Summary. Phenomena taking place at nanometer dimensions
may be theoretically described with a relatively obscure
relationship that has been known for at least 40 years. The paper
will describe the derivations in detail and relate experimental
measurements of forces and dimensions to predicted bands.
Boundary conditions for nanotechnology will be defined.
Theory. The classical radiius of an electron, r_{0},
is related to the Compton wavelength, r_{2}, and the Bohr
radius, r_{4}, by the formula r_{n} = r_{0}/^{n/2},
where n = 0, 1, 2, 3, 4..., and is the fine structure
constant. It appears that no one has looked at the significance
of n>4. This paper looks at relationships from n = 0 to n =
12, which actually separates natural phenomena into a series of
unique bands. Calculations of r_{n} from n = 0 to n = 12
are summarized in Table 1.
The various bands, when related to the Coulomb force, leads to
a series of stairsteps between the Coulomb force and some
hypothetical new force, F_{h}. The Coulomb force, F_{e},
at r_{n} is equal to to F_{h} at r_{n+1}.
For example, F_{e} at r_{2} is equal to F_{h}
at r_{3}, or F_{e} at r_{10} is equal to
F_{h} at r_{11}. Note this recurring relationship
in Table 1 between the Coulomb force and the hypothetical force.
It is relatively easy to show that this hypothetical force has a
surprisingly simple force law of F_{h} = hc/2r^{2}.
Observations. The last column in Table 1 summarizes
some of the observations about the different dimensions and
forces. When n = 0, r_{0} is the classical radius, which
has more theoretical value than any physical significance. The
case of n = 1, or r_{1}, has not been observed. As
already stated, when n = 2, r_{2} defines the Compton
wavelength and the region where the scattering of particles by
waves has been observed. Another non-observed region is at n = 3
and r_{3}. Bohr derived the result at n = 4, and r_{4},
which has been observed as the minimum orbit of electrons around
nuclei. It would appear, from what has already been stated, that
even magnitudes of n = 0, 2, 4 are favored; however a transition
may take place at n = 5 and r_{5}. Nothing appears to be
prominent about n = 5 and r_{5}, except that this is the
beginning of the region of nanotechnology. Experimental
measurements with atomic force microscopy/spectroscopy fall in
the band between n = 5 and n = 6. The much used Rydberg number
plays a role at n = 6 and r_{6}. Experimental
measurements have been made at n = 7 and r_{7} of the
zero point radiation force, or the so-called Casimir effect,
between two very close, neutral plates. The hypothetical force
derived above, F_{h}, is very similar to the empirical
formula for the zero point radiation force. A very interesting
phenomenological pattern emerges from n = 7 to n = 11. The
average feature sizes of various electronic devices fall in the
middle of different bands rather than at specific r_{n}
magnitudes. Consequently, the following bands exist: Very Large
Scale Integrated circuits (VLSIC) between r_{7} and r_{8};
Large Scale Integrated Circuits (LSIC) between r_{8} and
r_{9}; Small Scale Integrated Circuits (SSIC) between r_{9}
and r_{10}; and, transistors between r_{10} and r_{11}.
One last observation should be noted: the hypothetical force in
Table 1 has an observed range from 10^{-16} to 10^{-6}
m. This force is exactly equal in magnitude to the
measured strong-color force (7.14 x 10^{5} N) when F_{h}
is calculated at the Compton wavelength of a proton, r = 2.11 x
10^{-16} m.
Predictions. One nanometer falls approximately in the
middle, near n = 5 and r_{5}, in Table 1. This region is
at the boundary of atomic structure and the beginning of
molecular structure. The hypothetical force (zero point radiation
force) should have a major effect on phenomena in this region.
The Rdyberg length at n = 6 and r_{6} may be the maximum
Bohr orbit. Large atoms may force nanotechnology explorations
above r_{6}. Measurements of forces should all fall
between the Coulomb force and the hypothetical force for the
dimensions indicated in Table 1. In other words, phenomena are
bounded between F_{e} and F_{h}. Different
thresholds may exist for nanotechnology. The pattern set by
electronic devices indicates that it may be easier to develop
nanotechnology devices within the bands of r_{5} to r_{6}
and r_{6} to r_{7}, not specifically at r_{5},
r_{6}, or r_{7}, although r_{6} might be
preferred. The odd levels, n = 1, 3, 5, 7, 9, should be examined
for any unique experimental transitions.
Table 1.
Various bands of phenomena are defined theoretically starting
with the classical radius of the electron. Calculations are
based upon the mass of the electron using the equation r_{n}
= r_{0}/^{n/2}.
Band
n
Dimensions ..
r_{n}, meters ..
Coulomb ..
Force, N ..
Hypothetical ..
Force, N ..
Phenomena .. .. ..
.. ..
.. ..
.. ..
..
0
2.82 x 10^{-15}
2.91 x 10^{1}. .
3.98 x 10^{3} . .
Classical radius of electron, r_{0}
1
3.30 x 10^{-14}
2.12 x 10^{-1}..
2.91 x 10^{-1}
..
Particle structure, r_{0} to
r_{1}
2
3.86 x 10^{-13}
1.55 x 10^{-3}..
2.12 x 10^{-1}
..
Compton wavelength, r_{2}
3
4.52 x 10^{-12}
1.13 x 10^{-5}..
1.55 x 10^{-3}
..
Particle interactions, r_{2}
to r_{3}
4
5.29 x 10^{-11}
8.25 x 10^{-8}.
1.13 x 10^{-5}
..
Bohr radius, r_{4}
5
6.19 x 10^{-10}
6.03 x 10^{-10}
8.25 x 10^{-8}
.
Nanotechnology, r_{5} to r_{7}
6
7.25 x 10^{-9}..
4.39 x 10^{-12}
6.03 x 10^{-10}
Maximum atomic orbit, r_{6}
7
8.49 x 10^{-8}..
3.20 x 10^{-14}
4.39 x 10^{-12}
Molecular structure, r_{5} to
r_{7}
8
9.94 x 10^{-7}..
2.34 x 10^{-16}
3.20 x 10^{-14}
VLSIC, r_{7} to r_{8}
9
1.16 x 10^{-5}..
1.72 x 10^{-18}
2.34 x 10^{-16}
LSIC, r_{8} to r_{9}
10
1.36 x 10^{-4}..
1.25 x 10^{-20}
1.72 x 10^{-18}
SSIC, r_{9} to r_{10}
11
1.59 x 10^{-3}..
9.14 x 10^{-23}
1.25 x 10^{-20}
Transistors, r_{10} to r_{11}
12
2.00 x 10^{-2}..
5.78 x 10^{-25}
9.14 x 10^{-23}
Macroscopic
*Corresponding Address:
Dr. Robert J. Heaston, 220 Arlington Avenue, Naperville, IL
60565, ph: 630-416-8338, fax: 630-416-9203, email:
75630.3424@compuserve.com
^{a}DR. ROBERT J. HEASTON (principal author) has a BS
and MS from the University of Arkansas and a Ph.D. from Ohio
State University, all in chemical engineering. He is retired from
the Department of Defense (DOD), where he organized the DOD
Microwave and Millimeter Wave Monolithic Integrated Circuit
(MIMIC) program and assisted witht he organization of the Very
High Speed Integrated Circuit (VHSIC) program. Dr. Heaston is
currently a member of the Board of Army Science and Technology
under the National Research Council.
^{b}DANIEL F. VAN DER WERKEN, JR., has a BS from
Virginia Tech and an MS from the Air FOrce Institute of
Technology, both in electrical engineering. Microsoft employs him
as an Escalation Engineer.