In summary, to calculate bulk modulus and volume expansivity of a liquid, it is
necessary to determine
m, the molecular weight
nv, the number of volumetric units
s1, geometric factor
s2, geometric factor
nt, the number of temperature units
np, the number of pressure units
Example Calculations and Comparisons with Experiment5,6
I selected four important liquids: acetic acid, carbon tetrachloride, ethyl acetate, and
water. Here are the results, in table format.
Chemical Formula M
Nv Nt Np P atm
T ok Balc atm Bobs atm
Acetic Acit CH3CO2H 60.05 .9046 .7820 4 1.0 7 1 288.16 11441.503 11279.014
Carbon Tetrachloride
CCl4 153.81 1.0 .9183 6 1.0 5 1 250.26 12334.317 11878.218
Ethyl Acetate CH3CO2C2H5 88.10 .9818 .9818 6 1.0 6 1 293.16 8687.0274 8733.6283
Water H2O 18.0153 .8707 .8707 1.5 2.0 9 1 273.16 19697.992 19698.877
Chemical calc k1 obsK-1
acetit acid 1.1377x10-3 1.269x10-3
Carbon Tetrachloride
1.240x10-3 1.2987x10-3
Ethyl Acetate 1.24398x10-3 1.304x10-3
Water 7.72383x10-4 7.992x10-4
(The values of have not yet been determined, which explains the descrepancy between
calc and obs.)
The np values are easy to understand. In acetic acid, the CH3 contributes 3 units and the
CO2H contributes 4. In carbon tetrachloride, each atom contributes 1 unit. In ethyl
acetate, each volumetric group contributes a unit. In water, 3 molecules of 3 atoms each
act against the external pressure, for a total of 9. All values of nv, nt, and np are integral
or half-integral, as required by the nature of the Reciprocal System. This is very different
from the empirical correlations used by other investigators.7
In the coming years I hope some member of ISUS will calculate the results for thousands
of liquids following the equations given here.
References:
1. M. Abbott, H. Van Ness, Thermodynamics (New York: McGraw-Hill Book
Company, 1972), p. 105.
2. R. Satz, "The Liquid State in the Reciprocal System: The Volume/Temperature
Relation, a Contemporary Mathematical Treatement," Reciprocity, Vol. XXIII, No. 2,
Autumn 1994. Incidentally, the normal function should have been denoted by , not
erf.
(The numerical results of the paper do not change, because was actually used.)
3. R. Satz, "The Equation of State of Solid Matter," Reciprocity, Vol. X, No. 2, Spring-
Summer 1980.
4. D. Larson, Nothing But Motion (Portland, Oregon: North Pacific Publishers, 1979), p.
160.
5. Handbook of Chemistry and Physics, 72nd Edition (Cleveland: The Chemical Rubber
Company, 1991-1992), pp. 6-108 to 6-110.
6. American Institute of Physics Handbook (New York: McGraw-Hill Book Company,
1972). values are difficult to find. If you know the volume at temperature i and
temperature f (and the pressure is constant), then from equation 1, .
7. R. Reid, J. Prausnitz, B. Poling, The Properties of Gases and Liquids, 4th Edition
(New York: McGraw-Hill, Inc., 1987).
8. D. Larson, The Liquid State, privately circulated series of papers on the liquid state,
circa. 1960-1964. Note: for this work, I made use of his paper IV. Larson used the
semi-empirical value 415.84 atm for the liquid natural pressure unit. My derivation of
Plnu is unique.
PERMITTIVITY, PERMEABILITY AND THE SPEED
OF LIGHT IN THE RECIPROCAL SYSTEM
Introduction
Physics textbooks do not provide a theoretical derivation of Newton‗s law of gravitation
or Coulomb‘s laws of electrostatics and magnetostatics; they are simply stated as
empirical truths. By contrast, books on the Reciprocal System, such as Ref. [1], [2], [3],
do provide a theoretical derivation of these laws. This paper will take a closer look at the
terms of the electrostatic and magnetostatic equations--the terms of the gravitational
equation have already been discussed in detail, most recently in Ref. [3]. Unlike the force
of gravitation the forces of electrostatics and magnetostatics can be reduced by the
intervening media between the charges. The question to be answered is: what are the
dimensions of all the terms of Coulomb‘s laws?
Permittivity
Coulomb‘s law of electrostatic attraction is expressed as
Q1 Q²
Fe = ke ——— (1)
r²
where Q1 and Q² are the electric charges, r is the distance between them, Fe is the force,
and ke is the proportionality constant. In the Reciprocal System all physical quantities are
expressed in terms of space and time only; there are no separate dimensions for mass or
charge. So, electric charge is not taken as a fundamental entity and given an independent
unit. Larson has deduced that the dimensions of force and charge are
Fe = [t/s²]
Q = [t/s]
In the gravitational expression, the second mass and the distance are considered to be
dimensionless ratios. Such a procedure could be used in analyzing the electrostatic
expression; however I find it more fruitful to treat the second charge and the distance as
having dimensions. In this case Coulomb‗s law expressed in dimensions is
[t/s] [t/s]
[t/s²] = ke ———– (2)
[s²]
For this equation to be dimensionally correct, the dimensions of ke must be
ke = [s²/t] (3)
These are the dimensions of permittivity (in the Reciprocal System). However, the
conventional expression for the coefficient of the law is
ke = 1/(4 ) (4)
where is the permittivity, expressed in farads/meter. Thus the derivation gives a result
that is the inverse of the usual coefficient. But in the Reciprocal System the farad is
reducible to a length. So in the conventional units that are used, permittivity turns out to
be dimensionless whereas physically it is not. It should not then be surprising that the
numerical values of permittivity are inverted. Instead of saying that the permittivity of air
is 1.0006 times that of free space, I would say that it is 1/1.0006 = .9994 times that of free
space. This actually sounds better! (The other part of the coefficient (1/4 ) of the
conventional expression was put in for practical reasons having nothing to do with basic
physics; there is no point to keeping it in the Reciprocal System). Of course, the end
result-the calculated force--must be the same in both systems.
Permeability
Coulomb‗s law of magnetostatics is
M1 M²
Fm = km ——— (5)
r²
where M1 and M² are the magnetic charges, Fm is the force, r the separation distance, and
km the proportionaiity constant. Larson has deduced that the dimensions of magnetic
charge are
M = [t²/s²] (6)
Then, expressed dimensionally, Coulomb‗s law for magnetostatics is
[ t²/s² ] [ t² /s² ]
[t/s²] km ——————— (7)
[s²]
For this equation to be dimensionally correct, the dimensions of km must be
km = [s4/t³] (8)
But in the Reciprocal System magnetic permeability (symbol ) has the dimensions t /s .
Thus km = 1/ . This time the derivation from the Reciprocal System is in exact accord
with the conventional Kennelly system, with magnetic charges or poles expressed in
webers (volt-sec). The only thing awkward here is the name ―permeability‖ . On the basis
of the equation, a better name for this quantity would be ―impermeability‖ . Besides, as
Larson has pointed out ―permeability‖ is the magnetic analog of electric resistance
(t² /s³ * t/s). Perhaps for parallelism with the revised electrostatic expression we should
put the reciprocal of what is now called permeability into the numerator of the law but
call it by the same name. Thus the higher the electric force between charges, the higher
the (reciprocal) ―permittivity‖ ; and the higher the magnetic force between magnetic
charges, the higher the (reciprocal) ―permeability‖ .
Permittivity, Permeability, and the Speed of Light
One way to confirm the identification of the dimensions of permittivity and permeability
is to use them in the same expression. One such expression is Maxwell‗s famous result
from electromagnetic theory:
c = 1/( o o )½ (9)
where c is the speed of light, o is the permittivity of free space, and o is the
permeability of free space. In the dimensional terms of the Reciprocal System, the
equation is
[s/t] = 1/[(s²/t)(t³/s4)]½ (10)
As expected, the dimensions check out fine. These new results should help clarify
electrostatics and magnetostatics for both students and working scientists and engineers.
References
1. Dewey B. Larson, The Structure of the Physical Unieerse (Portland, Oregon: North
Pacific Publishers, 1959). Note: in this, the first presentation of the Reciprocal System,
the permittivity and permeability were treated as dimensionless (p. 82).
2. Dewey B. Larson, Nothing But Motion (Portland, Oregon: North Pacific Publishers,
1979).
3. Dewey B. Larson, Basic Properties of Matter (Salt Lake City, Utah: International
Society of Unified Science2 1988). Note: the dimensions of permittivity are stated as
[s²/t] on p. 172; the dimensions of permeability are stated as [t³ /s4] on p. 222.
APPENDIX: SAMPLE CALCULATIONS
1. Electrostatics
What is the force exerted by a charge of one coulomb on another charge of one coulomb
one km away, in air?
From eq. 3, the value of the permittivity in free space is
snat ²/tnat = (4.558816*10-6
)²/1.520655*10-16
= 136670.11 cm²/sec.
In air, the value is .9994 times this, or 136588.11 cm²/sec. Now a coulomb is defined as
the electrostatic charge which when placed at a distance of 1 meter from an equal charge
of the same sign produces a repulsive force of 8.98755*10-9
N. In space-time terms, this
force is
8.98755*109 N * 10
5 dynes/N * (7.316889*10
-6 sec/cm²) / (3.27223*10² dynes)
= 20096664 sec/cm² .
Then from Coulomb‗s law (for a vacuum) we have
20096664 sec/cm² = 136670.11 cm²/sec * Q² sec²/cm² / 10000 cm².
Solving for Q gives 1212.6213 see/em per coulomb. So for the problem at hand we have
Fe - 136588.11*1212.6213²/100000² = 20.084604 sec/cm².
Converting back to conventional units we have
20.084604 sec/cm² * 3.27223*10² dynes/7.316889*10-6
sec/cm² * 10-5
N/dynes
= 8982.1567 N.
This agrees with the value from experiment.
2. Magnetostatics
What is the force exerted by a magnetic pole with a strenath of one weber against another
magnetic pole of equal strength one km away, in vacuum?
The value of the permeability of free space is
t³/s4 = (1.520655*10
-16)³/(4.558816*10
-6)4 = 8.1411073*10
-27 sec /cm
4 .
Now a weber may be defined as the strength of a magnetic pole which exerts in a vacuum
a force of 63325.74 N upon another magnetic pole of the same strength one meter away.
In space-time terms this force is
63325.74 N * 105 dynes/N 2 (7.316889*10
-6 sec/cm²)/(3.27223*10² dynes) = 141.59989
sec/cm .
From Coulomb‘s magnestatic law we get
141.59989 sec/cm² = (1/8.1411073*10 -27 sec3/cm4) * M²sec4/cm4/10000 cm².
Solving for M gives 1.0736759*10-10
sec²/cm² per weber. So for the problem at hand we
have
Fm = (1/8·1411073*10-27
sec³/cm4) * (1.0736759*10-10 sec /cm ) / (100000 cm) =
1.41599890 sec/cm.
