Natural Units and Atomic
Units
by Charles William Johnson
Abstract:
The author examines the mediating factors implied in the relationship
of natural units
to atomic units as listed in the CODATA recommended fundamental
physical constants.
At some point, the theme of natural
units and Planck units must be analyzed in depth from a theoretical
perspective. Before that may be accomplished, however, it is necessary
to consider the fundamental physical constants that are expressed
in the CODATA as natural units and atomic units.
In certain cases of physical constants,
it would appear that there exists a conversion factor implied between
the stated natural units and the atomic units as presented in Table
VI, "The values in SI units of some nonSI units based on the 2006
CODATA adjustment of the values of the constants". [Published in
"CODATA Recommended Values of the Fundamental Physical Constants:
2006", by Peter J. Mohr; Barry N. Taylor; and, David B. Newell,
published in the CRC, 2009  2010 Handbook of Chemistry and Physics,
Boca Raton, Florida, 2010: pages 19 and 110.]
The manner in which natural units
and atomic units relate to one another may be obvious to some, but
in my case, I have been unable to discern their exact relationship.
Therefore, I present an elementary analysis of a possible relationship
among some of the natural unit constants and some of the atomic
unit constants as presented in the CODATA.
Essentially, Table VI of
the CODATA presents the Quantity, Symbol, Numerical Value, Unit
and Relative standard uncertainty of the natural units (n.u.) and
the atomic units (a.u.) of certain fundamental physical constants.
Several constants are repeated in both categories as shall be noted
below.
The electron volt is presented
outside of the natural units, but is presented within the atomic
units as a.u. of charge: elementary charge (1.602176487). The concern
in this analysis concentrates upon the mediation factor employed
between the same nomenclature term presented as a natural unit and
as an atomic unit, but with different numerical values. There is
no apparent definition in the table that would suggest which are
the numerical factors that serve as a conversion method between
natural units and atomic units.
Each of the constants listed on
the table present a symbol or symbolic formula identifying their
derivation. However, that symbolic representation does not explain
why or how the different numerical values are presented for apparently
the same constant. For example, consider why the natural unit of
energy is given a fractal numerical value of 8.18710438 while the
atomic unit of energy appears with a fractal numerical value of
4.35974394. (Note that the numerical values appearing in the
analytical part of this essay are written in their fractal form,
with a floating decimal place, without reference to scientific notation,
for purposes of comparison.)
Such a question as posed above
may appear to a physicist or chemist as being unaware of the relationship
between these different natural/atomic units. Yet, from the layperson's
viewpoint, such a query seems valid in my mind. My first impression
is that a certain ambiguity exists in the presentation of natural/atomic
numerical values as some constants maintain the same value in both
categories (mass, action), while others offer different values for
the same nomenclature (velocity, energy, momentum, length, time).
As mentioned, atomic unit of charge is somewhat of a hybrid, since
it appears as an atomic unit, but not as a natural unit (electron
volt).
For the sake of comparison, the
order in which the natural units are presented is maintained in
the order in which the atomic units are presented here. Although
in Table VI of the CODATA, the order varies.
There are fifteen remaining atomic
unit constants that are not treated in this analysis, and which
have no apparent corresponding natural unit constant on Table
VI.
Also listed together with the
electron volt at the beginning of Table VI, is the (unified)
atomic mass unit:
u 
1.660538782 x10^{12}

kg 5.0x10^{8} 
This particular constant does
not appear to have relevancy for the analysis at this time in considering
the relationship between natural units and atomic units.
I began my query about the relationship
between the natural units and the atomic units because of an idea
that has struck me throughout my studies of the CODATA recommended
fundamental physical constants. The question that keeps arising
in my mind is why some numerical values have their inverse expression
listed in the CODATA and others do not. From the point of view of
consistency and from that of how spacetime/motion exists in terms
of matterenergy, I would suspect that if one inverse (reciprocal)
relationship is significant then they are all significant. If one
is listed, then all of them should be listed.
This idea arises when one considers
the first two entries on Table VI and their possible relationship:
the electron volt as of the (unified) atomic mass unit.
