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2 The Energy Correction Term Source Components.2.1 Derivation of the Central Coulomb Force Taking Into Account the Distributed Nature of the Spinning Electron Charge.In this derivation it is to be noted that because of the high spin rate of the electron at the point of orbital transition, account must be taken of the Lorentz - Fitzgerald contraction of circumferential dimensions. The derivation here will be restricted to the electron, similar effects generated by the nucleus are discussed in Section 2.4. Consider Fig. 2.1 below as representative of a spinning electron in any orbital.
The charge on the nucleus is +Ze.
The area of the electron elemental at rest is
L is the surface area of the electron and all other parameters are as shown in Fig 2.1.
With the electron shell spinning at the high angular rate of ewsp as shown, all circumferential dimensions will be subject to
Lorentz-Fitzgerald contraction. Thus the surface area of the electron
elemental, becomes
c is the velocity of light. In [8] it was proposed that electron orbit shell transitions took place via the emission of a photon because the surface velocity of the circumference of the spinning electron tended to exceed the terminal velocity, ( ~ the velocity of light), in the Relativistic Domain D0, (Pseudo-Euclidean Space-Time), (see [10] for an explanation of this). Thus in (2.2) above, at the point of an orbit shell transition
Therefore the coulomb coupling energy between the nucleus and the electron elemental is
It is now necessary to determine the distance l1 which must also take
account of Lorentz-Fitzgerald contraction of circumferential dimensions
associated with the electron. From Fig. 2.1
r is the distance between the centre of the two particles.
In these last two equations, (2.10) and (2.11), the effects of Lorentz-Fitzgerald contraction of circumferential dimensions have been incorporated. Details of this are shown in Appendix C. Substitution of (2.9), (2.10) and (2.11) into (2.8) gives
Substitution of (2.13) and (2.7) into (2.6) then gives
In (2.16), because r >> Ge, the denominator can be binomially expanded retaining only terms up to 1/r3. This then yields
Integrating over the complete electron spherical shell
Where in the integral with respect to q, advantage of symmetry has been taken to simplify the integration process. The integral with respect to f is simple and is taken first to yield
In Appendix B it is shown that the surface area of a sphere, spinning such that its circumferential velocity tends to the velocity of light is exactly half that when at rest. Incorporating this into (2.19) yields
Evaluation of this final integral gives
This relationship will be further embellished in the next Section via the derivation of the parameter r.
2.2 Derivation of the Effect of the Dynamic Distance of the Nucleus From the Centre of Orbital Motion.It was shown in [9] that due to the manner in which spin-orbit coupling energy was distributed, the precession rate of the nucleus about the central point of orbital rotation would be considerably greater than that of the electron. The geometry is shown in Fig. 2.2 below.
In view of the increased angular rate of the orbit of the nucleus, with the electron frozen in the position shown in Fig. 2.2, Eq.(2.23) can be restated as
wp is the angular rate of the proton nucleus. To determine the effect of this dynamic distance between the two particles, the average of (2.24) around the proton orbital is taken thus
Expanding the square root binomially, (re >> rp), and retaining terms
only up to 1/re2 gives
Inserting this into (2.21) gives for the central coulomb force coupling
energy
Again because re >> rp both terms inside the main bracket can be
binomially expanded retaining only terms up to 1/re2 thus
In Section 3.0 this expression will be converted to a form suitable for numerical computation of values, and therefore also suitable for addition to [9], Eq. (3.39), to produce the complete expression for orbital energy of the fine structure. Prior to that, Ge must first be determined.
2.3 Derivation of the Electron Spin Matter Wave Radius, to Meet the Spin Angular Momentum Quantum Criteria.The reason for the necessity of this parameter, which is primarily apparent in the first two orbit shells, is explained as follows. When, via the mechanism proposed in [8], a spinning electron makes a transition into any circular orbital, because there is no spin induction in a circular orbital, the value of angular spin rate carried by the electron will remain unchanged. At the same time if the spin angular momentum does not meet the quantum criteria, the only parameter that can vary in order to meet this is the electron spin matter wave radius, and it is proposed that this parameter thereby increases accordingly towards meeting this criteria. In any elliptic orbital, exactly the same effect will take place but in addition, due to spin induction, the magnitude of spin angular rate will also increase, both parameters thereby contributing to meeting the spin angular momentum quantum criteria. Also it will be seen that this effect is not only significant in these orbital energy correction terms, but is also central to the manner in which orbital transitions are initiated. These issues will be further discussed in detail in Section 5.4. To determine the magnitude of the spin matter wave radius in each orbital, it is necessary to derive a precise expression for the electron spin angular momentum, which can then be equated to the quantum criteria. The resulting expression can then be solved for the spin matter wave radius. This is pursued as follows. Consider again Fig. 2.1. If the mass surface density of the electron shell is Dme/ , then the rest mass of the elemental is
The spin velocity of the elemental is
and so the relativistically adjusted mass of the elemental is
Finally, the relativistically adjusted spin angular momentum of the elemental is then given by
In evaluating the integral of (2.35), two problems arise, first, the Lamb Shift is, in the low orbit shells, very sensitive to this particular process and it is therefore necessary to obtain as precise a solution as possible. Because the integration of (2.35) over the surface of the electron shell involves a version of the complete elliptic integral of the first kind, its exact evaluation in terms of analytic functions is not possible. Secondly, in order to determine values, it is clearly necessary to know the value of ewsp and a method of determining this parameter to the required accuracy and precision is not clear. Consequently, because of these problems, an empirical method of completing the solution must be adopted as follows. First perform a binomial expansion of (2.35) and write its integration as
Perform the first simple integral with respect to f and, taking advantage of symmetry, write the result as
Now integrate with respect to q retaining all terms to give
and this can be written
From Appendix B it is seen that when ewspGe ® c, the surface area of the spinning electron approaches half that of its rest mass value. Therefore incorporating this and putting ewspGe=k//c in (2.39) gives
and the restrictions on k// such that the solution converges to a positive mean value are that (i) its value must be close to unity and, (ii) it must, when (2.40) is equated to the spin angular momentum quantum criteria, enable a unique solution for Ge such that the Lamb Shift, in all orbitals in which it is apparent, is correctly determined. Thus equating (2.40) to the spin angular momentum quantum criteria and solving for Ge gives
In (2.41) the summation, with suitable empirical values of k// must be taken to sufficient terms such that it has completely converged. This will then provide a value of Ge for each orbital for insertion in (2.31). This final evaluation is developed in Section 3.0.
2.4 The Distributed Nature of the Charge on the Nucleus.To some extent the effect of the nucleus has already been accomplished in that account has been taken of the distance of it from the central point of orbital rotation. However, the nucleus also carries a charge and the question arises as to whether the distributed nature of this charge affects orbital energy in the same manner as that of the electron. There is no doubt that this must be so but because, in the case of hydrogen, the nucleus is so much smaller than the electron, its charge distribution contribution is not apparent as a fine structure correction. This would not necessarily be the case in other atoms/ions in which the nucleus contained additional protons, as well as neutrons. Consequently, the theoretical relationships developed herein from this point on are strictly only applicable to hydrogen.
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P.G.Bass, April 2008
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