Converting back to conventional units we have
1.4159989*10-4 sec/cm² * 3.27223*10² dynes/(7.316889*10-6
sec/cm ) * 10-5 N dyne =
0633257 N.
This agrees with the value from experiment.
SUMMARY:
1. permittivity of free space = 136670.11 cm²/sec
2. permeabilit of free s ace = 8.1411073 10 sec /cm
THE UNIT OF MAGNETIC CHARGE
In terms of the egs system, the unit of electron charge (and quantity) is calculated by
multiplying the Faraday constant by the mass equivalent of unit atomic weight:
2.89366x1014 esu/g-equiv * 1.65979x10-24 g = 4.80287x10-10 esu
(ref. [1]). Of course, 4.80287x10-10 esu is equal to 1.602062x10-19 coulombs.
In space-time terms, the dimensions of electron charge (or electric charge in general) are
t/s. The magnetic charge is a two-dimensional form of the electric charge; its space-time
dimensions are t2 /s2 . The numeric value of the magnetic charge must therefore be the
value of the electric charge divided by the the natural value of s/t, or the speed of light. In
the egs system, this results in
4.80287x10-10 esu / 2.99793x1010 cm/sec = 1.602062x10-20 ―esu‖
(ref. [2]). The ―esu‖ here are the magnetic units of the electrostatic system. According to
ref. [3], 1 ―esu‖ of magnetic flux (equivalent to charge in the Reciprocal System) equals
299.8 webers. Thus one unit of magnetic charge equals 4.802982x10-18 webers.
Each atom has two rotating systems; if one system acquires a magnetic charge, the other
system must also acquire a charge if there is to be stability and permanence. Henee each
atom has two poles, or centers of magnetie effect--it is dipolar, not monopolar.
Consider a simple ―bar magnet‖ of four iron atoms. The poles would be arranged in this
manner: N-S - N-S - N-S - N-S. Only the end atoms are not neutralized; therefore, in
general only the surface atoms of a bar magnet contribute to its effective magnetic charge
(one unit of charge per surface atom). This means that it would take 2.0820399x1017
magnetically charged surface atoms to generate one weber of magnetic flux. Since iron
has a mean atomic weight of 55.847, the total mass of these surface atoms would come to
1.9301763x10-5 grams.
References:
1. Dewey B. Larson, Basic Properties of Matter (Salt Lake City, Utah: International
Society of Unified Science, 1988), p. 110.
2. Dewey B. Larson, The Structure of the Physical Universe (Portland, Oregon:
North Pacific Publishers, 1959), p. 211 (except that the numerical value has been
updated in ref.1).
3. Robert Resnick and David Halliday, Physics (New York: John Wiley & Sons,
Ine., 1966), p. 33, Appendix G, of the supplement.
PHOTOIONIZATION AND PHOTOMAGNETIZATION
Introduction: the Reciprocal System vs. Present Theory
Consider a group of atoms in an electric field and bombarded with ultra-violet photons or
a group of atoms in a magnetic field and bombarded with radio photons. What happens?
Two theories exist that can give an answer: Quantum Mechanics and the Reciprocal
system. Both are quantized, but the first is a matter-structure theory, whereas the second
is a motion-process theory. Quantum Mechanics considers atoms to consist of various
subatoms which have intrinsic charge, magnetic moment, and angular momentum; the
atom‘s charge, moment, and momentum are derived from that of its subatoms. The
Reciprocal system views atoms as composed of two photons, each having rotational
motion in three dimensions; the atom has no intrinsic electric charge or magnetic
moment—electric and magnetic effects result from additions of rotational vibratory
motions to the base rotational motions.
Quantum Mechanics‘ explanation of electric ionization is that previously 0existing
charged particles, the protons and electrons, are separated; the Reciprocal System‘s
explanation is that the positive and negative charges 0are created in the process, and thus
have no prior existence. Quantum Mechanics‘ explanation of the magnetic resonance
experiments is that the experimenters have found intrinsic magnetic moments of nuclei;
the Reciprocal System‘s explanation is that the experimenters have induced temporary
magnetic charges in their material. Quantitative details of both theories will now be
examined. (Full comprehension of this paper requires previous reading of two of D. B.
Larson‗s books, Refs. 1 and 2, and one of my papers, Ref. 3.)
I. Photoionization
A. Subatoms
1. Present theory.
According to present thought, the electric charge is unanalyzable and undefined, except
operationally. Either a particle has or does not have an intrinsic electric charge—there is
no possibility of ionizing an uncharged subatom.
Present photoelectric theory states that, upon absorption of a sufficiently energetic
photon, a pre-existing charged electron is ejected from its atom to move in an external
circuit. [4,5] The energy necessary to tear 0the electron loose is called the work function
of the material. No commonly accepted equation for the work function, based on
Quantum Mechanics, exists.
2. Reciprocal System
For details of subatom and atom motions, see Refs. 1, 2, and 3. In the Reciprocal System,
electric charge is not an intrinsic feature of a subatom; rather, charges may be created or
destroyed, not necessarily in pairs, and 0thus charge conservation in a process does not
always hold true. However, total motion displacement is conserved in each process.
As with all other phenomena in the Reciprocal system, electric charge is a motion, in this
case a simple harmonic rotational vibration, as shown in Figure I.
FIGURE I: ELECTRIC CHARGE
An equation for this motion will now be derived. Let q be the rotation angle in radians,
be its frequency in Hz, and t be the time in seconds. From the figure, the amplitude of the
motion is radians and the angular distance traveled each cycle is 4 radians. Hence the
equation is
q = cos(4 t) (1)
As shown in a previous paper of mine [3], a negative electric charge has the frequency
-elec. = R/2 (2)
where R is the Rydberg frequency (3.288 * 1015 Hz).
Electrons exist within matter, but not as intrinsic features of atoms. 0Also, these electrons
are ordinarily uncharged. To travel outside of matter 0the electrons must become charged
or ionized. The energy for the charge and 0the kinetic energy of the charged electron
come from absorption of a photon, 0thus producing the photoelectric effect. A rigorous
equation for this 0effect, slightly modified from Ref. 6, can now be given. Let
h = Planck‗s constant phot.
phot= photon frequency
v = electron velocity (outside of matter) 0
m = electron mass
eV = electric potential surrounding the matter
Wo. = work function of the matter
Uk. = energy of the electron before the process begins
Ul = energy lost by charged electron in moving to surface
The equation for the electron‗s kinetic energy outside of matter is then
1/2 mv² = h phot. + eV - Wo. + Uk - Ue (3)
According to the Reciprocal System the work function is the energy necessary to charge a
uncharged electron.Since the rotational vibration is scalar, like the linear vibration,
Planck‘s law holds for electric charges 0as well as for photons:
EI,e. = h -elec. = h * R/2 = 2.17 eV (4)
Note: any observed values of work function less than 2.17 eV emply previous electron
energy, Uk). The value of EI,e given in (4) is modified by the environment of the electron,
i.e., by the atom in which it currently exists. The electron‗s charge may be in the same
dimension as one of the atom‗s magnetic rotations or in the same dimension as the
electric rotation. In the Reciprocal system, charge is energy, t/s, the inverse of velocity.
The atom‗s magnetic rotation velocity is vmag, its electric rotation velocity is v; the
inverse of these in natural units is c/vmag and c/velec , where c is the speed of light. If the
atom has only one electric time displacement unit (velec= 1/2c), the ionization energy of
the electron is not increased, hence a 1 must be subtracted from c/velec. Finally, the atomic
motions take place in the time region, whereas we want the energy as measured in the
time-space region—so the square root of the inverse velocity expressions 0must be taken.
Thus
wo. = 2.17 * [c/vmag.]½ eV. -and/or (5a)
wo. = 2.17 * [c/velec. -1]½ eV
These equations, theoretically derived, are nearly identical to the ―empirical‖ equation
given by Larson in Ref 1 (p. 118), eq. (142)). The set of 0values of Wo. is
Wo={2.1 7, 3.07, 3.76, 4.34, 4.82} <5b>
Table I compares the theoretical results with those observed.
TABLE I
WORK FUNCTION AND IONIZATION ENERGY
Wo EI
Element c/vmag c/velec calc. obs calc. obs.
Li
2 2.17 2.28 4.34 5.39
Be 3
3.76 3.92 7.52 9.32
B 3
3.76 4.4 8.68 8.296
C
5 4.34 4.341 4.34 11.264
Na
2 2.17 2.25 7.52 5.138
Mg 3
3.76 3.78 7.52 7.644
Al
4 3.76 3.43 8.68 5.984
Si
5 4.34 4.2 4.34 8.149
K
2 2.17. 2.12 6.14 4.39
Ca
3 3.07 3.20 7.52 6.111
Ti 3
3.76 3.95 7.52 6.83
V 3
3.76 3.95 8.68 6.83
Cr 4
4.34 4.37 7.52 6.764
Mn 3
3.76 3.76 7.52 7.432
Fe 3
3.76 3.91 7.52 7.90
Co 3
3.76 3.9 7.52 7.86
Ni 3
3.76 3.67 7.52 7.633
Cu 3
3.76 3.85 7.52 7.724
Zn 3
3.76 3.89 7.52 9.391
Ga
4 3.76 3.80 7.52 6.00
Ge
5 4.34 4.29 8.68 7.88
As
6 4.82 5.11 9.64 9.81
Se 4
4.34 4.42 8.68 9.75
Rb
2 2.17 2.16 4.34 4.176
Sr
3 3.07 2.74 6.14 5.692
Zr 3
3.76 3.73 7.52 6.835
Nb 3
3.76 3.96 7.52 6.88
Mo 3
3.76 4.08 7.52 7.131
Ru 4
4.34 4.52 8.68 7.36
Rh 4
4.34 4.57 8.68 7.46
Pd 4
4.34 4.49 8.68 8.33
Ag 4
4.34 4.33 8.68 7.574
Cd 3
3.76 3.73 7.52 8.991
Sn 3
3.76 3.62 7.52 7.332
Sb 4
4.34 4.14 8.68 8.64
Te 4
4.34 4.70 8.68 9.09
Cs
2 2.17 1.96 4.34 3.893
Ba*
2 2.17 2.11 4.34 5.810
La* 2
3.07 3.3 6.14 5.61
Ce* 2
3.07 2.84 6.14 6.91
Pr* 2
3.07 2.7 6.14 5.76
Nd* 2
3.07 3.3 6.14 6.31
Sm* 2
3.07 3.2 6.14 5.6
Hf* 2
3.07 3.53 6.14 5.5
Ta 3
3.76 3.96 7.52 7.7
W 4
4.34 4.35 8.68 7.98
Re 4
4.34 5.0 8.68 7.87
Os 4
4.34 4.55 8.68 8.7
Ir 4
4.34 4.5 8.68 9.2
Pt 4
4.34 4.09 8.68 8.96
Au 4
4.34 4.46 8.68 9.223
Hg 4
4.34 4.5 8.68 10.434
Tl 3
3.76 3.84 7.52 6.106
Pb 3
3.76 3.94 7.52 7.415
Bi 4
4.34 4.31 8.68 7.287
The agreement is excellent (the correlation coefficient rcorr = .964).
