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Theoretical Predictions of Nanometer Phenomena

R.J. Heaston* and D.F. Van der Werken, Jr.

Dr. Robert J. Heastona
Daniel F. Van Der Werken, Jr.b

This is an abstract for a poster to be presented at the Fifth Foresight Conference on Molecular Nanotechnology.

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, r0, is related to the Compton wavelength, r2, and the Bohr radius, r4, by the formula rn = r0/(alpha)n/2, where n = 0, 1, 2, 3, 4..., and (alpha) 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 rn 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, Fh. The Coulomb force, Fe, at rn is equal to to Fh at rn+1. For example, Fe at r2 is equal to Fh at r3, or Fe at r10 is equal to Fh at r11. 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 Fh = hc/2(pi)r2.

Observations. The last column in Table 1 summarizes some of the observations about the different dimensions and forces. When n = 0, r0 is the classical radius, which has more theoretical value than any physical significance. The case of n = 1, or r1, has not been observed. As already stated, when n = 2, r2 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 r3. Bohr derived the result at n = 4, and r4, 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 r5. Nothing appears to be prominent about n = 5 and r5, 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 r6. Experimental measurements have been made at n = 7 and r7 of the zero point radiation force, or the so-called Casimir effect, between two very close, neutral plates. The hypothetical force derived above, Fh, 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 rn magnitudes. Consequently, the following bands exist: Very Large Scale Integrated circuits (VLSIC) between r7 and r8; Large Scale Integrated Circuits (LSIC) between r8 and r9; Small Scale Integrated Circuits (SSIC) between r9 and r10; and, transistors between r10 and r11. 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 105 N) when Fh 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 r5, 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 r6 may be the maximum Bohr orbit. Large atoms may force nanotechnology explorations above r6. 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 Fe and Fh. 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 r5 to r6 and r6 to r7, not specifically at r5, r6, or r7, although r6 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 rn = r0/(alpha)n/2.
rn, meters
Force, N
Force, N
0 2.82 x 10-15 2.91 x 101 3.98 x 103 Classical radius of electron, r0
1 3.30 x 10-14 2.12 x 10-1 2.91 x 10-1 Particle structure, r0 to r1
2 3.86 x 10-13 1.55 x 10-3 2.12 x 10-1 Compton wavelength, r2
3 4.52 x 10-12 1.13 x 10-5 1.55 x 10-3 Particle interactions, r2 to r3
4 5.29 x 10-11 8.25 x 10-8 1.13 x 10-5 Bohr radius, r4
5 6.19 x 10-10 6.03 x 10-10 8.25 x 10-8 Nanotechnology, r5 to r7
6 7.25 x 10-9 4.39 x 10-12 6.03 x 10-10 Maximum atomic orbit, r6
7 8.49 x 10-8 3.20 x 10-14 4.39 x 10-12 Molecular structure, r5 to r7
8 9.94 x 10-7 2.34 x 10-16 3.20 x 10-14 VLSIC, r7 to r8
9 1.16 x 10-5 1.72 x 10-18 2.34 x 10-16 LSIC, r8 to r9
10 1.36 x 10-4 1.25 x 10-20 1.72 x 10-18 SSIC, r9 to r10
11 1.59 x 10-3 9.14 x 10-23 1.25 x 10-20 Transistors, r10 to r11
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:

aDR. 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.

bDANIEL 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.

4625 Highcroft Lane, Charlotte, NC 28269, Bus. ph: (704) 582-8290, Home ph: (704) 548-8333, fax: (704) 357-1170, email:

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