MILLENNIUM SCIENCE LETTERS (MSL)

July 5, 2001

Volume 1 - Number 1

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Tracking The Fine Structure Function

By Thomas S. KaKovitch

  The quantum and relativity theories do correct each other along the dimensionless constant that enters into the local laws of physics as:

    The term arises from electron spin and relativity corrections. The Sommerfeld equation of the energy bound electron after deducting the rest energy, developed in 1916 before the introduction of wave mechanics, produces with great exactness, the values of hydrogen terms as calculated by Dirac's relativistic wave mechanics.  The energy of the bound electron after deducting the rest energy, can be stated in terms of the principal and azimuthal quantum numbers, and also in terms of the challenging pure number referred to in physics as the fine structure constant.  The transition between the energy levels of the upper state and the lower state produces spectral lines.  This is called the fine structure of the spectral lines.  The spectral lines themselves are made up of a system of finer lines which determines the fine structure splitting.  The fine structure splitting corresponds to a precessional motion of the electron about the nucleus, taking into account the relativistic variability of mass, and partly compensating it with its electronic spin while maintaining discrete amounts of the fine structure splitting.  The amount of the fine structure splitting, is called (after Sommerfeld) the fine structure constant().  The fine structure constant is an empirical pure number, the only quantity indeed of zero dimensions which can be formed from the electronic constants of e, h, and c.  This empirical value seems to be closely connected with charged elementary particles in general and it compensates the relativistic variability of mass with charge spin.  

    There seems to be little doubt that the existence of this dimensionless number indicates a deeper relation between electrodynamics and quantum theory than the current theories provide, and the theoretical determination of its exact numerical value () is a challenge to physics.  All attempts have so far been in vain.  The search will focus on whether a function describing the fine structure does exist.  Will this function have at least one of its solutions tracking the fine structure constant?  Are there more solutions hitherto unknown associated with this function?

    The empirical data which led to () come from electron/photon interractions.  To track a function which describes the fine structure one must track the scattered light quantum.  One parameter which stands out in the scattered light quantum equations is the angle of scattering () produced by the collision of a light quantum with an electron.  Since no angle of scattering is distinguishable from another, all angles of scattering are acceptable.  The mechanism which produces the scattered light quantum is the precessional motion of the electron about the nucleus.  The fine structure splitting corresponds to a precessional motion of the electron about the nucleus where light quantum and electron interactions are numerous.  The fine structure function must show traces of light scattering at all angles.  

    The first track is focused on the Compton effect - a light quantum on colliding with an electron transfers part of its energy to the latter, and its wave-length becomes greater after the scattering.  The Compton formula for the change of wave-length of the light quantum due to the scattering process runs:
    

				(1)	(ref. 2)
where		=	the Compton wave-length
and		=	the angle of scattering.

    The relation thus formulated can be introduced into the Sommerfeld equation of the energy bound electron after deducting the rest energy.  Numerous equations are generated.  After a series of algebraic operations we developed the following equation:

				(2)
where				(3)
and		=	the fine structure constant
		=	the angle of scattering.

    When we plotted the angle of scattering   versus the function (, ) for the range = to we obtained hyperfine splitting always leading to a constant value for (, ) =  0.0000532504 (fig. 1).    (ref. 3)

  The value of (, ) turns out to be equal to ().  The square root of (, ) is the -function and can be written as:

    
The equation (4) describes  () as a function.
    
The function () depends on the angle of scattering produced by the Compton effect.

When the angle of scattering () is plotted vs the function () for the range = to we obtained hyperfine splitting leading to the constant ()=0.00729729. (fig. 2)  The result shows that one of the solutions of the () function is tracked as the fine structure constant .

         () = 0.00729729 =             (5)    (ref. 4)


is an empirical value which depends on other empirical values such as e, h, and c.  The fine structure value is limited within the standard error of ± 8.0 X .

Fig. 2.
The two dimensional graph of the function () vs the angle of scattering .

    Further analysis of equation (2) shows there were more solutions in the imaginary plane.  The second track is to focus on solving equation (2)for  (, ) when (, )=() and equating the result to its identical equation (3).  This resulted in producing the following equation:

        
Solving for (), the equation (6) tracked the following four imaginary solutions:

        

Shown together, all four imaginary solutions appear as:
    
            

The four functions:
          (),
         (),
         (),
         (),
are the solutions of equation (6) and are imaginary.

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