As discussed by Ref. 7, the number of electrons emitted per incident pho-ton depends on
both the nature of the emitter and the frequency of the incident radiation.At frequencies
lower than that at which maximum yield is obtained, reflectivity of incident photons is so
high that only a few photons take part in the emission process.At frequencies higher than
that at which maximum yield is obtained, the photons penetrate to such a depth that the
electrons, newly charged at that depth, lose too much energy in coming to the surface.
Of the remaining subatoms only the positron, proton and H can take an electric charge.As
shown in Ref 3, a positive electric charge has the frequency
+elec. = 2R/ (6)
*Asterisk denotes entry of rotation into second space-time unit in the 4a and 4b groups;
also where value 3 appears in magnetic rotation, this is the inverse of actual rotation.
Values of c/vmag'
c/velec and Wo obs. are taken from Table.
The ionization energy of a free positron or proton (not including force-coupling effects)
is then
EI,p. = h +elec. = h * 2R/ = 8.68 eV (7)
Now consider the ionization of the intermediate particle, H. Here everything is unity: H
has one natural unit of primary mass and one unit of electric space displacement; one
positive charge is created and one negative charge. So the required photon energy must
also be unity by the Principle *Asterisk denotes entry of rotation into second space-time
unit in the 4A and 4B groups; also where value 3 appears in magnetic rotation, this is the
inverse of actual rotation. Values of c/vmag, c/velec., and Wo obs. are taken from Table CIX
of Ref. 1. of Equivalence (see Ref. 1, p. 21):
EI,H1 = h * R = 13.595 eV (8)
as observed.
B. Atoms
1. Present Theory
In present theory, ionization is thought to be the ejection of an elec-tron from its orbital,
leaving a net positive charge on the atom. No generally accepted equation has been
developed to calculate the energy of ionization from Quantum Mechanics.One might
expect the ionization energy to be practically the same as the work function— but they
are not.Indeed, there is no evidence that matter becomes ionized in the photoelectric
effect.
Gas and liquid ionization is currently thought to be the break-up of previously existing
oppositely-charged units.
2. Reciprocal System
In the photoelectric effect, only charged electrons are created, not charged atoms.But
generally in atomic ionization both positive and negative electric charges are created
(they do not exist previously).Where negative ions can form, as with the electronegative
elements, they do so. But usually negatively-charged electrons serve as the second
component of the force couple creating the charges. Thus in most cases a positively-
charged ion and a negatively-charged electron are the results of ionization. So, rather than
the ionization energy being the binding energy of an elec-tron to a nucleus, it is the
energy required to create two charges, one 1 -positive and one negative.
Equation (5) gave the energy necessary to create a negative charge on an electron. From
mechanical considerations it is obvious that the energy necessary to create a positive-
negative charge pair is twice that needed to create the negative charge on the
electron.Hence for the first ionization level the energy is
EI,atom = 4.34 * [c/vmag.]½ ev (9a)
EI,atom = 4.34 * [c/velec.-1]½ ev
The set of values of EI,atom is
EI,atom = {4.34, 6.14, 7.52, 8.68, 9.64} (9b)
Table I compares calculated values with observed values of ionization [8]. 0Agreement is
very good. (Avg. Wo calc. =
3.706; avg. Wo. obs.=3.779. Avg. EI. calc. =7.412; avg. EI.
obs=7.366. Avg. EI. calc./avg. Wo. calc.=2.000; avg. EI.
obs./avg. Wo. obs.=1.95. For EI. obs. and EI.calc. rcorr.=.803.
Individual discrepancies are most likely due to experimental error.)
Ordinarily, solids are not ionized; the resulting forces would overcome the cohesive
energy and break apart the solid.Liquids and gases are more readily ionized, with the
energy often being supplied by photons.An electric field is of course necessary to prevent
the charges from recombining. For a good discussion of liquid ionization, see Ref. 1.
Gaseous ionization depends on both electric field strength and gas pressure. An equation
for the saturation electric field will now be derived. Let
I = primary current (ions created by photons/time)
Ephot./t = photon energy absorbed per unit time
EI. = ionization energy
€f. = electric field strength
€fnat.= natural unit of electric field strength
P = gas pressure Pnat. = natural unit of pressure
The equation for the primary ion current is then
I =(Ephot./t) * (1/EI.) * ( €f./€ fnat.) * (Pnat./P) (10)
Clearly if
Ephot. = EI.,€ f =€ f nat, and P = Pnat., one ion
per time t will be created. For a fixed energy input, I can be increased either by
0increasing €f. or decreasing P until
( €f. / €fnat.) * (Pnat./P) = 1 (11a)
Solving for €f we have
f. = €fnat. * (P/Pnat.) (11b)
With €fnat. = 2.04133 * 1016 v/m and Pnat. = 1.5539 * 107 atm.
and letting P = 1 * 10-5 atm., then
€fsaturated = 13137 v/m
The newly created ions and charged electrons can themselves cause further ionization,
which is called secondary ionization.
The reverse of ionization is the addition of a negatively charged electron to a singly
charged positive ion, which results in a neutral atom. Thus the ―electron affinity‖ of a
singly charged positive ion is just the negative of the ionization energy of the
corresponding neutral atom.[9] Where current theory gets into difficulty is in
understanding the ―electron affinity‖ of neutral atoms. According to the Reciprocal
System, the electron loses its charge upon absorption in matter, resulting in a reverse
photoelectric effect.
II. Photomagnetization
A. Subatoms
1. Present Theory
According to present thought, the magnetic resonance experiments detect the magnetic
moments intrinsic to subatoms and atoms (nuclei).The magnetic moment is considered to
result from the angular spin of the electric charge of a particle.It is given in units of the
Bohr magneton or the nuclear magneton, as derived from Dirac‗s theory.
A number of problems exist with this theory.The neutron has a magnetic moment, but no
electric charge. Pions and alpha particles do have electric charges but no magnetic
moment; present theory claims that these particles have zero spin— but that seems
equally strange.The magnetic moment of the proton has been obtained from experiments
with ice and water; thus the magnetic moment could actually be that of H itself,
regardless of what theory says.The magnetic moment of the anti-proton has been found
not to equal the magnetic moment of the proton (or rather, H1 ).
More fundamentally, the value of the magnetic moment is not measured + directly; it is
inferred from the data. To see this, let
o= resonant photon frequency
h o. = absorbed photon energy
B = magnetic field strength
µ = magnetic moment in field direction
L = angular spin no.
The equation (from Ref. 10) is
h o = [µ /L] * B (12)
The energy h o and field B are measured, L is inferred from other data (usually
spectroscopic), and then is calculated. If L is bogus, then µ is bogus. All that the
experiments tell us is that for a given B there exists a certain photon frequency at which
great amounts of energy are absorbed by he sub-atoms and atoms. The conclusion that
the relation between h o and B is µ /L is purely hypothetical.
2. Reciprocal System
My interpretation of the magnetic resonance experiments, on the basis of the Reciprocal
System, is radically different.
Here the sub-atoms and atoms have no intrinsic magnetic moments or magnetic charges.
(Note: the isotopic charges in atoms cancel the magnetic effect of the magnetic charges of
the neutrinos contained within). But under certain circumstances, such as in the magnetic
resonance experiments, temporary magnetic charges can be induced. A magnetic charge
is a rotational vibration of one of the inner magnetic rotations of the sub-atom or atom.
As given in Ref. 3, the vibrational frequency of a unit magnetic charge is
vmag. = 2R/ (13)
The required energy to produce this vibration depends on the environment: the magnetic
field and the velocity of the principal or subordinate magnetic rotation which is modified
by the charge. The resulting equation is related to, but different from, the equation for
energy of electric ionization, eq. (5). In the Reciprocal system, magnetic effects are the
square of electric effects—so the square root of eq. (5) is eliminated. Also, as in the
equations for force and distance in the Reciprocal System, the effect of the magnetic
velocity is inverse to that of the electric velocity. Thus, instead of the energy being
proportional to c/vmag., the energy is proportional to vmag./c. The complete
expression is
h o = [h * (2R/ * (vmag./c * 1/Bnat.] * B (14a)
Larson [1] has previously identified magnetic susceptibility as proportional to vmag./c.
This provides additional support for that term in the above equation.
FIGURE II: MAGNETIC CHARGE
As shown in Figure II, the sole difference between a magnetic charge and an electric
charge is that the magnetic charge is an electric charge that is given an extra angular spin
by either the subordinate magnetic rotation or the electric rotation (depending on whether
the charge is placed on the principal or subordinate magnetic rotation).
If no magnetic field is present, and the photons are not reflected, the photon energy is
simply transformed to thermal energy of the particle. For a given magnetic field B, o can
be calculated for each kind of sub-atom and atom.As discussed in Ref. 3, vmag./c can take
on the following values:
vmag./c = {.20, .22, .25, .29, .33, .40, .50} (14b)
Setting B equal to 1 Tesla and knowing that Bnat = 6.813*107 Tesla, the following
resonance frequencies are obtained:
o. in MHz = {6.14, 6.83, 7.68, 8.78, 10.24, 12.29, 15.36} (14c)
The calculation assumes that no other energy is present that can be utilized in the creation
of the magnetic charge.Unfortunately, magnetic resonance experiments have been done at
room temperature rather than at temperatures close to degrees K, so the absorption of
thermal energy is a definite possibility.
Nearly all atoms have magnetic resonance frequencies at or below 15.6 MHz (with B = 1
Tesla), in accord with eq.(14). But the intermediate parti- cles, the neutron and H ,have
higher observed frequencies,29.16 MHz and 42,57 MHz, respectively. One theoretical
explanation is that these particles require multiple magnetic charges if they are to have
any at all.The neutron is comprised of two rotational systems:a proton rotational system
and a cosmic neutrino rotational system. In terms of total rotational speed (in natural
units) the notation is
{
1/3 - 1/2
neutron
1
2 - 2
(See refs. 1 and 3 for details). Suppose each rotational system takes a magnetic charge on
its subordinate magnetic rotation. For the proton rotational system, the energy required is
h * (2R/ ) * (1/2) * (B/Bnat.)
For the cosmic neutrino rotation (which takes an inverse charge) the energy required is
h * (2R/ ) * 2 * (B/Bnat.)
The combined energy is
h * (2R/ ) * (B/Bnat.)
and the resonance frequency is 30.70 MHz (for B and Bnat.) as before. The small
discrepancy between the observed and cal- culated values may be due to the absorption of
thermal energy.
The notation for H1 is
{
1/3 - 1/2
H1
2
1/2 - 1/2
In this case, though, each rotational system apparently takes two charges, so that the
frequency is 3R/ rather than 2R/ . The energy required is then
[h * (3R/ ) * (1/2) + h * (3R/ ) * (1/2)] * (B/Bnat.)
giving a resonance frequency of 46.05 MHz. Again the discrepancy between that
observed and that calculated may be due to the utilization of thermal energy.
