MILLENNIUM SCIENCE LETTERS (MSL)

August 20, 2001

Volume 1 - Number 2

www.mrti-usa.com

Argand Diagrams of The Fine Structure Function

By Thomas S. KaKovitch

ABSTRACT

This work aims to connect the scattered light quantum of the Compton effect which depends only on the angle of scattering , with the precessional motion of the electron around the nucleus of a hydrogen atom as described by Sommerfeld's energy of the bound electron after deducting the rest energy to produce the fine structure splitting. This paper discusses the mathematical analysis which will numerically produce the amount of the fine structure splitting (called after Sommerfeld) the fine structure constant. The narrative shows how the fine structure constant depends on the angle of scattering and the theoretical determination of its numerical value is a straight forward approach to the Compton effect.

KEYWORDS

The fine structure constant, The fine structure function, The angle of scattering, The Compton effect, An Argand diagram, The complex variable, The complex conjugate, ArgZ.

INTRODUCTION

The mathematical approach considered to connect the fine structure function to different Argand diagrams is a series of equations which lead to angular relations. The detailed development of mathematical expressions shows dependencies between the angle of scattering of the Compton effect and the polar angle arg of an Argand diagram. These equations can directly track the numerical value of the fine structure constant in an Argand diagram in terms of the angle of scattering .

MATHEMATICAL ANALYSIS

Consider the complex variable Z=X + iY where i^2 = -1. There are two geometric representations of the complex number Z = X + iY:

	(a) as a point in the xy-plane, or
	(b) as a vector traced from the origin.

Either representation is called an Argand diagram.

We were able to prove that equation (4)

	==	(4)	Ref. 1

tracked the angle of scattering

of the Compton effect to the fine structure function

Similarly, we were also able to prove that equation (6)

		(6)	Ref. 2
tracked the angle of scattering of the Compton effect to the four imaginary parametric solutions (), (), (), ().			Ref. 3

The equations (4) and (6) produced a total of five solutions. To account for all these five solutions we construct an Argand diagram to represent the complex variable

			(7)
where	=		(8)
and	=	()	(9)

When we plotted the angle of scattering versus the function Re[] for the range = to we obtained (fig. 1):

Re[Z1(Alpha, Theta)]

Fig. 1.
The fine structure constant

(

) vs the angle of scattering

When we plotted the angle of scattering versus the function Im[] for the range = to we obtained (fig. 2):

Im[Z1(Alpha, Theta)]
Fig. 2.
The two dimensional graph of the function vs the angle of scattering .

	From Fig. 1., the analyses show that the empirical value of the fine structure constant
	( =0.00729729) is the real part of a complex variable function expressed by
	or	Re[] =

	This equation claims that the fine structure constant can be represented by an Argand diagram.

The complex conjugate of is:

			(10)

Similarly, we can construct a second Argand diagram to represent the complex variable

such that:

				(11)

and its complex conjugate:

				(12)

The two Argand diagrams which contain the two complex variables and and their respective complex conjugates and involve all of the five solutions which were obtained from equations (4) and (6).

Further analyses show that the absolute values of the complex variables

and

are:

					(13)

					(14)

We can choose the polar coordinates

and

where	=	the length of a vector from the origin

and	=	the angle that makes with the x-axis.

We can write:

					(15)

and					(16)

We can rewrite the complex variable

in polar coordinates, as:

					(17)

or					(18)

We insert equation (7) into (18) and obtain:

					(19)

We match the left hand side and the right hand side of the equation (19) for the real and imaginary values and obtain:

					(20)

and					(21)

We insert equation (13) into (15) and obtain:

					(22)

We insert equation (22) into (20) and obtain:

					(23)

or					(24)

We solve the parametric equation (24) for arg

in terms of its parameters,

and

and obtain equations (25) and (29). (Click here to examine equation (25)).

The equation (25) is the first of two solutions to

represented by equation (16).
The parametric complex variable equation (25) could be written in terms of its real and imaginary parts.

Let us define the parametric complex variable equation (25) as:

					(26)
where					(27)
and					(28)

	When the parametric function is plotted vs the angle of scattering for the range =0° to 720° and for = 7.29729 * 10^(-3) we obtained (fig. 3):

	Fig 3. The parametric function vs the angle of scattering .

	When the parametric function is plotted vs the angle of scattering for the range =0° to 720° and for = 7.29729 * 10^(-3) we obtained (fig. 4):

	Fig 4. The parametric function vs the angle of scattering .

The equation (29) is the second of two solutions to represented by equation (16). (Click here to examine equation (29)).