1.602176487 divided by
1.660538782 equals fractal 9.64853398 (the Faraday constant)
Yet, one does not find the inverse
expression of 9.64853398 in the CODATA: fractal 1.036426883.
Nevertheless, one does find for example such expressions as in the
fine structure constant (7.2973525376) and the inverse fine structure
constant (1.37035999679) which appear in the CODATA, but do not,
however, appear on the list in Table VI of the CODATA. In the
case of the cited example, one must ask why is e/u significant,
yet u/e not apparently so. In my mind, they are both materially
viable, i.e., they both exist.
The other question that arises
in my mind, as I ponder the difference between the numerical values
of the same category/event of matterenergy, concerns a possible
conversion factor as they are expressed in either natural units
and in atomic units. The symbolic expressions listed along side
each particular constant have different numerical values that suggest
the possibility that a conversion factor exists for deriving natural
units from atomic units or deriving atomic units from natural units.
A basic procedure in considering this aspect is to divide one of
the units by the other. In this case, I began by dividing the numerical
values of the natural units by those of the atomic units. [For
a commonly held distinction between natural units and atomic units
consider the citations at the end of this essay.]
Where the numerical values are
the same in one set of constants or the other, there is no factor
other than unit 1.0, if one wishes to make such an observation.
In these two cases, there is no
analytical problem other than the question regarding the repetition
of terms, a certain redundancy of values with different nomenclature.
No matter which name is given to the constant, the numerical computations
shall be the same since the numerical values are the same for each
pair of values.
Now, consider each of the cited
pairs for either natural units or atomic units that have the same
nomenclature term but distinct numerical values. And, then, consider
their relation of division between themselves.
The inverse fine structure constant
[1.37035999679] mediates the natural unit of velocity with the atomic
unit of velocity as follows.
299792458 divided by 2.1876912541
equals 1.37035999681 [ = inverse fine structure constant]
An unrecognized constant value
[1.87788652101] mediates the natural unit of energy with the atomic
unit of energy as follows.
8.18710438 divided by 4.35974394
equals 1.87788652101
The inverse fine structure constant
[1.37035999679] mediates the natural unit of momentum with the atomic
unit of momentum as follows.
2.73092406 divided by 1.992851565
equals 1.37035999467 [= inverse fine structure constant]
The fine structure constant [7.297352537516]
mediates the natural unit of length with the atomic unit of length
as follows.
3.8615926459 divided by
5.2917720859 equals 7.297352537516 [= fine structure
constant]
An unrecognized constant value
[5.32513540596] mediates the natural unit of time with the atomic
unit of time as follows.
1.2880886570 divided by
2.418884326505 equals 5.32513540596
From the previous computations,
it becomes obvious that the fine structure constant and its inverse
expression are conversion factors between some of the fundamental
physical constants expressed as natural units and/or atomic units.
However, at first glance, it was not obvious to me at least, what
the significance of the numerical values of 1.87788652101
and 5.32513540596 represented as they do not appear elsewhere
in the CODATA listing of fundamental constants.
It did not take long, however,
to find out what those two numerical values represent. Consider
the following computation that involves the fine structure constant
and its inverse expression.
the fine structure constant  the inverse
fine structure constant
1 divided by 7.2973525376 equals 1.37035999679

Now consider the following computation,
whereby the fine structure constant is divided by its inverse expression:
7.2973525376 divided by
1.37035999679 equals 5.325135405801
Then consider when the inverse
fine structure constant is divided by the fine structure constant
per se.
1.37035999679 divided by
7.2973525376 equals 1.877886520802
Therefore, the conversion factor
for n.u.a.u. of energy and time is the ratio
between the fine structure constant and the inverse fine structure
constant [5.325135405801  1.877886520802].
Of the five fundamental physical
constants that maintain the same term nomenclature but present different
numerical values for either their natural unit expression or their
atomic unit expression, three pairs are based on the fine structure
constant and its inverse expression; and, two pairs are based on
the ratio of the fine structure constant and its inverse
expression.