Both the real proton and anti-proton (inverse proton) and the material and cosmic
neutrinos and the material and cosmic massless neutrons should have resonance
frequencies of 15.36 MHz, unless they take multiple charges.
The electron and positron have no subordinate magnetic displacements at all and thus
cannot take magnetic charges. All magnetic effects of these particles (and also the muon),
whether uncharged or charged, result from their being in translational motion. To quote
Larson [2]:
As we have seen, the electric charge is a one-dimensional modification of the rotational
motion of an atom or sub-atomic particle and the magnetic charge is a similar two-
dimensional modification. The characteristic effects of the magnetic charge originate
because the one-dimensional forces are distributed over two dimensions by the second
rotation. But for this purpose it is not necessary that the motion in the second dimension
be rotational. We can see why this is true if we examined the behavior of the axes of
rotation. The axis of the electric rotation of an atom is a line: a one-dimensional figure. A
stationary electric charge thus has no two-dimensional rotational effects. For a magnetic
charge the locus of all positions of either axis is a disk: a two-dimensional figure and the
magnetic charge has two-dimensional properties. But if we move the electric charge
translationally, the locus of all positions of the axis is again a two-dimensional figure, and
hence a moving electric charge has a two-dimensional distribution of forces comparable
to that of a magnetic chargeÉ If an uncharged electron or positron is given a translational
motion, this again is motion in two dimensions and it produces electromagnetism, a
magnetic effect.
Particles heavier than the electron and positron would show a similar magnetic effect if
they could be accelerated to the same high velocities.
B. Atoms
1. Present Theory
Present theory regards the magnetic moment of nuclei to result from a combination of the
moments of its constituent sub-atoms. Even-even nuclei are regarded as having zero net
spin and thus zero moment. According to Segre‗s account of current theory [11], adding a
neutron to an even-even nucleus is supposed to yield (* + 1/2)[ -3.826/(2* + 1)] nuclear
magnetons for the magnetic moment, where* is the spin angular momentum of the added
neutron; subtracting a neutron is supposed to yield (* - 1/2)[3.826/(2* +1)] nuclear
magnetons. Adding a proton is supposed to yield (* + 1/2)[1+ 4.586/(2* +1)] nuclear
magnetons, whereas subtracting a proton is supposed to yield (* - 1/2)[1-4.586/2 *+ 1)]
nuclear magnetons. These expressions do bracket the data, as Segre points out, but that is
all: they do not work in detailed application.
The sign given for the moment is an inference, not a result from experiment (which
measures only photon energy and magnetic field strength at resonance). In present theory,
both the magnetic moment and angular spin are vectors.If they are aligned the magnetic
moment is said to be positive; if anti-aligned, negative.
2. Reciprocal System
In the Reciprocal System all basic motions — including the electric and magnetic
charges— are scalar. In addition, the magnetic charge is intrinsically dipolar: the
magnetic rotation of the one-dimensional rotational vibration can be viewed both
clockwise and counter-clockwise (see Figure 2). Generally in the magnetic resonance
experiments, the atoms are induced to take only a single magnetic charge and thus have
resonance frequencies of 15.36 MHz or less. The exceptions, such as F and T1, are
apparently induced to take multiple charges
Table II compares the observed resonance frequencies [12] (of stable isotopes) with the
theoretical results from eq. (14). (Note: Thermal energy and cohesive energy are not
taken into account; numerous isotopes (usually unstable) given in Ref. 12 have resonance
frequencies less than 6.14 MHz—these are not given in the table below. Further
theoretical work is necessary to include thermal energy, cohesive energy, and instability
effects in magnetic resonance).
TABLE II
MAGNETIC RESONANCE FREQUENCIES
vmag./c [B=1 Tesla]
Isotope Displ. Mag. Speed calc.(MHz) obs.(MHz)
5 B11 2-1-3 1/2 15.36 13.66
21.Sc45 3-2-3 1/3 10.24 10.36
25.Mn55 3-2-7 1/3 10.24 10.57
27.Co59 3-2-9 1/3 10.24 10.12
29.Cu61 3-3-(7) 1/3 10.24 10.12
31.Ga69 3-3-(5) 1/3 10.24 10.24
33.As75 3-3-(3) 1/4 7.68 7.31
34.Se77 3-3-(2) 2/7 8.78 8.14
35.Br79 3-3-(1) 1/3 10.24 10.70
41.Nb93 3-3-5 1/3 10.24 10.45
47.Ag104 4-3-(7) 1/5 6.14 6.10
48.Cd111 4-3-(6) 2/7 8.78 9.07
49.In113 4-3-(5) 2/7 8.78 9.35
51.Sb121 4-3-(3) 1/3 10.24 10.24
63.Eu151 4-3-9 1/3 10.24 10.56
65.Tb159 4-3-11 2/7 8.78 8.64
67.Ho165 4-3-13 2/7 8.78 8.93
70.Yb171 4-3-16 1/4 7.68 7.52
73.Ta181 4-4-(13) 1/5 6.14 5.14
78.Pt195 4-4-(8) 2/7 8.78 9.24
82.Pb207 4-4-(4) 2/7 8.78 8.99
The equation appears to work well for the majority of atoms studied (rcorr = .956 for the
table). In those cases where the equation does poorly, the thermal effects may be the
culprit. Ideally, the experiments should be repeated with the atoms widely separated and
at close to 0 degrees K in temperature — then the effects of thermal energy and cohesive
energy would be eliminated. Under such conditions all frequencies for the induction of a
single unit of magnetic charge in a stable isotope should be between 6.14 MHz and 15.36
MHz when the field B is 1 Tesla.
Atoms with an even number of neutrino-induced isotopic charges (half on each rotational
system) have no need to acquire another charge for balance. Hence in the current jargon
these atoms do not have a ‗magnetic moment.‘
In the series of isotopes of an element, the placement of the magnetic charge appears to
alternate with the placement of the isotopic charge. A quantitative check on this is
difficult to do because in most such series there are unstable isotopes —this instability
adds another variable to the problem. Apparently if the number of isotopic charges is
greater than that allowed by the magnetic ionizat ion level, the energy required to induce
a magnetic charge is decreased. Once the magnetic field is turned off and the photon
bombardment ceases, the magnetic charges are transformed to radio photons and lost.
Only a few elements, such as Co, Ni, and Fe, are able to retain a magnetic charge.
Summary. Electric and magnetic charges are not unanalyzable; they are one and two-
dimensional rotational vibrations of a sub-atom or atom. They are also not permanent and
inviolate; they may be created or destroyed, so long as overall motion displacement is
conserved.
1a. The work function of a material is not the energy required to remove the least tightly
bound electron; it is the energy required to charge an uncharged electron in that material.
1b. The ionization energy of an atom is not the energy required to separate pre-existing
charged protons and electrons; it is the energy required to create a negative charge (on an
electron) and a positive charge (on an atom) and is equal to twice the work function.
2. The magnetic resonance energy of a sub-atom or atom does not indicate a previously
existing intrinsic magnetic moment; it is the energy required to induce one or more
dipolar magnetic charges in sub-atoms or atoms.
References
1. D. B. Larson, The Structure of the Physical Universe (Portland, Oregon: North
Pacific Publishers, 1959) pp. 21, 61-89, 116-131. Larson‗s Nothing But Motion
(Portland,Oregon: North Pacific Publishers, 1979) is the first of a series of
volumes of the second edition of The Structure of the Physical Universe.
2. D. B. Larson, New Light on Space and Time (Portland, Oregon: North Pacific
Publishers, 1965) pp. 165-196.
3. R. W. Satz, ―Further Mathematics of the Reciprocal System,‖ Reciprocity, Vol. X,
No. 3, Autumn 1980.
4. F. Bueche, Introduction to Physics for Scientists and Engineers (New York:
McGraw-Hill, 1969) pp. 541-559, 741-745, 813-815.
5. R. Cautreau and W. Savin, Modern Physics (New York: McGraw-Hill, 1978) pp.
56-61.
6. E. L. Chaffee, ―Electronics‖ in Fundamental Formulas of Physics, ed. D. H.
Menzel (New York: Dover Publ., 1960) p. 353.
7. R. Rose, L. Shepard, J. Wulff, The Structure and Properties of Materials:
Electronic Properties (New York: John Wiley & Sons, 1966) pp. 26-31.
8. V. W. Finkelnburg and W. Humbach, ―Ionisierunggenergien von Atomen und
Atomionen,‖ Die Naturwissenschaften, Heft 2, Jg. 42, 1955, pp. 35-37.
9. T. L. Brown and H. E. Lemay, Jr., Chemistry: The Central Science (Englewood
Cliffs, New Jersey: Prentice-Hall, 1977) pp. 196-198.
10. E. R. Andrew, ―Nuclear Magnetic Resonance,‖ Encyclopaedic Dictionary of
Physics, Vol. 5, ed. J. Thewlis (New York: Macmillan, 1962) pp. 70-73.
11. E. Segre, Nuclei and Particles, second ed. (Reading Massachusetts: W. A.
Benjamin, 1977) pp. 274-279.
12. 12. E. U. Condon and H. Odishaw, eds., Handbook of Physics (New York:
McGraw-Hill, 1967) pp. 9-93 to 9-101. (The table presented gives and L, from
which the resonance frequency can be retrocalculated
THEORY OF ELECTRONS AND CURRENTS
This paper will present the Reciprocal System theory of electrons and currents and
compare it with the conventional theory
1. The Electron
a. conventional theory
According to present theory1 electrons are classified (along with muons and neutrinos) as
leptons, meaning that they are not affected by the strong interaction of nuclear forces but
suffer the weak interaction that causes beta decay. These subatoms are all considered to
be fermions: they obey Fermi-Dirac statistics, have spin s =½, and have spinor-wave
functions that satisfy the Dirac equation. The present theory does not yield equations
enabling the calculation of electron mass, charge, and magnetic moment. The empirical
values are:
mass: m = 9.109*10-31 kg (1)
charge: e = -1.601*10-19 coulombs (2)
magnetic moment: ue = 9.28*10-24 joule/tesla (3)
Also no size or shape is definitely specified. The closest we have is the following:
It is obviously tempting to picture an electon as a spinning sphere of electric charge
whose radius is determined by the dimensional relation e2/a = mc2 at which the
electrostatic self-energy of the charge distribution is comparable with the relativistic
energy of the rest mass. This classical electron radius, a = 2.81785*10-15 m, is an
important scale parameter in physics; but the uniqueness of e, the arbitrariness of the
quantization rules, and the difficulty of making it properly relativistic, forbid such a
purely classical model.
Note that for this radius, and for a spin angular momentum of ½ Ã3h, the angular
velocity of the electron must be 2*1025 rad/sec — giving an equatorial speed of about
200c!
b. Reciprocal System
The Reciprocal System is much more specific on the details of electron attributes than
conventional theory. My previous papers3 4have described the shape, size, and all
motions constituting the electron.
