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in Relation to hc, the Product of Planck's Constant and the Speed of Light |
Newton's Law of Gravitation is that the force between masses m_{1} and m_{2} separated by a distance r is given by
where G is a constant equal to 6.67259×10^{-11} m^{3}/kg.
The mass of any body is essentially equal to the number of nucleons it contains times the mass of a nucleon; m_{i}=m_{n}n_{i}. Thus the force formula could be represented as
Thus the force constant in units of kg*m^{3}/s^{2} is Gm_{n}² = (6.67259×10^{-11})(1.6749×10^{-27})^{2} =1.871855×10^{-64} kg*m^{3}/s^{2}. The ratio of this constant to the product of Planck's constant and the speed of light in a vacuum, hc=1.9865×10^{-25} kg*m^{3}/s^{2}, is 9.42288×10^{-40}.
Coulomb's Law of Electrostatics is that the force between two charges q_{1} and q_{2} separated by a distance r is given by
where (1/(4πε)) is a constant equal to 9×10^{9} kg*m^{3}/s^{2}.
The charge of any body is essentially equal to the net number of elementary charges it contains times the value of the elementary charge; q_{i}=q_{e}n_{i}. Thus the force formula could be represented as
Thus the force constant is in units of kg*m^{3}/s^{2} is
(1/(4πε))q_{e}²=(9×10^{9})(1.60218x10^{-19})²
= 2.3103×10^{-28} kg*m^{3}/s^{2}.
The ratio of this constant to hc=1.9865×10^{-25} kg*m^{3}/s^{2}
is 1.163×10^{-3} or approximately 1/860. If h-bar, h=h/2π,
had been used to compute the ratio the result would have come out 1/137, the familiar
fine structure constant.
Suppose the force between two nucleons is given by the formula
The justification for this formula is that the nuclear force is carried by particles subject to decay; i.e., the π mesons. The population of remaining particles is a negative exponential function the time since emission which translates into a negative exponential function of distance. These remaining particles are spread over an area of 4πr². The intensity is thus proportional to e^{−λr}/r². For more on this model see Nuclear Force.
In that previous work an estimate was made of the magnitude of the constant H* based upon the physical characteristics of the deuteron. It is 1.92570×10^{-25} kg*m^{3}/s^{2}. These are the same units as those of hc. The magnitude of hc is 1.986259×10^{-25}. The ratio is 0.96876=1/1.03225. This is close enough to unity to suggest that the actual value of H* is hc. This would make the nuclear force formula
The near-equality of the nuclear force constant with the value of hc may be merely an accident but it is certainly an intriguing one. It is notable that the mass of the π0 meson is about 3.4 percent less than the masses of the π± mesons.
[Update: The above estimate of H, 1.92570×10^{-25} kg*m^{3}/s^{2}, was based upon the mass deficit of the deuteron being 2.22457 MeV. It appears that the mass deficit is considerably greater than this value. An alternate estimate of H based upon the separation distance of the nucleons in a deuteron being 3.2 fermi is 3.392372×10^{-26} kg*m^{3}/s^{2}. This makes H equal to 0.17079hc=(1/5.855)hc.]
The estimate of λ is 6.57×10^{14} m^{-1}=1/(1.522 fermi). This makes ρ_{0}=1/λ=1.522 fermi, a natural unit of length. This length divided by the speed of light would be a natural unit of time,
Planck's constant is a natural unit of action, energy×time, so h/τ_{0} would be a natural unit of energy. Its value is
Through the Einstein relation, E=mc², there would be a natural unit of mass equal to
This is notably close to the masses of the proton and the neutron; i.e., 1.6726×10^{−27} and 1.6749×10^{−27} kg, respectively. There is only about a 15 percent difference betweeen the computed m_{0} and the masses of the proton and neutron.
The explanation for the computed natural unit of mass being close to the mass of the
proton and neutron is prosaic. The scale length ρ_{0} is computed from
Yukawa's relation which involves h-bar, h, whereas in the above computation
of the natural unit of energy Planck's constant h was used. This introduced a factor
of 2π. If the same version of Planck's constant had been used in both places the
mass of the pi meson would have been the result of the computation of a natural
unit of mass. Instead what resulted was 2π times the mass of the pi meson. By
coincidence (?) the mass of the proton is approximately 2π times the mass of the
pi meson.
(To be continued.)
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