Note further, the two values derived
thereof said ratio are themselves reciprocal expressions of one
another.
1 divided by 5.325135405801
equals 1.877886520802
The conversion factors, then,
for the seven repeat fundamental constants as natural and atomic
units, are: a) the unit 1.0, b) the fine structure constant and
its inverse expression, and, c) the ratio of these latter two expressions.
One may question whether it is
a valid procedure to employ the fine structure constant and its
inverse expression as a conversion factor as of the natural units.
More significantly, one must question whether it is valid to employ
the ratio of these two constants as a conversion factor between
the natural units and the atomic units.
Let me illustrate the query by
using a similar procedure with an apparently unrelated numerical
value, as in the boiling point of water [373.15 kelvin].
Take an apparently constant value
[3.7315] and find its reciprocal expression [2.679887444].
1 divided by 3.7315
equals 2.679887444
Now, divide the first value by
the second, as was done in the previous relationship for the fine
structure constant and its inverse expression.
3.7315 divided by 2.679887444
equals 1.392409225
1 divided by 1.392409225
equals 7.181796716407
The numerical values 1.392409225
and 7.181796716407 thus represent the ratio of the boiling point
of water to its own reciprocal. According to the computational procedure,
these numbers would serve in the same capacity as the numerical
values derived above regarding the conversion between natural units
and atomic units:
The first set of numerical values
would be relational in procedural terms as the second numerical
values. Computationally 7.2973525376 is to 3.7315
as 1.37035999679 is to 2.679887444; and so on for
the two sets of numbers. In terms of general logic, a ratio resulting
from two constant values must represent a constant value in itself.
In such case, if the ratio of
numerical values between a constant and its inverse expression represent
correct measurements, then they should be listed in the CODATA with
their corresponding explanation, along side the other derived constants
reflecting ratios. The question remains, however, whether it is
possible to have the ratio [inverse or not] of a pair of constants
[inverse or not fine structure constant] set as a conversion factor
for deriving natural and atomic units.
In summary, in their conversions,
the natural/atomic units of momentum, length and velocity
employ the fine structure constant and its inverse expression. While
the natural/atomic units of energy and time employ
the ratio of the fine structure constant and its inverse
expression. And, the natural/atomic units of charge and mass
employ unit 1.0. The question remains whether the computational
procedure is valid if based upon distinct conversion factors or
relationships. In other words, how does one know which particular
conversion factor to employ for which particular pairs of natural/atomic
constant units.
From here, one would have to analyze
the remaining atomic unit constants as of their symbolic formulae
which present multiples [powers greater than 1] of some of
the cited terms. Such an analysis would question why, for example,
a particular term is taken to the second, third, fourth or fifth
powers.
One of the analytical purposes
behind deriving natural units is to find the minimum expression
of matterenergy in terms of its units of measurement. Yet, the
cited symbolic formulae present an apparent theoretical contradiction
in that they express multiples or powers of some of the terms. Consider
a single example in this regard.
a.u. of electric polarizability
times a.u. of first polarizability
1.6487772536 times 3.206361533
equals 5.286575962428 halves to 2.5813359191 [near
the von Klitzing constant 2.5812807557].
This becomes more obvious is one
limits the number of decimal places in order to comprehend the fractal
multiples occurring within these values.
1.6487 times 3.2064 equals
5.28639168 halves to 2.581245938.
This value is somewhat low for
the von Klitzing constant, but adjustments may be made accordingly.
The idea is to illustrate how certain constants together produce
fractal multiples of other constants, which apparently are part
of their original terms. Additional examples of the relationship
of multiples among pairs of constants have been forwarded in some
of my previous essays on the Planck constants.
Inasmuch as the measurements of
the fundamental physical constants are correct, then one may find
relationships infinitely so. For the identified fundamental physical
constants reflect a specific aspect of spacetime/motion in its varied
forms of matterenergy, which are relational from the outset. In
searching for the minimum expression of the natural units,
then, one would expect to view the analysis upon the minimum expression
of the numerical values of the different physical constants.