The electron is a spherical particle resulting from the rotation of a single photon. The
frequency of the photon is
phot = 2R = 6.576115*1015 cycles/sec (4)
(Here R is the Rydberg frequency). The rotational speeds in revolutions per second
around the three axes are r/ - 2R/ - 4R/ or in terms of rev/sec
elec= 1.0466212*1015rev/sec. - 2.0932424*1015 rev./sec -4.1864848*1015 (5)
The electron may be charged or uncharged. If charged, the electron has an added
rotational vibratory motion of
-elec = R/2 = 5.233106*1014 cycles/sec (6)
The diameter d of the electron is one natural space unit, reduced by the appropriate inter-
regional ratio (142.22 here). Thus,
d = 4.55884*10-8/142.22 = 3.2054 Å (7)
2. Electron Flow
a. conventional theory
According to present theory, conduction in metals takes place by movement of the
electrons in the outermost shells of the atoms making up the crystalline structure of the
solid. These electrons reach an average drift velocity which is directly proportional to the
electric field intensity
vd = E (8)
where µ, the mobility, has the units m2/V*s. For a conductor of length l, conductivity
ó(siemans per meter), and cross-sectional area A, eq. (8) may be rewritten as
vd = ( *1/( *A))*I m/s (9)
EXAMPLE: For a copper conductor 100 mm long and 3 mm in diameter, what is the
average drift velocity of the electrons if the current is 10 amps?
For copper,
= 5.8*107 S/M -
= 0.0032 m2/V*s
Here
A = ¼ (3*10-3)2 = 7.0686*10-6 m²
Thus,
vd = (.0032*.1/(5.8*107*7.0686*10-6))*10
= 7.805*10-6 m/s
b. Reciprocal System
In the Reciprocal System, the natural unit of velocity is 2.99793*108 m/s (the speed of
light) and the natural unit of current, which is also a velocity, is 1.0535*10-3 amperes.
The conversion is thus
2.99793*108 m/s/1.05353*10-3 amps = 2.8456048*1011 m/s/amps.
Hence the ―drift‖ velocity of electrons (here uncharged and massless) in the Reciprocal
System is
vd = 2.846*1011*I m/s (10)
EXAMPLE: For the case of the previous example,
vd = 2.846*1011*10 = 2.846*1012 m/s (11)
The answer of the Reciprocal System is 3.646*1017 times the answer of conventional
theory!
Of course, the number of electrons passing a given point per second must be the same in
both theories.
In the conventional theory,
N = (10 C/s)(1 electron/1.6*10-19C) = 6.25*1019 elec/s
In the Reciprocal System,
N = 3.15842*106 esu/s*1 electron/4.80287*10-10esu *10 amps/1.05353*10-3 amps
= 6.24*1019 elec/s
The difference in ―drift‖ velocities must therefore be due to vastly different numbers of
electrons in the matter of the two theories. More about this in another paper.
References
1. Encyclopedia Britannica, Vol. 6, pp. 665-672.
2. Ibid., p. 667.
3. R. Satz, ―Further Mathematics of the Reciprocal System,‖ Reciprocity, Vol. X,
No. 3.
4. R. Satz, ―Photoionization and Photomagnetization in the Reciprocal
System,‖Reciprocity, Vol. XII, No. 1.
HUBBLE‘S LAW AND THE RECIPROCAL SYSTEM
The conceptual basis for Hubble‘s Law in the Reciprocal System has been discussed by
Mr. Larson in a number of his works. This paper will present some additional
mathematical details.
Hubble‘s Law is commonly written as
v = Hr (1)
where v is the velocity of a distant galaxy, in km/sec, r is the distance to the galaxy, in
Mpc, and H is Hubble‘s constant, in km sec-1 Mpc-1 . In differential form, the equation is
dr/dt = Hr (2)
However, as shown in Larson‘s The Structure of the Physical Universe,¹ the recession
starts at the gravitational limit of our galaxy, denoted by r . Thus the correct expression is
dr/dt = H (r - ro) (3)
Clearly the velocity is zero when r = ro.
The equation is a first order linear differential equation² and can be easily solved for r.
The result if
r = ro + (ri - ro)eHt
(4)
where ri is the initial position of the external galaxy.
Of great interest is the determination of Hubble‘s constant from first principles.
According to the Structure, the ratio of effective to total gravitational units is 1/156.4444.
At the distance a galaxy recedes at the speed of light, the effective gravitational force
drops betow the value of unity and vanishes. Thus
1/156.4444 · M
G/r1² =1 (5)
(equation 159 of Structure). Solving for r1, the limiting distance, yields
r1 = MG½ /12.51 (6)
(equation 160 of Structure).
Putting this vatue of r1 in Hubble‘s Law, one can solve for the constant:
H = c/r1-ro c/r1 (7)
where c is the velocity of light.
The value of the constant thus depends on the value of the mass of the Galaxy, MG.
According to reference three, this is
MG = 2.587 1041 kg
(8)
With this value the constant is
h = 114.522 · km/ sec Mpc (9)
However, according to Sandage the value of H is
H = 75.0 · km/sec Mpc (10)
This implies that the actual value of the mass of the Galaxy is
MG = 6.032 * 1041 (11)
or 2.33 times that estimated.
It seems to me that Hubble‘s constant has been more accurately determined than the mass
of the Galaxy. Thus the analysis leads to the conclusion that the mass of our Galaxy is
greater than supposed—probably because of a white dwarf galactic core that still remains
difficult to observe.
References
1. Dewey B. Larson The Structure of the Physical Universe, First Edition (Portland,
Oregon: North Pacific Publishers, 1959.
2. Richard Bronson , Modern Introductor Differential Equations (New York:
McGraw-Hill Book Company, 1973.
3. Martin Harwit, Astrophysical Concepts (New York: John Wiley & Sons, 1973).
GLOBULAR CLUSTER MECHANICS IN THE
RECIPROCAL SYSTEM
This paper discusses the forces on stars in a globular cluster. Consider Figure 1; the
symbols are defined as follows:
Mg = mass of the stars of a globular ciuster Internal to g that of a particular star
m = mass of that particular star
mp = mass of the nearest neigboring stars
xg = distance of the star from the mass center of the globular g cluster
xp = distance of the star from the mass center of the nearest neighboring stars
xpg = distance of the mass center of the nearest neighboring stars from the mass center of
the globular cluster
xpo = equilibrium distance of the star from the mass center of the nearest neigboring stars
x = distance of the star from the mass center of the nearest neighboring stars, relative to
the equilibrlum distance
Recall that in the Reciprocal System two forces are acting on the star:
1. Gravitation of the star by the cluster as a whole—this produces an inward motion.
2. Progression of the star away from its nearest neighbors—this produces an
outward motion.
My goal in this paper is to derive the expression for the net force acting on the star, to
find the equilibrium position (xpo) of the star, and to determine whether or not this
position is stable.
Nehru‘s recent paper [1] provides the starting point. Some additional symbols are needed:
dog = gravitational limit of the globular cluster
dop = gravitational limit of nearest neigboring star:
yg = non-dimenslonal distance of the star from the mass center of the globular cluster
yp = non-dimensional distance of the star from the mass center of the nearest neighboring
stars
vog = ―zero-point speed‖ of the star relatlve to the globular cluster
vop = ―zero-point speed‖ of the star relative to the nearest neighboring stars
vng = net inward gravitational speed of the star
vnp - net outward progression speed of the star
vn = net speed of the star
G = ―universal‖ gravitational constant
Mo = mass of the sun
ag = acceleration from gravitation of the globular ctuster
ap = acceleration from progression away from the nearest neigbors
an = net acceleration of the star
In this notation,
dog = 3.77 * (Mg / Mo)½
(ly) (1)
yg = xg /dog = (x + xpo + xpg ) / dog (2)
vog = (2 * G * Mg / dog )½
(3)
vng = vog * (1 / yg ½ - yg½) (Inward) (4)
dop = 3.77*(mp / Mo)½ (ly) (5)
yp = xp / dop = (x + xpo) / dop (6)
vop = (2 * G * mp /dop )½ (7)
vnp = (½) * vop * (yp - 1/yp ) (outward) (8)
vn = vnp - vng (9)
Differentiating the velocity expressions with respect to time gives the accelerations:
ag = G * Mg * (1/xg² - 1/dog ²) (inward) (10)
ap = G * mp * (½) * (xp / dop³ - dop / xp³) (outward) (11)
an = ap - ag (12)
At equilibrium,
an = 0 (13)
Let
h = mp / (2 * dop³) (14)
i = Mg * (1/dog² - 1/xg²) (15)
j = (1/2) * mp * dop (16)
Then, in terms of xpo, at equilibrium,
h * xpo 4 + i * xpo ³ - 1 = 0 (17)
a quartic equation.
The appendix gives a simple computer program written in BASICA to solve equation 17
numerically. (An attempt to solve the equation analytically using the MU MATH AI
program failed). A sample run with Mg = 200*Mo , mp = 2*Mo , xg = 40 ly, dog = 53.32
ly, and dop = 5.33 ly produced xpo = 9.29 ly.. Another sample run with Mg = 30000*Mo ,
mp = 200*Mo ,xg = 400 ly, dog = 652.98 ly, and dop = 33.32 ly produced xpo = 178.94 ly.
Input parameters that are physically impossible produce negative distances.
Now let‘s turn to the question of the stability of this positlon, xpo The net force acting on
the star in terms of the distance from equilibrium, x, is
F = m * G * ((1/2) * mp * ((xpo + x)/dop³ - dop / (xpo + x)³)
- Mg * (1 / (xpo + x + xpg )² - 1 / dog²)) (18)
Differentiating F with respect to x gives
dF / dx = m *G * ((1/2) * mp * (1/dop³ + 3 * dop / (xpo + x)4)
+ 2 * Mg / (xpo + x + xpg )³) (19)
If x is positive, dF / dx is positive and hence F increases with x.
If x is negative, dF / dx is still positive. Thus
- dF / dx < 0 (20)
This is the definition of instability. Hence, xpo is a point of unstable equilibrium. But
there is one saving grace: the forces near this point are quite small, so sudden changes in
position are precluded.
Globular clusters continually grow by accretion until eventually being absorbed into
galaxies. The stars in the clusters must keep changing their temporary equilibrium
positions.
Reference
1. K. V. K. Nehru, ―The Gravitational Limit and the Hubble‘s Law,‖ presented at the
1986 Convention of ISUS.
STELLAR ENERGY GENERATION
IN THE RECIPROCAL SYSTEM
The theory of stellar energy generation in the Reciprocal System is stated qualitatively in
various works by Mr. Larson, such as Quasars and Pulsars. For the benefit of new
readers of Reciprocity, I quote Mr. Larson in full:
Inasmuch as a charge is a modification of the basic rotation, the number of charges that
an atom can acquire, the degree of ionization, as it is called, is limited by the number of
rotational units of the appropriate space-time direction that exist in the atomic structure;
the number of units available for modification. Negative ionization is confined to low
levels, as the effective negative rotation is never more than a few units. The limit of
positive ionization is the atomic number, which represents the net total nurnber of units
of rotational time displacement in the atom.
