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A.1 Preliminary Analysis Involving the Solution to the Orbital Equation of Motion.
The relativistic orbit of the electron about the proton nucleus can be
pictorially represented as in Fig. A.1
The solution of the orbital equation of motion was, for a general case,
effected in [2], Eq.(5.40) and is repeated here for convenience
Here, for conformity with the nomenclature used in this paper, F in [2],Eq.(5.40) has been replaced with j. Eq.(A.1) is of the form
Where
In order to obtain the required terms in the nomenclature of atomic structure as used in this series of papers, it is necessary to obtain suitable expressions for m0, h and L. Note that h in the above equation (A.1), is not Planck's constant but the swept area constant of the orbit as defined in [2]. Also note that in all of the ensuing analysis, all terms involving the velocity of light, c, will, where necessary, be binomially expanded to retain only those of relativistic second order. First for m0, this can be determined from the eccentricity terms in (A.1) and (A.3), thus
Now, substitution of this into the semi-latus rectum half of (A.1), then
gives after reduction
To determine a suitable expression for h, from [2], Eq.(5.22)
and from (A.2) this becomes
Solving (A.9) for h gives
Substitution of this into (A.6) then gives for L,
and also from (A.2) and (A.10)
Eqs.(A.11), (A.12) and (A.13) are the required subsidiary equations to determine orbital energy and can be converted to atomic structure
nomenclature by putting F0 = Ze2, thus
These results are used in the derivation of electron orbital energy in the
main text, Section 2.4, and in the quantisation of sample orbits thereafter.
A.2 Derivation of Sommerfeld's Full Relativistically Mass Corrected Elliptic Orbit Energy Equation. In Section 3.5, the expanded version of Sommerfeld's equation for a relativistically mass corrected electron orbit energy level was derived. The full version is derived here for interest. Multiplying out (2.24), adding the rest mass energy and binomially contracting gives
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