Now, whether it is possible to
derive minimum natural units, and/or atomic units, as illustrated
in the literature, constitutes another level of analysis. With the
idea of physical quanta arises the idea of a minimum
expression of a unit of measurement. That particular subject remains
to be discussed. To date, I have only been attempting to consider
the internal logic of the fundamental physical constants as of their
numerical expressions presented in the literature. One must still
analyze existing relationships of the terms before attempting to
analyze the defining concept of a natural unit of matterenergy.
In my view, the varied numerical
relationships among the different pairs of natural/atomic units
of certain constants, as analyzed in this essay, do not afford a
theoretical understanding of the concept of natural units. In order
to comprehend the theoretical conceptualization of natural units,
one would have to venture beyond the numerical values of the fundamental
physical constants.
0
Citations:
"Natural
units are intended to elegantly simplify particular algebraic
expressions appearing in physical law or to normalize
some chosen physical quantities that are properties of universal
elementary particles and that may be reasonably believed
to be constant. However, what may be believed and forced to be constant
in one system of natural units can very well be allowed or even
assumed to vary in another natural unit system.
"Natural
units are natural because the origin of their definition comes only
from properties of nature and not from any human construct.
Planck units are often, without qualification, called "natural
units", when in fact they are only one of several systems of natural
units, albeit the best known such system. Planck units might be
considered unique in that the set of units are not based on properties
of any prototype, object, or particle but are solely
derived from the properties of free space.
"As with any set of base units or
fundamental units the base units of a set of natural units
will include definitions and values for length, mass,
time, temperature, and electric charge. Some
physicists do not recognize temperature as a fundamental physical
quantity, since it simply expresses the energy per degree of freedom
of a particle, which can be expressed in terms of energy (or mass,
length, and time). Virtually every system of natural units normalizes
Boltzmann's constant k_{B} to 1, which can be thought
of as simply a way of defining the unit temperature." [Source: www.wikipedia.
com, under Natural Units].
0
"Atomic Units {Hartree}... First proposed
by Douglas Hartree to simplify the physics of the Hydrogen
atom. Duff calls these "Bohr units". The unit energy
in this system is the total energy of the electron in the
first circular orbit of the Bohr atom and called the Hartree
energy, E_{h}. The unit velocity is the velocity of
that electron, the unit mass is the electron mass, me, and
the unit length is the Bohr radius, .
They can be obtained from "Schrödinger" units with the substitution:
This removes the speed of light (as
well as the gravitational constant) from the definitions and its
numerical value in atomic units is the reciprocal of the finestructure
constant, 1/?. Hence any observed variation over space or time in
the value of ? is attributed to variation in the speed of light."
[Source: Wikipedia.com, under Natural Units]
March, 2010 ©Copyrighted by
Charles William Johnson. All rights reserved.
Reproduction prohibited.
Earth/matriX: SCIENCE TODAY
P.O. Box 231126, New Orleans, Louisiana 701831126, USA
johnson@earthmatrix.com










The physics paradigm today is based mainly upon the concept of csquare, the squaring of the speed of light in a vacuum. Numerous fundamental physical and chemical constants provided in the physics literature [CODATA] reflect numerical values based upon powers of c, the speed of light in vacuo. The speed of light in a vacuum is determined to represent the upper limit of movement of massenergy by physicists. 

The upper speed limit for a light photon is 299792458 meters/second. The square of that number produces a numerical value that does not exist in any form of matterenergy. The csquare actually represents a number that corresponds to a near massless event: a light photon. The author goes beyond a critique of Albert Einstein’s famous formula based upon this unreal number. The rejection of Einstein’s formula is explored through basic math, the summation of powers in the equation’s terms. 

A common procedure followed in deriving many of the CODATA recommendations is to divide certain fundamental physical constants by the value of the elementary charge, e, 1.602176487. With regard to the Planck constants and units of measurement, the case is argued that Max Planck may have simply reversed engineered this procedure in order to derive his natural units. 