Electric ionization may be produced by any one of a number of agencies, inasmuch as the
requirement for this process is essentially nothing more than the availability of sufficient
energy under appropriate conditions. In the universe at large the predominant process is
thermal ionization. Thermal or heat energy is linear motion of material particles, and it is
therefore space displacement. In the ionization process this linear space displacement is
transformed into rotational space displacement: positive charge. As the temperature
increases, more and more space displacement becomes available for ionization, and the
degree of ionization rises until the atom finally reaches the point where it is fully ionized;
that is, each of its units of time displacement has acquired a positive charge.
If the temperature of the fully ionized atom continues to rise, a destructive limit is
ultimately reached at the point where the total space displacement, the sum of the
ionization and the thermal energy, is equal to the time displacement of one of the
magnetic rotational units. Here the oppositely directed rotational displacements neutralize
each other, and both revert to the linear basis, destroying this portion of the atomic
structure. Since the maximum ionization increases with the atomic number, the amount of
thermal energy required to bring the total space displacement of a fully ionized atom up
to the destructive limit is less for heavier atoms, and the effect is to establish a
temperature limit for each element that is inversely related to atomic number. As the
temperature of an aggregate rises the heaviest elements are therefore the first to
disintegrate.1
To sum up, when the destructive thermat limit is reached, the following word equation
holds true:
ionization energy of atom + thermal energy of atom = energy
equivalent of one unit of magnetic time displacement (la)
Let EI be the ionization energy, ET be the thermal energy, and EM be the oppositely
directed magnetic rotational energy. Then in symbols,
EI + ET = EM (lb)
Each of the terms in the equation will now be discussed.
Equivalent energy of one unit of magnetic time displacement
Before we can find the energy equivalent of one unit of magnetic time displacement, we
must find the mass equivalent. According to deductions previously made from the
postulates of the Reciprocal System the electric equivalent of a magnetic displacement n
is 2n²; this does not refer to the total from zero to n—it is the equivalent of the nth term
alone. Each electrical unit is equal to two atomic mass units, and each atomic mass unit is
equivalent to 931.48 MeV. For n equal to 3 and 4, the following table results:
n 2n² amu EM MeV
3 18 36 33533.28
4 32 64 59614.72
Thus, the third magnetic time displacement is equivatent to 33533.28 MeV, and the
fourth unit to 59614.72 MeV.
Ionization energy
At the present stage of development of the Reciprocal System we do not have a
theoretical equation giving the energy needed to completely ionize at atom—but then
neither does quantum mechanics. An empirical equation will have to do for now.
Reference three has the most comprehensive table of ionization values available, giving
the complete ionization energy for the first twenty elements. From the values, I have
derived the following empirical equation:
EI - 13.595 + 52.148(Z-1)² 10-6 MeV (2)
where Z is the atomic number. Of course, other equations are possible.4 Extrapolating any
empirical equation to high values of Z is risky, but this will have to do. For thorium, at
no. 90, eq. 2 gives
EITh = .413 MeV
Thermal energy
Let k be Boltzmann‘s constant in MeV/°K and T be the temperature of an atom in °K
Then the standard equation for the thermal energy (based on the ideal gas taw) is
ET = 3/2 kt (mv²/2) = 1.292 x 10-10 (3)
Calculation of critical temperature and velocity
From eq. lb,
ET = EM - EI
Then,
TCRIT = EM - EI / 1.292 x 10-10 °K (4)
For thorium, EM is 59614.72 MeV and EI is .413 MeV, so
TCRITTh = 4.614110449 x 1014 °K
This is fantastically high from our view as spectators on the earth, but in terms of natural
units, the temperature is ―only‖ 127.44.
With k in J/°K, equation 3 can be solved for the velocity at the critical temperature.
vCRIT = (3kTCRIT/m)½ (5)
For thorium, this amounts to
vCRITTh = 2.5289 x 108 m/sec
This is 84% of the speed of light:
For iron, the critical temperature is
TCRITFe = 4.61413989 x 1014 o
K
and the critical velocity is
vCRITFe = 2.7165 x 108 m/sec
This is 91% of the speed of light! No wonder atoms are accelerated to velocities above
the speed of light during a supernova explosion:
Most likely the motion of the atoms in the core of a star is circular. The greater the
temperature, the higher the velocity—thus as theoretically expected O and B type stars
have much greater rotational velocities than G and K type stars.
Rate of energy generation
Since both the unit of magnetic time displacement and the opposing space displacement
revert to linear motion, the total energy radiated per critical atom is
ERAD = 2xEM =119229.44 MeV (6)
for n = 4.
The rate of energy generation depends on the number of atoms at the critical temperature,
NCRIT. This in turn depends on the total mass of the star M, the average mass per critical
atom, m, and on the fraction FCRIT of the mass M that is critical. Thus
NCRIT = FCRITxM /mCRIT (7)
Let PST be the total power output of a star. Then assuming no accretion whatever, the
litetime of a star can be calculated as follows:
LST = NCRITERAD/ PST sec (8)
For the sun,
M = 2 x 1030 kg
PST = 2.43x1039 Mev/ sec
Taking thorium as representative of the critical elements,
m = 2.988 x 10-25 kg/atom
Assuming various values of FCRIT we can calculate the lifetime of a star with no
accretion. The following table results.
FCRIT LST (no accretion) years
.01 1.0389 x 1011
.001 1.0389 x 1010
.0001 1.0389 x 1009
.00001 1.0389 x 1008
.000001 1.0389 x 1007
According to the Reciprocal System, net accretion does occur over the life of a star, but
there may be periods where there is a net loss. Since such a period may last as long as a
billion years, I believe we are on good ground assuming that FCRIT is equal to .0001. At
present we have no way of deducing theoretically the fraction of the mass of a star that is
critical. Certainly, observation is no help; observation can only indicate the composition
of the stellar atmosphere, not that of the central core.
Rate of accretion
The sun appears to be one-third along its way on the Herzsprung-Russell diagram. Since
the sun has been estimated to be in existence for about 5 billion years, we can roughly
assume that the average lifetime of a star is 15 billion years. According to the theory, a
star slowly increases in temperature until the critical temperature of the iron group of
elements is reached, at which point the life of the star is terminated in a supernova1
explosion. From the equations presented in this paper, the critical temperature of iron is
3,091,400,000 oK above that of uranium. Thus the rate of change of temperature with
time can be roughly expressed as follows:
dT/dt / T/ t = 3,091,400,000/15 x 109 = .206 °K (10)
(Even if L were only 7.5x109 years, the increase in T per year would be less than .5 °K).
Thus stars are for most of their lives very stable energy generators. From this we can
conclude that the rate of accretion is just slightly greater than the rate of mass lost
through burning. For calculating the rate of accretion we can assume that for the short
term they are identical.
Let RACC be the rate of accretion in kg/sec. Then
RACC = PSTmCRIT / ERADFCRIT (10)
Using previous values of PST, ERAD, mCRIT and FCRIT equal to .0001,
RACC = 6.099 x 1013 kg/ sec = 1.925 x 1021 kq/ yr
This amounts to .000000096% of the mass of the star per year. It would take over 3108
years for the accretion to amount to the mass of the earth:
Clearly we cannot observe this small rate of accretion. Observation cannot tell us whether
the mass of the sun is remaining constant or slowly increasing, as we believe, or whether
the mass of the sun is slowly decreasing, as present theory suggests.
Conclusion
The current theory of stellar energy generation has been criticized elsewhere, and a
summary of that criticism is presented in reference five. The basic differences between
the new theory and the current one are as follows:
1. In the new theory, energy is generated by disintegration of heavy elements; in the
current theory, energy is generated by fusion of light etements.
2. In the new theory the temperature of the stellar core is of the order of 4.6 x 1014
°K in current theory, it is 3x107 °K for the first phase, and 109 for later phases.
3. In the new theory, ordinary stellar energy generating processes do not give rise to
neutrino emission, but in current theory they do. So far, no neutrinos have been
found to emanate from the sun.
4. In the new theory, one method for energy generation serves all types of stars;
current theory proposes that various stars have different energy schemes: proton-
proton reaction: the CNO bi-cycle; helium burning; (y, ) reaction of C12 , O16 ,
Ne20 nuclei; e-process; r-process.
Thus, though observation (other than neutrino counts) cannot at present decide in favor of
one theory over the other, Occam‘s Razer can: the new theory wins hands down.
References
1. Dewey B. Larson, Quasars and Pulsars (Portland, Oregon: North Pacific
Publishers, 1971), p. 60.
2. Dewey B. Larson, New Light on Space and Time (Portland, Oregon: North Pacific
Publishers 1965) p. 234
3. Von W. Finkelnburg and W. Humbach, ―Ionisierungsenergien von Atomen and
Atomionen,‖ Die Naturwissenschaften, Heft 2, Jg. 42, 1955, pp. 35-37.
4. For instance a Polynomial equation in Z has been worked out by computer by
Frank V. Meyer: EI = 78.6411 - 72.8213 x Z + 33.675 x Z² + .801221 x Z³ .
5. Ronald W. Satz, The Unmysterious Universe (Troy, NY: Troy Printers, 1971),
p. 11.
6. R. Davis, Jr., D. S. Harmer, K. C. Hoffman, ―Search for Neutrinos from the Sun,‖
Phys. Rev. Let., 20, 1205 (1968).
Author‘s Note: This paper is not meant to be the last word on subject of stellar energy.
Rather it is meant only to be the second word. Constructive criticism would be welcome.
THE GRAVITATIONAL ATTRACTION OF
THE GALAXY
In a previous paper1 I worked out the general form of Newton‘s Law of Gravitation and
applied it to the special case of a planet orbiting the sun. In this case Newton‘s Law was
modified by the factor
1/(1 - v²/c²) ½
For the case of an object moving directly toward another object rather than orbiting, the
genetal equation reduces to Newton‘s Law multiplied by the factor
(1 - v²/c²)
This is exactly of the same form as Lorentz‘s modification of Coulomb‘s Law.
Before applying the new factor, it is important to realize that the galaxy cannot be
represented as a pofnt mass; rather it should be represented as a flat disk. The Newtonian
actraetion of a flat disk for a point mass has been worked out before², but will be repeated
here.
In Figure I let the radius of the disk be r and let its surface density be . I aim to find the
attraction of the disk for a polnt mass located at P on the perpendicular line passing
through the center of the disk. Let 0 be the origin of a system of polar ccordinates p and
, and let z be the distance along the line to the attracted location P.