Similarly, the parametric complex variable equation (29) could be written in terms of its real and imaginary parts.

Let us define the parametric complex variable equation (29) as:

					(30)
where					(31)
and					(32)

When the parametric function is plotted vs the angle of scattering for the range =0° to 720° and for = 7.29729 * 10^(-3) we obtained (fig. 5):

Fig 5.
The parametric function

vs the angle of scattering

When the parametric function is plotted vs the angle of scattering for the range =0° to 720° and for = 7.29729 * 10^(-3) we obtained (fig. 6):

Fig 6.
The parametric function

vs the angle of scattering

Similarly we could obtain the solutions to . Due to the complexity of this topic, we will deal with at some future date.

ANGULAR RELATIONS

The figures (3, 4, 5, and 6 above) show the mathematical relations between the angle of scattering of the Compton effect and the polar angle arg of an Argand diagram.

These mathematical expressions show the dependencies between the angle of scattering of the Compton effect and the polar angle arg of an Argand diagram in terms of the fine structure constant .

The complexity of equations (25) and (29) shows that the parametric relations between arg and the angle of scattering is very detailed and non-linear. The figures (3, 4, 5, and 6 above) show that arg is represented by modulating waves both along the real and the imaginary axes of an Argand diagram. The angle of scattering of a Compton effect has a physical significance. The polar angle arg has a mathematical significance. The equations (25) and (29) which connect arg to the parameter via show that the physical interactions between photons and charged particles, as defined by the Compton effect, generate mathematical dynamics in an Argand diagram defined by the parametric complex variables and .

The mathematical dynamics of Argand diagrams which represent the complex variables

and the physical dynamics of the Compton effect represented by the angle of scattering

are related to the fine structure function

as shown:

					(33)
					(34)

When the function of equation (33) is plotted vs the angle of scattering for the range =0° to 720° and for = 7.29729 * 10^(-3) we obtained (fig. 7):

Fig 7.
The fine structure function

vs the angle of scattering



	When the function of equation (33) is plotted vs the angle of scattering for the range =0° to 720° and for = 7.29729 * 10^(-3) - 8 *10^(-8) we obtained (fig. 8):

	Fig 8. The fine structure function vs the angle of scattering .

	When the function of equation (33) is plotted vs the angle of scattering for the range =0° to 720° and for = 7.29729 * 10^(-3) + 8 *10^(-8) we obtained (fig. 9):

	Fig 9. The fine structure function vs the angle of scattering .

	When the function of equation (34) is plotted vs the angle of scattering for the range =0° to 720° and for = 7.29729 * 10^(-3) we obtained (fig. 10):

	Fig 10. The fine structure function vs the angle of scattering .

	When the function of equation (34) is plotted vs the angle of scattering for the range =0° to 720° and for = 7.29729 * 10^(-3) - 8 *10^(-8) we obtained (fig. 11):

	Fig 11. The fine structure function vs the angle of scattering .

	When the function of equation (34) is plotted vs the angle of scattering for the range =0° to 720° and for = 7.29729 * 10^(-3) + 8 *10^(-8) we obtained (fig. 12):

	Fig 12. The fine structure function vs the angle of scattering .

	CONCLUSIONS

	The numerical values of the dimensionless fine structure function can be obtained from Argand diagrams which represent the parametric complex variables and where the angle of scattering of the Compton effect and are its two parameters. The fine structure constant is an empirical value which depends on other empirical values, such as , e, h, and c. The numerical value of is limited within the standard error of + or - 8 * 10^(-8). Furthermore, there exists an angular relation between the polar angles arg and arg of Argand diagrams which represent the parametric complex variables and and the angle of scattering of the Compton effect.





	Thomas S. KaKovitch is the president and CEO of MRTI. He is the inventor and patent holder of five issued U.S. Patents and several foreign Patents. He is also a tenured professor in the College of Math and Sciences at the University of the District of Columbia, Washington, DC, U.S.A.




	Copyright © 2001-2007 by Professor Thomas S. KaKovitch All rights reserved under Pan American and International Copyright Conventions.

MILLENNIUM SCIENCE LETTERS (MSL)

August 20, 2001

Volume 1 - Number 2

www.mrti-usa.com

Argand Diagrams of The Fine Structure Function

By Thomas S. KaKovitch

ABSTRACT

KEYWORDS

INTRODUCTION

MATHEMATICAL ANALYSIS

ANGULAR RELATIONS

CONCLUSIONS

All rights reserved under Pan American and International Copyright Conventions.