Since pdp is the area of an element in polar coordinates, the mass of such an element is
dm = pdpd (1)
The dietance of the element dm from P is
R = (p² + z²)½ (2)
and the attraction of the mass dm for the mass at P is
- G dm = G pdpd
— ———— (3)
R² p² + z²
and the component of the attraction along the exis is
- G pdpd . z G pdpd
———— – = - ———— (4)
p² + z² R (p² + z²)3/2
The total intensity of attraction of the disk for the point P mass is
n
2 pdpd
I = G z ————–
o o (p² + z²)3/2
r pdp
I = 2 G z ————–
o (p² + z²)3/2
[
z - z
]
I = 2 G ———— —— (5)
(z² + r²)½
(z²)½
Assuming z positive,
[
z
]
I = 2 G ———— - 1 (5¹)
(z² + r²)½
Now with the modifying factor included, the acceleration of the point masa toward the
disk is
dv
[
z
]
[
1 - v²
]
— = 2 G ———— - 1 —
dt (z² + r²)½
c²
dv 2 G
[
z
]
[ c² - v² ]
— = —— ———— - 1 (6)
dt c² (z² + r²)½
dv = dv dz = v dv
— — — — (7)
dt dz dt dz
The crucial deduccion in Larson‘s gravitational theory is that the gravitational force of
any mass extends outwatd only a finite amount the gravitational force does not extend out
to ―infinity‖, as commonly assumed. At the gravitational limit of the galaxy, which will
be denoted by do, the attracted velocity of a mass is zero. This velocity becomes larger to
the degree that the mass is loeated closer to the galaxy. Let the velocity be v at distance z.
Then, separating the variables in equation 6 and integrating between the limits, the result
is
v vdv
z
2 G zdz
z 2 G dz
————— = ———– ———— ———— (8)
o c² - v² do c² (z² + r²)½
do c²
The outcome of this result is that
v
[
- 4 G [(z² + r²)
½ - (do² + r²)
½ + do - z]
½
]
– = 1 - e ——–– ½ (9)
c c²
For our galaxy the constants in the equation are as follows:
G = 6.67 x 10-11 N - m²/kg²
c² = (3 x 108)² m²/s²
= .2975 kg/m²
r = 4.626 x 1020 m
do = 2.177 x 1022 m
With these values equation 9 becomes
v
[
- 2.771 x 10-27[(z² + 2,140 x 10 41)½ - 9.087 x 1013 - z]
]
–
= 1 - e ½ (10)
c
Speed in km/sec vs. distance ia kiloparsecs is plotted in the graph (Fig. 2). Great caution
must be used in applying equation 10 to real masses:
1. A globular cluster or a small galaxy associated with the Milky Way galaxy is not
really a point mass; in fact, observation shows that the near side stars of such
objects are attracted at the expense of the farside stars.
2. Globular clusters are not falling directly toward the galactic center; rather they are
orbiting.
3. Small local galaxies are at a distance close to the gravitational limit of the galaxy
— their veloclties aze difficult to measure and compare with theory.
Even so, the calculated velocities, neverthaless, agree in a very general vay with those
observed for the local group of objects.
References:
1. Satz, R.W. REClPROCITY Vol. IV, No. 2, p. 25, July 1974,
2. MacMillan, W., The Theory of the Potential (New York: McGraw Hill Book
Company, 1930),pp. 15-16.
DISCUSSION OF LARSON‘S
GRAVITATIONAL EQUATION
As brought out at the recent convention, some confusion has arisen over Larson‘s
gravitational equation, eq. (2) of the original edition of the Structure of the Physical
Universe:
F = mm‗/s² (1)
The correct expanded version of this equation is
(m × 3.7115 x 10-32) × (m‘ × 3.7115 x 10-32)
F = ————————————————— (2)
1521 × 10-15 × (s/1)²
where 3.7115 x 10-32 sec³/cm³ is the value of the natural unit of mass and m and m‘ are
simply numbers. The number .1521 x 10-15 sec is the natural unit of time. From equation
(2), the natural value of the gravitational constant can be determined:
Gn.u.= (3.7115 × 10-32)²/.1521 × 10-15 = 9.0567 x 10-48 (3)
Thus equation (1) might better be written as
Fn.u = 9.0567 × 10-48 mm‘/s² (4)
where all values are in natural units. The expression for G in equation (3) can be
converted to conventional units. First, the cgs system:
Fn.u.sn.u.² dynes
Gcgs = 9.0567 ×10-48 ———— × 109.7 ——–
mn.u² Fn.u
(.456 x 10-5 cm mn.u²
× —————— × ——————
sn.u² (.5565 x10-24 g
= 6.67 x 10-8 dynes cm²/g² (5)
The MKS system
Fn.usn.u² N
GMKS = 9.0567 ×10-48 ———– 109.7 10-5 ——
mn.u² Fn.u
(.456 × 10-7 m)² mn.u²
× —————— × ———————
sn.u² (.5565 × 10-27 kg)²
= 6.67 × 10-11 N-m²/kg² (6)
Both check. The importance of this cannot be overestimated. These equations completely
confirm Larson‘s identification of all the fundamental units.
Note that if in equation (2), the value 3.7115 × 10-26 sec³/m³
is used then the correct time value must be .1521 × 10-3 sec for the denominator (or 1012
units of time).
THE INTERACTION OF ALPHA PARTICLES
AND GOLDS ATOMS
A New Explanation of Rutherford Scattering
Introduction
Nearly all present-day physicists are convinced of the truth of the assertion in the
following quotation from Weidner and Sells‘ Elementary Modern Physics(1)
:
It was by the alphaparticle scattering experiments, suggested by Rutherford, that the
existence of atomic nuclei was established.
However, when we study the literature of Rutherford‘s era, we find that he and his
associates, Geiger and Marsden, did not in fact discover the atomic nucleus. Geiger and
Marsden‘s paper, ‖The Laws of Deflexion of a Particles through Large Angles, ‖(2)
does
present strong experimental evidence of a central repulsive force originating from atoms,
but the paper does not prove that this force is electrical in nature. What their experiments
did prove is that the number of particles scattered through an angle q is proportional to
1/sin4 (q/2) and to the inverse of the square of the kinetic energy of the particles, 1/Ek
2.
Of course, the experiments did disprove the Thomson ‖plum pudding ‖ atom model,
which did not predict a strong central repulsive force; but it is one thing to disprove a
theory; it is quite another to prove one. If the Rutherford model were the only alternative
left, we might have to conclude that it is correctÑbut there are always other alternatives.
This paper will present one such alternative: the Reciprocal System of physical theory. A
new scattering equation will be derived and compared with the experimental facts as
found in an uptodate version of the original experiment, that conducted by Prof. Adrian
C. Melissinos and his students.(3)
The originator of the Reciprocal System is Dewey B.
Larson; for full comprehension of this paper, the reader should first study Larson‘s
books.(4,5,6)
I. Theory
A. The Repulsion Force: F
1. The Reciprocal System
In the Reciprocal System, nonionized and nonmagnetized matter is subject to only two
primary forces: the spacetime progression and gravitation. In the timespace region, the
progression is outward and gravitation inward, whereas in the time region (inside unit
space) the progression is inward and gravitation outward -- a repulsion. Right at the
boundary the progression is zero, but the net gravitation is not zero. Compared with the
repulsive gravitational force, the attractive gravitational force is negligible. Thus at the
boundary only the repulsive gravitational force is effective. Now consider what happens
when an atom A, which is moving towards an atom B, reaches this boundary. According
to Larson,(6)
When atom A reaches point X, one unit of space distant from B, it cannot move any
closer to B in space. It is, however, free to change its position in time relative to the time
location occupied by atom B. The reciprocal relation between space and time makes an
increase in time separation equivalent to a decrease in space separation, and while atom
A cannot move closer to atom B in space, it can move to the equivalent of a spatial
position that is closer to B by moving outward in coordinate time. . . . No matter what the
spatial direction of the motion of the atom may have been before unit distance was
reached, the temporal direction of the motion after it makes the transition to motion in
time is determined purely by chance.
A previous paper of mine, ‖Time Region Particle Dynamics, ‖(7)
dealt with the situation
in which atom A (say an alpha particle) is assumed to continue to move directly toward
atom B (say a gold atom) in the time region. This paper will consider the general case in
which no assumption is made as to the actual motion that takes place in the equivalent
space of the time region. All that will be considered here is the equivalent motion that
takes place at the boundary, i.e., in actual space. Here, with the atoms separated by so, the
repulsive gravitational force F is (from Ref. 4)
F = KG/s4 = KG/so
4 (if s = so) (1)
The repulsion coefficient KG is expressed by
KG = [Fpso4/(156.44)
4] * [ln
4 teff/ln
²t‘eff] (2)
where Fp is the natural unit of force and the number 156.44 is the interregional ratio. The
dimensionless variables teff and t‘eff are material constants determined by the
characteristics of the interacting particles, and will be discussed further later.
2. Conventional Theory
In present theory, the alpha particles somehow avoid interacting with the cloud of
electrons supposedly surrounding each gold nucleus. The only force involved comes from
the presumed nucleus. At low energies this is a Coulombic force, given by
F = zZe²/4 os
² (3)
where z is the atomic number of the alpha particule (helium), Z is the atomic number of
gold, e is the value of electric charge, eo is the permittivity constant of free space, and s is
the separation distance. This is an inverse square force, rather than an inverse quartic
force as in the Reciprocal System. Why this Coulombic force should act between
particles but not within nuclei is a question completely unanswered by current theory.
B. The Impact Parameter: b
MOTION IN ACTUAL SPACE: so remains constant, but angle f changes from o to -
where is the deflection angle observed in the time-space region.
The figure shows a typical collision process. The impact parameter is the distance that the
alpha particle would have passed the gold atom if there had been no force between them.
1. Reciprocal System
Let m be the mass of the alpha particle and vo be its initial velocity. Referring to the
figure, we have
impulse = (mv)y
Fdt = mvo sin( )
(KG/so4) sin(f)dt = mvosin( )
(4)
The alpha particle passes from the timespace region, through the time region, and back
into the timespace region. For the general case, we cannot write an equation for the actual
motion in the equivalent space of the time region, but we can write an equation for the
equivalent motion in the actual space of the timespace region. Throughout this motion,
neither the angular momentum, nor the actual spatial separation, changes. But the angle
does change. Therefore,
mvob = mso²(d /dt) (5)
Separating dt from d in eq. (5) and substituting in eq. (4), we have
(KG/so4) * (so
²/vob) sin( ) d( ) = mvo sin( )
(6a)
or
(KG/so²mvo
²b) sin( ) d( ) = sin( )
Since the kinetic energy, Ek, is (1
/2) mvo², eq. (6a) can be rewritten as
[KG/(so² * 2 * Ek * b)] sin( ) d( ) = sin( ) (6b)
Before the collision, = 0 and after the collision, = , so these are the limits on the
integral.
sin( ) df = -cos( ) o o
= cos ( ) + cos(o)
= cos( ) + 1
Thus, [KG/(so² * 2 * Ek * b)] * (cos( ) + 1) = sin( ) (7a)
But sin( )/(cos( ) + 1) = tan ( /2)
so KG/(so2 * 2 * Ek * b) = tan( /2) (7b)
Solving for b we finally obtain
b = [KG/(2 * so² * Ek)] * cot( /2) (7c)
2. Conventional Theory
Refs. 1, 3, and 8 all have derivations for b in current theory. The result is
b = [zZe²/(8 oEk)] * cot ( /2) (8)
C. Target Cross-Section:
The target Cross — Section is defined as = b².
1. The Reciprocal System
From eq. (7c), using the above, we obtain
= [ KG²/(4 * so4 * Ek
²)] * cot
² ( /2) (9)
2. Conventional Theory
With b from eq. (8),
= [z²Z
²e
4/(64 o
²Ek
²)] * cot
²( /2) (10)
D. Differential Cross-Section: d
The differential cross-section is
d = 2 bdb
1. Reciprocal System
Here, db = [-KG/(so² * 4 * Ek)] * [d /sin
²( /2)] (11)
Thus d = 2 * [KG/(2 * so² * Ek)] * [cos( /2)/sin( /2)]
* [-KG/(4 * so² * Ek)] * [d /sin
²( /2)]
= [-2 KG² cos( /2) d ] / [8 * so
4 * Ek
² * sin
³( /2)]
= [ KG² cos( /2) d ]/[4so
4 * Ek
² * sin
³( /2)] (12)
(with the minus sign dropped).
The angles and + d define two cones with the horizontal line through the gold atoms
as their axis. The differential solid angle d between the two cones is
d = 2 sin( ) d
Since the cosine term in eq. (12) can be expressed as
cos( /2) = sin( )/[2 * sin( /2)]
eq. (12) becomes
d = [ KG² (sin ( )) d ]/[8so
4 * Ek
² * sin
4( /2)]
= [KG² d ]/[16 * so
4 * Ek
² * sin
4( /2)]
or d /d = KG²/[16 * so
4 * Ek
² * sin
4( /2)] (13)
The units of d /d are meter squared per steradian, m²/sr.
2. Conventional Theory
Similarly, for conventional theory,
d /d = z²Z
²e
4/[256 *
² * o
2 * Ek
² * sin
4( /2)] (14)
E. The Scattering Constant: Ks
It is immediately seen from eqs. (13) and (14) that for both
theories,
(d /d ) * sin4( /2) = Ks
a constant.
1. Reciprocal System
Here,
K = KG²/(16 * so
4 * Ek
²) (15)
2. Conventional Theory
Here,
Ks = z²Z
²e
4/(256 *
²* o
² * Ek
²) (16)
F. The Number of Particles Scattered per Minute at the angle : Is
So far we have looked at the situation involving only one alpha particle and only one gold
atom. For the situation in which a beam of alpha particles strikes a gold foil, we would
like to compute the number of particles scattered through a certain angle q. Let
Io = number of incident alpha particles/minute
d = detector solid angle
N = area density of scatterers (number of gold atoms/m2)
The value of N is determined from this equation:
N = * * [No/A) * (104 cm
²/m
²)] (17)
where
d = thickness of foil (cm)
r = density of scatterer (g/cm3)
No = 6.02 * 1023
, Avogadro‘s number
A = atomic weight of scatterer
Then the number per minute, Is, of alpha particles scattered into the detector at the angle
is
Is = Io * N (d /d ) * d
= Io * * * (No/A) * 104 * (d /d ) * d
1. Reciprocal System
Here,
Is = Io * * * (No/A) * 104 * KG
²* d /[16 * so
4 * Ek
2 * sin
4( /2)] (18)
2. Conventional Theory
Here,
Is = Io * * * No/A * 104
* (z²Z
²e
4 * d )/[256 *
² * o² * Ek
² * sin
4( /2)] (19)
Note: Both equations are based on the assumption that E is sufficiently low such that
‖relativistic ‖ effects can be neglected and such that the gold atoms remain stationary
during the interaction -- this would not be the case at very high alpha particle energies.
II. Experiment
A. The Experimental Set-up
Professor Adrian C. Melissinos carried out a modern version of the original
GeigerMarsden experiment and described his findings in Ref. 3. In this experiment
Io = 1.1 * 105 incident alpha particles/minute
d = da/L2 = .8786/(6.67)
² = 0.197 sr
[da is differential area of detector and L is distance of detector from foil]
= 19.3 g/cm3 (gold)
= .00025 cm
A = 197 (gold)
Thus,
N = * * (No/A) 104
= (.00025)(19.3)(6.02 * 1023
/197)*104
N = 1.4744 * 1023
gold atoms/m2
So the equation for Is is
Is = Io * N * (d /d ) * d
= 1.1 * 105 * 1.4744 * 10
23 * (d /d ) * .0197
Is = 3.1950 * 1026
* d /d
Is is measured, and then (d /d )is computed. For each angle q the product (d /d ) *
sin4( /2) can be found and the results plotted. From leastsquares error analysis, the best
fit experimental value of Ks can be obtained.
Melissinos states that the above value of Io, and hence also Is, is subject to at least a ± 20
percent error in view of the approximations used and the nonuniformities in beam density
and direction. One other uncertainty is the energy of the alpha particles. The incoming
energy is 5.2 MeV, but since the particles lose a considerable amount of energy in
traversing the target, Melissinos believes that it is more appropriate to use a mean value
of Ek for the calculations. He calculates the mean value to be
Ek = 4.39 MeV = 7.03 * 10-13
J
B. Calculation of Ks for the Experimental Set-up
1. Reciprocal System
Here,
Ks = KG²/(16 * so
4 * Ek
²)
KG = [Fp * so4/(156.44)
4] * [ln
4 teff/ln
² t‘eff]
Fp = 3.27223 * 10-3
Newtons
so = 4.558816 * 10-8
meters
ln² t‘eff = 1, since helium has no electric dispacement -- it is ‖inert. ‖
The difficult part in calculating Keff for gold helium. The (tentative) method used here
will be different from that used in my previous paper, Ref. 7. Gold has 3 active rotational
dimensions with t = 4.5 in all (see Ref. 4 for more details). Helium has only 1 active
dimension, with t = 3. The other two dimensions have t = 1. In the first dimension the
mean value for gold and helium is
t = (4.5 * 3)1/2
= 3.67
In the other two dimensions, the full rotational force of the gold atom is present, so
instead of (4.5 * 1)1/2
= 2.1, we simply have t = 4.5. The mean over all three dimensions
is
teff = (3.67 * 4.5 * 4.5)1/3
= 4.2
Thus (tentatively),
KG = [3.27223 * 103 * (4.558816 * 10
8)4/(156.44)
4][ln
4(4.2)]
= 1.00 * 1040
nm4
The (tentative) value of Ks is then Ks = (1.00 * 1040)2/[16 * (4.558816 * 10
8)4 * (7.03 *
1013)2] Ks = (2.93 * 10
28 (m
2/sr)
2. Conventional Theory
Here,
Ks = z²Z²e
4/(256 *
² * o
² * Ek
²)
z = 2
Z = 79 e = 1.602 * 1019 coulombs
eo = 8.85 * 1012 coulombs/N-m²
Ek = 7.03 * 1013 J
Thus,
Ks = [(2²)(79
²)(1.602 * 10
19)4] / [256 *
2 * (8.85 * 10
-12)² * (7.03 * 10
-13)²] Ks =
1.68 * 10-28
(m²/sr)
3. Experiment
The best fit experimental value, according to Melissinos, is
Ks = 2.70 * 1028 (m²/sr)
Thus the theoretical and experimental results can be summarized as follows:
Ks(m²/sr)
Experiment: 2.70*10-28
Reciprocal Systems 2.93*10-28 (tentatively)
Conventional Theory 1.68*10-28
Appraisal
It is well known that Rutherford‘s theory of scattering fails at high energy. On the basis
of Melissinos‘ experiment, we must also reject this theory at low energy. But, given the
present climate of thought, Melissinos himself could not come to this conclusion. He says
(Ref. 3, p.250):
The difference between the observed and theoretical constants, while at first sight large,
can be traced to the limited sensitivity of the apparatus and mainly to
(a) Uncertainty in incoming flux
(b) Uncertainty in foil thickness [and, to a lesser extent, to]
(c) Extended size of the beam and lack of parallelism
(d) Extended angular size of the detector
(e) Plural scattering in the foil (for the data at small angles)
(f) Background (for the data at large angles)
However, given the close result of the Reciprocal System, it now appears that Melissinos
is too modest. Even with some uncertainty in the theoretical value of KG and the
experimental values of Io and Ek, it appears that the Reciprocal System is consistent with
the observed result, whereas conventional theory is not.
Can an appeal to more sophisticated mathematics rescue the current theory? It cannot.
From Ref. 1 (p. 223) we have this statement:
It is a remarkable fact that a thoroughgoing wavemechanical treatment of scattering by an
inversesquare force yields precisely the same result as that yielded by the strictly classical
particle analysis discussed here.
Thus, despite the thousands of books, the thousands of papers, and the thousands of
lectures on the nuclear theory of the atom, the physicists are going to have to discard their
cherished concept.
The scattering equation of the Reciprocal System will next have to be applied to other
pairs of incident and target particles and to other energy levels.
References
1. R. Weidner and R. Sells, Elementary Modern Physics (Boston: Allyn & Bacon,
1968), pp. 222-223.
2. H. Geiger and E. Marsden, ―The Laws of Deflexion of a Particles through Large
Angles, ‖ Philosophical Magazine, 25 (1913), 604-628.
3. A. Melissinos, Experiments in Modern Physics (New York: Academic Press,
1966), pp. 231-252.
4. D. Larson, The Structure of the Physical Universe (Portland, Ore.: North Pacific
Publishers, 1959). (The revised edition is being published in several volumes, the
first of which is Nothing But Motion, 1979).
5. D. Larson, The Case Against the Nuclear Atom (Portland, Ore.:North Pacific
Publishers, 1963).
6. D. Larson, New Light on Space and Time (Portland, Ore.: North Pacific
Publishers, 1965), pp. 115-116.
7. R. Satz, ―Time Region Particle Dynamics, ‖ Reciprocity IX.2 (Summer, 1979).
8. H. Enge, M. Wehr, J. Richards, Introduction to Atomic Physics (Reading, Mass.:
Addison Wesley Publishing Co., 1972).
A PROPOSAL FOR A CRUCIAL EXPERIMENT
Rutherford‘s nuclear theory of the atom has held sway in the scientific community for 70
years. I now propose a test which may disprove it.
In the original scattering experiment, charged helium atoms (alpha particles) were
beamed at a gold foil; the resultant scattering was claimed to be due to Coulombic
repulsion by charged nuclei. Now suppose that that non-charged helium atoms are
beamed at a gold foil. The Reciprocal System of Theory predicts the same scattering in
this case. The Rutherford theory predicts no scattering.
Of course, experimenting with non-charged helium atoms is more difficult than with
charged ones. The scattering apparatus will have to incorporate a means to inject
electrons to neutralize the alpha particles and a different material to detect the particles
after scattering.
Numerous physics laboratories in the world have the capability to carry out this
experiment. Which will be the first?
See
THE INTERNATIONAL SOCIETY OF UNIFIED SCIENCE
LINKS
The Collected Works of Dewey B. Larson
Reciprocal System Dynamics by Ronald W. Satz
Glimpses of a New Paradigm by K.V.K. Nehru
Toward a Unified Cosmological Physics by Arnold Studtmann
Kinetons and Quanta by Jan Sammer
Special Thanks to the folks who set up the web site at;
http://www.reciprocalsystem.com/rs/links.htm
Recommended